Introduction to the Finite Element Method...
Introduction to the Finite Element Method This section presents the general steps included in the finite element method. Typically, for the structural stress analysis, it is required to determine the stresses and deformation (strains) throughout the structure which is in equilibrium and is subjected to applied loads. The finite element method involves modeling of the structure using small units (finite elements). A displacement function is associated with each finite element. The followings are the steps used in finite element method. This will be followed by illustration of the application of these steps on springs and plane stress cases. Step 1. Discretize and Select Element Types Divide the structure into an equivalent system of finite elements with associated nodes. The simplest line elements, Fig.1.a has two nodes, one at each end. The basic twodimensional elements, Fig. 1.b are loaded by forces in their own plane (plane stress). They are triangular or quadrilateral elements. The common three dimensional elements, Fig.1.c, are tetrahedral and hexahedral (brick) elements. They are used to perform three dimensional stress analysis in 3-D solid bodies. Step 2. Select a Displacement Function Choose a displacement function within each element using the nodal values of the element. Linear, quadratic, and polynomials are frequently used functions. Step 3. Define the Stress/Strain Relationships = du/dx = E Step 4. Derive the Element Stiffness Matrix and Equations The stiffness matrix and element equations relating nodal forces and displacements are obtained using force equilibrium conditions or the principle of minimum potential energy. Step 5. Assemble the Element Equations to obtain the Global Equations Step 6. Solve for the Unknown Displacements Step 7. Solve for the Element Strains and Stresses Step 8. Interpret the Results The final goal is to interpret and analyze the results for use in the design process.
Stiffness (Displacement) Method in Finite Element Analysis The linear spring is introduced to provide a simple yet generally instructive tool to illustrate the basic concepts of stiffness method. The principle of minimum potential energy is introduced and applied derive the spring element equations, and use it to solve a spring assemblage problem. This principle will be illustrated using the simplest of elements (those with small numbers of degrees of freedom) so that it will become a more easily understood concept when necessarily applied to elements with large numbers of degrees of freedom in subsequent discussions.
Derivation of the Stiffness Matrix for a Spring Element The symbol k is called the spring constant or stiffness of the spring. Analogies to actual spring constants arise in numerous engineering problems.A prismatic uniaxial bar has a spring constant k = AE/L, where A represents the cross-sectional area of the bar, E is the modulus of elasticity, and L is the bar length. Similarly, a prismatic circular-cross-section bar in torsion has a spring constant k=JG/L, where J is the polar moment of inertia and G is the shear modulus of the material. For one-dimensional heat conduction, k=AKxx/L, where Kxx is the thermal conductivity of the material and onedimensional fluid flow through a porous media, k=AKxx /L, where Kxx is the permeability coefficient of the material. Consider the linear spring element shown in Figure1. The reference points 1&2 are called the nodes of the spring element. The local nodal forces are d1x and d2x for the spring element. These nodal displacements are called the degrees of freedom at each node.
f1x , d1x
f2x , d2x
Figure 1 Linear spring element with positive nodal displacement and force conventions
A relationship between nodal forces and nodal displacements for a spring element need to be developed. This relationship is the stiffness matrix. The nodal force matrix is related to the nodal displacement matrix as follows:
f1x k11 k12 d1x f 2 x k 21 k 22 d 2 x
Where the elements kij of the matrix in Eq. (1) are to be determined.
Step 1 Select Element Type Consider the linear spring (Fig.2) subjected to resulting nodal tensile forces T. The original distance between nodes before deformation is denoted by L.
Step 2 Select a Displacement Function A displacement function u is assumed. Here a linear displacement variation along the x axis of the spring is assumed. Therefore
d2x L x
Figure 2 Linear spring subjected to tensile forces
u a1 a2 x (2) In general, the total number of coefficients a is equal to the total number of degrees of freedom associated with the element. Here the total number of degrees of freedom is two – an axial displacement at each of the two nodes of the element. In matrix form, Eq. (2) becomes
a u 1 x 1 (3) a 2 Evaluating u at each node and solving for a1 and a 2 from Eq.(2) as follows:
u(0) d1x a1 (4) u( L) d 2 x a2 L d1x (5) Or, solving Eq. (5) for a 2 , a2
d 2 x d1x (6) L
On substituting Eqs.(4)and (6) into Eq. (2), we have u(
d 2 x d1x ) x d1x (7) L
In matrix form, we express Eq. (7) as
x d1x ] (8) L d 2 x
u [ N1
d N 2 ] 1x (9) d 2 x x L
x (1) L
are called the shape functions because the N i s express the shape of assumed displacement function over the domain of element when the ith element degree of freedom has unit value and all other degrees of freedom are zero. In addition, they are often called interpolation functions because we are interpolating to find the value of a function between given nodal values.
Step 3 Define the Strain/Displacement and Stress/Strain Relationships The tensile forces T produces a total elongation (deformation) of the spring. For the linear spring, T and are related through Hooke’s law by T k (11)
where is the deformation of the spring,
d 2 x d1x (12) Step 4 Derive the Element Stiffness Matrix and Equations Nodal forces are
f 2 x T (13)
Using Eqs. (11), (12), and (13), we have
T f 1x k (d 2 x d1x ) (14) T f 1x k (d 2 x d1x ) f1x k (d 2 x d1x ) (15) f 2 x k (d 2 x d1x )
Now expressing Eqs.(15) in a single matrix equation yields
f 1x k f 2 x k
k d1x (16) k d 2 x
k Stiffness Matrix is k k
k (17) k
Step 5 Assemble the Element Equations to Obtain the Global Equations and Introduce Boundary Conditions N
K [K ] k e 1
F [F ] f
Where k and f are now element stiffness and force matrices expresses in global reference frame. Step 6 Solve for the Nodal Displacements The displacements are then determined by imposing boundary conditions and solving a system of equations, F K d simultaneously. Step 7 Solve for the Element Forces Finally, the element forces are determined by back-substitution, applied to each element.