Spring 2012 CVEN 6525 Nonlinear Structural Analysis Geometric Non Linearity Due: March 21, 2012 You need to program the geometric nonlinearity in a computer program. More specifically, you need to: 1. Impl Implemen ementt the geometric geometric stiffness matrix of b eam columns. columns. 2. Perform Perform a bifu bifurcati rcation on analy analysis sis and dete determine rmine the elast elastic ic load mul multipl tiplier ier λ which would cause instability. 3. Perf Perform orm an incre incremen mental tal second order nonlinear nonlinear elastic analysis of a fram frame. e. and this can be achieved in one of three ways: 1. Modify (and streamline) the simple program listed below to perform geometric geometric non-linear non-linear analysis. More specifically: 2. Modify Modify the educational educational version version of Mercu Mercury ry by compl completin eting g the blanks (actually (actually XXX XXX)) in the code given to you. 3. Use your own own program. program. Once completed, test your program through the following problems: 1. Dete Determine rmine λ for the following frame
P
P
T2 u1
T3
B
2
C
4
A2 34.6 in , I2 200 in
2
u1
4
A3 18.0 in , I3 50 in
10 ft
A1
2
6 ft
4
24.5 in , I1 100 in
D
E 29,0 ,00 00 ks ksi
A
15 ft
2. Determine the internal forces for the following following member; compare results for axial compression and axial tension. 50 kN
80,000 kN
80,000kN
6m
6m
1
3. Consider the following cantilever beam with the following properties: E = 800 kN/mm2 , L = 4,000 mm, I = 1e7 mm4 , A=50 mm2 , p= 5,000 kN. Discritize the bar into 5 members of equal length and using increment of -10kN plot u and v vs P . z
Following is an exact dimensionless solution.
and this is a solution obtained by Mercury Load−Disp 1 0.9 0.8 0.7 t n 0.6 e m e c 0.5 a l p s i D 0.4
We note that the internal forces are consistent with the reactions (specially for the second node of element 2), and amongst themselves, i.e. the moment at node 2 is the same for both elements (8.0315).
