Meriam, Kraige & Bolton — Distributed Forces: Centroids and Centers of Gravity: Problem 5_121

⚡ Mecademy AIENG정역학 · ch5  Problem Statement Draw the shear and moment diagrams for the linearly loaded simple beam shown. Determine the maximum magnitude of the bending moment . Problem 5/121 (a) Determination of Support Reactions 1. Formula: By symmetry of the loading and supports, the reactions at both ends must be equal. The total load is equal to the area under the load distribution curve. 2. Substitution: The load distribution is a triangle with a base length and a peak intensity at the center. Since the structure is symmetric: 3. Calculation: 4. Result: The vertical reactions at supports and are: ● Final Conclusion: Each support reaction is upward with a magnitude of . (b) Shear and Moment Diagrams (Analysis) M W F = ∑ y 0⟹R + A R = B w (x)dx ∫ 0 l l w 0 W= ⋅ 2 1 base⋅height= ⋅ 2 1 l⋅w 0 R = A R = B 2 W R = A R = B w l = 2 1 ( 2 1 0 ) 4 w l 0 AB R = A R = B 4 w l 0 4 w l 0 1. Formula: For the left half of the beam (), the load intensity is . The shear force and bending moment are found by: 2. Substitution: Boundary conditions at are and . 3. Calculation: Shear force for : This is a concave-down parabolic curve starting at and reaching at the center. Bending moment for : This is a cubic curve starting at and increasing to its peak at the center. 4. Result: The shear diagram consists of two parabolic segments symmetric about the center, where it crosses zero. The moment diagram consists of two cubic segments symmetric about the center, where it reaches its maximum value