We now come to the most outstanding and most versatile method of structural analysis: the Finite Element Method. It has made possible to analyze virtually all kinds of structures that human brain ever can imagine! If you have studied finite element before, you may skip this section. Those who did not, I present a
very very brief introduction of the subject. There exist more than 1001 books in this subject. But I warn you; the theory of finite element analysis is very complex!
What is meant by finite element? The answer is any element, which is not infinite. Don’t be exasperated; this is the real definition of finite element.
Did you play with mechano when you were a child? Just think how you built a model car or house by “Lego” parts? Now consider each part of mechano as “finite element”. A number of mechano elements were needed to build your model car or house.
Now consider a frame. It is made of a number of beam/column members or “bars”. Here the “bars” are “finite elements” of the “frame”.
I hope you have probably realized now that the frame analysis, so far what we have discussed in preceding sections, is actually finite element analysis in essence where each finite element is a “bar”.
Figure 23-1
This is the longitudinal section of a beam shown. That is, you are viewing a beam from its length side. Observe that here we consider the beam as 2- dimensional “Plane stress” structure. Don’t confuse this with 2D or 3D frame. By 2D or 3D frame we actually mean “Plane” and “Space” frame. In all previous cases, we treat all beams as “bars” like a “stick”. But in the above figure we are treating the beam taking into effect of its length as well as depth. That’s why it is 2D. Had we considered the width of beam in the analysis, it would have been termed as 3D solid. Pretty confusing! Look, there is a “cut” in
X Y
loaded by a uniformly distributed load. We like to find out the stresses at various points of the beam. Please note that analysis of this problem by classical method is close to impossible.
So, first we divide the beam into a number of “triangular finite elements”. Then we shall determine the member stiffness matrix [k] of each individual triangular element and ultimately we shall have to combine the member stiffness matrices into “global stiffness matrix” [K]; pretty much the way we did in case of frame analysis. Then we shall have to apply the boundary condition on [K] matrix.
Figure 23-2
After that we need to construct force matrix [P]. For this, distributed load must be converted to appropriate nodal loads by applicable equations. So, our problem can be represented by familiar equation [P] = [K][D]. From this equation we can solve for [D] and then we can find out nodal stresses form equation [σσσσ] = [C][εεεε] where [C] matrix differs in various cases like plane stress, plane strain etc. We are describing this problem as plane stress because we considered only 2 dimensions (X and Y) and stress variation along width (Z direction) has not been taken into account. That means we have taken care of only σx, σy and τxy. In this problem we considered the beam is made of
“triangular” finite elements, but we could have also considered it is made of “rectangular” finite elements as shown in figure 23-3.
Figure 23-3
If you analyze the beam with both triangular and rectangular elements as shown above, you will see that you get accurate answer when you use rectangular elements. It proves one very fundamental concept of finite element analysis: You must choose proper element for particular problem. You do get correct result with triangular finite element but you must use very fine mesh compared
X Y
Figure 23-4
As a crude rule, when you use triangular you will normally need much finer “mesh” than rectangular elements. The assembly of elements in finite element analysis is called “mesh”. Most powerful finite element programs can generate mesh automatically if you specify the boundary surfaces of the models. If you want to analyze the same beam in 3D then your model will look like as shown in figure 23-5.
Figure 23-5
In this case finite element will be 3D solid element like shown in figure 23-6.
Figure 23-6
This is an 8 noded finite element because it has 8 nodal points. If its each vertex has one additional point in the middle, then it would have been 20 noded finite element. Higher is the number of nodal points in an element better is the accuracy of the solution. But higher noded elements are difficult to calculate even with a computer since total number of nodes increases the size of global stiffness matrix. Whatever element you use, it must be compatible. Compatibility means there must not be any discontinuity or overlapping among the elements when the analysis model deforms under applied load. You can combine more than one kind of element in single structure. You should use more number of elements where you anticipate stress variation is more
One distinguishing feature of finite element method is that it does not provide “closed form” solution. Every problem in finite element analysis is unique. This probably needs little more explanation. Think of a simple beam. In classical method of analysis, you can make a program, which takes L, E, I and w as input and computes deflection at any point by solving the equation of elastic line, which can be easily formulated. But in case of finite element analysis, if you change the length of the beam, it becomes another new problem because the geometry of the model changes. Of course you can change E or w values or boundary condition without remodeling the whole structure.
Another aspect of Finite element analysis is that it almost always produces an approximate result. I used the word “almost” because finite element analysis does produce exact result only when the finite element is “bar” that is in “frame structures”.
You may be wondering that if finite element method can solve any structure, then what is the justification of studying classical methods of analysis. Aha! A real question indeed! You can realize it yourself. Just think of solving a simple beam in finite element method (this is presented just after this section). After you solve this beam by finite element method, you can easily check whether the result is correct or not by comparing the answer obtained by classical method. But now imagine the analysis of the fuselage of an airplane or the propeller of a ship. How do you check the correctness of these analyses? Therefore you must accept the finite element analysis result as exact result! That’s why it is so important that finite element analysis models must be created to simulate the actual structure as much as possible. You must use proper combination of finite elements, sufficiently accurate mesh, proper load and applicable boundary conditions. It is often a common practice to analyzing the structure first with a particular mesh and then repeating the whole analysis after doubling the mesh to see whether the result converges. But this method has drawbacks! Your program cannot analyze the structure if your number of mesh nodal points exceed the program’s capacity. Moreover, it is very difficult to predict beforehand what particular “finite element” will best simulate the structure. This is especially a demanding task for very complex structures.
Finite element method is nowadays widely used in all branches of aerospace engineering, bio-medical engineering, mechanical engineering and structural engineering etc. Some manufacturing companies spend millions of dollars every year in finite element analysis!
Sometimes even the most expensive finite element analysis programs produce wrong answer to complex problems. If you feel inclined to know more about this wonderful (?) tool of analysis, I strongly recommend that you to go through some standard finite element method textbooks.
One word of advice, many engineers tempt to use finite element analysis everywhere even when it is possible to analyze the particular structure using classical method of analysis. My main aim is to make you realize that finite element analysis is required only when it is absolutely necessary. Remember that finite element analysis programs are very expensive and they also demand great part of contribution from you for preparing input and interpreting output. Typically, a finite element analysis consists of following steps.
1. Defining the model (i.e. drawing it either in the finite element program’s graphical interface or importing it from a CAD program).
2. Creating the mesh (most programs can automatically generate mesh for best result).
3. Defining the boundary conditions. 4. Defining the loads.
5. Performing analysis (may take hours for complicated models!) 6. Interpreting the result (very important).
The steps are pretty straightforward. But there are many glitches!
In next page you will find an exercise of simply supported beam with uniformly distribute load analyzed by FEA method. This example is for your understanding of the basic concept of FEA only. In practice, this problem should be solved by simple flexure formula of σ = My/I. Remember this!
Exercise
A 5-m steel (E=200GPa) beam has width 200 mm and depth 500 mm. It is loaded by 10-kN/m uniformly distributed load. ν = 0.3. Its left end is hinged and right end is roller. Find deflection at mid point and maximum bending stress in the beam by finite element analysis. Try following modeling:
1. Plane stress analysis with 20 rectangular elements, each 0.5x0.25 m size. That means there are 10 elements in X direction and 2 elements in Y direction. You can convert the uniform load into nodal loads by applying 0.25 kN at extreme nodes and 0.5 kN at intermediate nodes.
2. Solid model analysis. Use standard solid brick or tetrahedral element. Most finite element analysis programs offer these elements.
Figure 23-7
In case of plane stress model formulation, you should use plate finite element whose thickness will be equal to the depth in Z direction. In this problem, this is equal to width of the beam. After performing the finite element analysis, you should get the answer: mid point deflection 1.95x10-4 m maximum stress 3.75 MPa. Your program may display slight different result due to numerical round off in calculation.
The deflected shape should resemble the following figure. Original shape is shown by dotted line. This 2D-beam analysis was performed in Visual Analysis.
For 2D analysis, after modeling your structure should look similar to this figure.
Figure 23-8
The figure 23-9 shows one of mid plane stresses, local σx distribution.
Your program should have the option to display other stresses e.g. σy , τxy etc.
Interpreting the finite element analysis result is very important. It is expected that you spend equal or more time in interpreting analysis result compared to the time previously spent in preparation of the model.
Later we shall see how finite element analysis can produce incompatible result. There you will realize why it is essential to learn some theory behind the finite element analysis.
To analyze the beam as 2D, you should not face any difficulty. However, you should take into account many other things when you analyze the same beam as 3D solid. Firstly, what will be the load? Look, here we’ve applied a total load of 10 kN/m x 5 m = 50 kN acting on the upper face area of 5 m x 0.2 m = 1 m. So, the applied load we have to specify as 50 kN/m² pressure normal to the upper surface. Be careful about the load’s direction. Now, comes the main hurdle, the meshing. If you are using a high-end FEA program, it will mesh the model itself. By default, the program will mesh it by using brick elements or tetrahedral elements. The next figure shows the beam with automatically generated tetrahedral mesh. You may note that, such high density meshing is not really required for this very problem. If you manually mesh with 20 numbers (2 elements along depth and 10 elements along length, similar to shown in fig. 23-7) 8-noded solid elements, (as in SAP2000) you will get exact result for this problem. Left end boundary condition is X, Y, Z translation fixed along bottom edge and Z translation fixed along bottom edge on right end.
Figure 23-10
Did you see that finite element analysis programs normally give you output in the form of nodal displacements and stresses. It does not show you bending moment or shear force diagram. Why? Well, why do you need bending moment and shear force values? To calculate stresses later, isn’t it? Finite element analysis programs directly give you the stress values!