Structural Analysis
Structural Analysis
6th Edition
ISBN: 9781337630931
Author: KASSIMALI, Aslam.
Publisher: Cengage,
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Concept explainers

Beams
In structural engineering, a beam is a component made of a variety of materials (including steel alloys, and wood) that can resist loads applied laterally to the beam axis. Members, elements, rafters, shafts, and purlins are all terms used to describe be…
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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
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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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