Meriam, Kraige & Bolton — Distributed Forces: Centroids and Centers of Gravity: Problem 5_124

⚡ Mecademy AIENG정역학 · ch5  Problem Statement For the beam and loading shown, determine equations for the internal shear force and bending moment at any location . State the values of the internal shear force and bending moment at and . The distributed load follows the equation and reaches a maximum intensity of at a distance of from the fixed support at . The total length of the cantilever beam is . Problem 5/124 (a) Equations for the internal shear force and bending moment 1. Formula: First, determine the constants and using the given loading conditions: Then, the internal forces are derived from the differential relations: 2. Substitution: From the maximum conditions: So, . Total load (shear at support ): V Mx x=2 mx=4 mw=k x− 1 k x 2 2 2 kN/m3 m A5 m VM k 1 k 2 w(x)=k x− 1 k x 2 2 = dx dw k − 1 2k x= 2 0 at x=3 m (location of maximum) w(3)=2 kN/m (maximum value) = dx dV − w (x)and = dx dM V(x) k − 1 2k (3)= 2 0⟹k = 1 6k 2 (6k )(3)− 2 k (3)= 2 2 2⟹18k − 2 9k = 2 2⟹9k = 2 2⟹k = 2 k = 1 6 = ( 9 2 ) kN/m 3 4 2 w(x)= x− 3 4 x 9 22 A R = A w (x)dx= ∫ 0 5 x− x dx= ∫ 0 5 ( 3 4 9 2 2 ) x− x = [ 3 2 2 27 2 3 ] 0 5 − 3 502 Moment at support : 3. Calculation: Integrating for shear force : Integrating for bending moment : 4. Result: ● Final Conclusion: The general equations for internal shear and bending moment are and , where is in meters. (b) Internal shear force and bending moment at 1. Formula: Substitute into the equations derived in sub-question (a). 2. Substitution: 3. Calculat