Image transcription text

Figure 3 Imagine that the horizontal lower bar in Figure 3 is rigid and is constrained to remain horizontal
as load P and displacement v increase. The three vertical bars are elastic-perfectly,»r plastic, with A :1, E
:1, and L = 2. The three bars have the respective yield-point loads El" : 2,

Image transcription text

Ff] : 4, and FF : 6. Starting frcm zerc lead. apply successive lead increments M3: = 6, all": : 4, and 5P; : 2.
Determine the currespcnding values of the incremental displacements av! and total displacements vr by
solving the incremental equations step by step. Remember to update the tangent stiffness ... 

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incr. load it. total resid. tangent it. no. level no. disp. force stiffness disp. fn 1 0.5 0 0.00000 0.50000
5.00000 0.10000 1 0.10000 0.04000 4.20000 0.00952 2 0.10952 0.00038 4.12384 0.00009 3 0.10961
0.00001 2 1.0 0.10961 0.50001 4.12312 0.12127 1 0.23088 0.05882 3.15296 0.01866 2 0.24... 

Image transcription text

iteration disp. residual tang. iterative number value stiffness disp. di Ad? - 5di + f Kri = 5 - 8d; 6d = Kyri 0
0.00000 1.00000 5.00000 0.20000 0.20000 0.16000 3.40000 0.04706 0.24706 0.00885 3.02352
0.00293 3 0.24999 0.00003

This is for a course in non-linear element analysis. Note that the images are NOT related, only for illustration. This means that the calculations is just an example to make my question and topic more clear.

 

Previously we have been given a function for the residual forces, such as shown in the last image with ri = 4d^2 -di +f.

 

My question is as follows : 

1. How can I establish an such an equation for the problem in the first image, in order to do iterational computation as illustrated in the latter images?

2. What happens with the equation once the material enters the plastic zone?