Meriam, Kraige & Bolton — Distributed Forces: Centroids and Centers of Gravity: Problem 5_104

⚡ Mecademy AIENG정역학 · ch5  Problem Statement The transition between the loads of 10 kN/m and 37 kN/m is accomplished by means of a cubic function of form , the slope of which is zero at its endpoints m and m. Determine the reactions at and . Problem 5/104 (a) Determination of the Distributed Loading Function and Equivalent Resultants ● Calculation Process 1. Present Final Formula: The beam loading consists of three segments. We first determine the cubic loading function for the transition region . Let , such that the cubic function is for . The boundary conditions are: At (): kN/m, At (): kN/m, The total resultant force is and the total moment about is . 2. Substitute Values: From boundary conditions at : and . For : Substituting : , then . Thus, for , . 3. Partial Operations: Calculate resultant forces and moments for each segment: Segment 1 (): kN/m kN kNm w=k + 0 k x+ 1 k x+ 2 2 k x 3 3 x=1x=4AB w(x)1≤x≤4 u=x−1w(u)=a + 0 a u+ 1 a u+ 2 2 a u 3 3 0≤u≤3 u=0x=1w(0)=10w(0)= ′ 0 u=3x=4w(3)=37w(3)= ′ 0 R=w(x)dx∫AM = A x⋅∫w(x)dx u=0a = 0 10a = 1 0 u=3 10+9a + 2 27a = 3 37⟹a + 2 3a = 3 3 6a + 2 27a = 3 0⟹2a + 2 9a = 3 0⟹a = 2 −4.5a 3 a 2 −4.5a + 3 3a = 3 3⟹−1.5a = 3 3⟹a = 3 −2 a = 2 91≤x≤4w(u)=10+9u− 2 2u 3 R i M Ai 0≤x≤1w=10R = 1 10×1=10M = A1 10×0.5=5⋅ Segment 2 (): where kN kNm Segment 3 (): kN/m kN kNm 4. Final Calculation: Total Force: kN Total Moment about A: kNm ● Final Conclusion: The total equivalent concentrated load is kN acting at a distance of m from point . (b) Calcu