(Solution) - a A simple beam AB with length L and height -(2025 Original AI-Free Solution)

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(a) A simple beam AB with length L and height h supports a uniform load of intensity q (see the first part of the figure). Obtain a formula for the curvature shortening ? of this beam. Also, obtain a formula for the maximum bending stress ?b in the beam due to the load q.
(b) Now assume that the ends of the beam are pinned so that curvature shortening is prevented and a horizontal force H develops at the supports (see the second part of the figure). Obtain a formula for the corresponding axial tensile stress ?t.

(c) Using the formulas in parts (a) and (b), calculate the curvature shortening ? l, the maximum bending stress ?b, and the tensile stress ?t for the following steel beam: length L = 3 m, height h = 300 mm, modulus of elasticity E = 200 GPa, and moment of inertiaI = 36x106 mm4. Also, the load on the beam has intensity q = 25 kN/m.
Compare the tensile stress ?t produced by the axial forces with the maximum bending stress ?b produced by the uniform load.