Beam Deflection (9.1-9.5) MAE 314 – Solid Mechanics Yun Jing Beam Deflection 1 Introduction Up to now, we have been primarily calculating normal and shear stresses. In Chapter 9, we will learn how to formulate the deflection curve (also known as the elastic curve) of a beam. Beam Deflection 2 Differential Equation of Deflection d ds d ds 1 dy ds Recall from Ch. 4 that 1/ρ is the curvature of the beam. dy θ dx dx dx y Beam Deflection cos ds dy Slope of the deflection curve tan sin ds 3 Assumptions Assumption 1: θ is small. 1 1. ds dx d ds 2. dy dx tan d dx d 2 dx d y dx 2 1 Assumption 2: Beam is linearly elastic. 2 d y dx 1 2 M EI Thus, the differential equation for the deflection curve is: 2 d y dx 2 M EI Beam Deflection 4 Diff. Equations for M, V, and w Recall from Ch. 5: dV w dM dx So we can write: dx 2 EI d y dx 2 3 M EI d y dx 3 4 V EI d y dx 4 w Deflection curve can be found by integrating V Bending moment equation (2 constants of integration) Shear-force equation (3 constants of integration) Load equation (4 constants of integration) Chosen method depends on which is more convenient. Beam Deflection 5 Boundary Conditions Sometimes a single equation is sufficient for the entire length of the beam, sometimes it must be divided into sections. Since we integrate twice there will be two constants of integration for each section. These can be solved using boundary conditions. Deflections and slopes at supports Known moment and shear conditions V B 0 M B 0 M A Beam Deflection 0 M B 0 6 Boundary Conditions Section AC: yAC(x) Continuity conditions: Displacement continuity y AC ( C ) y CB ( C ) Slope continuity dy AC dy CB ( C ) AC ( C ) ( C ) CB ( C ) dx dx Symmetry conditions: Section CB: yCB(x) dy 0 dx Beam Deflection 7 Example Problem For the beam and loading shown, (a) express the magnitude and location of the maximum deflection in terms of w0, L, E, and I, (b) Calculate the value of the maximum deflection, assuming that beam AB is a W18 x 50 rolled shape and that w0 = 4.5 kips/ft, L = 18 ft, and E = 29 x 106 psi. Beam Deflection 8 Statically Indeterminate Beams When there are more reactions than can be solved using statics, the beam is indeterminate. Take advantage of boundary conditions to solve indeterminate problems. Problem: Number of reactions: 3 (MA, Ay, By) x=0, y=0 x=0, θ=0 x=L, y=0 Number of equations: 2 (Σ M = 0, Σ Fy = 0) One too many reactions! Additionally, if we solve for the deflection curve, we will have two constants of integration, which adds two more unknowns! Solution: Boundary conditions Beam Deflection 9 Statically Indeterminate Beams Problem: Number of reactions: 4 (MA, Ay, MB, By) x=0, y=0 x=0, θ=0 x=L, y=0 x=0, θ=0 Number of equations: 2 (Σ M = 0, Σ Fy = 0) + 2 constants of integration Solution: Boundary conditions Beam Deflection 10 Example Problem For the beam and loading shown, determine the reaction at the roller support. Beam Deflection 11 Beam Deflection: Method of Superposition (9.7-9.8) MAE 314 – Solid Mechanics Yun Jing Beam Deflection: Method of Superposition 12 Method of Superposition Deflection and slope of a beam produced by multiple loads acting simultaneously can be found by superposing the deflections produced by the same loads acting separately. Reference Appendix D (Beam Deflections and Slopes) Method of superposition can be applied to statically determinate and statically indeterminate beams. Beam Deflection: Method of Superposition 13 Superposition Consider sample problem 9.9 in text. Find reactions at A and C. Method 1: Choose MC and RC as redundant. Method 2: Choose MC and MA as redundant. Beam Deflection: Method of Superposition 14 Example Problem For the beam and loading shown, determine (a) the deflection at C, (b) the slope at A Beam Deflection: Method of Superposition 15 Example Problem For the beam and loading shown, determine (a) the deflection at C, and (b) the slope at end A. Beam Deflection: Method of Superposition 16 Example Problem For the beam shown, determine the reaction at B. Beam Deflection: Method of Superposition 17

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