Fall, 2021 ME 323 – Mechanics of Materials Lecture 19 – Deflection of beams Reading assignment: 7.1 – 7.4 Instructor: Prof. Marcial Gonzalez Last modified: 8/16/21 9:20:26 AM Deflection of beams Beam theory (@ ME 323) J. Bernoulli L. Euler - Geometry of the solid body: Longitudinal Plane straight, slender member with of Symmetry constant cross section Longitudinal Axis that is design to support transverse loads. - Kinematic assumptions: Bernoulli-Euler Beam Theory - Material behavior: isotropic linear elastic material; small deformations. - Equilibrium: 1) relate stress distribution (normal and shear stress) with internal resultants (only shear and bending moment) 2) find deformed configuration 4 Deflection of beams Why do we study deflection of beams? Lateral deflection of H/500 Design of jumping poles Design of fishing poles Atomic force microscopy (commercially available since 1989) + Solving statically indeterminate beams!! 5 Deflection of beams Moment-curvature equation From Lecture 15: Relationship between the deflection inclination and the inclination angle (~slope): angle (~slope) deflection Relationship between the slope and the radius of curvature : Moment-curvature equation Note: second-order, ODE 6 Deflection of beams Load-deflection equation From Lecture 13: Using the moment-curvature equation Shear-deflection equation Load-deflection equation Note: fourth-order, ODE (constant cross-section and material properties) Shear-deflection equation Load-deflection equation Note: fourth-order, ODE 7 Deflection of beams Example 29: The uniformly loaded beam shown in the figure is completely fixed at end B. Determine an expression for the deflection curve . (a) Use the second-order method. (follow sign conventions) 8 Deflection of beams = | = | Boundary conditions (follow sign conventions) 9 Deflection of beams = | = | Continuity conditions >0 >0 10 Deflection of beams Example 29 (cont.): The uniformly loaded beam shown in the figure is completely fixed at end B. Determine an expression for the deflection curve . Plus boundary conditions Q: Maximum deflection? Q: Slope at free end? (follow sign conventions) 11 Deflection of beams Any questions? 12.