Results Interpretation: Total, Deformation, Nominal and Engineering Strain

Description

Strain is a dimensionless quantity that describes the deformation of an object with respect to a reference size. The deformation in the finite element analysis comes from the relative movement of the nodes. The reference size depends on whether the Total Strain or the Deformational Strain is requested, and is affected only by the presence of pre-load attributes (pre-stress, pre-strain, pre-tension, pre-curvature, thermal expansion, temperature gradients). In the absence of these attributes, the two strain measures are the same.

Deformational Strain
This strain is measured with respect to the initial position of the nodes as defined in the mesh. A beam element fully fixed at both ends with an applied axial pre-strain attribute will produce zero deformational strain (the nodes do not move), even though there will be a resultant axial force. Conversely, the same axial pre-strain attribute on a cantilever beam will produce a non-zero deformational strain (equal to the pre-strain) but no resultant axial force.
Total Strain
This strain is measured with respect to the position of the nodes after all pre-load attributes have been taken into account (i.e., after an unrestrained element is allowed to fully expand/contract due to the applied pre-loads). A beam element fully fixed at both ends with an applied axial pre-strain attribute will produce a total strain equal to the pre-strain in addition to a resultant axial force. Conversely, the same axial pre-strain attribute on a cantilever beam will produce a zero total strain and no resulting axial force.

From these two descriptions we can see that the deformational strain reports how much the nodes in the mesh have moved, whereas the total strain reports the strain that corresponds to the element stress for that strain on the stress-strain table of the material. The total strain is sometimes referred to as the mechanical strain.

Strain Definitions

Strand7 provides a number of strain definitions including nominal strain, engineering strain and Green's strain, depending on the element. The nominal and engineering strains are the more commonly used measured for structural analysis.

To illustrate the above concepts, we look at a simple uniaxial example and follow its strain history as it responds to different loading actions.

A steel solid round bar is modelled. The bar is 36 mm in diameter and 1.0 m long. It is pre-tensioned to 700 kN and is anchored to a 1000 kN/mm spring. It is then subjected to a temperature rise of 50 °C. In the Strand7 model, the bar is modelled as 1.0 m long (i.e., the initial mesh dimension).

Strain Results using the Nominal Strain Definition

If we designate the initial length of the bar as , and its current length as , the nominal strain can be expressed as:

We begin with a bar that is slightly shorter than 1.0 m and then stretch it to fit the 1.0 m current length to generate the required pre-tension of 700 kN. To achieve this, we need:

This means that we have started with a stress-free bar length, , of 996.5615 mm and stretched it by 3.4385 mm to reach this pre-tension level at the meshed model length of 1.0 m. At this stage, we have the following result output:
The bar is now anchored to the spring; the bar-spring system re-adjusts to reach a new static equilibrium. As a result, the bar has contracted by 0.5816 mm due to the extension of the spring, which means that the current length (i.e., the deformed size) of the bar, , becomes 999.4184 mm. The result output now changes to:
Next, the bar undergoes thermal expansion of 50 °C × thermal expansion coefficient = 50 × 1.15×10^-5 = 5.75×10^-4. The bar extends and reaches a new equilibrium position with the spring at 0.09726 mm from the previous position, which means that the current length of the bar, (i.e., the new deformed size), becomes 999.5157 mm. As thermal expansion also modifies the stress-free size of the bar, it is considered as a type of pre-strain. The result output now changes to:

Strain Results using the Engineering Strain Definition

If we designate the initial length of the bar as and its current length as , the engineering strain can be expressed as:

We repeat the example and look at the resulting strain values in the engineering strain definition.

The process is still the same in that we begin with a bar that is slightly shorter than 1.0 m and then stretch it to fit the 1.0 m current length, , to generate the required pre-tension of 700 kN. To achieve this, we need:

This means that we have started with a stress-free bar length, , of 996.5615 mm and stretched it by 3.4385 mm to reach this pre-tension level at the meshed model length of 1.0 m. At this stage, we have the following result output:
The bar is now anchored to the spring; the bar-spring system re-adjusts to reach a new static equilibrium. As a result, the bar has contracted by 0.5816 mm, due to the extension of the spring, which means that the current length (i.e., the deformed size), of the bar, , becomes 999.4184 mm. The result output now changes to:
Next, the bar undergoes thermal expansion of 50 °C × thermal expansion coefficient = 50 × 1.15×10^-5 = 5.75×10^-4. The bar extends and reaches a new equilibrium position with the spring at 0.09726 mm from the previous position, which means that the current length (i.e., the new deformed size) of the bar, , becomes 999.5157 mm. As thermal expansion also modifies the stress-free size of the bar, it is considered as a type of pre-strain. The result output now changes to:

If we run this analysis in Strand7 and extract the stress in the bar, we obtain a stress of 475.83 MPa. Dividing this stress by the total strain recovers the Young's modulus of 200 000 MPa.