# Dynamics of structures: combined plane stress and plane strain conditions

Problem statement

For an engineering project in school, we are checking the stability in the longitudinal direction of an apartment building, taking into account the dynamic effect of earthquakes. For this, we are making a two-dimensional finite element model in Matlab. We have to assume that:

• There is a plane stress condition in the columns
• There is plane strain condition in the floors and walls.

The stability of the building is guaranteed:

• In the transverse direction: membrane action in the columns and the walls.
• In the longitudinal direction bending stiffness of the connections between the columns, the floors and walls.

The picture attached shows an apartment building with properties:

• Nine bays (each 5 m width and 0.25 m thick)
• Six floors (each 3.5 m height and 0.25 m thick)
• Depth of the building is 12 m.
• The building is supported by twenty columns (each 1.6 m width and 5 m height)
• The thickness of the columns has still to be determined but should lie between 0.2 m and 0.8 m.
• E = 35 GPa
• Coefficient of Poisson = 0.25
• Mass density = 2500 kg/m³
• Finishing layer of 350 kg/m²

Plane strain and plane stress can be modeled by means of beam elements, provided that in the elements with plane strain condition adjusted values E' and ν' for the Young’s modulus and the coefficient of Poisson are provided:

E'=E/(1-ν^2) and ν'=ν/(1-ν)

Questions

We are struggling with the implementation of the plane stress and plane strain conditions. We have tried to model this as follows:

• Plane strain: We modelled the building with a thickness of 1 m for the superstructure. Therefore, we adjusted the mass density from 2500 kg/m³ to 12 m*2500 kg/m³. In this way the mass of each element is obtained by multiplying by the thickness and length of the element.
• Plane stress: We doubled the stiffness of the column to include both columns at the base.

This implementation gives not the results we are hoping for. Are there any suggestions for improvement?

• I do not understand why you are scaling mass and stiffness properties... A plane strain assumption is simply that epsilon_33=0 while the plane stress is that sigma_33=0. You have to account for these assumptions when calculating the deformations (epsilon) and stresses (sigma) associated to your loading conditions. May 20, 2015 at 17:00
• First of all, thanks for your response! Our idea was to look at it as two frameworks connected to each other like a mass-spring-mass system. The goal of scaling is to combine the stiffness properties of the two frameworks. Your advice is to take into account 1 m of depth without any scaling of the mass and stiffness properties?
– user1711
May 20, 2015 at 21:37
• Exactly. And once you obtain the displacements associated to whatever load you are studying, you can derive the strain and stresses for each part of the structure based on the different assumptions May 22, 2015 at 14:01
• @CJT, Looks like you might have created two "CJT" Accounts. I rejected an edit that was initiated from the second account. You can edit your own post. So I suggest that you use account used to post the question in order to edit to question. May 31, 2015 at 11:28