exs_spring

Purpose:

Show the basic steps in a finite element calculation.

Description:

The general procedure in linear finite element calculations is carried out for a simple structure. The steps are:

  • Define the model

  • Generate element matrices

  • Assemble element matrices into the global system of equations

  • Solve the global system of equations

  • Evaluate element forces

Consider the system of three linear elastic springs, and the corresponding finite element model. The system of springs is fixed in its ends and loaded by a single load \(F\).

_images/exs1_1.png
_images/exs1_2.png
Example:

The computation is initialized by defining the topology matrix Edof, containing element numbers and global element degrees of freedom:

>> Edof=[1  1 2;
         2  2 3;
         3  2 3];

The global stiffness matrix K (3×3) of zeros:

>> K=zeros(3,3)

K =
     0     0     0
     0     0     0
     0     0     0

And the load vector f (3×1) with the load \(F=100\) in position 2:

>> f=zeros(3,1);  f(2)=100

f =
      0
    100
      0

Element stiffness matrices are generated by the function spring1e. The element property ep for the springs contains the spring stiffnesses \(k\) and \(2k\) respectively, where \(k=1500\):

>> k=1500;  ep1=2*k;  ep2=k;  ep3=2*k;
>> Ke1=spring1e(ep1)

Ke1 =
        3000       -3000
       -3000        3000

>> Ke2=spring1e(ep2)

Ke2 =
        1500       -1500
       -1500        1500

>> Ke3=spring1e(ep3)

Ke1 =
        3000       -3000
       -3000        3000

The element stiffness matrices are assembled into the global stiffness matrix K according to the topology:

>> K=assem(Edof(1,:),K,Ke1)

K =
        3000       -3000           0
       -3000        3000           0
           0           0           0

>> K=assem(Edof(2,:),K,Ke2)

K =
        3000       -3000           0
       -3000        4500       -1500
           0       -1500        1500

>> K=assem(Edof(3,:),K,Ke3)

K =
        3000       -3000           0
       -3000        7500       -4500
           0       -4500        4500

The global system of equations is solved considering the boundary conditions given in bc:

>> bc= [1 0; 3 0];
>> [a,r]=solveq(K,f,bc)

a =
         0
    0.0133
         0

r =

  -40.0000
         0
  -60.0000

Element forces are evaluated from the element displacements. These are obtained from the global displacements a using the function extract:

>> ed1=extract_ed(Edof(1,:),a)

ed1 =
      0      0.0133

>> ed2=extract_ed(Edof(2,:),a)

ed2 =
      0.0133      0

>> ed3=extract_ed(Edof(3,:),a)

ed3 =
      0.0133      0

The spring forces are evaluated using the function spring1s:

>> es1=spring1s(ep1,ed1)

es1 =
       40

>> es2=spring1s(ep2,ed2)

es2 =
      -20

>> es3=spring1s(ep3,ed3)

es3 =
      -40