Understanding Simply Supported Beams and Varying Loads

This question asks about the nature of the shear force diagram for a simply supported beam subjected to a linearly varying load. A simply supported beam is supported at both ends, allowing rotation but preventing vertical displacement. A linearly varying load is one whose intensity changes uniformly along the length of the beam, typically from zero at one end to a maximum at the other, or from one value to another.

Relationship Between Load, Shear Force, and Bending Moment

In structural analysis, there are fundamental relationships between the applied load, shear force, and bending moment along a beam:

• The shear force at any point is the integral of the load distribution up to that point. Mathematically, this is represented as \( V(x) = - \int w(x) \, dx \), where \(w(x)\) is the load intensity as a function of position \(x\), and the negative sign accounts for standard conventions.
• The bending moment at any point is the integral of the shear force up to that point. Mathematically, this is represented as \( M(x) = \int V(x) \, dx \).

This means that the degree of the curve representing the shear force diagram is one higher than the degree of the curve representing the load distribution. Similarly, the degree of the bending moment diagram is one higher than the degree of the shear force diagram.

Analyzing the Linearly Varying Load

The problem states the beam is subjected to a linearly varying load. This means the load intensity \(w(x)\) can be described by a linear function of the position \(x\) along the beam. A linear function has a degree of 1. For example, if the load varies from 0 at \(x=0\) to \(w_0\) at \(x=L\), the load intensity at any point \(x\) can be written as:

\( w(x) = \frac{w_0}{L} x \)

This is a linear equation in \(x\).

Determining the Nature of the Shear Force Diagram

Since the shear force \(V(x)\) is the integral of the load intensity \(w(x)\), and \(w(x)\) is a linear function (degree 1), integrating \(w(x)\) will result in a function one degree higher than linear. The integral of \(x\) is \(x^2/2\). Therefore, the shear force \(V(x)\) will be described by a function of degree 2.

Integrating \( w(x) = \frac{w_0}{L} x \) gives:

\( V(x) = - \int \left(\frac{w_0}{L} x\right) \, dx = - \frac{w_0}{L} \int x \, dx = - \frac{w_0}{L} \left(\frac{x^2}{2}\right) + C \)

Where \(C\) is the constant of integration, determined by boundary conditions (like the reaction forces at the supports). The resulting expression for \(V(x)\) contains an \(x^2\) term, making it a quadratic function.

A quadratic function is represented graphically by a parabola.

Conclusion on Shear Force Diagram

Based on the integration relationship, if the load is linear (degree 1), the shear force will be parabolic (degree 2). If the load were uniform (constant, degree 0), the shear force would be linear (degree 1). If the load were a concentrated force (represented by a Dirac delta function), the shear force would have a step change (discontinuous).

For the given problem of a simply supported beam with a linearly varying load, the shear force diagram will follow a parabolic curve.

Analysis of Options

• Elliptic: Elliptic variation is not typically observed in shear force or bending moment diagrams resulting from standard polynomial load distributions.
• Parabolic: This corresponds to a quadratic relationship, which is obtained by integrating a linear load distribution. This matches our analysis.
• 3^rd degree curve: A 3^rd degree curve (cubic) would result from integrating a parabolic shear force diagram, which corresponds to a parabolic load distribution (degree 2). Since our load is linear (degree 1), the shear force is parabolic (degree 2), and the bending moment would be cubic (degree 3).
• Linear: A linear shear force diagram results from a uniform (constant) load distribution. Since the load is linearly varying, the shear force cannot be linear.

Therefore, the nature of the variation of the shear force diagram for a simply supported beam subjected to a linearly varying load is parabolic.

Revision Table: Shear Force and Bending Moment Diagrams

Additional Information: Different Load Cases on Simply Supported Beams

Understanding the shape of shear force and bending moment diagrams for different load types is crucial in structural analysis. Here are a few common cases for simply supported beams:

• Concentrated Load at Midspan: The shear force diagram is rectangular, with a constant positive value from one support to the load, a sudden drop at the load, and a constant negative value from the load to the other support. The bending moment diagram is triangular, with the maximum moment at the point of the concentrated load.
• Uniformly Distributed Load (UDL): The load intensity is constant along the beam. The shear force diagram is linear, varying from a positive value at one support to a negative value at the other, crossing zero at midspan. The bending moment diagram is parabolic, with the maximum moment at midspan where the shear force is zero.
• Linearly Varying Load (Triangular Load): As discussed in this problem, varying from zero at one support to maximum at the other. The shear force diagram is parabolic. The bending moment diagram is cubic (3^rd degree curve).

These standard cases help predict the behavior of beams under various loading conditions and are fundamental concepts in mechanics of materials and structural engineering.