Engineering Design Handbook - Ballistic Missile Series, Structures

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AD NUMBER AD830267

NEW LIMITATION CHANGE

TO Approved for public release, distribution unlimited

FROM Distribution authorized to U.S. Gov't. agencies and their contractors; Critical Technology; SEP 1963. Other requests shall be referred to Army Materiel Command, Washington, DC.

AUTHORITY USAMC ltr, 14 Jan 1972

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SEPTEMBER 1963

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ENG IEERING DESIGN HANDBOOK SERIES Tkw Engineering Design Handbook Series is inzended to provide a com21latio= of principles And fundamental data to in assstng engineers ir the evolution of new designs which will meet tactical and technlcai supplement experiem needs while also embod)ing riatlsfacto.-y prod.ibillty and mainainability. ULo.ed . elow are the hlandbooksw.-b hale been publishedor submitted forpublicatioh. Handooke wnh prbLicatlon dates pr'ior to I August 196Z were published as Z-rerles Ordnance Corps pamphlets. AMC Circular 310-38, 19 .Tuly 1963. redesignated those publications :,a706-series AMC pamphIt,- (i.e., ORDP 1:0-138 was redesignated AIACP 706138); All new, reprinted, or revised nandbooks are being publr.hcd as 706-series AMC pamphlets. General and Miscellaneous Subjects Title Number Elements of Armament Engineering, Part One, 106 Sources of Energy Eleents of Armament Engineering. Part Two, 107 Ballistics 108 Elements of Armimenl Engineering, Part Three, Weapon Systems and Components lie Experimental Statistics, Section 1, B.sic Concepts and Analysis of Measurement Data III Experimental Statistics, Section Z, Analysis of Enumerative and Classificatory Data 112 Experimental Statiytics, Section 3, Planning and Analysis of Comparative Experiments Experimental Statistics, Section 4, Special 113 Topics 114 Experimental Statistics, Section 5, Tables Maintenance Engineering Guide for Ordnance 134 Design Inventions, Pztents, and Related Matters 135 Section 1, Theory 136 Servomechaennsm, Servomechanisms, Section 2, Measurement 137 and Signal Converters Servomechanisms, Secd-n,3, Amplification 138 Servomechanisms, Section 4, Power Elemento 139 and System Design 17G(C) Armor and Its Applicatim to Vehicles (U) 270 Propellant Actuated Devices 290(C) Warheads--Genera. (U) 331 355

j

Compensating Elements (Fire Control Series) he Automotive Assembly (P.Atomotive Series)

Ammunition and Exploclves Series Solid Propellants, Part One 175 176(C) Solid Propellants, Part Two (U) Properties of Explosives of Military Interest, 177 Section 1 1781C) Properties of Explosives of Military Interest, Section 2 (U) 21Z0 Fuzes, General and Mechanical 211(C) 212(S) 213(S) 214(S) 215(C) 244

245(C) 246 ~acteristics 247(C) 248 249(C)

Fuzes. Proximity, Electrical, Part One (U) Fuzes, Proximity, Electrical, Part Two (U) Fuzes, Proximity, Electrical, Part Three (U) Fuzes, Proximity. Electrical, Part Four (U) Fuzes, Proximity, Electrical, Part Five (U) Section 1, Artillery Ammunition--General, with Table of Contentq Glossary and Index for Series Section 2, Design for Terminal Effects (U) Section 3, Design for Control of Flight Char-

Ballistic Missile Series Number Z81(S-RD) 282 284(C) Z86

Title Weapon System Effectiveness (U) Propulsion and Propellants Trajectories (U) Structures

Ballistics Sries Trajectories, Differential Effects, and Data 140 for Projectiles Elements of Terminal Ballistics, Part One, 160(5) Introduction, Hill Mechanisms. and Vulnerability (U) Elements of Terminal Ballistics, Part Two, 161(S) Collection and Analysis of Data Concerni=g Targets (U) 162(5-RD) Elements of Terminal Ballistics, Part Three, Application to Aissile and Space Targets(U) (Xrriages and Mounts Series Cradles 341 34, Rectil Systems 343 Top Carriages Bottom Car iages 344 Equilibrators 345 Elevating Mechanisms 346 Traversing Mechanisms 347 Materials Handbooks 301 302 303 305 306 3C7 308 309 310 311

Aluminum and Aluminum Alloyc Copper and Copper Alloys Magnesium and Magnesium Alloys Titanium and Titanium Alloys Adhesives Gasket Materials (Nonmetallic) Glass Plastics Rubber and Rubber-Like Materials Corrosion and Corrosion Protection of Metals

Surface-to-Air Missile Series Part One, Syttem integration 291 Part Two, Weapon Control 292 293 Part Three, Computers 294(5) Part Four, Mibsile Armament (U) 295(S) Part Five, Countermeasures (U) 296 Part Six, Structures and Power Sources Part Seven, Sample Problem (U) 297(S)

Section 4, Design for Projection (U) Section 5, Inspection Aspects of Artillery Ammunition Design Section 6, Manufacture of Metallic Cornponenta of Artillery Ammunition (U)

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-*HEADQUARTERS

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SlILUNITED) STATES

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ARMY MATERIEL COMMAND

WASHINGTON 25, D.C.

30 September 1963

AMCP 706-286, Structures, forming part of the Ballistic Missile Series of the Army Materiel Command Engineering Design Handbook

Series, is published for the information and guidance of all concerned. (AHCRD) FOR THE COMY14DER:

SELWYN D. SMIT , JR. Major General, USA

Chief of Staff ')FFICIAL:

R. 0. DAYONC

Colonel,

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Chief, Adiistrative

DISTRIBUTION:

Office

Special

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WOEACE

"-Tishandbook has been prepared as one of a series on the design of ballistic missiles. It presevis selected data on the properties of materials of construction including the effects of elevated temperatures and other operational conditions on ballistic missile structures. ParticuJar attention has been given to the stress analysis of thin shells. in addition the special design considerations applicable to ballistic missile structures are discussed.. The handbook was prepared for the Office of Ordnance Res'-:cn, Ordnance Corp3, U. S. Armyn. The text and illustrations were prepared by Vitro Laboratories under contract with Duke University, wvth the technical assistance of Army Ballistic Missile Agency and the Special Projects Branch of Navy Bureau of Ordnanve.

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STRUCTURES

CONMEtTS Page

Chapt7.-

1 1- . P urpcse and Scope ...................... ....................... 2 1-2. Ballistic Missile Structural Considerations ----......... 1-3. Elevated Temperature Effeci ...-..... .

2 Properties of Materia

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...........--------.. 2-1. Mechanical Behavior ... ior 2-2. Creep e 2-3. Heat Transfer -...

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2-4. Selcted Data an Properties of Materials of Construc16 tion ---38 2-5. References and Bibliography ............-............................

3 Stress Analysis- .. -

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41 3-1. General Considerations ............................................... 3-2. Stresses and Deformations in Thin Shells Under Uniform Internal Pressure ............. ..................

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3-3. Disccntinuit Stresses in Thin Sheis

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3-4. Buckling of Thin Circular Cylindrical Sheic

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35. Buckling of Thin Spherical Shells

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3-6. Thermal Stresses .............-................. 3-7. Thermal "Buckling

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3-8. Creep Effects ..................

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3-9. References and Bibliography............

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4 Design Considerations ...............................

57 ..................... 4-1. General Comments ..............-....... 4-2. Aerodynamic Influence on External Configuration

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4-3. Aerodynamic Heating ............................ 58 ..... 62 _..-......... .-.............. 4-4. Design for Applied Loads ..... 66 ...... ....... .......... ....... 4-5. Structural Failure ..............4-6. Time and Temperature Dependence of Design Criteria 69 70 4-7. Miscellaneous Factors Affecting Structural Design ...... 73 4-8. References and Bibliography .........-...............................

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STRUCTURES

LST OF ILLUS1 Figure

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T;tle

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2-1.

Typical Tensile Stress-Strain Diagrams for Steel and Aluminum Alloys 2-2. DtAtortion of an Element by Normal and Shear Strains 2-3. Fatigue Leading and Typical Stress-Cycles to Fa~iure -. (S-N) Curve .... 2-4. Typical Presentation of Creep Dat . 2-5. 2-6.

8 9 11 12

Effect of Temperature on Modulus of Elasticity for Several Typical Materials.-_ _ Effect of Temperature on Yield Stress for Sevewal Typical Materials 26

2-7.

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Effect of Temperature on W21ght-Strength Ratio for Several Typiczl Mterials- .... 27 2-8. Effect of Temperature on Cost-Stren th Ratio for 1everal Typical Materials_ _ 23 2-9. Compressive Stress-Strain Curves for 2024-T3 Aluminum A31o. Clad Sheet Stock at Various Temperatures--.. 28 2-10. Compressive Stress-Strain Curves for 17-7 PH (TH 950) Stainless Steel Sheet Stock at Various Temperatures

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-11. Tangent bodulus Curves for 2024-T3 Aluminum Alloy Alcad Sheet in Compression.............. ... 29 2-12. Stress for 0.2" Creep-Set of X2020-T6 and 2024-T6 Aluminum Alloys as a Function of Time and Temperature .............................-........

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2-1.. Rupture Stress for 2024-T6 Aluminum Alloy as a Function of Time and Temperature ---.......---------------........ 30 2-14. Stress-Rupture and Creep Characteristics of X2020 Aluminum Alloy at 300F ......................................... 31 2-15. Tension Creep Curves for 2024-T4 Aluminum Alloy.. 32 2-16. Creep Rate and Time to Rupture Curves for 3S-H18 Aluminum Alloy .................... _............................. .... 33 2-17. Master Creep Curves for 2024-T3 Aluminum Alloy Clad Sheet Stock .............................. ........................ . 34 2-18. Fatigue Properties of 2024-T4 Aluminum Alloy (Extruded Bar) at Elevated Temperatures ..... ------...... 34 2-19. Thermal Conductivity of Selected Materials .................. 35 2-20. Specific Heat of Selected Materials................................... 36 2-21. Mean Coefficient of Thermal Expansion for Several M aterials..................................................... .... ......... 37

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$VflUCTIJRES

IST OF ILLUSTRATIONS (WCmtised). Title Page Figure 3-1. Theoretical Deformations of Pressure Vessel under 42 Membrane Stresses Alone-.... 3-2. Deforn-ations of Thin Circular Cylindrical Shell under 42 Beriding Stresses 3-3. Determination Of Tangent Modulus and Secant Modur 46 lus from Stress-Stfain 47 3-4. Bar Heated between Two Insulated Walls 3-5. Assumed Temperature Distribution in Long Rectan47 gular Flat Plate 3-6. Ring Frame with Temperature 'Varying Linearly 48 through Depth 3-7. Ring-Reinforced Circular Cylindrical Shell ----49 3-8. 3-9.

Simply Supported Plate Subjected to Thermal Stresses . . . . 49 in One Direction .. Assumed Stress Distribution in Rectangular Flat Plate 50 with Free Edges ...

... 3-10. Tensile Creep Curve 3-11. Deflected Column 3-12. Stress-Strain Curve at Elevated Temperature ... 4-1. Effects of Aerodynamic Heating on Model Nose Cone, Stagnation Temperature 1995OR ------4-2. Effects of Aerodynamic Heating on Model Nose Cone, Stagnation Temperature 2030'R -................ -3, Tank Section, Jupiter Missile ....-.................-----------. 4-4. Top Section, Redstone Missile ...................-..................

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51 53 54 59 59 4 65

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STR UCTURES

LST OF TABLES Tabie

Title

Page ____I__1

1-1. Principal Notations

2-1. Mechanical Properties of Some Typical Alloy Steels.. 18 2-2. Mechanical Properties of Some Typical Aluminum Al19 Ioys..__

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2-3. Mechanical Properties of Some Typical Alloys of Magnesium, MlyJbeenum, and Niobium 2-4. Mechanical Pioperties of Titanium and Some of its Typical AllIoIyb 2-5. Mechanical Propertien of Some Typical Cermets, Metallic .21 Refractorie and Ceramics 2-6. Physical Properties of Some Typical Materials -----2-7. Approximate Costs of Structural Materials (1955)______ 2-8. Mechanical Properties of 2024-T6 Aluminum Alloy Products at Varicus Temperatures and after Heating .-....... 4-1. Weight Breakdown for Redstone A Missile ---------.

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22 23 24 72

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tapter I~

INTROD UCTICN 14. PURPOSE AND SCOPE

'TABLE 1-1. PRlCli(zL O y.k-mJ

The information presented in this hand-

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book has been selected to convey ,an appreciation of the problens -whicai face the missile sLructural design 4pecialist and to provide a basis Vir understanding the considexations -which leal to the final sttuctural coafiguraion. Since a complete o'itline ,ofthe theory and a-ractice of tructm'al lesign.and analysis can(, be included in a single volume, the more elementary coniderations have been omitted. It is a.sumed that the reader has a technical background .and a reasonable knowledge of the fundamentals of the strength of materials and structural principes. However, the 'book has been designed for the use of graduate engineers -ixth little or no direct experience in this field -ather than for the experienced structural design engineer or structural analyst. Some of the problems facing the ballistic missile structual design engineer are outlined in this introductory chapter. In Chapter 2, Properties of Materials, elementary and basic considerations in the behavior of materials are introduced, mechanical, creep and thermal properties are discussed, and data on physical properties are presented. Chapter 3, Stress Analysis, gives formulas of stress and strain for basic structural elements under selected lo-ds and boundary conditions. Preliminary stress calculations can be made by the use of the information presented in these two chapters. In the final chapter, Design C(.nsiderations, some of the problem- facing the ballistic missile design engineer are presented and various possible

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ten,.th oflat plate Cross-sectinal area Radiatiug :area, surface area and area normal to heat flow in radiation, convection and conduction modes of heat transfer, resp-ctively Width of-flat plate Distance to -extreme fibers from neutral plane in a pla'e Brinell hardness Constant, creep laws British thermal unit Thermal capacity Distance from neutral axis to extreme fiber Constant, creeup laws Temperature, Centigrade scale Diameter Internal energy Young's Modulus Tangent modulus Secant modulus Effective modulus Temperature, Fahrenheit scale Compressive yield stress Shear modulus Torsional rigidity Heat transfer coefficient Beam deptht Shell or plate thickness Moment of inertia Thermal conductivity

a A

b

B Btu c C

°C D E E, E, Em, F F1. Q GC h

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Constants, creep laws

solutions are discussed.

k j , k2

The principal notation used in this volume "sshown in Table 1-1. This volume was w.ritsen by S.V.Nardo and N.J.

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.,off, Polytechnic Institute of Brooklyn, Brooklyn, N. Y.. and edited by C. D. Fitz, Vitro Laboratories, West Orange, N. J.

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STRUCIURES

TABLE 1-1. (Continued)

1-2- BALLISTIC MISSILE STRUCTURAL CONSIDERATIONS

Critical pressure End load on column Rate of cooling Rate of heat generation in elemental volume Rate ,.f het-t flow Cornstant, creep laws Quantity of heat Q Muliiplying factor r Rockwell hardness R Radius Rc,, R~, R, Stress ratios, interaction formulas PTemperature, Rankine scale Room temperature R.T. Thickness t Time t~,Critical time T Temperature u, v, w Displacements in x, y, z directions respectively VVolume Shear force wSpecific weight x, y, z Space variables yDsncfrm euriaiina

The structural designer of a missile or an airplane h&-, much les,, freedom in selecting the external shape of -his design than the designc'r of a bridge or building strufcture. The form is largely prescribed for him other requirements. As a consequence his by t-zrk is a more difficult one. The designer hast to work within the space available and inust ensure that sufficient strength is provided with the minimum amount of weight. The conservation of weight is of particuiar importance in ballistic miss"le 111_ ue to the magnitude of the growth factor. This factor is the additional weight of 'the fuel anid struetural rnaterial requiredi by the addition of one pound of n)ay!oad to the 3riginal payload. The most important difference between the structure of a ballistic missile and mrost other structures is that the re-entry portion of the tyliss~le structure must be designed to withstand high rates of hcatinir This aerodynarnic lheatng raises tl~e temperature of the structure and influences -adically its loadcarrying capacity. Of course, aerodynamic healing is present in all supersonic aircraft, but it is h~i the ballistic missile that it a9ttains -its maximum importance. For this _7eason a new parameter, namely 4 enperaturL, enters the analysis of the structure. The elevated temperature, in turn, in-

GREEK VITTTERS Thermal diffusivity a Coefficient of thermal expansion Normal strain e Emissivity Half ape;: angle of cone ly Constant, creep Jaws vPoicson's Ratio Rate of radiant energy emissiop or absorption Radius of gyration p stress a'Normal Stpfaii-Boltzmann constant Criti&. normal stress Shear st;-ess TlCritical shear stress

must also be considered carefully in the analysis of missile structures. While 'the structure of a conve ntional ground-based subsonic airplane in ,irindcpie lasts for an almost indefinite per.od, the hot sti uctural ukuremets of missiles change their shape continuously under load until they become 'o deformed that they are incapablie of performing their structural duties. Every structuare that has to perform at very high temperature has, therefore, a definite lifetime beyond wh~ich it cannot be utilized. The consideration of this lifetime muakes the task of designing and analy:,sing a missile strticture much more complicated than similar iasks Performed itn connectidn with more conventional structilres, T~te "one-

Constant, creep lav.s Prefsure Cceflcient of first harmonic of thermal stress distribution

p

-external

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P q

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INTRODUCTION

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shot" nature of the ballistic missile intro. duces n design philosophy which cannot be tolerated in other structure..

structures and the thermal stresses of nonuniformly heated structures. Thermal stresses are of the nature of the secondary stresses of conventional structural engineering. They are a consequence of the deformation. and not of the externally applied loads. is unlikely they will cause failure except,It perhaps, in fatigue. However,

1-3. ELEVATED TEMPERATURE EFFECTS 1-3.1. Effects on Material Properties. The most immediate effect of elevated temperature -s the deterioration of the load-carrying failur excelt e p a fati e H er, he material. ateial.Man tets ave they may cause unduly large plastic deformacapacity of capaityof the Many tests have tions and, in particular, compressive thermal been carried out to obtain detailed informastresses might cause buckling. tion on this effect. The collected data indiThermal buckling may lead to the developcates the deleterious effect elevated temperaof bulges on the surface of the body or ment properties a material's physical which aftect the rigidity, deformation, ulticontrol surfaces of the missile which coild interfere seriously with the aerodynamic mate strength, fatigue and buckling of structhe accuracy of particular, performance. for that shows also tural members. The data be .....-, ....... For these m,a, pathIn figl 4he ........ the........... every structural material there exists a maxithe the missile miigit structural su r designer d and reasons mum temperature beyond which the loadanalyst must be conversant with the calcucarrying capacity is reduced to such an lation c' thermal stresses and with the extent that the material cannot be advanA I lormuis predicting thermal buzkling. tageously utilized.1-3.3. Croep and Creep Buckling. A third 1-3.2. Thermal Stresses and Thermal effect of elevated temperatures, important of effect Buckling. The second ;mportant little known in conventional comparatively of aerodynamic heating is the development is creep. This phemachinery, iterasteesi teeracture. risen in ahetdrdisujcetoaosantnil and structures in Even structure. internal stresses in the instance, when for observed, be can eometuericnomenon of uniform hecase tensile constant a to subjected is rod heated a cnstructural element, externaltureeric Under this load the rod elongates con-mnaxirnaof different eomdieetiateonload. strauctsrora a combination iusywthie.lhogladfrex matestraints tiiuously with time. Although lead, for examrials will induce it ternal stresses. T'hese pi., exhibits creep at room temperature, with temperature called stresses are sometimes employed structural materials the commonly problem, practical every almost s!resses. In p enomenon is practically non-e-istent at however, uniform temperatures in a heated becomes important, temperature. unlikely to o room rather structural element strutura are re eleent rtherunliely wever, aial ma- Creep even lead to the elimnina.e in tera to ma ic a of I the that are reasons occ,.r. Two principal rticular material from ie in tion of a aerodynamic heing itself is non-uniform structures, when the temperature is raised over the exterior surface of the structure, sufficiently. and that the heat capacity of the structure first structural effect of creep is the v e fThe varies from point to point. In consequence of deformation created. It is important that the analyst be the non-uniform temperature distribution,and the material would expand non-uniformly msiesrcua einradaaytb deformathe when able to predict the time y ia ossibl geoetrcally.Asuthe weeth were this possible geometrically. As the nattions will become so large that they substanural 4eformations of the material are incompatible, internal stresses are set up in the tially change the aerodynamic shape of the missile or that they interfere with the proper non -uniformly heated st.uctural elements to parts of the eemeableof the functioning re-establish the continuity of the deformasystem. as tions. These stresses are known properly thermal stresses. However, in this section the A second effect of creep is the occurrence term thermal stresses shall be 'used for both of two entirely new types of structural instathe temperature stresses of uniformly heated bility. The first is instability in tension. When

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V

STRUCTURES

P

a straight bar is attached to a rigid structure at its upper end and is loaded by a weight at its lower end, the bar elongates continuously with time. As creep deformations of metals take place substantially without any change in the total volume of the material, ;. one percent increase in the length of the bar must be accompanied by a one percent decrease in its cross-setiona area. However,

action of these bending moments the column deforms in creep bending. The increasing. curvatures in turn lead to increasing lever arms which again result in increasinir hpnd-± ing moments. This is a vicious circle which can end only in the deveiopment o very large curvature and compiete collapse of the column. When the behavior of the materiA is such that the creep rates increase niore rap-

constant load applied to a !:.-aller crosssectional area causes an increased stress in the bar which in turn increases the rate at which creep deformations take place. The creep xtrain rate is generally a non-linear function of the stress, increasing much more rapidly than the stress. The consequence is a very rapid acceleration ,f the creep proce;s which can end only in a reduction of the cross-sectional area to the value at which the material will rupture instantaneously under the applied load. The time at which this rupture takes place is known as the critical tinie. This characteristic exista for all structural materials at elevated temperatures and must be known to the structural designier. A second instability of importance is known as creep buckling. This type of buckling differs ini many respects from elastic and inelastic buckling as these phenomena are known to the structural engineer. The main

idly than th-bnn~r, -

difference is that in creep buckling one cannot utilize the concept of a efriical load. Buckling takes place under any compressive load however small if one waits long enough after the Application of the load. On the other hand, jL.St as in the case of tension, a new concept must be introduced, namely that of a critical ime. An explanation of this phenornenon can be givea without ditlhcuity. Every practicai column, as distinguished from the 'dealized ones underlying some theoretical calculations, is imperfect. This means that the column's axis is never coinpletely straight and the load is never applied

therefore take the necessary rrecautions in this regard. Some of these precautions are of an entirely aerodynamic nature; if the contour.; of the missile are properly shaped, the heating that the missile will experience can be minimized. A second precautionary measure is more in the realm of the structural designer; it consists of provisiorr for thermal protection through shielding the structural materivl with insulation, through cool ing the structure by means of the evaporation of a coolant, or. through other means. In some designs the external metal covering of the missile is used as a heat sink. When

to it completely centrally, Consequently the column is subjected to bending motmenii. A bending moment can be calculated as the applied !oad multiplied by its lever arm, or the distance of the centroid of the cross-section from the line of load application, Under the

this is the case, information is reqLired azout the thermal conductivity, the thermal diffusivity, and the specific heat of the material. The form.r quantities determine how rapidly heat will 1.e transmitted from the surf;)e of the missile Vxits interior, while the last item

the time -_ necessary for creep buckling to occur is eways finite (not infinitely long). Consequently we can properly talk of a criticP'1* - e . Depending upon the material, the 1oda, . 4nd the temperature, this critical time may be a few years, a few months, a few days, a few minutes or even a few seconds. It is therefore most important that the engineer be conversant with the facts of creep buckling when designing missile structure-. 1-3.4. Melting and Ablation. Other phenomena which must be treated, if a complete picture of the problems of ballistic missile structures is to be give,, are even farther removed from conventional structural analysis. The heating rates experienced by ballistic missiles on re-entry are so large that the surface of the material may melt, sublime or even burn. The designer of the missile must

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INTRODUCTION

indicates how much heat can be stored in the material, Re-entry has also been accomplished suecessfully through 1eting ithe surface of the missile melt and the molten metal be carried away by the windstream. If sufficient material is available in the structure, the integrity of the missile may not be impaired by this melting. However, the rates of melting and ablation must be calculated, and the designer must also have information about the sta-

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bility of the process; in particular, whether ridges or pits will be formed or whether the material will be consumed uniformly over the entire surface. Similarly it has been suggested that a missile miAght be allowed to burnL during re-entry just as in the case of ameteor. Again, a great deal of information is needed on this problem before the designer can employ burnifig as a means of maintaining the int.grity of the interior of the structure.

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Chapter 2 PROPERTIES OF MATERtALS

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2-1. MECHANICAL SEHAV*_ In the analysis and design of any structure, the physical propertiesof available ms are of primary importance. With relatively few exceptions, structural engineers in the past were concerned only with those properties which might be classed as purely mechanical. 'ertain elastic constants, yield and breaking strengths, and fatigue characteristics might be cited as examples. The relatively recent developments in jet and rocket propulsive systems, high-performance aircraft, and missiles have greatly increased the complexity of structural analysis and desig~n problems,. Designers of 'today's high-speed in the material properties already mentioned,

Data on the properties of some of the more pertinent engineering materials used in mise design are collected and presented in taii.-lar and graphical form at the end of this

but also in the thermal, creep and plastic characteristics. Indeed, it is certain that this interest will extend to those physical characteristics which were heretofore in the domain of solid-scate physics. A major reason for this extension of interest ir materiat properties is that ballistic missiles are subject to high rates of heating. This heating occurs upon re-entry into the atmosphere and (under certain trajectory conditions) durinr the ascending phase of the Bight. Unless the heat is dissipated the temperature will rise. The loas! carrying capacity of the strecture is affected by the temperatures and also by the length time at an attained, elevated temperature.

a unit length to obtain the strain. When the data are presented graphically in the form of stress as a function of strain, the result is a s ress-strain curve. Typical curves are shown ip Figure 2-1 for steel vnd aluminum alloys. Before the information derivable from these curves is discusseul, a brief review of the concepts of stress and strain will be presented. Stress is the intensity of the force per unit area acting on a plane. This property is a vector quantity; the normal component of the vector is the normal or direct stress, and the component in the plane, the sfr ar .stress. In a cylindrical test specimen, the quantitj obtained by dividing the force by actthe cross-sectional area .is the total normal stre.szs

As a genera) rule, it may be said that elevated temperatures have a deleterious effect on the mechanical properties of materials, For example, the modulus of elasticity, yield stress, and the ultimate stress usually decrease with increasing temperature. It will also be se, n that the rupture stress in fatigue. or the number of cycles to failure generally decreases with an increase in temperature.

ing on a plar.e perpendicular to the axis of the cylinder With a material that can be considered homogeneous and isutropic, and with a cen'.rally applied load, this definition is reasonzbly consistent and accurate. Inasmuch as this normal stress is the only stress acting in this simplified case, the stress condition is said to be uni-axial. The stress picture is accompanivd by a strain picture with normal and shear strains

2-1.1. Stress-Strain Curves. Probably the mechanical data most familiar to engineers are stress-straifn curves. This information is obtained experimentally by testing a prescribed standard specimen of the material in tension or compression and simultaneously recording the load and elongation, in thsimplest of thene tests, a cylindrical *pecimen is loaded in tension ;r :atesting machine, the observed load is divided by the original s'.ess, and the elongation is measured over

7

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STRUCTURES

developing at each point in the material. Imagine that parallel planes are passed through two neighboring points. The normal strain is the relative displacements of these points normal to the planes divided by the

Figure 2-2 illustrates the separate effects of normal and shear strains on the deformation of a square element of the material. In part (a) of the figure, the element is subjected only to normai strains. This type of

original distance between the planes. The disrelativeparalas the is defined measured two points of the "placement shear slrain lel to the planes, divided by the original

deformation would occur, for example, if a onlya stress in normal .11vie the that element, were anthe thex direction stress acting strain in the y direction accompanies the

distance between the planes. Normal strains have the effect of distorting a cubical element of t.e material without altering the orthogonality between tie sides of the element, whereas shearing trains are a measure of the change from the original right angles between the side-.

strain in the x direction due to stress in the x direction. Part (b) of the figure illustrates the manner in which the element would be distorted if it were subjected to a condition of pure shear stresses. In the most general case there would also be u displacements (x-direction) of the end points of the element.

]1

A

iW

B

/-PROPORTIONAL

0 I

MAXIMUM STRESS

YL

YIELD STRESS

U

IT

STRAIN

c

ALUMINUM ALLOYS

VEELD STRESS

7

UJ

'ROPOW'1ONAL

MAXIMUM STRESS LIMIT

a:

//

,STRAIN

STEELS FiSu.,e 2-1. Typical Tensile Stjss-Strnin Dogrumzl

8

for Steel and Alum'nuzi Alloys

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PROPERTIES OF MATERIALS

~~dx

dx

i

du

-

-[d-

L

dyj

(a)

.

NORMAL STRAIN

du

SHE.A

dx

STRAIN

I

-.

=dx

Figure 2-2. Disfortion of an Elernenst by Nos-mal vdSIear Strains

2-1.2. Propart'onal Limit. Notwithstanding the relotively simple nature of"the test that yields the stress-strain curves of Figure 2-1, some of the most useful and basic properties of the material are deduced from it. It will be noted that both curves exhibit an initial linear relationship between stress and strain, The value of the stress corre-sponding to the end of this ~rroortina linear called the hmt. relationship rovied heis tresesye-

spring constant or Young's Modulus can be assumed to be independent of both pesition and direction within the material. Clearly, there0vre, this one physical property is of major importance to the structural designer when the design criterion is a rigidity requirement. 2-I.4. Y'ield.

Consider a simn!;'e tensile

specield. im limit ite to specimen loaded Conde beyond the aelastic

proportionallimit. Provided the stresses remain below thel ,

a s r s

o r -p n i g t

on

n F g

a stress corresponding to point B in Fiure 2-1a. Upon the reo d of the load, the rial remains elastic, i.e., it assumes its origidecr;ses approximately linearly with stress load. the of removal the upon nal shape line BC because by deformations as illustrated strain It Furter, he mteril i linarlyelasic.i elastic. ne elastic part of the canonly be linearly is Further, the material might be well to point out that in reality, t nereaic par t e tiosc n e material canmateialcanbe be stressed the r-oporsresed above bovethe~i~orregained. A permanent set is observed when tionai limit and still remain elastic. However, the load is completely removed. The unloadfor most engineering materials, the proporing line BC is approxi.: ely parallel to the straight linelus.OA whose slope is equal to tional and elastic limits are frequently relatively close s O If f he e pr per. miaterial were the sltei Modulus. Young's used -o each other and are in fact interchangeably by engineers. fectly elastic, it would have unloaded along the original loading line OAB. For materials 2-1.3.. Young's Modulus. The slope of the stress-strain curve, that is the rate of change of the stress with increasing strain in the linear region is calied Young's Modulus. Physically, it may be considered a spring constant because Young's Moduius is proportional to the force per unit elongation. Although the stress-strain curve gives the value of the spring constant in only ornu direction, most engineering material, Are fairly

whose stress-strain curve has the charocteristics shown in part (a) of tne figure, t:p yield strpss ii defined a$ that value of the stress c.arre.ipwndirg to an offnet, c,, of 0.002 -ur materials whose -tress-strain curves are similar to that shown in part (b) of the figure, the first point of horizontal tangency is taken as the yield point. 2-1.5. Ultimate Stress. The maximum value

homogeneous and isotropic, and hence the

of the stress reached in the simple tensile

9

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STRUCTURES

test is called the ultiniate sirfs. Since all stresses are . sed on the original crossectional area of the specimen, the ultimate stress corresponds to the maximum load, and it may be considered as indicaive of the strength of the material. Relatively few members in airplane and missile structures, however, are critical in tension. The ultimate strems of a material in compression is net quite as clear-cut because of the column action. Even when the column action is prevented by lateral constraints or by the proportions of the test specimen only some materials such as wood and stone will exhibit a definite value of the ultimate stress. For most metals, an arbitrary criterion must be used to define this quantity. When the design criterion is associated with failure or rupture t SL

ratio of the lateral strain to the axial strain in a uniaxially stressed material. As mentioned earlier, a cylindrical tensile test specimen subjected to an axial load produces an axial strain. This axial strain is always accompanied by a lateral strain smaller in magnitude, and opposite in sign [Figure 2-2(a) ]. For most engineering materials Poisson's Ratio has a value of approximately 0.3.

stress assumes major iUmprtance as a material propeyty. 2-1.6. Tarigent and Secant Modulus, Modu2-1.of TaRgetIndsestucatMualsnalyses lus of Rigid. In some structural analyses (column buckling, for example), a so-called tangent modulus is used. This quantity is simply the slope of the stress-strain curve at any point. Below the proportional limit, the tangent modulus is everywhere equal to the Young's Modulus. Some investigators use the concept of a secant modulus, which is the ratio of stress to strain at any point. Young's Modulus and the secant modulus are also equal below the proportional limit. Torsion stress-strain diagrams can also be plotted from torsion tests on round tubes or round solid sections. The initial slope of this curve, analogous to Young's Modulus, yields the modulus of elasticity in shear, referred to as the Modulus of Rigidity.

2-1.8. Fatigue. The physical properties of materials discussed so far have been based on static tests in which the load is slowly applied without repetition. It is well known that some structural elements are subject to vibrations which cause the stresses to fluctuate. Under these conditions, failure can occur at a stress level which may be considerably lower than the ultimate stress previously discussed. This type of failure is known as fatigue failure. The fatigue strengtl. of a structural element is probably the least amenable to analysis and hence the most difficult of the various design strengths to evaluate. Unfortunately, its determination is not confined !o a knowledge of the nature and magnitude of L%-, fluctuating stresses and some simply physical properties. it is known that the "atigue strength of a specimen is affected by size, local discontinuities, and heat treatment, to mention a few of the parameters. Investigators in this field have nevertheless made significant strides by their analytical studies and extensive experimental programs. The data used by design engineers in determining fatigue strength are based on tests cf standar.1 specimens (smooth or notched) subject to controlled fluctuating stresses, in

2-1.7. Poisson's Ratio. For homogeneous, isotropic materials, the Modulus of Rigidity is related to Young's Modulus in the following manner: (2-1) G = E/2( ± v) where E is 'Young's Modulus G is Modulus of Rigidity v is Poisson's Ratio Pqisson's Ratio is the absolute value of the

which the number of cycles to failure is recorded. Part (a) of Figure 2-3 shows a periodic variation of stress to which a fatiguc specimen may be subjected and defines the important stress parameters affecting fatigut life. In part (b) of The figure, a characteristic curve is presented which gives results of fatigue tests on a specimen. The curve is drawn for a mean stress value of zero, and relates the stress to the number of cycles to

oh suctuall nirner, then the ultimate

10

f -a

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PROPERTIES OF MATERIALS

4"failure. Curves of

-

-with

this type are commonly q;aued S-N diagrams. The value of the greatest. stress which the material can withstand for an- infinite number of cycles without failurt is called the endurance limnit. This is a simple and admittedly crude measure of the fatigue streftgth of the material. If the endurance limit so defined cannot be detected a reasonable amount of testing, a practical endurance limit is often introduced as the stress at which 107 load cycles do not cause failure. For the case of zero mear, stress, it will be

noted that the mnaximnum ntress is equal to the stress amplitude. The maximnumr and minimum stresses are equal in value, -)pposite in sign, and correspond to) a case of completely reversexd loading. When the mean stress is other than zero, the stress amplitude core-esponding to fatigue decreauses with inereasing mean stress. Trypical curves at room and elevated temperatures are presented in the literature.'4 *Superscript numbers refer to the bibliography at the end of thi chgpter.

b

K

OTrEAN U)

ENDURANCE LI MIT

TIME

A

NUMBER OF CYCLES TO FAILURE (LOG SCALE)

Figure 2-3. Fatigue Loading and Typical Strevs. Cycles to Failure (S-N) Curve

2-1.9. Other Mechanical Properties. Of great importance to the missile designer is the plastic behavior of materials. Plasticity is related to the ability of a material to sustamn a permanent deformation without fracture, A perfectly plastic material, for examFle, would require no increase in stress to produce a continuing plastic deformation oree [z 7ipld stress was reached; it would be repr;s-aiid by a segment of the stress axis and a horizontal straight line on the stress,.strain diagrini. The importance of plasticity

Ican

be illu.,iialrd by copsidering a structure

which is su~ije, lowaly to a high stress level,

*

If the material is ductile, plastic flow occurs, and the load is ridistributed more uniformly. Neighboring points in the material absorb the load which is shed at the point of stress concentration, thus lowering the intensity orA the maximum stress and lessening the possibility of early failure. A brittle material does not readily flow plastically and hence there is the danger of brittle fracture. Quantitative

measures of the ductility of a material are the percent elongation in 2 inches of a test specimen and the reduction of area at fracture in a tensi or test. The hardne,, of a material is a measure of its ability to i-esist permanent surface deformation under applied concentrated lowds. There are several standard methods of incasuring tis quantity. Values of hardness for several materials are given in the tabies at the end of this chapter. Approximate corr-lation between the various hardness mcales

are given in t2~e literature." It can be shown

that for many steels thie tensile stftength is proportioial to the hardness. 22 RPBHVO 22.CE BAVO Creep may be defined as the contirwing deformation tWat occur's in a material subjected t(; a constant stress. At room temperatures, the cre4op of engineering mat!rals is unimportant, but at eievated temperatuires

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STRUC URES

creep effects mnay yield important design criteria. Creep data are generally presented in the form of st:ain as a function of time as shown

~-EL

in Figure 2-4. Specimens are subjected to a

I____________

conslant load, which cozresponds to a con. stant stress level based on the original cross-

C

STRAIN

.

JPRIMAR\ -

S

sectionl area of the specimen. Thie instantaneous value of the strain, t. corresponds to the elastic deformation calculable from the load. original cross-sectional area and Young's Modulus. With the passage of tLime,J the strain increasts; the strain rate is high at first and gradually decreases to a constant value. The portion of this creep curve inI

RGE

__

STAGV

I

which the creep rate is changing is called the pi inzary stage or primary e 'eep. The portion corresponding to a constant ate of strain is called the seeondary stage, or secondary creep. In the tertiary stage, the strain rate suddenly increases until the specimenSAG ruptures. Remova-l of the load during the secondaryj stage results in the recovery of the elastic

SEODR STNAE

strain r, and sometimes also of that portion of the creep strain it,obtained by extrap~iating the straight line of the secondary creep stage to t = 0. Some solid state physicists attribute this to a release of internal stresses which occurred during the loading and creep processes. it should b,.. noted that the true physical nature of cr esp has not yet been esZabhshed with any &gree of finality. Elongat-s i rn the specimen that occur contraction of the specimen in the lateral direction. This decreases the cros~s-sectional area and increases the -value of the true stress whici' is the loadO divided by the true cross-sectional area. This process acceleratesTIE

-

-

-

TERTIARY STAGE

RtP7URE

rapidly a.2d culmipates in rupture of thet-TM sppcinei when the greatly reduced crosssecticty e,, area can no longer 3upport the applied load. The portion of the creep curve corresponding to increasing rates of creep is the tert-ary stage. it should be mentioned that -met,,ilographic- changes in the material (for instance those caused by annealing during the test) can also lead to increasing rates

of

creep

Figure 2-4. Typical Presentation of Creep Ocita

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PROPERT'SES OF MATERIALS

I!

A so-called f/ray body will reflect some of th rain nryrcevd ec h misf agra boy wlldepend ulon siv peer

2-3. HEAT TRANSSMl

Iiscustomar todstinguish threebasic paii anoter cond (3)

I

ion hil

th

iectu

absorbed and reflected. It shouid be noted

energy ismntcmtl under stying ththea pheomeon s osered o b c istnt ith ~certain physical laws pustulated by v;%rious Th paticuar aw hic is invetigors

tion to being proportional to ihe fourth pow-.?r of the temnperature, is also dependent onhehycaprptisotetodad varies with the wave length. It is usual to

applicable can be established by recourse to experimental methods.onteweleg.ThcrepndgeqWhatever the wrid~ of heat trancfi~, the miissile designer is bask'.ly interested in ifUtmperzture distribution -ei.her as a function f sgae or of space nnd time. The temjperaure distribution is required to deterin heat transfer rates, to make possible the cots,putation of thermal stresses, to deterinine the change lin material physical properties, or to cGP vith a problem ;Sinmefting, abla Lion or burhing. 2-3.1. Radiatic-a. The transfer of heat energy by radiation is probably best exerr plifled by considering the transfer of heat rom the sun to the earth. Thermal energy F Lrom a hot body, the sun, is converted into __/ eleetromagnetic energy which is transr-itted thrug spceto a cold body, the earth, wharc iisreconverted into thermal eniergy. The radiant energy reaching the cold body can be absorbed, reflected or transmitted throuf.h the body. Since most e;;gineering materials are opaque, the tronsmiss~vity of these materials is zero, hence the cf i.ctivity and absorptivity only need be consiide. The concept _f a black body is the limiting case of a body with 100O/1 absorptivity and zero reflectivity. For a given absolute temperature, no body emits or absorbs more heat than a black body. The r-ate at kvhich the radiant energy is emitted or absorbed by a black body is given by the Stefan-Boltzmann

I

-

=rate

=

' diepdencTe ortespradint poer

tion for 1)for a gray body is written (2 3) 'cAT, 4", wherw t is the emissivity of the gr-y body. The emissivity foi, some engineering materials -1.gi'ven in the literature.'" Two final relations should be considered. The first is that radiant interchange between 5i',res is governxed by the aiove principles if there is a non-absorbing medium between the surfaces. Second, the radiant interchange also dtepends upon tho.. iverse square of the distanze between the surface-s and the relative orientation of one surface with respect to another. If it is assumed that the radiation fromn a heated surface has the same flux in all directions, the orientation or "view" becomes purely geometrical in nature. Terv factors are also given in the literature.'-2-3l.2. Convection. 'The trainsmoission of heat by tonvection (liffens materi-illy from the transmission (if heat by radiation. In the case of conve-tive heat transfer, heat is conveyed from one point to another by the movement of a fluid Atream; i.e., ihe transport process is &ssocidetd w~th the actual movemneat of a fluid streai-n. Convective heat transfer may bentta or forced depending upon whether the fluid motion oc-curs by virtue of the nattendency of the he,.ted, lighter elerneit 10-111

blower or othct means are used to produce or abof radia.-,L enryem~s tine hee he fluid to Itsha,or pl T bofy Bt (2 2)ac so tionby here that natural convective cooling of a surface area A a0 the b')n dary of a~ s-olid is proportionail to (T',, -T_) I %%here T,, arnd T..

a ~tefan-13olt?mann

A

ongl th

constant 0.174 x 10-4 Btu1 ( ir)(Wt)(F') radiating area, ft"

are the surface ind arnbiont fluid tempera-

T =absolute temperature, F 13

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4"

STRUCTURES

tures respectively. In practice, a linearized approximation can usually be taken and tie analytical expression for the convective cooling of a surface can be written as where q = rate of cooling, Btu/hr h = heat transfer coefficiet, Btu/(br)(ft )(F) Strictly speaking, the convective cooling of a surface is a combination of heat transfer br th by coivectic n and conduction within the fluid. The 'eader will note that- the above equation is similar to Fourier's basic heat conduction Equation (?-5). The reason for its appearance here lies n a concept which essentially converts the convective cooling problem to a conduction problem. This concept considers that there is a film adjacent to the surface throtigh which there is a linear temperature drop from TW to T,. The effective film thickness is difficult to measure and is incorporated in the heat transfer coefficient h which also is a function of the physical and dys'amical properties of the media invoived. 2-3.3. Conduction. Of greatest interest to the missile structural designer is the tkansmission ;f heat by conduction. This process is generally interpreted as a simple molealar interchange of kinetic energy from one molecule to an adjacent molecule by virtue of elac-tic impacts. The heat flow is observed to pass frurn the molecules at high temperature to those at a relatively low:er temperature. it has been custo ary ;n Lh azialysis of the temperature distribution in a structure to consider convective and radiant heat transfer as the mneans by which the external air flow heats the structure, and to disregard these modes of beat transfer when the flow of heat in the interior of the structure is sought.

p

to higher thermal stresses than those basd on a consideration of all three modes. In a solid structure the oniy mode f heat transfer is conduction; in hollow structure, however, and radiation can significantly conv Tmoify ion the temperature distribution calculated on the be basis of conduction It might mentioned that alone. a relativly complete analysis of simultaneous effects of conduction, convection, and radiation is possible through the use of computers. The basic law of heat conduction has ts origin in the fundamental work of the F.ench mathematician Jean Fourier. Fourier's Law states that the instant-neous rate of heat flow in a one-dimen:lienuO booy is given by the prduct of the area perpendicular to the flow, the rate of change of temperature in the direction of f!c'- .nd the thermal conductivity. dQ aT ( q (25) kA t In t above expresio, Q =. uqntity of heat, Btu t =time, hr q =rate of heat flow, Btu/hr k =therm co.iductivily, the a.nount of heat per unit time flowing through a unit area under a unit temperature gradient, Btu-in.ift2 hr F A =area normal to heat flow, ft 2 T temperature, F x =space zoordinate in 'tow direction, ft The units uised are British Gravitational units, and the minus sign is arbitrarily chosen to make Q a positive- quantity. In the formi 2itown, the temperature T is.assumed to bc a function of space and time (transient conduction), although it may be a function of space alone isteady conduction). Taking the case of steady conduction in one dimension and one for which A is not a function of x, Fourier's Equation is

Two reasons can be mentioned in defense of

q

-,k

A

("r,

this iimplified approach; first, the equations of conduction have been solved rigorously for many more cases than those governing the o'her two modes of heat transfer; and second, the resuits obtained from an analysis

where L = total path length (x2 T7 = temperature at x2

based on conduction alone always correspond

T, = temperature at x,

4

T (2 6)

14

-

xi) /

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PROPERTIES OF MATERIALS Cniefor example, a r~rd of unfo01rm crm-settional area A and length L~. F r a

zontstt vlueof the heat flow q, the tern-

assuming k, c. and w eomstant. Utder these conditions, the most genefal heat conduction

equation reda-es to

perature diffarence (Tz - TI) is inversely 13T Of eT q 0 +' . a + proportional to the Ithermal conductivity k. Thus -amaterial of high thermal conduCtIvity 29 where(29 will allowi s given heat flow at a lower ternperature difference than a material of lower k_ a= rmal diffusivity ftl/hr thermal conductivity. The missile designer inteestd ineonfrcing he eatinpu onthe For systems that do not contain heat sources snurface of a ntructur3 to the intrirwt a or sinks, q =0, and Equation (2-9) becomes minium tmpeatur els atthe urfae, he FoarierEquation. If sources and sinks areI will choose a mnatearial hazvin~g high thermal isa T present 4at Lkeg temperature iied,-t=0 conductivity (assuing all other maicerial properties are the samt-).a In the transient state of heat conduction, an Eqain(-)rdcstPiso' Equation. Finally, for steady conduction in the quantity of heat entering and leaving a systems free of sources and sinks, the wellvolurne element at any instant is not equal. known Laplace Equation is obtained. This The increase in internal energy, dE is linear partial-differential equation appears frequently in many branches of science. (2?7) dE = c wVa dt at Solutions are discussed in the literature. The thermal diffu-iivity is a derived, ratewhere rial property important to the missile dec thermal capacity of the rnaternal, th;_% signer as a measure of the ability of a of heat energy the naterial iraterial to absorb or diffuse heat. As defined can absorb per unit weight for a unit by Equation (2-9) thermal diffusivity is temperature rise Btu/ib./*F essentially the ratio of th.- thermal conducw = specific weight of the material, lbs,/ft2 tivity to the specific heat. Thue tirse required V = volume, 11,3 to bring a material to a specified temperature level varies inversely with the thermal difA missile designer interested in using the fusivity and directly with the square of the structaral material as a heat sink, would -

.--

)amount

I

material of high thermal capacity, choose~ a

conducting path length.

in three dimensions c~n be written

upon the material and the temperature rise.

again assuming that other material propertes were equal. The equation of continuity fr the heat flow thrcugh a volume element

(2 8)

For the single dimension case the ratio of the unit elongation per unit temperature rise is called the coefficient of thermnal expansion.

where the effect of the heat input q, Btu/fthr, has been added. In the most general case, c, w and q are functions of x, y, z and T; q may also be time dependent. As written, Equation (2-8) is valid for an inhomogeneous, but isotropic solid. Moreover, the equation is non-linear if k, c, or w depend on T.

d ( 0 where acoefficient of thermal expansiop I F per unit length ~ Engineers find it more convenient to define a mean coefficient of thermal expansion as j follows:

aTl T 0 ax (k ax)+ -a.(ka O

+ ew

rk,

a

2-3.4, Thermal Expansion. When an engineering material is heated, it expands in all directions in a manner that is dependent

+ q

OT oz (I+ (I

by taking q as a function of T only, and

Man caes ar f pactcalintres coere

-elongation

a -

L

~dT(2

iI'l.

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0

STRUCTURES where T. is a reference temperature. The

efficient of linear expansion. Coefficients of

uvit elongailM can be easily calculated after . is computed arid plotted ws a function of temperature from e=a-=. 1.) (2 12) ternfor a is required elorgation If the unit perature rise other than from the reference temperature, say T, to T., then : a,..(T..1.) - a,,,(T - '') (2 13)

thermal expansion for some engineering materials are given in the tables and graphs at the end of this chapter.

Consider the case of a slender rod of original length 4.,, heated to a temperature T, from a reference temperature value T., The increase in the length of the rod due to the effects of thermal expansion is required. Taking T, - T,. = T = 100 0F as the ternperature rise, L,, = 1 ft, and assuming L, for the material ha i value of 10 - 7-,the unit elongation from (2-12) is The ea of rod'~) 10-~ is 1material The elongation of the rod is

2-4. SELECTED DATA ON PROPERTIES OF MATERIALS OF CONSTRUCTION Collected in the pages that follow are selected data on material properties pertinent to missile design. Data are included on materials that appear to have promise for application in designs of the future. These data have been gleaned from a number of references which are listed in the bibliography. One of the principal reasons for collecting the information has been to give the reader a quantitative feeling for the order of magnitude of the physical properties involved. in sore instances, the variations in these physical properties for different alloys of the same are presented, but this is not con-

-he

(2-14)

sidered of paramount interest. In addition to

It might be well to point out the implicit

the objetive mentioned, th curves are set

assumption that the rod is not subject to any external oastraint. Ifthe rod isconstrained extrna ~~isrants I th ro i coisraied to remain at its original length, then the temperature rise would produce normal

up to afford an easy comparison between one material and another, The frequency with which~ time and temperature appear as parameters may be noted. Another objective in the presentation of the is tofinds point up the form presentationdata which favor with theofmissile de-

AL

L. = 10 - 2 ft

stress stresseqire rodcesanpeqondi top-oprequiredteto produce an equal and

posite unit elongation. The value of this (thermal) stress is obtained by multiplying the unit elongation by Young's Modulus:

tion w nd s avoryth th m isilces signer and stress analyst. in many instances datc of a particular kind are shown only for one alloy of one material. This is particularly true in the graphical presentation of creep data, and stress-strain and tangent modulus curves as functions of temperature. Tables 2-1 through 2-5 cover several alloys each of steel, aluminum, titanium, magnesium, and a few ceramics. In this compilation the room temperature mechanical properties are given for each alloy and for a selected form and condition. Young's Modulus, yield and ultimate strengths are presented on the basis of tensile test data. The yield stress is based upon a 0.2%. permanent offset condition. The elongation is the percent change in length measured over 2 inches of a test specimen. Table 2-6 summarizes the physical properties of various materials: density, melting

a = Ee (2-15) Hence if Young's Modulus for t1be rod used in the above example is 107 psi, the stress can easily be calculated: = 101(10 - 1) = 10,000 psi Partial restraints at the ends of the rod produce thermal stresses which are between the Values of zero thermal stress for unrestrained ends and the thermal stress calculated above for fully restrained ends. Expansion of the rod in the lateral directions may be calculated in a similar manner. The coefficient of thermal expansion is usually assumed to be independent of direction. Since most engineering material3 can be considered isotropic, the assumption is fairly good. The quantity a is also called the co-

16

I

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PROPERTIFS OF MATERIALS tion with the interpretation of the curves point, specific heat, thermal conductivity and shown in these Pi/ures. The stress-strain -coefficientof thermal expansion. In this corncurves shown P -e obtained at a:constant ternpilatijn, only one or two alloys of steel, are perat.me given by the intersection of the magnesium, aluminum and titanium curve with the temperature axis. The stresssuch of materials listed, but thi. properties strain curves are spread in their correct as copper, tungsten, molybdenum, oxides, posti*;t on the temperature axis (abscipsa) carbides and borides are presented because in order that curves at intermediate ternof the attractiveness of one or more of their Mean peratures; m'fy be interpolated. For an.y given elevated tcrmperatures. properties at stress-strain curve, the strain at any stress values of the thermal properties are given level may ;ie obtained by interpolation betogether with the temperature range within tween constant strain lines or by using the which they are applicable. For a number of strain smle (shown only in Figure 2-9) and materials, the, thermal properties are plotted measurinr .from the point of zero stress. An as a function of temperature and a reference illustrativ3 example :s given in Figure 2-9 to the corresponding figure is given. fo further clarifi-a;. n. In Figure,, 2-5 and 2-6 the modulus of ata at various stresses nio4u elasticity anSi the yield stress are plotted a±angent J . _niinum alloy in Figure are shown fot" are shoTh valuf . nninum as io F functions of the temperature for a selected T e value, ,xe 4,owp s Mos of number of typical materials. As alreAdy menMaulus, of , b A Moulus at tioned, the principal purpose is not only to of A tabulation various pronerties ternper.'n at a room and elevated mechanical show the variation of these properties for a temperatres fir an aluminum alloy is given material, but: also to indicate the relative a s e Creep aalu iue . t-r magnitudes and the behavior between differincluded. ent materials. It is significant to point out incle that these properties are affected by the rate by F pres inh ree Th of heating, soaking time at test temperature, through 2-17 are in the forms preferred by asthe ctrain rate. Most of the mechanical structural designers and analysts. In Figure properties shown are for moderate heating reader should note that the elastic 2-15, and strain rate, and are soaked for one-half strainthe is subtracted from the total strain to hour or more at the test temperature. Short time test results, which are not quite as plenobtain the curves. The log-log plot of minitiful, indicate an increase in the modulus of mum creep rate as a function of stress in elsticityl, dyiedstress with an increase Figure 2-16 is particularly useful in the elasticity end yield srswihainese formulr'~nn of creep laws for engineering he master aster creerves mtr ina rate and with and strain heating rate in the creep curves in the Finally, in materiai.. time. soaking the in a decrease are total parameters the 2-17, of Figure strain, and a so-called Larson-Miller parameter which includes both the time and temperature effect. The comments made in connection with Figures 2-9 and 2-10 also hold for this figure except that temperature is replaced by the Larson-Miller parameter. Inasmuch as fatigue considerations are relatively unimportant in missile type structures, only one set of curves is given for an alloy of aluminum in Figure 2-18. The final set of figures, 2-19 through 2-21, show the thermal properties of a wide variety of materials as functions of temperature. To enable a comparison between the various materials, each figure shows the variation of a particular thermal property.

The weight-strength and cost-strength ratios of one alloy each of aluminum, magnesium, titanium, steel and nickel are presented in Figures 2-7 and 2-8. Note how effectively the curves give an approximate idea of the maximum useful temperatures of the various alloys based upon these criteria. Cost data are listed in Table 2-7 and strength data are based on the tensile yield strength given in Figure 2-6. The reader is cautioned against arriving at any broad generalization based upon criteria of this type. Stress-strain curves at various temperatures are shown for an aluminum alloy and a steel alloy in Figures 2-9 and 2-10 respectively. A word of caution is made in connec17

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