Kiến trúc xây dựng - Mechanics of materials

MECHANICS OF MATERIALS Third Edition Ferdinand P. Beer E. Russell Johnston, Jr. John T. DeWolf Lecture Notes: J. Walt Oler Texas Tech University CHAPTER © 2002 The McGraw-Hill Companies, Inc. All rights reserved. Analysis and Design of Beams for Bending © 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALS Third Edition Beer • Johnston • DeWolf 5 - 2 Analysis and Design of Beams for Bending Introduction Shear and Bending Moment Diagrams Sample Problem 5.1 Sample Problem 5.2 Relations Among Load, Shear, and Bending Moment Sample Problem 5.3 Sample Problem 5.5 Design of Prismatic Beams for Bending Sample Problem 5.8 © 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALS Third Edition Beer • Johnston • DeWolf 5 - 3 Introduction • Beams - structural members supporting loads at various points along the member • Objective - Analysis and design of beams • Transverse loadings of beams are classified as concentrated loads or distributed loads • Applied loads result in internal forces consisting of a shear force (from the shear stress distribution) and a bending couple (from the normal stress distribution) • Normal stress is often the critical design criteria S M I cM I My mx ==−= σσ Requires determination of the location and magnitude of largest bending moment © 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALS Third Edition Beer • Johnston • DeWolf 5 - 4 Introduction Classification of Beam Supports © 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALS Third Edition Beer • Johnston • DeWolf 5 - 5 Shear and Bending Moment Diagrams • Determination of maximum normal and shearing stresses requires identification of maximum internal shear force and bending couple. • Shear force and bending couple at a point are determined by passing a section through the beam and applying an equilibrium analysis on the beam portions on either side of the section. • Sign conventions for shear forces V and V’ and bending couples M and M’ © 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALS Third Edition Beer • Johnston • DeWolf 5 - 6 Sample Problem 5.1 For the timber beam and loading shown, draw the shear and bend- moment diagrams and determine the maximum normal stress due to bending. SOLUTION: • Treating the entire beam as a rigid body, determine the reaction forces • Identify the maximum shear and bending-moment from plots of their distributions. • Section the beam at points near supports and load application points. Apply equilibrium analyses on resulting free-bodies to determine internal shear forces and bending couples • Apply the elastic flexure formulas to determine the corresponding maximum normal stress. © 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALS Third Edition Beer • Johnston • DeWolf 5 - 7 Sample Problem 5.1 SOLUTION: • Treating the entire beam as a rigid body, determine the reaction forces ∑ ∑ ==== kN14kN40:0 from DBBy RRMF • Section the beam and apply equilibrium analyses on resulting free-bodies ( )( ) 00m0kN200 kN200kN200 111 11 ==+∑ = −==−−∑ = MMM VVFy ( )( ) mkN500m5.2kN200 kN200kN200 222 22 ⋅−==+∑ = −==−−∑ = MMM VVFy 0kN14 mkN28kN14 mkN28kN26 mkN50kN26 66 55 44 33 =−= ⋅+=−= ⋅+=+= ⋅−=+= MV MV MV MV © 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALS Third Edition Beer • Johnston • DeWolf 5 - 8 Sample Problem 5.1 • Identify the maximum shear and bending- moment from plots of their distributions. mkN50kN26 ⋅=== Bmm MMV • Apply the elastic flexure formulas to determine the corresponding maximum normal stress. ( )( ) 36 3 36 2 6 12 6 1 m1033.833 mN1050 m1033.833 m250.0m080.0 − − × ⋅×== ×= == S M hbS B mσ Pa100.60 6×=mσ © 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALS Third Edition Beer • Johnston • DeWolf 5 - 9 Sample Problem 5.2 The structure shown is constructed of a W10x112 rolled-steel beam. (a) Draw the shear and bending-moment diagrams for the beam and the given loading. (b) determine normal stress in sections just to the right and left of point D. SOLUTION: • Replace the 10 kip load with an equivalent force-couple system at D. Find the reactions at B by considering the beam as a rigid body. • Section the beam at points near the support and load application points. Apply equilibrium analyses on resulting free-bodies to determine internal shear forces and bending couples. • Apply the elastic flexure formulas to determine the maximum normal stress to the left and right of point D. © 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALS Third Edition Beer • Johnston • DeWolf 5 - 10 Sample Problem 5.2 SOLUTION: • Replace the 10 kip load with equivalent force- couple system at D. Find reactions at B. • Section the beam and apply equilibrium analyses on resulting free-bodies. ( )( ) ftkip5.1030 kips3030 : 2 2 1 1 ⋅−==+∑ = −==−−∑ = xMMxxM xVVxF CtoAFrom y ( ) ( ) ftkip249604240 kips240240 : 2 ⋅−==+−∑ = −==−−∑ = xMMxM VVF DtoCFrom y ( ) ftkip34226kips34 : ⋅−=−= xMV BtoDFrom © 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALS Third Edition Beer • Johnston • DeWolf 5 - 11 Sample Problem 5.2 • Apply the elastic flexure formulas to determine the maximum normal stress to the left and right of point D. From Appendix C for a W10x112 rolled steel shape, S = 126 in3 about the X-X axis. 3 3 in126 inkip1776 : in126 inkip2016 : ⋅== ⋅== S M DofrighttheTo S M DoflefttheTo m m σ σ ksi0.16=mσ ksi1.14=mσ © 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALS Third Edition Beer • Johnston • DeWolf 5 - 12 Relations Among Load, Shear, and Bending Moment ( ) xwV xwVVVFy ∆−=∆ =∆−∆+−=∑ 0:0 ∫−=− −= D C x x CD dxwVV w dx dV • Relationship between load and shear: ( ) ( )221 0 2 :0 xwxVM xxwxVMMMMC ∆−∆=∆ =∆∆+∆−−∆+=∑ ′ ∫=− = D C x x CD dxVMM dx dM 0 • Relationship between shear and bending moment: © 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALS Third Edition Beer • Johnston • DeWolf 5 - 13 Sample Problem 5.3 SOLUTION: • Taking the entire beam as a free body, determine the reactions at A and D. • Apply the relationship between shear and load to develop the shear diagram. Draw the shear and bending moment diagrams for the beam and loading shown. • Apply the relationship between bending moment and shear to develop the bending moment diagram. © 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALS Third Edition Beer • Johnston • DeWolf 5 - 14 Sample Problem 5.3 SOLUTION: • Taking the entire beam as a free body, determine the reactions at A and D. ( ) ( )( ) ( )( ) ( )( ) kips18 kips12kips26kips12kips200 0F kips26 ft28kips12ft14kips12ft6kips20ft240 0 y = −+−−= =∑ = −−−= =∑ y y A A A D D M • Apply the relationship between shear and load to develop the shear diagram. dxwdVw dx dV −=−= - zero slope between concentrated loads - linear variation over uniform load segment © 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALS Third Edition Beer • Johnston • DeWolf 5 - 15 Sample Problem 5.3 • Apply the relationship between bending moment and shear to develop the bending moment diagram. dxVdMV dx dM == - bending moment at A and E is zero - total of all bending moment changes across the beam should be zero - net change in bending moment is equal to areas under shear distribution segments - bending moment variation between D and E is quadratic - bending moment variation between A, B, C and D is linear © 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALS Third Edition Beer • Johnston • DeWolf 5 - 16 Sample Problem 5.5 SOLUTION: • Taking the entire beam as a free body, determine the reactions at C. • Apply the relationship between shear and load to develop the shear diagram. Draw the shear and bending moment diagrams for the beam and loading shown. • Apply the relationship between bending moment and shear to develop the bending moment diagram. © 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALS Third Edition Beer • Johnston • DeWolf 5 - 17 Sample Problem 5.5 SOLUTION: • Taking the entire beam as a free body, determine the reactions at C. ⎟⎠ ⎞⎜⎝ ⎛ −−=+⎟⎠ ⎞⎜⎝ ⎛ −==∑ =+−==∑ 33 0 0 02 1 02 1 02 1 02 1 aLawMMaLawM awRRawF CCC CCy Results from integration of the load and shear distributions should be equivalent. • Apply the relationship between shear and load to develop the shear diagram. ( )curveloadunderareaawV a xxwdx a xwVV B a a AB −=−= ⎥⎥⎦ ⎤ ⎢⎢⎣ ⎡ ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ −−=∫ ⎟⎠ ⎞⎜⎝ ⎛ −−=− 02 1 0 2 0 0 0 2 1 - No change in shear between B and C. - Compatible with free body analysis © 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALS Third Edition Beer • Johnston • DeWolf 5 - 18 Sample Problem 5.5 • Apply the relationship between bending moment and shear to develop the bending moment diagram. 2 03 1 0 32 0 0 2 0 622 awM a xxwdx a xxwMM B a a AB −= ⎥⎥⎦ ⎤ ⎢⎢⎣ ⎡ ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ −−=∫ ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ −−=− ( ) ( ) ( ) ⎟⎠ ⎞⎜⎝ ⎛ −=−−= −−=∫ −=− 32 3 006 1 02 1 02 1 aLwaaLawM aLawdxawMM C L a CB Results at C are compatible with free-body analysis © 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALS Third Edition Beer • Johnston • DeWolf 5 - 19 Design of Prismatic Beams for Bending • The largest normal stress is found at the surface where the maximum bending moment occurs. S M I cM m maxmax ==σ • A safe design requires that the maximum normal stress be less than the allowable stress for the material used. This criteria leads to the determination of the minimum acceptable section modulus. all allm M S σ σσ max min = ≤ • Among beam section choices which have an acceptable section modulus, the one with the smallest weight per unit length or cross sectional area will be the least expensive and the best choice. © 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALS Third Edition Beer • Johnston • DeWolf 5 - 20 Sample Problem 5.8 A simply supported steel beam is to carry the distributed and concentrated loads shown. Knowing that the allowable normal stress for the grade of steel to be used is 160 MPa, select the wide-flange shape that should be used. SOLUTION: • Considering the entire beam as a free- body, determine the reactions at A and D. • Develop the shear diagram for the beam and load distribution. From the diagram, determine the maximum bending moment. • Determine the minimum acceptable beam section modulus. Choose the best standard section which meets this criteria. © 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALS Third Edition Beer • Johnston • DeWolf 5 - 21 Sample Problem 5.8 • Considering the entire beam as a free-body, determine the reactions at A and D. ( ) ( )( ) ( )( ) kN0.52 kN50kN60kN0.580 kN0.58 m4kN50m5.1kN60m50 = −−+==∑ = −−==∑ y yy A A AF D DM • Develop the shear diagram and determine the maximum bending moment. ( ) kN8 kN60 kN0.52 −= −=−=− == B AB yA V curveloadunderareaVV AV • Maximum bending moment occurs at V = 0 or x = 2.6 m. ( ) kN6.67 ,max = = EtoAcurveshearunderareaM © 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALS Third Edition Beer • Johnston • DeWolf 5 - 22 Sample Problem 5.8 • Determine the minimum acceptable beam section modulus. 3336 max min mm105.422m105.422 MPa160 mkN6.67 ×=×= ⋅== − all M S σ • Choose the best standard section which meets this criteria. 4481.46W200 5358.44W250 5497.38W310 4749.32W360 63738.8W410 mm, 3 × × × × × SShape 9.32360×W