Meriam, Kraige & Bolton — Distributed Forces: Centroids and Centers of Gravity: Problem 5_92

⚡ Mecademy AIENG정역학 · ch5  Problem Statement Determine the force and moment reactions at the support A of the built-in beam which is subjected to the sine-wave load distribution shown in the figure. The beam has a length and the maximum intensity of the distributed load is . Problem 5/92 (a) Vertical Force Reaction at Support A ● Calculation Process 1. Present Final Formula: The vertical force reaction must equal the total downward load for equilibrium. The load intensity is defined by the function . 2. Substitute Values: 3. Partial Operations: Performing the integration: Factor out the constant: Integrate: , where . Evaluate at boundaries: Compute limits: 4. Final Calculation: ● Final Conclusion: The vertical force reaction at support A is acting upwards. l w 0 R A W w(x)=w sin 0 ( l πx ) R = A w (x)dx ∫ 0 l R = A w sin dx ∫ 0 l 0 ( l πx ) R = A w sin dx 0 ∫ 0 l ( l πx ) sin(kx)dx=∫− cos(kx) k 1 k= l π R= A w − cos 0 [ π l ( l πx )] 0 l R = A − cosπ−cos0 = π w l 0 () − −1−1 π w l 0 () R = A − (−2)= π w l 0 π 2w l 0 R = A π 2w l 0 (b) Bending Moment Reaction at Support A ● Calculation Process 1. Present Final Formula: The counter-clockwise moment reaction at support A must balance the clockwise moment created by the distributed load. 2. Substitute Values: 3. Partial Operations: Using integration by parts : Let Let Applying the formula: Evaluating the first term: Integrating the second term: Evaluating second term limits: 4. Final Calculation: ● Final Conclusion: The moment r