9 The cgrw-hill Compnies, Inc. All rights reserved. Fifth SI Edition CHAPTER 5 ECHANICS OF ATERIALS Ferdinnd P. Beer E. Russell Johnston, Jr. John T. DeWolf Dvid F. zurek Lecture Notes: J. Wlt Oler Texs Tech University Anlysis nd Design of Bems for Bending
ifthechanics OF ATERIALS Contents Beer Johnston DeWolf zurek Introduction Sher nd Bending oment Digrms Smple Problem 5. Smple Problem 5. Reltions Among Lod, Sher, nd Bending oment Smple Problem 5. Smple Problem 5.5 Design of Prismtic Bems for Bending Smple Problem 5.8 9 The cgrw-hill Compnies, Inc. All rights reserved. 5-
ifthechanics OF ATERIALS Introduction Beer Johnston DeWolf zurek Objective - Anlysis nd design of bems Bems - structurl members supporting lods t vrious points long the member Trnsverse lodings of bems re clssified s concentrted lods or distributed lods Applied lods result in internl forces consisting of sher force (from the sher stress distribution) nd bending couple (from the norml stress distribution) Norml stress is often the criticl design criteri σ y σ I c I x m S Requires determintion of the loction nd mgnitude of lrgest bending moment 9 The cgrw-hill Compnies, Inc. All rights reserved. 5-
ifthechanics OF ATERIALS Introduction Beer Johnston DeWolf zurek Clssifiction of Bem Supports 9 The cgrw-hill Compnies, Inc. All rights reserved. 5-4
ifthechanics OF ATERIALS Sher nd Bending oment Digrms Beer Johnston DeWolf zurek Determintion of mximum norml nd shering stresses requires identifiction of mximum internl sher force nd bending couple. Sher force nd bending couple t point re determined by pssing section through the bem nd pplying n equilibrium nlysis on the bem portions on either side of the section. Sign conventions for sher forces nd nd bending couples nd 9 The cgrw-hill Compnies, Inc. All rights reserved. 5-5
ifthechanics OF ATERIALS Smple Problem 5. Beer Johnston DeWolf zurek SOLUTION: Treting the entire bem s rigid body, determine the rection forces For the timber bem nd loding shown, drw the sher nd bendmoment digrms nd determine the mximum norml stress due to bending. Section the bem t points ner supports nd lod ppliction points. Apply equilibrium nlyses on resulting free-bodies to determine internl sher forces nd bending couples Identify the mximum sher nd bending-moment from plots of their distributions. Apply the elstic flexure formuls to determine the corresponding mximum norml stress. 9 The cgrw-hill Compnies, Inc. All rights reserved. 5-6
ifthechanics OF ATERIALS Smple Problem 5. Beer Johnston DeWolf zurek SOLUTION: Treting the entire bem s rigid body, determine the rection forces from F : R 46kN R 4kN y B Section the bem nd pply equilibrium nlyses on resulting free-bodies F y kn ( kn)( m) + B D kn F y kn kn ( kn)(.5m) + 5kN m + 6kN 5kN m 4 + 6kN 4 + 8kN m 5 4kN 5 + 8kN m 6 4kN 6 9 The cgrw-hill Compnies, Inc. All rights reserved. 5-7
ifthechanics OF ATERIALS Smple Problem 5. Beer Johnston DeWolf zurek Identify the mximum sher nd bendingmoment from plots of their distributions. m 6kN 5kN m m B Apply the elstic flexure formuls to determine the corresponding mximum norml stress. S 6 b h 6 8. (.8 m)(.5 m) 6 m σ m S B 5 8. N m 6 m σ m 6. 6 P 9 The cgrw-hill Compnies, Inc. All rights reserved. 5-8
ifthechanics OF ATERIALS Smple Problem 5. SOLUTION: Beer Johnston DeWolf zurek Replce the 45 kn lod with n equivlent force-couple system t D. Find the rections t B by considering the bem s rigid body. The structure shown is constructed of W 5x67 rolled-steel bem. () Drw the sher nd bending-moment digrms for the bem nd the given loding. (b) determine norml stress in sections just to the right nd left of point D. Section the bem t points ner the support nd lod ppliction points. Apply equilibrium nlyses on resulting free-bodies to determine internl sher forces nd bending couples. Apply the elstic flexure formuls to determine the mximum norml stress to the left nd right of point D. 9 The cgrw-hill Compnies, Inc. All rights reserved. 5-9
ifthechanics OF ATERIALS Smple Problem 5. SOLUTION: Beer Johnston DeWolf zurek Replce the 45 kn lod with equivlent forcecouple system t D. Find rections t B. Section the bem nd pply equilibrium nlyses on resulting free-bodies. From A to C : Fy 45x 45x kn ( 45x)( x) +.5 knm x From C to D : Fy 8 8 8kN ( x.) + ( 9.6 8 )knm x From D to B : 5kN ( 5. 5x)kNm 9 The cgrw-hill Compnies, Inc. All rights reserved. 5-
ifthechanics OF ATERIALS Smple Problem 5. Beer Johnston DeWolf zurek Apply the elstic flexure formuls to determine the mximum norml stress to the left nd right of point D. From Appendix C for W5x67 rolled steel shpe, S.8x - m bout the X-X xis. To the left of D : σ 6.8.8 To the right of D : σ m m S S 99.8.8 - - Nm m Nm m σ m σ m 9P 96 P 9 The cgrw-hill Compnies, Inc. All rights reserved. 5-
ifthechanics OF ATERIALS Beer Johnston DeWolf zurek Reltions Among Lod, Sher, nd Bending oment Reltionship between lod nd sher: : ( + ) w x F y w x C d dx D w C xd w dx xc Reltionship between sher nd bending moment: : ( + ) d dx D x x + w x x w C x x D C ( x) dx 9 The cgrw-hill Compnies, Inc. All rights reserved. 5-
ifthechanics OF ATERIALS Smple Problem 5. Beer Johnston DeWolf zurek SOLUTION: Tking the entire bem s free body, determine the rections t A nd D. Apply the reltionship between sher nd lod to develop the sher digrm. Drw the sher nd bending moment digrms for the bem nd loding shown. Apply the reltionship between bending moment nd sher to develop the bending moment digrm. 9 The cgrw-hill Compnies, Inc. All rights reserved. 5-
ifthechanics OF ATERIALS Smple Problem 5. SOLUTION: Beer Johnston DeWolf zurek Tking the entire bem s free body, determine the rections t A nd D. F A ( 7.m) ( 9kN)(.8m) ( 54 kn)( 4. m) ( 5.8 kn)( 8.4 m) D D 5.6 kn y A A y y 9kN 54kN + 5.6 kn 5.8 kn 8. kn Apply the reltionship between sher nd lod to develop the sher digrm. d dx w d w dx - zero slope between concentrted lods - liner vrition over uniform lod segment 9 The cgrw-hill Compnies, Inc. All rights reserved. 5-4
ifthechanics OF ATERIALS Smple Problem 5. Beer Johnston DeWolf zurek Apply the reltionship between bending moment nd sher to develop the bending moment digrm. d dx d dx - bending moment t A nd E is zero - bending moment vrition between A, B, C nd D is liner - bending moment vrition between D nd E is qudrtic - net chnge in bending moment is equl to res under sher distribution segments - totl of ll bending moment chnges cross the bem should be zero 9 The cgrw-hill Compnies, Inc. All rights reserved. 5-5
ifthechanics OF ATERIALS Smple Problem 5.5 Beer Johnston DeWolf zurek SOLUTION: Tking the entire bem s free body, determine the rections t C. Apply the reltionship between sher nd lod to develop the sher digrm. Drw the sher nd bending moment digrms for the bem nd loding shown. Apply the reltionship between bending moment nd sher to develop the bending moment digrm. 9 The cgrw-hill Compnies, Inc. All rights reserved. 5-6
ifthechanics OF ATERIALS Smple Problem 5.5 SOLUTION: Beer Johnston DeWolf zurek Tking the entire bem s free body, determine the rections t C. F y C w w + R C L + C R C C w w L Results from integrtion of the lod nd sher distributions should be equivlent. Apply the reltionship between sher nd lod to develop the sher digrm. B B A w w x dx w x x ( re under lod curve) - No chnge in sher between B nd C. - Comptible with free body nlysis 9 The cgrw-hill Compnies, Inc. All rights reserved. 5-7
9 The cgrw-hill Compnies, Inc. All rights reserved. ifthechanics OF ATERIALS dition Beer Johnston DeWolf zurek 5-8 Smple Problem 5.5 Apply the reltionship between bending moment nd sher to develop the bending moment digrm. 6 w x x w dx x x w B A B ( ) ( ) ( ) 6 L w L w L w dx w C L C B Results t C re comptible with free-body nlysis
ifthechanics OF ATERIALS Design of Prismtic Bems for Bending The lrgest norml stress is found t the surfce where the mximum bending moment occurs. σ m I c mx mx S A sfe design requires tht the mximum norml stress be less thn the llowble stress for the mteril used. This criteri leds to the determintion of the minimum cceptble section modulus. σ S m σ min ll σ mx ll Among bem section choices which hve n cceptble section modulus, the one with the smllest weight per unit length or cross sectionl re will be the lest expensive nd the best choice. Beer Johnston DeWolf zurek 9 The cgrw-hill Compnies, Inc. All rights reserved. 5-9
ifthechanics OF ATERIALS Smple Problem 5.8 Beer Johnston DeWolf zurek SOLUTION: Considering the entire bem s freebody, determine the rections t A nd D. A simply supported steel bem is to crry the distributed nd concentrted lods shown. Knowing tht the llowble norml stress for the grde of steel to be used is 6 P, select the wide-flnge shpe tht should be used. Develop the sher digrm for the bem nd lod distribution. From the digrm, determine the mximum bending moment. Determine the minimum cceptble bem section modulus. Choose the best stndrd section which meets this criteri. 9 The cgrw-hill Compnies, Inc. All rights reserved. 5-
ifthechanics OF ATERIALS Smple Problem 5.8 Beer Johnston DeWolf zurek Considering the entire bem s free-body, determine the rections t A nd D. D( 5m) ( 6kN)(.5m) ( 5kN)( 4m) F A A D 58. kn y y A y 5. kn + 58. kn 6kN 5kN Develop the sher digrm nd determine the mximum bending moment. A B B A y A 8kN 5. kn ( re under lod curve) 6kN ximum bending moment occurs t or x.6 m. ( re under sher curve, Ato E) mx 67.6 kn 9 The cgrw-hill Compnies, Inc. All rights reserved. 5-
ifthechanics OF ATERIALS Smple Problem 5.8 Beer Johnston DeWolf zurek Determine the minimum cceptble bem section modulus. S min σ mx ll 4.5 67.6 kn m 6P 6 m 4.5 mm Shpe W4 8.8 W6.9 W 8.7 W5 44.8 W 46. S 67 474 549 55 448 mm Choose the best stndrd section which meets this criteri. W 6.9 9 The cgrw-hill Compnies, Inc. All rights reserved. 5-