Computer Analysis and Design of Earthquake Resistant Structures: A Handbook (Advances in Earthquake Engineering) 1853123749, 9781853123740

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The present book was written to fill a gap in the literature on seismic analysis and design of structures. It is characterized by two special features. Firstly the book is unique in that no other book on earthquake engineering, old or recent, combines tutorial and state-of-the-art aspects of the subject from the modern viewpoint of computerized methodologies. Secondly, it is comprehensive in that most types of structures such as buildings, bridges, lifelines, storage tanks, dams and offshore structures are described. The book also covers materials such as concrete, steel and masonry. Related subjects, such as engineering seismology, site effects and seismic hazard analysis are also included, while deterministic and stochastic methods are presented. More specifically, the book covers numerical methods in earthquake engineering. stochastic analysis methods, engineering seismology, strong ground motion and site effects, seismic hazard analysis and design earthquake loads, soil-structure interaction, principles of e rthquake resistant design, buildings, reinforced concrete structures, steel structures, masonry structures and monuments, bridges, earth and concrete dams, offshore structures, storage tanks, underground and Dfeline structures, seismic isolation and control and seismic retrofit of concrete structures. The present handbook will serve well the present and future needs of both research and practising structural engineers engaged in seismic analysis and design.

Books of Related Interest The Kobe Eart uake - Geodynamical Aspects Edited by: C.A. Brebbia ISBN: 1853124303; 1562523457 (US, Canada.Mexico) 1996 160pp

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Earthquake Resistant Engineering Structures Edited by: G.D. Manolis, D.E. Beskos, C.A. Brebbia ISBN: 1853124567 1996 752pp

ail: [email protected] ://www .cmp.co.uk

ISBN: 1853123749 ISSN: 1361-617X

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Computer Analysis an Design of Earthquake Resistant Structures . A Handbook

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Computer Analysis and Design of Earthquake Resistant Structures A Handbook

International series on Advances in Earthquake Engineering Objectives: The objectives of this series are to provide clear accounts of both basic and applied research in the various fields of earthquake engineering with particular reference to earthquake resistant analysis and the design of structural systems. The series consists of books concerned with state-of-the-art developments in earthquake engineering and as such comprises several volumes every year covering the latest developments and applications. Each volume is composed of authored works or edited volumes of several chapters written by leading researchers in the field. The scope of the series covers almost the entire spectrum of earthquake engineering and as such the following topics will be discussed: engineering seismology, strong ground motions and site effects, seismic hazard evaluation and design earthquake loads, soil-structure interaction, numerical methods in earthquake engineering, stochastic analysis methods, principles of earthquake resistant design, reinforced concrete structures, steel structures, masonry and masonry infill structures, historical buildings and monuments, bridges, earth and concrete dams, underground and lifeline structures, storage tanks, silos and other industrial structures, offshore structures, seismic isolation and control, vulnerability and risk assessment of structural systems, repair and retrofit, seismic code regulations and case studies in earthquake engineering.

Series Editors: Professor D.E. Beskos Department of Civil Engineering University of Patras GR-26500 Patras Greece

E. Kausel Department of Civil & Environmental Engineering Massachusetts Institute of Technology Cambridge MA 02139 USA

Honorary Editors: C.A. Brebbia Wessex Institute of Technology Ashurst Lodge, Ashurst Southampton S040 7AA UK J.T. Roesset Department of Civil Engineering Texas A & M University College Station Texas 77843 USA

A.S. rlc,

where

{t;}

and

{!}

transform domain, acting on the receiver node p or the source node q. The (3x3)influence matrices

Substitution of equation ( 161) into the constitutive equation yields the cognate traction tensor TIJ(x,lm) in the form

are the (3xl) displacement and traction vectors, in the

(U]

and

uij =L f N,U9(N,x,xP,co)dB.

-pq

A

[r]

for the source node q are given by

(166)

a.:=l Ba -pq

Tij

=.Lf NJy(N,x,xP,co)dB. A

a.=J Ba

-

(167)

56

Computer Analysis and Design of Earthquake Resistant Structures

For non-singular elements, i.e. when the nodes p and q do not belong to the same element, standard 2-D Gauss quadrature can be used for the evaluation of the above integrals. However, in the case of singular elements the integrations indicated by equations (37) and (38) should be interpreted in the sense of Cauchy principal value and evaluated using either special mapping or other techniques [ 190, 191]. More on the subject of time and frequency domain BEM's can be found in the book of Wolf [192] and the recent review article ofBeskos [17].

10 Special BEM's for linear dynamic analysis Among the various special BEM' s mentioned in the review articles of Beskos [16, 17], two of them have been applied to earthquake engineering problems and are briefly described here. These are the indirect BEM and the dual reciprocity BEM (DR/BEM). According to the indirect BEM, introduction of fictitious sources of unknown density on the boundary (or close to it to avoid singularities) permits one to express displacements and tractions as integrals, involving these fictitious sources and the fundamental solution or Green's function of the problem. Employment of the boundary conditions of the problem results in two boundary integral equations in terms of these unknown sources. Once these sources are found, displacements and tractions are easily determined. Indirect BEM' s have been formulated in the frequency (mostly) as well as the time domain. One can mention here the frequency domain works of Wong [ 193 ], Dravinski and coworkers [194-197], Sanchez-Sesma and co-workers [198-201] and Luco and co-workers [202,203] dealing with seismic wave diffraction by two- and threedimensional canyons or alluvial valleys and the time domain works of Antes and Trondle [204], Wolf[192] and Crouch and co-workers [205,206]. In the conventional BEM use is made of the elastodynamic fundamental solution and this results in a boundary-only discretization. If the much simpler elastostatic fundamental solution is employed, an inertial volume integral is created. A second application of the reciprocal theorem (the first pne was used to achieve an integral representation of the solution) converts this volume integral into a surface integral. Thus, the DR/BEM is created which has the advantages of computational simplicity and boundary-only discretization, although some interior collocation points are usually necessary for increased accuracy. The DR/BEM, first introduced by Nardini and Brebbia [207], has been successfully used for free vibration analysis of two- and three-dimensional structures (Nardini and Brebbia [207,208], Ahmad and Banerjee [209], Wang and Banerjee [210], Coda and Venturini [211] and Davies and Moslehy [212]) and forced vibration analysis of two- and three-dimensional structures in the time domain (Nardini and Brebbia [208], Loeffier and Mansur [213], Pekau et al. [214], Fedelinski et al. [215], Chirino et al. [216] and Agnantiaris et al. [217,218].

Computer Analysis and Design of Earthquake Resistant Structures

57

11 Hybrid BEM/FEM scheme for linear dynamic analysis In some classes of problems, especially those involving soil-structure and fluidstructure interaction, coupling of the BEM with the FEM in time or the frequency domain results in a hybrid BEMJFEM scheme, which combines the advantages of both methods and reduces or eliminates their disadvantages. This coupling is achieved through equilibrium and compatibility or by the application of variational principles at the interface of the two domains modelled by the two methods. The system equations are solved in the frequency domain and the time domain .response is obtained through numerical inversion (Fourier or Laplace} or the time domain by stepwise time integration (e.g., Newmark type). Among the plethora of references on the subject mentioned in Beskos [ 16, 17], one can mention here the following works: i) using time domain formulations for soil-structure interaction those of Karabalis and Beskos [219], Spyrakos and Beskos [220], Fukui and Ishida [221],Von Estorff et al [222-224], Abe and Yoshida [225], Hayashi and Takahashi [226], Takemiya et al [227], Belytschko and Lu [228] and Pan and Atluri [229]. ii) using frequency domain formulations for soil-structure interaction those of Kobayashi and co-workers [230,231], Mossessian and Dravinski [232], Gaitanaros and Karabalis [233], Datta and co-workers [234,235], Schmid and co-workers [236,237], Aubry and Clouteau [238], Mita and Luco [239] and Zhang et al. [240]. iii) using time domain formulations for dam-reservoir systems those of Von Estorffand Antes [241], Nowak and Hall [242], Touhei and Ohmachi [243] and Guan et al. [244]. iv) using frequency domain formulations for dam-reservoir systems those of Clouteau and Aubry [245], and Tan and Chopra [246].

12 Introduction to nonlinear dynamic analysis Realistic earthquake engineering problems involve material and geometric nonlinearities. Material nonlinearities are due to the inelastic constitutive material behavior of the structure and/or its foundation soil, while geometric nonlinearities are usually due to unilateral contact conditions between the structure and its foundation. Nonlinear dynamic problems are classified as wave propagation and inertial or structural dynamic problems with seismic problems belonging to the later category, exactly as in the case of linear analysis. Analysis of these nonlinear dynamic structural systems is usually done by the time domain FEM. The BEM has been successfully applied to small scale problems of _inelastic dynamic structural analysis (without seismic forces) as described in the review ofBeskos [18] and to a simple inelastic soil-structure interaction by Pavlatos et al. [ 19]. However, the BEM alone or in combination

t 58

Computer Analysis and Design of Earthquake Resistant Structures

Computer Analysis and Design of Earthquake Resistant Structures

Mii+ Cu+ F= R

with the FEM has been extensively applied to unilateral contact analysis problems encountered in soil-structure interaction. Actually the FEM discretizes the systems in space (semi-discretization) and time integration techniques are usually employed to solve the resulting nonlinear matrix equations of motion. Some techniques other than direct time integration are also used for solving these equations approximately but ~ore efficiently. In subsequent sections, various methods for nonlinear dynamic analysis as applied to earthquake engineering problems are presented and critically discussed. Direct time integration methods for solving the inelastic dynamic equations of motion are first presented. Both implicit and explicit integration schemes are presented but the emphasis is on implicit techniques best suited for inertial or structural dynamics problems. Some other techniques involving partitioning, reduction, operator splitting etc. are also briefly presented. A discussion on · methods other than time integration· is provided next. These include modal superposition, equivalent linear analysis and use of inelastic spectra for approximate, yet efficiently obtciined results. Dynamic unilateral contact problems encountered in soil-structure interaction are also discussed in a separate section. These nonlinear problems are solved in either the time domain or the hybrid frequency-time domain in conjunction with the FEM or mainly the BEM. The interested reader can consult a number of excellent books and review articles on the subject of nonlinear dynamic analysis for further details. One can mention here the books of Belytschko et al. [247], Donea [248], Belytschko and Hughes [3], Liu et al. [249], Zienkiewicz and Taylor [5], Bathe [9] and Chopra [8] and the review articles of Weeks [250], Wilson et al. [46], Bathe et al. [251], Belytschko [252], Stricklin and Raisler [253], Adeli et al. [254], Park [255,256], Hughes and Belytschko [257], Belytschko [31 ], Hughes [32], Noor and Atluri [258], Wood [259], Dokainish and Subbaraj [260] and Subbaraj and Dokainish [261]. Among the available general and special purpose FEM computer programs for analysing nonlinear structural systems under seismic and general dynamic excitations one can mention here ANSYS, ASK.A, MARC, NASTRAN, NONSAP, ADINA etc. (Brebbia [27], Fredriksson and Mackerle [28]).

13 13.1

(168)

where M is the mass matrix, C is the damping matrix, F is the vector of the nodal internal forces, R is the vector of the externally applied nodal loads and ii, u and u are the displacement, velocity and acceleration vectors of the finite element assemblage. The usual approach for solving the above system of nonlinear differential equations of the second order is an incremental step-bystep direct time integration. Thus assuming that the solution for the discrete time t is known, the solution for the discrete time t+ L1t is obtained, where Lit is an appropriately chosen time increment. Equation (168) written at time t+Lit takes the form (169) and since the solution is known at time t one can write (170) where F is the increment of nodal internal forces due to the increment in element displacements and stresses during the time interval Lit. This vector is approximated by (171) where u is the vector of incremental displacement and 'K is the tangent stiffuess matrix defined as (172)

Substitution of F by its expression ( 171) into equation ( 169) and solution for u, results in an approximate value of the displacement (173)

Time integration of materially nonlinear dynamic equations Implicit and explicit methods

Consider a materially nonlinear (inelastic) structural system discretized spatially by the FEM. The governing equations of motion for that system read (Bathe [9])

59

'

I

and corresponding approximate values of stresses and nodal forces at time t+ Lit. Then one proceeds to the next time step calculations. The approximation introduced by the use of equation ( 171 ), i.e., the use of the tangent stiffuess matrix instead of the unknown secant stiffuess matrix, may lead to considerable error and iterations are employed within every time step to reduce that error. The most popular iteration methods for this problem are

60

Computer Analysis and Design of Earthquake Resistant Structures

Computer Analysis and Design of Earthquake Resistant Structures

based on the classical Newton-Raphson algorithm. Thus, knowing an increment of displacements, defining new total displacements, the solution procedure previously described can be repeated using the currently known total displacements instead of those at time t, i.e., fork= 1,2,3, ... ,

(174) t+M (k) U

=

t+M (k-1) U

A

+ ull

(k)

with initial conditions (175) In the above K denotes the effective tangent stiffiiess matrix, which is a function of K, M and C and F denotes the effective vector of internal forces, which is a function of M and C both being defined by the particular time integration algorithm used. However, the above iterative approach is very expensive because it requires the evaluation and factoring of a new tangent stiffiiess matrix within every time step. A more efficient iterative scheme is the modified Newton-Raphson one in which the tangent stiffiiess matrix is calculated at time t, the beginning of the time step, and is used during all iterations within that time step (from t to t+ Llt). Thus, in the modified Newton-Raphson iteration, equation (174) is replaced by (Bathe [9]) t ~ 0 (k) =t+MR_t+MF(k-1)

_F

(176)

For the particular case, of the Newmark method ~1/4, ;=112) of time integration as an example, one can prove that (Bathe [9]) t t 4 2 K= K + - M + - C M2 M

(177)

4 2 F='+t.t u(k-1J ( - M + -C)-

M2

t

M

4 2 t 4 u(M+-C)- u(-+C)-M'ii 2 ~t M At

(178)

Newton, modified Newton and especially quasi-Newton algorithms are discussed in detail in the review articles of Papadrakakis [262] and

61

Papadrakakis and Pantazopoulos [263] while applications of these algorithms to nonlinear structural dynamics can be found, e.g., in Geradin et al. [264], Bathe and Cimento [265], Herting [266] and Dyka and Remondi [267]. As it was mentioned in the case of linear dynamic analysis, direct time integration methods can be classified into two major categories: explicit and implicit methods. In seismic problems where the response is usually dominated by the lower modes (inertial or structural dynamics problems) implicit methods are mainly used because due to their inherent numerical damping characteristics eliminate the effect of higher modes on the response. However, explicit methods are also sometimes used in seismic problems, especially those involving extended soil media. Explicit methods involve no matrix equations and hence are simpler than implicit ones, which require solution of matrix equations during each time step. In general, direct time integration algorithms have to be accurate and stable. Explicit schemes require small time steps for stability (conditionally stable), often smaller than necessary for accuracy. Implicit schemes are usually unconditionally stable and the time step required is usually larger than the one in explicit schemes. As it was mentioned in linear analysis, the selection of the appropriate time step M is crucial in ensuring accuracy and stability of solution. In nonlinear problems, frequencies and wave velocities change with time. However, the practical guidelines for selecting the appropriate time step Llt described in connection with linear analysis are also used in nonlinear analysis although, strictly speaking, they are applicable only to linear problems (Bathe [9]). For. reasons of computational efficiency in nonlinear dynamics the time step Lit is desired to be large, without, however, jeopardizing accuracy. Special techniques to accomplish this are reported in Argyris et al. [52], Boise et al. [268] and Chen and Robinson [269]. A method to compute Llt for preventing interstep events has been devised by Bernal [270]. The implicit time integration methods of Houbolt [37], Newmark [39] and Wilson [271,46] described in section 3 dealing with linear dynamic analysis can also be used for solving nonlinear dynamic problems. Tables 5 and 6 provide the necessary steps for time integration of the nonlinear equations of motion (168) according to the Newmark method for /J=l/4, ;=112 (also known as trapezoidal rule or average acceleration method) and the Wilson 8 method, respectively and have been taken from Chopra [8] written in the present notation. Table 7, also taken from Chopra [8] and written in the present notation, presents the steps of modified Newton-Raphson iteration used within every time step of the above two time integration algorithms. Wilson's 8 method is unconditionally stable for 8:::_1.37 and usually provides very good results for 8=1.42. When 8=1 the method reduces to the linear acceleration method, which is conditionally stable, i.e., if Lit < 0.551 TN, where TN is the shortest natural period of the system.

62

Computer Analysis and Design of Earthquake Resistant Structures

Computer Analysis and Design of Earthquake Resistant Structures

Table 5. Newmark method (,8:=1/4, r-=1/2) for nonlinear dynamic analysis

A.

Initial calculations I. Solve M 0 ii = 0 R - C 0 il- °F 2. Select Lit 4

3. Compute A= At M+2C

B.

and

Table 6.

A. for

0

ii

B=2M

Calculations for each time step 1. sR. =Lill.+ A 1 u + B 1 ii

B.

2. Determine the tangent stiffness matrix t K

3. tK=

At

At

2

4. Solve for Liu from t K and Raphson iteration of Table 7.

7. C.

u +Liu,

1

K

ou

• /).t(.. Af A .. 6 . iiU = u+-uu 2

t+At u= 1 u +du,

t+At ii= t ii+ Liu

7 . t+M u = t u + iiu,

Repetition for the next time step. Replace t by t+ Lit and implement steps 1 to 7 in B for the next time step. C.

Other implicit and unconditionally stable time integration algorithms applicable to nonlinear dynamic problems are the stiffly stable algorithm of Park [272] the HHT or a-method of Hilber et al. [47] also used by Herting [266], the extended Newmark method of Klein and Trujillo [273] and the generalized time FEM of Kujawski and Desai [274], Desai et al. [275] and Woo and Desai [276]. The review articles of Owen [277] and Geradin et al. [278] on implicit algorithms applied to linear and nonlinear dynamics should also be mentioned. Among, the various explicit time integration schemes for dynamic nonlinear problems one can mention those of the central difference operator [9], the fourth-order Runge-Kutta method [279,280], the algorithms of Kujawski and Gallagher [281], Kujawski et al. [282] and Kujawski [283] and those of Hoff and Taylor [284] and Chung and Lee [285]. However, most of them have not been tested in seismic problems.

13.2

Calculations for each time step I. bR = (}Lill_ + A' il + B 1 ii

Newton-Raphson iteration of Table 7. 6 6 t . t .. d A •• 1 . 111)

V = 0.025 I ---v=0.0501 --·-·· V = 0. /00 I

.'111'.

I

--

,'I I I'.

------~-----~r I

L~------------1

,'I I I',

I

I

,'I I I '.

(157)

1' ~

Q,

I

0.50

,'I

I

I

,'

I

\

I

------~----;~ :

I

,'

0.25

,'

I

I

I

,'

'1'

\

I

',

I

I

_\_ - _',-.:'.., - - - - - - - l '!. \ I "-.._

I

x· ,

0.00

I

I

------'1--7--,--r I

',

~~-~---~-------

- j

I

-----1

+,..,...,.....;-~~.;:;::;:~,rtrnm:;:;::;=t=rrT'FfmT1

0. 0

0.50

1.00

1.50

2.00

P= ri>tf Ci>; Figure 6: Modal correlation coefficient [eq. (159)] as function of the frequency ratio /3 and of the modal damping.

References [l] T. T. Baber. Nonzero mean random vibration of hysteretic frames.

(159)

With the last approximation, all terms in eq. (157) are functions of the modal properties of the structure and of the response spectrum only. The approach allows to determine (approximately) the peak value of any desired response quantity without the need of rigorous random vibration methods, but using the customary and much simpler response spectrum technique coupled with modal response. Eq. (159) is plotted in Fig. 6, where the p1h are given as functions of f3 and for different values of 11. It is seen that for lightly damped systems (small v's) the correlation between modes decreases rapidly for f3 different from unity, i.e., when the modal frequencies are well separated. If all modes have well separated frequencies, it is sufficiently accurate to disregard correlations (i.e. Pih = 0 (j =/:- h) and PH = 1), and then eq. (157) coincides with the well known SRSS rule (Square Root of the Sum of the Squares): Rxk =

I

:' 1 11".

l~

= wh/w1. For 111 = llh = 11, eq. (158) simplifies to: 8112 (1 + (3) (33/2

------,------~.--------------1

0.75

where Rk(wJi 111) = q1kR(wj, vi) indicates the contribution of j-th mode to the k-th response component of the quantity of interest. Eq. (157) is the modal combination formula known as CQC (Complete Quadratic Combination) [18]. In eq. (157) the correlation coefficients p1h are functions of the form of PSD of the input process. To eliminate this dependency, one could assume the input to be a white noise. This assumption can be acceptable when the input is wide-banded, as it is normally the case for ground motion models, and when the system is lightly damped. In this case, the integral in eq. (154), which is now function of the frequencies and damping factors of the modes involved only, can be evaluated analytically, resulting in [18]: 8v1VJVh(111 + f311h)f3 3 12 (158) Plh - (1 - (32)2 + 411111hf3(l + {3 2) + 4(11J + 11~)/3 2 where

1.00

149

(160)

1

One should note that the SRSS rule may lead to significant errors if the structure has closely spaced important modes, for which p1h is significantly different from zero. The use of eq. (157) is therefore to be recommended in general.

Computers and Structures, 23(2):265-277, 1986. [2] T. T. Baber and M. N. Noori. Modeling general hysteresis behavior and random vibration application. Jnl Vibr., Acoust., Stress, and Rel. Desig., 108:411-417, 1986. [3] T. T. Baber and Y. H. Wen. Random vibration of hysteretic degrading systems. Jnl Eng. Mech. Div. - ASCE, 107(EM6):1069-1087, 1981.

[4] T. T. Baber and Y. K. Wen. Stochastic response of multistorey yielding frames. Earth. Eng. Struct. Dyn., 10:403-416, 1982. [5] A. N. Beavers and E. D. Denman. A new solution method for the Lyapunov matrix equation. SIAM J. Appl. Math., 29(3):416-421, 1975. [6] R. Bouc. Modele mathematique d'hysteresis. ACUSTICA, 24:16-25, 1971. [7] G. Q. Cai and Y. K. Lin. On exact stationary solution of equivalent non-linear stochastic systems. Int. J. Non-Linear Mechanics, 23(4):315-325, 1988. [8] T. K. Caughey. Response of a nonlinear string to random loading. Jnl Appl. Mech. - ASME, 26:341-344, 1959. [9] T. K. Caughey. Equivalent linearization techniques. Jnl Acoust. Soc. Amer, 35(11):1706-1711, 1963.

150

Computer Analysis and Design of Earthquake Resistant Structures

Computer Analysis and Design of Earthquake Resistant Structures

[10] T. K. Caughey and Fai Ma. The exact steady-state solution of a class of non-linear stochastic systems. Int. J. Non-Linear Mechanics, 17(3):137-142, 1982.

[26] K. Kanai. Semi-empirical formula for the seismic characteristics of the ground. Technical Report 35, Univ. Tokyo Bull. Earthquake Res. Inst., 1957.

[11] R. W. Clough and J. Penzien. Dynamics of Structures. McGraw-Hill, New York, 1982.

[27] Y. K. Lin. Probabilistic Theory of Structural Dynamics. McGraw-Hill, New York, 1967. Reprint: Krieger Publishing Company 1976-1986.

[12] F. Colangelo, R. Giannini, and P. E. Pinto. Seismic reliability analysis of r.c. frames structures with stochastic properties. Structural Safety, 1996. Accepted.

[28] Q. Liu and H. G. Davies. Application of non-gaussian closure to the nonstationary response of a Duffing oscillator. Int. J. Non-Linear Mech., 23(3):241-250, 1988.

[13] U.S. Atomic Energy Commission. Regulatory Guide 1.60. Design Response Spectra of Nuclear Power Plants, 1973.

[29] J.E. Luco and H. L. Wong. Response of a rigid foundation to a spatially random ground motion. Earth. Eng. Struct. Dyn., 14:891-908, 1986.

[14] H. Cramer. On the intersection between the trajectories of a normal stationary stochastic process. Arkiv. Math., 6:337, 1966.

[30] G. Monti, C. Nuti, and P. E. Pinto. Non linear response of bridges under multi-support excitation. Jnl Struct. Eng. - ASCE, 122(10), 1996.

[15] S. H. Crandall. Non-gaussian closure for random vibration of non-linear oscillators. Int. J. Non-Linear Mech., 15:303-313, 1980. [16] S. H. Crandall. Non-gaussian closure techniques for stationary random vibration. Int. J. Non-Linear Mech., 20(1):1-8, 1985. [17] S. H. Crandall and W. D. Mark. Random Vibration in Mechanical Systems. Academic Press, New York, 1963. [18] A. Der Kiureghian. A response spectrum method for random vibration analysis of MDF systems. Earth. Eng. Struct. Dyn., 9:419-435, 1981. [19] A. Der Kiureghian and A. Neoenofer. A response spectrum method for multiple support seismic excitation. Earth. Eng. Struct. Dyn., 21:713740, 1992. [20] Eurocode 8 - Design Provisions for Earthquake Resistance of Structures. ENV 1998-1-2. Brussels, 1994. [21] D. A. Gasparini and E. H. Vanmarke. Simulated earthquake motions compatible with prescribed response spectra. Technical Report R76-4, MIT, 1976. [22] A. Giuffre, R. Giannini, and R. Masiani. Seismic reliability analysis of structuress and piping systems in nuclear power plants. In Proc. of 8th WCEE, volume 7, pages 23-30, S. Francisco, 1984. [23] M. Grigoriu, S. E. Ruiz, and E. Rosenblueth. Nonstationary models of seismic ground acceleration. Earthquake Spectra, 4:551-568, 1988. [24] W. D. lwan and A. B. Mason. Equivalent linearization for systems subjected to non-stationary random excitation. Int. J. Non-Linear Mechanics, pages 71-82, 1980. [25] R. N. Iyengar and P. K. Dash. Study of random vibration of nonlinear systems by gaussian closure technique. J. Appl. Mech. ASME, 45:393399, 1978.

151

[31] N. C. Nigam. Introduction to Random Vibrations. MIT-Press, Cambridge MA., 1983. [32] A. Papoulis. Probability, Random variables, and Stochastic Processes. McGraw-Hill, New York, 1965. [33] F. Perotti. Structural reponse to non-stationary multiple support random excitation. Earth. Eng. Struct. Dyn., 19:513-527, 1990. [34] M. B. Priestley. Evolutionary spectra and nonstationary processes. J. Royal Statist. Soc., B(27):204-237, 1960. [35] S. 0. Rice. mathematical analysis of random noise. In N. Wax, editor, Selected Papers on Noise and Stochastic Processes, pages 133-249. Dover, 1954. [36] J. B. Roberts and P. D. Spanos. Stochastic averaging: an approximate methods of solving random vibration problems. Int. J. Non-Linear Mechanics, 21(2):111-134, 1986. [37] J. B. Roberts and P. D. Spanos. Random Vibration and Statistical Linearization. J. Wiley & Sons, Chichester, 1990. [38] R. Y. Rubinstein. Simulation and Monte Carlo Method. J. Wiley, New York, 1981. [39] F. Sabetta and A. Pugliese. Simulation of nonstationary time histories scaled for magnitude, distance and soil conditions. In Proc. 10th Europ. Conj. on Earth. Eng., pages 247-253, Vienna, 1994. Balkema. [40] H. B. Seed, C. E. Ugas, and J. Lysmer. Site dependent spectra for earthquake resistant design. Bull. Seis. Soc. Am., 66(1), 1976. [41] M. Shinozuka. Monte Carlo solution of structural dynamics. Comps. Struct., 2:855-874, 1972.

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Computer Analysis and Design of Earthquake Resistant Structures

[42] T. T. Soong. Random Differential Equations in Science and Engineering. Academic Press, New York, 1973. [43] T. T. Soong. Probabilistic Modeling and Analysis in Science and Engineering. J. Wiley & Sons, New York, 1981. [44] T. T. Soong and M. Grigoriu. Random Vibration of Mechanic.al and Structural Systems. Prentice Hall, Englewood Cliffs, New Jersey, 1993. [45] H. Tajimi. A statistical method of determining the maximum response of a building structure during an earthquake. In Proc. 2nd WCEE, volume 2, pages 781-798, Tokyo and Kyoto, 1960. [46] E. H. Vanmarcke. Properties of spectral moments with applications to random vibration. J. Eng. Mech. Div. ASCE, 98(EM2):425-446, 1972. [47] E. H. Vanmarcke. Structural response to earthquake. In C. Lomnitz and E. Rosenblueth, editors, Seismic Risk and Engineering Decisions, pages 287-337. Elsevier, 1977. [48] E. H. Vanmarke. On the distribution of the first-passage time for normal stationary random processes. J. App. Mech. ASME, 42:215-220, 1975. [49] E. H. Vanmarke, editor. Proc. Int. Workshop on Spatial Variation of Earthquake Ground Motion, Buffalo, N.Y., 1988. National Center for Earthquake Engineering Research. [50] Y. K . Wen. Methods of random vibration for inelastic structures. Appl. Mech. Rev. - ASME, 42(2):39-46, 1989. [51] Y. K. Wen. Methods for random vibration of hysteretic systems. Jnl Eng. Mech. Div. - ASCE, 102(EM2):249-263, 1976. [52] Y. K. Wen and C. H. Yhe. Biaxial and torsional response of inelastic structures under random excitation. Structural Safety, (6):137-152, 1989. [53] W. F. Wu and Y. K. Lin. Cumulant neglect closure for non-linear oscillators under random parametric and external excitations. Int. J. Non-Linear Mech., 19(4):349-362, 1984. [54] C. Y. Yang. Random Vibration of Structures. J. Wiley & Sons, New York, 1986. [55] A. Zerva. Effect of spatial variability and propagation of seismic ground motions on the response of multiply supported structures. Prob. Eng. Mech., 6:212-221, 1991.

Chapter 3 Engineering seismology A.S. Papageorgiou Department of Civil Engineering, Rensselaer Polytechnic Institute, Troy, NY 12180-3590, USA

Abstract Engineering (or Strong Motion) Seismology is in a period of remarkable and promising developments. These advancements along with recent destructive earthquakes (i.e., the 1989 Loma Prieta, 1994 Northridge, and 1995 Hyogoken Nanbu (Kobe) earthquakes) have precipitated important changes in the philosophy of Building Codes.(" Performance-Based Codes"). In order to achieve the more stringent design requirements imposed by the revised codes, it is necessary to synthesize reliable time-domain realizations of the ground motions that a structure may experience during its lifetime. This chapter presents an overview of some of the most important developments in Engineering Seismology in order to make them more readily accessible to structural and geotechnical engineers. The mechanics of earthquake fault rupture are discussed with due emphasis on the physics of the various pertinent phenomena. Some of the most effective numerical techniques for strong motion simulation are reviewed followed by an example of forward modeling using the deterministic kinematic modeling approach developed by seismologists.

1

Introduction

Earthquake Seismology deals with the study of the generation, propagation, and recording of elastic waves in the earth, and of the physical processes occurring at the source of an earthquake. By the term Engineering (or Strong-Motion) Seismology we mean that part of seismology dealing with earthquakes close enough to the causative source where ground motion is strong enough to pose a threat to engineering structures. The principle

154

Computer Analysis and Design of Earthquake Resistant Structures

Computer Analysis and Design of Earthquake Resistant Structures

problem of engineering seismology is the estimation of strength, frequency content, duration and spatial variability of the most destructive (in terms of its effects on a particular structure) ground shaking that is likely to occur at a site. This estimation should be based on the physics of the generation and propagation of seismic waves. According to Aki [13], the ultimate objective of current research efforts is to "compute seismic motion expected at a specific site of an engineering structure when the fault mapped by geologists breaks." In the early days of earthquake studies, before the developments of sensitive seismographs, all seismology was of necessity "strong motion seismology,'' as this is evident in the work of Robert Mallet (1810-1881) (see Muir Wood [150]; Bolt [43]), who established the basis of observational field seismology in his detailed study of the destructive Neapolitan earthquake of 16 December 1857 in Italy (Mallet [142]). In this chapter we will survey some of the most significant developments in the field of engineering seismology. The presentation is substantially based on the contributions of Aki [14, 15, 18], Madariaga [139], Spudich and Archuleta [172], Papageorgiou and Aki [155, 156, 157]. Given the limited available space, this survey cannot be exhaustive. [A more extensive survey may be found in a monograph published by the Architectural Institute of Japan [29]]. For instance, no discussion of the subject of local site effects on strong ground motion (Roesset [163]; Sanchez-Sesma [166]; Aki [17]) will be presented. However, key references are provided, and the chapter may be used by the serious students as a "road map" through the voluminous literature in order to select a course of more detailed-study that is most suitable for each one individually. For completeness of the presentation and in deference to engineers who may not be familiar with the subject, we start with a brief discussion of the elastodynamic representation theorem which is the basis of all numerical simulations of strong motion generated by realistic earthquake sources. Then, we proceed to discuss the mechanics of fault rupture (the kinematics and dynamics of rupture of a heterogeneous fault plane). In the last section we review some of the most effective numerical techniques for strong motion simulation. We start by presenting the engineering (stochastic) approach for ground motion simulation, followed by a discussion of the deterministic kinematic modeling approach developed by seismologists. We close the discussion by presenting an example of forward modeling using the deterministic kinematic modeling approach.

2

155

Tectonic Processes and the Mechanics of Earthquake Rupture

On the basis of overwhelming evidence it is now widely accepted that earthquakes are caused by the dynamic spreading of shear rupture on a fault plane (Aki [10]). This model of earthquake source is the "fault model," initially proposed by Reid [161] in his "elastic rebound theory". On the basis of deformations observed on the surface or measured by geodetic methods and seismic data obtained at local and distant stations, Reid proposed that the San Francisco earthquake of 1906 was the release of strain energy stored in the vicinity of the San Andreas fault by a slip a.long the fault. This theory stirred a lot of controversies. Aki [12] gives an historical account. of the controversies which the fault model has survived from its early days until it was firmly established in the mid 1960s when a quantitative test of the model became possible with the use of the global network of calibrated stations, the advent of large-scale digital computers and the development of an appropriate mathematical ,framework, the so-called dislocation theory, which relates the observed seismogram with the slip motion a.cross a fault plane. Furthermore, the success of the theory of plate tect~nics provides the strongest support for the fault model. The theory of plate tectonics, which describes the kinematics of the upper layer of the earth, was implicit in Reid's elastic rebound theory. It is based upon the assumption that the upper part of the crust, called the lithosphere, is much more rigid than the underlying asthenosphere. The lithosphere is composed of a number of plates which move relative to the mantle and to each other. Indeed, the consistency of plate motions with the direction and amount of slip during earthquakes everywhere is remarkable.

2.1

Kinematics of Fault Rupture

We have already pointed out above that earthquake ground motion results from unstable slip accompanying a sudden drop in shear stress on a geologic fault. Therefore, an earthquake is primarily a mechanical process. During the short span of this process, the Earth, except in the earthquake source, behaves as an elastic body. Consequently, seismic waves are linear elastic waves propagating in a very complex, nonhomogeneous, dissipative, prestressed medium (the Earth is in a prestressed state due to internal deformation and its own gravitational field). Therefore, the basic analytical tool for studying earthquakes is classical elastodynamic theory (e.g., Gurtin [83]; Achenbach [2]; Eringen and Suhubi [75]; Miklowitz [146]) supplemented with fracture mechanics (e.g., Freund [80]; Kostrov and Das [130]). 2.1.1 Source Representation Theory In order to express mathematically the ground motion induced by an earthquake, we need a formula for

Computer Analysis and Design of Earthquake Resistant Structures

Computer Analysis and Design of Earthquake Resistant Structures

the displacement - at a general point in space and time - in terms of I.he physical parameters that originated the motion. This formula is provided by I.he elastodynamic representation theorem. As noted by Aki and Richards [24), the representation theorem is a bookkeeping device by which the displacement from real~stic source models is synthesized from the displacement produced by the simplest of sources - namely, the unidirectional unit impulse, which is localized precisely both in space and time. The displacement response due to such a singular source is referred to as Green's function. [The counterpart of the elastodynamic representation theorem in the engineering discipline of Structural Dynamics is referred to as Duhamel 's integral and the corresponding Green's function is referred to as the unit impulse response (Clough and Penzien [65]; Chopra [63]).]

to be a Green's function, then a representation of the other displacement field is given by the following expression,

156

For notational purposes, let us start by introducing a few of the fundamental equations of elastodynamics. Let u, t:, r be linearized displacement strain and stress measured from some initial prestress configuration. Th~ prestress does not affect the propagation of waves, but it affects the level of strain energy liberated by an earthquake event (see Aki and Richards [24]; Box 3.4). The linearized equation of motion and the constitutive law (i.e., Hooke's law) for a general linear elastic, anisotropic body are respectively pu; = f; Tij

=

+ Tij,j

(1)

(2)

CijpqO), k= S wave number and Oi=(N-2j-l)7r/2N. Figs. 7 & 8 illustrate some numerical results. This model gives physical insight on the basic mechanisms of local surface waves generation and allowed a simple approximation for the response of a class of alluvial valleys of triangular shape [57]. Examples of application are presented by Faccioli [4]. In almost all analytical solutions use should be made of the computer to evaluate the solution i.e. the spectra and synthetics. There are some approaches that require some more computation but still are named analytical. That is the case in the study of a 30 alluvial valley of hemispherical shape under incidence of elastic waves. Lee [58] succeeded in obtaining power series of the eigenfunction coefficients and the corresponding boundary conditions. This gives an infinite linear system which once truncated can be solved. This approach is limited to the studied hemispherical shape and to low frequencies. For shallow circular geometries various analytical solutions of this type have been recently obtained (e.g. Todorovska and Lee [59-61]) for incident P and SY and Rayleigh waves.

212

Computer Analysis and Design of Earthquake Resistant Structures

Computer Analysis and Design of Earthquake Resistant Structures

8

approaches, discretization is required for the media and the boundaries, respectively.

_y_ or u 110

213

Uo

10

6

(

.........

// /

2

?

/II /II tV

'±: - -, , ,,,

lv"ol

?

/

4

v= 1/4

-... ........

8

--SH

- - N=3 - - - N•5 -·-·- N=7

-.-sv 6

'- '-'-

'~~

--.~~------_J

\ \ /// ~uf

~~

4

"

O'-~~.._~~-'-~~...J....~~--'-~~---'~~--'

60

90

120

150

180

210

e

240

Fig. 5 Top amplifications versus wedge internal angle for incidence of plane SH and SV waves.

16

20

kx= 2h 1T

/3

Fig. 7 Normalized surface amplitudes versus 2fx/{3 for a dipping layer. Rigid moving base with antiplane motion vrpiwf.. Different angles 7r/2N, N=3, 5, 7 (After Sanchez-Sesma and Velazquez [56]).

Displacement Amplitude

11=2/3 x

,·

I

I

,, /

/."'\'. \

\

\

N

\

\

\ ·, I/ '-

.,._.

I

I I

300/I I I I /

-0.5

0

0.5

1.5

2

kx/"

Fig. 6 Normalized surface amplitudes versus kx/7r. Incidence of plane SH waves upon a wedge with internal angle of 2 7r/3. Different incidence angles alm,r)dmdr

(21)

Ml

The integrals in Eq.(21) are performed in the upper region of M-R relationships at a contour line being the intensity of "a" in Fig.9. If many focal regions having different seismicity exist in the space with sources effective to a hazard, the average occurrence rate becomes the sum of individual occurrence rates from the characteristics of Poisson process, Mui

n(a)= ~ n i

As is mentioned above for Poisson process, the probability at a site that the intensity, A, of earthquake ground motions such as PGA, PGV and response spectra is in excess of some value, a, is given by using the average annual occurrence rate n(a) to be A>a,

0

;

S S fM;(m)fR;(r)P(A>alm,r)dmdr Di

(18)

The intensity determined from an attenuation law is dependent on the magnitude of earthquake and focal distance as shown in Eq.(8), therefore

(22)

Ml

When the seismicity is described by the maximum magnitude model due to the characteristic earthquake as mentioned in Sec.3.3, the average occurrence rate due to active faults is simplified as follows, n(a)= ~ n S fR;(r)P(A>alM;,r)dr

P(A>a, t)=l-exp[-n(a)t]

(19)

where A* is the median value of intensity by attenuation law in M=m and R=r, and Fu(·) is a cumulative distribution function expressing a random error as uncertainty of observed intensities of earthquake ground motions, often given by a log-normal distribution function. In the case that the seismicity within a focal region is uniform and the variables of M and R are probabilistically independent, Eq.(19) is rewritten using the probability density functions of each M and R, fM(m) and fR(r), n(a)=n 0

5.2 Probabilistic Procedures for the Assessment of a Hazard

fMR(m,r)P(A>alm,r)dmdr

Ml

where n 0 is the average annual occurrence rate of earthquakes determined from Eq.(5), and fj.,IR(m,r) is the joint probability density function in terms of variables of magnitude, M, and distance, R in a focal region. The P(A>alm,r) is the conditional probability of "A" greater than "a" predicted in M=m and R=r, and that is obtained as follows.

(17) where (' f indicates the inverse transform operator. Using the return period T,, Eq.(13) is rewritten as 1-exp[-t/T,], therefore it can be seen that taking the expectancy of an annual hazard variable with a return period of T, the same as the life-time t, has a probability of 0.63 in which a hazard exceeds the expectancy.

257

i

0

;

(23)

Di

where M; means the deterministic maximum magnitude in a fault (i). The cumulative distribution function of focal distance, R, in space in the above equation equals the probability to be Rm 'Ir) with m' =ln[Ag(R)/qi]/q 2 , therefore Eq.(21) becomes, (25) Substituting Eq.(6) into Eq.(25),

within an interval of Y-years are the maxima in the interval for each one-year. They are extracted and arranged from larger value in order and the j-th sample is named as Ai. The following annual non-exceedance probability is given for Ai by the method of Hazen plot. (28) Considering the uncertainty of observations, observations are extracted by simulating through the probability function of random errors. The well-known empirical distributions are the type-1 (Gumbel), type-2 (Frechet) and type-3 (Weibull) extreme value distribution functions. The type-2 distribution is most commonly used for seismic hazard, which is expressed in the form, k

FA(a)=exp[-(C/a) ] n( a)=n 1-B+B( a/q 1)-b'/qZ e'Ffo~b'M 1 ] S [g(r)fb q fR(r)dr} 0

(29)

{

D

(26)

If no upper boundary of earthquake magnitude exists, B is equal to 1. In

this case, the cumulative distribution function of annual maxima by Eq.(15) can be expressed by the form, b'/ 2

FA(a)=exp[-C(af q ]

(27)

where C means the constant term in Eq.(26). The distribution of Eq.(27) is the type-2 extreme value distribution function described in Sec.5.3. From the form of Eq.(27), it is found that the non-exceedance probability in terms of an intensity varies with the exponent of b'/q 2 , in other words is affected by the 'b-value' of Eq.(4) and the scaling factor q 2 for magnitude in Eq.(8). 5.3 Assessment of Annual Cumulative Distribution Function of a Hazard by Fitting Extreme Valoe Distributions

Then the following form as an extreme value distribution is proposed as to have the upper limit of annual maxima, A. (Kanda [20]). FA(a)=exp[-C{(A.-a)/a} k]

(30)

If the extreme value distribution function described by Eqs.(29) and (30) is obtained, it is easy to inversely calculate the expectancy of annual maxima ·from Eq.(17). In addition, when the empirical distribution conforms to the Frechet distribution of Eq.(29), the expectancy with an average return period of T,, AT,, can be transformed to

(31) where the subscript of (o) indicates some standard values.

6 Examples of Seismic Hazard Analyses 6.1 Objective Site and Given Conditions

Statistical analyses for wind speed etc. are frequently made in order to assess the annual cumulative distribution function by means of extracting samples of annual maxima from observations and setting their corresponding probability, and then fitting an extreme value distribution to a sample probability. For a seismic hazard, observation means the ground motion intensity estimated by an attenuation faw derived from historical earthquake data. Samples of annual maxima (extreme values)

The target site is at the center of Tokyo in Japan. Since the southern Kanto district surrounding Tokyo is suspected having some active faults hidden under the thick alluvium deposits, the historical earthquakes are used for source data, which occurred in the interval from 1585 to 1984 with the epicentral distance shorter than 150km. The locations of epicenters of selected earthquakes with a magnitude larger than 6.5 are shown in Fig.13. It is assumed in probabilistic procedures that for the

260

Computer Analysis and Design of Earthquake Resistant Structures

Computer Analysis and Design of Earthquake Resistant Structures

148.G•E

sake of convenience the seismicity is uniform within the region. The intensity of earthquake ground motions for a hazard is peak ground acceleration (PGA) at a firm ground. The attenuation law is derived from Eq.(11) for PGV at hard rock site which is converted to PGA by multiplying by a soil amplification factor of 2.0 and a conversion factor for PGA of 15 (1/sec). The hypocentral distance is calculated by using constant focal depth of 30km on an average sense.

138.S•E

141.S•E

:..:..;'.~-----...,..,...-~=------;-----,

Q 0

37. S• N

M~7.5 7.S>M~6.5

0 0

26 l

6.2 Analyses by Probabilistic Procedures

In order to obtain the magnitude-frequency relationship, the historical earthquakes with a magnitude larger than 5.5 which is the lower boundary, M1, in the 100-years interval from 1885 to 1984 are used, since smaller earthquakes before the public instrumental observation have not been recorded. The cumulative number of earthquakes larger than a magnitude divided by the interval of 100-years is plotted by circles in Fig.14. The earthquake with the largest magnitude of 7.9 is the great Kanto earthquake in 1923. The G-R equation of Eq.(4) derived from least square fitting results in Figure 13 Locations of Tokyo and epicenters of historical earthquakes with magnitude larger than 6.5. s~--~--~--~--~

a:

El. 5

>...... >-

El. 2

UJ

u

z

UJ

:::>

0

El. 1

UJ

:!:

El.EIS I

El. 92 El.Ell

(32)

Therefore the 'b-value' is 1.05 and the average annual occurrence rate, n of earthquakes with M>5.5 is 2.85. From the largest magnitude by tectonic structures shown in Fig.3, the upper boundary of magnitude, M of 8.0 is given, of which such earthquake should imply interplate earthquake in the subduction zone. Then the spatial distribution function, FR(r), in terms of epicentral distance is (7r r2 )/(7r R/) with R.=150km. Under the conditions described above, probabilistic hazard analyses are performed by using the method introduced in Sec.5.2. The resulted annual cumulative distribution functions versus PGA, called annual hazard curves are illustrated in Fig.15, where the left- and righthand-side axes show the annual non-exceedance probability in doubled logarithms and the average return period defined by Eq.(16), respectively. In Fig.15, the hazard curve shown by a solid line does not consider the uncertainty of observations by random errors. On the other hand curves shown by dashed and dotted lines consider the uncertainties expressed as log-normal distribution with the logarithmic deviation of 0.4 and 0.6, respectively. The effect of uncertainty appears on the hazard curve in the larger non-exceedance range of engineering interest. Sensitivity analyses for hazard curves are supplementary carried out using Eq.(26) without considering the uncertainly to make clear the effects of the b-value and the upper boundary of magnitude. As the 0

0

2

a:

log N(M)/Y =6.23 - 1.05M

-

-

-

I -1- -

-

-

I -1-

e

I I

-

------------6

7 MAGNI TUOE

I -1- -

I

8

-

-

---9

Figure 14 Relative frequency of earthquake occurrence from 1885 to 1984 around Tokyo. The G-R equation of Eq.(32) is shown by solid line.

,

,

t 262

Computer Analysis and Design of Earthquake Resistant Structures

Computer Analysis and Design of Earthquake Resistant Structures

263

;:: 99. 9 ~-r-r-r.------,--.-----.--,~-1888 , 1

I

:: 99. 8

I

I

cc

/

a> ~99.5

I

I

I

Lr.I

- - -

""1 -

-

/

I

6.3 Direct Estimation of Annual Hazard Curve

I I

I 1/ • /I

,'I/

I

I

~

58

l

::::::.

,'

I-

~I,/ I A T - - - ~ - - - - - ~ - - 1 I I 1 I/

z 95

cc

::::>

I

z

,'f

I I

I ,' I

results are shown in Fig.16, it is recognized that by decreasing the b-value, the expectancy at the same return period becomes larger because the frequency of earthquakes with larger magnitude relatively increases, and the upper boundary of magnitude acts at a hazard curve as if that has the upper limit of PGA in proportion to given Mu.

- -

~

I,'

0

z

I

1

289 ~

I

/

r -: r- - - - - - r- - - -

z

cc

I

cc

I

/I /

J

Lr.I

...I

,'

'

,'

f -:'- - - - r- -

-

l

98

a::

I

I

'

~ -'- - t- - - -

I

I I

J

99

0

x

/

I

Q...

u

589

I

CD

Lr.I Lr.I

1,'

1-,f- - - 'I I /I

- - - ..., - - - - r - -

...I

u z cc

,'

I I :

>-

28

I

981........4.......L..oL...L...l_ _-..J..._.....L..........1.........L.....L...L....L...LJl8 58 199 299 589 1899

ACCELERATION !GALI

Figure 15 Hazard curves for PGA in Tokyo analyzed by probabilistic

Based on the method described in Sec.5.3, annual maximum samples arranged in order with non-exceedance probability given by Eq.(28) are plotted by circles in Fig.17, which are calculated by an attenuation law without uncertainty against historical earthquakes in the interval of 400years from 1585 to 1984. The annual probability distribution functions expressed by Eq.(29) or Eq.(30) are estimated by tail-fitting to 20 samples from the largest one, of which Eq.(30) has the upper limit of 765 Gals determined from the assumption of short distance site by the largest magnitude of 7 .75 in Fig.3. The equations for extreme value distribution are as follows.

procedures. The solid line does not consider the uncertainty and the dashed and dotted lines consider the uncertainties with the logarithmic deviation of 0.4 and 0.6, respectively.

F A(a)=exp[ -{ 13.9/a} 1. 91 ]

(33)

F A(a)=exp[ -0.0012{(765-a)/a} 1.56 ]

(34)

-99.9r-r-rr-rr------,.--........,...----.-,.~,,--.,-.,.....,.....,1999 N

I

>-

:: 99. 8

I

cc

I

CD

Q...

I

99

u

x

I

••

I

//

98

-

-

-

cc

299 ~

I

0 0

I

fj/ .,A f/1

z 95

cc

'I

I

I

I

-

I

I

I

- - - - r- - - -

I I

! I I

I

I

I

I~/ , I

0

cc

I

I

If - r- -

I/

I

z z

7

l / ""1 ,

Lr.I

::::>

a::

I

---~-;/ 1:-----~--- 199 -a:: UJ Q... t/

z

...I

I

I

I '/I

0

Lr.I Lr.I

ffr--- 599

---~---1,,.t 1---~---

Lr.I

u z cc

I

I

I

a>

f

fl

I

I/ I '/' 1: ,' I: y'' t1/ I.

1

~99.5

I

I

/

---i----r~7

...I

I

,'

;

I ,'

I

-

-

~ I

-

-

-

-

~ I

59

~

where the variable "a" is in Gal. The expected PGAs corresponding to average return periods in the range of years from 20 to 300 assessed by both Eqs.(33) and (34) are almost consistent with those of the annual hazard curve shown by the solid line in Fig.15. This tendency can be understood from the fact that the exponent of 1.91 of Eq.(33) expressed as Frechet distribution is nearly equal to that of 1.72 computed by b'/q 2 from the b-value and q 2 as mentioned in Eq.(27) 6.4 Seismic Hazard Map in Japan

~

,_

~ -

-

29

99LL....a......J......L...!....'-----l--L--L-1........1......LL..LJ19 59 199 299 599 1999

ACCELERATION !GRLl

Figure 16 Sensitivity of b-value (b) and the largest magnitude (Mu) for hazard curve. The solid line has b=l.05 and Mu=8.0. The dashed and dotted lines show the variation by b of 0.95 and 1.15, and the 1- and 2dotted-chained lines by Mu of 8.5 and 7.5, respectively.

Following a similar method as that described in 6.3, seismic hazard analyses at many sites over Japan have been performed by A.1.J. [15]. Taking the PGA with an average return period of 100-years for the characteristic value, called 100-years maximum, the contour lines of sites having the same 100-years maximum are drawn as shown in Fig.18. This map points out that the seismicity at the northern region along the Pacific and the central region in Japan is high. The relation to convert 100-years maximum, A 100 , to an expectancy, AT,, with arbitrary return period is formulated by Eq.(35) on the basis of the average tendency at seven large cities distributed over

264

Computer Analysis and Design of Earthquake Resistant Structures _99,9~~~~~~~~~~~~~1888

,1

N

>-

::::: 99. 8

I I

I

I

I

I I

- - - ..., - - - - r - - ,_ - r - - -

...J

v

CD

Japan shown in Fig.19, i.e., (35)

58fil

I

a::

where T, is an arbitrary return period in years.

a:

CD

288 ~

~99.S ~

-r------r----

99

0

0

ILi ILi

01 I

I I

188 ~ ILi ~

I

u

58

x

~

:::::> .... ILi

ILi I

z

a::

0

z 95

a:

28

:::::>

z z

a:

Figure 17 Hazard curves by direct estimation using the Hazen plot. The dashed and solid lines show the relations by Eqs.(33) and (34), respectively.

Figure 18 Seismic hazard map for 100-years maximum of PGA(Gal) at firm ground in Japan (A.l.J. [15]).

6.S. Design Response Spectra

0 0

ILi

...J

265

I

a:

u z a:

Computer Analysis and Design of Earthquake Resistant Structures

Peak values such as PGA estimated by seismic hazard analysis are available to evaluate design response spectra on a probabilistic basis, if the spectra are constructed by the product of spectral shape and peak values. One such design response spectrum has been already presented by Eq.(12) in Sec.4.2. On the basis of this proposal, one example of design response spectra is presented below, which is discussed with an average return period that is the occurrence probability. Let us consider the case of the seismic design of a structure constructed in Tokyo, and apply the results of seismic hazard analyses mentioned in Sec.6.2 and 6.3. It is expected for the 100-years maximum that a firm ground site in Tokyo experiences a PGA of about 180 Gals and a PGV of 180/15=12cm/s. It is noted here that the uncertainty by random errors is ignored. From Eq.(12), design response spectra with a return period of 100-year become as shown in Fig.20. Then the spectra at soft and medium soil ground sites are evaluated by substituting PGA and PGV into Eq.(12) which are obtained by multiplying the soil amplification factor from peaks at firm ground, shown in Table 2 (A.l.J. [15]). Next, we evaluate design response spectra at a medium soil ground site in Tokyo which is assumed to satisfy the acceptable exceedance level of 10% probability during the life time of SO-years of a target structure. This condition yields that the peak values to assess design response spectra become the expectancies with a return period of 475 years from Eq.(13) of the Poisson process. Equation (35) gives (475/100) 0 ·54 =2.3, therefore 475-years PGA at a medium soil ground site is l 80x2.3x l.2=5001 Gals. The design response spectrum for 4 75 years return period is shown by the solid line in Fig.21. The dashed line represents the acceleration response spectrum expected from seismic loads at similar ground sites in Tokyo for the safety level by the Japanese national building code. Although the code regulates only shear force, we transform a base shear to an acceleration response assuming the effectiveness of structural modal mass of 0.8 to the total. Both response spectra are almost identical except in the sort period range. Thus it can be said that the seismic load at a medium soil ground site in Tokyo for the safety level by the Japanese national building code corresponds to

266

Computer Analysis and Design of Earthquake Resistant Structures

Computer Analysis and Design of Earthquake Resistant Structures

267

h=S.SS 688

..

II

.....I

.....

g

a:

4QQ

......

0 0

x 8

Sapporo Nii gala Nagoya Fukuoka

!:::. Sendai

288

ll

Tokyo

e

Ohsaka

s.2~=--~__J~~---'-~~-L~~~-'-~---.J

2s

se

1es

2ee

RETURN PERIOD CYERRI

see

1see

Figure 19 The relation to convert from A100 to ATr.

e.s

h=S.SS ISQQ

-------

.....I

:31eee

Medium

forPGA 1.2

forPGV 2.0

5

2

Figure 20 Design response spectra with the return period of 100-year in Tokyo. The spectra for firm, medium and soft soil ground sites are shown in chained, solid and dashed lines, respectively.

Table 2 Soil amplification factor from peaks at firm ground

Soil Type

1

PERIOD CSECI

a: (f")

see

---- ....

"'

''

''

''

''

''

''

................. -~~

Soft

1.2

3.0

s Figure 21 Comparison of the design response spectra for 475 years return period with ones by the Japanese national building code, which are shown by solid and dashed lines, respectively.

r Computer Analysis and Design of Earthquake Resistant Structures

Computer Analysis and Design of Earthquake Resistant Structures

the seismic load for the average return period of 475-years, or in other words, the arrival is one chance during 475 years on the average.

Seismological Society of America, 1981, 71, 2011-2038 14. Kanai,K. and Suzuki,T. (1968); Expectancy of the maximum velocity amplitude of earthquake motion at bedrock, Bulletin of Earthquake Research Institute, University of Tokyo, 1968, 663-666 15. Architectural Institute of Japan ; Recommendation for loading in buildings and its commentary (in Japanese), A.l.J., 1993 16. Watabe,M., Tohdo,M., Chiba,O. and Fukuzawa,R. ; Peak accelerations and response spectra of vertical strong-ground motions from near-field records in USA, Proceedings of the 8th Japan Earthquake Engineering Symposium, 1990, 301-306 17. Watabe,M. and Tohdo,M. ; Research on the design earthquake ground motions, Part-2 Analysis on the characteristics of earthquake ground motions for practical generation (in Japanese), Transaction of Architectural Institute of Japan, 1982, 312, 63-71 18. Hisada,T., Ohsaki,Y., Watabe,M. and Ohta,T. ; Design spectra for stiff structures on rock, 2nd International Conference on Microzonation, Vol.3, 1978, 1187-1198 19. Cornell,C.A. ; Engineering seismic risk analysis, Bulletin of the Seismological Society of America, 1968, 58, 1583-1606 20. Kanda,J.; A new extreme value distribution with lower and upper limits for earthquake motion and wind speed, Theoretical and Applied Mechanics, University of Tokyo Press, 1981, 31, 351-360

268

References 1. Usami,T. ; Extensive list of earthquakes accompanied by damages in Japan (in Japanese). New edition, University of Tokyo Press, 1987 2. Research Group for Active Faults (edited) ; Active faults in Japan ; Sheet maps and inventories (in Japanese), University of Tokyo Press, 1991 3. Matsuda,T. ; Magnitude and recurrence interval of earthquakes from a fault, Zisin, Journal of Seismic Society of Japan, 1975, 28, 269-283 4. Omote,S., Ohsaki,Y., Kakimi,T. and Matsuda,T; Japanese practice for estimating the expected maximum earthquake force at a nuclear power plant site, Bull. of the New Zealand National Society for Earthquake engineering, 1980, 13, 37-48 5. Kanamori,H. ; The energy release in great earthquakes, Journal of Geophysics Research, 1977, 82, 2981-2987 6. Utsu,T ; Relationships between earthquake magnitude scales (in Japanese),Bulletin of Earthquake Research Institute, University of Tokyo, 1982, 57, 465-497 7. Gutenberg,B. and Richter,C.F. ; Frequency of earthquakes in California, Bulletin of the Seismological Society of America, 1944,34, 185-188 8. Utsu,T. ; Seismology (in Japanese, 2nd ed.), Kyoritsu Publication, 1984 9. Wesnousky,S.G., Scholz,C.H., Shimazaki,K. and Matsuda,T.; Integration of geological and seismological data for the analysis of seismic hazard: A case study of Japan, Bulletin of the Seismological Society of America, 1984, 74, 687-708 10. Shimazaki,K. and Nakata,T. ; Time-predictable recurrence model for large earthquakes, Geophysics Research Letters 7, 1980, 279-282 11. Kakimi,T. ; 3.2 Faults and earthquakes, in Seismic design of nuclear facilities edited by Ohsaki,Y. and Watabe,M. (in Japanese), Sangyo-gizyutu Publication, 1987, 98 12. Watabe,M. and Tohdo,M. ; Research on the design earthquake ground motions, Part-1 Literature survey and maxima of earthquake ground motions (in Japanese), Transaction of Architectural Institute of Japan, 1981,303,41-51 13. Joyner,W.B. and Boore,D.M. ; Peak horizontal acceleration and velocity from strong-motion records including records from the 1979 Imperial Valley, California, earthquake, Bulletin of the

269

Chapter 6 Soil-structure interaction H. Antes & C.C. Spyrakos Institute ofApplied Mechanics, Technical University of Braunschweig, D-38106 Braunschweig, Germany, and Department of Civil Engineering, National Technical University of Athens, GR-15700 Athens, Greece

Abstract This chapter focuses on computerized analysis and design of above-ground structures under dynamic loads, such as earthquakes with special regard to the effects of soil-structure interaction (with the exception of dam and tank problems where the fluid plays an important additional role). First, a detailed state-of-the-art review of various methodologies that ranges from experimental techniques and quasi-static, one-dimensional semi-analytical approaches to numerical procedures for transient, three-dimensional problems is given. The review also discusses special effects like uplifting phenomena. Second, currently used popular numerical methods are presented: namely, the Finite-Element method (FEM), the boundary element formulation (BEM), and hybrid approaches which combine the last two methods (e.g., FEM-BEM). Then, some numerical examples illustrate the effectiveness of the techniques and demonstrate the importance of soil-structure interaction. Finally, a short report on available commercial software that points out their respective capabilities in taking soil-structure interaction into account is presented.

1

Introduction

Vibrations of soils and dynamic interaction between soils and structures are two subjects that have received considerable attention in recent years. There is a variety of practical engineering problems associated with these general areas, e.g., design of foundations and retaining walls; response of structures to blast loading or earthquakes; dynamic interaction between adjacent foundations or piles and structures through the soil; and the as-

Computer Analysis and Design of Earthquake Resistant Structures

Computer Analysis and Design of Earthquake Resistant Structures

sessment of vibrations caused by transient waves. Extensive studies conducted in the past three decades have demonstrated that - in general - soil-structure interaction (SSI) has the following effects: (1) reduction of the resonant frequencies of systems in comparison to those of the fixed-base structure; (2) partial dissipation of the vibrational energy of the structure through wave radiation into the soil; and (3) modification of the actual foundation motion from the free field motion. There is a great range of above-ground structures where the dynamic interaction plays an important role, e.g., multi-storey buildings ([1], [2], [3], [4]), nuclear power plants [5], towers [6] or chimneys [7], and bridges ([8], [9]). But, in any case, the dynamic system whose response is to be determined consists of two distinct parts with different properties: the generalized structure with bounded dimensions that consists of the actual structure with its foundation and possibly an irregular adjacent soil region, and the unbounded soil extending to infinity. The structure is modelled either by simple single- or multi-degree-of-freedom equivalent oscillators or by finite elements. Modelling of the infinite soil must contain a representation of its material and topography, and, in particular, of the radiation condition [10]. Both interacting parts, the structure and the soil can behave linearly or non-linearly [11]. Hence, rational and successful design of this kind of complex systems requires both the verification of soil-structure interaction data and the availability of accurate and efficient methodologies for dynamic analysis.

et al [17] on that of sliding gravity-type retaining walls, and Tamori et al ([18], [19]) on the effects of plastic deformation of soils to soil-pile-building interaction. Shaking tables mounted in geotechnical centrifuges are used in order to study, e.g., liquefaction of horizontal ground [20], degradation of stiffness in quay walls retaining saturated fill [21], free-standing cantilever retaining walls [22], pile cap response dependence on the fundamental frequency of the soil-pile system [15], or failure and settlement characteristics of deep and shallow foundations in sand [23]. When shaking table experiments are not available or possible, forced vibration tests are very often conducted mainly to determine the dynamic characteristics of small- or full-scale models, but also to analyze soil-structure interaction effects. In these tests, small vibration generators are installed on the structure to produce both horizontal and vertical oscillations, and the response is recorded by displacement controllers and load measurement cells. Due to its simplicity and rather easy application, this method has been widely used, e.g., for a five-story steel frame building [24], for a large massive structure embedded in quarternary ground [25], for a reactor building [26], for a frame-footing model [27], and for an embedded foundation supported by pile group [28]. Kobori et al [24] studied also the effects of the vibrator position on the structural response. When using models, the scale effects in dynamic testing of structures are very important. Some scaling laws and a state-of-the-art report of those effects have been given by Krawinkler [29], and for centrifuge shaking tables by Schofield et al [15]. More detailed studies on specific structures and aspects may be found in the papers of, e.g., Sabnis et al [30] on size effects in concrete or of Wallace et al [31 J in the simulation of cracks, and of Tomazevic et al [32] about scale effects on stress-strain properties of masonry. As an alternative means for laboratory evaluation of the seismic performance of structures, the so-called pseudodynamic test method has recently attracted much attention. This method is a hybrid methodology based on a quasistatic test procedure in which the dynamic response of a structure is simulated numerically in an on-line computer by means of a direct stepby-step time integration algorithm. Comprehensive reviews of the current advances of the method have been provided by Takanashi and Nakashima [33] and Mahin et al [34]. However, there are three major sources of inaccuracies associated with the test procedure: The first is related to the analytical idealization of the structural specimen [35]; the second is the numerical errors associated with a time integration algorithm ([35], [36]), and the third is the experimental feedback errors introduced in the test procedure ([37], [38]). Nevertheless, comparison of the results of the pseudodynamic test with those of shaking table tests [39] have indicated that such effects are insignificant for steel structures under seismic load conditions. The conventional pseudo-dynamic testing method has been improved

272

1.1

Experimental Methods of Analysis

When soil-structure interaction (SSI) analysis method is considered in design calculations, the method should assure for its capability of predicting reliable results or otherwise rather conservative results for structural response. Such assessment for the methodology can be made by comparative examination of measured results with their corresponding predicted results. This requires some good measured data available from either seismic observations or experiments. Most of the presently available data from seismic observations evaluated from the viewpoint of SSI are given by Tajimi [12] in a state-of-the-art report and consist of embedded structures and pile-supported buildings, e.g., of a reactor building [13] and of a reinforced concrete tower [14]. Since these data are not sufficient for general applications, more often tests are performed by either laboratory experiments or in-situ tests. The main equipment for experimental study of geotechnical models is the shaking table, recently also in combination with geotechnical centrifuges [15]. There exists a large number of relevant reports on such experiments either for small- or for full-scale models and in studying linear as well as nonlinear effects from which only a small number are mentioned here. Magenes et. al [16] report on the seismic behaviour of unreinforced masonry, Uwabe

273

274

Computer Analysis and Design of Earthquake Resistant Structures

Computer Analysis and Design of Earthquake Resistant Structures

by Toki et al [40] by introducing a time-dependent pseudo-forcing function which is derived from the frequency-dependent dynamic characteristics of the system.

analyses [60]. As applications, one can mention here in the case of discrete structural modelling, i.e., using some analogy with a single- or multi-degree-of-freedom fixed-base oscillator, Bielak [61] studying buildings on a prismatic embedded foundation, Ganev et al [62] analyzing the effects of separation of the structure, a reinforced concrete tower from the soil, Anagnostopoulos et al [63] investigating the pounding between adjacent buildings. b) analytical or semi-analytical methods which are developed for wave propagation and scattering problems based on the theory of linear elasticity or viscoelasticity, and are restricted to idealized models. Hence, only simple problems like different types of foundations, piles and buried structures (see, the chapter on underground structures) that act as an obstacle in an otherwise homogeneous medium are considered. In a state-of-the-art paper, Gazetas [64] reported on several solutions for foundation vibrations, e.g., of Novak for circular foundations embedded in a half-space [65] and for piles in layered media [66]. More recently, framed structure-layered soil interaction [67] and pile-soil-pile interaction ([68], [69], [70]) and the response of piles during the passage of Rayleigh waves [71] have been studied by an analytical method based on a dynamic Winkler foundation model. In this context, one can also mention the use of displacement Green's functions to study the interaction between several flexible strip foundations [72]. Here, one should also register the so-called cone (wedge) model of Wolf (e.g., [73], [75]) to determine dynamic-stiffness coefficients of surface foundations which are truncated semi-infinite rods (bars) with one-dimensional wave propagation occurring along their axes in vertical direction, and are accurate enough for many practical applications. Applying this model, Wolf studied many interaction problems, e.g., the effects of modelling surface foundations in 2D instead of in 3D [74]. Besides, there exist also some analytical results for incident and scattered fields in the far-field region [76] which can be combined with numerical models for the near-field, e.g., with the Finite-Element Method [77]. c) numerical domain methods that require interior discretization of the domain in addition to the boundary one, such as the Finite Difference Method (FDM) and especially the Finite-Element Method (FEM) formulated in the frequency domain or in the time domain. These methods are used for modelling the structure as well as the soil medium under two or three dimensional conditions and involving or not nonlinearities of the structure and/or the soil. While the FDM is rarely applied to dynamic soil-structure interaction problems since it is not suitable for complex geometries, the FEM is a versatile technique since it can handle arbitrary structural geometry, medium inhomogeneities and complex material behaviour. The only disadvantage is the difficult realization of 'non-reflecting' boundaries when infinite soil domains are involved.

1.2

Semi-Analytical and Numerical Methods

In general, as it was classified by J. Wolf [41], the existing methods for studying the relevant interaction phenomena in dynamic analysis of structures lead to either a substructure method or in a direct method. The main difference is that the substructure method uses the radiation condition formulated at infinite which results in rigorous boundary conditions being global in space and time, while in the direct method approximations are used in introducing transmitting boundaries, i.e., the radiation condition is formulated on the interaction horizon between the generalized structure and the infinite soil in such a way that a highly absorbing boundary condition results which is local in space and time. Based on various mathematical principles many apparently different local transmitting boundaries have been developed [41] in the frequency as well as in the time domain: the viscous damper ([42], [43]), the paraxial approximation ([44], [45]), the extrapolation algorithm [46], and the superposition boundary ([47], [48]) to name just a few. It has been shown ([50], [49]) that all the transmitting boundaries mentioned above are actually mathematically equivalent in the continuum limit. They can be classified as being of first order, second order, etc., whereby higher-order schemes may lead to dynamic instabilities. A review of transmitting boundaries and a discussion of their limitations has been given by Mengi et al. [52] and by Bettes ([53], [54]), and a stability criterion has been derived by Liao [51]. Besides this classification in a direct method or a substructure method, the existing methodologies to study dynamic interaction can be classified as follows: a) the so-called dynamic method that employs concentrated masses, springs, and dashpots for simulating the soil or a rigid massless foundation on the soil and a mostly discrete system for modelling the structure. In this approach, the main problem is to define an adequate representation of the mass-spring-dashpot system. In the beginning, only three frequencyindependent parameters had been used, e.g., in the models described by Barkan [55] and Lysmer and Richart [56], but later studies, e.g., of Lysmer [57] and Luco et al. [58] showed that the properties of the springs and dashpots are required to depend on the frequency of the excitation. Nevertheless, the desire for simple models have motivated to search for improved discrete frequency-independent models, e.g., the five-parameter model of Barros and Luco [59] gives a good approximation of the vertical impedance function for circular, square and rectangular rigid foundations resting on the surface or embedded in an elastic half-space over a wide frequency range. Moreover, an improved two-spring model has been developed also for foundation uplift

275

Computer Analysis and Design of Earthquake Resistant Structures

Computer Analysis and Design of Earthquake Resistant Structures

Hence, there is a plethora of FEM works on SSI from which only some are mentioned here to show the great variety of those studies, e.g., foundations resting on an elasto-plastic half-space [78] or on two-phase saturated porous soil (79]; frame structures with a raft foundation on layered elastic soil [80], on an elastic-perfectly plastic homogeneous subsoil (81] with strain hardening characteristics (82] and on soil with a hyperbolic stress-strain behavior [83]; reinforced retaining walls [84] and the effect of backfill on it (85]; and long-span cable-supported bridges (86]. Obviously, the realistic constitutive modelling of the soil has gained much attention in FEM approaches. In this context, besides the above SSI applications, there exist several inelastic soil models ([ll], (87]), e.g., the cap-plasticity-model ((88], (89]) or the Prevost-Iwan multiple-yield-surface theory of plasticity (90]. d) the Boundary Element Method (BEM), a numerical method which for linear problems and homogeneous media requires only a boundary discretization of the considered domain. Moreover, in dynamics [91], it represents exactly all wave scattering and refraction phenomena as well as wave radiation in infinite domains. A significant limitation of this method is, however, that only linearly elastic or visco-elastic homogeneous domains can be treated without increasing the computational effort considerably, e.g., by using subdomain techniques or interior elements. An excellent, rather complete review on applications of the BEM in dynamic interaction between foundations as well as underground structures and the soil has been given by Beskos [92]. Other use of the BEM for both the structure and the soil medium is rather rare in the frequency domain as well as in the time domain since structures are often non-homogeneous and behave non-linearly. Only in the field of dam-reservoir problems, Dominguez and Maeso ((93], (94]) presented a 3D BEM analysis for the complete system, i.e., the arch dam, the water, and the soil. Additionally, one can mention some works considering special, but for SSI important, aspects like vibration reduction by trenches in the soil, e.g., by Beskos et al [95], Klein et al (96], and Mateo et al [97], or unilateral contact, again mainly between foundations and the soil ((98], (99]). e) hybrid numerical methods that combine a numerical domain method like the FEM for the structure or for the structure and a soil portion surrounding it, and an analytical or a numerical boundary method for the remaining soil medium which extends to infinity. More specifically, there are hybrid methodologies in the frequency domain using the FEM for the structure and its near soil surrounding and analytical methods for the remaining soil, e.g., for two-dimensional problems by Eilouch et al (100], and in 30 by Gupta and Penzien [101] and by Kundu et al (77]. Recently, more often, the FEM and the BEM have been combined to a hybrid methodology where their coupling can be performed either pointwise or via variational conditions; a comprehensive literature

survey on this coupling can be found in Pavlatos and Beskos [102]. There are applications of this approach in the frequency domain, e.g., by Huh and Schmid (103], Kobayashi and Mori [104], Nishimura et al [105], Auersch and Schmid (106], Klein et al (107], and by Capuani et al. (108], as well as in the time domain, either as a direct time-stepping scheme , e.g., by Karabalis and Beskos (109], Spyrakos and Beskos [llO], von Estorff and Kausel [111], Spyrakos and Patel [112] assuming elastic behaviour of the structure and the soil, and Pavlatos et al [113] for elastoplastic structures, or via a causal FFT treatment as, e.g., by Hayashi and Takahashi (114], and using a step-by-step method in the Laplace domain [115], respectively. As it is evident from the aforementioned five methodologies, the analytical ones are of very restricted application, and hence, numerical methods have to be used for the solution of realistic problems in an accurate and inexpensive manner. It is also evident that the most appropriate method for structural analysis is the FEM, especially in three dimensions. This is attributed to shortcomings of FDM in considering problems with complex boundary conditions, while the BEM is not so efficient for finite bodies with a high surface to volume ratio. In addition, the FEM can very easily take into account anisotropy, nonhomogeneities and nonlinearities of the structure and a portion of the surrounding soil, which are difficult to handle with the BEM. On the other hand, dynamic analysis of infinite or semi-infinite bodies, such as soil media, is handled ideally by the BEM which takes into account automatically the radiation condition and requires discretization of only the soil-structure interface or - in addition - a finite portion of the free soil surface depending on the kind of the employed Green's function. Use of either the FEM or the FDM for the soil would require not only a discretization of the interior and the artificial boundaries of the soil body but also use of special devices in order to avoid wave reflections on these boundaries. Even though quite a number of non-reflecting or absorbing boundaries have been proposed over the years (see above, e.g., (42], (43], [44], [45], (47], [48], (49], [52], (53], [54]), none of them covers exactly all the cases of wave propagation, especially in three dimensions, while an increase of the computational cost is always observed. It is also obvious that use of accurate or approximate analytical methods to modell the soil is not as good as use of the BEM with respect to generality and accuracy. Thus, one reaches the conclusion that the hybrid methodology that employs the FEM for the structure and the BEM for the soil medium is the best way for dynamically analyzing soil-structure interaction (SSI) problems since it exploits their respective advantages and minimizes their disadvantages.

276

277

278

2

Computer Analysis and Design of Earthquake Resistant Structures

Computer Analysis and Design of Earthquake Resistant Structures

The FEM, the BEM, and the BEM-FIS Formulation in Elastodynamics

w. In this case, the time dependence can be removed by transforming the governing equations via Laplace transform. This transforms the equations of motion (1) into the equation

Assuming homogeneity and linearly elastic material, the dynamic behaviour of structures as well as wave propagation in domains is governed by the equations of motion

(C12 - £:22) Ui,ij

2 .. + £:2Uj,ii - Uj

bj p

(1)

= --

where ui = ui(x, t) is the displacement vector at point x and time t, bi is the body force vector per unit mass, c1 and c2 are the dilatational (pressure) and shear wave velocities, respectively, which are given either in terms of the Lame constants ..\ andµ or of Young's modulus E and Poisson's ratio v for three-dimensional domains as well as for plane strain state by 2 Ci

..\ + 2µ

E(l - v) = p(I

+ v)(l - 2v)

E C2 = 2p(l + v) 2

= -p-

µ

p

and for a plane stress state by

E

2 -

Ci -

p(I

E

2

+ v)(l - v); C2

= 2p(l

(3)

+ v)

with p being the mass density of the body. Latin indices, e.g., i, j receive the values 1, 2, and 1, 2, 3 in 2-D and 3-D, respectively, where summation convention is implied over repeated indices, and commas and overdots denote spatial and temporal differentiation, respectively. Furthermore, the motion of a structure or of the soil medium needs to satisfy the following conditions on the boundary r = r 1 + r 2:

u;(x, t) = ii;(x, t) fort > 0onr 1 p;(x,t)=p;(x,t) /ort>Oonf 2 and initial conditions at time t = Oon

(4) (5)

nu L

u;(x, 0) = ii;0 (x) and u;(x, 0) = ii;0 (x)

(6)

where the prescribed values are indicated by overbars. p; denotes the tractions along the boundary with the outward normal vector ni, i.e., p; = CJ;ini, where the stresses are defined as displacement derivatives according to

CJ;i = p [o;i ( ci - 2~) = P [o;i

+ 2~t;i] {::} u = Cee ( ci - 2~) uk,k + ~ (u;,j + Uj,i)] {::} u

(7)

Ckk

= CeDuu.

(CJ:2 - C22)·Ui,ij +2· £:2Uj,ii

(8)

When the dynamic behaviour under harmonic, i.e., sinusoidally varying excitations is predicted, the response is a function of the angular frequency

-

2·

s Uj

- +-) 1. = - (lb· p j + SUjo Vjo = -µqj

(9)

where quantities marked now by '?indicate the transformed variables, and s is the complex transformation parameter of the Laplace transform. The initial values are grouped in equation (9) together with ~bi in a modified body force term ~qi. The transformed equations must be solved together with the transformed pertinent boundary conditions on r1 and r2. The special case of harmonic excitations is contained in equations (9) when s is replaced by iw, and the initial conditions are neglected.

2.1 (2)

279

Elastodynamic FEM equations

The starting point for the numerical treatment of elastodynamic problems using the FE approach is an integral variational formulation arising from the equations of motion (1) (for details, see Bathe [117]). According to the assumed linear behaviour of all systems considered in this paper, a straindisplacement relation for infinitesimal small motions and Hooke's material law are used. It should be mentioned, however, that either of these equations also might be of nonlinear nature, for instance, Hooke's law could be replaced by an elastoplastic material law, which is useful, i.e., to simulate a more realistic behaviour of soil or parts of the structure. The displacement FE method is applied to determine the spatial distribution field, while the time dependence is either removed in the case of harmonic excitations or the time integration is treated with a finite difference method (FDM).

2.1.1

Frequency domain analyses

For three-dimensional domains n with the surface r, the principle of virtual work including the inertia effects for time-harmonic motions with frequency w may be used as a basic variational formulation. It is defined as

According to the displacement FEM, the domain is discretized into a finite number L of elements ne over which the displacement vector ui(x) is approximated as a

uf(x) =

L

N~(x)u?e

(11)

a=l

ure

where N~(x) are the shape functions and are the nodal displacements. Then, via the relation (11), the principle of virtual displacements

t 280

Computer Analysis and Design of Earthquake Resistant Structures

Computer Analysis and Design of Earthquake Resistant Structures

(10) yields the element stiffness matrix K(e), the element mass matrix M(e), and the element body and surface load vector rgl and f~e), respectively, to be (12)

the natural frequencies of the system affect the response due to large values of the dynamic magnification factor. The spatial variation of the load has an effect on which modes will be mostly excited (see Spyrakos [118]). Besides the useful insight into the dynamic behaviour of a structure by modal analysis, there is an additional advantage: the size of the system may be reduced by a transformation to the modal subspace which may be advantageous for soil-structure analysis. Although depending on the frequency range of interest, the number m of determined natural frequencies w; is generally much smaller than the size n of the system. Hence, the matrices K and M, the state vector u, and the 'load' vector f may be reduced to the modal subspace. Multiplying by the modal matrix (16)

(13) where Du denotes the differential operator relating strains to displacements, C 0 contains the material law, and b and p are body and surface forces, respectively. The stiffness matrix K, the mass M, and the load vector f of the total discretized domain can be obtained by assembling all element matrices. Finally, the FEM equations describing harmonic vibrations of elastic or viscoelastic domains are

(14) FEM equations for other types of elastic structures may be derived similarly, e.g., for elastic shells modelling tank walls [119] or for coupled shear walls modelling tall buildings [108]. The choice to perform either static or dynamic analysis, that is, whether to include or neglect the inertia effects and the time variation of the loads, is usually based on engineering judgement. Generally, when the loadmg is slowly applied, the dynamic effects can be ignored, e.g., when the rise time of a ramp load is at least three times longer than the fundamental period of a basically vibrating with the first mode structural system. However, when the frequency of the excitation exceeds about one-third of the structure's fundamental frequency, inertia effects become important and the problem should be treated as dynamic. Regardless of what type of analysis is selected to solve a linear dynamic problem, modal analysis should be the first step. Natural frequencies and mode shapes are required for time history modal superposition, frequency response, response spectrum, and random vibration analysis, i.e., it provides useful insight into the dynamic behaviour of a structure. The objective of modal analysis is to calculate the natural circular frequencies w; and the corresponding i-th mode shape ;, which is performed by searching for non-zero solutions of the homogeneous eqns (14), i.e., for zero loadings. This requires that the determinant of eqns (14) is zero, that is (15) det (K -w;M) = 0. For each solution w~ of this so-called characteristic equation of the system, we can solve the homogeneous eqns (14) for ;- As a general rule, the number of modes to request in modal analysis depends of the natural frequencies of the system as related to the frequency content and spatial variation of the applied loads. The loads with frequencies that are close to

281

containing the m calculated right latent eigenvectors, this transformation can be easily carried out:

where the matrix A contains in its diagonal all the determined natural frequencies (w;) 2 , i = 1, 2, .. , m. For closed domains, one obtains the reduced system (18) (K* - w2M*) z = f* which has exactly only those (not more) natural frequencies and right latent eigenvectors that have already been determined. Moreover, the reduced matrices K* and M* are diagonal due to the orthogonality of the eigenvectors with respect of the stiffness matrix K and the mass matrix M. It should be mentioned that these reduced matrices are related to the modal states and can also be used for direct time integration procedures

[119]. 2.1.2

Time history direct integration

The main advantage of direct integration over modal superposition is that direct integration can be used for both linear and nonlinear dynamics problems. Direct integration should be preferred when many modes must be included and the response is required for a short duration of time, such as in shock vibration problems. This method uses time step-by-step integration algorithms to solve the FEM system of coupled equations of motion which are obtained, as described above, but through a time domain variational formulation or a Galerkin weighted residual formulation and using time-dependent displacement approximations a

uHx, t) =

L a=l

N~(x)ufe(t),

(19)

f' 282

Computer Analysis and Design of Earthquake Resistant Structures

Computer Analysis and Design of Earthquake Resistant Structures

and an additional collocation at time ntlt,

(24) (20)

where u

0

-0 :::; Q)

Table 1: Parameter relationships for strong ground motion

rJl

a..

1 L---'"--'--'-'-...._.......__ 0.01 0.1

_.__._.........................._

__.___..__,_...............,

10

Rock Alluvium

~

vg I ag (cm/sec/g)

agug Iv;,

71.0 122.0

5.9 6.0

Period (sec)

Figure 4: Elastic response spectra of the 1940 El Centro record (comp. NS) in tripartite logarithmic plot

To obtain the NH spectral ordinates, designated as A, V and D in Figure 5 for the constant acceleration, velocity and displacement regions, respectively, the

380

Computer Analysis and Design of Earthquake Resistant Structures

Computer Analysis and Design of Earthquake Resistant Structures

parameters ag, vg and ug of the ground motion envelope must be multiplied by the corresponding amplification factors given in Table 2, reproduced here from Newmark and Hall [49].

spectrum is referred to as elastic response spectrum, is symbolized by Se and is given by the following functional form:

I Table 2: Spectrum amplification factors for elastic horizontal response Damping (% Critical) 0.5 1 2 3 5 7 10

20

p

( I

One Sigma (84.1%) A D v 3.04 5.10 3.84 4.38 3.38 2.73 3.66 2.92 2.42 2.24 3.24 2.64 2.30 2.71 2.01 2.36 2.08 1.85 1.99 1.84 1.69 1.26 1.37 1.38

v 2.59 2.31 2.03 1.86 1.65 1.51 1.37 1.08

D 2.01 1.82 1.63 1.52 1.39 1.29 1.20 1.01

These factors are given for eight values of damping and for two levels of non exceedance probability: 50% and 84.1%. For periods below 0.12 sec, the design spectrum starts slopping towards the constant acceleration brach of the ground motion envelope and becomes identical to it for periods less than about 0.03 sec. For structures with longer periods, say greater than 1. 0 sec, these spectra are not conservative enough for the low modes of vibration in a modal type of analysis and definitely not conservative enough for an equivalent static analysis. This is due to uncertainties in the combination of modal responses as well as due to other factors, such as inadequate representation of ground motions for distant earthquakes. For this reason, a modification of these spectra has been suggested by Newmark and Hall, for the period range between 1.0 sec and 6.0 sec. Although the NH spectra are considered fixed shape spectra, they are not quite so. They are fixed only as far as the constant amplification factors for the three spectral regions are concerned. The ground motion envelope, on the other hand, is not fixed but can be drawn to reflect local soil conditions, e.g. in accordance with the ratios v g I ag and ag ug Iv~ in Table 1, or using the more detailed values for these ratios from other studies that may reflect the influence of earthquake magnitude or focal distance on the ground motion parameters. The smooth spectra developed by Seed et al [64] were based on strong ground motions recorded on sites classified in four broad categories with respect to the local soil conditions: rocks, stiff soils, intermediate soil sites and soft - deep soil sites. These smooth spectra, as well as those by Newmark et al [48], Newmark and Hall [49] and Mohraz [42], were used as the basis for the ground motion spectra specified in the well-known ATC-3 recommendations [71]. Subsequently, they were adopted by the UBC code [73] and the European prestandard EC8 [18]. In EC8, the ground motion elastic acceleration design

ll

T

Se= agSll +Ta (2.5ri- l)

s, = 25 ••s~

Median (50%) A 3.68 3.21 2.74 2.46 2.12 1.89 1.64 1.17

s, = 25 ••s~(T~) Se= 2.5

381

l

JI

r

I

(13)

agSri(~~J(T~ j

where Ta, Tc, To= corner periods given in Table 3, ag = design ground acceleration in g's, S = soil parameter from Table 3, ri= ~7 I (2 +I;) ~ 0.7 damping correction factor with reference value equal to 1. 0 for a damping ratio I;= 5 (%). These spectra for ri = 1 are shown in Fig. 6. Table 3: Parameters of Elastic Response Spectra for EC8 Soil Class A B

c

s

TR(s)

Tc(s)

T0 (s)

1.0 1.0 0.9

0.10 0.15 0.20

0.40 0.60 0.80

3.0 3.0 3.0

From the foregoing presentation it follows that for each of three broad categories of local soil conditions (or four according to some codes), the code elastic design spectra are specified by a fixed shape and a scaling factor: the peak ground acceleration that characterizes the seismic hazard of the area. This description, however, fails to account for any influence that earthquake magnitude or source-to-site distance might have on the frequency content of the motion and consequently on the shape of the spectrum. Thus, for sites affected by large distant earthquakes, such a procedure will understimate spectral velocities, except for high frequencies (McGuire [41 ]). As a consequence, it can be concluded that this approach does not provide a constant nonexceedance probability for all spectral regions. To alleviate this problem, an approximate procedure has been introduced in ATC-3, which allows the specification of a variable spectral shape through the introduction of a second variable related to the peak ground velocity. This procedure requires two seismic hazard maps, one for the effective peak ground acceleration and another for the effective peak ground velocity.

382

Computer Analysis and Design of Earthquake Resistant Structures

r

Computer Analysis and Design of Earthquake Resistant Structures

383

The increasing number of strong ground motion records worldwide has provided the means for statistical evaluations of the effects of earthquake magnitude and distance on spectral shapes and for evaluation of the so-called uniform hazard or probabilistic spectra. Derived by regression analyses performed directly on response spectral values of recorded ground motions, such spectra are given, as their name suggests, for uniform probability of nonexceedance, constant for all periods (McGuire [40) , Trifunac and Anderson [72], Joyner and Boore [29], Katayama [32]). Uniform hazard spectra are the basis of the design spectra of the New Zealand code [47].

1000.0.------....----.-----....,......,.--,..----~----.

~ -..

E

u

>-

1(J

0

...J

w

>

2.4 Inelastic response and design spectra

The inelastic response spectra are defined in a manner analogous to the elastic response spectra, but are computed for SDOF oscillators with non-linear resistances. In the simplest and most usual case, the resistance of the oscillator is assumed elasto-plastic with an initial stiffhess k, yield displacement Uy and maximum or yield force Fy (Fig. 7). The equation of motion is the same as eqn (3) for an elastic system, except that the term olu in the left side is now replaced by F(u)/m, where F(u) is the resistance force, function of the displacement u. In addition, a new parameter is introduced, the ductility factor µdefined asµ= umaxluy. We note that for such systems the maximum force that the earthquake can cause in the oscillator is Fy, a known property of the oscillator. The quantity of interest now is Umax and even more so the ductility factor µ that characterizes non dimensionally the inelastic deformation demand

l.0'-------''""---+------'"'---1------->~

0,01

10.0

1.0

0.1

PERIOD (sec) i·' i

Figure 5: Newmark - Hall design spectrum (Ilg= 1.0 g)

2.5 'I

F

.\\\ \ \

A

\"\/ B} ELASTIC (.... u 0 ...J w

>

0 0

ii3 10.or.,,___-----t'-*'-+---

lfl Cl..

0.1

1.0

PER I OD

10.0

(sec)

Figure 9: N~k-Hall inelastic acceleration (Sa) and displacement (Sd) design spectra for µ = 4

388

Computer Analysis and Design of Earthquake Resistant Structures

3 Elastic analyses of buildings for earthquake actions 3.1 Modelling and analysis considerations

~'I'

I

~·

!

~

I

,,I

11'

~ t. ~·

,,,,I

'

The first step in designing an earthquake resistant building is the selection of an appropriate structural form and construction material, within the architectural constraints dictated by functional requirements. This is part of the preliminary design phase, in which initial member sizing for the selected structural form is also done based usually on experience or on simplified calculations. For optimum results, the structural designer should follow some basic and time tested principles of conceptual design, e.g. selection of a well defined lateral load resisting system, avoidance of discontinuities and irregularities in plan and in elevation, symmetric placement of stiff elements if possible on the perimeter of the structure, proper consideration for safe transmission of the earthquake generated loads to the soil medium through a strong, stiff and continuous foundation etc. Decisions on these issues can affect the safety and lifetime perfomance of a building under seismic conditions. They are addressed in more detail elsewhere in this book. The second step in the design process is to create an appropriate model of the building that will adequately represent its stiffness, strength and mass distribution, so that its response to an earthquake could be predicted with sufficient accuracy. This model and its degree of sophistication are normally dictated by the analysis and verification requirements specified in the applicable code. Most modern codes permit static analyses for buildings that meet certain regularity criteria and limitations on period or height and require dynamic analyses if such criteria and limitations are not met. The following Table 4 from EC8 is quite specific in this respect and indicative of the current trend towards more sophisticated analyses. The codes (or pre-standards) permit elastic Table 4: Model and analysis requirements per EC8 ReS!!lari!Y Plan Elevation YES YES YES NO NO YES NO NO

Allowed Model PLANAR PLANAR SPATIAL SPATIAL

SimQlification Analysis SIMPLIFIED MULTIMODAL MULTIMODAL MULTIMODAL

Behavior factor REFERENCE DECREASED REFERENCE DECREASED

dynamic analyses and 3-D models to be used in every case. EC8 in particular, specifies the multimodal response spectrum method as the reference method of analysis and permits even more advanced methods, such as nonlinear dynamic analyses and random vibration solutions, under the condition that the latter lead

T I

Computer Analysis and Design of Earthquake Resistant Structures

389

to a base shear not less than 80% of the base shear given by the reference method. If this is not met, all response variables from such methods must be scaled proportionately by the scale factor that will bring the base shear to the 80% level of the base shear computed by the reference method. It must be noted here that the continuously increasing capacity of personal computers has made possible the use of powerful analysis and design software, even by the smallest design offices in their routine work. With such software, static and dynamic analyses of detailed structural models are easily performed and thus some of the simplifications still permitted by the codes tend to become obsolete. Moreover, this capacity permits the investigation of alternative solutions that can lead to better and more economical designs. As a result of this progress, the state of practice today in seismic analysis and design of buildings is to use detailed, 3-D space-frame models for static solutions and reduced, 3-D models for dynamic solutions. For the majority of buildings whose floor diaphragms are sufficiently rigid in their planes, the most widely used model for dynamic analyses is based on the following simplifications: 1. The floors are rigid in their planes having 3 D.O.F., two horizontal translations and a rotation around a vertical axis. 2. The masses of the building and mass moments of inertia are lumped at the floor levels at the corresponding degrees of freedom. 3. The vertical component of the earthquake motion is ignored. 4. The inertia forces or moments due to the vertical or rotational components of joint motion are negligible and hence ignored. For this reason the corresponding degrees of freedom are eliminated from the dynamic model through static condensation. The above simplified model, shown in Fig. 10, makes the problem very easy to solve through the drastic reduction of the dynamic degrees of freedom and yet produces quite accurate results. If the floor diaphragms are not sufficiently rigid in their planes, e.g. in buildings with very stiff vertical resisting elements (such as elevator cores) and with diaphragms having irregular shapes, large openings etc., the in plane rigid assumption may not be valid any more. In such cases, a more complicated model will be required with additional degrees of freedom per floor, in order to properly introduce the in plane flexibility. Soil flexibility should also be included in the model and so should those infill walls built in full contact with the boundary structural members. However, uncertainty in infill wall behavior and lack of a widely accepted model for analysis and design, are reasons for which infill walls are either not addressed at all by codes or their treatment is limited to a change in the estimated fundamental period of the building and perhaps in the value of the behaviour factor. The flexibility of the soil is usually modelled by inserting discrete springdashpot elements between the foundation members (footings, grade beams or piles) and the soil medium, with the spring and dashpot constants determined

390

Computer Analysis and Design of Earthquake Resistant Structures

Computer Analysis and Design of Earthquake Resistant Structures

Vn

Un r - - U n I I

I

I

I

I I I

----

+

U2

+-

11 I

,I

~

Ug(t)

-

+

y a

-

fn

391

from elastic, static, half-space solutions. In such cases, however, solution of the problem by modal analysis requires determination of equivalent modal damping (Roesset et al [58]). In cases of stiff, massive foundations, the effects of soil structure interaction, addressed in another chapter of this book, must also be considered. For the response spectrum, modal analysis type of solutions presented herein, the approximate method in ATC-3 [71], or its successor, NEHRP [46], may be used for this purpose. 3.2 Reduction of degrees of freedom

f2

The detailed structural models with 6 D.O.F. per joint that are often used for static analyses and design of non-symmetric buildings, could also be used for dynamic analyses of buildings subjected to 3 - component earthquake motions. Except for the difficulties associated with problem size, this type of solution is, in principle, straightforward and has been applied for seismic analysis of irregular buildings with flexible floor diaphragms and for space frame structures in general (e.g. Anagnostopoulos [4]). For the vast majority of buildings, however, the simplified model described above, with 3 D.O.F. per floor, is typically used. Moreover, this model is usually analyzed for the two horizontal components of motion, because this is only what the codes normally require. The effects of the vertical component of motion are considered to be within the safety margins of dead load design and only for horizontal cantilevers and prestressed horizontal members the codes specify an additional load due to the vertical motion component. The reduced model of the building for seismic response analysis is obtained by static and kinematic condensation of its stiffiless matrix, through which a large number of dynamic degrees of freedom is eliminated. This reduction in problem size is dramatic. For example, a 25 storey building with 20 columns per storey has a total of 25 x 20 x 6 = 3000 static D.O.F., compared to 25 x 3 = 75 dynamic D.O.F. of the reduced model. Of course, the derivation of the lateral stiffiless matrix and the recovery of member forces are both based on the full model with all the static D.O.F. For the formulation of the equations of motion, we will consider that the structural system consists of space and plane frames, with 6 and 3 D.O.F. per joint, respectively, interconnected by the floor diaphragms. The static and kinematic condensation will be applied to the complete stiffiless matrix of each frame to obtain its lateral stiffiless matrix, which will subsequently be used to assemble the total lateral stiffiless matrix of the complete building. 3.2.1 Static condensation

Figure 10: Building model with 3 D.O.F. per floor

Static condensation is a general method for eliminating degrees of freedom from a static or dynamic problem. Such degrees of freedom are those with zero or

392

i1 I ,i

Computer Analysis and Design of Earthquake Resistant Structures

Computer Analysis and Design of Earthquake Resistant Structures

negligible external loads associated with them. In the case of earthquake response, the external loads are inertia forces, products of joint accelerations with the corresponding masses or mass moments of inertia. Compared to the forces generated by the translational motion in the horizontal planes, the moments due to rotational joint accelerations are negligible and so the corresponding degrees of freedom can be eliminated from the dynamic model. Moreover, due to reasons given earlier, the vertical component of the seismic motion is neglected in typical building analysis and so the vertical joint translations can also be eliminated from the dynamic mode~ leading to further reductions in D.O.F. Note, however, that in this case, the reasoning for neglecting the vertical action is not that the induced member forces (mainly axial forces in columns) are negligible, but rather that the available safety margins from dead and vertical live load design can compensate for them. This may not be so when the ground motion has a strong vertical component, as it may happen at sites close to the fault rupture zone. Consider a space or plane frame that is part of the structural system of a building. Let k be the complete stiffness matrix of this frame and let us denote with u and Uo the vectors of D.O.F. with and without external loads respectively. For the present application, u contains only horizontal translation~ and Uo the vertical translations and all joint rotations. In case of a plane frame, u contains only one horizontal translation per joint and u0 a vertical translation and a horizontal rotation (per joint). Let p be the load vector of external forces associated with u (for dynamic solutions, elements of p will be of the form mil(t) where m = mass, il(t) = acceleration, function of time t). By proper partitioning of the stiffness matrix k, the equilibrium equations take the form:

(18)

Written separately, the two sets of equations are:

Solving the second of the two equations for u0 we obtain: Uo=-

k -lkt 22 120

and substituting in the first equation we get:

(19)

393 (20)

The matrix: (21)

is the statically condensed stiffness matrix of the frame, relating the horizontal joint loads to the respective degrees of freedom. For the dynamic problem, these loads are generated from corresponding masses. The eliminated D.0.F. in Uo have zero mass associated with them and hence are not used as dynamic D.0.F. They are only maintained as static D.O.F. that are computed from the dynamic ones by means of eqn (19). Notice that kr is 1/3 the size of k. However, kr is a full matrix while k is highly banded and thus the computational savings from this reduction in size are not what one would expect from a comparable reduction in building size (in terms of number of joints). It is also noted that there is no approximation involved in this procedure, other than how close to zero are the external loads acting on the eliminated D.O.F. The method is exact if such loads are exactly zero.

3.2.2 Kinematic condensation This is applied on the statically condensed stiffness matrix (or matrices) to further reduce the degrees of freedom of a static or dynamic problem. The method involves introduction of kinematic constraints on different degrees of freedom, which are expressed in terms of a few others (master and slave D.O.F.). Thus, in a building with floor diaphragms rigid in their planes, all horizontal translations of the joints in a floor plane can be expressed by simple geometric relations in terms of three master D.O.F.: two horizontal translations and one vertical rotation of any reference point in that plane (rigid body motion in the plane of the diaphragm). This reference point is taken to be the point where the masses assigned to the given floor are lumped, i.e. the center of gravity of these masses. Let us drop for simplicity of notation the index r and call k the reduced stiffness matrix of a space frame. This relates the horizontal joint displacements [ut vt] with the corresponding external horizontal forces [f! f~], where t indicates the transpose of a matrix or vector. Vectors u and v contain the displacements of the frame joints along the horizontal x and y axes, respectively, grouped by floor level. If we assume that the frame has fl. joints per floor and that the total number of floors is n, the joint displacement vectors are:

394

Computer Analysis and Design of Earthquake Resistant Structures

Computer Analysis and Design of Earthquake Resistant Structures

395

It is now assumed that all floors are rigid in their planes. If the displacements of the center of mass of floor i are ui, vi and Si (Si= rotation about the vertical z axis), the displacements uij and Vij ofjointj at floor i (Fig. 10) are given by:

}

uii=ui-Yiisi vii=vi+xiisi Thus, the reduced force-displacement equations of the space frame m partitioned form are:

(25)

or in matrix form for all floors: u=Tur+Ty Sr}

lkxx kyx

l

kxy l {u} {fx} kyy v = fy

J

(26)

v = Tvr + Tx Sr

(22) where:

. i' .1

•,

The static equilibrium of every floor in the frame requires in addition to equations (22), the moment equilibrium of external and internal forces in the plane of the floor. Ifwe define a vector of joint torsional moments fz having f elements per floor for a total of n floors, the third set of equations required for torsional moment equilibrium of all floors is, in matrix form, the following: (23)

Dyfx + Dxfy = fz I

...

where : Dx, Dy diagonal matrices (n.f x n.f), with elements the Xij and - Yij coordinates of joint j in floor i in a system at the center of mass (C.M.) of floor i (Fig. 10) , as follows:

r I

I Irx·1 Xi2 d. I I Dx = iagl I I I l

l I I I

I

l

Xjf

i = 1,2, ... n

J

l I I I I I I

J

r I

I 1-Yil I -Yi2 Dy= diagl I I I I l

l

l I I I -Yit j

i = 1,2, ... n

l I I I I I I

J

are the three vectors of floor horizontal displacements and vertical rotations. Matrices T, Tx and Ty are transformation matrices with dimensions (n.f x n):

1xu

0

IXJ2

0 0

0 0

kxxu+kxyv=~ k~u+kyyv=~ (Dykxx + Dxkyx) u + (Dykxy + Dxkyy) v = fz

j~

0 0

0

0

0

0

.~.1

0

1

Tx =

I o

0

.'..I

l

0 0

0

1

Substituting fx and fv in eqn (23) from eqn (22), we obtain the following set of three matrix equations:

~l ... I

0

oI

0

0

0

I

0

(24)

Xn]

0

Xn2

I0

0 0

0 0

Xn£

J n

-Y2e

0

n

Ty

=1

0

x2e

0

0

I o

J

I

l

o

0 n

,....

"

396

Computer Analysis and Design of Earthquake Resistant Structures

Computer Analysis and Design of Earthquake Resistant Structures

Notice that:

Substituting u and v in eqns (24) from eqns (26) and rearranging we get: kxx Tuf + kxy Tv f + (kxx Ty+ kxy TX) sf

= fx l

kyxTuf +kyyTvf +(kyxTy +kyyTx)Sf

=fy

I

+ [(Dykxx + Dxkyx) TY + (Dykxy + Dxk))') Tx]Sf = f,

J

:::::::::::::::::::::: 1

,

j

(28)

where: t

kr,xx = T kxx T,

kr,yy

kr vz I

kr:~J

(29)

If the building is modeled as a single space frame, then its lateral stiffness matrix is given by eqn (29). Quite often, however, the problem is simplified by breaking the space frame into plane frames and other (plane) elements, interconnected by the floor diaphragms. This approximation, although it does not enforce compatibility of the condensed D. 0 .F. that are common to more than one plane frame, it gives often good results, adequate for design applications. Only for irregular buildings, in which the identification of plane frames or elements cannot be easily done, this approximation may become questionable. For the sake of generality, we will assume here that the building consists of one or more space and plane frames. The lateral stiffness matrices of separate space frames in the form of eqn (29) are simply added together, provided that the kinematic condensation for each of them was carried out with reference to the same points, i.e. the mass centers of the floor diaphragms. Referring to Figure 10, we assume that the plane frame (e), at an angle (a) with respect to the floor x axis, has a lateral stiffness matrix k (n x n) and lateral displacements, Oi (i = 1, 2, .. ., n, where n = number of floors). The x component of the lateral force by frame (e) on floor i is the following:

i1 ·I.

·l:

kt(SF) =I k~,xy

k~ ,_vz

l

3.2.3 Contribution of plane frames to the building lateral stiffness matrix

Condensation of the above equations is achieved by premultiplying them on both sides by Tt. This is in effect a replacement of then equations per floor and per direction by their sum. The resulting set of equations is:

...•'

kr,xz I

with its submatrices explained under eqns (28). In the case of plane frames, the kinematic condensation is achieved by simply adding the rows and columns of kr corresponding to the horizontal joint displacements of each floor. The resulting lateral stiffness matrix kt relates the horizontal floor displacements to the respective lateral forces, being equivalent to the kr,xx (or kr,yy) submatrix of k £(SF) in eqn (29) above.

~

(Dykxx +Dxkyx)Tuf +(Dykxy +Dxkyy)Tvf +

kr,xy

lk~,XZ

(27)

Tx=DxT, Ty= Dy T

II kr,xx

397

t

kr,xy = T kxy T,

kr,zz =T~ (kxxTy +kxyTx)+T~ (kyxTy +kyyTx) fxr, fyf and fzr = external force vectors containing the x and y total floor forces and the z torsional moment, respectively.

n

fxi = cosaL kijo j

(30)

1

For the above expressions, eqns (27) were also used to indicate the symmetry of the stiffness matrix. Thus, the lateral stiffness matrix of the space frame ke(SF)relating the floor D.O.F. Of, Vf, Sr to the corresponding total lateral floor forces fxr, fyf and torsional moments fzr is:

Consider a point A(Xei, Yei) on the axis of frame (e) on floor i, with displacements llei, Vei· The frame displacement Oi in terms ofllei and Vei is:

I

398

Computer Analysis and Design of Earthquake Resistant Structures

Computer Analysis and Design of Earthquake Resistant Structures where:

or, by making use of the rigid body motion offloor i:

2

(31)

where Ui, Vi and 0i are the lateral translations and rotation of the mass center of floor i. Substituting in eqn (30) we obtain: 2

fxi = cos a

n

n

l

l

L kijU j + cosa · sina L

399

L kijS j

kxy = (cosa.sina)k

k}'?( = k~

kyy = (sin a)k

kxz = (cosa)Dk

2

kyz = (sina)Dk

k zx = k 1xz

kzz= DkD

D = diag [di],

n

kijv j + dicosa

kxx = (cos a)k

i = 1, 2, ... , n,

n =number of floors.

(32)

l

where: (33)

The total lateral stiffness of the building corresponding to the three D.O.F. per floor is finally formed by a direct summation of lateral stiffness matrices of the form of eqns (29) and (36) combining space and plane frames, respectively. The procedure outlined above requires modification in cases where one or more floor diaphragms do not extend over the whole building layout.

3.3 Modal time history analysis

. 11

is the distance of the frame axis from the reference point (center of mass of floor i). In a similar manner we obtain:

!, n

n

n

l

l

l

2 f yt· -- cosa · sina"" £..... k IJ.. u J· + sin a"" £..... k IJ··v J· + d I· sina "" £..... k IJ..3 J·

(34)

Let the building model of Fig. 10, which has 3 D.O.F. per floor, be subjected to the horizontal ground motion components Ugx(t) and ugy(t). Ifwe call Ua(t) and U(t) the vectors of absolute and relative to the ground horizontal translations and rotations, Ua, Va, Sa and u, v, S, respectively, we can write: (37)

As in the case of the space frame, we call fzi = - fxiYei + fyiXei and using the above expressions for fxi, fyi we get:

fzi = dicosa

n

n

n

l

l

l

L kijUj + disina L kijVj + df L kijSj

(35)

On the basis of eqns (32), (34) and (35), it follows that the contribution of the plane frame (e) to the lateral stiffness matrix of the building is the following:

(36)

where e1 and ez = vectors of dimension 3n, having units in the locations corresponding to the u and v D.O.F., respectively, and zeros elsewhere. For simplicity of notation we will drop the time variable t, but it will be implied that the Ua and U vectors and their derivatives with respect to t (velocities and accelerations) are all functions oft. By applying D' Alembert's principle we can write 3 equations of motion per floor mass expressed as dynamic equilibrium equations. These equations are similar to eqn (2) for the SDOF system, except that the damping and elastic forces are of the form:

Fd

n

2n

3n

l

l n

n+l 2n

2n+l 3n

I

n+l

2n+l

=-(L Cijllj + L CijVj + L CjjSj) I

Fe = -

(L l

kijuj +

~

L kijv j + L kijs j)J

400

Computer Analysis and Design of Earthquake Resistant Structures

Computer Analysis and Design of Earthquake Resistant Structures

where Cij and kij are elements of the damping and lateral stiffness matrices, respectively. In matrix form, the equations of motion of the building model are:

of the effective modal masses (see below). For building models with 3 D.O.F. per floor, the eigenvalue problem is small and presents no difficulty. It can be solved easily by any of a number of methods. After the mode shapes

1, below which some codes require

., t

406

Computer Analysis and Design of Earthquake Resistant Structures

Computer Analysis and Design of Earthquake Resistant Structures

407

consideration of closely spaced mode effects, is 1.10. As a final note on modal combination rules, we must also mention the sum of the absolute modal values, which is not used for design but rather as an upper bound indicator.

where Sai is the ordinate of the design acceleration spectrum at Ti = 27t/C>j. Ri,max can be computed as the value of R for a static loading of the building by the elastic forces of mode i, which now become (see eqn 49): (59)

Similarly, based on eqn ( 47), we see that the design (maximum) floor displacements for mode i are:

0.8

~

·a

(60)

0.6

~0 u

c

.Q

1ii

and also, based on eqn (50), the design (maximum) base shear component in mode i is:

0.4

~0

(

()

(61)

0.2

1.0 Frequency ratio

1.5

2

A;i =w/wi

Figure 11: Correlation coefficient Pij for the CQC rule Returning now back to eqn (53) we notice that \Jli,max, the maximum of \Jli(t) from eqn (46), is nothing else than the spectral displacement Sdi of the ground motion iigx(t) for ro = C>i (or T = Ti = 27t/C>j). Therefore, if the response spectrum of the earthquake motion ii gx ( t) is available, we do not even have to solve the modal equations (46), but can read the values of Sdi directly from the spectrum, find Ri,max from eqn (53) and finally Rniax from eqns (54) or (55). We recall that for design applications, we do not use the response spectra of some specific earthquake motions but rather the smooth design spectra specified by the applicable code (e.g. such as the EC8 spectra given by eqn (17)). Thus the Sdi values are readily available for all the building circular frequencies corresponding to the R. modes in the modal combination eqns (54) or (55). Therefore, in terms of the design acceleration spectrum Sa, the modal component Ri,max of Rmax is: (i = 1, 2, ... R.)

(58)

The modal components of the response variables in eqns (58) to (61) need be combined either by the SRSS or by the CQC rule to give the design value of the respective variable. It must be noted here that application of the modal combination rule should be the last operation for determining the design value of any response variable. This means that it is incorrect to determine some response variable as function of other response variables already determined by modal combination (e.g. it is incorrect to determine interstorey drifts as differences of the final computed floor displacements, or stresses from the final values of section forces). Rather, the modal components of any desired variable must first be determined and then the combination rule must be applied. In summary, the modal response spectrum analysis involves the following steps: 1. Idealization of the building and formation of the mass and stiflhess matrices through static and kinematic condensation. 2. Determination of a sufficient number of mode shapes i and corresponding circular frequencies C>i by solving the eigenvalue problem. The required minimum number of modes is usually determined from code specified criteria (typically only few modes are sufficient for each direction of motion). Determination of the quantities Lix, Mi, fjx and M~ (eqns 44, 45, 43 and 51). 3. For each modal period Ti, determination from the applicable design spectrum of the corresponding spectral ordinates Sai. Adjustment of spectral values for damping if the selected modal damping is different than

'I

'

,.,

f 408

Computer Analysis and Design of Earthquake Resistant Structures

Computer Analysis and Design of Earthquake Resistant Structures

the damping associated with the design spectrum (codes usually include the necessary adjustment fonnulas). 4. Detennination of the complete set of D.O.F. corresponding to each mode shape i, from which

+-- u, --t

35 KN/m

I

ti +-

Level

m,

Level 1 2 3

3.50.

2.80

t

Level 3

Mode 1 2 3

--

35X35

4.00

i

m2

Level 2 30X30

5

12

14

13

.J.---

7.00

s.oo

15

s.oo-4-

Ca I

Figure 12: Example 3-storey frame

-2.L

~

1.000 0.620 0.400

1.000 -0.822 -0.801

_ro_

6.577 14.628 28.289

T(sec) 0.955 0.430 0.222

JL 0.120 -0.657 1.000

L 137.495 -56.028 18.077

~ 61.72 50.22 25.55 M 103.08 157.44 99.74

(

~ 61.72 -66.58 -51.17

r 1.334 -0.356 0.181 SUM

.MS2. 7.41 -53.21 63.89

•

M 183.399 19.938 3.276 206.613

•

M IM112t 0.888 0.097 0.016 I.000

( b)

We notice that 'L,M" = Mtot (a check of our calculations) and that nearly 90% of the effective mass is in the first mode. Using the T values above, we compute from eqn (17) for soil class Band q = 5 the spectral ordinates Sa for each mode.

~··

,. Computer Analysis and Design of Earthquake Resistant Structures

Computer Analysis and Design of Earthquake Resistant Structures

Subsequently, we form the products rSa, rsd and from eqn (59) we compute the elastic modal forces. All these are summarized below:

Having computed the response of the building in one of its principal directions, say the horizontal x direction, the same type of analysis must be performed for the other principal direction (y). Subsequently, the maximum value of any response variable for the combined action of the two motion components is usually approximated either by the SRSS rule or by the (1+30%) rule suggested by Rosenblueth and Contreras [61]. The two directional combination rules are:

410

2 Sa (m/s ) 2 rSa(mls ) rsd(cm)

1.080 1.441 3.331

Mode 2 1.470 -0.523 -0.244

Level 1 2 3

3 1.470 0.266 0.033

The computation of Qi as a summation of the elastic modal forces and from eqn 61 was carried out for checking purposes.

Qi (sum) Qi (eqn 61) Qmax (SRSS) Q11Qmax

Mode 2 -32.28 34.82 26.76

88.94 72.37 36.82

3 1.97 -14.15 17.00

29.30 4.82 29.31 4.82 200.26 198.07/200.26 = 0.989

Mode 3 2 Urnax{SRSS} ~ Level 0.997 3.34 0.004 3.33 -0.24 1 0.995 2.07 2 2.06 0.20 -0.022 0.995 1.34 0.033 3 1.33 0.19 ---------- ----------------------------------- ----------------------0.918 1.34 0.026 1.27 -0.44 u1 - u2 Notice that the interstorey drift (u1 - u2)max = 1.34 cm is different than u 1max - u2max = 1.27 cm. Finally, the modal components of the gross member forces Ma, Ou, Na are computed by multiplying the elastic modal forces by the influence matrix given earlier:

Mode

Ma Oa Na

37.72 21.42 12.28

2 -13. 76 -7.92 -4.37

3 0.90 0.54 0.25

(a)

2 2 ] 112 R =± [Rex) + R(y)

(62)

(b)

R(x) ± 0.30 R(y) R=+max { 0.30 R(x) ± R(y)

(63)

198.13 198.07

Next we compute floor displacements from eqn (60) and interstory drift for levels 1 and 2.

Member force

411

Rmax {SRSS} 40.16 22.84 13.04

R11Rrnax 0.94 0.94 0.94

In this example we see that the first mode contribution to any of the response variables examined is dominant and would alone be a sufficient approximation for design purposes. Of course, the above calculations are fully automated in the various computer programs for dynamic earthquake response analysis.

Both these rules will be found in the codes and give comparable results. A third level of approximation is introduced when one applies design code equations that include more than one component of stress or design equations for determining required amounts ofreinforcement in R.C. members, etc. Given that modal analysis gives estimates of maximum values for each component, not all of which occur at the same instant, one is faced with the problem of what values to combine for code checks. The problem becomes more complicated if one considers that the signs of response variables are lost in modal analysis and, furthermore, in situations where a design variable is determined from a max min combination of two or three different response variables (e.g. areas of reinforcement in columns). Due to such difficulties, the modal combinations are often performed at the level of gross member forces, which are subsequently used for the design checks. This problem is addressed in detail for reinforced concrete buildings in another chapter of this book. An assessment of modal and directional combinations, as well as design stress approximations for the response spectrum method, applied to steel offshore structures (3-D braced space frames) under three-component earthquake actions, can be found in Anagnostopoulos [4].

4 Inelastic analyses of buildings for earthquake actions A basic philosophy of codes for earthquake resistant building design, reflecting the need for safety at a reasonable cost, is that under design level earthquakes the response of the building will be inelastic. This has permitted specification of seismic design forces that are only fractions of what elastic behaviour would have required, thus leading to substantially smaller member sizes. At the same

l 412

I ~'

Computer Analysis and Design of Earthquake Resistant Structures

time, in order to enable the building to respond safely and in a controlled manner, modem codes have adopted the so called capacity - design philosophy [53,54], according to which some members are intentionally made stronger than others, so that yielding is forced into pre selected locations that are specially detailed for it. Under this philosophy, design provisions that lead to strong columns - weak beams, force most of the yielding· into the beam ends, keep most of the columns elastic and greatly reduce the danger of mechanism formation during a strong earthquake. Buildings so designed and detailed possess ductility, so that in a strong earthquake they may undergo several cycles of inelastic deformations without any noticeable loss of their strength. The extent, however, of the expected inelastic action in terms of its distribution, intensity and number of cycles under a design level event, is something for which no verification is required by the codes, because this needs inelastic dynamic analyses. For reasons explained below, such analyses have not been introduced into the codes but are still used mostly for research and also by code makers for calibration purposes and for verification of typical code applications. Inelastic dynamic, earthquake response, analyses of buildings with realistic MDOF models date back into the mid - sixties, when Clough and Benuska [16] published an extensive study on the earthquake behavior of plane frames, based on inelastic dynamic solutions. Since then, such analyses have been used mostly as a research tool to carry out a great deal of parametric studies, which together with a large volume of experimental work and field observations have increased considerably our knowledge on this subject and provided a better understanding of inelastic building behavior in strong earthquakes. This knowledge has been filtering into the codes and reached a point that some of them (e.g. the New Zealand code [4 7]) include explicit specification of the intended level of inelastic action by means of a ductility factor µ. In the EC8 and UBC codes, the values of the behavior or response reduction factor (q in EC8, R in UBC), which determine the level of the design seismic forces and hence the level of intended inelastic behavior for various structural types and construction materials, have also been calibrated by means of inelastic analyses. Yet, the type of analysis the codes specify for design is still elastic (equivalent static or dynamic, response spectrum method). And while inelastic dynamic response analyses could have been specified for design verification, their complexity and mainly the lack of well established, widely accepted models for member behavior under inelastic cyclic loadings, not to mention the even greater lack of such models for infill walls, has kept them out of the building codes. Given that the much simpler elastic dynamic analysis has been introduced into the codes only in the past 10 years, it should not be surprising that inelastic dynamic analysis is still, and will be for the years to come, primarily a research tool for academics and code writers. On rare occasions, it may also be used for design verification of unusual buildings or buildings of high importance (e.g. Fintel and Ghosh, [22]).

I

Computer Analysis and Design of Earthquake Resistant Structures

413

4.1 Inelastic member idealization As mentioned above, one of the main obstacles preventing the introduction of non linear, dynamic analyses of buildings into the codes is the large number of available idealizations at different levels of sophistication and the lack of a standard model for which a consensus among the experts could be established. In general, there are three levels of sophistication that have been used: At the first level, a complete building may be idealized as a condensed structural model, in which a single non linear element replaces an entire frame story or even the complete structure. In this last case the entire building is reduced to a SDOF system with a multi linear force - deformation relationship, determined from non linear, static, pushover analyses with gradually increasing lateral forces (Pique [55]). The model in which frame stories are idealized by a single non linear element, is based on assumed shear beam type of behavior that most frames exhibit. The assumption is not valid for shear walls, which are far coupled systems and must be modeled accordingly, e.g. by plastic hinge elements, with one element per shear wall segment between consequtive floors. The non linear frame and shear wall elements can be interconnected by the floor diaphragms to create a 3-D model of the entire building (Anagnostopoulos [2]). Such models cannot, obviously, predict the inelastic response of individual frame members. Today, the wide availability of computers that are orders of magnitude more poweful than the machines for which these idealizations were aimed and the subsequent development of more detailed and sophisticated models for full 3-D solutions, make these simplified models nearly obsolete. The second level of sophistication corresponds to models in which structural members, beams or columns, are idealized as distinct elements and all inelastic deformations are concentrated at their two ends. This is the most widely used idealization for 2-D analyses. Beam and beam - column models that limit all inelastic action at the two ends belong to the so - called lumped plasticity idealizations and are known as point hinge models. There are two such models, developed originally for inelastic beams (and used also for beam columns in combination with a moment-axial force interaction diagram giving the yield moment as function of the axial force) in two dimensions: the one component and the two - component models (Giberson [24], Anagnostopoulos [3]). The first consists of an elastic beam with a nonlinear rotational spring at each end, while the second consists of two beams, one elastic and the other elastoplastic (or bilinear) acting in parallel. Their formulations are derived on the basis of four possible yield states: fully elastic member, yielding at one or the other end and yielding at both ends. By assuming a certain state of end rotations, it is possible to relate the post-yield stiftbess parameters of the two models to get a partial match between the two. It is impossible, however, to match them in all yield states and as a result differences in response predictions

414

Computer Analysis and Design of Earthquake Resistant Structures

Computer Analysis and Design of Earthquake Resistant Structures

may be expected. A detailed comparison of inelastic response predictions by the two models for a 4-storey frame has shown average differences in maximum rotational member ductilities between 2% and 30%, while the differences in specific members were up to 100% (Luyties et al [38]). Figure 13 from Ref [38] shows such a comparison of maximum girder ductilities in the four story frame due to an artificial earthquake at 0.3g_ and 0.5g peak ground accelerations. Irrespective of which of the two models is employed, it is necessary to use appropriate values for the post - yield stiffness and yield moment, compatible with the expected level of inelastic action. Such values have been derived for beams with bilinear moment - curvature characteristics (Fig. 14) on the basis of spread plasticity considerations (Anagnostopoulos [3]). They are plotted here in Figure 15, where the upper graph gives the ratio of the post yield stiffness K to the corresponding elastic stiffness K = 6EIJ f ( f = beam length) and the lower graph the ratio of the equivalent yield moment MP to the nominal yield moment of the section MP.The abscissa is the

4

I

""',, I

3 0

E-
~

CZl

I

415

I

2

0.33g-\

0

2

0.50g

3

4

5

6

7

8

ROTATIONAL DUCTILITY FACTOR (µr)

rotational ductility factor, which must be estimated or assumed beforehand for entering the curves, and the parameter p is the post-yield stiffness ratio of the bilinear moment-curvature relation of the beam sections. The effects of postyield stiffness ratio on inelastic response is shown in Figures 16 and 17 for a 10 - storey frame, subjected to NS component of the 1940 El Centro record. Columns were kept elastic to have a strong column - weak beam design. Analyses were carried out using the two - component beam model of the DRAIN - 2D program [31], first for a post yield ratio p = 0.03, equal to that of the moment - curvature relationship, and subsequently with a ratio g = s I k and a yield moment Mp, both estimated from the graphs of Fig. 15. In fact, a

(a) (b)

one-component model two-component model

Figure 13: Rotational ductilities in beams of a 4-storey frame predicted by two different point hinge models [38]

second iteration (3rd analysis) was carried out, for which maximum ductility factors from the second analysis were used to enter the two graphs, but gave practically the same results with the 2nd analysis (for which the g ratio was based on the ductility factors computed in the first analysis with g = p = 0.03). Figure 16 shows the effects of the variation in the post - yield stiffness ratio on the top floor displacement response and Figure 17 shows the effects on the rotational ductility factors (curves I vs 2 and 3 vs 4). The curves marked "curvature" pertain to definitions of the ductility factor and will be discussed in a subsequent section. We see that substantial differences can result, especially in local response variables such as member ductilities. Between the two models, the one - component is usually preferable as it permits modeling of joint flexibility and straightforward handling of any desired moment - rotation hysteretic relationships for cyclic load reversals. A variety of hysteretic rules, developed on the basis of experimental data, are available, especially for reinforced concrete (RIC) members (e.g. Takeda [70], Park et al [51], etc). For 3-D analyses the problem becomes much more complicated especially for column members for which the interaction of Mx - My - Mz and N must be considered. A large number of models have been developed, a few as extensions of the 2-D point hinge models, especially for reinforced concrete members.

M

y

Figure 14: Bilinear moment-curvature relationship for beams

J

416

Computer Analysis and Design of Earthquake Resistant Structures

Computer Analysis and Design of Earthquake Resistant Structures

0.5 , . . . . . - - . , . . , . - - - - - - - - - - - - - - - - - - - , I ~ 1. 0

417

15

e

.;:, 10

0.A

z ~

w

5

u

.... .... u

16

~ ~

where My is the yield moment, dM the plastic moment beyond My corresponding to 1 is an "overstrength "factor (essentially an additional partial safety factor), and LMsb, LMsc denote the sum of beam or column moments around the joint, as obtained from the analysis for the design seismic forces and quasi-permanent gravity loads. The minimum required ratio LMRciLMSc, which in eqn (1) equals YRd/8, is called Capacity Design magnification factor for column moments. Eqn ( 1), with YRd= 1.2, is the format of the Capacity Design rule imposed on "Special Moment RC Frames" by the U.S. Uniform Building Code [l], by Appendix A of the ACI-318 Building Code Requirements [2], by the SEAOC Recommendations [3] and by the NEHRP Provisions [4]. The same format is adopted by the New Zealand standards [5,6], with a YRd factor equal to an "overstrength factor" of 1.25/0.85:::::1.47 times a "dynamic magnification factor" co> 1, which aims at protecting the columns against increases of end moments due to higher mode inelastic responses, but essentially works as an additional safety factor against development of inelasticity in the columns rather than in the beams. In columns framing with beams only in one horizontal direction and responding to seismic actions only in this direction, the value of co is taken equal to 1.3, giving an aggregate overstrength factor YRd in eqn (1) equal to 1.9. The seismic response of columns belonging to more than one frame is characterized by biaxial bending, due to the bidirectionality of the horizontal

447

seismic action. For reasons of simplicity the New Zealand standards do not consider simultaneous seismic action in two orthogonal horizontal directions (explicitly considered by most other modern Codes), and attempt to cover such a possibility by increasing the value of co from 1.3 to 1. 5. Higher mode effects, to which the co-factor is presumably aimed at, are more important in long period structures. Accordingly, ifthe fundamental period Tis above 0.75-0.80sec, the New Zealand standards increase the value of co in the upper 70% of the structure, in which higher mode effects mainly develop, from 1.3 to 0.6T+0.85~ 1.8 in "one-way" columns, and from 1.5 to 0.5T+l.1~1.9 in "two-way" ones. The upper limits correspond to T=l .6sec and to an aggregate overstrength factor of 2.65 and 2.80 respectively. These co-values apply for frame structures with full ductility, designed for a global displacement ductility factor µ 0 of 6. Strong shear walls impose a first-mode-like linear heightwise distribution of lateral drifts, suppressing higher mode effects and rendering formation of a soft storey kinematically impossible (Figs. l(b) and (c)). Accordingly in dual systems, with shear walls resisting at least one-third of the seismic base shear, the value of the co factor is reduced to 1.2, regardless of the natural period. The aggregate overstrength factor, YRd, in eqn (1) is reduced then to 1. 76. In the socalled frames of "limited ductility", designed for a global displacement ductility factor µ 0 between 1.25 and 3.0 (instead of µ 0=6 for fully ductile frames), the requirement for no plastic hinging in the columns is relaxed, at the expense of higher design lateral forces and strength. Accordingly, the co-factor of the New Zealand standards is reduced to 1.1 for one-way columns designed against the formation of plastic hinges, and to 1.3 for two-way ones, giving aggregate values of the overstrength factoryRd in eqn (1) equal to 1.62 and 1.91 respectively. Development of plastic hinges at the base-section of the bottom storey columns and at the top-section of top storey columns is kinematically compatible with the beam-sway mechanism of Fig. l (a) (and in the case _of the bottom storey columns a prerequisite, unless significant rotations of the foundation elements take place). Accordingly, the Capacity Design magnification of column moments, Ms0 , required by the New Zealand Standards, is limited to multiplication by l/0.85=1.18 for "one-way" columns and to 1.l/0.85=1.29 for "twoway" ones, i.e. those belonging to frames in more than one horizontal direction. Columns "dominated by cantilever action", i.e. which in some storeys (typically the lower ones) do not develop a point of inflection within the storey height under the horizontal seismic action, represent another special case for the application of the Capacity Design rule, eqn (1), according to New Zealand standards: As in those storeys the magnitude of column moments is not dictated by the moment input from the beams at the individual intermediate nodes, but by the action of the column as a whole, from bottom to top, the magnification factor on column moments from the analysis, Msc, is taken equal to the ratio of the design flexural capacity of the ground storey bottom section, MRc, to the corresponding seismic moment according to the analysis for the design seismic action, MEc, times an overstrength factor of (1.25+2(v-0.1) 2)/0.85. This factor re-

T 448

Computer Analysis and Design of Earthquake Resistant Structures

Computer Analysis and Design of Earthquake Resistant Structures

fleets the enhancement of concrete strength due to confinement, as a function of the column axial load ratio, v=N/AJcd (Ac=column cross-sectional area, f0 d=design value of concrete strength). In addition, column design moments are multiplied by an ro-factor value obtained by linear interpolation between the minimum value of 1.0 (or 1.1 for two-way columns) at the base and the normal value at the storey where a point of inflection first appears. Eurocode 8 [7] introduces a different column Capacity Design rule:

two-way columns, columns dominated by frame- v. cantilever-action, ductile frames v. frames of limited ductility with many special cases, etc.), is computationally inconvenient. (Indeed, the New Zealand approach is hand-calculations oriented). On the contrary, the Eurocode 8 approach of eqn (3) applies uniformly to all types of structural systems and columns (gravity- or seismic-action dominated ones, dual systems or frames, frame- or cantilever-action dominated columns, etc.), talcing automatically into account the special cases handled by other standards through special rules. Moreover, it applies simultaneously to both horizontal directions of bending, giving the appropriate weight to the principal bending direction. So, it is computationally very convenient. In dual systems we rely on the lateral stiffuess of walls to impose a linear drift distribution with height and avoid localisation of inelastic deformations in a single, "soft" storey. To mobilize all beams into inelastic action and minimize the local rotation and ductility demands for given global displacement ductility or top drift ratio, wall rotations should be constrained to its base region. Moreover, such an intentional localisation of wall deformations at the base results in a simpler and possibly more economical wall design, as it relieves the designer from the need to detail the rest of the wall for high ductility and allows him to proportion it in shear considering it as elastic. To this end, the New Zealand standards and Eurocode 8 (but not the U.S. standards), require proportioning the wall sections in bending for a linear envelope over the height of the positive and negative wall moments, as these result from the analysis for the seismic action (Fig. 2). Such an envelope includes also the often quoted increase in bending moments due to higher modes and wall inelastic action, which affect only the region above the base after the development of plastic hinging there 10 . Moreover, the New Zealand standards define the control point of the linear moment envelope near the base to be not the seismic wall moment from the analysis, MEw, but the design flexural capacity of the wall at its base as detailed, MRw, times an overstrength factor of 1.25/0.9=1.38. In this way, -the possibility that a base overstrength will delay hinging at the base and cause a proportional increase of wall moments over the full height, is accounted for. U.S. standards [l-4] do not include provisions for Capacity Design of walls, relying on their high stiffuess and on the satisfaction of eqn ( 1) by the v ••. 12 columns for avoidance of soft-storey formation.

In eqn (3) the overstrength factor i'Rd is taken equal to 1. 3 5 for RC structures of the highest ductility level in Eurocode 8 (Ductility Class or DC High, H) and to 1.20 for the intermediate ductility level of DC Medium or M. If the seismic action controls proportioning of the beams, as in frame structures in high seismicity regions, then LMRb~LMsb i.e. 0~1, and eqn (3) degenerates into eqn (1) If proportioning of beams is not controlled by the seismic action (as, e.g., in long-span beams with high gravity loads, or in "wall-equivalent" dual structures in which the frames take up only a small fraction of the seismic forces, or in the direction of column bending which is normal to the principal direction of the horizontal seismic action), then LMRb»LMsb and otends to zero. In such cases, the moments for which columns are proportioned are ( 1+yRd) times those from the analysis, but still are very low in comparison to the beam strengths. To prevent column hinging at the base of the bottom storey earlier than beam hinging in the storeys, Eurocode 8 [7] applies a moment magnification at the bottom sections of ground storey columns equal to that at their top section. For simplicity the New Zealand standard applies the Capacity Design magnification of eqn ( 1) on the column moments resulting from the analysis for the horizontal seismic action acting alone, while the U.S. Codes [1-4] and Eurocode 8 [7] include in Msc the contribution of the quasi-permanent gravity loads, G+\ll2Q (as well as that of an appropriately reduced simultaneous horizontal component of the seismic action in the orthogonal direction). U.S. standards require satisfaction of eqn ( 1) separately in each horizontal direction of bending but in every column of a "Special Moment Frame". If eqn ( 1) is not satisfied at a single level of a column of such a frame, the contribution of that column to the lateral strength and stiffness of the frame must be neglected and the column has to be proportioned for gravity loads alone, respe~t­ ing though all the requirements for minimum longitudinal and transverse remforcement of "Special Moment Frames", so that it can sustain the ductility demands imposed by the rest of the lateral-force-resisting system, the displacements of which it shares. Obviously, considering the strength and stiffuess of a column for one set of actions (the gravity loads) and neglecting it for another, is inconvenient from the modelling and analysis point of view. Similarly, the application of eqn (1) within the framework of the New Zealand standards [5,6], with the many different special cases (dual systems v. pure frames, one-way v.

449

b

(b)

(a)

Vs MRw

MRw

VSd.b _ _ __,__ _. .

(from analysis)

Figure 2: Capacity-design determination of wall design a) moments and b) shears

450

I•,

Computer Analysis and Design of Earthquake Resistant Structures

Computer Analysis and Design of Earthquake Resistant Structures

2.2.2. Capacity design in shear. Ductile behaviour of RC members under inelastic cyclic loading is only possible when inelasticity is controlled by the tension steel. This is the case in flexural members underreinforced in tension, with compression zone protected against concrete crushing and bar buckling through closely spaced transverse reinforcement. Load transfer by shear does not lend itself to ductile cyclic behaviour, as: a) after tensile yielding of the transverse bars, shear defonnations are associated with slippage along wide-open diagonal cracks and hence dissipate very little energy, and b) diagonal tension failure is inherently brittle. So, as in a single member the modes of load transfer and failure in shear and in flexure are in series, modern seismic design standards adopt a Capacity Design approach of RC members in shear, which strives to preclude member shear failure through proportioning them for a shear force higher than that associated with flexural yielding at member ends. In beams framing at both ends into stronger vertical members satisfying the Capacity Design rule, eqn (!)or eqn (3), a negative moment plastic hinge is expected to develop under the design seismic action at one end and a positive moment one at the other. If the development of a plastic hinge in negative bending at end 1 and in positive bending at end 2 correspond to a positive shear force in the beam, the absolutely maximum and minimum design shears at point x along the beam are given by the following Capacity Design rule in shear:

"limited ductility", one or both ends of which are expected to remain elastic in flexure during the design seismic action, the New Zealand standard still requires computing the Capacity Design shear forces from eqns (4) and (5), assuming though that positive plastic hinges develop not at beam ends but elsewhere along the span (e.g., at the point where the available flexural capacity in positive bending first becomes equal to the demand seismic moment, under the combination of quasi-pennanent gravity loads with the seismic action that causes beam or column yielding, whichever comes first, around the two end joints of the beam). As the distance between these more likely plastic hinge locations is less than the clear span In of the beam, eqns (4) and (5) produce in this way higher Capacity Design shears for beams in frames of"limited ductility". Strong beams framing into weak columns (i.e. which do not satisfy the Capacity Design rule, eqns (l) or (3)), are unlikely to develop plastic hinges at the ends before the columns to which they frame into. Then, if at end 1 of the beam the sum of beam design flexural capacities exceeds that of the corresponding ones of the columns in the sense associated with negative beam moment at end 1, i.e. if c~:::MRb)1.>( LMrc)I- (where subscripts denote the end of the beam and the sign of beam moment there), then MRhI- in eqn (4) should be replaced with the beam moment at column hinging both above and below the joint at end 1. Assuming that the moment input from the yielding columns to the elastic beams is shared by the two beams framing into the joint in proportion to their end moments from the linear elastic analysis for the design seismic action, MEb, the beam moment at end 1 at column yielding there can be assumed equal to IMEb1ILMRc/ILMEbl1- (with seismic beam moments, MEb, around the joint are considered with signs). Similarly for end 2 and for the two directions of bending of the beam. So a rational generalization ofeqns (4) and (5) is:

M;bl+M~b2 , a. IVE (x )I) . max Vsd (x) = Ys.G+1j1 20 (x) + mm(hd 1n

(4)

. . M~bl +M;b2 , a.Vdx) I I) mmVsd(x)=Vs.G+ljl 20 (x)-mm(hd 1n

(5)

In eqns (4) and (5) Vs,G+ljl 2Q(x) denotes the shear force at x due to the quasi-permanent transverse loads on the beam, computed considering the beam as simply-supported; YRd is an overstrength factor (essentially an additional partial safety factor); MRbt, MRhi. (i=l,2) denote the design flexural capacities of end sections 1 and 2 as detailed, in positive and negative bending, respectively; In is the distance of sections 1and2 along the beam; VE(x) is the shear force at x due to the design seismic action according to the linear elastic analysis and a.> 1 is a code-specific factor, aiming at limiting the magnitude of Capacity-Design computed shears. Eurocode 8 [7] requires the application of eqns (4) and (5) only in DC High beams, with yRd=l .25 and a. equal to the behaviour factor q. (For the other Ductility Classes Eurocode 8 essentially specifies for beams a.=l and YRd>>l). The Japanese standard [8] specifies for beams YRd=l and a.=1.5. For the beams of"Intermediate Moment RC Frames", which, as explained in the next section, correspond to a lower required ductility level than the "Special Moment Frames'', U.S. standards [1-4] specify YRd=l/0.9=1.11 and a.=2, while for those of"Special Moment Frames" the requirement is for YRd=l.25/0.9=1.39 and a.>>l (implying that the last term in parenthesis in eqns (4) and (5) does not apply). The provision of the New Zealand standards for the beams of fully ductile frames is similar: YRd=l.25/0.85=1.47 and a.>> 1. For beams in frames of

451

hd . . _ min(LMR0 ,LMRb) I I maxV5d(x) = Ys.G+ljl 20 (x)+-mm[mm(MRbl,( I" J )1- MEbl ln L.,MEb

+min(M~b2 ,(

min(LMRc•LMRb) I I I I IIMEbl ) 2+MEb2), aVE(x)]

(6)

YRd · · + min(LMRc'LMRb) · mmVsd(x)=Vs.G+ljl 20 (x)--mm[mm(MRbl•( I" I )1+IMEb1I Jn L.,MEb min(LMRc'LMRb) I I I I . _ -mm(MRb2,( IIMEbl )2_MEb2), a. VE(x)]

(7)

In eqns (6) and (7) subscripts 1 and 2 denote beam ends and - or+ the direction of beam bending there. The simplest way to establish Capacity Design shear forces for columns is to assume that both column ends, 1 and 2, develop plastic hinges in opposite bending (+ or -) and compute the resulting shear force from equilibrium. As typically no intermediate transverse loads act on columns, Capacity Design shears are constant along the column height and equal to:

452

1

Computer Analysis and Design of Earthquake Resistant Structures

I

. v- = mm(y M

M

M- +M· Rel Re2

h

•

a.IV

j) E

(8)

n

v;d = min( hd

M. +MRel h Re2 ' a.jVE

i)

(9)

n

In eqns (8) and (9) h0 denotes the clear height of the column, considered in general equal to the distance between the top of the beam or slab at the base of the column and the soffit of the beam at the top, within the plane of bending. If the column shear capacity is independent of the direction of the shear force, then only the maximum of the shear forces V sd- and Vs/ in eqns (8) and (9) is of interest. The design flexural capacities MRc1 and MRc2 of the column should be the maximum possible within the range of fluctuation of its axial force under the combination of quasi-permanent gravity loads and the design seismic action. If the column shear capacity depends on axial force as well (shear capacity increases with increasing axial compression), more than one possible axial force values should be considered for the calculation of MRci (i=l ,2) in eqns (8) and (9), in search of the most critical condition for the shear verification of the column and the proportioning of its transverse reinforcement. Eurocode 8 adopts for columns of Ductility Class High (H) and Medium (M) the Capacity Design format of eqns (8) and (9), with a. equal to the behaviour factor q and with YRd=l.35 for DC H and YRd=l.2 for DC M. The U.S standards [1-4] adopt the same format for the columns of "Intermediate Moment Frames", with a.=2 and YRd=l/0.7=1.43. For the columns of"Special Moment Frames" the same standards specify YRd=l.25/0.7=1.79, do not employ the term alVEI, but adopt a format analogous to eqns (6) and (7) to limit the design shear of columns framing into weaker beams and satisfying the Capacity Design rule of eqn (1). The same approach, expressed in eqns (10) and (11) below, is followed by the Japanese standards [8], with YRd=l and a.=1.5:

(10)

+

Vsd

hd

.

.

+

= -mm[mm(MRel ,( hn

-

min(IMRe•LMRb) I" )1.IMEc1\ L.,MEb 1

(11)

. _ min(LMR"LMRb) \ \ \ \ -mm(MRc2,( IIMEbl )2- MEc2 ), a VE(x)] The New Zealand standards adopt the general Capacity Design format of columns in shear in a slightly simplified form. For "one-way" columns of "fully ductile" frames, designed for a global displacement factor µs of 6.0, YRd is taken

Computer Analysis and Design of Earthquake Resistant Structures

453

equal to 1.91, and for "two-way" ones YRd is equal to 2.35, while the a.JVEI term does not apply. The corresponding YRd values in frames of"limited ductility" are 1.62 and 1.91, while a. is taken equal to 6/1.25=4.8, corresponding to essentially elastic response. Column flexural capacities, MRci (i=l,2), are multiplied by 2 1+1.6(v-O. l ) , to account for enhancement due to confinement of concrete which increases with increasing axial load ratio, v=N/ Acfcd· ' The higher mode and inelastic action effects on wall moments above the base, mentioned at the end of Sect. 2.2.1, increase also the value of the wall shear forces at the base and higher up, above the values corresponding to plastic hinging at the base according to the elastic analysis predictions. The taller and more slender the wall, the larger is such an increase. For this reason and to provide for an overstrength in shear at least equal to that in bending at the base, the New Zealand standards magnify wall shears from the analysis for the seismic action, VEw, by the afore-mentioned ratio of YRdMRwlMEw times a dynamic magnification factor of min(l.8, 0.9+0. lnstor, l.3+n,10,/30) (n,10, denotes the number of storeys, which is roughly proportional to the fundamental period of the structure, Tl). For the same reasons Eurocode 8 magnifies seismic shears VEw of walls in the upper two Ductility Classes of buildings by the square-root of the sum of (YRdMRwlMEw) (with YRd equal to 1.25 for DC High and to 1.15 for Medium) and O. l(q~oag!S.(T 1)) 2, in which ~ 0ag=2.5ag is the spectral acceleration in the constant acceleration range of the 5%-damped elastic design spectrum, q the behaviour factor and S.(T 1) the elastic spectral acceleration at the fundamental period in the direction parallel to the wall (strictly speaking, the period of the mode with the largest modal participation mass in the direction of the computed wall shear force). This second term is neglected for walls with slenderness (i.e. height-to-horizontal-dimension) ratio less than 2 and becomes significant for flexible dual systems on relatively hard soil (for which the fundamental period T1 lies well into the constant spectral velocity range). Finally, to improve further protection against underestimation of shear demands in the analysis, the New Zealand and the European standards require the design shear force in the upper two-thirds of the wall to follow a linear envelope, defined by a top value equal to half the design shear at the base of the wall, and a value at one-third of the wall height equal to the corresponding value from the analysis, magnified as above, but not less than half the value at the base (Fig. 2(b)).

2.2.3. Capacity design of foundation structures: Because of the importance of the foundation for the transfer of gravity loads to the ground, and due to the difficulty and cost normally associated with detection and repair of damage in foundation structures, the latter should be protected from significant inelastic deformations during earthquakes. In other words, the foundation part of ductile structures should be designed to remain elastic during the design seismic action. This can be achieved through the Capacity Design concept: foundation elements are not designed to transfer to the ground just the seismic action effects resulting from the linear elastic analysis, but the maximum internal forces that can be

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Computer Analysis and Design of Earthquake Resistant Structures

Computer Analysis and Design of Earthquake Resistant Structures

developed in the as-detailed vertical element they support (column or wall) at overstrength. This implies that seismic action effects resulting from the linear analysis of the response to the design seismic action for the foundation elements, and the forces to be transmitted to the ground, should be multiplied by the, magnified by an overstrength factor YRd, ratio of the flexural capacity of the bottom section of the supported column or wall, MR, to the corresponding elastic seismic moment from the analysis, ME, before superposition with the computed effects of the quasi-permanent gravity loads. The multiplication by the overstrength ratio YRdMR/ME should extend to all relevant seismic action effects, such as bending moments, shears and axial forces transferred from a vertical element to its footing, to the associated soil reactions, to the end moments, shears, etc. of tie-beams or foundation beams framing into the footing, etc. This Capacity Design concept for foundation elements can be easily applied to isolated footings, with or without tie-beams. Often though more than one vertical elements share a common foundation element, e.g. a combined footing, a foundation beam or wall, a raft foundation, etc. Then a (weighted by MR or ME) average of the overstrength ratio YRdMR/ME, pertaining to all the associated vertical elements, may be used to amplify all seismic action effects of the relevant foundation elements and soil reactions, as these result from the linear elastic analysis of the superstructure-foundation-soil system for the design seismic action. If such an operation is computationally inconvenient or raises questions regarding its implementation, a constant but conservatively large overstrength factor YRd could be used instead of the average YRdMR/ME value.

the structure is nearly independent of the magnitude and the extent of inelastic action. Last but not least, a structure with ample ductility supply is more tolerant to the magnitude and the details of the seismic action, and, in view of the large uncertainty of the extreme seismic action in the lifetime of the structure and of the structural response to it, it represents a better earthquake-resistant design. An opinion widely shared by experts [4,10] is that increasing the design lateral forces and the strength of a structure beyond a certain level does not improve materially its seismic performance. According to this view, performance can only be improved through better detailing for ductility. There are also strong arguments in favour of less ductility and more strength in seismic design: the higher the lateral strength of a structure, the smaller is the structural damage during moderate but more frequent earthquakes, or even due to the design seismic action. Moreover, from the construction point of view, detailing a RC member for strength is easier and simpler than detailing it for high ductility. More importantly, a large percentage of structures to be designed for earthquake resistance possess significant lateral strength, due to their force-based design against normal actions. Gravity-dominated low-tomedium-rise structures in low-to-moderate seismicity regions, wind-dominated structures, such as cooling towers or tall, flexible buildings, etc., are examples of structures possessing significant resistance to earthquake forces, without even being designed for them. For such structures it makes sense to profit from the available margin of lateral strength to avoid complex and expensive detailing of members for ductility. Last but not least, if the configuration of the structural system is unusually complex and irregular and falls outside the framework of normal structures mainly addressed by seismic design standards, the designer may feel more confident for his design if he narrows the gap between the results of the linear elastic analysis, on which proportioning of members is based, and the nonlinear seismic response under the design seismic action. This is achieved by using a lower value of the force reduction or behaviour factor R or q, implying lower global and local ductility demands. The reduction in global ductility demand at the expense of increased lateral strength, is often the result of a drastic relaxation, or even a waiver, of Capacity Design requirements. Capacity Design rules for columns in bending and for beams and columns in shear aim at avoiding overstrength in the ductile modes of behaviour and (possible) failure, such as those (mainly of beams) in flexure, relative to the more brittle ones, such as that of all elements in shear and of columns in flexure. Such overstrengths may occur if the resistance of the more ductile modes is controlled by gravity loads or minimum reinforcement, while that of more brittle ones is controlled by the design seismic action. In lower ductility structures design seismic internal forces are in the order of 50% or more of those resulting from purely elastic response to the design ground motion. For so high design seismic forces, it is expected that the seismic action will control proportioning of every member against all failure modes and that undesirable overstrengths will not take place. Moreover, the demand values of the member curvature ductility factors associated with the low global displace-

2.3. Strength versus ductility in RC seismic design As already stated in Sect. 2.1 the magnitude of lateral seismic forces for which an earthquake resistant structure has to be designed is about inversely proportional to the global displacement ductility factor, µo, that the structure is prepared to sustain. Therefore, there seems to be an economic incentive in increasing the available global ductility supply of a RC structure, through proper application of Capacity Design procedures and detailing of the members for development oflocal ductility. Moreover, limiting the magnitude of lateral forces that a structure can sustain facilitates the design and verification of the foundation, as it limits also the magnitude of the forces to be transmitted to the soil, and reduces the likelihood of permanent soil deformations or of uplifting and rocking of the structure relative to the ground. In addition, setting an upper limit on the level of lateral forces that can be developed, reduces response accelerations and hence protects better any equipment mounted on the structure, or nonstructural parts which are sensitive to accelerations (such as masonry infill panels in the out-of-plane direction). Setting an upper limit on lateral forces, by allowing structures to yield, does not affect adversely deformation-sensitive nonstructural parts (such as infill panels and partitions in the in-plane direction}, since, according to the "equal displacement rule" which applies in good approximation in the frequency range in which the predominant frequency of the seismic response of most structures lies, the magnitude of lateral displacements and deformations of

455

456

';

Computer Analysis and Design of Earthquake Resistant Structures

ment ductility factors of low ductility structures, are relatively low, even if inelastic deformation demands are not uniformly distributed in the structure. These low local ductility demands can be easily accommodated through detailing of beams and columns which is only slightly more demanding than in nonearthquake resistant structures. Accordingly, Capacity Design requirements can be waived. As explained in the following sections, the application of Capacity Design rules greatly complicates the entire phase of proportioning/detailed design of members. The design process is complicated further by the stringent member detailing requirements for high local ductility. Accordingly, the selection of a higher or lower ductility level for a structure has important implications on design computations: if the designer opts for a higher ductility for his structure, he needs to have at his disposal more advanced computational tools and he should have the experience and expertise necessary for their use. Most modern seismic codes provide for more than one combinations of strength and ductility, sometimes leaving the choice to the designer, sometimes not. U.S. standards are an example of this latter case: The official standards [13) allow the use of "Ordinary Moment Frames", with no requirements for ductility whatsover, in the lower two seismicity zones of the U.S. In the two zones with the highest seismicity only "Special Moment Frames" can be used, which enjoy very good global ductility, through the application of Capacity Design rules for columns in bending and for beams, columns and joints in shear, as well as high local ductility, through the use of stringent detailing of longitudinal and transverse reinforcement of all types of members. In the intermediate zone of moderate seismicity, the use of "Intermediate Moment Frames" is allowed, which do not have to satisfy the Capacity Design rule of columns in bending, eqn (1), or of joints in shear, and follow a more relaxed Capacity Design rule of beams and columns in shear and less stringent requirements for the longitudinal reinforcement of beams and the transverse bars of columns. The NEHRP recommendations [4] adopt the same typology of RC frames according to global and local ductility supply, but specify their use not according to seismicity alone, but to the desired "Seismic Performance", which is an increasing function of both seismicity and "Seismic Hazard Exposure" (a term used to describe the type of occupancy and the importance of the structure). The force reduction factor R specified by the U.S. standards [1-4) for the structural systems of different ductility are also different: An R factor value of 8 [4] to 8.5[1,3) is allowed for "Special Moment RC Frames", in combination with RC walls, coupled or not, while dual systems of RC walls and "Intermediate Moment RC Frames" are assigned an R value of 6 [4] or 6.5 [1,4). In the absence of walls the R factors of these latterframes are equal to 4 [4] or 5 [1,3), while those of "Ordinary RC frames" are equal just to 2 [4] or 3.5 [1,3). Eurocode 8 allows trading ductility for strength through the provision of the three alternative Ductility Classes. Actually explicit allowance is made for an additional Ductility Class, namely that of structures in very low seismicity regions, which are allowed to follow only the provisions of Eurocode 2 and are designed for a behaviour factor q slightly higher than 1.0. In Ductility Class

1

Computer Analysis and Design of Earthquake Resistant Structures

457

(DC) Low (L) the behaviour factor q is half of that of DC High (H), while in DC Medium (M) the q-factor is 75% of that of DC H. In return, the lower the Ductility Class the less demanding are the detailing requirements for local member ductility. Moreover, the higher the Ductility Class, the more are the columns overdesigned in flexure and the beams, columns and walls in shear. As quantified by F ardis [ 11] through the full design of 26 RC structures to Eurocodes 2 and 8, the end result is that, although the total quantity of steel and concrete is essentially independent of the Ductility Class for which the structure is designed, the higher the DC, the larger is the share of transverse reinforcement and of column reinforcement to the total quantity of steel. All the provisions ofEurocode 8 and 2 for the proportioning and detailing of RC beams, columns and walls of the three different DCs are conveniently summarized and compared by Fardis [12) in the form of three Tables. The selection among the DCs of Eurocode 8 is left to national authorities. It is expected that in low seismicity regions DC L will predominate, as more economic and easier to apply, whereas in high seismicity ones the economic incentives for applying DC H will be reinforced by the available professional tradition and expertise in computationally demanding seismic design and in on-site execution of complex detailing for ductility. It is hoped though that the engineer will have the option to select himself the most appropriate DC for his design. The New Zealand standards [5,6) allow the designer to select a global displacement ductility factor µc; less than the value of 6 corresponding to fully ductile frames, if he decides so, e.g. in the cases listed at the beginning of this section. Design lateral forces are increased then, as the design response spectrum of the New Zealand standards is an explicit function of µc;. In return, some (or even all) of the columns (especially internal ones) are allowed to develop plastic hinges. The ductility factor to be used for the design (typically between 1.25 and 3 in these "limited ductility" frames) is a function of the fraction of storey shear carried by those columns which are protected from plastic hinge development through the application of the associated Capacity Design rule. This rule though is applied with lower dynamic magnification factors ro than in fully ductile frames, namely with ro=l. l or 1.3 for "one-way" and "two-way" columns respectively, regardless of the value of the fundamental period (these values correspond to aggregate overstrength factors YRd of 1.62 and 1.91 respectively). Capacity Design of beams and columns in shear is essentially as in fully ductile frames, with the generalized expressions, eqns ( 10) and ( 11 ), applied to columns, depending on whether they are expected to develop plastic hinges or not. In addition to the relaxation of Capacity Design in flexure for the columns, column detailing requirements are also relaxed, in return for the increase in design forces: in columns expected to remain elastic, stirrup spacing for the prevention of bar buckling is as in detailing for normal (i.e. non-seismic) actions, while ifthe design global ductility factor µc; is less than 3, the confining steel in columns allowed to develop plastic hinges is computed on the basis of a curvature ductility factor, ~.of 10, instead of20 in plastic hinges of columns of

l 458

.., ••1

Computer Analysis and Design of Earthquake Resistant Structures

Computer Analysis and Design of Earthquake Resistant Structures

fully ductile frames, which means that it is finally reduced by about 30%. Clearly the New Zealand design provisions for alternative strengthductility combinations give considerable flexibility and freedom to the designer, without unduly complicating the design process, to the point of making it computationally inconvenient or very difficult to implement in a software code. The current Japanese standard [8) offers the maximum flexibility and choice to the designer, by providing for 4 different ductility classes (called "Performance Rating") at the structure, storey and member level, and by allowing mixing in the same storey members of different ductility classes, as well as designing different storeys in the same structure for different ductility classes and associated force reduction or behaviour factors. Specifically lateral forces are defined at the storey shear level, instead of the storey (inelastic) load level preferred by most seismic Codes: the behaviour factor dividing the elastic shear force of a storey is a function of the percentage of storey shear carried by walls and of the "Performance Rating" (i.e. the ductility class) of the storey members, regardless of what happens in the overlying stories: If the Eurocode 8 notation ofDCs is used also for the "Performance Ratings", i.e. H, M, Land 0, a storey is classified as H if, according to the analysis, at least 50% of its shear force is carried by class H elements, and L if at least 50% of its shear is carried by class L ones. Otherwise, the storey is classified as M. In the presence of any element with Performance Rating 0, the storey is classified as 0 and requires special care. If more than 70% of the storey shear is carried by columns or walls, the structural system of the storey is characterized as "frame" or "wall" respectively, with intermediate cases classified as "dual" storeys. The lower value of the inverse of the behaviour factor is 0.3 and applies to "frame" storeys of "Performance Rating" H. The value of the inverse of q increases by 0.05 in going from "frame" to "dual" to "wall" storeys, and in decreasing the storey "Performance Rating" by one level. The highest value of the inverse of q is 0. 55 and corresponds to class O "wall" storeys. (To compare with the corresponding behaviour factors of the other Codes, the values quoted above should be multiplied by 0.9 before inversion. Then the maximum q-factor value, applying to a frame system of the highest ductility level is about 3.8, and the lowest, for a "zero"-ductility wall system is 2). Elements are classified as H, M, L and 0 depending on their failure mode (with a non-flexural failure mode implying classification as 0), and of their normalized to bdfcd design seismic shear. The classification of columns takes also into account the magnitude of the maximum, normalized to Acfcd, axial compression, the slenderness ratio and the resemblance to a short column, and the longitudinal steel ratio. This system of trading storey strength for ductility; although very flexible and liberal, is extremely inconvenient from the computational point of view, as, among others, it requires information from the analysis of the structure for the seismic action and even from the proportioning and verification of members, for the determination of the magnitude of lateral forces for which the structure needs to be analysed and designed. Apparently this Code approach lends itself only to storey-by storey hand calculations, necessitating rough approximations for the distribution of storey

459

shears to the storey elements, almost regardless of what happens beyond the immediately adjacent storeys. In summary, U.S. and European standards provide for a few "discrete" strength-ductility combinations, each with its own well-defined rules for member proportioning and detailing. They are, therefore, most convenient for computational implementation and routine application, although they limit significantly the choices available to the designer. At the other extreme, the very flexible Japanese approach is almost impossible to implement computationally, at least in its full range and potential. Despite its general hand-calculationsoriented spirit, the flexible and liberal New Zealand approach of trading strength for ductility, seems fairly well suited for computational implementation.

3.

Conceptual design of earthquake resistant RC structures

The governing consideration in conceptual design of RC structures controlled by normal actions, is the minimization of cost and the maximization of the functionality of the facility. Safety is seldom a major concern in this phase of the design of such structures, as it is essentially guaranteed by the subsequent design phases, i.e. by the application of State-of-the-Art analysis methods and by conformance to Codes during member proportioning and detailing. On the contrary, structural configuration is a key factor for the safety of seismic-controlled structures, as it limits deviations of the actual response to the design seismic action from that assumed in design and used as the basis of member proportioning: A key postulate of codified seismic design is that peak structural displacements and deformations due to the design seismic action are approximated reasonably well, at the local and global level, by the results of an elastic analysis, even though members are not proportioned but for a small fraction of the internal forces resulting from it. Abrupt and drastic changes in stiftbess and strength along the height may invalidate this analysis and design postulate, -causing localisation of deformations and early failure of some members. Staggered floor arrangements or staggering of wall panels in elevation, or large openings in critical regions of elements, may also cause localisation of inelastic deformations in the transition regions. Low redundancy systems lack the ability of redistribution of internal forces in case of premature failure of regions or elements, and possess limited overstrength over the design seismic action. Very irregular and asymmetric arrangements of structural elements in plan may cause strongly torsional response and/or coupling oflateral and torsional vibrations, which may not have been considered in design, and hence may precipitate failure due to localisation of deformations in a few perimeter elements of the structure. Structural elements non uniformly distributed over a plan with a large aspect ratio, or floor slabs with large internal openings and/or many re-entrant comers, may invalidate the common rigid diaphragm assumption and lead to unexpected modes of vibration and to force distributions different from those considered. Weak connection of all elements at all floor levels impairs the ability of the structure to act as a whole, limits its redundancy and redistribution capacity and invalidates

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Computer Analysis and Design of Earthquake Resistant Structures

Computer Analysis and Design of Earthquake Resistant Structures

many modelling assumptions. Foundation of vertical elements at different levels, without a strong horizontal connection between them, raises questions regarding the validity of the support conditions assumed in analysis. These are a few examples of cases in which structural configuration increases the uncertainty of the actual response relative to that considered in design. This most important, especially for earthquake resistant structures, phase of design, gives the opportunity to the designer to use his judgement, creativity and experience, but does not lend itself to computer-based automation. Research and development efforts have long been underway for the production of non-algorithmic, heuristic computational tools which supplement the engineer's experience and judgement with "expert knowledge", and assist him in exploring and comparing many alternative configurations. These efforts have focused on transferring know-how from the Artificial Intelligence field of Knowledge-Based

(Expert) Systems (KBES), to structural design. A few examples of this effort are found in [13-20]. Knowledge-based systems seem to hold promise as a flexible and expandable interactive tool offering assistance to the designer during conceptual and preliminary design: they can propose to him alternative solutions, rate them according to certain criteria, accept and elaborate, or even propose, modifications to these solutions at the request of the designer, perform a few calculations for preliminary sizing of key members and cost estimations, etc. As noted by Fenves [21 ], case-based systems, in which past designs, previously stored and indexed, are retrieved, if they are "similar" to the new problem according to a certain similarity metric (Maher [15]), in order to serve as a starting for a new design, hold particular promise, in that they reflect the individual style and idiosyncrasy of the designer or his organization. Despite the more than one decade of R&D efforts on KBES in structural design, and the already available know-how and experience in other areas of engineering and science, results to date are not impressive. At present KBESs do not seem capable of producing and recommending anything more than routine, almost standardized, conceptual designs, which do not go much beyond the capabilities of a rather inexperienced designer. Possibility due to the increased complexity of the problem, there is a clear lack of KBES for conceptual design of earthquake resistant structures. The recent work by Avramidis et al [20] is a notable exception in this respect. In particular for earthquake resistant design, the knowledge-base should include, in addition to "expert" knowledge, the rules and recommendations of design codes regarding the uniformity and regularity of the distribution of stiffness, mass, etc. in plan and in elevation, the provision of lateral stiffness near the perimeter for torsional rigidity, the configuration of floor diaphragms and foundations, etc., along with the capacity for the corre~nding required calculations. Despite the limited fruits of the relevant efforts to date, KBESs seem to be the proper framework for providing in the long run interactive assistance to the engineer in this very important stage of design, without falling into the pitfall of design automation and abolition of the engineer's judgement and initiative, which characterises the algorithmic/procedural computational approaches.

4.

461

Modelling and analysis of RC structures for seismic actions

4.1. Overview of codified seismic analysis approaches 4.1.1. General framework. Most modem seismic design standards provide for two alternative methods of linear-elastic seismic analysis: "Multimodal response spectrum" analysis [7], also called "modal" analysis [4) or "dynamic lateral force" [1,3] procedure, and "lateral force" [1,3] or "equivalent lateral force" procedure [4], or "simplified modal" analysis [7]. Differences between Codes in this respect are not limited to terminology: they extend to the general attitude towards the method of analysis. Although most codes permit application of the "multimodal response spectrum" analysis to all kinds of structures, and require its use in structures which are irregular in elevation or flexible enough for the higher mode response to be important, only Eurocode 8 considers this method as the reference method, and respects fully its rules and results. Other codes essentially consider the "static" or "equivalent lateral force" procedure as reference and adapt "modal" analysis results and its rules of application to conform to it [1,3,4]. The attitude of these latter codes towards analysis, shared by the majority of the international community of seismic design experts and practitioners, is that linear elastic analyses performed within the framework of earthquake resistant structural design are of relatively little relevance and value, as their results apply up to a small fraction (1/q or 1/R) of the design seismic action. Consequently, moderate earthquakes, expected to occur at least once in a structure's lifetime will cause the strength of several members to be exhausted and the structure to enter the inelastic range. Moreover, even in very regular structures designed according to modem seismic standards, peak inelastic deformation demands under the design seismic action will differ significantly, both in magnitude and in distribution, from predictions based on the results of linear elastic analyses. It makes little sense therefore, according to this point of view, to apply a sophisticated, complex and computationally demanding analysis procedure, liable to misuse, misinterpretations or even design errors due to lack of relevant experience and expertise from the part of the designer. It makes more sense, instead, to select the configuration of the structure so that it lies well within the field of application of the time-tested and almost fool-proof"static" of"equivalent lateral force" analysis procedure. This attitude towards analysis approaches is reflected in the view that the value of the "multimodal response spectrum" procedure lies only in its ability: a) to provide a distribution of lateral inertia forces with height which, for a certain class of structures (those with irregularities of geometry, mass, stiffness or strength in elevation, or the flexible, higher-mode-dominated ones), is more representative of the actual dynamic response; and b) to account better for coupling of torsional and translational vibrations in structures with strong asymmetries and irregularities in plan. Consistently with perception a), it is suggested by its supporters to apply the "modal" analysis procedure not directly, i.e. by computing modal contributions to the seismic action effects of interest (such as member

462

·;,,,

·'·•·· ,. •·•I

Computer Analysis and Design of Earthquake Resistant Structures

internal forces) from the natural mode shapes themselves and combining them according to the appropriate rules, but indirectly, i.e. by applying storey modal lateral forces from the modal analysis as external loads and computing therefrom through static analysis the modal seismic action effects of interest. Moreover, the magnitude of the resultant of these modal lateral forces (i.e. the base shear) is scaled up to match that computed from the design spectrum at the fundamental period of the structure, fully in irregular structures or up to 90% for regular ones [1,3): In [4] scaling is to the base shear determined from the design s?ectrum at a pen~d equal to 1.4~ to 2.05 times an empirically determined penod value, depending on the magnitude of the design ground acceleration. Most seismic design standards require all structural members to be included in the model of the lateral-load-resisting system, and to be proportioned for the seismic action effects computed from the analysis. U.S. standards, though, allow the designer to designate members or subassemblies, such as frames .or walls, a~ part of the lateral-load-resisting system or not. Those parts not assigned to this latter system should be included in the structural model for the analysis for normal actions, such as gravity and wind loads, but are negl~cted co~pletely. in the ~ode! used for the seismic analysis. From the computational pomt of view, this requires the ability to designate some parts of the overall structural system as inactive for the analyses for certain actions. This can be achieved by setting their stiffuesses equal to very small values, if this does ~ot create ill-conditioning problems, or, preferably, by inactivating the elements m. th~ assembl~ of the global stiffuess matrix and in the subsequent phases of se1sffilc analysis. The same standards require these parts of the structure not only to be designed for the normal action effects computed from the corresponding analyses and to be detailed for ductility (through the provision of some mi~mum longitudinal and transverse-confining reinforcement, and through Capacity-Design determination of member design shear forces, etc.), but also to remain below ultimate strength under the lateral displacements induced by the design seismic action, including P-~ (second-order) effects. In [1,3] these displacen:ients are computed by multiplying the elastic displacements from the analysis for the seismic action by 3/8 of the force reduction factor R, while in [4] the multiplicative factor is more realistically chosen to be only 10% to 30% lower than R, depending on the inherent ductility of the RC structural system. From the computational point of view this check is non-trivial. It can be done through the following modification of the phase of internal force calculation from j.oin~ disp!acements, for those parts considered inactive in the analysis for the se1srruc action: The regular calculation of the inactive member end forces is either omitted or done with essentially zero member stiffuesses, so that computed seismic action effects for these members are zero. A separate internal force calculation phase follows for these members, in which the actual member stiffness matrices are used and the already computed member end displacements are first multiplied by the amplification factor above (3R/8, etc.) and further increased for P-~ effects resulting from these amplified end displacements (e.g. by dividing everything by 1-0, where e is the ratio of storey second-order to first-

Computer Analysis and Design of Earthquake Resistant Structures

463

oments). Resulting member end moments are compared then with the J h s order m h · d · corresponding flexural capacities. If found highe'., t. e ana1ys1s- es1gn eye e a to be repeated, activating these members and de~1gmng them full~ for the result. · rnic action effects. It is obvious that this freedom provided to the de. . · d 1·t f mg se1s · by the U S standards is at the expense of the s1mp11c1ty an genera 1 Y o signer · · . . . · dd · the entire computational procedure, mcludmg modeling, anal~s1s an es~gn .. U.S. standards [1,3,4) introduce additional comput.a~1onal co~phcat1ons with their requirement that in dual systems, frames, in addition to bemg proportioned on the basis of the results of the analysis of the total structural system for the seismic action, should be designed to act as a back-up syst~m, able to. sustain at least 25% of the design base shear of the entire structure m each honzo~­ tal direction. In a computational environment this requires a separate static analysis in each direction, of a structural model which include~ only th~ frames. This analysis is performed for equivalent lateral forces amounting _to 25 Yo of the base shear of the entire structure, as determined from the analysis of the latter considering its global dynamic characteristics. The most adverse of the two seismic action effects, calculated for the frame members from the glob~! (possibly multimodal) analysis of the dual system a!1d from ~he .separate static ones for the frames alone, should be used then in thelf proport1omng. . As the Code-specified "equivalent lateral forces" represent essent1~lly conservative envelopes of storey inertia forces for the purpose of the calculation of storey shears, peak overturning moments at various hori~ontal lev~ls of the structure may be grossly overestimated, if calculated. as statically eqmvalent to these "static" or "equivalent lateral forces". Accordmgly, the .NEHRP recommendations [4] allow reduction of the so-computed overturmng moments by 25% at the foundation level and by smaller percentages at storey levels. The r;duction is specified as zero up to the 1Oth storey from the top and equal to 2 Yo for every storey below that, up to a maximum reduction of 20% at and belo~ the 20th storey from the top. Modal overturning moments .from .modal analysis reflect realistically the distribution of modal inertia forc~s With he1g~t, and hence their final combination through the SRRS or other eqmvalent rule mto peak dynamic storey overturning moments is considered to be e~sentially free" of ~h~ conservatism associated with an equilibrium-based calculation of storey static or "equivalent lateral" forces. Accordingly overturning mon:ients from :nodal analysis can be reduced only at the foundation level b~ a maximu.m of lOYo [4). It should be emphasized, though, that in a computer-aided analysis ~f structures there is no easy way to distinguish between the effects o~ o~erturmn~ ~oments and those of storey shears. However, it is certainly w1thi~ the spint of ~he NEHRP recommendations -to implement the allowed reduction of overt~rmng moments through a corresponding reduction of the seismic axial forc~s m the vertical elements, leaving all other seismic action effects the sam~. This reduction leads to a violation of equilibrium, at the level of member mternal forces and at that of reactions and external loads, and should be effected at the end, just prior to the use of axial forces for the proportioning of vertical elements.

l Computer Analysis and Design of Earthquake Resistant Structures

Computer Analysis and Design of Earthquake Resistant Structures

4.1.2. Multimodal response spectrum (or "dynamic") analysis method. The multimodal response spectrum approach for the seismic response analysis of an elastic structure (often called "dynamic" analysis) has been described in detail in other parts of this book, and especially in Chapter 8 on Buildings. In that latter chapter details have been given also on the application of static and dynamic condensation to reduce the number of static DOFs into 3 dynamic DOFs per floor (two horizontal translations and one rotation about the vertical axis), taking profit of the behaviour of horizontal slabs as rigid diaphragms in their plane and of the small inertia forces normally associated with vertical translations and nodal rotations about the horizontal axis, for horizontal seismic action. This technique is especially suited to RC buildings, as cast-in-place RC slabs, monolithically connected to the floor beams, etc., satisfy best the rigid floor assumption. As described in Chapter 8, for each normal mode m of the reduced dynamic model, the design response spectrum in entered with the natural period T of the mode, to determine the corresponding spectral acceleration Sa,m· Then, for each one of the two horizontal components of the seismic action, Ex and Ey, two horizontal forces and a moment with respect to the vertical axis is computed for each mode, m, at each floor level i, Fxim, Fyim and Tim. These forces and moments are computed as the product of: a) the participation factor r Xm Or r Ym Of mode m tO the response to the seismic action along direction X Or Y, whichever is of interest, b) the mass associated with the corresponding floor DOF, mXi, myi or lei, c) the corresponding component of the mode eigenvector, Q>Xim, eim, and d) Sa,m. A static analysis is then performed for each mode and horizontal direction of the seismic action, of the full structure according to the static model in 3D, subjected to static forces and moments FXim, FYim and Tim at the corresponding dynamic DOFs of each floor i. Response quantities, like member stress resultants, nodal displacements, member deformations, such as the difference in horizontal translations at the top and bottom of each vertical member in a storey, etc., are computed separately for each mode-and combined for all modes according to the familiar SRSS rule [22], or preferably, according to more accurate alternatives reflecting better the statistical combination of the effects of closely-spaced modes (e.g. the CQC rule [23,24] or alternatives [25]), to yield the best estimate of these peak response results, for the design seismic action acting in horizontal directions X or Y. This "indirect" approach to the estimation of peak dynamic seismic action effects is not possible in structures which do not possess rigid-floors at storey levels, or if the vertical component of the seismic action is of interest or importance in analysis and design. Moreover, with recent (and future) advances in computer hardware, the complexity in analysis software for the reduction of the static DOFs into a much smaller number of dynamic DOFs at floor levels, outweighs the increased demands in computer time and memory, even for largesized structures which fulfil the conditions for static and kinematic condensation. For these reasons the emphasis here is placed on the more general and computationally simpler approach of estimating peak dynamic response quantities directly from the eigenmode results: According to the "direct" approach,

for each normal mode m the spectral displacement, Sd,m, is calculated from the design (pseudo-)acceleration spectrum, as (Tm/27t )2 Sa,m, and used to compute the nodal displacement vector of the entire structure for mode m, Um, as the product of sd,m times the modal participation factor, r Xm, r Ym or Cm, times the eigenvector,

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Computer Analysis and Design of Earthquake Resistant Structures

Computer Analysis and Design of Earthquake Resistant Structures

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supposed to be resistant with respect to seismic action if it can successfully bear events in two different orthogonal directions. If one considers a simple box system constituted by four walls, it is reasonable to think that, for each seismic event in one of the main directions of the building, only two of the walls will give resistance to the building, while the other two will be only supported by them. This is evident in Figure 10 showing the seismic actions and resistant walls in a simple box system. If the structure is either more complex or not symmetric, the analysis is obviously not so simple, and the total force will be distributed among all the elements, in a way that depends upon the characteristics of the horizontal diaphragms. Figure 11 describes the seismic actions on a building with variable stiftbess walls. It is then possible to single out two different limit conditions: infinitely stiff in-plane behavior of horizontal diaphragms or infinitely weak one. Referring to figure 11, the following behaviors can be put into evidence: a) if the horizontal diaphragms are infinitely stiff, all the building moves in the same way, that is each wall possesses the same displacement, and the two more stiff side walls will provide the same amount of resistance to the whole building. The horizontal action is then distributed according to the stiftbess of each element, as explained in section 7.1.2. Even in the presence of torsional effects induced by the seismic action, the approach is still the same and the same way of distribution of forces still holds. b) on the contrary, if the horizontal diaphragms are supposed to be infinitely weak, horizontal forces will be distributed among all the elements according to the part of vertical loads that they bear. The internal pillars will then receive the maximum horizontal forces and, consequently, they will exhibit large deformations, while the two stiff side walls will only bear a very small part of the total horizontal action. It should be clear that the way in which the horizontal force is distributed among all the different elements does not depend upon the mechanical model used for each single element but only on the in-plane stiftbess of the horizontal elements. These two kinds of approach are extremely distinct and lead to completely different results, as it will be shown in following paragraphs. Real cases are of course intermediate to these limit cases, and so the correct design should be done having in mind both of them, trying to extrapolate from them useful information about the real behavior of the flooring system. As a matter of fact, the total bearing capacity of the whole system could be evaluated by hypothesizing an infinitely stiff behavior of horizontal elements, while local verifications should be done by following the opposite hypothesis.

7.2.1 Infinitely weak horizontal diaphragms - The distribution of horizontal forces is proportional to the part of vertical load borne by each single elements. The study must then proceed by analyzing the vertical load acting on each floor, distributing it among the supporting elements, and considering an horizontal force to each element proportional to the vertical one. Of course, because of the negligible out-of-plane stiffness of all the walls, the part of the

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Computer Analysis and Design of Earthquake Resistant Structures

Computer Analysis and Design of Earthquake Resistant Structures

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Computer Analysis and Design of Earthquake Resistant Structures

Computer Analysis and Design of Earthquake Resistant Structures

horizontal force due to the weight of non-resistant walls (i.e. of those orthogonal to the seismic direction) should also be distributed among the resistant element. On the whole, the distribution of horizontal action will follow the scheme reported in figure 12. The analysis of each resistant element will be then performed considering an horizontal action given by the relation Fh =a·Fv where Fh and Fv, respectively, indicates the horizontal and vertical force to be taken into account, while a is the proportional factor derived from the maximum acceleration induced by the seismic event. It should be noted that several real cases are not so far from to this one. When floors consist of wood elements as well as brick vaults, the in-plane stiffiless of horizontal elements is so small that it can be often considered as negligible. [ 18] 7.2.2 Infinitely stiff horizontal diaphragms - As illustrated in section 7 .1.2, the distribution of horizontal forces can be done by simply referring to the stiffiless of each element, eventually considering the plastic range for each one of them. For this method to be applied, some control must be done about the effectiveness of the restraint between horizontal and vertical elements. Because of the stiffness of the horizontal diaphragm, the motion of the floor can be thought of as a rigid body and, if a rigid connection is present between vertical and horizontal elements, the component of the motion of the upper edge of each vertical element is the same as the horizontal one. The effectiveness of the restraint of the upper edge of the walls as well as their lengths (defined by not taking into account the eventual presence of openings in them) will contribute to their stiffiless and, consequently, to the portion of horizontal forces to be taken into account in their verification.[18] 7.3 Finite elements modeling of structural elements 7.3.1 General aspects - In several cases the simple approach described in section 7.2, is not applicable when analyzing complex buildings, i.e. those where a box system is not easily recognizable or those where the simplified hypotheses illustrated above do not properly describe the real behavior of single elements.[6, 12, 42] In such cases, an approach based on numerical modeling of all the elements by finite elements techniques could appear as more suitable, and the seismic analysis can still oe performed by the "equivalent" static one. Before the description of the method, some preliminary considerations have to be made, regarding the use of the technique and, in a general way, the use of models in the history of the structural study of masonry buildings. As it is well known, the "geometric" description of the real structure obtained by a finite element modeling is normally very accurate, because even very small elements can be suitably described and visualized. Nevertheless,

Figure 12: Distribution of horizontal forces referring to a single wall. (infinitely weak horizontal diaphragms).

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T 590

Computer Analysis and Design of Earthquake Resistant Structures

Computer Analysis and Design of Earthquake Resistant Structures

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results often do not possess the same degree of accuracy, because of the great uncertainties previously described and of the inherent complexity of the "material" used. 7.3.2 "Physical" and "numerical" modeling - Architects as well as engineers have always built "physical" models during the study of their designs, both for formal verification of the structure and for technological and structural control of their work. In the past centuries, and particularly until the age of Galileo, the role of the model was essential because, on the basis of the "proportionality principle", the stability of the model was thought to represent the stability of the real construction; the use of the same ratio between the same elements in the model and in reality seemed to guarantee the resistance of the real building. Although this assumption is wrong, physical models have always kept a relevant role in the study of the stability of buildings, and only after the complete development of the theory of structures, numerical models have obtained a more relevant role in the design procedure. Figure 13 shows Gaudi's structural model made with sandbags but looking like a finite element numerical model.

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7.3.2.1 Finite elements models - While Finite Element Methods (F.E.M. in the following) are widely and successfully used when analyzing both steel and reinforced concrete structures, their utilization in the analysis of masonry structures still provoke some disputes in the scientific community. Together with some scientists which diffusely use them, other designers completely refuse their application in this specific field. As a matter of fact, the F.E.M. can lead to very accurate description of the mechanical behavior of masonry structures, and the obtained results are often not obtainable by other ways of analysis. However, the correctness of the results is not controllable, and there could be no agreement between theoretical and real behavior.[l, 3, 7, 41) The main differences between "real" structures and numerical modeling arise because several F.E.M. based computer programs do not adequately describe the complex mechanical behavior inherent to masonry elements, especially the loosening of resistance with respect to tensile stresses. On the other hand, even if this aspect can be taken into account by some programs, it is still very difficult to introduce correct constitutive laws for the basic material and to ensure the convergence of such highly nonlinear problems. In the field of analysis of masonry structures, F.E.M. must be used very carefully, and the control on the obtained results must be higher than the one usually adopted when dealing with other kind of materials. It is also to be noted that, unlike linearly behaving structures, the problem to be tackled is often an "inverse" one, in the sense that several numerical models have to be made before finding the one capable to closely describe the real behavior. For existing monumental structures, built over the centuries along different erection stages and using different materials, the modeling can only

Figure J3: A Gaudi 's structural model made with sandbags but looking like a Finite Elements mnnerical model.

592

Computer Analysis and Design of Earthquake Resistant Structures

Computer Analysis and Design of Earthquake Resistant Structures

roughly describe the real situation, and the model can be made, of course, only after a very detailed survey of the real situation. Following the observations made in the preceding paragraphs, it is also necessary to single out such "hidden" elements (such as pillars, arches and so on) which could substantially increase the total resistance of the construction. Because of different contribution to the total resistance of the building due to different elements, it is also often necessary to "emphasize" the structural role of some elements with respect to others. As an example, in order to achieve results more adherent to reality, the resistance of non structural walls could be notably decreased if not even zeroed. The situation is even more complex if the dynamic analysis of masonry structures is considered. In this case, the usual linear analysis based on modal superimposition is not applicable, and the real behavior is far away from the numerical one. As it has been previously noted, due to the presence of cracks as well as the decrease of resistant section due to of tensile stresses, the real dynamic behavior can not be constant during the motion, and only a more accurate analysis could lead to more consistent results. Unfortunately, computer programs capable of performing this nonlinear analyses are nowadays only confined to scientific research purposes, and so the only applicable way in most usual cases is the "equivalent" static analysis, with all its described limitations.

7.3.3 Some remarks on the use of finite element models -In a first step, masonry could be modeled as a linear elastic, homogeneous and isotropic continuum. This approach is acceptable until the first tensile stresses (and, consequently, first cracks) appear in the structure. The elastic model (even if it represents a situation that the structure will possess for a very short period of its life) is however very useful for a "qualitative" comprehension of the mechanical behavior of the structural elements, because it can frequently lead to understanding the cause of the formation of first cracks as well as of other noticed structural problems. Figure 14 shows the FEM model and the results of analysis of a church with belltower under horizontal seismic forces (areas with max. values of compression are blue and brown and areas with probable damages because of tensile forces are pink and yellow) as reported in who have used the general purpose computer code PROSAP V.2.0 ALGOR SUPERSAP- F. Pannella and A. Vanitelli. Figure 15 shows the finite element model of a damaged arch. taken from C.Blasi who employed the program ANSYS [3] Elastic models can also represent the first step in a "structural identification" pattern, that is of all the set of procedures necessary to give a numerical model as close to the reality as possible. After having singled out all those zones where tensile stresses are present, other linear as well as nonlinear models can be developed. As an example, the crack can be taken into account even in a second linear elastic model, just breaking the material continuity of the masonry along the length of cracks. On the contrary, if a nonlinear model is developed, obtained results must be carefully interpreted, and constitutive laws must be correctly defined, so that is

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Computer Analysis and Design of Earthquake Resistant Structures

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Computer Analysis and Design of Earthquake Resistant Structures

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preferable to perform several different analyses. Figure 16 portrays a non-linear structural model exhibiting clearly the evolution of the cracks during an earthquake (Vignoli et al [40]). Figure 17 shows the deformation pattern of a damaged old bridge in Florence analysed on the basis of a non linera FEM model. Concerning with the modeling of the ground, different models for the mechanical characteristics of the base ground can lead to completely different results in the tensional pattern of the masonry. As a concluding remark, it is easily understandable how it is preferable to use the simplest possible models, in order to have a constant "control" on the obtained results and to "check" the "reliability" and "reality" of the model with respect to the observed situations.

7.3.4 Some useful finite elements in masonry structural modeling - In a three-dimensional (3D) modeling of masonry structures, the most widely used finite element is represented by the 3D eight-nodes element, derived from the spatial deformation of a parallelepiped element, that is from any spatial disposition of eight points in space. This element, normally referred to as "brick" element, is normally present in almost all F.E.M. codes and it represents the base for more complex elements which will be briefly described in the following. Its mechanical characteristics vary depending upon the specific code used. Apart from the simpler isotropic elastic element, some codes provide for nonlinear constitutive mechanical laws (i.e. elastic/plastic behavior-) as well as for other particular aspects (such as creep). The most widely used nonlinear elements are essentially the following two: • Smeared cracks element. This element possesses eight nodes and only traslational degrees of freedom (D.O.F.), so that 24 D.O.F. are taken into account. With respect to the linear element, the presence of cracks is reproduced in a "smeared" way, and it is possible to simulate both cracks due to tensile stresses ("cracking") and cracks due to excessive compressive stresses ("crushing"). The nonlinear analysis is performed by step-by-step algorythms; when cracking (or crushing) is reached, it is smeared on the whole surface of the element, and stresses are re-distributed among other non- cracked elements. It is often possible to define a multi-layered element, with different mechanical characteristics for each layer as the one used for the analysis shows in Figure 16. • Three-dimensional "gap" element. This element is able to reproduce the unilateral contact within adjacent surfaces. By this element, the two surfaces transmit only compressive and shear stresses, but not tensile stresses. Once the friction limit is reached, relative sliding is also allowed. If this element is used together with other elements, it is also possible to give a certain tensile resistance to the element, and no tensile stresses will be allowed once the limit resistance is reached. Each "gap" element possesses two nodes with three D.0.F. each, and the link between the surfaces is assumed as orthogonal to the direction given by the two nodes. The element is thus characterized by the normal stiffiless Kn, the tangential stiffiiess K 1 and the "gap" variable which

594

Computer Analysis and Design of Earthquake Resistant Structures

Computer Analysis and Design of Earthquake Resistant Structures

595

preferable to perform several different analyses. Figure 16 portrays a non-linear structural model exhibiting clearly the evolution of the cracks during an earthquake (Vignoli et al [40]). Figure 17 shows the deformation pattern of a damaged old bridge ih Florence analysed on the basis of a non linera FEM model. Concerning with the modeling of the ground, different models for the mechanical characteristics of the base ground can lead to completely different results in the tensional pattern of the masonry. As a concluding remark, it is easily understandable how it is preferable to use the simplest possible models, in order to have a constant "control" on the obtained results and to "check" the "reliability" and "reality" of the model with respect to the observed situations.

Figure 15: Finite-elements model of a damaged arch

7.3.4 Some useful finite elements in masonry structural modeling - In a three-dimensional (3D) modeling of masonry structures, the most widely used finite element is represented by the 3D eight-nodes element, derived from the spatial deformation of a parallelepiped element, that is from any spatial disposition of eight points in space. This element, normally referred to as "brick" element, is normally present in almost all F.E.M. codes and it represents the base for more complex elements which will be briefly described in the following . Its mechanical characteristics vary depending upon the specific code used. Apart from the simpler isotropic elastic element, some codes provide for nonlinear constitutive mechanical laws (i.e. elastic/plastic behavior1 as well as for other particular aspects (such as creep). The most widely used nonlinear elements are essentially the following two: • Smeared cracks element. This element possesses eight nodes and only traslational degrees of freedom (D.O.F.), so that 24 D.O.F. are taken into account. With respect to the linear element, the presence of cracks is reproduced in a "smeared" way, and it is possible to simulate both cracks due to tensile stresses ("cracking") and cracks due to excessive compressive stresses ("crushing"). The nonlinear analysis is performed by step-by-step algorythms; when cracking (or crushing) is reached, it is smeared on the whole surface of the element, and stresses are re-distributed among other non- cracked elements. It is often possible to define a multi-layered element, with different mechanical characteristics for each layer as the one used for the analysis shows in Figure 16. • Three-dimensional "gap" element. This element is able to reproduce the unilateral contact within adjacent surfaces. By this element, the two surfaces transmit only compressive and shear stresses, but not tensile stresses. Once the friction limit is reached, relative sliding is also allowed. If this element is used together with other elements, it is also possible to give a certain tensile resistance to the element, and no tensile stresses will be allowed once the limit resistance is reached. Each "gap" element possesses two nodes with three D.0 .F. each, and the link between the surfaces is assumed as orthogonal to the direction given by the two nodes. The element is thus characterized by the normal stiffitess K the tangential stiffitess K1 and the "gap" variable which 0,

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Computer Analysis and Design of Earthquake Resistant Structures

Computer Analysis and Design of Earthquake Resistant Structures

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Computer Analysis and Design of Earthquake Resistant Structures

Computer Analysis and Design of Earthquake Resistant Structures

represents either the initial interference between the two surfaces (if positive) or the movement between the two surfaces (if negative). Kt represents the shear stiffuess before sliding and it is dependent upon the normal force Fn. Sliding can occur when tangential force Ft is higher than µ·Fn, with µ being the friction coefficient. The element is a nonlinear one (from a geometrical point of view instead of a mechanical one) and convergence to solution is obviously still reached after a step-by-step procedure with displacement control. As output, the "gap" variable will indicate the final state of the two surfaces: if it is negative, the two surfaces are still in contact, while if positive, a movement in the direction normal to the two surfaces has occurred. The "gap" element is very useful in the study of masonry structures, because it partially allows the modeling of possible cracks formation, and, if crack occurs, the opening of the "gap" will simulate it as shown in Figure 17.

with the presence of seismic actions) some tensile stresses can occur, and so the resistant section can be reduced. The use of "gap" elements in horizontal planes allow one to reproduce the real situation and to simulate the possible presence of cracks.

7.3.5 Problems related to "mesh" definition - Besides the aforementioned aspects, the modeling of masonry structures requires some remarks about the "geometrical" discretization of the structure itself. The first step in any F.E.M. modeling has to do with the choice of the structural nodes as well as of the elements to be used. On a general basis, because of the importance of this step in the subsequent analysis, some criteria should be used: • the geometrical discretization ("mesh" in the following) should be dense in those zones where either high gradients of tension are expectable or high tensile stresses will occur, thus leading to a crack formation mechanism; • the elements should be as much regular as it is possible, so that is preferable not to use elements with sensibly different dimensions. Apart from previous aspects, in a masonry structure the "optimal" mesh is not easy to be determined, unlike what happens in framed structures. Some computer codes allow the automatic mesh generation, and so it is possible, by using different meshes, to single out the optimal one from certain points of view. It is however to be noted that, in cases not so simple as the one of a single element, the automatically generated mesh is often not satisfactory because of the inherent complexity of most of the masonry structures to be investigated. As for other aspects, some "general" rules are not identifiable, and it is preferable to investigate, for each analyzed case, the accuracy of the solution by varying some of the involved parameters. In the following paragraphs, some examples will be reported and a critical analysis of the obtained results will be performed, in order to give some criteria on the "best" modeling procedure to be adopted.

7.3.6 Masonry pillars - Pillars are very simple structural elements, where the main stresses are only due to axial loads, so that no particular problems arise in their numerical modeling. The more obvious choice for the structural mesh is the superimposition of blocks, both in the vertical and in the two transversal dimensions. When the axial force is eccentric with respect to the center of gravity of the section or when horizontal loads are applied on the element (as it is the case

599

7.3.7 Masonry walls - The F.E.M. modeling of walls substantially presents the same characteristics as those reported for the modeling of pillars. The same criteria can be adopted, especially when horizontal loads are present. As it was pointed out before, the numerical modeling in a completely elastic range can lead to results far away from the real behavior, so that a nonlinear modeling must be adopted using bricks as well as "gap" elements. When the horizontal forces acting on a wall are known, the F .E.M. discretization of the element can lead to good results which are also easy to be interpreted. However, this is only the case when horizontal diaphragms can be assumed to be infinitely weak in their plane. On the contrary, the amount of horizontal force acting on a single wall can only be determined by a complete modeling of the whole structure, and thus the results are not so easy to understand as before. [IO, 11, 16, 20, 21, 22, 23, 36] 7.3.8 Complex structural masonry systems - The observations reported for simple structural elements can lead to some useful remarks when dealing with the numerical modeling of complex structural systems. The two most importants aspects of the problem are: • Choice of the mechanical model. The reported examples have shown how far, the "resistant" geometry of masonry structures can be from the "apparent" one. It is then necessary to develop a model not only capable of describing the geometry of the building but also capable of describing the structural "hierarchy" of resistant elements. In order to obtain a simplification on the modeling, the some hypotheses adopted when studying reinforced concrete or steel structures can be used: some elements (such as non resistant walls) can be thought of only as dead weight, neglecting their contribution to the total resistance of the building, even if this assumption is not realistic especially when considering a serviceability condition of the structure. Once the main structural elements (pillars, vaults, arches, structural walls) have been singled out, the other elements can either not be modeled or taken into account with very low contribution to the overall resistance(e.g. by reducing their modulus of elasticity). As an example, floors as well as other horizontal elements often possess such a low tensile resistance that, especially under strong earthquakes, it can not be assumed that they can properly transfer the horizontal action to all the vertical structural elements. In this case, it is preferable not to model the horizontal diaphragms (unless they have been reinforced by a concrete slab) and to distribute horizontal actions to each single element according to the part of vertical loads acting on them. When dealing with the choice of the mechanical model, the "kind" of model is to be chosen, that is whether or not to use a nonlinear modeling of the structure. The examples reported before could thus be used as a "guide" in this sense.

600

Computer Analysis and Design of Earthquake Resistant Structures

• Choice of models regarding only sub-parts of the buildings. It is recommended to perform both the analysis of single elements (under horizontal actions proportional to the vertical ones) and the analysis of sub-parts of the whole structure, in order to improve the knowledge of the real behavior only in those zones believed to be more "critical" or more "complex". In this way, the whole model can be simpler, according to the fact that more complex parts have been studied separately. Slender monumental buildings: the problem of "rocking" - When dealing with the seismic analysis of slender monumental buildings, the approach is very close to the one usually adopted when considering the dynamic response of structures formed by the superimposition of independent rigid blocks.[2, 5, 8, 9, 15, 19, 27, 28, 29]. As a matter of fact, in this kind of buildings (such as bell towers) the collapse tends to be similar to the one occurring in a set of independent elements partially linked between them, where collapse can occur by excessive relative rotations ("rocking") as well as by excessive relative horizontal movements ("sliding"), without reaching the ultimate resistance of the constituting material. In the following, some developed techniques will be briefly illustrated, refering to the most specific field in which this approach has been utilized, that is the dynamic response under seismic actions of ancient columns, even in the presence of a superimposed "rigid beam", like the one shown in Figure 18 depicting the temple of Saturn in Rome. The study of the oscillations of a rigid block has been initially performed by Housner [27], where the contact between each block was supposed to be punctual, i.e. in a point only. In this way, the block, during its motion, is alternatively supported on one edge or on the other, this leading to a nonlinear equation of motion dependent upon the instantaneous position of the block itself The problem can be partially modified and simplified by introducing the hypothesis of an elastic continuous restraint between two adjacent blocks, i.e. the hypothesis of a fictitious deformable Winkler monolateral cushion between the blocks. In this way, a double effect is obtained: the influence of effective restraint between blocks can be analyzed, and some evaluations upon the effectiveness of consolidation works (if present) can be made. This extension of the Housner's model is due to Blasi and Spinelli [8, 9], where several different conditions have been taken into account. It is to be noted that the approach reported in [8, 9] (and in several other works) presents the advantages of allowing the study under several different situations, including the possible presence of viscous damping, of prestressing in the columns and of sliding between adjacent blocks. Moreover, when dealing with a columnade, it is also possible to consider different behaviors for each constituting columns. A second extension of the model could be made by considering not only a deterministic approach (the response to seismic actions is determined by the integration of the equations of motion in time-domain [31]) but considering the

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Computer Analysis and Design of Earthquake Resistant Structures

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Computer Analysis and Design of Earthquake Resistant Structures

inherent stochasticity of the problem, and determining the probability of collapse of the system (see [19] for further details). In this case the problem is obviously still a nonlinear one, and it can be tackled only after a linearization of the problem itself In the referenced work, the study is performed by an equivalent linearization technique. In this kind of problem, however, it is not possible to perform either "equivalent" static analyses or linear analysis, because of the complexity of the problem itself From this point of view, it is a "special" case with respect to the other illustrated ones, and the adopted techniques are also quite different.

8. Conclusions In short, procedures to be used in the construction of a numerical model of a masonry building can be summarized as follows.

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Computer Analysis and Design of Earthquake Resistant Structures

603

_ comparison between obtained results and observed situation.

• Definition and use of a nonlinear model - when using "gap" elements: verification of their correct positioning in order to avoid both incorrect modeling of the structure and the presence of some weak zones; - when using elements with nonlinear mechanical behavior: verification of the convergence of the procedure as well as of the reliability of the qbtained results; - evaluation, by other simplified approaches, of the global reliability of the obtained results; - definition of more nonlinear models, using different hypotheses and discretization of the structure; - comparison between obtained results and observed situation.

References

• Preliminary surveys - knowledge of the geometry; - knowledge of erection stages; - knowledge of mechanical characteristics of all kinds of masonry used and evaluation of the effectiveness of linking between different elements; - singling out of main structural elements (pillars, structural arches, vaults, ties) and of their mechanical characteristics; - evaluation of effectiveness of horizontal diaphragms and of the connection with resistant vertical structures; - survey of cracks, elements in a low degree of conservation, sinking and other ground movement.

• Construction and use of a preliminary elastic model - subdivision of the whole structure into independent structural subsystems, perfonning a partial analysis instead of a "global" analysis (this is necessary when elements are not well connected to others); - modeling of the geometry, including cracks, sinking and whatever can be significant from a structural point of view; - modeling of mechanical characteristics, including a strong differentiation between structural and non-structural elements; - modeling of the ground and the differential sinking between different parts of the building; - modeling of the horizontal diaphragms (when effective) or substitution of them with horizontal actions pertaining to each single element; - modeling of drifting elements (such as vaults or arches) or substitution of them with the actions that they transmit to the other structural elements; - evaluation of "equivalent" static forces to be applied in order to simulate the seismic action; - considerations about the purely "qualitative" description obtained, singling out only those zones where tensile stresses can occur during seismic events; - definition of more elastic models, in order to evaluate the influence of the adopted hypotheses and of the adopted parameters in the obtained results, and to delimit a "range" within the "real" structural response.

1. ADINA, User's manual, ADINA Engineering AB, Vasteras, Sweden, 1982. 2. Allen, R.H. & Duan, X., Effect of linearizing on rocking-block toppling, Journal of Structural Engineering, ASCE, 1995, 121,1146-1149. 3. ANSYS, Swanson Analysis System Inc., Houston PA, USA, 1986. 4. Anthoine, A., Magonette, G. & Magenes, G., Shear - compression testing and analysis of brick masonry walls, pp.1657-1662, Proceedings 10th European Conference on Earthquake Engineering, Vienna, A.A. Balkema, Rotterdam, 1995. 5. Augusti, G. & Sinopoli, A., Modelling the dynamics of large block structures, Meccanica, 1992, 27, 195-211. 6. Bathe, K.J., Finite elements procedures in engineering analysis, PrenticeHall, Englewood Cliffs, New Jersey, USA, 1982. 7. BEASY, Boundary elements analysis system, User's manual, Computational Mechanical Publications, Southampton U .K., 1989. 8. Blasi, C. & Spinelli, P., Dynamic analysis of stone block systems, in Numeta 85 - Numerical Methods in Engineering: Theory and Applications, J. Middleton and G.N.Pande, Editors, pp. 645-652, A.A. Balkema, Rotterdam, 1986. 9. Blasi, C. & Spinelli, P., Dynamic analysis of ancient stone monuments for restoration design, Proceedings of 7th ASCE/EMD Specialty Conference, Blacksburg, Virginia, USA, May 1988 10. Beskos, D.E., Use of finite and boundary elements for the analysis of monuments and special buildings I (in Greek), Bulletin of the Greek Society of Civil Engineers, 1994, No 216, 31-43. 11. Beskos, D.E., Use of finite and boundary elements for the ..analysis of monuments and special buildings II (in Greek), Bulletin of the Greek Society of Civil Engineers, 1994, No 217, 15-32. 12. Beskos, D.E., (ed)., Boundary Element Methods in Structural Analysis, ASCE, New York, 1989.

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Computer Analysis and Design of Earthquake Resistant Structures

Computer Analysis and Design of Earthquake Resistant Structures

13. Brebbia, C.A., (ed)., Structural Repair and Maintenance of Historical Buildings, Computational Mechanics Publications, Southampton U.K., 1989. 14. Brebbia, C.A., Dominguez, J. & Escrig, F., (eds)., Structural Repair and Maintenance of Historical Buildings, Vol. I : General Studies, Materials and Analysis, Vol. II : Dynamics, Stabilization and Restoration, Computational Mechanics Publications, Southampton U.K., 1991. 15. Cai, G.Q., Yu, J.S. & Lin, Y.K., Toppling of rigid block under evolutionary random base excitation, Journal of Engineering Mechanics, ASCE, 1995, 121, 924-929. 16. Calderani, B. & Marone, P., Modelli per la verifica statica degli edifici in muratura in zona sismica, Ingegneria Sismica, 1987, 4, 19-27. 17. Castley, A. C. & Abrams, D .P ., Response of building systems with rocking piers and flexible diaphragms, Proceedings of the ASCE Structures Congress IX, Chicago, April 1996. 18. Del Piero, G., Le Costruzioni in Muratura, International Centre for Mechanical Sciences, Collana di Ingegneria Strutturale No2 - CISM, Udine, 1984. 19. Facchini, L., Gusella, V. & Spinelli, P., Block random rocking and seismic vulnerability estimation, Structural Engineering, 1994, 16, 412-424. 20. Ferrari, A., Galano, L. & Vignoli, A., Resistenza a taglio di pannelli murari con aperture, influenza degli interventi di cerchiatura, pp. 13-20, Bollettino degli Ingegneri, Nol0-11, Firenze, 1994. 21. Gambarotta, L., & Lugomarino, S., Static and dynamic analysis of large scale brick masonry walls, pp. 882-893, Proceedings of the 7th North American Masonry Conference. Notre Dame University, Indiana, USA, June 1996. 22. Gambarotta, L. & Lagomarino, S., Damage in brick masonry shear walls, pp. 463-472, Proceedings Europe - U.S. Workshop on Fracture and Damage in Brittle Structures: Experiment, Modeling and Computer Analysis, Prague, Czech Republic, E & FN Spoon, London, U.K., 1994. 23. Giannattasio, G. & Gilberti, C., Un procedimento semplificato per la verifica a rottura di edifici in muratura in zona sismica, Ingegneria Sismica, 1989, 6, 34-39. 24. Guidobodoni, E., Catalogue of ancient earthquakes in the Mediterranean area up to the 10th century, Istituto Nazionale di Geofisica - Roma, 1994. 25. Guidobodoni, E. & Ferrari, G., Historical cities and earthquakes: Florence during the last nine centuries and evaluation of seismic hazard, Annali di Geofisica, Vol. 38, No 5~6, S.G.A. Bologna, Italy, 1995. 26. Guidobodoni, E., Riva, P., Petrini, V., Madini, A., Moretti, & Lombardini, A.: A structural analysis in seismic archaeology: the walls of Noto and the 1693 earthquake, Annali di Geofisica, Vol. 38, No 11-12, S.G.A. Bologna, Italy, 1995.

27. Housner, W., The behaviour of inverted pendulum structures during earthquakes, Bulletin of the Seismological Society of America, 1963, 53, 403-417. 28. Koh, AS. & Mustafa, G., Free rocking of cylindrical structures, Journal of Engineering Mechanics, ASCE, 1990, 116, 35-54. 29. Manos, G.C. & Demoshenous, M., Comparative study of the global dynamics of solid and sliced rigid bodies, in Proceedings 10th European Conference on earthquake engineering, Vienna, G .. Duma, Ed., A.A. Balkema, Rotterdam, 1995, 1087-1092. 30. Mechanics and Structural Restoration of Masonry Structures (1992). Proceedings of International Conference, June 1-4, 1992, National Technical University of Athens, Athens, Greece, 1992. 31. Newmark, N.M., A method of computation for structural dynamics, Journal of the Engineering Mechanics Division, ASCE, 1959, 85, 67-94. 32. Page, AW., Finite element model for masonry, Journal of Strustural Division, ASCE, 1978, 104, 1267-1285. 33. Rondelet, G., Trattato Teorico e Pratico dell'Arte di Edificare, Trad. Italiana 1832, Ed. Caranenti, Mantova, Italy, 1832. 34. Structural Conservation of Stone Masonry: Diagnosis, Repair and Strengthening, Proceedings of International Conference, Greek Ministry of Culture, Athens, Greece, 1989. 35. Structural Preservation of the Architectural Heritage, Proceedings of IABSE Symposium, Rome, Italy, September 15-17, 1993, International Association for Bridge and Structural Engineering (IABSE), Zurich, Switzerland, 1993. 36. Ting-Bin, M., Guo-Bin, S., Quin-Lin, W. & Wen-Zong, Y., The behaviour and strenght of brick and reinforced concrete composite walls beams with door opening, pp. 384-394, Proceedings of the 8th International BrickBlock Masonry Conference, Dublin, Ireland, September 19-21, 1988. 37. Vignoli, A., Chiostrini, S. & Galano, L., Modellazione agli elementi finiti de! comportamento sperimentale a rottura di murature di pietrame, Atti de! VII Convegno Italiano di Meccanica Computazionale, Universita degli Studi di Trieste, pp. 134-139, Giugno 1993. 38. Vignoli, A. & Chiostrini, S., In-situ determination of the strength properties of masonry walls by destructive shear and compression tests, Masonry International, Journal ofthe British Masonry Society, 1994, 7, 87-96. 39. Vignoli, A. & Chiostrini, S., Modellazione numerica di prove distruttive di pareti in muratura sottoposte ad azioni orizzontali, Atti de! 7° Convegno Nazionale "L'Ingegneria Sismica in Italia", vol. 1, pp. 301-310, Siena, Settembre 1995. 40. Vignoli, A., Galano, L. & Consorti, C., Miglioramento di edifici rurali in zona sismica; esempio su una colonia tipica fiorentina, Bollettino Ingegneri di Firenze, pp. 3-9, Novembre 1995. 41. Wilson, E.L. & Habibullah, A., SAP90, User's manual, Computers and Structures Inc., Berkeley, California, USA, 1990.

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Computer Analysis and Design of Earthquake Resistant Structures

42. Zienkiewicz, O.C. (1977). The Finite Element Method, 3'd Edition, McGraw-Hill Book Co., London, UK, 1977.

Chapter 12 Bridges K. Kawashima Department of Civil Engineering, Tokyo lnstitutute of Technology, Jvfeguro-ku, Tokyo, 152,Japan

Abstract This chapter presents seismic response analysis procedures for bridges, together with linear and nonlinear analysis of a variety of bridges subjected to severe earthquake ground motions. Because of the unique nature of the structural response of bridges, an appropriate analytical method should be employed. In analyzing the response of simple/multi-span bridges and c..ble-supported bridges, multiple excitation at their supports and the discontinuous behavior of expansion joints are considered. The seismic response of isolated bridges and cable-stayed bridges is discussed, as is the application of variable dampers. ''I

1 Introduction Bridges are unique in their structural response. First, they are longitudinally lengthy, and consist of many structural components which contribute to the overall resistance capability of the system. Decks are often skewed and curved, and intermediate expansion joints divide a bridge system into several structural segments with different natural periods. Second, there are various structural types with complex geometries and dynamic response characteristics. Suspension bridges and cable stayed bridges generally display a very complex structural response with long natural periods, often exceeding 10 seconds. Many modes with closely spaced natural periods contribute to the complexity of the structural response. Third, bridges are generally constructed at soft soil sites such as rivers and bay areas. Because ground motions are amplified at these sites, greater attention should be paid to seismic design for large ground motion. Failure of foundations associated with _the instability of surrounding ground is a common occurrence. Fourth, the degree of statical indeterminacy is smaller in bridges than in buildings, and therefore ductility of piers/columns needs to -be carefully examined to prevent failure during strong earthquakes. Various analytical methods have been developed to predict the seismic response of bridges. This has enabled to construct bridges which were difficult to design when computer analysis was not available. For example, precise linear and nonlinear seismic response analysis is essential for long span bridges, bridges with complex geometric features, cable supported bridges, and tall bridges.

608

Computer Analysis and Design of Earthquake Resistant Structures

Computer Analysis and Design of Earthquake Resistant Structures

Computers have also greatly assisted in the analys!s of bridges w_hich have failed during past earthquakes, and have greatly contnbuted to the improvement of seismic design methods. . . This chapter describes the seismic response analytical procedures and gives numerical examples for various types of bridges. In Sections 2 and 3, the standard analytical modeling of bridges, and linear and nonlinear seismic response analysis procedures are presented, respectively. Sections 4:-8 show how computers are used in dynamic response analyse~ so as to clarif~ the complex structural behavior of bridges. A general conclusion on the effectiveness of the seismic response analysis of bridges is presented in Section 9.

609

large cyclic inelastic deformations. Therefore, nonlinear finite elements for the mathematical model, which have realistic nonlinear hysteretic force - deformation characteristics, must be chosen. The stiffness of these elements are time dependent and are functions of element deformations and deformation histories. Usually, they are linearized for analysis in a piecewise fashion using tangent stiffnesses at discrete times. Thus, the total stiffness matrix for the entire structure may be written as N

(2)

Kr =Ikn i-1

2 Analytical Modeling of Bridges

where, K 1 = total stiffness matrix at time t, and kn= stiffness matrix for element i at time t Nonlinearity arising from large geometry changes is not generally included as it is negligible.

2.1 Structural System

a) Decks

Generally bridges consist of girders, piers/columns, abutments, foundations, bearing supports, and expansion joints. In addition, special types of bridges such as arch bridges, suspensions bridges and cable stayed bridges have arch members, towers, anchorages, cables, hangers and links. Energy dissipating devices and active mass devices are used in passive and active control. Because the structural characteristics of bridges depend on their types, emphasis is placed here on showing analytical modeling for girder type bridges, based on Tseng and Penzien [34]. The structural system of this type of bridge consists of a multiple-span continuous deck supported by bearings on or rigidly connected to reinforced concrete piers/ columns and abutments. The deck may be straight, curved or skewed, and is supported at discrete locations along its longitudinal ax!s. Intermediate expansion joints divide the deck into several segments. The entire structural system generally exhibits the characteristics of a continuous space frame. Its dynamic response to earthquake excitations is of a lower mode type; hence, a mathematical model of discrete form can be used to approximate the continuous system This form of modeling leads to a system with a finite number of degrees of freedom Following the standard finite element procedure, these degrees of freedom are chosen as the nodal displacements of the discrete finite element model. For a three dimensional model, each nodal point usually has 6 degrees of freedom, i.e., 3 translation components and 3 rotation components. Internal constraints may reduce this number at some nodal points. For dynamic response analysis, the stiffness, mass and damping properties of each finite element must be realistically defined.

2.2 Stiffness Idealization The finite element idealization of a complete bridge system results in a stiffness matrix which is an assemblage of the generalized stiffness matrices for individual elements as (1) i-1

where, K= total stiffness matrix for the entire bridge system, k;= stiffness matrix for element i, and N= total number of elements in the bridge system. For small amplitude response, the bridge system may be modeled by a set of linear elements; however, when subjected to high amplitude response as occurs during severe earthquakes, certain critical regions of the system may undergo

When determining internal stress distributions in the constituent flanges and webs of box girders under localized loadings, elaborate methods of analysis, such as finite element analysis, must be used. However, when the external loadings are relatively uniformly distributed, and when only resultant forces on transverse cross-section, i.e., 3 components of force and 3 components of moment, are required, a simple beam analysis as shown in Fig. 1 is usually sufficient to yield accurate results. Since a typical deck is extremely stiff and strong in comparison with its supporting columns and abutments, the high amplitude bridge response produced during severe ground shaking will be caused primarily by deformations in the columns, abutments and expansion joints. The deck will remain elastic and, therefore, can be modeled by linear elastic elements. Nonlinear elements must, however, be used for columns, abutments and expansion joints.

z (a) Typical bridge superstructure

(b) Idealized model

Fig. 1 Analytical model of bridge superstructures

b) Columns The structural behavior of columns can be adequately modeled using simple beam elements. Because of large amplitude response, coupled inelastic deformations may occur in these membe~s. For example, Fig. 2 shows lateral force vs. lateral displacement hysteresis loops which were obtained by cyclic loading tests of circular reinforced concrete columns (Priestley, Seible and Chai [29]). Fig. 2 (a) shows the hysteretic behavior of an as-built column while Fig. 2 (b) shows the hysteretic behavior of a column strengthened with a steel cylinder jacket Confinement of concrete by hoops is important to increase the ductility and energy dissipating capacity of reinforced concrete columns (Park [27], Priestley and Park

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damage. In idealizing abutments by beam elements, it is usual to assume equivalent linear springs in longitudinal and transverse directions to simulate the restraints on the superstructure provided by any abutment It is important to select the spring stiffness accurately so as to allow correct distribution of seismic loads throughout the structural systems. For this purpose, the spring stiffness must reflect the dynamic behavior of the soil behind the abutment, the structural components of the abutment, and the interaction between the soil and the structural components of the abutment Substantial nonlinear behavior is expected in the abutment because some of the elements constituting the abutment may be subjected to significant yielding (Maroney and Chai [19]).

d) Foundations Various idealizations have been developed for foundations. Complex idealization uses nonlinear finite element models. A simpler and more appropriate model consists of 3 translational and 3 rotational soil springs, as shown in Fig. 4, to connect the base of each column and abutment to a rigid foundation where the seismic excitation is fully prescribed. For linear analysis, the stiffness of these soil springs may be evaluated using linear elastic half-space theory (Penzien [28]). For large amplitude response, the foundation soils may undergo inelastic deformation of the hysteretic type. In this case, the six soil springs should be nonlinear hysteretic springs. Their stiffness can only be established through extensive experimental studies on the dynamic properties of foundation soils (Lam [17]).

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[30]). Fig. 3 shows the lateral force vs. displacement hysteretic loops of steel columns (MacRae and Kawashima [18]). Degradation of stiffness after the maximum load is achieved is generally larger in steel bridge columns than in well confined reinforced concrete columns. Because this tends to cause large residual displacement when subjected to large ground motion, such a feature must be carefully idealized in analysis. In realistically modeling the hysteretic behavior of reinforced concrete columns, stiffness degradation, strength loss and pinching are the key issues (Williams and Sexsmith [38]). Therefore, nonlinear beam elements which realistically characterize the inelastic hysteretic behavior of columns must be used.

c) Abutments The force-displacement relationship of abutments is a highly complex nonlinear problem. Failures are likely to be of the shear type causing excessive

Analytical model of bridge columns/ piers and foundations

2.3 Mass Idealization

The continuous mass of the bridge structural system is modeled in discrete form by lumping element masses at their end nodal points. Since ~ertia forces are associated with each of the six degrees of freedom at a nodal pomt, each lumped mass should be assigned an appropriate moment of i!1ertia a~ut its ow_n coordinate axes. It should also be noted that when conductmg nonlinear dynarmc analysis, the instantaneous stiffness matrix can become singular, ~which case it is required that a mass moment of inertia be assigned to each rotabonal degree of freedom. Following this procedure, a diagonal mass matrix mi is-established for each element i (i = 1,2, ·····,N). The diagonal mass matrix for the complete bridge system can then be assembled and expressed as N

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(3)

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Computer Analysis and Design of Earthquake Resistant Structures

Computer Analysis and Design of Earthquake Resistant Structures

In determining the overall dynamic response of bridges, this lumped mass method has been found to be quite adequate for analytical purposes.

2.4 Damping Idealization Velocity dependent damping in a bridge structural system is represented by a generalized damping matrix associated with the finite degree of freedom permitted in the analytical model. This matrix can be derived by consistent procedure similar to those used in deriving the stiffness matrix, provided the internal damping mechanism within each element is specified. The structural damping matrix for the complete bridge system would then be evaluated as N

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where C; is the damping matrix for the i-th element. In practice, however, it is difficult to establish the basic characteristics of damping in individual elements. It is therefore often assumed that the damping force consists of one set which is proportional to the velocities of each mass point and a set which is proportional to the rate of deformation. Thus, the structural damping matrix becomes C = aM +{JK (5) where a and f3 are the scalar proportionality constants. They are determined after assigning damping ratios to the first two natural modes of vibration (Clough and Penzien [7]). In the mode superposition method, modal damping ratios hk for each mode are required. Because energy dissipation occurs by specific mechanism, such as friction, the hysteretic behavior of structural components and radiation of energy from structures to soils, it may be possible to evaluate damping ratio at each element. Then, the modal damping ratios are approximated as (JRA [9]) n

bridges in which the hysteretic-type energy dissipating mechanism is predominant. It has been found that Eq. (6) provides accurate estimation for the modal damping ratios of seismic isolated bridges based on a model test (Kawashima, Hasegawa and Nagashima [13]). For nonlinear dynamic response analysis, the viscous damping properties of bridges are more difficult to access. As with the elastic case, it is often assumed as (8) C 1 = aM +{3~ In this relation, if degradation of the stiffness develops, the damping matrix decreases. However damping can increase in such a case due to hysteretic energy dissipation in the bridge. From such a point of view, Eq.(8) is only an assumption to make the analytical treatment easy. A more accurate idealization of damping considering the energy dissipating mechanism is required.

3 Analytical Procedures for Seismic Response of Bridges 3.1

Equations of Motion

The equation of motion for a n degree of freedom bridge system expressing dynamic equilibrium at time t can be expressed (Tseng and Penzien [34]) as Miit + Ctut + Ktut = R(t) (9) where, M, C 1 and K 1 are the mass, damping and stiffness matrices, respectively, and where R(t) is the applied dynamic load vectors. Vectors ii 1 , u1 and u1 are the absolute acceleration, velocity and displacement vectors, respectively. If the bridge system is subjected to prescribed support excitations, a complete set of nodal displacements u~ should be considered which include, in addition to the n free nodal point displacements, the nb prescribed non-zero support displacements. Thus, the complete nodal displacement vector can be expressed as

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0.5

1.0

15

TIME

Fig. 17

I

2.0

25

3.0

35

4.C

(SECOND!

Experimental and analytical response of model bridge subjected to a low intensity excitation

630

Computer Analysis and Design of Earthquake Resistant Structures

expansion joints of the type shown in Fig. 14 placed in symmetrical position. The fundamental natural frequency of the model was measured as 5 Hz, 6.6 Hz and 9-11 Hz in longitudinal, transverse and vertical directions, respectively. Fig. 17 shows the response of the model when it was subjected to a low intensity artificially generated ground motion with a peak accelerati~n of ~- 11 g in the transverse horizontal direction. Analytical responses which will be 0 75

r.

'I

~ w

050

~~

025

PREDICTED MEASURED

'\ \

::!!

....1-

!Zi 0

-0.25

lAl RESPONSE DISPLACEMENT AT CENTER PREDICTED

~w~ -

0.25r

-

----

MEASURED

'- ~A...,_l\.w-ff'-l.1~-¥· f\~~-f\L\l-f\H-f\~Jl-wt\44"/u.4\p~.....=.=-'-""~=--=--=V ~~"WV "tT V4N

~~

o~

CL

~ -o. 25

lBl RESPCJ-ISE DISPLACEMENT AT EXPANSION JOINT NO. I

0.50

I-

i_

-PREDICTED ---- MEASURED

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0.25

~::!!

~~

"' i5

_, -0.25

T

Computer Analysis and Design of Earthquake Resistant Structures

631

described later are also presented for comparison. During the test, collisions of the girder did not occur since the relative response displacements at the expansion joints remained below the initial joint gap. The joint restrainer tie bars however resisted joint separation. Since the tie bars resisted only joint separation, the displacement response was small in the outward or positive direction but large for the inward or negative direction. Fig. 18 shows the response of the model when it was subjected to high intensity horizontal and vertical excitations with peak accelerations of 0.47 g and 0.27 g, respectively. Analytical responses which will be described later are also presented for comparison. The same horizontal acceleration used for the low intensity excitation was adopted for this test by increasing the intensity. The vertical excitation was prescribed by artificially generated accelerograms with peak acceleration approximately one half of the pea.le horizontal excitation. Multiple collisions of the girders together with yielding of the joint restrainer bars took place at the expansion joints. As with the low intensity excitation, the responses of displacement, especially at the expansion joints, were unsymmetrical. However, they were of a type opposite to that developed in the low intensity excitation test, i.e., the motions were large in the outward direction and small in the inward direction. This type of response was caused by closure of the expansion joints during inward motion causing stiff arch action to take place between the abutments, while motion in the outward direction was resisted mainly by the more flexible tie bars.

5.2 Analytical Model of Expansion Joints Because the characteristics of expansion joints have a major influence on the seismic response of curved bridges, they must be correctly modeled. A nonlinear analytical model as shown in Fig. 19 well expresses the nonlinear

!Cl RESPONSE DISPLACEMENT AT EXPANSION JOINT N0.2

400 w u cc_ 300 0 200 w 100 ~

... ,.

0 -

400 w

~-

~~ w

~

!;1 E

0.50 :::c

0.40 ~-==riE-BAR FORCE ~:;~ ~ ~ 100 ---- TIE BAR YIELDING O.IO ,::;:l L _ _ . i l , __ ____._..L.Jl___.'----------'O ~ 0 IE l PREDIC1£0 TE BAR FORCE AND YELOHl OF EXPANSION JOINT N0.2

300

200

LOCAL NODAL COORDINATE

~YSTEM

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2.000~ L500

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JOtff N0.1

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(Al EXPANSION JOINT MlDEL

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~

~

w

u

m

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u

TIME ( SECOND l

Fig. 18

Experimental and analytical response of model bridge subjected to a high intensity excitation

(Bl DEFINITION OF COOROINATES

Fig. 19

Analytical model of expansion joint

I

632

Computer Analysis and Design of Earthquake Resistant Structures

Computer Analysis and Design of Earthquake Resistant Structures

behavior of the expansion joints. This model includes relative translational and rotational degree of freedom, elastoplastic joint restrainer tie bars, a~ng in tension, impact and Coulomb type friction with slippage (Tseng and Penzien[35], Kawashima and Penzien[ IO]). Longitudinal collisions are defined as taking place at points A and B (Fig. 19) when the relative displacement between the two end diaphragms uAx and uBx close the joint gap !::.G with a non-zero velocity. At the instant collision takes place, the longitudinal impact springs with large stiffness k1 , which are attached to one end diaphragm leaving a small gap !::. G with the other end diaphragm, start to resist the motion. A collision is completed when rebound occurs and the relative displacement between the two diaphragms becomes equal to the joint gap !::. G. The contact force acting at points A and B can be written as

where uE Ax and uE Bx are the current slippage at points A and B, respectively, and s and usBx are the elastic deformations at points A and B, respectively, as UAx given by uE Ax = viz:' 11· Az kC and uE Bx = viz:' 11· Bz kC where v is a constant coefficient of Coulomb friction, and

PAI= k1(uAx +t::.GXuAx +!::.G); PBI

=

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Figure 8: Infinitely long, circular cylindrical tunnel under Figure 9: Normalized shell response to P waves vs. e: (a) radial displacement; (b) axial displacement; (c) hoop stress. pressure (P) and shear (SV) time harmonic waves.

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Figure 11: Normalized (a) radial displacement; (b) axial Figure 10: Normalized shell response to SV waves vs. e: (a) displacement; (c) hoop stress vs. e for P waves. * present radial displacement; (b) axial displacement; (c) hoop stress. method, -e- Ref 130,-+- Ref 95, -e-Ref99.

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Computer Analysis and Design of Earthquake Resistant Structures

Computer Analysis and Design of Earthquake Resistant Structures

ros[99] (whose accuracy has been verified), while they disagree with those of

ing a real-site analysis [165] can be used in convolution with the impulse

Liu et al [95], which seem to be in error.

response to produce the correct anticipated motions at the cavity's walls. The

814

815

FEM analysis employs special cylindrical shell quadrilateral elements and is Example 5 - 2D BEM Model of a 3D Buried Pipeline The previous example is now re-solved by a reduced frequency domain BEM model, whereby the response of an infinitely long tunnel with a uniform circular cross section which is enveloped by seismic waves of arbitrary incidence is effectively reduced to a two dimensional problem, as discussed in Stamos and Beskos [131]. This reduction is achieved through a coordinate transformation, coupled with an intergration of the full space fundamental solutions entering the boundary integral equation statement along the direction of the tunnel axis. The modelling of a representative two dimensional slice of the problem (as shown in Fig. 8) is accomplished through the use of quadratic isoparametric line elements. In particular, 36 and 20 such elements are used to model the horizontal free surface and the inner/outer surfaces of the liner, respectively. By using the same data as in Example 4, the dimensionless radial and axial displacement amplitudes U, and Ux, respectively, as well as the dimensionless hoop stress amplitude l:00 are all plotted versus polar angle 0 for the case of an incident P wave in Fig. 11. Concurrently plotted are the results of Stamos and Beskos [130] as well as those of Liu et al [95] and of Luco and de Barros [99], which also appear in Example 4. We note the close agreement of the present results with those of Refs. [130] and [99]. Example 6 - Three Dimensional BEM!FEM Model

used to recover the dynamic strain and stress states within the pipeline. Pipeline cavity response The BEM mesh used for modelling the pipeline cavity consists of a surface part with 28 elements and a cavity part with 22 elements resulting in a total of 360 d.o.f. Following validation studies, a finite segment of length L=6D is considered, with traction-free boundary conditions along the cavity wall and fixed ends. The location of the CP, where unit displacement impulses of magnitude 1 mm along the three principal directions are applied, is at a distance of

JS· D from

the centerline of the cavity. The

critical time step is D.t=0.5 msec and analyses are carried out for 40 time steps. The results obtained are in the form of compliance functions at the cavity wall. A typical plot is given in Fig. 12 for four nodal points on the perimeter of the centrar cross-section and for the vertical impulse case. Complete plots can be found in Ref.[166]. Since this type of response at all cavity nodes will be used as input to the liner, it is worthwhile to note the following: (a) The total duration of the response T=40D.t=20 msec is adequate for allowing a decay of the signals due to radiation-type damping. (b) Nodes on the shadow zone of the cavity (e.g., 38) are less affected when compared to nodes close to the CP and to the surface of the ground (e.g., 5,19,52). (c) The more pronounced pipeline cavity vibrations are due to the

As discussed in section 4, a BEM formulation for three-dimensional elastody-

longtudinal impulse (max. observed longtudinal cavity motion was 0.13 mm)

namics and a FEM formulation for cylindrical pipes are combined in order to

followed by horizontal impulse perpendicular to the cavity axis (max. ob-

solve the buried pipeline problem. The problem is broken down into the fol-

served cavity motion in the same direction was 0.09 mm) and by the vertical

lowing three basic steps (1=_ig. 4): (a) Free-field motions at the control point,

impulse (max. observed cavity motion in the vertical plane was 0.08 mm). In

(b) Pipeline cavity analysis using the BEM and including all SSI effects and (c)

general, the maxinum cavity displacements are about one order of magnitude

Liner structural analysis by the FEM and using the results of the second step

less than the value of the applied surface displacements.

as input. The problem is solved for unit impulses in the principle directions at

Pipeline liner response The cavity response due to unit impulses at the CP is

the control point (CP). This way, realistic motions prescribed at the CP follow-

now used as input to the steel pipeline. The FEM mesh employs 256 elements

816

Computer Analysis and Design of Earthquake Resistant Structures

Computer Analysis and Design of Earthquake Resistant Structures

817

resulting 1200 nodal d.o.f. The displacement input was converted into generalized nodal forces in global coordinates for the FEM using the stiffness matrix

--

z

u

S2

I< ......

--

zt;

S2 [;oj '< 2!

uQ

Ot'! ...l ••

'

~

::;

:>

were computed at the element level and at the Gaussian intergration points. A

- ..

sponse is plotted in Figs. 14(a) and (b) along the mid-section circumference at

N::;

r-

;:

[)

and a late time 30.:lt=15 msec when the response starts to decay. Complete

,._ ,.._ ..._,, =

~

J- ,..-- ,..-I

z:;;>

two time steps, namely at 15M=7.5 msec when the response is most intense

plots ecrn be found in Ref. [166]. The following comments can now be made based on all the results: (a) For a prescribed horizontal displacement at the CP and parallel to the longitudinal axis of the pipeline, the near side of the liner is in compression while the far side is in tension, indicating bending towards the CP. The strains in the circumferential direction alternate sign, indicating distortion of the circumference. The rate of the decay of the circumferential vibrations is always faster than that of the longitudinal vibrations, indicating reverberations (or echo effect) in the latter case due to the finite length of the pipeline. (b) For a prescribed horizontal displacement at the CP perpendicular to and towards the pipeline axis, the basic picture (which changes with time) is that the central cross section is exhibits high tension at the top, less so at the

.. •

bottom and small compressive stresses at the left and right sides, which indi-

:~

cates a flattening of the liner. The hoop strains also alternate sign indicating

• "I

distortion of the cross-section, but they are primarily compressive indicating

• 8

: 'ii' =~.,

.

:i

that the liner is being pushed in . (c) For a prescribed vertical (downward) displacement at the CP, the situation is similar to that oJ case (b ). Scaling to real site motions The Extreme Value method for peak ground displacements with 90% probability of not being exceeded in the next 25 years (return period of 224 years) for the real site in Almyros region in Central Greece [165] was used for scaling the unit impulse results obtained by both

l I 818

Computer Analysis and Design of Earthquake Resistant Structures

Computer Analysis and Design of Earthquake Resistant Structures

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Composite fiber fabrics A recent development in construction materials technology is the introduction of composite fiber fabrics which can be used to provide external confinement to structural members such as beams and columns. Composite fiber fabrics are composed of various fibers, such as kevlar, aramid and glass, woven at some predetermined proportions. The fabric can be applied and bonded to concrete members using special epoxy resin (figure 27). This application which was proven by tests to enhance the displacement ductility capacity of common concrete members is relatively simple and cost effective. Plastic fibers In many situations where concrete provides the finished exterior surface a concern may exist due to spalling of cover. While this may not present a major structural vulnerability, it may create a life safety concern in such areas where concrete- cover inay spall off over sidewalks and other public areas. A possible approach to reduce spalling of concrete cover is to incorporate plastic fibers in the concrete mix. The fibers tend to improve the bond of the concrete cover to the core thus minimizing the potential for spalling. The use of plastic fibers is also advantageous because such fibers are not subject to corrosion which could stain exterior surfaces.

c..

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Composite design Conventional thickening of concrete shear walls and floor diaphragms may not always be possible due to space and weight limitations. In such situations it might be possible to employ a composite design solution using the combination of existing concrete and new steel plating (figure 26). This combination takes advantage of the higher steel strength and the superior concrete stiffness. A key design consideration for composite design is the achievement of combined action between the composite materiais. This can generally be accomplished by means of adequate shear transfer across the combined cross section using transverse dowels.

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Computer Analysis and Design of Earthquake Resistant Structures

Computer Analysis and Design of Earthquake Resistant Structures

9

909

Retrofit Implementation

Implementation of a seismic retrofit is often very challenging for a variety of reasons which include working in an occupied and functioning structure, and ensuring that the existing structure is not overstressed under construction loads. This section discusses some of the issues associated with seismic retrofit construction. SOFFIT

9.1 ...... EPOXY-SATURATED " " - . _ HIGH-STRENGTH FIBER WRAP OVER CONCRETE COLUMN

EXISTING FOOTING

COLUMN ELEVATION

EPOXY-sATURATED HIGH· STRENGTH FIBER WRAP OVER CONCRETE COLUMN

COLIJllNSECTION

Quality Assurance Quality assurance is a key element in the success of any project. This is particularly true for seismic retrofit projects, since added structural elements must perform well under the severe demands of a major earthquake. Quality assurance is performed at the prerogative of the owner and as mandated by the building code. Quality assurance is usually performed by a qualified special inspector employed by the owner. This individual must observe and spot-check the work to ensure conformance with the approved design drawings and specifications. For concrete building retrofit projects, special inspection usually includes the following: • Verification of reinforcing steel placement • Verification of proper concrete or shotcrete mixing and placement • Taking and performing strength tests on concrete or shotcrete specimens • Visual inspection of epoxy anchor installation • Load testing of epoxy anchors • Visual inspection of field welding The special inspection requirements for each project should be based on the requirements of the building code and other relevant design standards, as well as the added requirements of the engineer-of-record and building official. Active involvement by the engineer-of-record is of paramount importance to ensure adequate construction quality. Observations of the engineer-of-record at significant construction stages can prevent construction errors and ensure that the work is performed in accordance with the intent of the design. The engineer-of-record is the most qualified individual for this role due to his intimate familiarity with the project

9.2 Figure 27: Confining an existing column using a composite fabric wrap.

Constnlction Procedures

The method of construction must be well planned before engaging in a seismic retrofit. In fact, the _engineer-of-record should always keep in mind constructability issues throughout the development of the design. This is particularly true of retrofit construction, since existing conditions may present obstacles such as limited access for construction activities, limited capacity for construction loads, and potential disruption to occupants and existing equipment during construction.

910

Computer Analysis and Design of Earthquake Resistant Structures !ll

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It is often beneficial for the engineer-of-record to work closely with the contractor to develop a procedure which will enable construction under the particular project constraints. Typical coordination issues may involve material delivery, staging, and handling requirements as well as critical steps of the retrofit construction. For example, consider the case of the composite linkbeam described in Section 18. 7 .1. To ensure project success. the engineer-ofrecord, general contractor, and subcontractors teamed to identify the potential construction issues and to develop a detailed step-by-step installation procedure which addressed these issues (figure 28). This advanced planning allowed a unique and relatively difficult construction operation to proceed efficiently and effectively.

10

Conclusion

This chapter provides a general discussion of the state-of-the-practice in seismic retrofit design and construction of concrete structures. Seismic retrofit is similar in various respects to seismic design of new structures. However, retrofit design also presents unique challenges not encountered in the design of new structures, such as: • In-situ condition assessment is often required for older and damaged structures lacking adequate documentation • Retrofit design must overcome the various constraints of working in occupied environments • Performance assessments must consider the combined behavior of existing and new structural members Seismic design technology is rapidly evolving and the engineer must stay wellinformed in order to provide the most technologically suitable and cost effective solutions for each project.

References 1.

ATC. Proceedings of the Workshop to Resolve Seismic Rehabilitation Sub-Issues - July 29 and 30, 1993; Development of Guidelines for Seismic Rehabilitation of Buildings, Phase I: Issues Identification and Resolution, ATC 28-2, Applied Technology Council, Redwood City, California, 1993.

2.

ATC. Rough Draft, Product 1.2, Recommended Methodology for Seismic Evaluation and Retrofit of Existing Buildings, for the Seismic Safety Commission: Sacramento, California (unpublished), ATC-40, Applied Technology Council, Redwood City, California, 1995.

3.

ATC. Draft Guidelines for the Seismic Rehabilitation of Buildings, Volume I - Guidelines and Volume II - Commentary, Report No. ATC 33.03, Applied Technology Council, Redwood City, California, 1996.

912

Computer Analysis and Design of Earthquake Resistant Structures

Computer Analysis and Design of Earthquake Resistant Structures

913

4.

Chai, Y.H., Priestley, M.J.N. , and Seible, F. Seismic Retrofit of Circular Bridge Columns for Enhanced Flexural Performance, AC! Structural Journal, 1991 , Vol. 88, pp. 572-584.

15.

Maffei, J. & Cocke, D. Changes Needed to Earthquake Repair Codes, pp. 55-67, Proceedings of the 64th SEAOC Annual Convention , Indian Wells, California, 1995.

5.

Computers and Structures, Inc. SAP90™: A Series of Computer Programs for the Static and Dynamic Finite Element Analysis of Structures, Berkeley, California, 1994.

16.

6.

EERI. In Wait for the Next One, Proceedings of the 4th Japan/V.S. Workshop on Urban Earthquake Hazard Reduction, Earthquake Engineering Research Institute and Japan Institute of Social Safety Science, sponsors, Osaka, Japan, 1995.

State of California, Office of Planning and Research. Competing Against Time, Report to Governor George Deukmejian, Governor's Board of Inquiry on the 1989 Loma Prieta Earthquake, Sacramento, California, 1990.

17.

Wallace, J.W., and Moehle, J.P. BIAX: A Computer Programs for the Analysis of Reinforced Concrete Sections, Department of Civil Engineering, University of California, Berkeley, Report No. UCB/SEMM89/12, July, 1989.

7.

Ewing, R.D ., El-Mustapha, AM., & Kariotis, J.C. FEM/I: A Finite Element Computer Program for the Nonlinear Static Analysis of Reinforced Masonry Building Components. National Science Foundation Report No. 2.2-1, 1987.

8.

FEMA. NEHRP Handbook for Seismic Rehabilitation of Buildings, Phase I: Issues Identification and Resolution, ATC-28, Federal Emergency Management Agency, Report No. FEMA 237, Washington, D.C., 1992.

9.

FEMA. NEHRP Handbook for Seismic Evaluation of Existing Buildings, Building Seismic Safety Council, Federal Emergency Management Agency, Report No. FEMA 178, Washington, D.C., 1992.

10.

FEMA. NEHRP Handbook of Techniques for the Seismic Rehabilitation of Existing Buildings, Building Seismic Safety Council, Federal Emergency Management Agency, Report No. FEMA 172, Washington, D.C. , 1992.

11.

FEMA. Typical Costs for Seismic Rehabilitation of Buildings, Volume I Summary, Hart Consultant Group, Inc., Federal Emergency Management Agency, Report No. FEMA 156, Washington, D.C., 1994.

12.

FEMA. NEHRP Recommended Provisions for Seismic Regulations for New Buildings, 1994 Edition, Building Seismic Safety Council, Federal Emergency Management Agency, Reports Nos. FEMA 222A (Provisions) and FEMA 223A (Commentary), Washington, D.C., 1995.

13.

Freeman, S.A. A Review of Practical Approximate Inelastic Seismic Design Procedures - for New and Existing Buildings, pp. 311-331 , Proceedings of the 64th SEAOC Annual Convention, Indian Wells, California, 1995.

14.

ICBO. Uniform Building Code - 1994 Edition, International Conference of Building Officials, Whittier, California, 1994.