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9 beam deflection- Mechanics of Materials - 4th - Beer

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Nhan Tran
Nhan Tran

This document provides an overview of chapter 9 from the textbook "Mechanics of Materials" which covers deflection of beams. It includes sections on the deformation of beams under transverse loading, the equation of the elastic curve, statically indeterminate beams, and methods for determining deflection such as moment-area theorems, superposition, and examples of applying these methods to solve problems. Sample problems are provided throughout to demonstrate solving for beam deflection, slope, reactions and developing the equation of the elastic curve.

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MECHANICS OF MATERIALS Fourth Edition Ferdinand P. Beer E. Russell Johnston, Jr. John T. DeWolf Lecture Notes: J. Walt Oler Texas Tech University CHAPTER © 2006 The McGraw-Hill Companies, Inc. All rights reserved. 9 Deflection of Beams
© 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourth Edition Beer • Johnston • DeWolf 9 - 2 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 9.3 Method of Superposition Sample Problem 9.7 Application of Superposition to Statically Indeterminate ... Sample Problem 9.8 Moment-Area Theorems Application to Cantilever Beams and Beams With Symmetric ... Bending Moment Diagrams by Parts Sample Problem 9.11 Application of Moment-Area Theorems to Beams With Unsymme... Maximum Deflection Use of Moment-Area Theorems With Statically Indeterminate...
© 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourth Edition Beer • Johnston • DeWolf 9 - 3 Deformation of a Beam Under Transverse Loading • Relationship between bending moment and curvature for pure bending remains valid for general transverse loadings. EI xM )(1 = ρ • Cantilever beam subjected to concentrated load at the free end, EI Px −= ρ 1 • Curvature varies linearly with x • At the free end A, ∞== A A ρ ρ ,0 1 • At the support B, PL EI B B =≠ ρ ρ ,0 1
© 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourth Edition Beer • Johnston • DeWolf 9 - 4 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. EI xM )(1 = ρ • 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.
© 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourth Edition Beer • Johnston • DeWolf 9 - 5 Equation of the Elastic Curve • From elementary calculus, simplified for beam parameters, 2 2 232 2 2 1 1 dx yd dx dy dx yd ≈               + = ρ • Substituting and integrating, ( ) ( ) ( ) 21 00 1 0 2 2 1 CxCdxxMdxyEI CdxxM dx dy EIEI xM dx yd EIEI xx x ++= +=≈ == ∫∫ ∫θ ρ
© 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourth Edition Beer • Johnston • DeWolf 9 - 6 Equation of the Elastic Curve ( ) 21 00 CxCdxxMdxyEI xx ++= ∫∫ • Constants are determined from boundary conditions • Three cases for statically determinant beams, – Simply supported beam 0,0 == BA yy – Overhanging beam 0,0 == BA yy – Cantilever beam 0,0 == AAy θ • More complicated loadings require multiple integrals and application of requirement for continuity of displacement and slope.
© 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourth Edition Beer • Johnston • DeWolf 9 - 7 Direct Determination of the Elastic Curve From the Load Distribution • Equation for beam displacement becomes ( )xw dx yd EI dx Md −== 4 4 2 2 ( ) ( ) 43 2 22 13 16 1 CxCxCxC dxxwdxdxdxxyEI ++++ −= ∫∫∫∫ • Integrating four times yields • For a beam subjected to a distributed load, ( ) ( )xw dx dV dx Md xV dx dM −=== 2 2 • Constants are determined from boundary conditions.
© 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourth Edition Beer • Johnston • DeWolf 9 - 8 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 000 =∑=∑=∑ Ayx MFF The beam is statically indeterminate. ( ) 21 00 CxCdxxMdxyEI xx ++= ∫∫ • Also have the beam deflection equation, which introduces two unknowns but provides three additional equations from the boundary conditions: 0,At00,0At ===== yLxyx θ
© 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourth Edition Beer • Johnston • DeWolf 9 - 9 Sample Problem 9.1 ft4ft15kips50 psi1029in7236814 64 === ×==× aLP EIW For portion AB of the overhanging beam, (a) derive the equation for the elastic curve, (b) determine the maximum deflection, (c) evaluate ymax. SOLUTION: • Develop an expression for M(x) and derive differential equation for elastic curve. • Integrate differential equation twice and apply boundary conditions to obtain elastic curve. • Locate point of zero slope or point of maximum deflection. • Evaluate corresponding maximum deflection.
© 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourth Edition Beer • Johnston • DeWolf 9 - 10 Sample Problem 9.1 SOLUTION: • Develop an expression for M(x) and derive differential equation for elastic curve. - Reactions: ↑      +=↓= L a PR L Pa R BA 1 - From the free-body diagram for section AD, ( )Lxx L a PM <<−= 0 x L a P dx yd EI −=2 2 - The differential equation for the elastic curve,
© 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourth Edition Beer • Johnston • DeWolf 9 - 11 Sample Problem 9.1 PaLCLCL L a PyLx Cyx 6 1 6 1 0:0,at 0:0,0at 11 3 2 =+−=== === • Integrate differential equation twice and apply boundary conditions to obtain elastic curve. 21 3 1 2 6 1 2 1 CxCx L a PyEI Cx L a P dx dy EI ++−= +−= x L a P dx yd EI −=2 2               −= 32 6 L x L x EI PaL y PaLxx L a PyEI L x EI PaL dx dy PaLx L a P dx dy EI 6 1 6 1 31 66 1 2 1 3 2 2 +−=               −=+−= Substituting,
© 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourth Edition Beer • Johnston • DeWolf 9 - 12 Sample Problem 9.1 • Locate point of zero slope or point of maximum deflection.               −= 32 6 L x L x EI PaL y L L x L x EI PaL dx dy m m 577.0 3 31 6 0 2 ==               −== • Evaluate corresponding maximum deflection. ( )[ ]3 2 max 577.0577.0 6 −= EI PaL y EI PaL y 6 0642.0 2 max = ( )( )( ) ( )( )46 2 max in723psi10296 in180in48kips50 0642.0 × =y in238.0max =y
© 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourth Edition Beer • Johnston • DeWolf 9 - 13 Sample Problem 9.3 For the uniform beam, determine the reaction at A, derive the equation for the elastic curve, and determine the slope at A. (Note that the beam is statically indeterminate to the first degree) SOLUTION: • Develop the differential equation for the elastic curve (will be functionally dependent on the reaction at A). • Integrate twice and apply boundary conditions to solve for reaction at A and to obtain the elastic curve. • Evaluate the slope at A.
© 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourth Edition Beer • Johnston • DeWolf 9 - 14 Sample Problem 9.3 • Consider moment acting at section D, L xw xRM M x L xw xR M A A D 6 0 32 1 0 3 0 2 0 −= =−         − =∑ L xw xRM dx yd EI A 6 3 0 2 2 −== • The differential equation for the elastic curve,
© 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourth Edition Beer • Johnston • DeWolf 9 - 15 Sample Problem 9.3 L xw xRM dx yd EI A 6 3 0 2 2 −== • Integrate twice 21 5 03 1 4 02 1206 1 242 1 CxC L xw xRyEI C L xw xREI dx dy EI A A ++−= +−== θ • Apply boundary conditions: 0 1206 1 :0,at 0 242 1 :0,at 0:0,0at 21 4 03 1 3 02 2 =++−== =+−== === CLC Lw LRyLx C Lw LRLx Cyx A Aθ • Solve for reaction at A 0 30 1 3 1 4 0 3 =− LwLRA ↑= LwRA 0 10 1
© 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourth Edition Beer • Johnston • DeWolf 9 - 16 Sample Problem 9.3 xLw L xw xLwyEI       −−      = 3 0 5 03 0 120 1 12010 1 6 1 ( )xLxLx EIL w y 43250 2 120 −+−= • Substitute for C1, C2, and RA in the elastic curve equation, ( )42240 65 120 LxLx EIL w dx dy −+−==θ EI Lw A 120 3 0=θ • Differentiate once to find the slope, at x = 0,
© 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourth Edition Beer • Johnston • DeWolf 9 - 17 Method of Superposition Principle of Superposition: • Deformations of beams subjected to combinations of loadings may be obtained as the linear combination of the deformations from the individual loadings • Procedure is facilitated by tables of solutions for common types of loadings and supports.
© 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourth Edition Beer • Johnston • DeWolf 9 - 18 Sample Problem 9.7 For the beam and loading shown, determine the slope and deflection at point B. SOLUTION: Superpose the deformations due to Loading I and Loading II as shown.
© 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourth Edition Beer • Johnston • DeWolf 9 - 19 Sample Problem 9.7 Loading I ( ) EI wL IB 6 3 −=θ ( ) EI wL y IB 8 4 −= Loading II ( ) EI wL IIC 48 3 =θ ( ) EI wL y IIC 128 4 = In beam segment CB, the bending moment is zero and the elastic curve is a straight line. ( ) ( ) EI wL IICIIB 48 3 == θθ ( ) EI wLL EI wL EI wL y IIB 384 7 248128 434 =      +=
© 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourth Edition Beer • Johnston • DeWolf 9 - 20 Sample Problem 9.7 ( ) ( ) EI wL EI wL IIBIBB 486 33 +−=+= θθθ ( ) ( ) EI wL EI wL yyy IIBIBB 384 7 8 44 +−=+= EI wL B 48 7 3 −=θ EI wL yB 384 41 4 −= Combine the two solutions,
© 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourth Edition Beer • Johnston • DeWolf 9 - 21 Application of Superposition to Statically Indeterminate Beams • Method of superposition may be applied to determine the reactions at the supports of statically indeterminate beams. • Designate one of the reactions as redundant and eliminate or modify the support. • Determine the beam deformation without the redundant support. • Treat the redundant reaction as an unknown load which, together with the other loads, must produce deformations compatible with the original supports.
© 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourth Edition Beer • Johnston • DeWolf 9 - 22 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.
© 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourth Edition Beer • Johnston • DeWolf 9 - 23 Sample Problem 9.8 • Distributed Loading: ( ) [ ]xLLxx EI w y wB 334 2 24 +−−= ( ) EI wL LLLLL EI w y wB 4 3 34 01132.0 3 2 3 2 2 3 2 24 −=               +      −      −= At point B, Lx 3 2=
© 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourth Edition Beer • Johnston • DeWolf 9 - 24 Sample Problem 9.8 • Redundant Reaction Loading: EIL bPa yax 3 ,At 22 −== ( ) EI LR L L EIL R y B B RB 3 22 01646.0 33 2 3 =             = LbLa 3 1 3 2 andFor ==
© 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourth Edition Beer • Johnston • DeWolf 9 - 25 Sample Problem 9.8 • From statics, ↑=↑= wLRwLR CA 0413.0271.0 • For compatibility with original supports, yB = 0 ( ) ( ) EI LR EI wL yy B RBwB 34 01646.001132.00 +−=+= ↑= wLRB 688.0
© 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourth Edition Beer • Johnston • DeWolf 9 - 26 Sample Problem 9.8 ( ) EI wL EI wL wA 33 04167.0 24 −=−=θ ( ) ( ) EI wLL L L EIL wL EIL bLPb RA 32 2 22 03398.0 336 0688.0 6 =               −      = − −=θ ( ) ( ) EI wL EI wL RAwAA 33 03398.004167.0 +−=+= θθθ EI wL A 3 00769.0−=θ Slope at end A,
© 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourth Edition Beer • Johnston • DeWolf 9 - 27 Moment-Area Theorems • Geometric properties of the elastic curve can be used to determine deflection and slope. ∫ ∫∫ =− = == D C D C D C x x CD x x dx EI M dx EI M d EI M dx yd dx d θθ θ θ θ θ 2 2 • Consider a beam subjected to arbitrary loading, • First Moment-Area Theorem: area under (M/EI) diagram between C and D. =CDθ
© 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourth Edition Beer • Johnston • DeWolf 9 - 28 Moment-Area Theorems • 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. • Tangents to the elastic curve at P and P’ intercept a segment of length dt on the vertical through C. ∫= == D C x x DC dx EI M xt dx EI M xdxdt 1 11 θ = tangential deviation of C with respect to D
© 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourth Edition Beer • Johnston • DeWolf 9 - 29 Application to Cantilever Beams and Beams With Symmetric Loadings • Cantilever beam - Select tangent at A as the reference. ADD ADD A ty θ = = = θθ ,0with • Simply supported, symmetrically loaded beam - select tangent at C as the reference. CBB CBB C ty θ = = = θθ ,0with
© 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourth Edition Beer • Johnston • DeWolf 9 - 30 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.
© 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourth Edition Beer • Johnston • DeWolf 9 - 31 Sample Problem 9.11 For the prismatic beam shown, determine the slope and deflection at E. SOLUTION: • Determine the reactions at supports. • Construct shear, bending moment and (M/EI) diagrams. • Taking the tangent at C as the reference, evaluate the slope and tangential deviations at E.
© 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourth Edition Beer • Johnston • DeWolf 9 - 32 Sample Problem 9.11 SOLUTION: • Determine the reactions at supports. waRR DB == • Construct shear, bending moment and (M/EI) diagrams. ( ) EI wa a EI wa A EI LwaL EI wa A 623 1 422 32 2 22 1 −=         −= −=      −=
© 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourth Edition Beer • Johnston • DeWolf 9 - 33 Sample Problem 9.11 • Slope at E: EI wa EI Lwa AA CECECE 64 32 21 −−=+= =+= θθθθ ( )aL EI wa E 23 12 2 +−=θ       −−      −−−=             −            +      += −= EI Lwa EI wa EI Lwa EI Lwa L A a A L aA tty CDCEE 168164 44 3 4 224223 121 ( )aL EI wa yE +−= 2 8 3 • Deflection at E:
© 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourth Edition Beer • Johnston • DeWolf 9 - 34 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. L t AB A −=θ • The slope at other points is found with respect to reference tangent. ADAD θθθ += • The deflection at D is found from the tangential deviation at D. ABADD AB t L x tEFEDy t L x EF L HB x EF −=−= ==
© 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourth Edition Beer • Johnston • DeWolf 9 - 35 Maximum Deflection • Maximum deflection occurs at point K where the tangent is horizontal. AAK AKAK AB A L t θθ θθθ θ −= +== −= 0 • 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.
© 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourth Edition Beer • Johnston • DeWolf 9 - 36 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.

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9 beam deflection- Mechanics of Materials - 4th - Beer

  • 1. MECHANICS OF MATERIALS Fourth Edition Ferdinand P. Beer E. Russell Johnston, Jr. John T. DeWolf Lecture Notes: J. Walt Oler Texas Tech University CHAPTER © 2006 The McGraw-Hill Companies, Inc. All rights reserved. 9 Deflection of Beams
  • 2. © 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourth Edition Beer • Johnston • DeWolf 9 - 2 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 9.3 Method of Superposition Sample Problem 9.7 Application of Superposition to Statically Indeterminate ... Sample Problem 9.8 Moment-Area Theorems Application to Cantilever Beams and Beams With Symmetric ... Bending Moment Diagrams by Parts Sample Problem 9.11 Application of Moment-Area Theorems to Beams With Unsymme... Maximum Deflection Use of Moment-Area Theorems With Statically Indeterminate...
  • 3. © 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourth Edition Beer • Johnston • DeWolf 9 - 3 Deformation of a Beam Under Transverse Loading • Relationship between bending moment and curvature for pure bending remains valid for general transverse loadings. EI xM )(1 = ρ • Cantilever beam subjected to concentrated load at the free end, EI Px −= ρ 1 • Curvature varies linearly with x • At the free end A, ∞== A A ρ ρ ,0 1 • At the support B, PL EI B B =≠ ρ ρ ,0 1
  • 4. © 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourth Edition Beer • Johnston • DeWolf 9 - 4 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. EI xM )(1 = ρ • 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.
  • 5. © 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourth Edition Beer • Johnston • DeWolf 9 - 5 Equation of the Elastic Curve • From elementary calculus, simplified for beam parameters, 2 2 232 2 2 1 1 dx yd dx dy dx yd ≈               + = ρ • Substituting and integrating, ( ) ( ) ( ) 21 00 1 0 2 2 1 CxCdxxMdxyEI CdxxM dx dy EIEI xM dx yd EIEI xx x ++= +=≈ == ∫∫ ∫θ ρ
  • 6. © 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourth Edition Beer • Johnston • DeWolf 9 - 6 Equation of the Elastic Curve ( ) 21 00 CxCdxxMdxyEI xx ++= ∫∫ • Constants are determined from boundary conditions • Three cases for statically determinant beams, – Simply supported beam 0,0 == BA yy – Overhanging beam 0,0 == BA yy – Cantilever beam 0,0 == AAy θ • More complicated loadings require multiple integrals and application of requirement for continuity of displacement and slope.
  • 7. © 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourth Edition Beer • Johnston • DeWolf 9 - 7 Direct Determination of the Elastic Curve From the Load Distribution • Equation for beam displacement becomes ( )xw dx yd EI dx Md −== 4 4 2 2 ( ) ( ) 43 2 22 13 16 1 CxCxCxC dxxwdxdxdxxyEI ++++ −= ∫∫∫∫ • Integrating four times yields • For a beam subjected to a distributed load, ( ) ( )xw dx dV dx Md xV dx dM −=== 2 2 • Constants are determined from boundary conditions.
  • 8. © 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourth Edition Beer • Johnston • DeWolf 9 - 8 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 000 =∑=∑=∑ Ayx MFF The beam is statically indeterminate. ( ) 21 00 CxCdxxMdxyEI xx ++= ∫∫ • Also have the beam deflection equation, which introduces two unknowns but provides three additional equations from the boundary conditions: 0,At00,0At ===== yLxyx θ
  • 9. © 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourth Edition Beer • Johnston • DeWolf 9 - 9 Sample Problem 9.1 ft4ft15kips50 psi1029in7236814 64 === ×==× aLP EIW For portion AB of the overhanging beam, (a) derive the equation for the elastic curve, (b) determine the maximum deflection, (c) evaluate ymax. SOLUTION: • Develop an expression for M(x) and derive differential equation for elastic curve. • Integrate differential equation twice and apply boundary conditions to obtain elastic curve. • Locate point of zero slope or point of maximum deflection. • Evaluate corresponding maximum deflection.
  • 10. © 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourth Edition Beer • Johnston • DeWolf 9 - 10 Sample Problem 9.1 SOLUTION: • Develop an expression for M(x) and derive differential equation for elastic curve. - Reactions: ↑      +=↓= L a PR L Pa R BA 1 - From the free-body diagram for section AD, ( )Lxx L a PM <<−= 0 x L a P dx yd EI −=2 2 - The differential equation for the elastic curve,
  • 11. © 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourth Edition Beer • Johnston • DeWolf 9 - 11 Sample Problem 9.1 PaLCLCL L a PyLx Cyx 6 1 6 1 0:0,at 0:0,0at 11 3 2 =+−=== === • Integrate differential equation twice and apply boundary conditions to obtain elastic curve. 21 3 1 2 6 1 2 1 CxCx L a PyEI Cx L a P dx dy EI ++−= +−= x L a P dx yd EI −=2 2               −= 32 6 L x L x EI PaL y PaLxx L a PyEI L x EI PaL dx dy PaLx L a P dx dy EI 6 1 6 1 31 66 1 2 1 3 2 2 +−=               −=+−= Substituting,
  • 12. © 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourth Edition Beer • Johnston • DeWolf 9 - 12 Sample Problem 9.1 • Locate point of zero slope or point of maximum deflection.               −= 32 6 L x L x EI PaL y L L x L x EI PaL dx dy m m 577.0 3 31 6 0 2 ==               −== • Evaluate corresponding maximum deflection. ( )[ ]3 2 max 577.0577.0 6 −= EI PaL y EI PaL y 6 0642.0 2 max = ( )( )( ) ( )( )46 2 max in723psi10296 in180in48kips50 0642.0 × =y in238.0max =y
  • 13. © 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourth Edition Beer • Johnston • DeWolf 9 - 13 Sample Problem 9.3 For the uniform beam, determine the reaction at A, derive the equation for the elastic curve, and determine the slope at A. (Note that the beam is statically indeterminate to the first degree) SOLUTION: • Develop the differential equation for the elastic curve (will be functionally dependent on the reaction at A). • Integrate twice and apply boundary conditions to solve for reaction at A and to obtain the elastic curve. • Evaluate the slope at A.
  • 14. © 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourth Edition Beer • Johnston • DeWolf 9 - 14 Sample Problem 9.3 • Consider moment acting at section D, L xw xRM M x L xw xR M A A D 6 0 32 1 0 3 0 2 0 −= =−         − =∑ L xw xRM dx yd EI A 6 3 0 2 2 −== • The differential equation for the elastic curve,
  • 15. © 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourth Edition Beer • Johnston • DeWolf 9 - 15 Sample Problem 9.3 L xw xRM dx yd EI A 6 3 0 2 2 −== • Integrate twice 21 5 03 1 4 02 1206 1 242 1 CxC L xw xRyEI C L xw xREI dx dy EI A A ++−= +−== θ • Apply boundary conditions: 0 1206 1 :0,at 0 242 1 :0,at 0:0,0at 21 4 03 1 3 02 2 =++−== =+−== === CLC Lw LRyLx C Lw LRLx Cyx A Aθ • Solve for reaction at A 0 30 1 3 1 4 0 3 =− LwLRA ↑= LwRA 0 10 1
  • 16. © 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourth Edition Beer • Johnston • DeWolf 9 - 16 Sample Problem 9.3 xLw L xw xLwyEI       −−      = 3 0 5 03 0 120 1 12010 1 6 1 ( )xLxLx EIL w y 43250 2 120 −+−= • Substitute for C1, C2, and RA in the elastic curve equation, ( )42240 65 120 LxLx EIL w dx dy −+−==θ EI Lw A 120 3 0=θ • Differentiate once to find the slope, at x = 0,
  • 17. © 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourth Edition Beer • Johnston • DeWolf 9 - 17 Method of Superposition Principle of Superposition: • Deformations of beams subjected to combinations of loadings may be obtained as the linear combination of the deformations from the individual loadings • Procedure is facilitated by tables of solutions for common types of loadings and supports.
  • 18. © 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourth Edition Beer • Johnston • DeWolf 9 - 18 Sample Problem 9.7 For the beam and loading shown, determine the slope and deflection at point B. SOLUTION: Superpose the deformations due to Loading I and Loading II as shown.
  • 19. © 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourth Edition Beer • Johnston • DeWolf 9 - 19 Sample Problem 9.7 Loading I ( ) EI wL IB 6 3 −=θ ( ) EI wL y IB 8 4 −= Loading II ( ) EI wL IIC 48 3 =θ ( ) EI wL y IIC 128 4 = In beam segment CB, the bending moment is zero and the elastic curve is a straight line. ( ) ( ) EI wL IICIIB 48 3 == θθ ( ) EI wLL EI wL EI wL y IIB 384 7 248128 434 =      +=
  • 20. © 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourth Edition Beer • Johnston • DeWolf 9 - 20 Sample Problem 9.7 ( ) ( ) EI wL EI wL IIBIBB 486 33 +−=+= θθθ ( ) ( ) EI wL EI wL yyy IIBIBB 384 7 8 44 +−=+= EI wL B 48 7 3 −=θ EI wL yB 384 41 4 −= Combine the two solutions,
  • 21. © 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourth Edition Beer • Johnston • DeWolf 9 - 21 Application of Superposition to Statically Indeterminate Beams • Method of superposition may be applied to determine the reactions at the supports of statically indeterminate beams. • Designate one of the reactions as redundant and eliminate or modify the support. • Determine the beam deformation without the redundant support. • Treat the redundant reaction as an unknown load which, together with the other loads, must produce deformations compatible with the original supports.
  • 22. © 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourth Edition Beer • Johnston • DeWolf 9 - 22 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.
  • 23. © 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourth Edition Beer • Johnston • DeWolf 9 - 23 Sample Problem 9.8 • Distributed Loading: ( ) [ ]xLLxx EI w y wB 334 2 24 +−−= ( ) EI wL LLLLL EI w y wB 4 3 34 01132.0 3 2 3 2 2 3 2 24 −=               +      −      −= At point B, Lx 3 2=
  • 24. © 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourth Edition Beer • Johnston • DeWolf 9 - 24 Sample Problem 9.8 • Redundant Reaction Loading: EIL bPa yax 3 ,At 22 −== ( ) EI LR L L EIL R y B B RB 3 22 01646.0 33 2 3 =             = LbLa 3 1 3 2 andFor ==
  • 25. © 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourth Edition Beer • Johnston • DeWolf 9 - 25 Sample Problem 9.8 • From statics, ↑=↑= wLRwLR CA 0413.0271.0 • For compatibility with original supports, yB = 0 ( ) ( ) EI LR EI wL yy B RBwB 34 01646.001132.00 +−=+= ↑= wLRB 688.0
  • 26. © 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourth Edition Beer • Johnston • DeWolf 9 - 26 Sample Problem 9.8 ( ) EI wL EI wL wA 33 04167.0 24 −=−=θ ( ) ( ) EI wLL L L EIL wL EIL bLPb RA 32 2 22 03398.0 336 0688.0 6 =               −      = − −=θ ( ) ( ) EI wL EI wL RAwAA 33 03398.004167.0 +−=+= θθθ EI wL A 3 00769.0−=θ Slope at end A,
  • 27. © 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourth Edition Beer • Johnston • DeWolf 9 - 27 Moment-Area Theorems • Geometric properties of the elastic curve can be used to determine deflection and slope. ∫ ∫∫ =− = == D C D C D C x x CD x x dx EI M dx EI M d EI M dx yd dx d θθ θ θ θ θ 2 2 • Consider a beam subjected to arbitrary loading, • First Moment-Area Theorem: area under (M/EI) diagram between C and D. =CDθ
  • 28. © 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourth Edition Beer • Johnston • DeWolf 9 - 28 Moment-Area Theorems • 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. • Tangents to the elastic curve at P and P’ intercept a segment of length dt on the vertical through C. ∫= == D C x x DC dx EI M xt dx EI M xdxdt 1 11 θ = tangential deviation of C with respect to D
  • 29. © 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourth Edition Beer • Johnston • DeWolf 9 - 29 Application to Cantilever Beams and Beams With Symmetric Loadings • Cantilever beam - Select tangent at A as the reference. ADD ADD A ty θ = = = θθ ,0with • Simply supported, symmetrically loaded beam - select tangent at C as the reference. CBB CBB C ty θ = = = θθ ,0with
  • 30. © 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourth Edition Beer • Johnston • DeWolf 9 - 30 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.
  • 31. © 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourth Edition Beer • Johnston • DeWolf 9 - 31 Sample Problem 9.11 For the prismatic beam shown, determine the slope and deflection at E. SOLUTION: • Determine the reactions at supports. • Construct shear, bending moment and (M/EI) diagrams. • Taking the tangent at C as the reference, evaluate the slope and tangential deviations at E.
  • 32. © 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourth Edition Beer • Johnston • DeWolf 9 - 32 Sample Problem 9.11 SOLUTION: • Determine the reactions at supports. waRR DB == • Construct shear, bending moment and (M/EI) diagrams. ( ) EI wa a EI wa A EI LwaL EI wa A 623 1 422 32 2 22 1 −=         −= −=      −=
  • 33. © 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourth Edition Beer • Johnston • DeWolf 9 - 33 Sample Problem 9.11 • Slope at E: EI wa EI Lwa AA CECECE 64 32 21 −−=+= =+= θθθθ ( )aL EI wa E 23 12 2 +−=θ       −−      −−−=             −            +      += −= EI Lwa EI wa EI Lwa EI Lwa L A a A L aA tty CDCEE 168164 44 3 4 224223 121 ( )aL EI wa yE +−= 2 8 3 • Deflection at E:
  • 34. © 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourth Edition Beer • Johnston • DeWolf 9 - 34 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. L t AB A −=θ • The slope at other points is found with respect to reference tangent. ADAD θθθ += • The deflection at D is found from the tangential deviation at D. ABADD AB t L x tEFEDy t L x EF L HB x EF −=−= ==
  • 35. © 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourth Edition Beer • Johnston • DeWolf 9 - 35 Maximum Deflection • Maximum deflection occurs at point K where the tangent is horizontal. AAK AKAK AB A L t θθ θθθ θ −= +== −= 0 • 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.
  • 36. © 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourth Edition Beer • Johnston • DeWolf 9 - 36 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.