Finite Element Analysis Refresher

When loading is applied to an object, the object undergoes displacements and stresses. A stress is simply a force per small unit of area. A strain is a change in length per unit of length. That is, a 1 % strain (which is very large for steel) means that a bar 100 inches long gets 1" longer.

Mathematically, the strain is proportional to the first derivative of the displacements. The stress is proportional to the strain. For a simple, one-axis thing, the stress is the strain times the Young's modulus, which is normally called E. For anything besides simple things, the stress also depends on strains in other directions, which involves the Poisson's ratio.

Two typical examples are rod (truss element) and a beam (beam element).

Rod (truss element)
Holding one end of a rod and pulling on the other end with a load P results in a change in length of the rod. For a length L and a cross-sectional area of A using a material of Young's modulus E, the stress is P/A. This is a tensile stress which acts along the entire rod. The displacement of the end is PL/EA.

Beam (beam element)

Embedding a beam with a section modulus Z and a Young's modulus of E into a wall, and then applying a load P at the end perpendicular to the length will result in a state of stress basically as follows: at a cross section, a distance X from the applied load, the stress will increase linearly from 0 at the centroid to the outermost part of the cross section. The maximum stress will be (P X)/Z.


Finite Element Analysis - Basic Concepts

If you were to do a stress analysis of a complicated shape with different types of loading, you would only have a few basic options:


1. Think of the complicated shape as a much simpler shape and solve the problem for the simpler shape.

2. Solve some very messy partial differential equations from theory of elasticity.

3. Use finite element analysis.

Finite elements are really neat. They are actually little bits and pieces (like a free-form puzzle) which can be made into any shape you desire. They are used to approximate the real-world structure. It is important to realize that finite elements, no matter how good, are only approximations. For example, the equations for elastic displacement are of the 4th order. Finite elements use "shape functions" which are of different orders. This simply means that there are basic assumptions about how the displacements change across an element from one node to another. A "linear" shape function, for instance, means that the displacements are assumed to be a straight line. A quadratic shape function means that the displacements are assumed to vary according to a quadratic (2nd order) equation. This means that there are built-in limits to the "quality" of each element's approximation of the real-world behavior

In finite element analysis, the displacements are generally computed first. Everything else is computed based on the displacements. Algor is no exception. The stresses in an element are computed based on the displacements.

Here's a discussion question: If the displacements are considered to vary
linearly in an element, the stress will be considered to vary:

A. Quadratically

B. Linearly

C. Not at all - they will be considered to be constant

The answer is found by recalling that the stress is proportional to the strain; the strain, in turn, is proportional to the first derivative of the displacement. The derivative of linear function is a constant. So the stress will be considered to be a constant if the "shape function" (defining the displacements) is linear.

There are some other useful implications of the theory of finite elements concerning vibration, but these will be deferred until vibrations are covered in detail at the appropriate place. The most important item is the effect of the lumped mass matrix.

What Each Type of Finite Element Does

There are many kinds of finite elements available in the Algor system. It is vital to know how to use an element before attempting something serious with it. These are the elements available to you, with a brief description.

Truss

A truss is a rod with 2 ends described in 3-D space. This only "understands" direct tension or compression. There is no check of compressive load with buckling load, which requires care when you use this. A common use of this element is to model some mechanical connections, as well as structures made of slender metal members which are built with connections which do not transfer moments,

Beam

This is a beam with 2 ends described in 3-D space. Beams have a third "pointer" node which is used to orient the beam's moments of inertia and shear areas in space. This element handles tension or compression and torsional moments, as well as shear deformation. No check of buckling is performed except with the eigennvalue Buckling processor, although torsional stresses are not computed. Torsional stresses in beams do strange things - a cantilever wide flange beam with a torsional moment at the free end will develop a transition occurring from shear stress to flange bending stress according to a decayed exponential function. Is it any wonder we do not compute shear stress?

Here are some of the common mistakes most people make with beams. These are not unique to the Algor finite element analysis system; they simply reflect a general lack of understanding of beam behavior.

An "open section" such as an I-beam, wide flange, channel, angle, or other such section has an extremely small torsional resistance to moment compared to what most people realize. Instead of the sum of the 2 bending moments of inertia, the torsional resistance is approximated by the sum of the individual torsional resistance's of the plates which make up the section. The torsional resistance to use is tabulated in standard manuals of steel construction for wide flange beams.

The second problem that people run into is caused by the fact that open sections have natural or "principal" axes for moments of inertia. For example, an angle with equal legs has a principal axis at an angle of 45 degrees to each leg, and the other axis is perpendicular to the first axis. This affects the calculation of bending stress, buckling strength, and stiffness of such members. The way to do this is to orient the "pointer" node in one of the principal directions instead of along a global coordinate axis.

The third problem is that with non-rectangular sections, the "worst sum" stress computed and placed in the ASCII file (filename.L) does not apply. This sum is meaningless unless the cross section is rectangular. However, it gets messy to handle in any other case. So, some care is required here.

Membrane

This is a 3 or 4 node element described in 3-D space. This element has no bending stiffness, but only membrane effects. This must therefore be used with a little care.



2-D Solid

This is a 3 or 4 node element described in the YZ plane. it comes in 3 very different varieties. These are:

1. Axisymmetric - To use this, you describe the cross-section of the axially symmetric part you desire to model. That is, to describe a can of vegetables, you would describe a vertical saw-cut cross section of the can. You would only describe the cross section from the middle of the boftom to the right side top'

The radius is always the Y direction and the axis is always the Z direction in the actual input.

2. Plane strain - This is used to describe "saw-cut" sections of long objects. The strain in the out-of-plane direction is taken to be zero, reflecting the assumption that the strain is all in one plane. The results are therefore reflecting a situation in which the "ends" of the piece are clamped at a fixed distance. Therefore, thermal stresses will be computed on this basis. If the ends of the piece being modeled are free, this will result in severe overestimates of the thermal stress. In these cases, the remedy is to layer a few bricks with the free end left to be free, which allows accurate computation of the stress.



3. Plane stress - This is the analogous type of situation to the above, except that the stress is considered to be only in the plane tinder consideration.


Bricks

Bricks are modeled in three dimensions, and can be modeled using either SuperDraw 11 directly using the Copy and Join command or can be layered from a Type 4 model. These are isotropic material elements, and the shape functions are such that it is desirable to make elements be as rectangular as possible.


The Algor finite element library contains other elements, such as boundary elements, pipes, rigid elements and direct stiffness elements, but the above list contains all the elements most people need.