University of Florida/Eml4507/s13.team4ever.Wulterkens

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Problem 2.1 Model 2-D Truss With 2 Inclined Elements and Verify Assembly

Given 2-D Truss Model and Constants[edit]

Consider the 2D truss system

Figure 1: 2D Truss system analyzed in this problem. ([1])


Use the following constants to very calculations



Find:[edit]

1) Explain the Assembly Process of the Above 2D Truss System[edit]

2) Verify With CALFEM[edit]

Solution: Assembly Process for 2d Truss System[edit]

Construct local stiffness matrices[edit]

The stiffness matrices will be constructed for individual elements 1 and 2. The rows and columns will be numbered to represent the force and degree of freedom. For element 1

For element 2

The variable K can be determined by

If the variables are plugged in, we get the following matrices:

For element 1

For element 2


It is clear that if we wish to combine the matrices, it will place coordinate 3,3 of element 2 on the location of 3,3 of element 1 due to them referencing the same point in the same direction. When combining the matrices, addition will be used since the matrices represent directional forces and forces always combine in the same direction by addition.

Combining local stiffness matrices into a global matrix[edit]

The local matrices from element 1 and 2 will be combined as described above to give


Solution: Verify With CALFEM[edit]

MatLab Code[edit]

EDU>> Edof = [1 1 2 3 4

2 3 4 5 6];

EDU>> K = zeros(6);

EDU>> f = zeros(6,1);

EDU>> f(4) = -7;

EDU>> E1 = 3; E2 = 5;

EDU>> L1 = 4;L2 = 2;

EDU>> A1 = 1;A2 = 2;

EDU>> ep1 = [E1 A1]

ep1 =

    3     1


EDU>> ep2 = [E2 A2]

ep2 =

    5     2


EDU>> ex1 = [-3.464 0]; ex2 = [0 1.414];

EDU>> ey1 = [-2 0]; ey2 = [0 -1.414];

EDU>> Ke1 = bar2e(ex1,ey1,ep1)

Ke1 =

   0.5625    0.3248   -0.5625   -0.3248
   0.3248    0.1875   -0.3248   -0.1875
  -0.5625   -0.3248    0.5625    0.3248
  -0.3248   -0.1875    0.3248    0.1875


EDU>> Ke2 = bar2e(ex2,ey2,ep2)

Ke2 =

   2.5004   -2.5004   -2.5004    2.5004
  -2.5004    2.5004    2.5004   -2.5004
  -2.5004    2.5004    2.5004   -2.5004
   2.5004   -2.5004   -2.5004    2.5004


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

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

K =

   0.5625    0.3248   -0.5625   -0.3248         0         0
   0.3248    0.1875   -0.3248   -0.1875         0         0
  -0.5625   -0.3248    3.0629   -2.1756   -2.5004    2.5004
  -0.3248   -0.1875   -2.1756    2.6879    2.5004   -2.5004
        0         0   -2.5004    2.5004    2.5004   -2.5004
        0         0    2.5004    -2.5004  -2.5004    2.5004


EDU>> bc = [1 0;2 0;5 0;6 0];

EDU>> [a,r]=solveq(K,f,bc)

a =

        0
        0
  -4.3519
  -6.1268
        0
        0


r =

   4.4378
   2.5622
        0
  -0.0000
  -4.4378
   4.4378

Conclusion[edit]

By comparing the value from the hand calculation and the value from the CALFEM program, we can see that the two matrices are identical. This verifies the assembly process of using addition when combining local matrices by addition.

--EML4507.s13.team4ever.Wulterkens (discusscontribs) 07:21, 6 February 2013 (UTC)