Mechanics of Materials – MAE 243 (Section 002) Spring 2008 Dr. Konstantinos A. Sierros

Problem Calculate the shear force V and bending moment M at a cross section just to the left of the 1600-lb load acting on the simple beam AB shown in the figure.

Problem A simply supported beam AB supports a trapezoidally distributed load (see figure). The intensity of the load varies linearly from 50 kN/m at support A to 30 kN/m at support B. Calculate the shear force V and bending moment M at the midpoint of the beam.

Problem The simply-supported beam ABCD is loaded by a weight W = 27 kN through the arrangement shown in the figure. The cable passes over a small frictionless pulley at B and is attached at E to the end of the vertical arm. Calculate the axial force N, shear force V, and bending moment M at section C, which is just to the left of the vertical arm. (Note: Disregard the widths of the beam and vertical arm and use centerline dimensions when making calculations.)

5.1: Stresses in beams - Introduction Stresses and strains associated with shear forces and bending moments of beams The loads acting on the beam cause it to bend. The initially straight axis (fig 5-1a) is bent into a curve which is called the deflection curve (fig 5-1b) All loads act in the xy plane, known as the plane of bending The deflection of the beam at any point along its axis is the displacement of that point from its original position, measured in the y direction

5.2:Pure bending and non-uniform bending Pure bending refers to flexure of a beam under a constant bending moment (figs 5-2 and 5-3) Non-uniform bending refers to flexure in the presence of shear forces, which means that the bending moment changes as we move along the axis of the beam

5.3: Curvature of a beam When loads are applied to a beam, its longitudinal axis is deformed into a curve. The resulting strains and stresses in the beam are directly related to the curvature of the deflection curve Point O ’ is the center of curvature and ρ is the radius of curvature κ For small deflections

5.3: Curvature of a beam The sign convention for curvature depends upon the orientation of the coordinate axis Positive when the beam is bent concave upward Negative when the beam is bent concave downward

5.4: Longitudinal strains in beams The longitudinal strains in a beam can be found by analyzing the curvature of the beam and the associated deformations Cross-sections of the beam mn and pq remain plane and normal to the longitudinal axis Surface ss is called the neutral surface of the beam. Its intersection with any cross-sectional plane is called the neutral axis of the cross section Normal strains ε x are created from planes that either lenghten or shorten Strain-curvature relation