MEE 320 Chapter 9, Spring 2018

MEE 320: Strength of Materials Chapter 9 Deflection of Beams MEE 320: Strength of Materials 2 Introduction A new aspect in the design of beams will be considered in this chapter. Previously we designed beams for strength. In this part we will consider design of beams including the maximum deflection of the beam in the design specifications. The photo shows a cable-stayed bridge under construction. Its design is based on both strength considerations and deflection evaluations. MEE 320: Strength of Materials 3 Deformation Under Transverse Loading Relationship between bending moment and curvature for pure bending. EI M x ( ) 1   Example: Cantilever beam subjected to concentrated load at the free end. EI   Px 1 Curvature varies linearly with x At the free end A, At the support B, Px M x )  (    A M  0 0) ( PL EI PL L M B     ( ) MEE 320: Strength of Materials 4 Curvature is zero at points where the bending moment is zero, i.e., at each end and at E. EI M x ( ) 1   Between A and E: M(x)>0 concave upwards Between E and D: M(x)<0 concave downwards Max moment Max curvature An equation for the beam shape or elastic curve is required to determine maximum deflection and slope. Deformation Under Transverse Loading Example: overhanging beam. MEE 320: Strength of Materials 5 Equation of the Elastic Curve The curvature of a plane curve at a point Q(x,y) can be written as 3 2 2 2 2 1 1               dx dy dx y d  2 2 1 dx d y  In a beam is small dx dy and EI M x ( ) 1   The governing differential equation for the elastic curve EI x M dx d y ) ( 2 2  EI is the flexural rigidity MEE 320: Strength of Materials 6 Equation of the Elastic Curve     2 1 0 0 1 0 C C x M x dx dx y EI C M x dx dx dy EI x x x              Integrate the governing differential equation twice with respect to x   x dx dy   tan ) ( 2 2 M x dx EI d y  The constants C1 and C2 are calculated from the boundary conditions Beam deflection   x y Beam slope MEE 320: Strength of Materials 7 Equation of the Elastic Curve Determination of C1 and C2 Three cases for statically determinant beams, – Simply supported beam 0 ,0   B A y y – Overhanging beam – Cantilever beam 0 ,0   A yA      2 1 0 0 1 0 C C x M x dx dx y EI C M x dx EI x x x               0 ,0   B A y y MEE 320: Strength of Materials 8 Example 1 9-1: For the loading shown, determine (a) the equation of the elastic curve for the cantilever beam AB, (b) the deflection at the free end, (c) the slope at the free end. MEE 320: Strength of Materials 9 Example 1 MEE 320: Strength of Materials 10 Example 1 MEE 320: Strength of Materials 11 Example 2 9-8: For the beam and loading shown, determine (a) the equation of the elastic curve for portion AB of the beam, (b) the slope at A, (c) the slope at B. MEE 320: Strength of Materials 12 Example 2 MEE 320: Strength of Materials 13 Example 3 9-15: For the beam and loading shown, knowing that a = 2 m, w = 50 kN/m, and E = 200 GPa, determine (a) the slope at support A, (b) the deflection at point C. MEE 320: Strength of Materials 14 Example 3 MEE 320: Strength of Materials 15 Direct Determination of the Elastic Curve Equation for beam displacement becomes   x w dx d y EI dx d M   4 4 2 2     4 3 2 2 3 1 2 1 6 1 C C x C x x C dx dx dx w x dx y x EI          Integrating four times yields For a beam subjected to a distributed load,     w x dx dV dx d M V x dx dM    2 2 Constants C1, C2, C3 and C4 are determined from boundary conditions. MEE 320: Strength of Materials 16 Statically Indeterminate Beams Conditions for static equilibrium yield 0 0 0       A y x M F F In a statically indeterminate beam, the deflection equations are used to solve for the reactions of the beam The beam deflection equations, which introduces 2 unknowns but provides 3 additional equations from the boundary conditions: 0 0 0 0 0          y L x x y x      2 1 0 0 1 0 C C x M x dx dx y EI C M x dx EI x x x               MEE 320: Strength of Materials 17 Example 4 9-21: For the beam and loading shown, determine the reaction at the roller support MEE 320: Strength of Materials 18 Example 4 MEE 320: Strength of Materials 19 Example 5 9-26: Determine the reaction at the roller support and draw the bending moment diagram for the beam and loading shown. MEE 320: Strength of Materials 20 Example 5 MEE 320: Strength of Materials 21 Example 6 9-33: Determine the reaction at A and draw the bending moment for the beam and loading shown. MEE 320: Strength of Materials 22 Example 6 MEE 320: Strength of Materials 23 Method of Superposition Procedure is facilitated by tables of solutions for common types of loadings and supports. For beams subjected to several concentrated or distributed loads: • Compute separately the deformation of each of the applied loads • The deformation of the combined load is found by adding the deformation of individual loads. MEE 320: Strength of Materials 24 Example 7 9-65: For the cantilever beam and loading shown, determine the slope and deflection at the free end. MEE 320: Strength of Materials 25 Example 7 MEE 320: Strength of Materials 26 Example 7 MEE 320: Strength of Materials 27 Example 8 9-80: For the uniform beam shown, determine (a) the reaction at A, (b) the reaction at B. MEE 320: Strength of Materials 28 Example 8 MEE 320: Strength of Materials 29 Example 8 MEE 320: Strength of Materials 30 Example 9 9-82: For the uniform beam shown, determine the reaction at each of the three supports. MEE 320: Strength of Materials 31 Example 9 MEE 320: Strength of Materials 32 Example 9