Calculation of displacements examples

In this guide, I will present the whole formula for calculating displacements taking into account external forces, temperatures, subsidence of supports and I will carry out an example of rotation calculation of the node in the frame on which all these elements operate.
With this guide, I will finish the topic of dislocations and move on to statically indeterminate structures in which a good understanding of the displacement method is necessary. Without a well-mastered method of displacement, please do not start the next steps!

Exercise.
Determine the rotation angle of node A.

Data:
HEB = 120
k = 4.43
EJ = 181
GA = 2.04 * 105
EA = 7.14 * 105
αt = 12 * 10-6

  1. Frame diagram with real load and internal force graphs.
  2. Determination of the displacement Φa M’a = 1 ‘
    Frame diagram with virtual load, virtual force charts and reactions resulting from virtual power.
    In the place and direction of the displacement being searched, we apply the virtual force M’a = 1 ‘.
    We only have a graph of bending moments. The remaining graphs of internal forces are zero.

    Graphs caused by temperature effects.

I could immediately start calculating the displacement, but as I wrote in the previous guide, it is best to write a whole pattern, so as not to make a mistake. The complete pattern including all elements looks as follows.

Even if you do not need the whole pattern, I still recommend that you write a complete pattern. Then you really minimize the chances of making a mistake.
Returning to this example, I will not need a member with internal forces taking into account the influence of shear forces and normal forces, because these charts from the virtual force are zero. I also do not need a formula for the average temperature.
The displacement will be affected by subsidence of supports, bending moments and temperature differences.
The angle of rotation of node A is 5.51 degrees and it is consistent with the direction of the virtual force applied to the structure.

In this way, we have reached the end of the subject of displacements. All the examples, examples and help will be in these guides whenever you need help, you can go back to them and look at the topic that interests you. Please also remember that I also help in the right place on the site. Traditionally, I remind you about exercise, exercise and once again exercise. After a few nights’ wasting, I can guarantee you that the method of displacement will not be difficult for you and I encourage you to solve many examples.