Understanding Bending Moment Diagrams

The question asks about the shape of the bending moment diagram (BMD) for a simply supported beam subjected to a uniformly distributed load (UDL) across its entire span. Let's break down the concepts involved.

A simply supported beam is a beam supported by a pin support at one end and a roller support at the other. These supports allow rotation but prevent vertical movement. A uniformly distributed load (UDL) is a load spread evenly over a length of the beam, typically represented by a constant load intensity per unit length (e.g., kN/m).

Relationship Between Load, Shear Force, and Bending Moment

There is a fundamental relationship between the load applied to a beam, the shear force developed within the beam, and the bending moment. This relationship is based on equilibrium principles and calculus:

• The rate of change of shear force along the beam is equal to the negative of the distributed load intensity: $$\frac{dV}{dx} = -w(x)$$ where $$V$$ is the shear force, $$x$$ is the distance along the beam, and $$w(x)$$ is the load intensity at $$x$$.
• The rate of change of bending moment along the beam is equal to the shear force: $$\frac{dM}{dx} = V(x)$$ where $$M$$ is the bending moment and $$V(x)$$ is the shear force at $$x$$.

Shear Force and Bending Moment for UDL

For a simply supported beam of span $$L$$ carrying a UDL of intensity $$w$$ over the entire span:

1. Shear Force Diagram (SFD): Since $$\frac{dV}{dx} = -w$$, and $$w$$ is constant, integrating this equation gives $$V(x) = -wx + C$$. This shows that the shear force varies linearly along the beam. The SFD for a simply supported beam with full UDL is a straight line that starts at $$+wL/2$$ at one support, crosses zero at the center, and ends at $$-wL/2$$ at the other support.
2. Bending Moment Diagram (BMD): Since $$\frac{dM}{dx} = V(x)$$, and we know that $$V(x)$$ is a linear function of $$x$$, integrating the linear shear force expression will give a quadratic function for the bending moment. $$M(x) = \int V(x) dx$$. Integrating a linear function ($$ax+b$$) results in a quadratic function ($$Ax^2+Bx+C$$).

Shape of the Bending Moment Diagram

A quadratic function plots as a parabola. Therefore, the bending moment diagram for a simply supported beam subjected to a uniformly distributed load over its entire span is a parabolic curve.

Key characteristics of this BMD:

• It is zero at both supports (because a simply supported end cannot resist bending moment).
• It is maximum at the point where the shear force is zero, which for a simply supported beam with full UDL is at the center of the span.
• The parabola opens downwards, indicating that the beam experiences sagging bending moment (tension at the bottom, compression at the top).

Analyzing the Options

• Rectangle/Square: These shapes represent a constant value, which is not the case for the bending moment under UDL.
• Right angle triangle: This shape represents a linearly varying function, like the shear force diagram for a UDL or the bending moment diagram for a point load on a cantilever/simply supported beam. The bending moment under UDL is not linear.
• Parabola: As derived from the integration of the linear shear force function, the bending moment function is quadratic, resulting in a parabolic shape.

Thus, the correct shape is a parabola.

Revision Table: Beam Diagrams

Additional Information on Bending Moment Diagram

The maximum bending moment ($$M_{max}$$) for a simply supported beam of span $$L$$ with a UDL of intensity $$w$$ over the entire span occurs at the center and is given by the formula:

Understanding the shape of the BMD is crucial in structural analysis and design as it helps determine the maximum bending stress the beam will experience, which is essential for selecting appropriate beam dimensions and materials.

The process of drawing SFD and BMD involves calculating reactions at supports, writing shear force and bending moment equations as functions of $$x$$, and then plotting these functions along the beam's length. For a UDL, the SFD is a straight line, and the BMD is a parabolic curve.