Understanding Influence Line Diagrams for Bending Moment

An Influence Line Diagram (ILD) is a graph showing the variation of a structural function (like reaction, shear force, or bending moment) at a specific point in a structure as a unit load moves across the structure. For a simply supported beam, the influence line for bending moment helps us determine the bending moment at a chosen point due to any load position or system of loads.

Influence Line Diagram for Bending Moment in a Simply Supported Beam

Let's consider a simply supported beam AB of length \(L\). We want to determine the shape of the influence line diagram for the bending moment at a specific point C located at a distance 'a' from support A and 'b' from support B, such that \(a+b=L\).

To construct the ILD for the bending moment at point C, we place a unit downward load (value 1) at a variable distance 'x' from support A, where \(0 \le x \le L\). We then calculate the bending moment at point C due to this unit load at position 'x'. The value of the bending moment at C for each position 'x' is the ordinate of the ILD at that position.

Calculating Bending Moment at Point C

Let \(R_A\) and \(R_B\) be the reactions at supports A and B respectively due to the unit load at distance 'x' from A. Using equilibrium equations:

• Sum of vertical forces = 0: \(R_A + R_B - 1 = 0\)
• Sum of moments about A = 0: \(R_B \times L - 1 \times x = 0 \implies R_B = \frac{x}{L}\)
• From the first equation: \(R_A = 1 - R_B = 1 - \frac{x}{L} = \frac{L-x}{L}\)

Now, we calculate the bending moment at C (\(M_C\)) due to the unit load at x. We consider the forces to the left or right of section C, depending on where the unit load is located relative to C.

• Case 1: Unit load is between A and C (\(0 \le x \le a\))

  The unit load is to the left of point C. The bending moment at C is the moment of forces to the left of C. Only \(R_A\) and the unit load are to the left.

  $$M_C = R_A \times a - 1 \times (a-x)$$

  Substitute \(R_A = \frac{L-x}{L}\):

  $$M_C = \frac{L-x}{L} \times a - (a-x)$$

  $$M_C = \frac{a(L-x) - L(a-x)}{L} = \frac{aL - ax - aL + Lx}{L} = \frac{x(L-a)}{L}$$

  Since \(L-a = b\), we get:

  $$M_C = \frac{bx}{L} \quad \text{for } 0 \le x \le a$$

  This is a linear equation in terms of x.

• Case 2: Unit load is between C and B (\(a \le x \le L\))

  The unit load is to the right of point C. The bending moment at C is the moment of forces to the left of C. Only \(R_A\) is to the left.

  $$M_C = R_A \times a$$

  Substitute \(R_A = \frac{L-x}{L}\):

  $$M_C = \frac{L-x}{L} \times a = \frac{a(L-x)}{L} \quad \text{for } a \le x \le L$$

  This is also a linear equation in terms of x.

Plotting the Influence Line Diagram

Let's examine the values of \(M_C\) at key points:

• At \(x=0\) (Load at A): Using \(M_C = \frac{bx}{L}\), \(M_C = \frac{b \times 0}{L} = 0\).
• At \(x=a\) (Load at C):
  • Using \(M_C = \frac{bx}{L}\), \(M_C = \frac{b \times a}{L} = \frac{ab}{L}\).
  • Using \(M_C = \frac{a(L-x)}{L}\), \(M_C = \frac{a(L-a)}{L} = \frac{ab}{L}\).
  The value is the same from both equations, and this is the maximum positive value.
• At \(x=L\) (Load at B): Using \(M_C = \frac{a(L-x)}{L}\), \(M_C = \frac{a(L-L)}{L} = 0\).

The function for \(M_C(x)\) is linear from \(x=0\) to \(x=a\) (slope \(\frac{b}{L}\)) and linear from \(x=a\) to \(x=L\) (slope \(-\frac{a}{L}\)). The ordinates are positive throughout \(0 < x < L\).

Plotting these values results in a graph with a straight line from (0, 0) to \((a, \frac{ab}{L})\) and another straight line from \((a, \frac{ab}{L})\) to (L, 0).

This shape is a triangle with its base along the beam (length L) and its apex at point C (at distance 'a' from A), having a maximum height (ordinate) of \(\frac{ab}{L}\).

Therefore, the influence line diagram for bending moment at a specific point in a simply supported beam is a triangle.

Conclusion on ILD Shape

Based on the derivation, the influence line diagram for bending moment at any specific point on a simply supported beam is indeed a triangular shape.

Revision Table: Simply Supported Beam ILD

Additional Information on Influence Lines

• Applications: ILDs are crucial for finding the maximum effect (like maximum bending moment or shear force) at a point due to moving loads, such as vehicles on bridges. By positioning the loads on the beam according to the ILD, engineers can determine the worst-case scenario.
• Müller-Breslau's Principle: This principle provides a qualitative method to draw influence lines. It states that the influence line for a function (like reaction, shear, or moment) is to the same scale as the deflected shape of the structure when a unit displacement corresponding to that function is introduced at the point where the function is desired. For bending moment at C, a hinge is introduced at C, and a unit rotation is applied, causing the beam to deform into a triangular shape.
• ILD vs. Bending Moment Diagram: It's important not to confuse an ILD with a bending moment diagram (BMD). A BMD shows the bending moment distribution along the entire beam for a *fixed* loading condition. An ILD shows the bending moment at a *fixed* point as a *unit load* moves across the beam.