# 5. Influence Lines

5.1 Introduction (Part 2) - Müller-Breslau Principle

This section repeats the creation of one of the influence lines of the previous section. This time we will use the principle of virtual displacement, instead of equilibrium.

Figure

The above figure shows the displaced shape used to determine the moment at point $c$. Recall how that was constructed:

  1. The constraint corresponding to the moment at $c$ was released.
    In this case, that constraint is on relative rotation of the beam either side of point $c$.
  2. A unit value of the displacement is imposed at the released constraint.
  3. A displaced shape is drawn where all other constraints are met.

The resulting virtual displaced shape is shown in red in the lower part of the figure.

Now, for the influence line, we wish to determine moment at $c$ due to a unit load applied a distance, $x$, from the left support. That load is shown in the top part of the figure.

To use the method of virtual work (virtual displacement), we need to know the displacements on the virtual displacement diagram corresponding to positions of the load. Here we show the single displacement, under the unit load, labelled as the distance, $y$, on the lower figure.

We can now write the virtual work expresssion and solve for the moment at $c$:

\[\begin{align} &M_c \times 1 - 1 \times y = 0\\ &M_c = y\\ \end{align}\]

This tells us that the moment at point $c$ due to a unit load at any point, is simply the ordinate on the displaced shape. In other words, the displaced shape is the influence line!!

Müller-Breslau Principle: The influence line for a structural response is given by the displaced shape of the structure when the constraint corresponding to the response is removed and a unit displacement is introduced in its place.

Procedure

  1. Relax the constraint corresponding to the response for which you wish to construct the influence line. Typical constraints are:
    1. Absolute translation for force reactions.
    2. Absolute rotation for moment reactions.
    3. Relative longitudinal displacement for axial forces (relative transverse displacements and relative rotations being maintained).
    4. Relative transverse displacement for shear force (relative longitudinal displacement being mainatained, as well as relative rotation (unless the moment is forced to be 0).
    5. Relative rotation for bending moments (relative longitudinal displacement being mainatained, as well as relative transverse (unless the shear force is forced to be 0).
  2. Impose a unit relative displacement in its place.
  3. Draw a displace shaped of the structure with all other constraints maintained.
  4. Compute ordinal values at key locations on the displaced shape.