Meriam, Kraige & Bolton — Distributed Forces: Centroids and Centers of Gravity: Problem 5_128

⚡ Mecademy AIENG정역학 · ch5  Problem Statement For the beam subjected to the concentrated couples and distributed load, determine the maximum value of the internal bending moment and its location. At , the distributed load is increasing at the rate of per foot. Problem 5/128 (a) Loading Function and Support Reactions ● Calculation Process 1. Present Final Formula: The distributed load is given by . From the diagram, at , , so . The rate of increase at is . At support (), the load is . 2. Substitute Values to find : So, for . 3. Calculate Resultant Force and its Moment: 4. Equilibrium for Reactions: Assume both concentrated couples are counter- clockwise ( and ) based on vector consistency. x=0 10 lb/ft w(x)=w + 0 k x+ 1 k x 2 2 x=0w(0)=200 lb/ftw = 0 200 x=0 = dx dw x=0 k = 1 10 lb/ft 2 Bx=24 ft 500 lb/ft k 2 500=200+10(24)+k (24) 2 2 500=200+240+576k 2 60=576k ⟹ 2 k = 2 ≈ 48 5 0.10417 lb/ft 3 w(x)=200+10x+ x 48 52 0≤x≤24 ft W= 200+10x+ x dx= ∫ 0 24 ( 48 5 2 )200x+5x+ x = [ 2 144 5 3 ] 0 24 480 M = x=0 load x⋅ ∫ 0 24 w(x)dx=100x+ x+ x = [ 2 3 10 3 192 5 4 ] 0 24 57600+46 2700 lb-ft1500 lb-ft ● Final Conclusion: The support reactions are and . (b) Bending-Moment Distribution ● Calculation Process 1. Present Final Formula: The internal bending moment for the span is: where (internal moment at the left end to balance applied CCW couple). 2. Substitute Values: The load moment integral is . 3. Final Expression: ● Final Conclusion: The bending moment equation for the main span is .