Kulak & Grondin Example 8.2

This example is from "Limit States Design in Structural Steel", Ninth Edition, CISC, by Kulak and Grondin. This example is slightly modified in order to compare the results with that obtained using the method of slope-deflection. The modifications to the model are:

a pin is inserted at the top of columns JK and IG, with the result that column stacks JKL and IGH offer no lateral resistance (in the original K&G example, the columns are continuous and there is a very small horizontal reaction at the base of those two columns).
the cross-sectional areas of all members are made very large so as to make axial changes in length neglible.



In [1]:

    
from salib import showImage
import salib.nbloader
import Frame2D_v03 as f2d

The notional lateral loads of 6.9 kN at node H and 8.4 kN at node G are added as before.



In [2]:

    
frame = f2d.Frame2D()
frame.loadStructuralData('KG82sd')   # read the CSV files in directory 'KG82sd.d'



In [3]:

    
frame.doall()
frame.saveStructuralData('KG82sd')









    



Time to solve = 0.01 sec.

ndof = 36, nfree = 26, nconstraints = 10



Nodes:
======

Node          X         Y  Constraints  DOF #s
----      -----     -----  -----------  ------
A         10000         0  FX,FY,MZ     26,27,28
B         10000      6500               0,1,2
C         10000     12000               3,4,5
D         20500         0  FX,FY,MZ     29,30,31
E         20500      6500               6,7,8
F         20500     12000               9,10,11
G         30500      6500               12,13,14
H         30500     12000               15,16,17
I         30500         0  FX,FY        32,33,18
J             0         0  FX,FY        34,35,19
K             0      6500               20,21,22
L             0     12000               23,24,25



Members:
========

Member   Node-J  Node-K    Length       dcx       dcy  Size                Ix           A  Releases
------   ------  ------    ------   -------   -------  --------      --------       -----  --------
AB       A       B         6500.0   0.00000   1.00000  W310x97       2.22e+08       1e+10  
BC       B       C         5500.0   0.00000   1.00000                2.22e+08       1e+10  
DE       D       E         6500.0   0.00000   1.00000                2.22e+08       1e+10  
EF       E       F         5500.0   0.00000   1.00000                2.22e+08       1e+10  
IG       I       G         6500.0   0.00000   1.00000                2.22e+08       1e+10  MZK
GH       G       H         5500.0   0.00000   1.00000                2.22e+08       1e+10  
JK       J       K         6500.0   0.00000   1.00000                2.22e+08       1e+10  MZK
KL       K       L         5500.0   0.00000   1.00000                2.22e+08       1e+10  
CF       C       F        10500.0   1.00000   0.00000                4.88e+08       1e+10  
BE       B       E        10500.0   1.00000   0.00000  W460x106      4.88e+08       1e+10  
FH       F       H        10000.0   1.00000   0.00000                4.88e+08       1e+10  MZJ,MZK
EG       E       G        10000.0   1.00000   0.00000                4.88e+08       1e+10  MZJ,MZK
KB       K       B        10000.0   1.00000   0.00000                4.88e+08       1e+10  MZJ,MZK
LC       L       C        10000.0   1.00000   0.00000                4.88e+08       1e+10  MZJ,MZK



Node Loads:
===========

Node  Dirn     Load
----  ----    -----
G     FX      43400
H     FX      21900

Member Loads:
=============

Member  Load
------  ----------------
CF      UDL(L=10500.0,W1=-45)
BE      UDL(L=10500.0,W1=-55)
FH      UDL(L=10000.0,W1=-45)
EG      UDL(L=10000.0,W1=-55)
KB      UDL(L=10000.0,W1=-55)
LC      UDL(L=10000.0,W1=-45)



Node Displacements:
===================

Node        DX         DY      Rotation
----      ------     ------   ---------
A          0.000      0.000   0.0000000
B         23.363     -0.000  -0.0071297
C         34.307     -0.000  -0.0072270
D          0.000      0.000   0.0000000
E         23.363     -0.000   0.0031089
F         34.307     -0.000   0.0057752
G         23.363     -0.000  -0.0019898
H         34.307     -0.000  -0.0019898
I          0.000      0.000  -0.0035943
J          0.000      0.000  -0.0035943
K         23.363     -0.000  -0.0019898
L         34.307     -0.000  -0.0019898



Reactions:
==========

Node        FX         FY         MZ  
----     -------    -------    -------
A         -0.371    995.932     49.908
D        -64.929   1054.068    189.782
I         -0.000    500.000      --   
J          0.000    500.000      --   



Member End Forces:
==================

          /----- Axial -----/   /----- Shear -----/   /----- Moment ----/
Member       FX J       FX K       FY J       FY K       MZ J       MZ K
------     -------    -------    -------    -------    -------    -------
AB         995.932   -995.932      0.371     -0.371     49.908    -47.494
BC         453.539   -453.539    -91.386     91.386   -250.526   -252.097
DE        1054.068  -1054.068     64.929    -64.929    189.782    232.254
EF         468.961   -468.961    113.286   -113.286    290.012    333.061
IG         500.000   -500.000      0.000     -0.000      0.000      0.000
GH         225.000   -225.000      0.000     -0.000      0.000      0.000
JK         500.000   -500.000      0.000      0.000      0.000      0.000
KL         225.000   -225.000      0.000     -0.000      0.000      0.000
CF          91.386    -91.386    228.539    243.961    252.097   -333.061
BE         -91.757     91.757    267.393    310.107    298.020   -522.266
FH         -21.900     21.900    225.000    225.000      0.000      0.000
EG         -43.400     43.400    275.000    275.000      0.000      0.000
KB           0.000      0.000    275.000    275.000      0.000      0.000
LC           0.000      0.000    225.000    225.000      0.000      0.000

Compare with results from slope-deflection



In [4]:

    
import pandas as pd

BM = [('AB', -49.9080183705027, 47.4941396356060),   # results from method of slope deflection
     ('BC', 250.526060428376, 252.096857589821),
     ('DE', -189.782099213905, -232.254022051198),
     ('EF', -290.011616822927, -333.061301195269),
     ('BE', -298.020200063982, 522.265638874126),
     ('CF', -252.096857589821, 333.061301195269)]

SD = pd.DataFrame({m:{'MZJ':-a,'MZK':-b} for m,a,b in BM}).T
SD









    Out[4]:






  
    
      
      MZJ
      MZK
    
  
  
    
      AB
      49.908018
      -47.494140
    
    
      BC
      -250.526060
      -252.096858
    
    
      BE
      298.020200
      -522.265639
    
    
      CF
      252.096858
      -333.061301
    
    
      DE
      189.782099
      232.254022
    
    
      EF
      290.011617
      333.061301



In [5]:

    
R = frame.get_mefs()       # get our member end forces for the same members
#R[['MZJ']] - SD[['MZJ']]
R = R.ix[SD.index]
R









    Out[5]:






  
    
      
      FXJ
      FYJ
      MZJ
      FXK
      FYK
      MZK
    
  
  
    
      AB
      995932.392434
      371.365815
      4.990802e+07
      -995932.392434
      -371.365815
      -4.749414e+07
    
    
      BC
      453539.100854
      -91385.984914
      -2.505261e+08
      -453539.100854
      91385.984914
      -2.520969e+08
    
    
      BE
      -91757.350586
      267393.291580
      2.980202e+08
      91757.350586
      310106.708420
      -5.222656e+08
    
    
      CF
      91385.983398
      228539.100854
      2.520969e+08
      -91385.983398
      243960.899146
      -3.330613e+08
    
    
      DE
      1054067.607566
      64928.635068
      1.897821e+08
      -1054067.607566
      -64928.635068
      2.322540e+08
    
    
      EF
      468960.899146
      113285.983887
      2.900116e+08
      -468960.899146
      -113285.983887
      3.330613e+08

% Difference in End Moments



In [6]:

    
m = R[['MZJ','MZK']]*1E-6
pd = (100*(m - SD[['MZJ','MZK']])/m).round(6)
pd

Max. difference is 0.0004%, which is quite good.



In [7]:

    
pd.abs().max()









    Out[7]:





MZJ    0.000002
MZK    0.000001
dtype: float64

	MZJ	MZK
AB	49.908018	-47.494140
BC	-250.526060	-252.096858
BE	298.020200	-522.265639
CF	252.096858	-333.061301
DE	189.782099	232.254022
EF	290.011617	333.061301

	FXJ	FYJ	MZJ	FXK	FYK	MZK
AB	995932.392434	371.365815	4.990802e+07	-995932.392434	-371.365815	-4.749414e+07
BC	453539.100854	-91385.984914	-2.505261e+08	-453539.100854	91385.984914	-2.520969e+08
BE	-91757.350586	267393.291580	2.980202e+08	91757.350586	310106.708420	-5.222656e+08
CF	91385.983398	228539.100854	2.520969e+08	-91385.983398	243960.899146	-3.330613e+08
DE	1054067.607566	64928.635068	1.897821e+08	-1054067.607566	-64928.635068	2.322540e+08
EF	468960.899146	113285.983887	2.900116e+08	-468960.899146	-113285.983887	3.330613e+08

	MZJ	MZK
AB	-0.000000	0.000001
BC	-0.000000	-0.000000
BE	-0.000000	-0.000000
CF	-0.000000	-0.000001
DE	0.000002	0.000001
EF	-0.000001	-0.000001