Influence lines

Influence lines – Movable load

The line of influence of a given static quantity (reaction, internal force, etc.) is a graph of the dependence of this magnitude on the variable position of the unitary force loading the system.

We can use the influence lines to calculate the values of static quantities (reactions, internal forces), to find: the most favorable load position, drawing an envelope of internal forces (especially useful in reinforced concrete constructions).
The influence lines of static quantities in statically determinate bar systems (beams, arcs, grates) are always linear functions.

In summary, using influence lines we can check the impact of applied force on structures.
To make it easier, we use unit power and observe how it works on the whole structure. Of course, we put the strength in a place that interests us.

Marking the influence line

Before discussing the basic examples, let’s get acquainted with the principles of constructing the influence lines.

  1. Multi-span beams, divided into dependent and independent elements.
  • The force acting on the independent element affects the dependent element (the chart will be on the dependent and independent element).
  • The force applied on the dependent element does not affect the independent element (the graph will be only on the dependent element).
  1. In the places of supports, the graph is zeroed, the only exception is the case in which we applied the unit force in the place of a given support.
  2. In the direction of the articulation the graph will rise (the value increases), after reaching the place where the joint is, the graph will start to fall. The joint is the place where the resulting triangle-shaped graph has the highest value.
    It can be said that this is such a tricked extreme value because it is an extreme value only in a given area. The value of the beam may be higher elsewhere.
  1. The values in individual places are calculated from the thesis of Tales.

Now let’s see what the IW graphs for straight beams look like. At the beginning, a single-span beam scheme with the cross-section A marked.
We will draw the lines of influence in the places of the supports (Va and Vb) and in the place of the cross-section A (Ma and Ta).

We see that the whole beam is just one element. An independent element.

Now I will discuss the individual lines of influence:

  1. The unit force is applied to the place of the support A. Here the second principle works. We see that the unit force evokes the value 1 and then falls to zero in the next support in place B.
  2. The impact line for the space in the B support is simply a mirror image of the above point.
  3. The influence line applied at the A section will produce the action shown in the figure above. It’s just a common part of the XA and XA ‘charts, the dotted gray line is part of the graphs that do not overlap.

In brief. By applying a unit load at both ends and then drawing their graphs, we get a moment graph. It’s enough to paint the boxes that overlap.

We can also use the formula used in internal forces to find the value of the apex of the triangle, and then guide the graph according to the rules.
Formula: \frac{(Unit \; strength '1' * X_{A} * X_{A} ')}{L}  , where L = X_{A} + X_{A}^{'}

The influence lines for the cut force are performed analogously, i.e. we use auxiliary lines or rules from internal forces.

Now let’s see how lines will look to the cantilever beam.

Everything works the same here. In the next tutorial we will deal with a more extensive multi-span beam, so that I can show how the charts work when there are dependent and independent elements.

Finally, I would like to add that we can see some analogy of the lines of influence to internal forces, but do not let us fool us.  The only thing we can use two patterns for moment and two for cutting forces, depending on the beam whether it is freely supported or fixed. In the next steps, stick to the rules regarding the impact line!