7. INTRODUCTION TO THE FINITE
ELEMENT METHOD
Engineers u a wide range of tools and techniques to ensure that the designs they create are safe. However, accidents sometimes happen and when they do, companies need to know if a product failed becau the design was inadequate or if there is another cau, such as an ur error. But they have to ensure that the product works well under a wide range of conditions, and try to avoid to the maximum a failure produced by any cau. One important tool to achieve this is the finite element method.
“The finite element method is one of the most powerful numerical techniques ever devid for solving differential (and integral) equations of initial and boundary-value problems in geometrically complicated regions.” (Red dy, 1988). There is some data that can not be ignored when analyzing an element by the finite element method. This input data is to define the domain, the boundary and initial conditions and also the physical properties. After knowing this data, if the analysis is done carefully, it will give satisfactory results. It can be said that the process to do this analysis is very methodical, and that it is why it is so popular, becau that makes it easier to apply. “The finite element analysis of a problem is
so systematic that it can be divided into a t of logical steps that can be implemented on a digital computer and can be utilized to solve a wide range of problems by merely changing the data input to the computer program.” (Reddy,1988).
The finite element analysis can be done for one, two and three-dimensional problems. But generally, the easier problems are
tho including one and two dimensions, and tho can be solved without the aid of a computer, becau even if they give a lot of equations, if they are handled with care, an exact result can be achieved. But if the analysis requires three-dimensional tools, then it would be a lot more complicated, becau it will involve a lot of equations that are very difficult to solve without having an error. That is why engineers have developed softwares that can perform the analys by computer, making everything easier. The softwares can make analysis of one, two and three dimensional problems with a very good accuracy.
A basic thing to understand how finite element works is to know that it divides the whole element into a finite number of small elements. “The domain of the problem is viewed as a collection of nonintercting simple subdomains, called finite elements…The subdivision of a domain into elemen
ts is termed finite element discretization. The collection of the elements is called the finite element mesh of the domain.” (Reddy,1988). The advantage of dividing a big element into small ones is that it allows that every small element has a simpler shape, which leads to a good approximation for the analysis. Another advantage is that at every node (the interction of the boundaries) aris an interpolant polynomial, which allows an accurate result at a specific point. Before the finite element method, engineers and physicians ud a method that involved the u of differential equations, which is known as the finite difference method.
The method of the finite element is a numerical technique that solves or at least approximates enough to a solution of a system of differential equations related with a physics or engineering problem. As explained before, this method requires a completely defined geometrical space, and then it would be subdivided into small portions, which together will form a mesh. The difference between the method of the finite element and the method of the finite difference is that in the cond one, the mesh consists of lines and rows of orthogonal lines, while in the method of the finite element the division does not
necessarily involves orthogonal lines, and this results in a more accurate analysis (Figure 38).
Figure 38 (taken from Algo r 13 ®)
diss是什么意思中文>winter是什么意思The equations ud for the finite element method are a lot, but they have the basison some single equations that describe a particular phenomenon. Tho equations are:
The elliptic equation is described by
22
22
0 dx dy
∂Φ∂Φ
mclub+=
The parabolic equation is described by
2
2
0 dx t
∂Φ∂Φ
-=
∂
The hyperbolic equation is described by
保留英文22
1 22
0 t x
episode∂Φ∂Φ
-=∂∂
No matter which is the cau of the internal forces and the deformation that they cau, there are three basic conditions that allow the finite element analysis: the equilibrium of forces, the compatibility of displacements and the laws of material behavior. “The first condition merely requires that the internal forces balance the external applied loads.” (Rocke t et. al., 1983). That is the most important condition, but the other two assure that the system will be
a statically determinate problem. Another condition that must be taken into account is that there exists a relationship between the load applied and the deformation, and this is given by Hooke’s law, as explained in past chapters, but only in the elastic range.
In order to achieve a structural analysis by matrix methods, there might be three ways: stiffness (displacement) method, flexibility (force) method and mixed method. In the first two methods, two basic conditions of nodal equilibrium and compatibility must be reached. In the first method, the once the displacement compatibility conditions are reached, then an answer can be given. In the cond method, once the conditions of nodal equilibrium are satisfied and then the compatibility of nodal displacement, forces are known in the members.
One of the principles that is the basis of the finite element method is the one known as principle of virtual work. “This principle is concerned with the relationship which exists between a t of external loads and the corresponding internal forces which together satisfy the equilibrium condition, and also with ts of joint (node) displacements and the corresponding member deformations which satisfy the conditions of compatibility.” This principle can be stated in terms of an equation of equilibrium of loads, where the work done by the external loads is equal to the internal virtual work absorbed by the element. Or expresd as an equation:
v F d δσε⋅ =⋅⋅∑⎰free money
where F are the external loads, the deflection, the system of internal forces, and the internal deformations.
A pin ended tie has similar characteristics to tho of an elastic spring that is subjected on one end, looking downwards, suffering the effects of gravity and the effect of an external load. The direct relationship between the force and the displacement of the free end is:5202
F k δ=⋅ the vale k is known as the stiffness of the spring. Once the value
of the applied force and the value of the stiffness, the equation can be inverted to find the displacement: 1
F k
δ= This is a simple example for systems that imply only a few data. But when the problem implies more complex systems, then the equations become a little more complicated. When a number of simple members are interconnected at a number of nodes, the displacement caud by the load can only be described by simultaneous equations. Then, the simpler equation en before becomes:
{}[]{}F K δ=
no one is therewhere K is the stiffness of the whole structure.
玷污的意思拾带重还For example, for a spring that has two pins, it generates two forces and two displacements. Therefore, the stiffness matrix would be of order 2 x 2:
111121221222F k k u F k k u ⎧⎫⎡⎤⎧⎫=⎨⎬⎨⎬⎢⎥⎩⎭⎣⎦⎩⎭
where u reprents the displacement.
Boundary conditions are the limitations t for the problem. The limitations are necessary in order to solve it, becau otherwi, the system would be taken as a rigid body. The limitations stated by the boundary conditions are like where does the element is likely to move, and were it is restricted. If there were not boundary conditions, the body would be floating in the space, and under the action of any load, it would not suffer any deformation, but it would move around the space as a rigid body. So, when assuming boundary conditions, it has to be assured that the element has enough of them in order to prevent moving as a rigid body. Once this is done, the values of the displacements are obtained and can be substituted in the last equation en, which will give that the displacement is equal to zero, becau the element can not move in any direction. Then, algebra is applied, and the values of the forces or of the stiffness are known.
