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Effect of shear and transverse compression on deflection of beams in elastic range Dhanju, Kulwant Singh 1963

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EFFECT OF SHEAR AND TRANSVERSE COMPRESSION ON DEFLECTION OF BEAMS IN ELASTIC RANGE by KULWANT SINGH DHANJU B. Tech. (Hons), Indian I n s t i t u t e of Technology, Kharagpur ( i 9 6 0 ) t A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF M. A. Sc.. in the Department of CIVIL ENGINEERING We accept t h i s t h e s i s as conforming to the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA September, I963 In presenting 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'requirements f o r 'an advanced degree at the U n i v e r s i t y • o f B r i t i s h Columbia, 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 reference and study. I further agree that per-mission for extensive copying of t h i s t h e s i s for s c h o l a r l y purposes'may be granted"by the Head of my Department or by h i s representatives,. I t i s understood that copying, or p u b l i -c a t i o n of t h i s t h e s i s for f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n permission. Department of The U n i v e r s i t y of B r i t i s h Columbia,. Vancouver 8, Canada. Date &f: I f , / <7 O . . ABSTRACT This t h e s i s presents a refinement of the c o n v e n t i o n a l theory of sheer def.lection of beams i n the e l a s t i c range. The author's theory allows f o r d i s c o n t i n u i t y of sheer d i s -t o r t i o n s of s e c t i o n s i n the v i c i n i t y of v e r t i c a l loads and a l s o f o r the e f f e c t of v e r t i c a l compression of the beam at the p o i n t of a p p l i c a t i o n of the v e r t i c a l l o a d . The w r i t e r does not i n v e s t i g a t e normal and shearing s t r e s s e s but deals only with the e f f e c t s c o n s i d e r e d as they i n f l u e n c e d e f l e c -t i o n . The theory was t e s t e d e x p e r i m e n t a l l y on an aluminum a l l o y I-beam and a r e c t a n g u l a r s t e e l beam. The beam was r e s t e d on three supports and the l o a d was a p p l i e d at two p o i n t s symmetrical to the c e n t r a l support, producing a s t a t i c a l l y indeterminate c o n d i t i o n . Due to smallness of sheer d e f l e c t i o n s , the comparison between the t h e o r i e s was made i n terms of s t a t i c a l l y unknown r e a c t i o n s r a t h e r than d e f l e c t i o n s themselves. These r e a c t i o n s were c a l c u l a t e d from s t r a i n s determined e x p e r i m e n t a l l y . The r e a c t i o n s com-puted i n conformity with the two t h e o r i e s were then compared with the ones found from the e l e c t r i c s t r a i n gage readings. Attempt was made to f i n d e x p e r i m e n t a l l y the e f f e c t of the p r o x i m i t y of the l o a d on the s t r a i n gage r e a d i n g s . v i i i ACKNOWLEDGEMENT The author wishes to express h i s s i n c e r e a p p r e c i a t i o n to Dr . A . H r e n n i k o f f f o r h i s i n s p i r i n g guidance and encourage-ment g iven throughout the p e r i o d of s tudy . Thanks are a l so extended to the t e c h n i c i a n s of the Department f o r t h e i r h e l p i n s e t t i n g up the exper iment . F i n a n c i a l support from the N a t i o n a l Research C o u n c i l of Canada in the form of a s t u d e n t s h i p i s g r a t e f u l l y acknow-l e d g e d . I l l Chapter I II I I I IV TABLE OF CONTENTS INTRODUCTION FORMULATION OF THE THEORY Shear deformations in the web Shear deformations i n the f l a n g e s Deformations produced by t r a n s v e r s e compression I n c o m p a t i b i l i t y of deformations between the web and the top flange . D i s c o n t i n u i t y i n the t r a n s i t i o n zone i D e f l e c t i o n due to angular d i s c o n t i n u i t y C o n ventional theory of sheer d e f l e c t i o n of beams EXPERIMENTAL VERIFICATION OF THE THEORY Apparatus and procedure R e s u l t s CORRECTION FACTORS FOR THE GAGE READINGS Theory DETERMINATION OF MODULUS OF ELASTICITY (a) From s t r a i n gage readings (b) From the l o a d - d e f l e c t i o n curve EXPERIMENT FOR DETERMINING CORRECTION FACTORS FOR THE GAGE READINGS AND THE MODULUS OF ELASTICITY Apparatus and procedure Results DERIVATION OF FORMULAS FOR THE TEST BEAMS Page 1 3 3 5 9 10 15 16 18 18 2 0 23 23 26 26 26 29 29 3 0 32 Chapter Page VI CONCLUSIONS 3$ TABLES 36 DIAGRAMS 42 GRAPHS 49 1 V LIST OF TABLES Table Page ( l a ) S t r a i n s i n aluminum a l l o y I-beam 3 ° . ( l b ) S t r a i n s i n r e c t a n g u l a r s t e e l beam 37 ( 2 a ) End r e a c t i o n s (aluminum a l l o y I-beam) 38 ( 2 b ) End r e a c t i o n s ( r e c t a n g u l a r s t e e l beam) 38 ( 3 a ) Average c o r r e c t e d s t r a i n s i n microinches per inch f o r AP = 7 5 0 0 l b s . determined by t e s t on I-beam compared with theo-r e t i c a l values 39 ( 3 b ) Average c o r r e c t e d s t r a i n s i n microinches per inch f o r AP = 1 5 0 0 0 l b s . determined by t e s t on r e c t a n g u l a r beam compared with t h e o r e t i c a l values 39 ( 4 a ) Measured s t r a i n s i n microinches per inch f o r d i f f e r e n t values of c e n t r a l l o a d (aluminum a l l o y I-beam) 4 0 ( 4 b ) Measured s t r a i n s i n microinches per inch f o r d i f f e r e n t values of c e n t r a l load ( r e c t a n g u l a r s t e e l beam) 4 0 ( 5 ) C o r r e c t i o n f a c t o r s f o r the gage readings 41 L I S T OF GRAPHS Page r i g s . (19) Comp a r i s o n o f e x p e r i m e n t a l and t h e o r e t i c a l v a l u e s of t h e r e a c t i o n R^i (a) R e c t a n g u l a r s t e e l beam 49 (b) Aluminum a l l o y I-beam 50 F i g s . ( 2 0 ) A v e r a g e s t r a i n i n m i c r o i n c h e s p e r i n c h v e r s u s ? a T f o r r e c t a n g u l a r s t e e l beam (a) Gages 1, 11 and 2, 12 51 (b) Gages 3, 9 and k. 10 51 F i g s . ( 2 0 ) A v e r a g e s t r a i n i n m i c r o i n c h e s p e r i n c h v e r s u s J a ? f o r aluminum a l l o y I-beam (c) Gages 1, 11 and 2, 12 52 (d) Gages 3, 9 and 4 / 10 53 F i g s . ( 2 1 ) (a) C o m p r e s s i v e s t r a i n v e r s u s c e n t r a l l o a d f o r aluminum a l l o y I-beam (Gages 2, k. 6, 8, 10, 12) 54 (b) T e n s i l e s t r a i n v e r s u s c e n t r a l l o a d f o r aluminum a l l o y I-beam (Gages 1, 3, 5, 1, 9, 11) 55 (c) T e n s i l e s t r a i n v e r s u s c e n t r a l l o a d f o r aluminum a l l o y I-beam (Gages 2, 4, 6, 8, 10, 12) 56 (d) C o m p r e s s i v e s t r a i n v e r s u s c e n t r a l l o a d f o r aluminum a l l o y I-beam (Gages 1, 3. 5, 7. 9. l l ) 57 F i g . ( 2 2 ) L o a d - d e f l e c t i o n c u r v e f o r aluminum a l l o y I-beam 5 8 F i g s . ( 2 3 ) (a) C o m p r e s s i v e s t r a i n v e r s u s c e n t r a l l o a d f o r r e c t a n g u l a r s t e e l beam (Gages 2, 4, 6, 8, 10, 12) 59 (b) T e n s i l e s t r a i n v e r s u s c e n t r a l l o a d f o r r e c t a n g u l a r s t e e l beam (Gages 1, 3, 5, 7, 9, l l ) 60 v i i Page F i g s . ( 2 3 ) (c) Compressive s t r a i n versus c e n t r a l load f o r r e c t a n g u l a r s t e e l beam (Gages 1, 3, 5, 7, 9, 11) 61 (d) T e n s i l e s t r a i n versus c e n t r a l l o a d f o r r e c t a n g u l a r s t e e l beam (Gages 2, 4, 6, 8, 10, 12) 62 F i g . ( 2 4 ) Lo ad-de f l e c t i o n s t e e l beam curve f o r r e c t a n g u l a r 63 1 CHAPTER I  I n t r o d u c t i o n A beam sub j e c t e d to concentrated loadshas an abrupt, change i n shearing f o r c e under the p o i n t of a p p l i c a t i o n of each l o a d . In order to s a t i s f y e q u i l i b r i u m and c o n t i n u i t y c o n d i t i o n s i n the t r a n s i t i o n region of abrupt sheer change, the warped s e c t i o n s on each side of the a p p l i e d load must f i t i n t o each other. According to the c o n v e n t i o n a l theory of sheer d e f l e c t i o n i n the e l a s t i c range, the c o n t i n u i t y between the p a r t s of the beam to the l e f t and r i g h t of the a p p l i e d load i s maintained simply by matching the c e n t r o i d a l elements at the s e c t i o n s of d i s c o n t i n u i t y . However, t h i s does not remove the disagreement between the two p a r t s a l l along the depth of the s e c t i o n s . There s t i l l remains a gap at the top and an o v e r l a p at the bottom of the beam in the t r a n s i t i o n zone. The author's theory i n developing expres-sions f o r the sheer d e f l e c t i o n of beams removes t h i s d i s -agreement by a p p l y i n g l o n g i t u d i n a l s t r e s s e s having zero r e s u l t a n t and moment about the n e u t r a l a x i s . The theory i s f u r t h e r r e f i n e d by a l l o w i n g f o r the s t r a i n s induced at r i g h t angles to the d i r e c t i o n of concentrated load due to P o i s s o n ' s r a t i o e f f e c t . The author's theory i s compared with the c o n v e n t i o n a l theory and checked f o r r e l i a b i l i t y by means of experiments with aluminum a l l o y I-beam and r e c t a n g u l a r s t e e l beam. The 2 d i f f e r e n c e of d e f l e c t i o n s by the two t h e o r i e s i s very small ( l e s s than 1 p e r c e n t ) . Therefore comparison i s made in terms of s t r e s s e s by making the beams work as s t a t i c a l l y i n d e t e r -minate. The d i f f e r e n c e of d e f l e c t i o n s r e v e a l s i t s e l f i n a d i f f e r e n c e of s t a t i c a l l y unknown r e a c t i o n s , which in turn a f f e c t the s t r e s s e s i n the beam. The s t r e s s e s are found e x p e r i m e n t a l l y by measuring normal f i b r e s t r a i n s with e l e c -t r i c r e s i s t a n c e s t r a i n gages a p p l i e d at the top and bottom su r f a c e s of the beam and then m u l t i p l y i n g them by a known modulus of e l a s t i c i t y determined i n a p r e v i o u s experiment. i In view of the smallness of the d i f f e r e n c e between the two t h e o r i e s , i t was important to f i n d the s t r a i n s very accu-rately. F o i l - t y p e s t r a i n gages made by Budd were used i n the experiment. Attempt was also made to detect the d e v i a t i o n of normal s t r e s s e s from the formula °*y= My/I i n the v i c i n i t y of con-c e n t r a t e d l o a d s . \ 3 CHAPTER II Formulation of the Theory The theory i s developed by c o n s i d e r i n g a simple I-beam having a constant c ro s s - s e c t i on throughout i t s length and sub j e c t e d to a concentrated load P a c t i n g at a d i s t a n c e from the l e f t support as shown i n f i g u r e ( l ) . F i r s t the sheer deformations of the t r a n s v e r s e s e c t i o n s are co n s i d e r e d . These deformations are c o n s i d e r e d q u i t e independently of deformations produced by moments. Sheer Deformations i n the Web The d i s t r i b u t i o n of v e r t i c a l sheer s t r e s s in the plane of a c r o s s - s e c t i o n away from the load P i s given by: q i r = VQ/t.I ' ( l . a ) w l where V = T o t a l v e r t i c a l sheer I = Moment of i n e r t i a of the s e c t i o n t ^ = Thickness, of the web Q = S t a t i c moment of area o f c r o s s - s e c t i o n (above y) about the n e u t r a l axis The I - s e c t i o n i s s i m p l i f i e d by c o n s i d e r i n g a l l the flang e area as concentrated at i t s centre and the web as extending f o r i t s f u l l height to the centres of the fl a n g e s as shown i n f i g . (2a). In the case of I - s e c t i o n : 0 = 0.,, + Q , flang e web where Q f l a n g e = ( b t 2 ^ ^ and = S t , , _ AIL In these expressions t ^ = Thickness of the flange h = Depth of the s e c t i o n between centres of flanges as shown in f i g . (2a) and y i s p o s i t i v e i n the upward d i r e c t i o n . S u b s t i t u t i n g Q,n + Q , f o r Q i n equation ( l . a ) : . f 1 ange web w Vbt 2h + Vh' 81 1 - ( l . b ) dy If AB i s the i n i t i a l p o s i t i o n of the c r o s s - s e c t i o n as shown in f i g u r e (2c), th en the h o r i z o n t a l displacement of a p o i n t d i s t a n t y from the n e u t r a l axis would be: Yw J 0 G where / G = Sheer Modulus S u b s t i t u t i n g the value of q as given by equation ( l . b ) , w the above equation on i n t e g r a t i o n gives 2 Y w V b t 2 h y + Vhl 2t,GI 8GI id 3h 2 ( l . c ) Thus under the a c t i o n of sheer s t r e s s q', the o r i g i n a l w plane s e c t i o n AB becomes the warped s e c t i o n A^B^ as shown in f i g u r e ( 2 c ) . It should be noted that Y w i s antisymmetrical about the centre 0 because v ( +y) =-Y (-y)» w ' w Sheer Deformations i n the Flanges The d i s t r i b u t i o n of sheer s t r e s s in the f l a n g e s i s given by: VQ V where - x t * 2 2 t ^ = Thickness of the flange x = Distance of the v e r t i c a l s e c t i o n from the centre of the flange as shown i n f i g u r e (2a) In t h i s formula the v a r i a t i o n of sheer s t r a i n through the t h i c k n e s s of the flange i s ignored • The s u b s t i t u t i o n of t h i s value of Q i n the above equation gives f l a n g e Vh 21 - x The h o r i z o n t a l movement of a point in the f l a n g e due to f l a n g e sheer alone would be I q f l a n g e , 2 Vhb 4GI x -x The h o r i z o n t a l displacement of the f l a n g e due to web sheer i s obtained by s u b s t i t u t i n g y = ^ i n equation ( l . c ) and i s equal to V b t 2 h 2 + Vh3 4 t x G I 24GI T h e r e f o r e , the t o t a l h o r i z o n t a l displacement of a point in the f l a n g e from the i n i t i a l p o s i t i o n AB would be Y f = AA 2 - Y l V b t . h 2 V 1 3 2 _ + VhT + Vhb 4 t 1 G I 24GI 4GI x x - k (l . d ) If = Sheer i n the beam on the l e f t of P = Sheer i n the beam on the r i g h t of P then sheer would deform a l l the s e c t i o n s to the l e f t and r i g h t of the a p p l i e d l o a d P as shown i n f i g u r e ('4)« Deformations Produced by Transverse Compression The lo a d P, assumed to be d i s t r i b u t e d i n the web over a small length 'a*, compresses the element T a ? i n the t r a n s -verse d i r e c t i o n . Due to Poisson's r a t i o e f f e c t , t h i s t r a n s -verse compression causes the element * a T to extend i n the l o n g i t u d i n a l d i r e c t i o n . The r e l a t i o n between t h i s extension in the web and the a p p l i e d load i s d e r i v e d below. R e f e r r i n g to f i g u r e (3a)# the v e r t i c a l compression at a s e c t i o n d i s t a n t y from the n e u t r a l axis i s equal to K. P where K i s a c o e f f i c i e n t to be determined l a t e r on. The v e r t i c a l compressive s t r e s s at t h i s s e c t i o n would be E P a t , • If ' u.' i s the Poisson's r a t i o f o r the m a t e r i a l of the beam, then the s t r a i n i n the l o n g i t u d i n a l f i b r e would be KP I t~E ^ The t o t a l s t r e t c h in t h i s f i b r e of length *a' w i l l t h e r e f o r e be 6 = KPu. ( l . e ) It should be noted that t h i s t o t a l s t r e t c h i s indepen-dent of the length of the element 'a*. The value of K i s found by c o n s i d e r i n g the e q u i l i b r i u m of the element i n f i g u r e (3b). Here i t i s assumed that the f u l l value of v e r t i c a l sheer s t r e s s develops on each v e r t i c a l face of the element 'a*. On t h i s assumption, the e q u i l i b r i u m of v e r t i c a l f o r c e s i s given by h/2 KP • 4 (q + q )dy - P = o w w where q and q are v e r t i c a l sheer s t r e s s e s in the web W l W2 j u s t to the l e f t and r i g h t of the a p p l i e d l o a d . S u b s t i t u t i o n of these values of q and q from equa-W l w2 t i o n ( l . b ) r e s u l t s i n K = 1 - 241 _ JiY. + All bht, 21 S u b s t i t u t i n g t h i s value of K i n equation ( l . e ) , we get 6 = h3 t. * 1 E 1 - 241 bht^ 21 _ 11 + - y T h i s displacement i n the web i s g r a p h i c a l l y i l l u s t r a t e d i n f i g u r e (2d). At y = 0 ° 2 t 1 E T r a n s f e r r i n g the y - a x i s to 0* r e s u l t s f6* i n t o two com-ponents as shown in f i g u r e (2d). The f i r s t component i s a constant displacement of magnitude ponent i s given by: ° 2 t 1 E 2 t x E and the second com-2t^E h ^ 121 bht 1 _ + A T h ^ 2 " y ( l . f ) Now p u t t i n g y = -y i n equation ( l . f ) , we get 6 » ( - y ) = 2 t x E 1 " h3 t. 121 bht. i + IX Ax N h3 T h i s may be w r i t t e n as 6» (-y) = - 2 t 1 E 1 -h 3 t . 121 k bht + 2Pu_ 2 t 1 E i - + A r > h h 3 - y l -h 3t 1+6bh 2 j t 1 121 h 3 t +6bh 2t - - 6 « ( y ) + 1 -h 3 t 1 + 6 b h 2 t 1 121 T h e r e f o r e 6 » ( - y ) - - 6 t ( y ) + | ^ l l - 1] = - 6 » ( y ) . 9 Hence 6 T ' i s ant isymmetr ical about the centre 0*. I n c o m p a t i b i l i t y of Deformations Between the Web and the Top F1ange In d e r i v i n g the e f f e c t of the t r a n s v e r s e compression due to the l o a d P, i t was assumed that the load P was d i s -t r i b u t e d only through the web of the element 'a'. T h e r e f o r e , the web and not the f l a n g e s of the element 'a f w i l l be de-formed l o n g i t u d i n a l l y due to the Poisson's r a t i o e f f e c t of the load P. This deformation, as d e r i v e d e a r l i e r , i s zero at the bottom but g r a d u a l l y i n c r e a s e s to a maximum at the top of the web and t h e r e f o r e produces i n c o m p a t i b i l i t y of deformations between the web and the top f l a n g e . To remove t h i s i n c o m p a t i b i l i t y , the top flange must be s t r e t c h e d the same amount as the top of web i s expanded l o n g i t u d i n a l l y by the Poisson's r a t i o e f f e c t of the load P. This s t r e t c h i s e f f e c t e d by a t e n s i o n f o r c e T x T a p p l i e d to the top f l a n g e as shown in f i g u r e (/+)• The magnitude of f x ' i s given by: A^E = ( 6 1 + 6 2 }max. = 6 max But T h e r e f o r e whe r e = hL.. E u-PA max t^x = ^ r - 1 ( l . g ) A^ = cro s s - s ect i on a l area of a f l a n g e . Subsequently t h i s t e n s i l e f o r c e *x T w i l l be n e u t r a l i s e d 10 in an a d d i t i o n a l step i n the d e r i v a t i o n . Thus f o r c e 'x* s t r e t c h e s the top f l a n g e from CC to mm on the l e f t and from GG to nn on the r i g h t of the element *a T as shown in f i g u r e (4)« D i s c o n t i n u i t y i n the T r a n s i t i o n Zone Just o u t s i d e the i n f l u e n c e of Poisson's r a t i o e f f e c t , the s h e a r i n g f o r c e s to the l e f t and r i g h t of the element T a ' deform the s e c t i o n s as shown i n f'ugure (4)» If the s e c t i o n s i n the t r a n s i t i o n zone were f r e e to deform, there would be a gap and o v e r l a p between the c e n t r a l element and each of the s i d e elements due to the f a c t that the sheer deformation i s not equal to the deformations of the c e n t r a l element. The disagreement of s e c t i o n s i s d i a g r a m a t i c a l l y i l l u s -t r a t e d i n f i g u r e s (5)« F i g u r e s (5a) and (5$) show the shear d i s t o r t i o n s i n the web on the l e f t and r i g h t of P respec-t i v e l y . F i g u r e (5b) i l l u s t r a t e s the d i s t o r t i o n s of the c e n t r a l element f a ' due to the Poisson's r a t i o e f f e c t of the l o a d P. To c a l c u l a t e the disagreement between the l e f t p art of the beam and the middle part and again between the middle pa r t and the r i g h t p a r t , b r i n g p o i n t s 0£ i n t o 0 and 0^ i n t o 0^ as shown i n f i g u r e s (5). The r e s u l t a n t disagreement in the web on the l e f t and r i g h t of the c e n t r a l element would be Y ~ ol and y - 61 r e s p e c t i v e l y . This i s i l l u s t r a t e d w^  1 2 d i a g r a m a t i c a l l y i n f i g u r e s (5d) and (5e)« But due to the i n -determinate d i s t r i b u t i o n of deformations i n the c e n t r a l e l e -ment, these r e s u l t a n t disagreements on the l e f t and r i g h t of / 11 *a r are not known. However, the t o t a l disagreement may be combined i n one as shown i n f i g u r e (5f)» In t h i s f i g u r e i f the l i n e AOB represents the extremity of the l e f t side of the beam and the l i n e COD that of the r i g h t s i d e , then the t o t a l gap between AOB and COD would be the sum of - &[ and - 6 £ , i . e . aw = • - - 6 J - 6 » = Y + Y - 6 * ( l . h ) W l W2 The t o t a l disagreement i n the top f l a n g e would be calcu-l a t e d by b r i n g i n g the l i n e AA to the l i n e C C * on the l e f t and the l i n e EE to the l i n e G fG f on the r i g h t as shown i n f i g u r e (k)* The t o t a l gap i n the top f l a n g e would be: % = L Y f i " 6 { ( m a x ) ] • - 6 » ( n a x ) = ( Y f i • Y ^ ) - ( 6 f + 6 » ) m „ " ^ + Y f 2 } " 6max But 6' from equation ( l . f ) i s equal to . max v 2t T h e r e f o r e a f t = Y f x + Y f 2 " 2 t ^ Since y* > Y f a n d 6 T are a l l antisymmetrical about th< r l X2 n e u t r a l a x i s , the t o t a l o v erlap i n the bottom f l a n g e would be : 12 Requirements of p h y s i c a l c o n t i n u i t y do not permit t h i s gap. The gap i s c l o s e d by superimposing a d d i t i o n a l d i s -placements to the l e f t side so that in f i g u r e (5f) AOB c o i n c i d e s with A^OB^. The a d d i t i o n a l displacements would be equal to the o r d i n a t e s between AOB and A^OB^ and are given by: 2z - — y i n the web ( l . k ) - Z Q i n the top fl a n g e ( l . l ) and + Z Q i n the bottom fla n g e (l.m) From equations ( l . h ) and ( l . k ) , the r e s u l t a n t d i s p l a c e -ment i n the web would be: 2z Y + Y - 6 1 - — ~ y w l w2 h From equations ( l . i ) and ( l . l ) , the r e s u l t a n t d i s p l a c e -ment i n the top flange would be: Y f l + Y f 2 ' 2 t l E " Z< S i m i l a r l y f rom ^ equations ( l . j ) and (l.m), the r e s u l t a n t displacement i n the bottom fla n g e would be: The o r i e n t a t i o n of A^OB^ i s such that the moment of the l o n g i t u d i n a l s t r e s s e s r e q u i r e d to produce the above r e s u l t a n t displacement about the n e u t r a l axis i s equal to zer o . Assuming the&Jlongi t udin a l s t r e s s e s to be p r o p o r t i o n a l to the displacements and equating t h e i r moment about the 13 n e u t r a l axis to zero r e s u l t s i n J A K ' ( Y W + Yw - 6 T - P)ydA 1 w 2 + 2 ! K' A Y f + Y f 1 r2 - z Pn h 2 t 1 E 2 dA = 0 ( l . n ) where Kf = Const, of p r o p o r t i o n a l i t y and [3 = Ordinate of l i n e A;0&, with respect to AOB as shown i n f i g u r e (5f) From equation ( l . c ) Y + Y w l w 2 ( V 1 - V 2 ) b t 2 h y ( V 1 - V 2 ) h ' 2t GI P b t 2 h y Phf 2t,GI + 8GI 8GI y - 3 h 2 3h' From equation ( l . d ) Y f , + Y f . ( v 1 - v 2 ) b t 2 h 2 ; ( v 1 - v 2 ) h -( v 1 - v 2 ) h b 4GI 24GI 2 x -P b ^ 2 h + Ph; 4 t 1 G I • Pbh 24GI 4GI x -x b 6' i s given by equation ( l . f ) . These values on s u b s t i t u t i o n i n equation ( l . n ) give .3 Ph-Z o 20(A+4A f)GI (A+3A f) 2 5A 2 5 A f b 2 A w A w h' Ph 3u A + 3 A f 20EI A+H.A, 2 t x E ( l . o ) w h e r e M = Moment of the f o r c e 'x* about the n e u t r a l • axi s and Now a = Length of the element. w h 2 On s u b s t i t u t i n g the value of f x ' from equation ( l . g ) , the above equation r e s u l t s i n at± 2 and t h e r e f o r e u-PA h A 9 = 2 E I t -The t o t a l value of 0 i s then given by: Of = o + A o 2z uPA £h o _,_ f / , \ = IT + 2Zlt[ ( l " r ) For r e c t a n g u l a r beam 2z 0 ' = -f ( l . s ) where Z q i s given by equation ( l . p ) . D e f l e c t i o n due to Angular D i s c o n t i n u i t y In F i g u r e (5f) the gap between AOB and COD i s now c l o s e d by r o t a t i n g the l e f t side i n r e l a t i o n to the r i g h t side through an angle 0 ' so that AOB c o i n c i d e s with A^OB^. Th i s r o t a t i o n would make the axis of. the beam on the l e f t 16 of P i n c l i n e d at an angle 0* to the h o r i z o n t a l as shown i n f i g u r e ( 6 ) . The beam i s now r o t a t e d about A so that both the ends A and B of the beam are at the same l e v e l as shown in f i g u r e (7). Thus i n a d d i t i o n to the bending d e f l e c t i o n , the angular d i s c o n t i n u i t y 9', developed at the l o c a t i o n of the a p p l i e d c o n c e n t r a t e d l o a d P, produces a d d i t i o n a l d e f l e c -t i o n of the beam. This a d d i t i o n a l d e f l e c t i o n i s given by: A = x9» - Y7 e fx and O'.x 0»L. when 0 < x < L. ( l . t ) 9». L, when L1 < x < L ( l . u ) where x = Distance from the l e f t support at which the d e f l e c t i o n i s measured. It should be noted that the l o n g i t u d i n a l s t r e s s e s pro-duced by r o t a t i o n 9' modify the law of sheer s t r e s s e s i n the v i c i n i t y of P and t h i s changes the manner of warping of the c r o s s - s e c t i o n r e q u i r i n g f u r t h e r c o r r e c t i o n s which are not attempted. Conventional Theory of Sheer D e f l e c t i o n of Beams The c o n v e n t i o n a l theory of sheer d e f l e c t i o n of beams does not allow f o r the Poisson's r a t i o e f f e c t of the load P 17 on the element 'a'. The angular d i s c o n t i n u i t y '0** i s c a l c u -l a t e d from the formula: Q» = w^  max . ( l . v ) The values of q w^max f o r an I-beam i s d e r i v e d from equation ( l . b ) by p u t t i n g y = 0 and V = and i s given by 2 V l b t 2 h + V _ ''w^ax 2 t 1 I 81 S i m i l a r l y q V 2 b t 2 h + VL w2max 2 t 1 I 81 On s u b s t i t u t i n g these values of q w^max and q i n equation ( l . u ) w2max ^ v ' 0» = (W h' SGI Ph 8GI 1 + ht. 1 + 4Aj w In case of r e c t a n g u l a r beam 2 0» = Ph' SGI (l.w) The sheer d e f l e c t i o n i s given by the equations (l.'t) and ( l . u ) . Now ac c o r d i n g to author's theory the value of 0* f o r a r e c t a n g u l a r beam loaded as shown i n f i g . ( l ) i s given by 9' = Ph 2(2 + (i)/10EI. For \i = 0.3 and E = 2 ( l + u.)G 0» = Ph 2/11.3G (1.x) Equations (l.w) and ( l . x ) show that f o r a simply sup-p o r t e d r e c t a n g u l a r beam 0* by co n v e n t i o n a l theory i s g r e a t e r than 0' by author's theory by 41,25 pe r c e n t . 18 CHAPTER I I I Experimental V e r i f i c a t i o n of the Theory The purpose of the experiment was to t e s t the accuracy of the author's theory and compare the r e s u l t s with the con-, v e n t i o n a l theory f o r shear d e f l e c t i o n of beams i n the e l a s t i c range. Attempt i s also made to determine the d e v i a t i o n of normal s t r e s s from the formula 6y= i n the v i c i n i t y of conce n t r a t e d l o a d s . Apparatus and Procedure S i n c e the d i f f e r e n c e of d e f l e c t i o n s by the two the o r i e s , i s very small ( l e s s than 1 percent) the comparison i s made i n terms of s t r e s s e s by making the beams work as s t a t i c a l l y i n d e t e r m i n a t e . The d i f f e r e n c e of d e f l e c t i o n s by the two t h e o r i e s m a n i f e s t s i t s e l f i n a d i f f e r e n c e of s t a t i c a l un-known r e a c t i o n s which i n tu r n a f f e c t the beam s t r e s s e s . The s t r e s s e s are found by experimental determination of normal s t r a i n s by e l e c t r i c r e s i s t a n c e s t r a i n gages and then m u l t i -p l y i n g them by a known modulus of e l a s t i c i t y determined e x p e r i m e n t a l l y by a previous experiment d i s c u s s e d i n the next chapter. In view of the smallness of the d i f f e r e n c e between the two t h e o r i e s , the s t r a i n s must be found very a c c u r a t e l y . F o i l type e l e c t r i c r e s i s t a n c e s t r a i n gages made by Budd, which are b e l i e v e d to be b e t t e r than the other types, were used f o r measuring the s t r a i n s . The manufacturer i n d i c a t e d the p o s s i b l e range of s c a t t e r i n the gage f a c t o r s by + 1.5 p e r c e n t . Therefore the gage readings, determined e x p e r i m e n t a l l y , were c o r r e c t e d by m u l t i p l y i n g them by c o r r e c -t i o n f a c t o r s . These c o r r e c t i o n f a c t o r s were determined by a previous experiment which i s d e s c r i b e d i n the next chapter. The gages were bonded s t r i c t l y a c c ording to the s t a n -dard s p e c i f i c a t i o n s . To e l i m i n a t e the e f f e c t of moisture, they were covered with m i c r o c r y s t a l l i n e wax. They were p l a c e d on top and bottom f l a n g e s along the centre l i n e i n l o c a t i o n shown in f i g . ( l l ) . Gages 1, 2 , 11 and 12 were p l a c e d s u f f i c i e n t l y f a r away from loads so t h a t there was no c o n c e n t r a t e d l o a d e f f e c t from the a p p l i e d loads and the r e a c t i o n s . The remaining gages were l o c a t e d c l o s e to the a p p l i e d loads and the c e n t r a l r e a c t i o n as shown in f i g u r e ( 1 1 ) . The apparatus was set up as shown in f i g u r e (18). The f i r s t specimen was a 65ST6 aluminum a l l o y I-beam 16 f e e t l o n g . The dimensions of the I - s e c t i o n are shown in f i g u r e ( l 3 a ) . Olsen Mechanical U n i v e r s a l T e s t i n g Machine was used f o r l o a d i n g . The f o r c e s at the s e c t i o n s A and C were d i s -1 3 t r i b u t e d by s t e e l bearing p l a t e s 3 wide, t h i c k and <• k extended across the e n t i r e length of the f l a n g e as shown in f i g u r e ( 1 5 ) . F i g u r e (16) shows the c e n t r a l r e a c t i o n where the f o r c e was d i s t r i b u t e d by a f l a t s t e e l b e a r i n g p l a t e 2" wide and 1 t h i c k . Loads were a p p l i e d symmetrically at two p o i n t s through a s t e e l l o a d i n g beam with attached k n i f e edge assemblies,shown in f i g u r e ( 1 7 ) . 20 A load of 2500 pounds was f i r s t a p p l i e d through a l o a d i n g beam at two p o i n t s 3" away from gages 3 and 9 as shown i n f i g u r e ( l l ) . The l o a d at each p o i n t was i n c r e a s e d i n 3750 pounds increment to 16250 pounds and a complete set of readings was taken at each l o a d change. The load was then r e l e a s e d to the i n i t i a l value and the gage readings were checked. The above procedure was repeated with loads a p p l i e d at p o i n t s 6" and 9" away from gages 3 and 9« The same experiment was repeated f o r s t e e l r e c t a n g u l a r beam except that the load increments were 7500 pounds and the f i n a l l o a d i n g was 65000 pounds. The dimensions of the r e c t a n g u l a r s e c t i o n are shown i n f i g u r e ( l 3 . b ) . R e s u l t s The gage readings were converted to microinches per inch and the r e s u l t s t a b u l a t e d i n Tables ( l a ) and ( l b ) . The s t r a i n readings f o r symmetrically p l a c e d gages were averaged under "Mean £ '* columns. These mean values were then cor-o r e c t e d u sing the c o r r e c t i o n f a c t o r s determined by a previous experiment and d e s c r i b e d i n the next chapter. The c o r r e c t e d mean values were t a b u l a t e d i n the same t a b l e under column "Mean £ » * . " . . c Tables (2a) and (2b) show the values of r e a c t i o n R^ computed from the c o r r e c t e d mean s t r a i n readings of gages 1, 11 and 2.12. These gages were s u f f i c i e n t l y f a r away from 21 t h e c o n c e n t r a t e d l o a d s . In t h e same t a b l e s , columns R and R show t h e t h e o r e t i -a c c a l v a l u e s o f R^ c a l c u l a t e d u s i n g f o r m u l a s b a s e d on t h e a u t h o r ' s t h e o r y and t h e c o n v e n t i o n a l t h e o r y . The d e r i v a t i o n of t h e s e f o r m u l a s i s g i v e n i n C h a p t e r V. These t h e o r e t i c a l and e x p e r i m e n t a l v a l u e s o f were, t h e n compared g r a p h i c a l l y i n f i g u r e s (19a) and (19b). I t can be seen f r o m f i g u r e (19a) t h a t f o r r e c t a n g u l a r x - s e c t i o n t h e o b s e r v e d v a l u e s o f R^ ag r e e v e r y c l o s e l y w i t h t h e v a l u e s computed by u s i n g the 1 a u t h o r ' s t h e o r y . Figure(19b) shows t h a t t h e r e i s a r e a s o n -a b l y g o od agreement between t h e computed and o b s e r v e d v a l u e s of R^. In the ca s e of I-beam t h e l i t t l e d e v i a t i o n o f t h e o b s e r v e d v a l u e s f r o m t h e t h e o r e t i c a l v a l u e s i s due t o c e r -t a i n i r r e g u l a r i t i e s i n t h e f l a n g e s . These g r a p h s c l e a r l y show t h a t t h e c o n v e n t i o n a l t h e o r y always g i v e s h i g h e r v a l u e s o f r e a c t i o n s . The e x p e r i m e n t s d e s c r i b e d h e r e a l l o w a l s o t o i n v e s t i -g a t e t h e e f f e c t of p r o x i m i t y of l o a d on t h e s t r a i n gage r e a d i n g s i n t h e v i c i n i t y o f t h e l o a d . T h i s i s shown g r a p h i c a l l y i n f i g u r e s (20). C u r v e s B and C r e p r e s e n t the m e a s u r e d s t r a i n s i n m i c r o i n c h e s p e r i n c h i n gages 3, Ui 9 and 10 as a f u n c t i o n o f t h e d i s t a n c e o f t h e a p p l i e d l o a d s f r o m gages 3 °r 9. On t h e same g r a p h , c u r v e s A show t h e t h e o r e t i c a l v a l u e s o f s t r a i n s i n t h e s e g a g e s . I t can be seen f r o m f i g u r e (20b) t h a t f o r beams o f r e c t a n g u l a r x-s e c t i o n t h e l o a d e f f e c t d i e s out at a d i s t a n c e e q u a l t o t h e d e p t h of t h e x - s e c t i o n . F i g u r e (20d) shows t h a t f o r beams of I - s e c t i o n t h e gages i n t h e b o t t o m f l a n g e are a f f e c t e d up t o a d i s t a n c e e q u a l t o 1-^  t i m e s t h e d e p t h o f t h e x - s e c t i o n . Due t o some i r r e g u l a r i t y i n t h e f l a n g e s , t h e t o p gages be-h a v e d i n an i r r e g u l a r manner. 23 CHAPTER IV Correct ion Factors f o r the Gage Readings The flange s t r a i n s were measured by f o i l type e l e c t r i c \ s t r a i n gages made by Budd. Since the s t r a i n s must be found .'i j very a c c u r a t e l y , i t was therefore decided to corr e c t the s t r a i n readings as described below. Theory : Let £ - an observed s t r a i n reading \i - c o r r e c t i o n f a c t o r M - bending moment at the secti o n where the s t r a i n i s measured then the corrected s t r a i n reading would be given by: H * - f £ (5.a) In case of a simply supported beam loaded with a con-centrated load at the centre, the bending moment at a sec-t i o n d i s t a n t ? d f from the l e f t support would be: M - | d Therefore from equation (5»a) M-e = |fj d (5.b) or UX «oP.d (5 .c) The t e n s i l e and compressive s t r a i n readings of gages placed symmetrically on e i t h e r side of the load are averaged s e p a r a t e l y . R e f e r r i n g to f i g u r e (12), i f 24 B = a v e r a g e s t r a i n r e a d i n g o f gages 1, 11 " 3 , 9 " 5, 7 » 2, 1 2 " 4, 1 0 and = " '« n " " 6 and 8 '5 ! C -t h e n i n view o f e r r o r s o f gage f a c t o r s , t h e t r u e v a l u e s of th e s t r a i n s w o u l d be ^^e^# ^2^2' 'J"3E3' ^6^6 r e s p e < t i v e l y . F o r i n c r e m e n t of l o a d AP A--e1 = K XAP *E2 " K 2 A P ( 5 - d ) A £ 6 = V P where , K^, are c o n s t a n t s . From e q u a t i o n ( 5 b ) , f o r any v a l u e of \ i l & e 1 ^ 2 A E 2 : L J , 3 A £ 3 H d l ! D 2 : d3 and W W ^ 6 A e 6 B dl : D 2 : d3 where d-^, d 2 and d^ are d i s t a n c e s of gages as shown i n f i g u r e ( 1 2 ) . T h i s c o u l d be w r i t t e n a s : ^ 1 ^ 1 T I 2 A E 2 l x3^ £3 ^UheU ^ 5 A £ 5 25 On s u b s t i t u t i n g the values of A E from equation ( 5.d), equation ( 5 i e ) r e s u l t s i n ! ^ 1 K 1 1 "2 "3 where C i s a constant. From the above equation ( 5 . f ) Ho = C d 3 d d 2 Assuming that the average gage f a c t o r of the gages used i s the t r u e value given by the manufacturer _ i _ 2 _ J _ A _ . _ i _ 6 = x ( 5 i g ) On s u b s t i t u t i o n of ' U . ' values from equation ( 5 . f ) , equation ( 5«g) r e s u l t s i n C = K l K 2 K 3 K 4 K 5 K 6 (5.h) In t h i s equation d^, and d^ are known. 26 ^1' ^2' 6^ a r e c a l c u l a t e d from e q u a t i o n s (5«d) by-p l o t t i n g s t r a i n r e a d i n g s as f u n c t i o n of the l o a d P. Thus knowing the va l u e of C, the c o r r e c t i o n f a c t o r s [i^, u.^, *'"» u.£ can be o b t a i n e d from e q u a t i o n s ( 5 . f ) . D e t e r m i n a t i o n of Modulus of E l a s t i c i t y The modulus of e l a s t i c i t y was c a l c u l a t e d from the s t r a i n gage r e a d i n g s and was then checked w i t h the v a l u e o b t a i n e d from the l o a d d e f l e c t i o n c u r v e . The f o r m u l a s used i n the two cases are d e r i v e d below. (a) From s t r a i n gage r e a d i n g s : True s t r a i n r e a d i n g = u-A.e • c i A E From e q u a t i o n (5.b) ^e = H i d u h F o r y = g E = - " (5.1) 4lM-£ But | = K T h e r e f o r e v = _JL_ d 41^ K In t h i s f o r m u l a h, I , \i and d are known. The v a l u e of K i s o b t a i n e d from graphs of £ vs P f o r those gages which are s u f f i c i e n t l y away from the e f f e c t of the c o n c e n t r a t e d l o a d . (b) From the l o a d - d e f l e c t i o n c u r v e : I-beam. For a s i m p l e I-beam l o a d e d at the c e n t r e by a c o n c e n t r a t e d l o a d P, the midspan bending 27 d e f l e c t i o n i s given by: 1 PL 3 b 48 EI Since ^ f o r the I beam i s 3 0 , the shear d e f l e c t i o n w i l l be n e g l i g i b l e as compared to the bending d e f l e c t i o n . However, i t w i l l be e v a l u a t e d by u s i n g an approximate formula which would be s u f f i c i e n t l y accurate f o r t h i s purpose. R e f e r r i n g to f i g u r e (2b), the average shear s t r e s s i n the web would be = _P qw 2t 1h qw Average Y w = ~ 2t 1hG where G i s the sheer modulus. Therefore mid-span sheer d e f l e c t i o n s 2t xhG 2 PL 4t 1hG S u b s t i t u t i n g G = 2 (i+'|i) and Lt = O.3 0.65PL fls t j h E The t o t a l midspan d e f l e c t i o n due to bending and sheer would be A . * _ i _ Pj2 . 0.65PL . v bh + A s ~ 48 ¥ T + t x h E ( 5 ' j ) 28 Rectangular Beam. The midspan bending d e f l e c t i o n f o r a r e c t a n g u l a r beam load with a concentrated l o a d P at the . centre i s given by = 1_ PL3. A b 48 EI For r e c t a n g u l a r beams, the author's formula f o r the sheer d e f l e c t i o n i s simple enough to be used here. From equations ( l . s ) and ( l . p ) 2z 0 ' = For [i = O.3 _ 0 . 2?Ph 2 EI Therefore midspan shear d e f l e c t i o n A s ~ 2 2 = 0.0575 ^ f l 1 But I = — t 2 h 3 T h e r e f o r e = 0 .69PL A s E ^ h T o t a l midspan d e f l e c t i o n would be A b + A s = 48 TT + "TT~rr ( 5 - k ) 29 Experiment f o r Determining C o r r e c t i o n F a c t o r s f o r the Gage  Readings and the Modulus of E l a s t i c i t y Apparatus and Procedure The aluminum a l l o y I-beam with gages i n l o c a t i o n as r e q u i r e d f o r the main t e s t was supported on two simple s t e e l rocker supports mounted at 15 f e e t apart on the a u x i l i a r y wings of the Olsen Mechanical U n i v e r s a l T e s t i n g Machine. A c e n t r a l l o a d at i t s midspan was a p p l i e d through an assembly shown in f i g u r e ( 1 4 ) . The fo rces at the r e a c t i o n p o i n t s were d i s t r i b u t e d by s t e e l b e a r i n g p l a t e s extending across the e n t i r e width of the f l a n g e . The f o r c e at the l o a d i n g p o i n t was s i m i l a r l y d i s t r i b u t e d by a wide be a r i n g p l a t e . S t r a i n s were measured at d i f f e r e n t s e c t i o n s shown in f i g u r e ( 1 2 ) . D e f l e c t ions at midspan and at the two supports of the beam were measured with d i a l gages which co u l d read a m i n i -mum d e f l e c t i o n of 0.001 i n c h e s . The maximum c a r r y i n g c a p a c i t y of the I-beam based on 30 /sq. inch y i e l d s t r e s s was found to be 4200 pounds. It was t h e r e f o r e decided to load the beam i n increments of 500 pounds up to a maximum of 3000 pounds. A complete set of s t r a i n s and d e f l e c t i o n readings was taken at each load increment. The beam was then i n v e r t e d and a s i m i l a r set of readings was taken. The r e s u l t s of the experiment are t a b u l a t e d i n Table ( 4 a ) . The same experiment was repeated w i t h x t h e r e c t a n g u l a r s t e e 1 be am. 3 0 Re s u i t s The r e s u l t s are presented i n the g r a p h i c a l form i n f i g u r e s ( 2 l ) . In f i g u r e ( 2 1 a ) , graph A represents average of the compressive s t r a i n readings r e g i s t e r e d i n the s t r a i n gages 6 and 8 as a f u n c t i o n of P. S i m i l a r l y graphs T B f and TC* represent the average of compressive s t r a i n readings r e g i s t e r e d i n the gages 4/ 10 and 2 , 12 r e s p e c t i v e l y . S i m i -l a r graphs are p l o t t e d f i g u r e ( 2 1 b ) f o r the s t r a i n readings r e g i s t e r e d i n gages 1 , 1 1 ; 3 / 9 and 5, 7« These graphs of £ vs P i n f i g u r e s ( 2 1 a ) and ( 2 1 b ) give s i x values of K which wh en s u b s t i t u t e d i n equations ( 5 f ) and ( 5 h ) would give the c o r r e s p o n d i n g values of the c o r r e c t i o n f a c t o r s . S i m i l a r v alues of c o r r e c t i o n f a c t o r s were found from the second set of experiments and the average values of the c o r r e c t i o n f a c -t o r s i s t a b u l a t e d i n Table ( 5 ) . The modulus of e l a s t i c i t y was c a l c u l a t e d from equation (5i) by s u b s t i t u t i n g the corresponding values of |X, d and K. The average value of the modulus of e l a s t i c i t y f o r the aluminum a l l o y determined from the two sets of experiments was found to be 1 0 . 0 2 y 10 l b s . / s q . i n c h . T h i s agrees very c l o s e i y with the value 9«99 x 10^ l b s . / s q . inch determined from the l o a d - d e f l e c t i o n curve f i g u r e ( 2 2 ) by the equation S i m i l a r sets of graphs were p l o t t e d f o r the r e c t a n g u l a r s t e e l beams and are shown i n f i g u r e s ( 2 3 ) . The c o r r e c t i o n f a c t o r s are t a b u l a t e d i n Table ( 5 ) and the modulus of e l a s -t i c i t y i s given below 3 1 E c a l c u l a t e d from the s t r a i n r e a d i n g s = 2 9 . 8 x 1 0 6 l b s . /•'» E from l o a d - d e f l e c t i o n curve s = 2 9 . 8 lbs./O". 32 CHAPTER V D e r i v a t i o n of Formulas f o r the Test-Beams The t e s t beam was set up as shown i n f i g u r e (8). To f i n d the t h e o r e t i c a l value of the r e a c t i o n at B, f i r s t con-s i d e r the I-beam loaded as shown i n f i g u r e (9)» The bending d e f l e c t i o n at B i s given by A B = 12^ U " K ) ( 2 + 2 K _ The shear d e f l e c t i o n at B i s given by Ag = L ( l - K ) 0 J where 0£ i s d e r i v e d from equation ( l . r ) by s u b s t i t u t i n g P = -z , l . e . 2z U-WA, u . g l _ PJr + 1 h 2EIt ' 2 where T z o T d e r i v e d s i m i l a r l y from equation (l-O) and i s given by Wh3 '61 40(A+4A f )GI (A+3A ) 2 5A 2 A f b 2 i — + £ + 5 _ J E — AT t A ,2 W w h ... 3 A+3A£ Wlrjx J f Wu. 4 0 E I A+ 4 A F ~4t 1E Therefore t o t a l d e f l e c t i o n at B: 3 A B X + A S = " l 2 E I ( l " K ) ( 2 + 2 K " + L ( l " K ) Q 1 Now c o n s i d e r the beam loaded as shown in f i g u r e (12). Bending d e f l e c t i o n at B 33 R B L  A B = 6ET~ S h e e r d e f l e c t i o n at B A S = LOJ h Q f . 2 Z o 2 + ^ B A f h where 9£ - - y ^ — + -and R Bh-'o2 20(A+4A f )GI (A+3A ) 2 5A 2 5A b 2 + — - + i A W R B h J t i A+3Af 20EI A + 4 A . A W 2t E 2z o l R, W B T o t a l d e f l e c t i o n at B A R + Aq B 2 S2' R B L 6TT + L 92 S u p e r i m p o s i n g t h e s e two c a s e s and e q u a t i n g r e s u l t a n t d e f l e c t i o n at B to z e r o 3 o 6EI + L 6 2 = l l E I ( l " K ) ( 2 + 2 K " k 2 ) + L ( l - K)OJ S u b s t i t u t i n g the v a l u e s of 9* and 0^ g i v e s R B - f ( i - r ) (2+2K-K ) L +6EI9*/W L 2 + 6EI0'/W where 9' = Wh 10(A+4A f)GI (A+3A f)* 5A f 5A fb' A ,2 h p . A+3A4 10EI A+4A. W. A W t 1 E h 34 For r e c t a n g u l a r beam: Q t = Wh2 _ Wh2u-10GI 2 Wh 20EI 10EI (2 + u.) A c c o r d i n g to the co n v e n t i o n a l theory For I - be am 0' = Wh SGI 1 + 4^ A w and f o r r e c t a n g u l a r beam 2 0» = Wh SGI " 35 CHAPTER VI  Conclusions The experiments show that the author's theory gives r e s u l t s which are lower and more accurate than the conven-t i o n a l theory. The d i f f e r e n c e between the r e s u l t s of the two t h e o r i e s i s very small and t h e r e f o r e the theory pre-sented has no p r a c t i c a l advantage. However, the author's theory o f f e r s a b e t t e r i n s i g h t i n t o the mechanics.of sheer deformation of beams i n the e l a s t i c range. It should be noted that the r e s u l t s as p r e d i c t e d by th author's theory have a c o n s i s t e n t d e v i a t i o n from the experi mental r e s u l t s . T h i s may be due to the f a c t that the author's theory does not allow f o r the e f f e c t of the l o n g i -t u d i n a l s t r e s s e s produced by the angular d i s c o n t i n u i t y at the p o i n t s of a p p l i c a t i o n of the concentrated l o a d s . TABLE 1(a)  S t r a i n s i n Aluminum A l l o y I-Beam 1 3 S 7 9 II ' I F " _I5 6 LJ* ^ — 4fl k 6 i P ' 2 —P_J 6 i 15 36 p i n Kips a Inch Gages 1 and 11 Gages 2 and 12 Gages 4 and 10 I Gages 3 and 9 Gages i 5 and 7 Gages 6 and 8 Obs Obs Me an £ Obs Obs Me an £ Obs Obs Me an £ pbs Obs Me an e Obs Obs Me an £ Obs Obs Me an £ 1 11 Obs e o Cor £ c 2 12 Obs e o Cor £ c 4 10 Obs Eo Cor e c 9 Obs e o Cor e c 5 7 Obs e o Cor £ c 6 8 Obs e o Cor £ c 10.0 3 6 9 300 237 179 294 234 177 297 236 178 297 236 178 300 237 180 301 236 180 300 236 180 300 236 180 381 329 251 370 320 247 375 325 249 376 326 250 |447 338 253 ' 447 334 251 447 336 252 446 335 251 245 237 212 246 241 219 245 239 215 248 241 217 226 220 197 228 228 202 237 224 200 236 223 199 17.5 3 6 9 586 466 350 583 459 347 585 462 348 585 46I 348 589 466 354 590 463 352 589 464 353 589 463 353 745 645 491 724 630 484 735 638 487 738 640 489 880 663 497 ^ 875 658 491 878 660 494 876 658 493 526 501 449 529 512 46O 528 506 454 535 512 458 488 470 419 493 483 432 490 476 425 498 474 423 25 .0 3 6 9 871 690 521 866 681 514 860 685 517 867 684 516 880 691 528 881 690 524 881 690 526 880 689 525 1108 958 732 1074 935 719 1091 946 725 1096 950 728 Jl31'2 989 ,740 1302 979 731 1397 984 735 1395 982 733 808 770 687 814 786 703 811 778 695 821 788 704 750 720 640 760 740 660 755 730 650 752 727 647 32.5 3 6 9 1160 917 691 1151 903 681 1156 910 686 1154 909 685 1170 917 701 1170 907 696 1170 912 699 1168 911 698 1470 1270 972 1428 1240 954 1449 1255 963 1454 1259 967 1748 1313 983 — 1 1734 1300 972 1741 13O6 977 1738 1304 975 1093 1040 930 1102 940 951 1097 990 940 1110 1000 950 1016 971 865 1030 1000 893 1023 985 879 1019 982 876 I 37 TABLE 1(b)  S t r a i n s i n Rectangular S t e e l Beam 5 — — . . r , „ 1 . i5 f k t . q , j 4 A" 4«" I p in Kips a Inch Gages 1 and 11 Gages 2 and 12 Gages 4 and 10 Gages 3 and 9 Gages 5 and 7 Gages 6 and 8 Obs Obs Me an £ Obs Obs Mean E Obs Obs Mean £ Obs "Obs Me an E Obs Obs Me an E Obs Obs Mean £ 1 11 Obs £ o Cor E c 2 12 Obs £ o Cor E c 4 10 Obs £ o Cor E c 3 9 Obs £ o . Cor £ c 5 7 Obs E o Cor £ c 6 8 Obs £ o Cor £ c 20 .0 3 6 9 102 83 61 101 80 61 102 82 61 102 82 61 101 80 62 101 80 61 101 80 62 102 81 62 149 116 88 147 116 88 147 116 88 146 115 87 135 111 89 133 111 88 134 111 89 135 112 90 98 87 80 99 91 83 98 89 82 96 87 80 103 94 83 104 97 86 103 96 85 103 96 85 35 .0 3 6 9 203 162 123 203 159 123 203 161 123 203 161 123 199 159 123 199 159 121 199 159 122 202 161 123 293 230 174 290 228 173 291 229 173 289 221 170 270 223 172 264 220 171 267 222 172 269 224 174 199 I84 162 205 191 170 202 188 166 198 I84 I64 209 194 171 213 200 175 211 197 173 212 198 174 50 .0 3 6 9 303 244 I83 303 237 182 303 241 I83 303 241 I83 299 239 I84 298 235 181 299 237 I83 302 239 185 440 341 259 435 339 259 437 340 •259 434 338 257 402 334 258 397 330 255 400 332 257 403 325 25 9 303 283 25 0 312 291 259 308 287 255 303 283 253 320 299 263 325 303 267 322 302 265 324 304 267 65 .0 3 6 9 403 322 243 400 317 242 402 320 243 402 320 243 399 315 242 396 312 239 398 314 241 401 317 244 586 453 342 579 449 343 583 451 342 581 449 340 534 442 341 528 436 336 531 439 339 535 442 342 410 38I 349 423 394 351 417 388 350 412 384 346 433 404 357 440 411 363 437 407 360 440 410 362 I i I i 38 Table 2 End Reactions P 2 4 6 8 10 12 ^ 15 , - i - b 1 1 6 i 5 7 * 6 _ b II 48 41 3" Table ( 2 .a) Aluminum a l l o y I-beam b Inch Lo ad P End Re act ions in l b s . By Strains By Theory RA % Me an Author* R a s Conven-t i o n a l R 24 7 . 5 1 5 . 0 2 2 . 5 3 0 . 0 1273 2 5 2 0 3 7 2 0 4 9 5 0 1 2 9 0 2 5 2 5 3 7 7 5 5 0 1 0 1282 2 5 2 3 3 7 4 8 4 9 8 0 1295 2 5 9 0 3 8 8 5 5 1 8 0 1329 2 6 5 8 3 9 8 7 5 3 1 6 27 7 . 5 15 . 0 2 2 . 5 3 0 . 0 1011 1975 2 9 3 0 3 9 0 0 1011 1985 2955 3 9 1 0 1011 1 9 8 0 2943 3905 1 0 6 1 2 1 2 2 3183 4244 1088 2 1 7 2 3 2 6 4 4352 3 0 7 . 5 1 5 . 0 2 2 . 5 3 0 . 0 7 6 4 1490 2215 2 9 4 0 7 7 2 1515 2 2 5 0 2 9 9 0 7 6 8 1503 2 2 3 3 2965 8 4 0 1 6 8 0 2 5 2 0 3 3 6 0 8 7 0 1 7 4 0 2 6 1 0 3 4 8 0 Table (2.b) Rectangular s t e e l be am 24 1 5 . 0 3 0 . 0 4 5 . 0 60 . 0 2 4 3 5 4 8 4 0 7235 9 5 8 0 2435 4 8 2 0 7200 9 5 6 0 2 4 3 5 4 8 3 0 7218 9 5 7 0 2 4 1 0 4 8 2 0 7230 9 6 4 0 2 4 4 0 4 8 8 0 7 3 2 0 9 7 6 0 27 1 5 . 0 3 0 . 0 4 5 . 0 60 . 0 1955 3 8 4 0 5 7 5 0 7630 1932 3 8 4 0 5 7 0 0 7 5 7 0 ' 1943 3 8 4 0 5 7 2 5 7600 1922 3 8 4 4 5766 7688 1965 3930 5885 7860 3 0 1 5 . 0 3 0 . 0 4 5 . 0 60 . 0 1455 3 8 4 0 4365 5 8 0 0 148O 2840 4 4 2 0 5 8 2 0 1468 2840 4 4 4 2 5 8 1 0 1463 2926 4 3 8 9 5852 . 1 5 0 0 3 0 0 0 4 5 0 0 6000 39 TABLE 3 Table (3 a) Showing the Average C o r r e c t e d S t r a i n s i n Microinches per Inch f o r AP = 7500 l b s . Determined by Test on I-Beam Compared with T h e o r e t i c a l Values Gages 1, 11 Gages 2, 12 Gages 4, 10 Gages 3, 9 Gages 5, 7 Gages 6, 8 a __ - —— - — — — — — £ e ~ i i E e e e e c t e c t e c t e c t e c t e c t 3«* 289 288 297 292 291 297 363 363 417 435 435 417 274 274 200 256 256 200 6" 227 226 244 228 227 244 314 314 342 327 327 342 248 248 179 247 247 179 9'* 171 171 193 175 174 193 246 246 270 244 244 270 235 235 148 220 220 148 Table (3b) Showing the Average C o r r e c t e d S t r a i n s i n Microinches per Inch f o r AP = 15000 l b s . Determined by Test on Rectangular S t e e l Beam Compared with T h e o r e t i c a l Values Gages 1,11 Gages 2, 12 Gages 4, 10 Gages 3, 9 Gages 5, 7 Gages 6, 8 3 ~£ £ ~ E £~ ~ £ Z~ ~~£ £ E £ £ £ , £ £ £ , e c t e c t e c t e c t e c t e c t 3«» "100 100 101 99 100 101 133 134 142 146 145 142 109 108 95 104 104 95 6" 79 79 81 79 80 81 110 111 113 113 112 113 102 101 89 97 97 89 9" 60 60 61 60 61 61 85 86 86 86 85 86 90 89 84 88 88 84 S t r a i n i n micr o i n c h per inch determined by t e s t C o r r e c t e d £ e S t r a i n i n micr o i n c h per inch determined t h e o r e t i c a l l y £ e £ c 1* 40 TABLE 4 Showing Measured S t r a i n s i n Microinches per Inch f o r D i f f e r e n t Values of C e n t r a l Load 1 i u 6. 7 - 7-6 9 6 57 7-6 Aluminum a l l o y I-beam Cen-t r a l S t r a i n i n Microinch/Inch Load G a a e s l - , 1 1 Gages2,12 Gages3,9 Gages4/10 Gages5,7 Gages 6,8 in Ten. Comp.Ten. Comp.Ten. Comp.Ten. Comp.Ten. Comp.Ten. Comp, Lbs . (a) Aluminum I-Be am 500 278 280 270 307 307 307 307 299 409 410 411 398 1000 495 496 493 488 549 538 541 539 733 735 723 717 1500 720 715 716 796 782 787 795 1062 IO45 IO46 IO46 1048 2000 941 935 939 939 1041 1025 IO32 1029 1394 1368 1378 1369 2500 1167 1148 H64 1152 1291 1271 1281 1275 1724 1695 1710 1693 30001396 1376 1398 1376 1542 1537 1537 1521 2055 2056 2050 2017 (b) S t e e l Be am 2000 63 4000 121 6000 180 8000 239 59 61 60 82 76 79 78 140 137 140 140 120 121 117 159 150 153 152 271 268 277 272 185 180 176 238 225 226 229 404 403 415 409 235 237 234 317 300 302 303 539 535 553 544 10000 300 298 295 295 399 377 377 382 680 670 692 684 1 ll 1 c" 3 5 6 6 7 9 4-fi - —1 •—. 4-'--6 >.. Rectangular s t e e l beam 41 TABLE (5) Showing the C o r r e c t i o n F a c t o r s f o r the Gage Readings Gage C o r r e c t i o n F a c t o r s \x I-Beara Rectangular Beam 1, 11 0.999 1.000 2, 12- 0.998- 1.014 3, 9 1.004 1.008 4, 10 0.998 O.993 5, 7 1.013 0.980 6, 8 0.998 1.005 f i g . (3a) Fig.(k) showing the deformations in the web and f l a n g e s due  to shear and t r a n s v e r s e compression Top flange A m c c 6 6 n , £ f i g s . (5) fig.(5a) Sheer fig.(5b) D i s t o r t i o n s fig.(5c) Sheer' d i s t o r t i o n s on of the c e n t r a l element d i s t o r t i o n s on l e f t of P 'a* r i g h t of P A C A, 9 ' / / o > / / y ' / :>;. 6, o 6 fig.(5f) Resultant disagreement' f i g . ( 7 ) |W A 1 1 U C 1 f i g . ( I ) A C I ^ S h e a r F o r c e Diagram W 2 S h e a r D e f l e c t i o n f i g . ( 9 ) 1 j j ! f r i \ » ,— >. — r ^ — . ~~_______SJvear D e f l e c t i o n ^ ______ f i g . (ToT^ 1,6 A i l l I B z 9 II C 1' I 1 1 _ i5_..J««_|a _ K c u 18* h c - 6 » 45 io » 12 » . a M 4 - / 5 ! D f i g . (11) _4, 5J_ 2 4 I 15 -7-6 f i g . ( 1 2 ) 10 12 .oU 84. J =57_ -ol^ 63-H - 3 - H 0.25 _ — oo| _ JL - 2 _ f i g . (13a) C r o s s - s e c t i o n o f t h e a luminum a l l o y I - b e a m f i g . ( 1 3 b ) C r o s s - s e c t , i o n o f t h e r e c t angu 1 s r s i.ee 1 be am f i g . ( 1 4 ) c e n t r a l l o a d i n g p o i n t 1 0 0 0 0 49 9 0 0 0 F j g . ( l 9 a ) Comparison of experimental and t h e o r e t i c a l  values of the r e a c t i o n R^ ( r e c t a n g u l a r s t e e l beam) 1 P 8000 7 0 0 0 1 j- - r. n 1 1 z A i -« 48 4 8* / / 6000 Y/ / x • H c o o (0 CU 5 0 0 0 4 0 0 0 / / 3 0 0 0 / 2000 fr 0 Experimental A Author's theory y Conventional theory 1000 15 3 0 45 60 Load P, i n Kips 50 F i g . ( l 9 b ) , Comparison of experimental and t h e o r e t i c a l values  of the r e a c t i o n (aluminum a l l o y I-beam) '5 W P 48 4-8 w / * / « / o X o * / £> / 0' fl X .c X V .0 / / X 0 Experimental A Author's theory x Conventional theory 7500 15000 22500 Load P i n l b s . 30000 32500 51 F i g . ( 2 0 ) Average s t r a i n i n microinches per inch versus T a T  f o r r e c t a n g u l a r s t e e l beam F i g . ( 2 0 a ) Gages 1 ,11 and 2 , 1 2 1 3 125 10C~ 75 O c u cu 0, w a> A u c •rH o u o c -H 5 0 9 M t - 4 6 L _ 48 X _ a _ J 6 U_ 15 J _.. 48" Theoret i c a l A Experimental 0 T h e o r e t i c a l rd u CO 150 125 100 75 3 " 6 " 9 " Distance of the l o a d from gages 3 o r 9 F i g . ( 2 0 b ) Gages 3 , 9 and k, 10 A Experiment a l 0 T h e o r e t i c a l 3 ti 6 tt 9 tt Distance of the load from gages 3 or 9 54 2000 1750 1500 -1250 x u CD ft w .OJ X u o O 1000 750 (0 CO 500 250 Fig.(21a) Compressive s t r a i n versus c e n t r a l load f o r  Aluminum a l l o y I-beam (Gages 2, 4, 6, 8, 10, 12) P 0 0 500 1000 1500 Load P i n l b s . J I I 2000 2500 3000 55 F i g . ( 2 1 b ) T e n s i l e s t r a i n versus c e n t r a l load f o r  aluminum a l l o y I-beam (Gages 1 , 3 , 5 , 7 , 9 , 11) / 5 0 0 1 0 0 0 1 5 0 0 Load P i n l b s . 2 0 0 0 2 5 0 0 3 0 0 0 56 2000 Fig.(21c) T e n s i l e s t r a i n versus c e n t r a l load f o r  aluminum beam (Gages 2, 4,6, 8, 10, 12) 0 1750 1500 1250 o c -H w OJ u -H o O 750 c. •H CQ •500 250 0 J_ 500 1000 1500 Lo ad P i n l b s . "2DUD 25UU 58 2.00 Fig.(22) L o a d - d e f l e c t i o n curve f o r aluminum a l l o y I-beam 0 1.75 ,0 1.50 / / co 1.25 CD A o / c 1.00 o 0 u CD rH CD (0 Is G CD O 0.75 0.50 0.25 or / / / 0 / 0 J_ 500 1000 1500 Load P in l b s . 2000 . 2500 3000 S t r a i n i n m i c r o i n c h e s per i n c h o o o o uo o o -p-o o Un O O Os O O O O t-o a 3 CO (V) o! o •p-o o o Os o. o o CO ot o| o ° o o o un vO F i g . ( 2 3 b ) T e n s i l e s t r a i n v e r s u s c e n t r a l l o a d f o r  r e c t a n g u l a r s t e e l beam (Gages 1, 3, 5, 7, 9, l l ) 700-600. A 500T" o c •ri u ID o. w 40C cu X, o -H o O e 30C_ •H id M CO 2 0 O 100-£ 4 6 1 P ' 8 10 12 i 1 1 L i Ll ) •- 11 " A 6 L _ Z , . . . . 4 I 21 _ 1 - 3 " S "] - f 6 U 21 J...6J..6 J . 21 1 I I I 6 2 0 0 0 4 0 0 0, 6000 Load P i n l b s . 8000 1 0 0 0 0 4 I i S ^ i 2 J c J C o j ^ r e ^ i y e _ g t r a i n versus c e n t r a l load f o r r e c t a n g u l a r s t e e l beam (Gage* 1, 3 , 5, 7, 9, l l ) 700 600 5 00 -lit OJ O d o o 400 a 300 ed u co 200( 100 -1 _L_ J_ 2000 4000 6000 Load P i n l b s . _ L _ 8000 10000 62 F i g . ( 2 3 d ) T e n s i l e s t r a i n versus c e n t r a l l o a d f o r r e c t a n g u l a r s t e e l be am (Gages 2, k, 6, 8, 10, 12) 2000 /+000 6000 8000 10000 Load P in l b s . v 63 F i g . (24) Load-def lect.i on curve f o r the r e c t a n g u l a r s t e e l be am p - /lift ~ A1- ft I 

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