Meriam, Kraige & Bolton — Distributed Forces: Centroids and Centers of Gravity: Problem 5_188

⚡ Mecademy AIENG정역학 · ch5  Problem Statement A prismatic structure of height and base is subjected to a horizontal wind load whose pressure increases from zero at the base to at the top according to . Determine the resisting moment at the base of the structure. Problem 5/188 (a) Determination of the resisting moment at the base ● Calculation Process 1. Present Final Formula: The total resisting moment at the base is equal to the moment produced by the distributed wind pressure about the base (where ). Since the structure is prismatic with height and width (assuming is the width of the face hit by the wind), the moment is given by: where is the differential force on an element of area . 2. Substitute Values: The pressure distribution is . At the top (), the pressure is , so: Thus, the pressure function is . The differential force is . Substituting these into the moment integral: 3. Partial Operations: Simplify the integrand and perform the integration: hb pp 0 p=k y M M y=0 hbb M= y d F∫ 0 h dFdA=bdy p=k yy=h p 0 p = 0 k ⟹hk= h p 0 p(y)= y h p 01/2 dF= p(y)dA= y (bd y ) ( h p 0 1/2 ) M= y y d y∫ 0 h ( h p b 0 1/2 ) M= y d y h p b 0 ∫ 0 h 3/2 Evaluate the definite integral: 4. Final Calculation: Substitute the result of the integral back into the expression for : ● Final Conclusion: The resisting moment at the base of the structure required to maintain equilibrium is This moment acts at the base to counteract the overturning effect of the wind load. ✨ Final Answer Summary (a)