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The following will be discussed in this chapter: Deformation of a beam under transverse loading, equation of the elastic curve, direct determination of the elastic curve from the load di, statically indeterminate beams, application of superposition to statically indeterminate, moment-area theorems,...
Third Edition CHAPTER MECHANICS OF MATERIALS Ferdinand P Beer E Russell Johnston, Jr John T DeWolf Deflection of Beams Lecture Notes: J Walt Oler Texas Tech University © 2002 The McGraw-Hill Companies, Inc All rights reserved Third Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Deflection of Beams Deformation of a Beam Under Transverse Loading Equation of the Elastic Curve Direct Determination of the Elastic Curve From the Load Di Statically Indeterminate Beams Sample Problem 9.1 Sample Problem 9.8 Moment-Area Theorems Application to Cantilever Beams and Beams With Symmetric Bending Moment Diagrams by Parts Sample Problem 9.11 Sample Problem 9.3 Application of Moment-Area Theorems to Beams With Unsymme Method of Superposition Maximum Deflection Sample Problem 9.7 Use of Moment-Area Theorems With Statically Indeterminate Application of Superposition to Statically Indeterminate © 2002 The McGraw-Hill Companies, Inc All rights reserved 9-2 Third Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Deformation of a Beam Under Transverse Loading • Relationship between bending moment and curvature for pure bending remains valid for general transverse loadings ρ = M ( x) EI • Cantilever beam subjected to concentrated load at the free end, ρ =− Px EI • Curvature varies linearly with x • At the free end A, ρ = 0, A • At the support B, © 2002 The McGraw-Hill Companies, Inc All rights reserved ρB ρA = ∞ ≠ 0, ρ B = EI PL 9-3 Third Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Deformation of a Beam Under Transverse Loading • Overhanging beam • Reactions at A and C • Bending moment diagram • Curvature is zero at points where the bending moment is zero, i.e., at each end and at E ρ = M ( x) EI • Beam is concave upwards where the bending moment is positive and concave downwards where it is negative • Maximum curvature occurs where the moment magnitude is a maximum • An equation for the beam shape or elastic curve is required to determine maximum deflection and slope © 2002 The McGraw-Hill Companies, Inc All rights reserved 9-4 Third Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Equation of the Elastic Curve • From elementary calculus, simplified for beam parameters, d2y ρ = dx 2 ⎤3 ⎡ ⎛ dy ⎞ ⎢1 + ⎜ ⎟ ⎥ ⎢⎣ ⎝ dx ⎠ ⎥⎦ ≈ d2y dx • Substituting and integrating, EI ρ = EI d2y dx = M (x) x dy = M ( x )dx + C1 EI θ ≈ EI dx ∫ x x 0 EI y = ∫ dx ∫ M ( x ) dx + C1x + C2 © 2002 The McGraw-Hill Companies, Inc All rights reserved 9-5 Third Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Equation of the Elastic Curve • Constants are determined from boundary conditions x x 0 EI y = ∫ dx ∫ M ( x ) dx + C1x + C2 • Three cases for statically determinant beams, – Simply supported beam y A = 0, yB = – Overhanging beam y A = 0, yB = – Cantilever beam y A = 0, θ A = • More complicated loadings require multiple integrals and application of requirement for continuity of displacement and slope © 2002 The McGraw-Hill Companies, Inc All rights reserved 9-6 Third Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Direct Determination of the Elastic Curve From the Load Distribution • For a beam subjected to a distributed load, d 2M dM = V (x) dx dV = = − w( x ) dx dx • Equation for beam displacement becomes d 2M dx = EI d4y dx = − w( x ) • Integrating four times yields EI y ( x ) = − ∫ dx ∫ dx ∫ dx ∫ w( x )dx + 16 C1x3 + 12 C2 x + C3 x + C4 • Constants are determined from boundary conditions © 2002 The McGraw-Hill Companies, Inc All rights reserved 9-7 Third Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Statically Indeterminate Beams • Consider beam with fixed support at A and roller support at B • From free-body diagram, note that there are four unknown reaction components • Conditions for static equilibrium yield ∑ Fx = ∑ Fy = ∑ M A = The beam is statically indeterminate • Also have the beam deflection equation, x x 0 EI y = ∫ dx ∫ M ( x ) dx + C1x + C2 which introduces two unknowns but provides three additional equations from the boundary conditions: At x = 0, θ = y = © 2002 The McGraw-Hill Companies, Inc All rights reserved At x = L, y = 9-8 Third Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Sample Problem 9.1 SOLUTION: • Develop an expression for M(x) and derive differential equation for elastic curve W 14 × 68 I = 723 in P = 50 kips L = 15 ft E = 29 × 106 psi a = ft • Integrate differential equation twice and apply boundary conditions to obtain elastic curve For portion AB of the overhanging beam, • Locate point of zero slope or point (a) derive the equation for the elastic curve, of maximum deflection (b) determine the maximum deflection, • Evaluate corresponding maximum (c) evaluate ymax deflection © 2002 The McGraw-Hill Companies, Inc All rights reserved 9-9 Third Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Sample Problem 9.1 SOLUTION: • Develop an expression for M(x) and derive differential equation for elastic curve - Reactions: RA = Pa ⎛ a⎞ ↓ RB = P⎜1 + ⎟ ↑ L ⎝ L⎠ - From the free-body diagram for section AD, M = −P a x L (0 < x < L ) - The differential equation for the elastic curve, EI © 2002 The McGraw-Hill Companies, Inc All rights reserved d2y a = − P x L dx - 10 Third Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Sample Problem 9.7 Combine the two solutions, wL3 wL3 θ B = (θ B )I + (θ B )II = − + EI 48 EI wL3 θB = 48 EI wL4 wL4 + y B = ( y B )I + ( y B )II = − EI 384 EI 41wL4 yB = 384 EI © 2002 The McGraw-Hill Companies, Inc All rights reserved - 20 Third Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Application of Superposition to Statically Indeterminate Beams • Method of superposition may be • Determine the beam deformation applied to determine the reactions at without the redundant support the supports of statically indeterminate • Treat the redundant reaction as an beams unknown load which, together with • Designate one of the reactions as the other loads, must produce redundant and eliminate or modify deformations compatible with the the support original supports © 2002 The McGraw-Hill Companies, Inc All rights reserved - 21 Third Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Sample Problem 9.8 For the uniform beam and loading shown, determine the reaction at each support and the slope at end A SOLUTION: • Release the “redundant” support at B, and find deformation • Apply reaction at B as an unknown load to force zero displacement at B © 2002 The McGraw-Hill Companies, Inc All rights reserved - 22 Third Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Sample Problem 9.8 • Distributed Loading: ⎤ w ⎡⎛ ⎞ ⎛2 ⎞ 3⎛ ⎞ ( yB )w = − ⎢⎜ L ⎟ − L⎜ L ⎟ + L ⎜ L ⎟⎥ 24 EI ⎢⎣⎝ ⎠ ⎝3 ⎠ ⎝ ⎠⎥⎦ wL4 = −0.01132 EI • Redundant Reaction Loading: 2 RB L3 RB ⎛ ⎞ ⎛ L ⎞ ( yB )R = ⎜ L ⎟ ⎜ ⎟ = 0.01646 EI 3EIL ⎝ ⎠ ⎝ ⎠ • For compatibility with original supports, yB = wL4 RB L3 = ( y B )w + ( y B )R = −0.01132 + 0.01646 EI EI RB = 0.688wL ↑ • From statics, R A = 0.271wL ↑ © 2002 The McGraw-Hill Companies, Inc All rights reserved RC = 0.0413wL ↑ - 23 Third Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Sample Problem 9.8 Slope at end A, wL3 wL3 (θ A )w = − = −0.04167 24 EI EI wL3 0.0688wL ⎛ L ⎞ ⎡ ⎛ L ⎞ ⎤ (θ A )R = ⎜ ⎟ ⎢ L − ⎜ ⎟ ⎥ = 0.03398 EI EIL ⎝ ⎠ ⎣⎢ ⎝ ⎠ ⎦⎥ wL3 wL3 θ A = (θ A )w + (θ A )R = −0.04167 + 0.03398 EI EI © 2002 The McGraw-Hill Companies, Inc All rights reserved wL3 θ A = −0.00769 EI - 24 Third Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Moment-Area Theorems • Geometric properties of the elastic curve can be used to determine deflection and slope • Consider a beam subjected to arbitrary loading, • First Moment-Area Theorem: area under (M/EI) diagram between C and D © 2002 The McGraw-Hill Companies, Inc All rights reserved - 25 Third Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Moment-Area Theorems • Tangents to the elastic curve at P and P’ intercept a segment of length dt on the vertical through C = tangential deviation of C with respect to D • Second Moment-Area Theorem: The tangential deviation of C with respect to D is equal to the first moment with respect to a vertical axis through C of the area under the (M/EI) diagram between C and D © 2002 The McGraw-Hill Companies, Inc All rights reserved - 26 Third Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Application to Cantilever Beams and Beams With Symmetric Loadings • Cantilever beam - Select tangent at A as the reference • Simply supported, symmetrically loaded beam - select tangent at C as the reference © 2002 The McGraw-Hill Companies, Inc All rights reserved - 27 Third Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Bending Moment Diagrams by Parts • Determination of the change of slope and the tangential deviation is simplified if the effect of each load is evaluated separately • Construct a separate (M/EI) diagram for each load - The change of slope, θD/C, is obtained by adding the areas under the diagrams - The tangential deviation, tD/C is obtained by adding the first moments of the areas with respect to a vertical axis through D • Bending moment diagram constructed from individual loads is said to be drawn by parts © 2002 The McGraw-Hill Companies, Inc All rights reserved - 28 Third Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Sample Problem 9.11 SOLUTION: • Determine the reactions at supports • Construct shear, bending moment and (M/EI) diagrams For the prismatic beam shown, determine • Taking the tangent at C as the the slope and deflection at E reference, evaluate the slope and tangential deviations at E © 2002 The McGraw-Hill Companies, Inc All rights reserved - 29 Third Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Sample Problem 9.11 SOLUTION: • Determine the reactions at supports RB = RD = wa • Construct shear, bending moment and (M/EI) diagrams wa ⎛ L ⎞ wa L A1 = − ⎜ ⎟=− EI ⎝ ⎠ EI ⎛⎜ wa ⎞⎟ wa (a ) = − A2 = − ⎜⎝ EI ⎟⎠ EI © 2002 The McGraw-Hill Companies, Inc All rights reserved - 30 Third Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Sample Problem 9.11 • Slope at E: θ E = θC + θ E C = θE C wa L wa = A1 + A2 = − − EI EI wa (3L + 2a ) θE = − 12 EI • Deflection at E: yE = tE C − tD C L⎞ ⎡ ⎛ ⎛ 3a ⎞⎤ ⎡ ⎛ L ⎞⎤ = ⎢ A1⎜ a + ⎟ + A2 ⎜ ⎟⎥ − ⎢ A1⎜ ⎟⎥ 4⎠ ⎝ ⎠⎦ ⎣ ⎝ ⎠⎦ ⎣ ⎝ ⎡ wa L wa L2 wa ⎤ ⎡ wa L2 ⎤ = ⎢− − − ⎥ ⎥ − ⎢− EI 16 EI EI 16 EI ⎥⎦ ⎢⎣ ⎥⎦ ⎢⎣ wa (2 L + a ) yE = − EI © 2002 The McGraw-Hill Companies, Inc All rights reserved - 31 Third Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Application of Moment-Area Theorems to Beams With Unsymmetric Loadings • Define reference tangent at support A Evaluate θA by determining the tangential deviation at B with respect to A • The slope at other points is found with respect to reference tangent θD = θ A +θD A • The deflection at D is found from the tangential deviation at D © 2002 The McGraw-Hill Companies, Inc All rights reserved - 32 Third Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Maximum Deflection • Maximum deflection occurs at point K where the tangent is horizontal • Point K may be determined by measuring an area under the (M/EI) diagram equal to -θA • Obtain ymax by computing the first moment with respect to the vertical axis through A of the area between A and K © 2002 The McGraw-Hill Companies, Inc All rights reserved - 33 Third Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Use of Moment-Area Theorems With Statically Indeterminate Beams • Reactions at supports of statically indeterminate beams are found by designating a redundant constraint and treating it as an unknown load which satisfies a displacement compatibility requirement • The (M/EI) diagram is drawn by parts The resulting tangential deviations are superposed and related by the compatibility requirement • With reactions determined, the slope and deflection are found from the moment-area method © 2002 The McGraw-Hill Companies, Inc All rights reserved - 34 ...Third Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Deflection of Beams Deformation of a Beam Under Transverse Loading Equation of the Elastic Curve Direct Determination of the Elastic... Application of Moment-Area Theorems to Beams With Unsymme Method of Superposition Maximum Deflection Sample Problem 9.7 Use of Moment-Area Theorems With Statically Indeterminate Application of Superposition... McGraw-Hill Companies, Inc All rights reserved - 20 Third Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Application of Superposition to Statically Indeterminate Beams • Method of superposition