Chapter 25: Deflection of Beams

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25.6:

Moment-Area Theorems

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Mechanical Engineering

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JoVE Core Mechanical Engineering

Moment-Area Theorems

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25.5: Method of Superposition

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25.7: Beams with Symmetric Loadings

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The moment-area theorem provides geometric properties of an elastic curve for determining deflection and slope at any point on the beam supporting a building floor. When plotted along the beam's length, the ratio M/EI creates a diagram similar to the bending moment diagram. The moment-area theorem derives the slope and tangential deviation equation from the beam diagram. The first moment-area theorem equates the angle between tangents at points P and Q to the area between these points under the M/EI diagram. The second moment-area theorem relates the tangential deviation of point P from Q to the first moment of the same area concerning the vertical axis through P. This theorem is also expressible in terms of the product of the area and the distance from the centroid of the area to the vertical axis through P. The tangential deviation is the vertical distance from Q to the tangent at P. It's calculated by multiplying the area under the M/EI diagram by the distance from its centroid to the vertical axis through Q.

25.6:

Moment-Area Theorems

The Moment-Area Theorem is crucial in structural engineering for analyzing beam bending, particularly in applications like building floor supports. This theorem utilizes the geometric properties of the elastic curve, which depicts how a beam deforms under load, to simplify the calculations of deflections and slopes.

The theorem is divided into two parts. The first part connects the angle between tangents at any two points on the beam's elastic curve to the area under a curve derived by plotting the quantity M/EI (where M is the bending moment, E is the modulus of elasticity, and I is the moment of inertia) against the beam's deflection along its length. The area under this curve directly corresponds to the total rotation occurring between these two points.

The second part of the theorem addresses the tangential deviation—or the vertical displacement—between any two points resulting from the beam's bending. It states that this deviation is equivalent to the first moment of the area under the M/EI curve about a vertical axis through one of these points, providing a measure of the beam segment's displacement from its original position. These theorems efficiently determine the slope and deflection at various points along a beam, essential for ensuring structural safety and performance under load.