Meriam, Kraige & Bolton — Distributed Forces: Centroids and Centers of Gravity: Problem 5_101

⚡ Mecademy AIENG정역학 · ch5  Problem Statement Determine the reactions at end A of the cantilever beam which is subjected to both linear and parabolic loads that act over the indicated regions. The slope of the distributed loading is continuous over the length of the beam. Problem 5/101 (a) Vertical reaction force at end A 1. Formula: The vertical reaction force is equal to the total area under the distributed load curve . For a cantilever beam fixed at (where ), the equilibrium equation is . 2. Substitution: Assuming the load starts from zero at the wall (), the linear load is . For the parabolic region, . Using , , and slope continuity , we find coefficients: , , and . 3. Calculation: Area of the linear region: . Area of the parabolic region: . Total vertical reaction: . 4. Result: ● Final Conclusion: The vertical reaction force at end is directed upwards. (b) Bending moment reaction at end A R A R A w(x)Ax=0 F =∑ y R − A w (x)dx= ∫ 0 L 0 R = A w (x)dx+ ∫ 0 9 1 w (x)dx ∫ 9 30 2 w(0)=0 w (x)= 1 x= 9 300 x 3 100 w (x)= 2 Ax+ 2 Bx+Cw (9)= 2 300w (30)= 2 100w (9)= 2 ′ w (9)= 1 ′ 3 100 A=− 49 100 B= 147 10300 C=− 49 8100 R = A x dx+ ∫ 0 9 ( 3 100 ) − x+ x− dx ∫ 9 30 ( 49 100 2 147 10300 49 8100 ) F = 1 x = [ 3 50 2 ] 0 9 (81)= 3 50 1350 lb F= 2 w (x)dx= ∫ 9 30 2 − x+ x− x = [ 147 100 3 147 5150 2 147 24300 ] 9 30 7350 lb R = A 1350+7350=8700 lb R = A 8700 lb A8700 lb M A 1. Formula: The reaction moment at the fixed support must balance the moment caused by the distributed load. S