Calculation of displacements including subsidence of supports and temperature

In this guide, we will deal with the next elements that affect the displacement, it is the temperature and subsidence of supports.

The principle of virtual work when subsidence supports.

Consider settling supports that are not related to the external load, but caused by external factors acting on the elements of the nature of the mainstay (supports), for example, ground washing under the foundation, earthquake, etc.

Example.
Calculate the horizontal displacement in point C. No external load is applied to the frame. The support at point A will move horizontally by 4 cm and the support at point D will settle to a depth of 2 cm.

We calculate the reactions caused by the virtual force applied in point C.

To account for the subsidence of supports or their displacement in the displacement method, it is necessary to multiply the actual subsidence of supports by reactions created by adding virtual force.

In this case, point C will move horizontally 6 cm to the side in which the virtual force is directed.
Remember the minus sign, which is in the pattern, very often many students forget about it.

The principle of virtual work with temperatures.

Taking into account the temperature, we have to calculate the difference and average temperatures on both sides of the rod and take into account the assembly temperature. Everything depends on what task you get from the teacher.

where:

M ‘- graph of moments bending from virtual force
αt – coefficient of material thermal expansion, for steel is 12 * 10-6 [1 / oC]
ΔT – temperature difference between the lower and upper fibers of the rod.
ΔT = Td (lower temperature) – Tg (upper temperature). The graph always on the warmer side, we treat like M
h – cross-section height
N ‘- graph of normal forces from virtual force
To – average temperature in the center of gravity of the bar section, taking into account the assembly temperature of the structure. No matter on which side, but with a sign. We treat like N
To = (Td + Tg) / 2 – Tm
Tm – assembly temperature

Example.
Calculate the horizontal displacement in point C. No external load is applied to the frame. Inside the structure, the temperature is -50 ° C, the outside temperature is + 20 ° C. The assembly temperature is + 10oC.
Let’s see what the temperature charts will look like.

Status of virtual workloads.

Having all the necessary data, graphs and formulas, we can calculate the displacement of the structure that was caused by the temperature.

It is very important to remember that the temperature difference is multiplied by M ‘, while the average temperature is multiplied by N’!

In this way, I’ve already discussed all the elements that affect the displacement. It is best to write a complete displacement pattern before the colloquium, which takes into account external forces, subsidence of supports and temperature. Then there is less chance of making a mistake.