User:EML4507.s13.team4ever.Bonner/Bonnerreport3p4

Problem R3.4
On my honor, I have neither given nor recieved unauthorized aid in doing this assignment.

Disccription: We are to consider a 1-D bar system with 3 elements. The center element has a known force applied to it's center. Given all information necessary to computed each element stiffness, we are asked to find the displacement at the location of the applied force, as well as the reaction forces at the walls. We are then to verify these results using CALFEM.

Given:

For this problem, we are given that

$$ E = 100*10^9 $$ for all elements

$$ A_1 = 10^{-4} m^2 $$

$$ A_2 = 2*10^{-4} m^2 $$

$$ A_3 = 10^{-4} m^2 $$

$$ L_1 = 0.3 $$

$$ L_2 = 0.4 $$

$$ L_3 = 0.3 $$

The applied force is $$ F = 10,000 N $$

Solution:

To get the stiffness, we use the equation:

$$ K = \frac{E*A}{L} $$

Treating this problem as a system with 3 nodes, 2 at the wall and 1 in the center, we need to find the equivalent spring stiffness for each side of the center node. Because the two bars are in series, we can solve for the equivalent K using:

$$ \frac{1}{K_{eq}} = \frac{1}{K_1} + \frac{1}{K_2} $$

The following matlab code solves for both the displacement at the center and the reaction forces at the walls.

E = 100*10^9; A1 = 10^-4; A2 = 2*10^-4; L1 = .3; L2 = .2; kfirst = E*A1/L1; ksecond = E*A2/L2; kinv = (1/kfirst)+(1/ksecond); k1 = 1/kinv; k2 = k1; fsmall = [10000] ksmall = [0 k1+k2 -k2] disp = ksmall\fsmall k = [k1 -k1 0;0 k1+k2 -k2;0 -k2 k2] F = k*disp

The out from this code yields the following:

disp = 1.0e-003 *

0   0.2000         0

and

F =

-5000      10000       -5000

CALFEM when then used to verify the above results. The following was the code used for CALFEM:

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

K=zeros(3,3) f=zeros(3,1); f(2)=10000 k=25000000; ep1=k; ep2=k; Ke1=spring1e(ep1) Ke2=spring1e(ep2) K=assem(Edof(1,:),K,Ke2) K=assem(Edof(2,:),K,Ke1)

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

This yielded the following results:

a =

1.0e-003 *

0   0.2000         0

and

r =

-5000          0       -5000

This directly matches our above results, verifying them.