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Inapplicability of limit design to structures made of some high strength aluminium alloys. Yu, Lawrence Kuang 1964

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INAPPLICABILITY OF LIMIT DESIGN TO STRUCTURES MADE OF SOME HIGH STRENGTH ALUMINUM ALLOYS by Lawrence Kuang Yu B.S. i n C i v i l Engineering, Taiwan Provincial Cheng Kung University, Tainan, Taiwan, China. 1961 A Thesis Submitted i n Partial Fulfillment of the Requirements for the Degree, of M.A.Sc. in the Department of C i v i l Engineering We accept this thesis as conforming to the required standard The University of British Columbia September, 1964 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the r e q u i r e m e n t s f o r an advanced degree at the U n i v e r s i t y of B r i t i s h C o l u m b i a , I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree that p e r -m i s s i o n f o r e x t e n s i v e c o p y i n g of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . I t i s unders tood that , c o p y i n g or p u b l i -c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l lowed without my w r i t t e n permission*, Department of C i v i l Engineer ing The U n i v e r s i t y of B r i t i s h C o l u m b i a , Vancouver 8, Canada Date September 1964  i ABSTRACT The basic assumption of l i m i t design, that moments are equalized by the formation of a mechanism has proven applicable to steel beams and cer-tain types of frames. It i s not known, however, i f the theory applies to light metal alloys. Steel possesses a considerable amount of strain har- . dening which i s essential to the formation of plastic hinges i n the beam, ' but some aluminum alloys which have l i t t l e strain hardening may not be suitable for l i m i t design. Two beam tests were carried out on a twice s t a t i c a l l y indeterminate beam made of Alcan 65S-T6 aluminum alloy to determine whether the mechanism predicted by the theory of l i m i t design i s realized before failure occurs i n the beam. Moments and deflections of the beam near failure are compared with theoretical predictions obtained from the theory of l i m i t design and the theory of inelastic bending. The la t t e r , developed by Dr. A. Hrennikoff i n 191Sj i s more "exact" than the theory of li m i t design. Test results showed that the beam failed at one of the early plastic hinges, before the mechanism was f u l l y developed. I t demonstrated that l i m i t design i s not always applicable to beams made of aluminum alloys which have very l i t t l e strain hardening. ACKNOWLEDGEMENT The author i s deeply indebted to Dr. A. Hrennikoff for his constant encouragement, inspiring guidance, criticism and supervision throughout the experimental work and the writing of this thesis. He also wishes to thank Mr. E.M. White for the help freely given during the tests. Research grants from the Department of C i v i l Engineering, the University of B r i t i s h Columbia, which made the author's studies i n Canada possible are gratefully acknowledged. i i TABLE OF CONTENTS Page INTRODUCTION - 1 CHAPTER I. THE THEORY OF LIMIT DESIGN 3 1. Stress Distribution under Flexure 4 2. Moment-Curvature Relation 5 3. Limit Design of St a t i c a l l y Indeterminate Structures 6 CHAPTER II. TEST ARRANGEMENT 8 1. General Arrangement of Test Beam 8 2. Supports and Loading Beam 9 3. Strain Measurements 9 4. Deflection 10 5. Buckling 10 6. I n i t i a l Assembly of Load Beam and Supports 12 7. Test Procedure 12 CHAPTER III. ADJUSTMENT OF STRAIN GAUGE READINGS AND COMPUTATION OF MOMENTS 14 1. Adjustment of Strain Gauge Readings 14 2. Computation of Moments 17 CHAPTER IV. TEST RESULTS * 18 1. Results of Beam Test No. 1 . 1 8 2. Results of Beam Test No. 2 ' 19 3. Summary of Test Results 20 4. Description of Failure Conditions 20 CHAPTER V. THEORETICAL ANALYSIS 23 1. Analysis by the Theory of Limit Design 23 2. Analysis by the Theory of Inelastic Bending 25 CONCLUSIONS 2 8 BIBLIOGRAPHY 31 APPENDIX 32 A - l . Relation between Load and End Reaction i n Loading 32 Beam of Testing Machine A-2. Determination of Modulus of E l a s t i c i t y 33 A-3, Elastic Solution of Test Beam . 3 6 A-4. Residual Stress Test 36 i i i LIST OF TABLES Page 1. Section and Mechanical Properties of the Material 38 2. Readings of S t r a i n Gauges and Deflections, Test No. 1 39 3. Readings of S t r a i n Gauges and Deflections, Test No. 2 40 4. Gauge Correction Factors and Moments, Test No. 1 41 5. Gauge Correction Factors and Moments, Test No. 2 42 6. Moments at Three C r i t i c a l Sections, Test No. 1 43 7. Moments at Three C r i t i c a l Sections, Test No. 2 44 8. Unit Functions f o r A l 65S-T6 Test Beam 45 9. Theoretical Predictions vs. Test Results 46 p 10. Ratio of g for Various Deflections 47 11. Readings of S t r a i n Gauges and Deflections, Simple Beam Test 48 12. Moduli of E l a s t i c i t y i n Simple Beam Test 49 i v LIST OF FIGURES Page 1. Stress-Strain Curve for Steel i n Tension 50 2. Idealized Stress-Strain Curve 50 3. Behaviour of the Beam under Various Stages of Bending 51 4. Geometrical Relation of Beam Moment and Curvature 52 5. Moment-Curvature of the Beam 52 6. General Arrangement of Test Beam 52 7. Additional M c due to Overhanging 53 8. Supports 54 9. Loading Beam and Loading Point 55 10. Preventation of Failure by Instability 56 11. The Spreaders 56 12. Compression Stress i n the Web 57 13. Spreader Acting as a Stiffener 57 14. Theoretical Buckling of Flanges 57 15. Inward Buckling of Flanges 57 16. Outward Buckling of Flanges 57 17. Desired Position of the Beam at Failure 58 18. I n i t i a l Assembly of the Test Beam 58 19. Adjustment of Strain Gauge Readings 59 20. P-€ Curves for a l l Gauges, Test No. 1 60 21. P-€ Curves for a l l Gauges, Test No. 2 61 22. Moment Diagrams for each P, Test No. 1 6 2 - 7 1 23. Moment Diagrams for each P, Test No. 2 72-83 24. Beam Load vs. Beam Moments, Test No, 1 84 25. Beam Load vs. Beam Moments, Test No. 2 85 26. Beam Load vs. Beam Deflection, Test No. 1 86 27. Beam Load vs. Beam Deflection, Test No. 2 87 28. Failure Conditions, Test No. 1 38 29. Failure Conditions, Test No. 2 89 30. Angle Changes of Test Beams After Failure 90 31. Indentations 91 32. Transmission of Forces at Failure 91 33. Compression at Tension Flange 91 34. Mohr's circles for Stress on Tension Flange at Failure 92 35. Key Locations of Moments . 9 2 36. Failure Mechanism of Test Beam 93 37. Deflection 6 C 93 38. Deflection and Angle Changes for Various Moment Diagrams 94 39. T r i a l Solution for Test Beam 95 40. Relation between Load and End Reaction of Loading Beam 96 41. Deflection of Simple Beam 97 42. Tension Test No. 1 98 43. Tension Test No. 2 99 44. Compression Test 100 45. Elastic Solution of. Test Beam 101 46. Residual Stress Test 102 INTRODUCTION 1 The theory of l i m i t design was f i r s t introduced more than forty years ago. In the last fifteen years, great progress has been made refining and enlarging the theory and i t s application. So far the theory have been mostly developed i n relation to flexural structures, that i s beams and r i g i d frames. Research has been mainly carried out on steel structures, and not much work has been published on the applicability of the theory of li m i t design to light metal alloys. Information i s lacking as to whether or not a structure made of some aluminum alloy with very l i t t l e strain hardening can f a i l by actual physical breaking at some of the earlier formed hinges, before the structure begins to act as a mechanism. Limit design has completely disre-garded the strain hardening property of the material and considered the duc-t i l i t y as the only characteristic of the beam contributing to the formation of a plastic hinge. The strain hardening property, however, i s i n fact necessary i n order to avoid an early failure of the beam. Without strain hardening the beam w i l l f a i l at values of load below those predicted by li m i t design. Most ductile metals, such as mild steel, are l i k e l y to have the neces-sary qualities, but some light metals alloys such as aluminum alloy 65S-T6 have very l i t t l e strain hardening and are probably unfit for limit design. One of the main purposes of this research is to attempt to prove that lig h t alloy structures are l i k e l y to f a i l at an early plastic hinge while the other hinges are s t i l l developing. A twice s t a t i c a l l y indeterminate beam of aluminum alloy 65S-T6 was set up for this purpose. Special devices were provided to prevent failure due to i n s t a b i l i t y . Moments were found by measuring normal fiber strain with electric resistance strain gauges (Budd foil-type) and then multiplying these strains by previously determined 2 section modulus and the modulus of e l a s t i c i t y . Readings of strain measure-ments were adjusted to eliminate most errors involved. A more exact theory of bending beyond the elastic range, developed by (2) Dr. A. Hrennikoff N , was also used to predict and analyze the problem. A second purpose of this research i s to test the above mentioned inelastic bending theory experimentally. \ CHAPTER I. THE THEORY OF LIMIT DESIGN 3 For many decades engineers have based their design and analysis of structures on the elastic theory. On the whole, i t has been proved satis-factory. Using Hooke,s Law as a basis of design, engineers have directed their attention only at the elastic behaviour of the material, and hence the elastic behaviour of the structure. Thus, the stress at any point in a structure, under any combination of loads should never exceed the elastic l i m i t . The fact that structures made of a ductile material such as steel can withstand stress much beyond the elastic limit has caused many to question the soundness of the concept that exceeding the elastic l i m i t stress at any point i n a structure i s equivalent to failure. Early i n 1920, N.C. Kist i n Holland f i r s t proposed the concept of limit design. Later, J.A. Van den B r o e k ^ i n the United States and J.F. B a k e r ^ i n England both advocated the theory of li m i t design. This theory, taking into account the "du c t i l i t y " of metals, explains the behaviour, not only of individual members but also of the entire structure when subjected to loads beyond the elastic l i m i t . Limit design, abandoning the unit stress as the criterion of design for steel members, has chosen the failure condition as the basis of design. I t promises to provide an adequate and uniform factor of safety on the basis of load rather than unit stress. I t assumes that failure takes place only when a sufficient number of sections of the structure reach the level of yield stress forming what are known as plastic hinges and transforming the struc-ture into a mechanism. As the structure becomes a mechanism, large deflec-tion occurs without further increase i n load. This i s the failure condition and the corresponding load intensity i s the failure load. The factor of safety may thus be defined as the ratio of failure load to working load or (1) where the working load, W^ , i s defined as the greatest load to be supported by the structure, while the failure load i s defined as the load of the same kind as the working load, but of such intensity a 3 to produce failure. The term failure i s used i n l i m i t design to signify a larger deformation. The logic of basing the theory of l i m i t design on a definite physical condition, the failure condition, and the absence of elaborate stress analysis are claimed to be the principal advantages of limi t design. 1. Stress Distribution under Flexure For a given beam, the moment and deformation limits for any stage of bending can be predicted from the known stress-strain curve of the material. A typical stress-strain curve for mild steel i n tension (Fig. l ) i s charac-terized by an i n i t i a l l y linear stress-strain relationship followed by a plastic region of considerable deformation with practically no increase i n stress, which i n turn i s followed by a region of strain hardening where increase i n deformation i s again accompanied by an increase i n stress. Limit design ignores the peak corresponding to the upper yield point and the strain hardening part of the stress-strain curve. I t assumes the plastic part, with the constant stress cr*, to extend indefinitely along the line C D. (Fig. 1). Thus the curve may be idealized as shown i n Fig. 2. During bending, the cross-sections of beams remain substantially plane irrespective of stresses. This results i n a linear distribution of strains across the section. I n i t i a l yielding at the extreme fibers occurs when the maximum stress, cr • ~, has just reached the yield point cr- . The cor-" y responding moment i s M - M. y cr Z y (2) As the moment i s increased to a value of M1, the strain i n the extreme fibers i s increased without increase i n stress, but a greater portion of the beam i s subjected to the yield stress. Increase i n moment results i n the spread of yielding u n t i l at some moment M , strain hardening starts to 0 Mr develop i n the extreme fibers. Further increase i n bending results i n rapid increase i n deflection. The beam may eventually f a i l by tensile rupture. The stresses corresponding to various stages of bending are shown i n Fig. 3. When the whole cross-section reaches the yield stress, as shown i n the limiting case i n Fig. 3, the corresponding moment i s Mp. Let the cross-sectional area of the beam be A and the distance of the center of gravity of one half of i t from the centroidal axis be y, then, by statics, M p - - A y o ^ - Zp0-y (3) where Z • A 7 « plastic section modulus (3a) From eqs. (2) and (3)> M Z _E „ _E M Z y (4) o i s called the shape factor of the section and i s the ratio of the bending moments M to M . P y 2. Moment-Curvature Relation Since l i m i t design assumes linear distribution of unit strain across a section of the beam during bending, the relation between moment and curva-ture may be derived from the geometrical considerations i n Fig. i+. Let h be the curvature produced by the bending moment M, e be the v It unit strain i n the outside fiber of the beam, and h be the depth of the section. Then Within the elastic l i m i t , relation (5) can be reduced to •5 " EI <6> Relation (5) can also be shown graphically as i n Fig. 5. Eq. (6) appears as the straight line OA. Beyond A the line curves sharply, approaching Mp asymptotically along AB. In the theory of limi t design, the curve OAB i s actually replaced by the broken straight line OFB. 3. Limit Design of Statically Indeterminate Structures There is no practical difference between limit and elastic design of sta t i c a l l y determinate structures, because i n s t a t i c a l l y determinate struc-tures, inelastic action does not modify the distribution of forces and moments i n the members. The limi t design of a st a t i c a l l y indeterminate structure, however, i s quite different from the elastic design. The basic design procedure i n lim i t design i s the calculation of the collapse load or failure load. Once the failure load i s known, i t can be divided by a suitable factor of safety and thus the allowable working load may be obtained. Determination of the value of the failure load i n a sta-t i c a l l y indeterminate structure, transformed into a mechanism, may be easily done by statics or by the principle of virtual work. In order to bring about the failure of a structure i t i s necessary to transform i t into a mechanism with an appropriate number of plastic hinges carrying the plastic moment M^ . Normally, the number of hinges i n the mechanism i s one greater than the number of static unknowns. This, however, i s only true i f the structure f a i l s as a whole, but i t i s also possible for a part of the structure to f a i l by forming a partial mechanism with a lesser number of hinges. Once the locations of plastic hinges are known, the failure load can be determined by statics or v i r t u a l work. For a structure of greater complexity, solution by the direct method may not be possible. Pro positions known as the static and kinematic principles are very useful i n handling these problems. CHAPTER II TEST ARRANGEMENT 8 In accordance with the theory of l i m i t design, a s t a t i c a l l y indeter-minate structure, under a gradually increasing load, f a i l s only when s u f f i -cient numbers of the sections form plastic hinges transforming the structure into a mechanism. This i s the failure condition which i s associated with large deflection. It was one of the main purposes of this investigation to see i f a structure made of a light metal such as aluminum alloy w i l l behave as visualized i n the theory of li m i t design. Two beam tests were carried out on 6 x 3 I-beams made of high strength aluminum alloy (Alcan 65S-T6, shape number 28008). Dimensions of the section and mechanical properties of the material are given i n Table 1. 1. General Arrangement of Test Beam The general arrangement of both tests were almost identical except that i n the f i r s t beam test, there was a length of 65" overhanging beyond the negative support E (Fig. 6a). The reason for this overhang was a desire to economize i n the use of material. The weight of the overhanging portion, of course, produced an additional negative moment, 1.75 Kip-in, at support D, Its effect on the moment at section C, however, was rather small (Fig. 7) i n comparison with those caused by the heavy test loads. The spacing of supports i n the tests was such as to permit a considerable amount of redistribution of moments to take place. It was desired to have the beam i n a nearly fixed condition at support B, without introducing excessively high shears i n parts AB and BC. At the same time, load point C was moved as far away as possible from the support B without defeating the purpose of the test. The beam"was twice s t a t i c a l l y indeterminate. As predicted by the theory of l i m i t design, the failure conditions i s a mechanism with three plastic hinges, which should occur under load point C and at supports B and D. 9 2. Supports and Loading Beam The test beams were supported by two inner rockers at B and D and assisted by two outer negative supports at A and E (Fig. 8a). The beams were tested i n a Tinius Olsen mechanical testing machine. In order that the test beam could f i t the te s t i n g machine, the loading beam of the te s t i n g machine was used. Details of these arrangements can be seen i n F i g . 8 and 9. At A and E a r o l l e r and a st e e l block, wide, resting on top of the flange of the beam, were used. Steel block, and 1 wide, respec-t i v e l y were placed on B and D. ( F i g . 8 (c) and (d) ) . Steel loading blocks were only wide at B and C, to reduce the length of p l a s t i c hinges. Shorter blocks would have caused crushing of the beam tested. Since no p l a s t i c hinge was expected at D, a s t e e l block, 1 J w wide, was used. Rockers and r o l l e r s at supports were used to prevent longitudinal components of reactions due to f r i c t i o n . No r o l l e r or rocker was present at point C, and hence the beam did not move l o n g i t u d i n a l l y with respect to the machine at t h i s point. At other points (A, B, D and E) the beam was not restrained l o n g i t u d i n a l l y and could move with respect to the supports. 3. S t r a i n Measurements Since the te s t beam was twice s t a t i c a l l y indeterminate, i t s moments i n p l a s t i c range could only be determined by s t r a i n measurements at each sec-t i o n . Extensive use was made of e l e c t r i c a l s t r a i n resistance gauges (Budd type C6-141B) i n determining the st r a i n s throughout the beam. S t r a i n gauges were placed symmetrically on the top and bottom of the flanges along the longitudinal centre l i n e of the beam. Locations of the gauges are shown i n Table 2 and Table 3. 10 4. Deflection Measurements of deflection were necessary to determine the ratio of the machine load, W, to the beam load, P, during the progress of the test. I t was also desirable to compare the actual deflection with those predicted by the theory of li m i t design and by the theory of inelastic bending. Federal dial gauges measured the deflections at the loading point, and at a distance of 1 f t . from the loading point. 5. B u c k l i n g Special measures (Fig. 10) had to be taken to prevent failure due to ins t a b i l i t y . These arrangements helped the equalization of moments i n the test beam and increased the failure load. No complexities i n stresses analy-sis nor i n moment computation were introduced by these special measures. The different purposes of these elements are explained as follows: (1) The spreaders (Fig. 10) made of alloy 65S-T6, 3/8" thick and of the same shape as shown i n Fig. 11 were placed on both sides of the web under loading point C and'over the inner support B. Two functions were provided by these spreaders: (a) As a web stiffener According to the A.I.S.C. Standard Specification, the compression stress i n the web of a steel beam, resulting from concentrated loads, shall not exceed f^ obtained as follows, for webs without stiffeners: (Fig. 12) (7) where P maximum interior load. a length of bearing. K distance from outer face of flange to web toe of the f i l l e t . t thickness of web. 11 2 For steel beams, the allowable f f e i s given as 24 K/in . Eq. (7) may also be used i n the test beam. Referring to Fig. 12 for the test beam, K = 1 3-" • t " I 1" a - 2 and as shown later i n Chapter V. P i s expected to be over 33 Kips, thus without the spreader acting as a stiffener, 12 * ^TS'JZ ' which i s rather high, and the web i s l i k e l y to cripple. With the spreaders, (Fig. 13), however, the effective bearing area, at section A-A. (Fig. 13b), i s increased to A - b t (8) s where b = 3", t - 3/8" for the test beam, s The compression stress i n the web at the web toe of the f i l l e t i s reduced to . -f •! - bV - ^  - 29^ K' in2 w The compression, P^, taken by the spreader i s distributed into the web by the f r i c t i o n a l force (Fig. 13c). (b) As a spreader As shown i n Fig. 14, the vertical component of the compressive stress tends to bend the flange inward. With spreaders i n position, the inward buckling i s thus prevented. (2) The bolt spreaders (Fig. 10) On both sides of the spreaders, the compression flange may also buckle inward for the same reason as i t would buckle inward i n the 12 absence of spreaders (Fig. 15). Bolt spreaders at these places thus prevent t h i s inward buckling. (3) The side frames (F i g . 10) As the load increases, however, v e r t i c a l shortening of the web immediately under the load, brought about by the load, creates a condi-t i o n as i l l u s t r a t e d i n F i g . 16 with the resultant tendency f o r the flanges to buckle outward. The side frames held the flanges against outward buckling. (4) A u x i l i a r y frames (Fig. 10) Additional frames were used, together with two blocks of s t e e l , to press the spreaders against the web of the beam. 6, I n i t i a l Assembly of Load Beam and Supports As predicted by the theory of i n e l a s t i c bending, the beam deflection under the loading point when f a i l u r e was reached, was about 2 inches. (See Chapter V). The desired locations of the te s t beam, at t h i s stage, i n r e l a -t i o n to the machine i s shown i n F i g , 17. The construction of the loading beam i s such that i t s two arms are equal when the beam i s horizontal, but are unequal when i t i s i n c l i n e d , and the r a t i o of the arms varies with the pro-gress of the t e s t as may be seen i n F i g . 17 and 18. I t i s desirable to have the loading beam horizontal when the test beam i s ready to f a i l . To achieve t h i s , the i n i t i a l assembly was made with the loading beam raised 2 inches at i t s l e f t end, and with the t e s t beam loaded at the th e o r e t i c a l point but d i s -placed bodily to the l e f t by the amount » 0.85". The rockers at B and D were also displaced a si m i l a r distance. The rocker supporting the load beam at the right end, L, was t i l t e d by a distance n 2 » 0.65". ( F i g . 18). 7. Test Procedure Both beam tests were carried out i n a si m i l a r manner. 13 Before proceeding with the test, the beam was checked for alignment. Screws on the tops of the outer negative supports A and E were tightened by hand so as to produce the least amount of negative moment at supports B and D. The strain gauges and dia l gauges were then set for zero readings. The sequence of loading i n the f i r s t beam test was i n approximate incre-ments of 5 Kips of beam load up to a failure load of 33 Kips. F u l l cognizance was taken of the creep effect, especially i n the inelastic range, and this may be seen i n Tables 2 and 3. Special care was taken to prevent the local buckling of flanges and crippling of the web at sections B and C under higher loads. The f i r s t beam test was halted at a beam load of 30 Kips after the f i r s t slight evi-dence of crippling of the compression flange under the loading point. The beam load was then reduced to 25 Kips and the test was stopped i n order to provide special measures, as mentioned earlier i n this chapter. The test was continued on the following day starting at a beam load of 25 Kips. At a l l stages of loading, readings of strain and deflection were recorded, except when the failure became imminent. At this moment, the readings of the strain gauges were incomplete due to the sudden rupture of the beam. The second beam test experienced less d i f f i c u l t y . The loadings were f i r s t taken at intervals of approximately 5 Kips i n beam load up to 25 Kips and then increased gradually u n t i l the beam f i n a l l y broke at 33.25 Kips. Readings of the strain gauges were improved by f i r s t taking the readings of key locations necessary for the determination of the beam moments. To accomplish t h i s , the sequence of readings was changed appropriately. 14 CHAPTER I I I . ADJUSTMENT OF STRAIN GAUGE READINGS AND COMPUTATION OF MOMENTS 1. Adjustment of S t r a i n Gauge Readings I t was mentioned early i n Chapter I I that the moments of the t e s t beams could only be found by s t r a i n measurements. Since a l l gauges were located at positions where only e l a s t i c moments occured, the moments at each gauges . lo c a t i o n (for locations of gauges see Table 2 and Table 3) could be computed by using the conventional e l a s t i c theory, thus M • eEZ (10) where e i s the true s t r a i n , E i s the modulus of e l a s t i c i t y which has been 2 3 found as 9,990 K/in and Z i s the section modulus, equal to 6.41 i n . The product of EZ i s a constant f o r the t e s t beam. The accuracy of the moment computed by Eq. (10) i s thus l a r g e l y dependent upon the value of c which was measured by the s t r a i n gauges. The manufacturer indicated that the possible range of scatter i n the gauge factors would be t \,%, Therefore, what indicator reads i s not true s t r a i n , e, but rather observed s t r a i n € . Hence the observed readings 6 should be corrected by multiplying them by a correction factor u. Eq. (10) then becomes M = u€EZ (11) The quantity u i s l i k e l y to remain constant for each gauge but w i l l be different f o r different gauges. As i s shown following; i t i s possible to f i n d i t s probable value by establishing a straight l i n e r e l ationship between load and s t r a i n when the beam i s s t i l l loaded within the e l a s t i c range. F i g . 19 shows the beam arrangement and the l o c a t i o n of the s t r a i n gauges i n general terms. The moments at the gauge locations i n the e l a s t i c regions are expressed by Eq. (11). When the load on the beam i s s t i l l very small, 15 the strains € and the corresponding moments M i n Eq. (11), may be not quite proportional, due to various minor irregularities of the beam shape or improper contact of the supports. However, as the load increases the pro-portionality between M and € i s improved. .In view of these circumstances the ratio ^ i n Eq. (11) should be replaced by the ratio |j| and thus the equation becomes A. End Spans AB and DE When the beam i s s t i l l i n the elastic range, for the end span DE, (fig. 19) M - K P d where K P i s the reaction of the support E, or AM » KAPd Eq. (12) now can be written as " • . # § . <»> For three pairs of gauges at distances d^, d^ and d^ from support E, >i - m\ I F .'. / 2 " ^\ E T - . ( 1 5 ) 3^ " ^\ l z ( l 6 ) Adding a l l n equations (three i n this case) we get Assuming a mean value of [i - 1, then 16 EZ n (18) Substituting Eq. (18) into Eqs. (14), (15) or (16) we get H i-1,2,3, (19) The value of K i s known by theoretical solution of the test beam i n AP the elastic range. The value of (—) i s found graphically by plotting load-strain diagrams for each gauge for the stage of loading in which the complete beam remains i n the elastic range. The span AB i s somewhat different to the span DE because there i s only one gauge i n the span and hence the correction factor can only be found from Eq. (14). B. Center Spans BC and CE Eq. (19) can be used for span CD except that d^ should be defined as the distance from the zero moment point,: found from the elastic solution of the test beam, (see A-3, i n Appendix) and K AP should be defined as the shear i n this part of the beam. Span BC i s again an exceptional case because not only i s one gauge present i n this span but also i t s location i s very close to the zero moment. The correction factor for this gauge i s taken as 1. This does not affect the moment diagrams drawn later. Since the beam i s a twice s t a t i c a l l y indeterminate structure, theoretically, the complete moment diagrams can be drawn on the basis of strains on the parts AB and DE, without regard for the parts BC and CD. \ 17 2. Computation of Moments After values of \i have been found for a l l gauges, by consideration of f u l l y elastic loading conditions, the same values can be used i n the inelastic range. A l l observed strains are multiplied by corresponding values of n for each P. By using Eq. ( U ) , M - n€EZ, converts 6 into moments for each gauge location. Once these moments have been found, they can be plotted for each cor-responding gauge locations. Straight line moment diagrams may be drawn so that they pass through A and E, where moments are equal to zero and through F and G (Fig. 19c). These may then be adjusted so that Pa Vi KL •» j- • ( S e e moment diagrams i n Fig. 22 and Fig. 23). The load-strain (P -€) diagrams for each gauge are given i n Fig. 20 for test No. 1 and i n Fig. 21 for test No. 2. Correction factors, u., for each gauge are presented i n Table 4 and 5. 18 CHAPTER IV. TEST RESULTS The results of the two beam tests were almost exactly the same. Both beams fail e d by rupture of the tension flange at the loading point C. Although, i n both tests, the section over the inner support B was not actually broken, considerable reduction of flange area was noticeable, i n d i -cating that the beam was ready to f a i l . Moment diagrams plotted later, showed that at the approach of failure, these two sections had developed almost the same amount of moment. However, the moment at the third c r i t i c a l section over the support D, at the time of failure, was far below that at the two other sections B and C. The maximum moments obtained for section D were 180.1 Kip-in for the f i r s t beam test and 178 Kip-in for the second. These values were less than 60% of those obtained for sections B and C. Of the data obtained from the two beam tests conducted i n this research, the following diagrams have been plotted i l l u s t r a t i n g the behaviour of the beam up to failure: (1) Summary of the test results: Beam load vs. beam moments at three c r i t i c a l sections B, C and D. (2) Beam load vs. beam deflections under the loading point C. 1. Results of Beam Test No. 1 Observed strain and deflection readings during test No. 1 are given i n Table 2. Locations of the strain gauges are also shown. The averages of the top and bottom gauge readings at the same section were taken for moment computation. Readings of strains i n those gauges with a gauge factor of 2 08 2.05 were multiplied by a factor of 2' oy si n c e the indicator gauge factor was set at 2.08. The effect of creep was considered when the beam was loaded up to 29.574 Kips by taking the readings of a l l gauges i n a complete 19 cycle twice while keeping the load constant. The beam failed at a load of 32.953 Kips with a deflection under the loading point C of 1.842 inches. Only readings of two gauge locations were taken while others were unobtain-able due to the sudden failure of the beam. These two locations are shown in Fig. 2 2j, as 7 and 8 . Mp was found without d i f f i c u l t y after and Mg were obtained by Eq. ( l l ) given i n Chapter I I I . To find Mg and Mg, an assumption that Mg equals M^  was necessary. This assumption i s just i f i e d by the physical condition over support D and by the fact that these two moments were very close to one another during the earlier stages of loading. 2. Results of Beam Test No. 2 The beam of test No. 2 behaved quite similarly to the beam of test No. 1. Loads were applied at the same increment up to a beam load of 24.57 Kips. Then i t was increased gradually u n t i l the beam fail e d at 33.25 Kips. The effect of creep on the readings of the strain gauges was also considered as i n test No. 1. ' Observed strain and deflection readings are given i n Table 3, For adjustment of the creep effect, readings of strain were reduced to a common time. As may be seen i n Table 3> strain readings for the gauge location 2 to 8 were reduced as i f they were measured at the same time as that of gauge location 1 . Although the readings of strain gauges were s t i l l incomplete as the beam fai l e d , the readings obtained from gauge locations 1, 2 , 3 and 4 suffice to solve the problem at this stage with the only disadvantage that the last set of readings could not be adjusted for any creep effect. Since the effect i s small, the error can be expected to be negligible. 20 3. Summary of Test Results As a summary of test results, tables and charts are provided showing the behaviour of the beams during each test. Table 6 gives the moments at three c r i t i c a l sections B, C and D for test No. 1; and Table 7 for test No. 2. Moment diagrams for intermediate loads are presented i n Fig. 22 and Fig. 23 for test No. 1 and test No. 2 respectively. The beam load versus moment at c r i t i c a l sections i s given i n Fig. 24 for test No. 1 and i n Fig. 25 for test No. 2. Finally, the beam load versus beam deflection i s given i n Fig. 26 and Fig. 27 for test No. 1 and No. 2 respectively. The extent of the equalization of moments under the load and over the inner supports can be easily visualized i n Fig. 24 and Fig. 25. 4. Description of Failure Conditions Both beams failed by the rupture at the tension flange under the loading point. Some reduction of flange area over the inner support B was also evident. These may be seen from the photographs shown i n Fig. 28, and Fig. 29. The angle changes at the two plastic hinges were later measured by a 12 i n . Starrett Protractor. Their values are shown i n Fig. 30. The tested beams exhibited compression dents at sections B and C as follows: dents were visible on the outside of the compression flanges at the contact with the bearing plates and on the inside of tension flanges at the contact with the spreaders. They are shown i n Fig. 31. The dent that occurred on top of the compression flange where the load was applied, needs no further explanation as i t was simply caused by the compressive force of the loads. At support D, the dent was caused by the reaction of support D. The cause of the dent which occurred at the contact surfaces of the tension flange and the spreaders can be explained i n the following manner. Fig. 32 shows the transmission of compression due to the applied load P. The compression was taken by the web alone during the early stage of loading. As the load P increased, a large part of i t was s t i l l taken by the web with another smaller part transmitted by the flange into the spreaders which in turn distributed i t into the web by f r i c t i o n . At this stage, v i r t u a l l y no force could be transmitted onto the lower flange on which the spreaders rested, (Fig. 32c). When later concentrated angle changes occurred at the points C and B, the transmission of force remained much the same except for a small portion carried by the spreaders into the tension flange to balance, passively, the transverse components of stresses i n the deflected flanges. The ultimate stress as found i n the tension test was 43.2 K/in . The largest angle changes occurred i n test No. 2 and were found to be 5°20', under loading point C, and 2°10* over the inner support B. The angle of 2°10» was chosen for the computation of the vertical component of tensile stress i n the flange (Fig. 30b) since i t i s closer to the actual angle just before beam failure. The angle of ^ ^O' under the loading point C, was, i n fact, the angle formed after the beam fa i l e d . Thus, the maximum vertical component of the stress i n the tension flange i s ; (Fig. 32b and Fig. 33) C - cr t W - 43.2 W t„ Sin 2°10« v s I <r' - — 43.2 Sin 2°10« .- 1.62 K/in 2 (t = t ) v t i s s This compressive stress was the only cause of the indentation of the tension flange where the spreader was seated. Acting perpendicularly to the stress i n the flange, this additional compressive stress certainly affected the stress condition as the beam approached failure. Its effect, however, i s minor as may be seen from the Mohr's circles drawn for pure tension and for tension and compression i n two perpendicular directions i n Fig. 34. I t i s evident, from Fig. 34(b), that the additional compressive stress has only slightly increased the maximum shearing stress. Thus, i t can be concluded that the passive compressive stress rj^, though causing indentation of the flange surface, had only a minor effect on the rupture of the beam flange. A tendency for outward buckling of the compression flanges was another feature of the behaviour of the beam at the approach of failure, but i t was effectively suppressed by the stabilizing side frames and was not a factor i n the failure of the beam. \ 23 CHAPTER V. THEORETICAL ANALYSIS 1. Analysis by the Theory of Limit Design A.. P l a s t i c Moment By d e f i n i t i o n , the p l a s t i c moment i s equal to Z p cr or M - Z cr (Eq. 3) P P y v 4 where Z^ i s the p l a s t i c section modulus and i s i n t h i s case found to be a constant equal to 7.524 i n (See Table l ) . Thus the value of M p depends upon the values of cr chosen. As may be seen i n Table 1, cr- was taken y y as 39.1 K/in as defined by a 0.2 per cent of f s e t i n the s t r e s s - s t r a i n curve of tension test No. 1. Thus M - Z cr - 7.524 (39.1) - 294.19 K i p - i n . p p y Since there i s no th e o r e t i c a l j u s t i f i c a t i o n to define cr as above, and since the value of M^ seems to be small, i t i s reasonable to use a higher value of cr l y i n g somewhere between the value of cr defined y y 2 2 above (39.1 K/in ) and the ultimate stress i n tension test No. 1 (43.2 K/in ), Taking cr - 42.5 K / i n 2 , we get, M p - 7.524(42.5) - 319.76 K i p - i n . B. Failure Load The structure i s a twice s t a t i c a l l y indeterminate beam with two redun-dant supports. The number of key moments i s three; they are at sections B, C and D. ( F i g . 35). The number of primary mechanisms are 3 - 2 •» 1, which i s the only possible mechanism with p l a s t i c hinges forming at B, C, and D and i s the only f a i l u r e mode of t h i s structure. -As shown i n F i g . 36 the f a i l u r e mechanism has three p l a s t i c hinges. I f one writes a v i r t u a l work equation f o r t h i s mechanism, one obtains, 2k p f 2k e - M p(e + e + f l e + ^e) r f n p  K12 w ' • r f 828 n p Substituting the M p value previously obtained, one has P f = ^ (319.76) - 36.9 Kips C. Deflection \ The deflection under load, 6 , as the beam approaches failure i s c found as follows. The moments at sections B, C and D just prior to the formation of the mechanism are a l l equal to Mp. But the stress condition everywhere else i n the beam i s elastic which j u s t i f i e s the application of the usual elastic equations. Referring to Fig. 37 the hinge at D i s known to be the last one to form. The deflection 6 • i s equal to c 6 c " & l + & 2 ' - 69 e1 + b2 But • »2 - i # M P • - W<y .-. 6 C - (18 x 69 • 793.5) - -^PHp 6„ - 3.39 inches \ i 25 2. Analysis by the Theory of Inelastic Bending For analytical purposes, functions ra, n, u and for the material must be known, A table of the above unit functions for the aluminum alloy 65S-T6 I-beam are presented i n Table 8, They have been obtained from the U.B.C. Master's thesis of Mr. D.E. Allen. Although the modulus of e l a s t i c i t y which was used i n preparing the above table (9,550 Kips/in ) is somewhat d i f -ferent from the one used i n the present research, the difference i s small and i s not expected to effect the results significantly. Formulas for the deflections and angle changes corresponding to various types of trapezoidal moment diagrams are summarized i n Fig. 38. These equa-tions form part of the equations derived by Dr. A. Hrennikoff i n his theory of inelastic bending and are particularly useful for this problem. The strain functions below the proportional l i m i t are determined by comparison with the values at the proportional l i m i t i n the following manner, When E - 9,550 K/in 2, K = 1.330 m - y + KeE - e E » 3.5.89 (IO*3) e K/in 2 (20) n - 4T22 e 2 E » 7.94 (103) e 2 K/in 2 (21) . 0 0 u - % ^ e 3 E 2 - 84.15 (106) £ 3 K 2 / i n 4 (22) O £. ( The solution of strains, moments and deflections, just before the beam f a i l s , requires estimating two moments or corresponding strains at once and solving the problem by t r i a l . Referring to Fig. 39, the solution of this problem consists of the following steps: Step 1, a maximum flange strain tQ » 88 (10 J ) i s assumed for section C. Assume the flange strain efi at B i s close to i t s maximum value; and the strain at D i s low, lying close to proportional l i m i t , where e •» 3.8 (10~3) (Fig. 39a) Step 2, Compute the deflections and angle changes for the four spans AB, BC, CD and DE and construct a l l four curves. (Fig. 39b). Step 3, Find the angles and a c at B and C. Since moment diagrams at B and C are flattened due to non-concentrated application of the load over the width , i t may be assumed that there exist constant moments over the length ^ " of inten-s i t i e s Mg and M^  with additional angle changes which may be computed by Eq. (d) i n Fig. 38. Step 4, F i t BC and CD together with the angle change a between — Q them, and t i l t them so that B and C are at the same lev e l . Step 5i Introduce angles o c c and ot^ so that B and C are s t i l l at the same lev e l . Step 6, Check to see i f . angle 9g - 6g' and 6 D - eD'. If they dis-agree, modify eg and and solve again. Two new t r i a l s may be needed before reaching a satisfactory solution. After that the result may be interpolated. (Fig. 39c). After successive t r i a l s following the above steps, a solution of e c - 88.0 (10"3) e B - 84.0 (10"3) and e Q - 3.3 (10"3) seemed to be very satisfactory (giving 6g - 239.3 jj x 10~ 3, 6g» - 239.2 i x 10" 3, 6 D - 119.1 jj x 10" 3, eD' - 118.8 i x 10"3) and are taken as the f i n a l solution. Values of the theoretical loads and moments just before failure are found without d i f f i c u l t y once the values of strain are known. From the results of t r i a l three, the maximum resisting moment corresponding to maxi-2 ' mum m_ (78.91 Kips/in ) occurred at load point C, and was equal to 27 1^ - ~ Awh (mc) « - | ( 1 . 4 2 2 ) ( 5 . 6 8 8 ) ( 7 8 . 9 1 ) - 319 Kip-in This value checks very closely v/ith the maximum test moment. Also, . . MB - | ( A W H ) (""B) - \ U.422) ( 5 . 6 8 8 ) ( 7 8 . 8 2 ) - 318 Kip-in which agrees well with the test moment obtained at B , For point D, where e Q i s found as 3 . 3 ( 1 0 '') the value of - 1 5 . 8 9 ( 1 0 3 ) e D = 52.4 K/in 2 (Eq. 20) and hence M D - § Awh (Dp) = ' 211 .5 Kip-in which i s greater than the value M ^ obtained from the test. The fialure load P f can be found by substituting value of Mfi, MQ and M^ into the expression of P " B + M C M C + M ' f 24 + 69 318 + 319 . 319 + 2 1 1 . 5 24 69 - 3 3 . 2 4 Kips. This value agrees perfectly with beam test No. 2 . The deflection 6 at failure i s found to be 1 . 8 2 inches which checks c closely with test No. 1 but i s a l i t t l e less than that obtained i n test No. 2 . The disagreement may be due to the shear and creep effects which are not taken into account by the theory. Thus, the theory agreed well with the experiment results. The values of the theoretical and experimental moments and deflections and capacity loads are presented i n Table 9. 28 CONCLUSIONS The assumption of l i m i t design that a redundant structure can only f a i l by the formation of a mechanism was not realized i n the beam tests. There i s a difference between the failure by rupture observed i n the test and the large deformation which constitutes failure i n l i m i t design. By the time failure occurred, the moments at the f i r s t two plastic hinges had equalized completely, but the moment at the third hinge was much less. The conclusion i s that i f the stress-strain curve of the material of the beam contains very l i t t l e strain hardening (for example aluminum alloy 655-T6) the equalization of moments before failure i s limited. The failure load predicted by the theory of limit design, taking o- = 39.1 Kips and y 2 providing a mechanism did occur, was 33.05 K/in , but, as mentioned before, 2 a more reasonable value of cr should be taken as 42.5,K/in . The predicted y failure load (36.9 Kips) i s higher than the test results (32.953 Kips for test No. 1 and 33.25 Kips for test No. 2). The deflection was predicted as 3.39 inches which i s rather large compared with the actual deflections of 1.842 inches and 2.00 inches obtained i n the tests. The reason i s that the mechanism i n connection with large deformation, was not f u l l y brought into action. More important i s the fact that the actual failure was entirely different from the li m i t design failure. The theory of inelastic bending predicted, however, that failure should occur at the tension flange. This f u l l y agreed with the actual condition at failure. The maximum moment occurred at the fractured section and i t s value agreed well with the value determined by the theory of inelastic bending. I t i s f e l t that due to the reduction of flange area at the loading point C, the resisting moment at this point might have been somewhat smaller than that at the support B whose section had not quite reached the state of f a i l u r e . Unfortunately, the strain readings at various sections as the beam was ready to f a i l were not obtained. The suspicion thus remains unverified. The theoretically obtained values of moments at the three sections agree with the test results with the exception of the moment over the support B which came out some 20$ higher than the experimental value. The failure load pre-dicted by the theory of inelastic bending as 33.24 Kips agreed exactly with the result of test No. 2 (33.25 Kips) and was a l i t t l e higher than that of test No. 1. The deflection was predicted as 1.82 inches which agrees closely with the observed deflection of test No. 1 (1.842 inches)and i s smaller than that of test No. 2 (2.00 inches). The theory of inelastic bending does not provide any prediction when the flange strain e exceeds the value correspon-ding to a maximum value of m. The two ^ " blocks used under the load point C and over the inner support B helped to equalize the moments. Spreaders, bolts and frames were used to prevent the failure of the beam due to i n s t a b i l i t y . There were some faults i n test No. 1. The arrangement, or more speci-f i c a l l y , the sequence of reading the strain gauges was not well planned beforehand. Test No. 1 was stopped overnight i n order to make some additional modifications to prevent the crippling of the compression flange. The incipient tendency for crippling was noticed early, and effective measures were taken to stop i t . Test No. 2 proceeded without any d i f f i c u l t y . The following conclusions may be drawn from the tests: 1. The results apply to aluminum alloy 65S-T6 which possesses l i t t l e strain hardening. The tests were arranged unfavorably to the theory of li m i t design. In other tests, the equalization of moments may very well materialize. 2. Strain hardening plays an essential part i n the equalization of moments at plastic hinges. 30 3. The theory of inelastic bending appears to describe more accurately than the theory of l i m i t design, the behaviour of the aluminum alloy members i n bending beyond the elastic range. 31 BIBLIOGRAPHY (1) Baker, Horne and Heyman, "Steel Skeleton", Vol. I I , Cambridge University-Press, 1956. (2) Hrennikoff, A., "Theory of Inelastic Bending with Reference to Limit Design", Trans. ASCE, Vol. 113, pp. 213, 1948. (3) Van den Broek, J.A., "Limit Design", John Wiley and Sons, Inc., 1948. 32 APPENDIX A-l Relation between Load and End Reaction i n Loading Beam of Testing Machine The arrangement of the tests of the aluminum alloy beams was such that the load had to be applied 30 inches off center of the testing machine. This necessitated the use of a special peice of equipment i n the testing machine, the loading beam, whose diagram i s shown i n Fig. 40. The load P applied to the aluminum beam was not the f u l l machine load W, but the reaction of the loading beam P which was equal to ^/2 when the loading beam was horizontal, but varied somewhat from this value when the loading beam was inclined. In the course of testing the end A of the loading beam did not move vertically, while the other end, resting on the aluminum beam moved down following the deflection of the beam. The purpose of the following derivation i s the determination of the ratio p i n terms of the deflection of the end of the loading beam. As shown i n Fig. 40, the loading beam was i n i t i a l l y raised up 2 inches at the l e f t end. To determine the relation between the machine load, W, and the load on the beam, P, when the deflection at C equals 6 , one can take moments about A of the load beam, and obtain 2 6 2*6 2*»»6 p ( 3 0 ^ d 1 + 3 0 - - ^ d 2 ) = W ( 3 0 . - - ^ d 2 ) 2-5 30 — 2TA~ ( d J 60 2_ ( w ) 2-6 c 6 0 + - 6 i r ( V d 2 } 1800 - 19.5 (2-5 ) 36,00 + 6 (2-6,) CW) (23) c \ 33 Eq. (23) was used to determine the beam loads when deflection, 6 , and c machine load, W, were known, as shown i n Table 10. A-2 Determination of Modulus of El a s t i c i t y The modulus of el a s t i c i t y was determined i n several ways. Beside that of tension and compression' tests, a simple beam test was carried out to check the modulus of ela s t i c i t y . a. Simple Beam Test (Fig. 41) The beam of 15 f t . span was simply supported at the ends and loaded i n the testing machine with a single concentrated load at the center. Budd Type C6-1A1B strain guages were symmetrically located, as shown i n Table 11. The moments i n the beam were determined by the relations M = eEZ where e i s the flange strain measured by strain gauges, Z i s the section modulus of P the beam and E i s the modulus of ela s t i c i t y ; also M «» — d^ where P i s the load and d^ i s the distance of the gauge from the nearest support. Since P and d^ are known, E can be easily found as follows: i , P d. T. M 1 eZ 2 eZor AP d In addition to the moment measurements, deflections at the center of the span and at the quarter points were also measured and used for determina-tion of the modulus of el a s t i c i t y . Center deflection caused by moments i s , 6 » P ^ 3 (25) m 48 EI Additional deflection caused by shear is given approximately, . K ' To <3> t ( 2 6 ) 34 where A, the web area, equals the product of the width of the web and the height of the beam, V i s the shear force, P/2, and G i s the modulus of shear. Thus the to t a l deflection at the center of the span i s then equal to P ! ^ P 9 " ' 6m * \ " B T B * TTo <27> E Since G « 2(1 + '^ ) * w n e r e V* Poisson's ratio, i s given as 0.33 by Alcan i n "The Strength of Aluminum", Eq. (27) becomes, 48 EI, 1 + 12 I (2.66) A 2 (28) with --Jt - 15(12) = 180 i n , I = 19.23 i n 4 and A = 0.25(6) =1.5 i n 2 , 6 = 6318 I (1 + 0.0126) - 6398 I Therefore E = 6,398 I p Due to various minor irregularities of the test, the — ratio was not quite constant i n the early stage of loading, and thus i t is desirable to replace •| i n the above equation by — E - 6,398 1| (29) The resultant experimental data are presented i n Table 12. 2 The value of E obtained from the strain readings, E = 9,942 K/in 3 and from the deflections, E d - 9,989 K/in 2 differs s l i g h t l y . In the following derivation the mean of these values w i l l be used. E - I (E s • E d) - \ (9,942 + 9,989) - 9,965 K/in 2 b. Simple Tension and Compression Tests Two tension and compression tests were also carried out for the deter-mination of the modulus of elas t i c i t y . Rectangular specimens of gauge length 2", cut from the web and flange of the beam were used for tension tests. The machine used was the 60,000 lbs Baldwin Universal hydraulic testing machine. During the early stage of loading, strains were measured by two Phillip's PR 9814 strain gauges, symmetrically located on both sides of specimens. Later when the gauges stopped registering at the strain of about 1 " 0.5%, direct measurement by spring calipers sensitive to was used. The stress-strain curves are presented i n the Figs. 42.and 43. As may be seen from the stress-strain curves the material taken from the flange speci-men exhibits some properties similar to those of steel. They are both per-fectly elastic up to a stress of 32.0 K/in and a strain of approximately 0.32 per cent. But when stressed past the elastic l i m i t , the material behaved differently from steel. The material has no upper or lower yield point as i s clearly evident i n the steel. The beginning of yielding may be defined as the stress at 0.2 per cent offset. Another distinction from struc tural steel is the virtual absence of strain hardening. The absence of strain hardening, as mentioned before, i s considered to be a decisive factor i n materialization of lim i t design. A compression test was conducted on a 12" length of the same I-beam again i n the Baldwin hydraulic testing machine. Strain measurements were obtained both by 16 Phi l l i p ' s PR 9814 gauges, arranged as shown i n Fig. 44 and also by the dial gauge mounted on the cross head of testing machine. The compression test i s less reliable because: (1) There i s no ultimate strength i n compression, and the specimen f a i l s by crippling soon after the proportional l i m i t i s reached. (2) The unstable character of loading tends to produce early crippling failure. (3) The dial gauge gave readings greater than expected due to crushing V I of the ends and supporting pads. The modulus of e l a s t i c i t y of 9,990 K/in i n tension test No. 1 was chosen since the specimen was from the flange and the beam fai l e d by rupture of the tension flange. A-3 Elastic Solution of Test Beam The elastic analysis of the test beam was carried out by a d i g i t a l com-puter, using the standard program for solution of planar structures, pre-pared by Dr. S. Tezcan. No detailed explanations about this analysis are necessary here. The results were obtained for a beam load of 10 Kips. A l l moments, shears and deflections are shown i n Fig. 45. The shear effect was included i n the program and the deflection obtained was somewhat less than that observed i n the test. However, the calculated moment over support B was greater than the test moment, the moment at C smaller, and the moment at D agreed closely. The above mentioned discrepancies i n moments and deflections are of l i t t l e importance, since the solution i s only used for the adjustment of strain gauge readings as described i n Chapter III. A-4 Residual Stress Test ' This test, unrelated to the main subject of the thesis, was performed in the interests of general knowledge concerning the residual stress i n extruded structural shapes. A section of one of 6" I beams used i n the above described research was u t i l i z e d i n this test. The particular section used came from a piece of the beam far from regions which had been deformed plast i c a l l y . Ten Phillip's PR 9814 strain gauges were used. Their locations are shown i n Fig. 46. The beam was later cut by an electric saw at a dis-tance of 3/8" from the center line of the gauges. New strain readings were taken and compared with those before cutting. The distribution of the stresses was found to be rather irregular around the beam section. The maximum compression stress was found to be 0.619 K/in at gauge 2 and the 2 maximum tension stress, 0.689 K/in at gauge 4 which were small when com-pared to those due to bending and can be considered as negligible. 38 a. Dimensions of Beam Cross-Section m I \—t. "A A" IP d b t f f A Iv - 6.000 i n . » 3.000 i n . • 0.250 i n . - 0.312 i n . - 0.375 i n . - 3.920 in.' - 19.23 i n J d - t. 5.688 i n . Aw • h *w " 1 , 4 2 2 in*' Af - h (A - A j A f P J r h / 2 Y , < p 0.959 i n . * - 1.348, Use K - 1.330 4 6 . U i n . .'dA - 7.524 i n . a - -f - 1.175 b. Mechanical Properties of Alcan 65S-T6 Alloy (From Tension Test No. l ) Modulus of E l a s t i c i t y Proportional Limit Yield Stress (0.2$ off set) Ultimate Stress Strain at Proportional Limit Strain at Maximum Stress % Elongation at Failure 9,990 K i p s / i n / 32.0 Kips/in. 2 39.1 Kips/in. 2 43.2 Kips/in. 2 0.00371 0.08 14$ Table 1 - Section and Mechanical Properties of the Material 7 7 ' 2 4 " P gr 6 9 " 5 4 ' 15* T B 7 C. 5 S ' (5) ( V S" Of) <S> 14" 39 g . f . : gauge f a c t o r t : t o p gauge r e a d i n g b : b o t t o m gauge r e a d i n g Z e r o l o a d r e a d i n g s : 4000 x 1 0 " ^ i n / i n I n d i c a t o r gauge f a c t o r s e t t i n g : 2 . 0 8 T i m e L o a d o n M a c h i n e W ( l b s ) ! L o a d ( T ) g . f . : 2.05 @ g . f . : 2 . 0 5 ( J ) g . f . : 2 . 0 8 (4) g . f . : 2 . 0 8 (T) g . f . : 2 . 0 8 ( 6 ) g . f . : 2 . 0 8 ( 7 ) g . f . : 2 . 0 5 (?)g.f.:2.08 D e f l e c t i o n 6 ( i n ) o n Beam P ( l b s ) t b A v e . e t j b A v e . 1 i e 1 : t b 1Ave. __.JJ£_ t b A v e . € t b f W e . e t 1 b 1 A v e . i j _ e t A v e . € t i b l A v e . 1 IO :Z8 /<0,000 5 6 6 3 4 3BI 4o4Sj 3 9 A 4 -42 11 3 7SSJ 4 / 2 2 ; 3 8 7 4 3 7 39; 4 2 o 9 , 3 7 16 J 4 2 7 o j 3772 ! 4 2 3 3 | 3S4-2! 4 1 6 o | a / 4 5 4.884- , 3 52 331 356 o 4 f i l O I6 0 - 5 2 a n ; 2 12 122 126 2 o l | <?o9j 2 8 6 J 27o! 223J 2 3 S ! 2 3 3 I5&! i (So| 3 6 1 1 ^ 0 3 2 ! 2)2 1 2 4 ! 1205 I 1 2 7 S ! i 2 3 6 i I - I S 9 JO: 5 6 2O.000 4-7S<S. 4.0 4-2 3 9 9 0 4 4 2 0 3 5 7 4 4 2 4 1 3755 • 3 6 o s i 4 4 o 3 ! 3 4 4 o | 4 5 3 3 i 3 5 3 0 [ 4 4 7 l ! 3 6 S 6 ; 4 3 \ 4 ' 0 . 2 5 4 9.7& 2 1 7 9 0 7661 7<S6 O i o ! o 2 6> 4 2 o I 426 241 24 5 3 9 s ! 4-03! 5 6 o ! 533i 4 -60J 4711 4 6 6 314 ; 3I4-! j 8 0 0 I o2<g, . ] 4 -23 2 4 3 ! 1 3 9 9 | . i 54 - 7 .! ! 4 7 2 314 10:45 3Or OOO /4>694 ! 28o& 5 1 SO j 40SO 3 9 9 6 _ o 4 2 _ o 4 3 4 6 2 7 1 3-565 4-3 59 3 6 3 6 =54-13; SS7 ; 4 -593: 3 I 6 S ; 4 7 9 I ! 3 50©; 4 6 9 6 3 5 3 2 . 44<S9: O . J 6 8 1 192 11 j 11-3 1 o f l o .oo4 6>27! 6 3 5 .... £.31 3 58 364 1 _593J_ _ . 1 5 9 0 <532 791 i 700; 696^ _.6SS I I JI20.3 36.2 1 ! S 1 2 •1 I 1 ! ! 1 4 6 9 /o:so 40,000 24 OO 5 5 9 7 i 4 0 7 6 3 9 9 4 4 8 3 5 : 3 1 5 3 44 -75 3 5 ) 3 3 2 1 8 ' 47S5--I 1 ! 26^4! 5055S 3 3 7 6 ; 4^620! I 6 O O ! o 4 1 S 3 5 i S 4 7 4 -75 4 37 7S5! 1 1 Q6 lo55J S 4 5 l 9 2 5 i 9 3 5 6 2 4 i <£2o! 162.1 0 4 2 t £ 4 1 4 6 1 • 1 7 3 4 1 M 0 8 O 1 1 9 4 7 ! ! 6 2 2 lo:S9 50,000 2 4 565 ] 2 0 0 4 ; I S 9 6 5 9 9 4 4 j o 9 l o g o 4 9 ; , 3 3 9 0 ; 1 3000; 2 5 9 0 J 5 3 4 S 2 S o o ; 5 1 S 3 j 1192 3 2 o 3 4 7 9 5 I 5 5 S Zo2c o_7_9. O S O 1 0 5 9 1 070 1 6 I O 1 o o o ; | o o 6 0 0 3 14 IO fl 3 4 9 1 200 ] 1 133 • 797^ 7 9 5 7 9 6 1 0 6 5 i <So& - 1 I ! 3 SO ! ;i?-io II: / 7 &0,000 2 9.S74-)69>o 6 3 0 0 2 300 4 2 3 ) 231 3 5 1 2 ISA 2 10 5 2 4 1 2 7 4 3 12 4-9 4 6 7 5 C7& 3 314 6*6 2 6&I 1 SI?) 5 3 2 0 1 3.2Q 1 3 2 0 2 176 I 6 2 4 5 7 3 3 1 7 33 2-441! 5225 1542 2 9 7 5 5020 C.790 124 I j l 2 57 1 5 5 9 15 25 | o 2 5 ! 2 3 5 5 2 1 3 . j - _ j I7AI 1 5 6 4 | o 2 3 l l :zo 60.000 2S,GOO 1695 <S>2 97 4 2 32 3 6 IO 5 2 4 I 1 2 4 1 2 7 4 5 4 6 7 5 3 3 1 5 2 6 3 0 1 32 ° 5 3 2 3 21.74 5 7 4 0 24-f-O 5 526 29TO I_030 j o 2 4 li>2 7 0.36G _Z_3£5 2 2 97 2 3?. 1 9 o I .25S 6 7 5! 6 5 5 1 3 2 2 J S 2 . 6 1740 17 , 3 5 1 526 15-43 1 2 3 3 3 1 1 2 4 8 i ' . & T h e f o l l o w i n g r e a d i n g s w e r e t a k e n o n t h e second I d a y o f t e s t 14-: AO 5Q.OOO 2.4,-735 2fo,2 5" 7 4 9 4&OS 2 7 J O 1 2 9 0 2p4«3 4 3 7 5 6 7 5 4 3 3 9 3 6 3 0 zs\s 1 5 8 5 5" 3 5 a 2 0 6 0 5 3 2 5 2 I 5 5 J 5 553: 2 9 3 o 5 o 6 5 1 /./SG 1 749 J 5 2 I &os 3 7 0 1 3 5 3 1 9 4 0 1 5 2 5 l<945'!| 553I I 7 9 A I o 7 o i | o !S2 - i 8 7 6 3 5 4 14-72 I S S 2 1 ! 1822 I 0 6 S /4:54- <<>o,ooo 29.-7Z4 1717 4 6 o 3 ; 2 6 - 9 0 X" .S03; 1 3 1 0 i ! _ J _ ! i|?.4 1 I4-0 SO&5 2931 4 4 4 c > 3 5 3 5 4 5 5 2 3 I O I C 9 0 5 5 6 3 1 5 63 I & 2 6 IS . 0 0 2 2 0 0 2 0 7 8 19.14. 2 0 S 6 5 7 7 4 1 7 7 4 I&27 2 7 3 1 1 5 2 c 8 1213 ! / 2 7 3 -2±3.5i22_o9 | o 6 7 ,_44p_465 1 V2 \ ^  1.1 1 1 21-39 14:51 6,0,000 2 3,770 17 15 6 1 2 9 • 1 22 17 4-6© ij &.?A.l AL? 5©64j 2 9 3 0 1 0 d 10 7 0 4 4 4 o ' 3 5 35 44Q : 4 6 5 •-- - 2 310 .1.690 5 5 6 3 J . 5 . 6 3 1 6 2 6 Go7§ 2c575 IOI 5 | 5 7 7 4 Z7S'i ' 5 2 IO 2 I2<5 1 1 2.4. 1 140 205.5' \JJ_AjiS2 j S ! 1352 1.22.45 i ! l o G 7 2.I4C j 12-14-IS:o<2> &&,ooo 32,953 . _ . : — i j_S6S_Ul63 2 4 3 2 | 2 | 6 3 2 -533 ' 5 4 6 2 /.84Z |4_6_l 14 62 . 1 T i ^ - 14 £-2 T a b l e 2 - R e a d i n g s o f S t r a i n G a u g e s a n d D e f l e c t i o n s , T e s t N o . 1 2 7 " V 31 2 4" & F * ' " " " c. t 2 " 1 2 " ; 2" — . — 2 1 " — — — —, 2 0 " 1 4 " 5 4 * ~7J 7 " 26" 22* AO g.f.: gauge factor t • top gauge reading b : bottom gauge reading Zero load readings: 4000 x 10^ in/i n Indicator gauge factor setting: 2,08 Time 09140 09:45 Load on Machine W (lbs) / o,ooo 20.000 Load on Beam P (lbs) 4.884 9.782 © 1 )g.f.:2.05 b Ave, 3 & | 2 j 4 3 8 1 j ,,388; . . S 8 l ! _S»? ! 3 9 0 3 2 13,4.780 737; . 780_ 784 ! 795 @ g.f.:2.05 4 o 3 4 o 3 4 4035 P ? 5 3970 O S O . 3 9 7 0 O 3 0 , Ave, ..032 p 3 2 .OS.2. 0 3 2 © g . f . : 2 . 08 3 8 3 8 6 2 3_685 S I 5 b 'Ave, € Ay s o 160 5 1 5 Si? (T)g.f.:2.05 4413 4-13 4_82o S 2 0 b Ave, e 3575 . -4-25 .3155 ! 8 4 5 -4.19 426 a 33 3 4 5 g.f. t:2.05 b:2.08 4 o 9 5 q 9 5 0 9 6 Ave. 3 8 7 7 1 2 3 \ I O 4 220 .22? 2 2 3 3 7 5 5 2 4 5 2 3 4 @g.f.:2.05 3 9 4 3 . 0 5 7 b \Ave, 4-/00 |OOJ 0 7 9 ! OSO 3 8 IO 190 4 j _ 9 5 i 1 9 5 , /92 ( 9 5 08 b Ave, 3 S O O 200 4-20O ZOO ZOO 3 6 o 3 ..•3.97 4392 -392.1 ! 3 9 5 © k . f . t:2.05 b:2.08 b Ave. - € ;S70;4-I2 5 I 3 0 132 1 2 5 123 5 7 4 6 4-2 5Q 25 4 ^ 2 5 S : 2 5 0 2 5 4 Deflection & c (in) C/3S 0.2S4-09:52 30,000 14.634 zsoa 5.185; I 1 9 2 t 1 3 5 ; 11.8? II20I I 4 0 3 0 , 3 9 7 0 0301 O 3 0 3528 44-75 o 3 Q 4-72 4 7 5 5.225 .1.225 2 7 38 I 2 6 2 4 7 4 1244 1261 43.40 _.3_40 3 4 5 3 6 3 8 3 6 2 3 5 3 3 7 2 0 280 43<3Qj .—3001_290 I 2 9 5 3 4 I O S 9 0 45,3? 5 8 5 5 6 2 1 4 3 7 Q 1 379: ! 5 3 7 3 3 4 : 3 7 0 i 3 7 7 0.371 09158 40,000 2 3 9 5 5 5 9 0 4-020;3365 I S o S . I 590\ l_59S ! ) 6 ? Q -5-370 J02S. 0 2 a 6 3 0 4 6 3 0 5 6 3 0 2 = 5 1 5 __6J?P X f e 3 P . I 6 8 5 6 3 0 ./6SO I 6 8 2 4 4 5 5 4 5 5 3 511 3 4 6 2 3 6 2 0 __s.eo. 4821 4 7 2 44-431 4 4 3 ! 4-12 7 3 0 4-fB 4 7 S O 3 4 9 5 4 4 9 5 ) 7 3 0 5»5: 7 3 0 5 7 a : 4-95! 5^3 0.484 10:05 SO, 000 24570 2 o | o I 99o to :oy 50.000 24510 Reduced for Creep Effect ? o 10 I 9 9 0 £J97pj I 9 7 0 i 1 9 3 0 1 2 0 I O 4 o 5 3 3 9 1 5 1 -r 3 1 9 5 4 S o l .. Q.53 0 8 5 j _ o 69. 1 ! Q 7 Q 8 o 5 _apj 6 0 6 0 2 0 6 0 9 7 0 461Q 3 3 9 0 2o30 2 o 4 5 6 I O .3515 4 S 4 Q 4 8 5 Z 9 9 S a«>3 2 0 7 5 6 i s 5 9 7 1 I 9 7 1 I 930 2oio 2 o l o 4054 ,o54 .3913 087 0 7 0 3 L ? 5 8o5_ 071 070 8 0 3 &03 6lO< CI4 2 o 6 0 1 9 6 3 a o B 2 2 o 4 6 4593 593 3 3 3 3 2 o 7 5 6 ° 7 6 o 7 2Y-75 3 5 I S 435 6 " 7 6<>7 54o: 5 1 2  5 ) 9 1 0 0 5 3J36o1 A65o\ I o o o j < 6 4 0 ' 1 oo2 6 4 9 ; 6 S C 6 4 0 4 5 4 0 5 4 0 -5/2 .573. 373 2 9 9 4 lOCMo 4 9 9 6 _ 9 9 6 tool 3-3.60 6 4 0 6 4 9 4 6 3 1 . 6 3 1 J & 4 0 ] 6 4 0 0.624 0.62.5 to :i5 to: u 55,000 27.011 55,000 27.O80 Reduced for Creep Effect IO: 25 10:2l 6 0.000 G 0,000 23,574 23.580 Reduced for Creep Effect 1 9 4 3 2057 6 1 3 2 41 o 4 1543. 2 1 3 2 2 ° 3 5 J o 4 I 5 2 2 1 1 5 J i Z 8 _ I 3 O 5 Q 9 4 9 o 6 A3PA. 9o4-1 9 4 5 6 ( 3 0 4 l l o 3 8 4 4 2 ? 55, .21.3 o .20.83 2 M Z ) IO I 5 6 9 o 7 4?_o| 9 o l 2 H 2 I 34 S 2 8 2 2258 9o5 6276 2276 16 34 2 3 6 6 1 ^ 3 7 4653 2 3 2 7 CS£> 2 3 6 o C 6 3 904 2 3 6 3 2 3 2 0 2 3 5 6 9o4 46.56. 6 5 6 2 S 5 9 ? r : 3 2 ' 2 3 o O 1 6 7 0 * 6 2 2 2 •2316. 2 3 5 0 4J5.3__3775 1 5 3 ; 2 a s 2 9 5 9 J.S9. .192 4 I 5 ^ : 3 7 7 1 2 5201 2 2 9 2 2 3 0 6 2 3 4 0 2 3 4 0 .» 5 6 . 229 192 195 194 L9-4.L 2 5 5 5 1 o 4 5 5TQ4Q J.54<2 5 £ 4 0 t o 4©: 6 4 7 7 | 4 2 3 I 0 4 0 2 4 7 7 2 5 7 7 64-73; | 4 2 3 2 4 7 S 1 2 5 7 7 -JA2L42 2 5 2 7 2 5 2 7 2 5 6 5 2 5 6 5 4 6 C 2 466.S 6 7 4 ; s 4 o 6 6 0 3340 6 6 0 3326 6 7 4 6 7 4 1445 555 4-6o7 _<&57 664 _ 6 6 3 6 & 3 3445 5 5 5 4.6.IO 6 IO, 33l4i475l & 7 2 6 3 6 3314 6 7 4 6 7 3 75" I -4754' 6 6 6 75.4 .58? „5S2 _5?0 539 2 3 61 j 5J2_b ( * I [ 3 9 i I I 2 5 3 ? Z 5 7 2 5 4-7 IO, I I 3 2 2 3 6 I.' 5 1 2 6 | I I 3 9 ' ) I 2 6 J i 735. 3 2 7 5 T i p ) . 7 2 5 H_ ii 4 7 1 3 1 ) 3 2 2664-J 7 J 8 7 2 8 I S 3 6 2<S63 _.7J9 -.725. 7 2 8 ( 3 3 7 ' t i I I 3 2 . _725. 7 3 5 J7I3;|_7_2-1. ! 723 5 3 2 Q J.32.P 5 321 / 321 J/56 3 2 a _ 8 4 4 _8_5<i 3 1 6 5 4-619 _-.<3l? S^S 4<S\9 8 35 1 329 I 3 2 8 347 819 8 3 3 3 3 7 0.7/1 0.7 IS 0,375 O.S84 I o:43 IO'AS 62.O0O 62,000 30.591 * 50.600 Reduced for Creep Effect |589j.(o.36S'. 24-1 I '2"2£-S I 5°>-1 :<,V570 2 3 7 0 2 388 2 4 3 2 2.3S8 -?4^.2 2 4 3 2 >|53 153 4-.!£4 I 6 4 3 7 7 3_ 2 2 7 3 7 6 1 2 3 9 J 9 0 1 9 3 39l 2oA 2 8 0 6 J J ° 4 28.S.6 1114. i j r / 2 o ! ! ! 2 C 6 5 3 5 I 3 7 4 -5 1 2 2 I 1 2 2 2 o 3 1112 I 1 t£> 2 5 3 5 \ 2 6 2 6 6 5 3 1 1 3 7 4 2 5 3 l ' 2 6 z 6 253.1 2 6 2 0 462.9 6 J 9 6 3 3 ! 6 3 6 3 7 2 5 7 9 2 6 2 0 4 6 3 6 6 4 6 3 3 6 1 6 3 9 2620 _3242 7 5 8 4 8 5 0 3 5 0 2 5 6 o 6 3 3 -5.°-4 a/6 1 4 3 Q 144-) 3 2 3 5 _7-65. 4 3 4 8 2.555 _<34_8 14-45 6 4 2 <Si8 6 4 0 S I 7 54-41 3 1 1 4 4 3 3 5 a&& 4 3 5 8 9 8 8 3 5 8 9 2 5 4 4 5 31 i o 4 8 8 5 . 1443 8 9 Q 1 4 4 4 9 o 2 8 8 5 S94 I 4 3 T 8 9 2 0.974 0.9S8 /o:5Q 11 :oo il'J2 11:15 63.500 63fo00 <h5,O00 65000 31, 32G> fs..1-4..! <s449 M 8 & : 2 4 4 9 2 4 6 1 2 5 I O 4.I.60 | 6 0 5JL49 2 5 1 2 0 5 . 2 7 7 7 I 2 2 3 5 2 2 6 I - 2 S 6 6 5 4 4 ; | 3 4 6 2 5 4 4 z o a 1225 3 / , 4 3 0 52.273 32.2S2 Reduced for Creep Effect 1/125 11130 ze.ooo &(o.000 32328 32.835 15/a 6 4 3 1 2 4 1 2 2 4 3 1 1 .4-3916490 2 -50, \ . 2 4 9 0 M±P . . tS459 4 1 5 9 2421. 2 4 5 5 .2525 2 5 6 2 5 2 £ 2 5 6 | 2 5 6 390. ?6IO Reduced for Creep Effect / / : 55 66,500 33.250 S5.30 2 5 3 0 1 3 9 I [ 6 5 2 8 2 6 0 9 I 3 G 4 2 6 3 6 2 5 2 8 6 5 4 9 2 5 4 9 2 5 7 p 2 6 o 5 2 S 6 9 2<So5 2tf.or 2 6 S O I S 3 4145 1 4 5 4 1 4 5 1 4 5 37-1£. 2 5 4 2 6 6 3 7 3 2 265 2 0 6 2 o 9 _2o. 2of i 2«5 2 o 9 4 1 3 3 . I 3 3 41: 1 3 3 4J2 . I 1 2 1 3 7 3 4 2 6 6 •3733 2 6 7 3735 2 6 5 2 0 0 2 o 3 2o3 1 9 6 2 7 7 Q 5 2 3 0 6336 I 2 3 0 I 2 3 0 2536 I 2 S O g629 ] 3 V 1 1.363. 2612J IS7! .53.72. I 3 7 S 2 5 9 0 I 4 I O 2 5 6 8 14-32 5"4/o I 4-10 54-31 1 4 3 .2525 14751 . 5475 1 4 7 5 6 5 2 4 2524 I 3 7 0 6 5 / 9 2519 2 6 5 4 2 5 9 9 2 6 3 5 4551 5SI SS3 [3 50 2 6 5 0 2 6 3 7 f 3 7 0 1 3 7 4 1 3 7 2 2 6 3 0 & 5 4 9 .2549 \4 IO 6 5 2 6 2 5 2 6 1 4 3 142(2. 6 5 / 6 25/<S 1475 1341 2 6 5 9 I 3 5 G 2 6 4 4 34-37 ;o36 Solo] 314 ) o t o S<S3 5 6 / 4 5 4 3 2 5 9 3 2 6 3 0 258_5 2 6 2 1 2575 26 IO 26,13 2 6 ° 4 2 6 4 0 2585 2 6 2 0 2 6 2 7 5 4 3 5 5 7 4 4 3 3 4 3 J 4 3 3 45_a6 3 3 6 394 4401 4 0 I 4 0 7 4 3 3 3 3 S 8 3 4 3 3 4 4 3 5 5 7 3_5<2S 4 3 5 -3.568 4 3 2 3 5 8 6 4 14 3 6 ) 2 3 S 3 .I3J34 2 6 o 6 2 5 6 1 2 6 00 3 o f l o 5oo8 9 2 0 IOO8 5 5 7 2 8 8 2 1118 436 -S8.7&. I 1 2 2 413 4 2 5 2 8 2 6 I I 7 4 4 I O 2 7 2 0 I 2 J O 5 2 2 0 (120 5 2 J 9 12 19 9 6 2 -23-.TQ 16 3 0 5162,4 3 o l 8 4 9 7 ) 1 6 2 4 9 3 2 9 7 6 1 6 2 3 3 3 6 9 7 1 2 3 6 6 964-- 0 7 7 ) 1 6 9 I 185 117 1 11 8 7 1 1 8 5 5 2 7 0 I 2 7 0 3 6 5 3 9 5 5308 I 3 0 8 L222. I 2 4 0 [ 2 3 9 1 2 7 8 1254 5-g SO 1 6 3 4 . 21 2 9 1871 2121 1 8 7 3 2 o 5 2 1 9 4 8 2 0 I O I 9 9 0 ) 6 3 Q 5 8 7 0 1 8 7 0 5 8 7 4 3 0 13 6 3 2 2 9 Q I (870 1 8 7 4 1 8 7 6 1 8 7 2 5"944 I 9 4 4 193! 1 9 4 6 1 9 5 7 9 5 4 4 9 7 1 S 3 7 o o l 9 7 ) l o 9 9 1 1 1 5 2 9 Q O I 1 16 50SO 1080 5 0 8 2 I O S 2 9 S 6 I0S7 |o99 l o 8 9 2 3 6 0 I 1 4 0 I 1 5 5 2 3 4 1 1 1_5;9 I 1 74 3 1 ? 0 1 I 20 5 1 3 9 ! 3 9 I 1 3 3 I 1 5 6 I I40 /. 190 1.208 1.451 1.472 I. 511 J.G08 2 . 0 0 0 Table 3 - Readings of Strain Gauges and Deflections, Test No. 2 41 • LOCLJ PUbs.) (2)-^4 "/.oo (3)^/3 =1.02 \©Mt = 0.35 liC^Vs- ~/.o4 u e, M, Mz Ms WIGI\ M7\U3€a Me 4,884 377 245 32 2.1 2 /6 14.1 117 7.6 2/2 I3.£> 2 7 6 18.0 223 14.5 153 JO.O 9.7 a 2 80O 54.4 26 -£52 28.1 231 15.0 4/3 26.9 542 35.3 445 29.0, 3o2 /a<s /4. 6 9 4 - 124,2 32.0 4-3 2.3 644 41.8 343 22.3 | <bll 38.5 &o<b 52.4 43.5\ 452 2 9 4 I9,G20 /695 tlo.3 42 2.3 358 54.8 455 29.6 | a/2 52.8 69./ <996 53.s\ j 598 39.0 24,565 Zl II 137.4 80 5.2 /OS 8 7o.8 574 37.4 lo3S C7.& / 3 6 6 59.0 /152 1 75.o\ 7 6 5 1 49.8 29 5 74 2440 158.8 2/3 /3.9 /2 75 83.0 42.0 /368 89.0 Z 7 6 5 - 1/4.8 14-75 II if 96.oS 984 C4.I 29,&oo 2433 I58.<b 2/3 /3.9 /275 83.0 41.9 /37Q 69.2 1768 //5.I 147 G J 96.o\ 988 <S4.3 24-,S10 1 1931 !25.<o IOS2 4,8.5 8 9 7 53.2 335 21.8 IS25 98.3 A3 6 6 121.3 1720 1 //2.o\/o26 1 (06.8 23.724 2340 /S2.3 II40 74.2 I090 70.9 4 2 9 233 J682 I09.S 2I20 Z37.9 1749 1/3.8 \//<S& jl ' 7'6.0 29.7 IO 2245 /S9.0 I/40 74.2 /090 70.9 429 23.9 1682 /OS.S 2122 /36.0 1749 //3.S J//69 76.0 32,953 2/39 1 142.81/4 08 9/.G © (D CD 0 © © © © € .=> Micro inches Per inch EZ=* 9. 9SOC6.4/) — <S 4yO 35: 9 K -in M =,yU€EZ G4,035.9 *-jn (£y.//) Table k - Gauge Correction Factors and Moments, Test No. 1 42 •Load (2)^4 ©<J3*/.o2 '(4)lU=/.o5 ©Ms =/.O4 =<2<37 (Z)l/7 = /.oC = CL38 Piths.) Mi Mz 4J3€S Ms Ms M-, Ms •• 4,384 424 27.2 32 2.1 I&3 • I05 446 23.6 /I4 7.04 69 5.1 213 I2.& / 2 6 8.2 9-7SZ | <966 3 2 2.1 320 20.5 884 56.1 24-2 A5-.5 14,8 IO.& 420 ' 26.9 249 /S.9 /4;634-J / 5 0 9 /.9 482 303 I320 84.6 365 23.4 254 Z6.3 6 2 4 40.0 370 23.7 / 7 6 2 1/2.8 2>a /. 7 4,39 4o3 1760 //2.8 488 31.2 36 1 23.1 33o £3.2 4 3 4 31.1 24.510 2/92 /40.3 70 4.5 316 52.3 2172 /3t>.0 Q35 40.7 4 4 a 28.7 106& £9.0 £23 40.2 zipao 147-2 9/3 538 2435 /S9.0 C86, 4J.d 5 0 7 324 /2o5 77.2 7/1 45.6 29530 j i 1 \2S4B\/63,Z 1 i /Q4 /2.4 /OSS 677 2685 /72.0 <S97 44.7 G27 4o.2 /4/2 So.S 322 52.7 30666 1 i /C9.3 2o3 / 3.o //33 725 2742 17 5.8 66"2 4Z3 703 45.0 1530 98.0 876 56.1 31430 | p675 17/.2 2o9 /3.4 /249 8O.0 2750 Z76.2 576 36.8 342 53.3 /738 /o/3 9 S3 62.o .32,232 2790 '76,7 2o8- A3,3 A ? 9 4 89.3 2733 I7S.O 4sa 28.1 /020 6<£>.0 l99o Z27.5 /070 £8.5 ' 32,835 2839 Id 1.8 203, /3.0 /-#47 94.G 2649 /&9.& 4o3 26.2 loao 69.2 2o32 /33.3 1120 7/-& 33,250 2S&5 I90.O I9<6 A?. 6 / 4 9 6 9S.8 2721 /T4.4 © <D © © © © CD.® €. = /4 Zero inches / ' e r Inch £ Z = 9.930 (6.41) = <Z4,o35. 9 K-in M = >ci€E Z = 64,035.9M e /<-/" /') Table 5 -Gauge Correction Factors and Moments, Test No. 2 43 p (lbs) K-in V K-in K-in Remarks 4,884 45 49 20 9,782 95 95 39 14,694 140 144 56 19,620 190 191 76 24,565 237 240 96 29,574 • 274 293 124 29,600 274 293 124 1 24,735 222 244 130 Test continued on the second day 29,724 273 294 144 29,770 275 .295 146 32,953 310 310 180.1 Computated values, see Results, Test 1. P • 1 A J B C Table 6 - Moments at Three C r i t i c a l Sections, Test No. 1 0 p Mc -(lbs) K-in K-in K-in Remarks 42 50 21 9,782 95 95 38 14,694 144 145 56 19,620 195 196 78 24,570 231 237 95 27,080. 252 270 106 29,580 280 297 128 30,600 282 307 137 31,430 300 311 140 32,282 308 311 160 32,835 311 .318 177 33,250 312 318 178 •A ' j~B~ C [~D E ' Table 7 - Moments at Three C r i t i c a l Sections, Test No. 2 3 . 8 60.3^ 114-72 4618 1.10 3.9 6lo76 120.03 4942 1.13 4.0 62.73 123-69 5182 1.19 4.2 64.31 130*37 5594 1.24 4.4 65 .33 134,-94 5890 1,32 4.6 66.25 138.89 6150 1.41 4.8 .66.6-6* 141.83 6346 lo46 5.0 67.53 145.02 6560 lo50 5.3 66.31 .149.01 6631 1*59 5.6 66.94 152.49 7070 1Z72 6.0 69.57 .156.12 7321 1.98 7.0 70.60 162 . 6 0 7790 2.17 8.0 71.33 166.26 8177 2.43 10.0 72.25 176.54 8772 2.70 12.0 72.79 182.51 9205 2.62 14,0 73.26 183.59 9649' Ml 16.0 73 . 51 192 .36 9926 3.98 16.0 73.64 194.52 10064 3.57 20.0 73.87 198.87 10405 2.72 22.0 74.06 203.33 10735 2.33 24.0 74.35 209.53 11195 2.03 26.0 74.61 216.11 11665 1.90 26.0 74.87 223.25 12219 1.80 30.0 75.13 230.72 12779 1.84 32.0 75.32 23606I 13222 1.76 36.0 75.76 251.52 14348 1.76 kO.O 76.05 262.67 15195 1.72 44.0. 76.41 277.42 16319 1.52 48.0 76.82 296.57 17787 1.46 52.0 77.17 313.92 19122 1.45 56.0'' 77.51 332.50 20559 1.41 60.0 77.84 351.51 22035 1.44 64.0 78.09 366.69 23219 1.66 66.0 78.19 323.55 23755 1.75 72.0 78.37 385.85 24718 1.56 76.0. 78.54 396.55 25715 1.55 60.0' 78.69 410.75 26674 , 1.59 64.0 78.62 420.99 27480 1.71 68.0 78.91 429.17 28125 2.10 Table 8 - Unit Functions for A l 65S-T6 Beams 46 Quantities Theory of Limit Design Inelastic Bending Theory Test Results Test No. 1 Test No. 2 Moment at B (Kip-in) 319.76 318 310 312 Moment at C (Kip-in) 319.76 319 310 318 Moment at D (Kip-in) 319.76 211.5 180.1 178 Failure Load P F (Kips) • 36.9 33.24 32.953 33.25 Deflection 6 c (in) 3.39 1.82 , 1.842 2.00 P E : t 21" B 2 4 " "c • •• • ' 6 9 " 9 3 " . D. 5-4" Table 9 - Theoretical Prediction vs* Test Results 47 Test P W 1800 - 19.5(2 - 6 ) C 3600 + 6(2 - 6 ) c (Eq. 23) Test No. 1 0.142 0.4884 0.254 0.4891 0.368 0.4898 0.486 0.4905 0.619 0.4913 0.866 0.4929 0.624 0.4914 1.156 • 0.4947 1.273 0.4954 1.700 0.4984 1.842 0.4990 Test No. 2 0.135 0.4884 0.254 0.4891 0.371 0.4898 0.484 0.4905 0.624 0.49H 0.711 0.4922 0.875 0.4929 0.974 0.4935 1.190 0,4949 1.457 0.4966 1.511 0.4969 1.608 0.4975 1.630 0.4976 2.000 0.5000 Table 10 - Ratio of ^ for Various Deflections 48 a. Strain Gauge Readings I 3 9 n 1 4 6 io i2 _ ' •S"7 63' 63" <54" • .94 ' JS'-joad (lbs) Gauges 1,000 2,000 3,000 Ave. Ae/1000 lbs 1 , 11 { 486 914 425 464 1318 444 2 / .12 { Ae 482 931 451 456 1387 454 3. 532 486 1018 1515 497 493 4 , . Id { Ae 530 1030 500 498 1528 499 707 651 1358 657 2015 654 8 { £ l-Ae 710 662 1372 665 2037 664 b. Readings of Deflections .ipad (lbs) Positions -500 1000 1500 2000 2500 3000 Center of span 0.360 0.694 1.015 1.335 1.653 1.975 45" from l e f t support 0.250 0.471 0.695 0.911 1.137 1.353 45" from right support 0.250 0.475 0.699 0.917 1.139 1.359 Table 11 - Readings of Strain Gauges and Deflections, Simple Beam Test 49 a. Modulus of E l a s t i c i t y by Strain Readings Gauges AP d Eq. (24): E - K / i n % (Z - 6.41 iir*) 1 a 11 3 a 9 5 a 7 iMfo (1°6) = 10>ou wMk (1°6) - 10'092 1 2 ^ 4 7 ( 1 ° 6 ) " 1 0 ' ° 1 8 E(Comp.) - 10,041 2 a 12 4 a 10 6 a 8 12 ^ 5 4 1 ( 1 ° 6 ) " 9 ' 7 9 3 1 ^ 9 1 ' ^ - 9 ' 8 6 7 wMui (1°6) " 9'867 E(Ten.) " 9 ' 8 A 2 - 3 E s - * <E(Comp.) * E(Ten)> " 9 ' 9 ^ b. Modulus of E l a s t i c i t y by Deflection Readings AP a A& Eq. (29): E - 6398 || K/in 2 When P increased from 1 Kip - 2 Kips A6 » 1.335 - 0.694 - 0.641 i n . When P increased from 2 Kips - 3 Kips A6 - 1.975 - 1.335 = 0.640 i n . E i 6 3 9 8 oia " 9 ' 9 8 1 . E2 e 6 3 9 8 oOT - 9' 9 9 7 ... •'. E d - \ (E x • E 2) - 9,989 K/in 2. Table 12 - Moduli of E l a s t i c i t y i n Simple Beam Test 50 51 Strain Distribution Stress Distribution Cross Section 2 JL 2 Fully Elastic A h h h rT Plastic Elastic J ~y Plastic M<MPI M = Mp (Limiting Case.) JEfr^ct of Sfrion Entirely Plastic (Idealized) Fig. 3 - Behavior of the Beam under Various Stages of Bending t # . M p *L _ 0 Bl . - M y - - M p 52 Fig. 4 - Geometrical Relation of Beam Moment and Curvature Fig. 5 - Moment-Curvature of the Beam 2 ' - 3 " 2 ' - o 6 2 ' - 3 " 5'- 9" A'-Q," 5'- 5' 7'- 9" A-' - <2>' (a) Theoretical Position of the Test Beam (b) General View Fig. 6 - General Arrangement of Test Beam 53 Fig. 7 - Additional Mc due to Overhanging \ Atyefot. Support t-oaj;^ /><,,„/ l&} ^LoaJinj Beam /tourer Base. of Test 5h Negative. Support" inj Machine. I vv//V////////////////''///>;s////////////'/////////' (a) Supports,and Loading Beam /Softer - Y2 Block: -Tes / Beam. Station G-Q — Base, of Test/ny . Mac/tine. -"=7 Base of 7e sting Moc-hine. (b) Outer Negative Supports A e E A s s / - ^ e 4 -1 p - ' ^ ' Direct ion Of Mac/iine Base. (c) Inner Support B MacA/ne. Base. (d) Inner Support D Fig. 8 - Supports (b) Loading Point C Fig. 9 - Loading Beam and Loading Point 1 t [VI 1111 tlUDUt 'Z 2. (a) (b) Fig. 12 - Compression Stress i n the Web Spreacle.r~ ( a ) Fig. 13 - Spreader Acting as a Stiffener P i (a) (b) Fig. 14 - Theoretical Buckling of Flanges I (o) B o l t Spreaders - Spreader Fig. 15 - Inward Buckling Fig. 16 - Outward Buckling 58 <t o-f Testing Machine 3 0 " 3 0 " B A = ^ 1 2 7 * AAachin*. Bo.se 2 4 * 69* 93" 5 4 " F i g . 17 - Desired Position of the Beam at Failure J 2 ' - </.-2 7 " 2>4* 69' 93" S4-" Fig. 18 - I n i t i a l Assembly of the Test Beam \ 59 P B1 (a) 0, C c 0. t> ' o >) E l ast Lc .J Solution D , a K B j C L — -(c); Final Moment Diagram Fig. 19 - Adjustment of Strain Gauge Readings 61 20 in IOOO 4 - ( £ t + 6 b ) l O f c i n / (a) Gauges 1, Span AB 20 in Q. 10 _ 2 0 0 0 \ in o o 50 100 ^ ( e t + e b ) i o 6 in/;n (b) Gauges 2, Span BC 20 in C L IO 6 " 5 20 a. 10 s o o IOOO (c) Gauges A, 5, 6 a 7, Span CD soo 1 0 0 0 (d) Gauges 3 » S, Span DE 6 7 3 6 i F i g . 21 - P- 6 Curves for a l l Gauges, Test No. 2 Moments (K - hn) 0 ro ro § 3 c* O • P-CO H <D CO c r as o /oo 50 c vi to - p C! 0) e o 2: 50 /OO F i g . 22(b)-Moment Diagram, P - 9.784 K i p s , Test No. 1 A isa too So c I m c § . T O /OO 150 F i g . 22(c) - Moment Diagram, P - 14.6% K i p s , T e s t No. 1 Moments (K - i n ) V / l A , 300 200 /Od /OO 200 300 \ t F i g . 22(e). - Moment Diagram, P. = 24.565 K i p s , T e s t N o . 1 F i g . 22(f) - Moment D i a g r a m , 'P - 24.570 K i p s , Tes t No. 1 t A 300 2oc /oo 100 Zoo soo t F i g . 22(g) - Moment Diagram, P - 29.574 Kips, Test No. 1 I c i F i g . 22(h) - Moment Diagram, P - 29.724 K i p s , T e s t No. 1 i 300 200 /oo 6 I » 03 g 5B 200 500 1 Fig. 22(i) - Moment Diagram, P • .29.77.0 Kips, Test No. 1 300 200 too o /oo 500 Fig. 22(J) - Moment Diagram, P. - .32,953 Kips, Test. No. 1 \ Fig. 23(a) - Moment Diagram, P - 4.884 Kips, Test No. 2 F i g . 23(b) - Moment Diagram, P - 9.782 Kips, Test No, 2 Fig. 23(c) - Moment Diagram, P - 1 4 . 6 % Kips, Test No. 2 F i g . 23(d) - Moment Diagram, P » 19.620 Kips, Test No. 2 Fig. 23(e) - Moment Diagram, P - 2A.570 Kips, Test No. 2 Moments (K-in) m Fig. 23(g) - Moment Diagram, P - 29.580 Kips, Test No. 2 F ig . 23(h) - Moment Diagram, P - 30.600 Kips, Test No. 2 F i g . 23(i) - Moment Diagram, P - 31.A30 Kips, Test No. 2 300 200 _ 100 _ 0 I0O _ •20OL 300 \ F i g . 23(j) - Moment. Diagram, P - 32.282 Kips, Test No. 2 Moments (K-in) 03 -P c CO 6 o Fig. 23(1) - Moment Diagram, P - 33.250 Kips, Test No. 2 100 200 300 Moments (Kip-in) Fig. 24 - Beam Load vs. Beam Moments, Test No. 1 30 o 20 10 0 86 Fig. 26 - Beam Load vs. Beam Deflection, Test No. 1 87 Fig. 27 - Beam Load vs. Beam Deflection, Test No. 2 (a) Kinks under Load Point C ( l e f t ) and at Inner Support B \ Loading Point (b) Test No. 2 Fig . 30 - Angle Changes of Test Beams after Failure 91 r Jer.1 4 See (b) Fig. 31 - Indentation |0Ti CTv N \ <Tu) "5/ p Spreaders TTT -Transmittecl •bl--fricTi'on Direct . transmission (b) (c) Fig. 32 - Transmission of Forces at Failure IH-c-o;tsw -n/i/nn. Mill,A Fig. 33 - Compression at Tension Flange 92 Fig. 35 - Key Locations of Moments Fig. 38 - Deflection and Angle Changes for Various Moment Diagrams 3 0 " US 30' 96 4=25.5* h^f—p P beam d, = !9.S' "base, of machine (a) P - When Loading Beam Horizontal d>*2S.s Z>-2 (b) Initial Raise of End B, D - 2 in -—: 30* A 2-Se. d,=25.5' <9 2-Sc , 2-Sc i 30+-so-'* ' dti/S£» dz-z/S.f" I. 2-&d 60 (c) During Test, When D - 2 - 6 Fig. 40 - Relation Between Load and End Reaction of Loading Beam Fig. 41 - Deflection of Simple Beam 2J4J 98 2J-/4-L. 12' 0 Dimensions of Spsc/rnen (not in scale.) Specimen: \" plain bar from flange of 6" I-beam Duration of Test: 1 1/4 hrs. Date: Aug. 15, 1963 (B) 0.1 Strain (Per Cent) Fig. 42 - Tension Test No. 1 2 '/A* 2 ^ / • Mi ' 1 1 Gauge. ' ' L :.. J /2" Dimensions of Spw'rnen (not in Sca/v.) Specimen: . p l a i n bar from web of I-beam ' Duration of time: 1 1 / 6 hrs. ! Date: Aug. 1 5 , 1 9 6 3 • Strain (Per Cent) Fig. 4 3 - Tension Test No. 2 uo <<P 30 c m co V u -p CO 20 10 100 11 ij 11 l l [j 'si : I 11 11 11 M 11 3* ..Dimensions of Specimen & Gauge . /)rranq&me.nt Specimen: section of 6" I-beam, 12 inches long Duration of Test: 2 3A hrs. Date: Aug. 14, 1963 0.2 a 3 0.4 Strain (Per Cent) 0.5 0.6 0.7 Fig. 44- Compression Test I :0 P = IO kips 3.64 *3&4 Shear art/ D ^ £ S<L = O./93" Deflection Fig* 45 - Elastic Solution of Test Beam 102 U - 3/6 " Ma-xirnum Tension 5frees /Compression Stress i 6 & i io 9 IZ" Fig. 46 - Residual Stress Test 

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