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Mechanics of Materials – MAE 243 (Section 002) Spring 2008

Mechanics of Materials – MAE 243 (Section 002) Spring 2008. Dr. Konstantinos A. Sierros. Problem 4.3-1 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 4.3-12

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Mechanics of Materials – MAE 243 (Section 002) Spring 2008

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

  2. Problem 4.3-1 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.

  3. Problem 4.3-12 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.

  4. Problem 4.3-14 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. 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

  6. 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

  7. 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 curvatureand ρ is the radius of curvature κ For small deflections

  8. 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

  9. 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 εx are created from planes that either lenghten or shorten Strain-curvature relation

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