Two-dimensional elastic skeleton geometric nonlinear analysis with Python

Program overview

• This program performs geometrical nonlinear analysis of a two-dimensional elastic plane frame under the assumption of element microdisplacement. As the
• element, a beam element with 1 element, 2 nodes, 1 node and 3 degrees of freedom (horizontal displacement, vertical displacement, rotation) is used.
• Only the nodal external force increment can be handled as the load.
• The arc length increment method is used as an analysis method to obtain the equilibrium condition. This makes it possible to track the phenomenon that the displacement increases but the load decreases. In the
• coordinate system, the right direction is the x-axis, the upward direction is the y-axis, and the counterclockwise direction is the positive rotation direction.
• Simultaneous linear equations are solved by numpy.linalg.solve (A, b).
• Input data file and output data file must be separated by spaces, and the file name is entered as a command line argument.

FEM analysis program

Script for executing FEM analysis

python3 py_fem_gfrmAL.py inp.txt out.txt nnmax

Input data format

npoin  nele  nsec  npfix  nlod  # Basic values for analysis
E  A  I                         # Material properties
    ..... (1 to nsec) .....     # 
node_1  node_2  isec            # Element connectivity, material set number
    ..... (1 to nele) .....     # 
x  y                            # Node coordinate of node
    ..... (1 to npoin) .....    # 
lp  fix_x  fix_y  fix_r         # Restricted node number
    ..... (1 to npfix) .....    # 
lp  df_x  df_y  df_r            # Loaded node and loading conditions (load increment)
    ..... (1 to nlod) .....     #     (omit data input if nlod=0)

Output data format

npoin  nele  nsec npfix  nlod  nnmax
    (Each value of above)
sec  E  A  I
    sec   : Material number
    E     : Elastic modulus
    A     : Section area
    I     : Moment of inertia
    ..... (1 to nsec) .....
node  x  y  fx  fy  fr  kox  koy  kor
    node   : Node number
    x      : x-coordinate
    y      : y-coordinate
    fx     : Load in x-direction
    fy     : Load in y-direction
    fr     : Moment load
    kox    : Index of restriction in x-direction (0: free, 1: fixed)
    koy    : Index of restriction in y-direction (0: free, 1: fixed)
    kor    : Index of restriction in rotation    (0: free, 1: fixed)
    ..... (1 to npoin) .....
elem  i  j  sec
    elem : Element number
    i    : Node number of start point
    j    : Node number of end point
    sec  : Material number
    ..... (1 to nele) .....
* nnn=0 iii=0 lam=0.0
node  fp-x  fp-y  fp-r  dis-x  dis-y  dis-r  dr-x  dr-y dr-r
    node  : Node number
    fp-x  : Total load in x-direction
    fp-y  : Total load in y-direction
    fp-r  : Total load in rotation
    dis-x : Displacement in x-direction
    dis-y : Displacement in y-direction
    dis-r : Displacement in rotation
    dr-x  : Un-balanced force in x-direction
    dr-y  : Un-balanced force in y-direction
    dr-r  : Un-balanced force in rotation
    ..... (1 to npoin) .....
elem  N_i  S_i  M_i  N_j  S_j  M_j
    elem : Element number
    N_i  : Axial force of node-i
    S_i  : Shear force of node-i
    M_i  : Moment of node-i
    N_j  : Axial force of node-j
    S_j  : Shear force of node-j
    M_j  : Moment of node-j
    ..... (1 to nele) .....
* nnn=1 iii=xx lam=xxx
node  fp-x  fp-y  fp-r  dis-x  dis-y  dis-r  dr-x  dr-y dr-r
    node  : Node number
    fp-x  : Total load in x-direction
    fp-y  : Total load in y-direction
    fp-r  : Total load in rotation
    dis-x : Displacement in x-direction
    dis-y : Displacement in y-direction
    dis-r : Displacement in rotation
    dr-x  : Un-balanced force in x-direction
    dr-y  : Un-balanced force in y-direction
    dr-r  : Un-balanced force in rotation
    ..... (1 to npoin) .....
elem  N_i  S_i  M_i  N_j  S_j  M_j
    elem : Element number
    N_i  : Axial force of node-i
    S_i  : Shear force of node-i
    M_i  : Moment of node-i
    N_j  : Axial force of node-j
    S_j  : Shear force of node-j
    M_j  : Moment of node-j
    ..... (1 to nele) .....

.... .... ....

* nnn=nnmax-1 iii=xx lam=xxx
node  fp-x  fp-y  fp-r  dis-x  dis-y  dis-r  dr-x  dr-y dr-r
    ..... (1 to npoin) .....
elem  N_i  S_i  M_i  N_j  S_j  M_j
    ..... (1 to nele) .....
n=(total degrees of freedom)  time=(calculation time)

Output example

Load-Displacement curve drawing program py_fig_gfrm.py

In the load-displacement curve creation program py_fig_gfrm.py, as a result of trying to make it versatile, the input from the command line became quite confusing.

Displacement mode diagram creation program py_fig_gfrm_mode.py

With the displacement mode diagram creation program py_fig_gfrm_mode.py, it is relatively easy to realize a versatile one only with the displacement mode, but it becomes quite complicated when trying to put boundary conditions. Therefore, since I want to create only 4 graphs, I specify the input file in the program and process each file individually.

(Arrow indicating load)

The arrow indicating the load is written as an arrow. with arrow ``` ax.arrow(x,y,u,v,head_length=hl, ....) ``` Write an instruction like

. For example, when drawing a vertical arrow, it should be noted that the length from the tail to the tip of the arrow is (v + head_length). The actual correspondence is as follows. ``` uu=0.0 vv=(ymax-ymin)*0.1 x1=xx[lnod-1] y1=yy[lnod-1]+vv hl=vv*0.4 hw=hl*0.5 ax.arrow(x1,y1,uu,-(vv-hl), lw=2.0,head_width=hw, head_length=hl, fc='#555555', ec='#555555') ```

(Symbol indicating fixed end)

The symbol indicating the fixed end drawn at the base of the vertical one-way beam is drawn using fill (... hatch ='///'). Here, the rectangular area is specified in fill, but since the border is unnecessary, linewidth = 0.0 is specified in fill to prevent the border from being drawn. The actual correspondence is as follows. ``` lp=1; x1=x[lp-1]; y1=y[lp-1] px=[x1-2*scl,x1+2*scl,x1+2*scl,x1-2*scl] py=[y1,y1,y1-1*scl,y1-1*scl] ax.fill(px,py,fill=False,linewidth=0.0,hatch='///') ax.plot([x1-2*scl,x1+2*scl],[y1,y1],color='#000000',linestyle='-',linewidth=1.5) ```

Script for FEM calculation execution / load displacement curve drawing program execution

# FEM calculation by Arc-Length method
python py_fem_gfrmAL.py inp_gfrm_canti.txt out_gfrm_canti.txt 70
python py_fem_gfrmAL.py inp_gfrm_cable.txt out_gfrm_cable.txt 290
python py_fem_gfrmAL.py inp_gfrm_arch.txt out_gfrm_arch.txt 310
python py_fem_gfrmAL.py inp_gfrm_lee.txt out_gfrm_lee.txt 160

# Drawing of Load-displacement curve
python py_fig_gfrm.py out_gfrm_canti.txt 11 11 -411.07 1000 1000 \$u/L\$ $\v/L\$ \$u/L\$\,$\v/L\$ \$P/P_{cr}\$ LL
python py_fig_gfrm.py out_gfrm_cable.txt 12 11 -1644.3 1000 1000 \$u/L\$ \$v/L\$ \$u/L\$\,\$v/L\$ \$P/P_{cr}\$ LL
python py_fig_gfrm.py out_gfrm_arch.txt 21 21 -666.4 -500 -500 \$u/R\$ \$v/R\$ \$u/R\$\,\$v/R\$ \$P\\cdot\(R^2/EI\)\$ UL
python py_fig_gfrm.py out_gfrm_lee.txt 13 13 -166.6 1000 -1000 \$u/L\$ \$v/L\$ \$u/L\$\,\$v/L\$ \$P\\cdot\(L^2/EI\)\$ LL

# Drawing of displacement mode
python py_fig_gfrm_mode.py

python py_fig_gfrm.py out.txt node-L node-D nd-L nd-u nd-v leg-u leg-v x-Label y-Label loc

Displacement mode and load-Displacement curve output example

that's all