Calculation of displacements

In this guide, we will deal with the more difficult thing than in the previous guides, it is the calculation of displacements in flat bar systems using the principles of virtual work. After reviewing all the guides and analyzing the examples I will put in the guides, I do not think you would have problems with that.

We are entering a new topic, there must be some theory and I will try to explain what the displacement method is all about.

As the name says, you will learn here to calculate displacements at selected points of the structure. We will check the vertical displacement (V), horizontal (U) and rotation (Φ). We will use the work of virtual forces for this.
The virtual force is a unitary force applied in the place and direction of the displacement being sought in a given structure. I will mark virtual forces as follows – 1 ‘.

The effect of external forces (applied to the construct) of virtual 1 ‘on real displacements (δi) is equal to the effect of internal virtual forces M’, T ‘and N’ on real deformations.

The Mohr-Wiereszczagin rule

Before presenting the scientific definition, I will describe what is going on in the Mohr-Wiereszczagin method.
The graphs (functions) in the shape of geometric figures: rectangle, triangle, parabola, etc., are multiplied as a result of real forces and virtual force.

DEFINITION: The integral determined by the product of two continuous functions, one of which is a linear function, is equal to the product of the area of the plot of a curvilinear function by the ordinate of the plot of the linear function corresponding to the center of gravity of the curvilinear plot.

After drawing the graphs of internal forces arising from real forces and virtual force, we obtain graphs of various geometric shapes.

  • Straight lines: rectangles, triangles.
  • Curvilinear: parabolas.

As it is written in the definition, we must keep the order of multiplication. If we want to multiply the straight (rectangle) and curvilinear (parabola) function, then the order is very important.
In this case, we multiply the area of the parabola by the center of gravity of the rectangle. The same situation occurs when the triangle is multiplied by a parabola.

Multiplying two rectilinear figures, for example: a rectangle and a triangle by itself is neutral, whether we take the area of the rectangle and multiply by the center of gravity of the triangle or area of the triangle and multiply by the center of gravity of the rectangle. The result will come out the same.

The formula to calculate the displacement looks as follows.

{1}'*Ua=\int \frac{{M}'*M}{EJ}ds+ \int \frac{k*{T}'*T}{GA}ds+\int \frac{{N}'*N}{EA}ds


1 ‘- virtual force
Ua – horizontal displacement in point A

The first is the displacement caused by the bending moment
M ‘- graph (function) of internal forces caused by virtual force
M – graph (function) of internal forces caused by real forces
EJ – bending strength


The second member is a displacement caused by cutting forces
k – shear factor
T ‘- graph (function) of internal forces caused by virtual force
T – graph (function) of internal forces caused by real forces
GA – shear strength


Third member, this is the displacement caused by normal forces
N ‘- graph (function) of internal forces caused by virtual force
N – graph (function) of internal forces caused by real forces
EA – stiffness

I will present the method and algorithm for calculating displacements in practice.
I would also like to point out that in this guide I will only consider internal forces M, T and N. Additional items such as displacement of supports and temperature will be included in the next guide.
Time on the example to present the rule of Mohr-Wiereszczagin.

I will solve the example in the points with the commentary so that everything is clear.

The content of the task.
Calculate the horizontal displacement of node A (Ua).

  1. Frame scheme with real load.

1.1. Graphs of internal forces from actual load.

  1. Scheme of the system (frame) with virtual load.
    In place and in the direction of the displacement being searched, we apply the virtual force Px ‘= 1’.

2.2. Graphs of internal forces from virtual load.

  1. Calculation of the Ua displacement (horizontal in point A) using the Mohr-Vereagagin rule.

    Required data:
    EJ = 1814,00\; kNm^{2}
    GA = 2,04 * 10^{5}\; kN
    EA = 7,14 * 10^{5}\; kN
    k = 4,43

Just in this example, we will only have rectilinear functions.
Parameters of individual geometric figures will be included in a separate guide, in which there will be exemplary multiplication of figures.
Here I will show it on numbers, I will not attach additional pictures, so as not to clutter up this guide.

At the beginning I will write the pattern in integrals, marking their compartments. Then I will substitute numbers and calculate the displacement.

The displacement value has been positive, i.e. the direction of displacement is consistent with the direction of the force assumed.
Let’s also consider the influence of individual internal forces.

The displacement:
…from the bending moment is up to 0,529m!
…from cutting force 0.0026m;
…from normal force 0.0001961m.

The influence of transverse and normal forces on the displacement is negligible, therefore in the design practice the influence of these forces is neglected.

In order not to extend this guide, I will end here.
I invite you to the next guide Calculation of displacements in flat systems – addition.