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Use singularity to find out what is the br equation for elastic curve y(x), slope at A and deflection at C? The cross section is W100X19.3 and E=75GPa. Did you notice something? A 1m C 42Nm 5m B

Use singularity to find out what is the br equation for elastic curve y(x), slope at A and deflection at C? The cross section is W100X19.3 and E=75GPa. Did you notice something? A 1m C 42Nm 5m B

Structural Analysis
Structural Analysis
6th Edition
ISBN:9781337630931
Author:KASSIMALI, Aslam.
Publisher:KASSIMALI, Aslam.
Chapter2: Loads On Structures
Section: Chapter Questions
Problem 1P
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### Understanding Elastic Curves and Deflections in Beams #### Problem Statement: Use singularity functions to find out what is the equation for the elastic curve \(y(x)\), the slope at point \(A\), and the deflection at point \(C\). The cross-section is specified as W100X19.3 and \(E = 75 GPa\). Did you notice something? #### Diagram Explanation: The provided diagram illustrates a beam supported at point \(A\) with a distance of 1 meter from support to point \(C\), where a moment \(42 Nm\) is applied. The distance from point \(C\) to the free end \(B\) is 5 meters. This setup forms a cantilever beam with an applied moment at point \(C\). #### Steps to Solve: 1. **Determine the Boundary Conditions:** - At point \(A\) (x = 0): The deflection \(y(0) = 0\) and the slope \(\theta(0)\) can be derived from the conditions of the supports and loadings. 2. **Write the Moment Equilibrium Equations:** - Integrate the moment equation considering the applied load, \(42 Nm\), at point \(C\), keeping in mind the location of \(C\) at 1 meter from the support \(A\). 3. **Apply Singularity Functions:** - Use singularity functions to express moments and shears in the beam. The moment at a certain location can be expressed using the Heaviside function \(H(x - a)\). 4. **Integrate the Moment-Curvature Relationship:** - The differential equation \(\frac{d^2y}{dx^2} = \frac{M(x)}{EI}\) governs the elastic curve, where \(E\) is the modulus of elasticity and \(I\) is the moment of inertia of the beam's cross section. 5. **Solve for Deflections and Slopes:** - Integrate the equation \(\frac{d^2y}{dx^2} = \frac{M(x)}{EI}\) twice to find the slope \(\frac{dy}{dx}\) and the deflection \(y(x)\). Apply boundary conditions to solve for integration constants. #### Cross-Section and Material Properties: - The cross-section is given as W100X19.3, which
Transcribed Image Text:### Understanding Elastic Curves and Deflections in Beams #### Problem Statement: Use singularity functions to find out what is the equation for the elastic curve \(y(x)\), the slope at point \(A\), and the deflection at point \(C\). The cross-section is specified as W100X19.3 and \(E = 75 GPa\). Did you notice something? #### Diagram Explanation: The provided diagram illustrates a beam supported at point \(A\) with a distance of 1 meter from support to point \(C\), where a moment \(42 Nm\) is applied. The distance from point \(C\) to the free end \(B\) is 5 meters. This setup forms a cantilever beam with an applied moment at point \(C\). #### Steps to Solve: 1. **Determine the Boundary Conditions:** - At point \(A\) (x = 0): The deflection \(y(0) = 0\) and the slope \(\theta(0)\) can be derived from the conditions of the supports and loadings. 2. **Write the Moment Equilibrium Equations:** - Integrate the moment equation considering the applied load, \(42 Nm\), at point \(C\), keeping in mind the location of \(C\) at 1 meter from the support \(A\). 3. **Apply Singularity Functions:** - Use singularity functions to express moments and shears in the beam. The moment at a certain location can be expressed using the Heaviside function \(H(x - a)\). 4. **Integrate the Moment-Curvature Relationship:** - The differential equation \(\frac{d^2y}{dx^2} = \frac{M(x)}{EI}\) governs the elastic curve, where \(E\) is the modulus of elasticity and \(I\) is the moment of inertia of the beam's cross section. 5. **Solve for Deflections and Slopes:** - Integrate the equation \(\frac{d^2y}{dx^2} = \frac{M(x)}{EI}\) twice to find the slope \(\frac{dy}{dx}\) and the deflection \(y(x)\). Apply boundary conditions to solve for integration constants. #### Cross-Section and Material Properties: - The cross-section is given as W100X19.3, which
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