Question

The ratio of area under the bending moment diagram to the flexural rigidity between two points of a beam gives the change in

• A. Shear Force
• B. Bending Moment
• C. Slope
• D. Deflection

Hint:

The area under the bending moment diagram divided by flexural rigidity \( E I \) gives the change in slope, as per the beam equation \( M = E I \frac{d^2 y}{dx^2} \).

Answer 1

The Correct Option is C

Step 1: Recall the relationship between bending moment and beam deflection.
The relationship between bending moment \( M \), slope \( \theta \), and deflection \( y \) in a beam is governed by the beam equation: \[ M = E I \frac{d^2 y}{dx^2}, \] where \( E I \) is the flexural rigidity (\( E \) is Young’s Modulus, \( I \) is the moment of inertia), and \( \frac{d^2 y}{dx^2} \) is the second derivative of deflection (curvature). Step 2: Relate bending moment to slope.
The slope of the beam is the first derivative of deflection, \( \theta = \frac{dy}{dx} \). Differentiate the beam equation: \[ \frac{dM}{dx} = \frac{d}{dx} \left( E I \frac{d^2 y}{dx^2} \right). \] Assuming \( E I \) is constant: \[ \frac{dM}{dx} = E I \frac{d^3 y}{dx^3}. \] But we need the slope: \[ M = E I \frac{d^2 y}{dx^2}, \] \[ \frac{d^2 y}{dx^2} = \frac{M}{E I}. \] Integrate with respect to \( x \): \[ \frac{dy}{dx} = \int \frac{M}{E I} \, dx, \] \[ \theta = \int \frac{M}{E I} \, dx. \] The change in slope between two points \( x_1 \) and \( x_2 \): \[ \Delta \theta = \theta(x_2) - \theta(x_1) = \int_{x_1}^{x_2} \frac{M}{E I} \, dx. \] If \( E I \) is constant: \[ \Delta \theta = \frac{1}{E I} \int_{x_1}^{x_2} M \, dx. \] The integral \( \int_{x_1}^{x_2} M \, dx \) is the area under the bending moment diagram between the two points. Step 3: Interpret the given ratio.
The problem states the ratio of the area under the bending moment diagram to the flexural rigidity: \[ \text{Ratio} = \frac{\text{Area under bending moment diagram}}{E I} = \frac{\int_{x_1}^{x_2} M \, dx}{E I}. \] From Step 2, this is exactly the change in slope: \[ \Delta \theta = \frac{\int_{x_1}^{x_2} M \, dx}{E I}. \] Step 4: Evaluate the options.
(1) Shear Force: The area under the bending moment diagram does not give shear force; shear force is the derivative of the bending moment (\( V = \frac{dM}{dx} \)). Incorrect.
(2) Bending Moment: The area under the bending moment diagram has units of moment times length, not bending moment directly. Incorrect.
(3) Slope: As derived, the area under the bending moment diagram divided by \( E I \) gives the change in slope. Correct.
(4) Deflection: To get deflection, we would need to integrate the slope again (\( y = \int \theta \, dx \)), so this ratio gives slope, not deflection directly. Incorrect.
Step 5: Select the correct answer.
The ratio gives the change in slope, matching option (3).