Truss Stabilty Investigation by the Matrix Method of Analysis

The following truss is unstable. Instead of determining that by finding a mechanism or by use of the zero-load test, this section will setup a full matrix analysis.

Import some utilities from a library for statically determinate 2D truss analysis.

from sdtruss import sdtruss, SDTError

Joint Coordinates

This section specifies the x-y coordinates of each joint. Each line of input specifies the id of the joint and its x and y coordinates:

jc = [ ('a', 0, 0),
       ('b', 0, 4),
       ('c', 2, 8),
       ('d', 6, 8),
       ('e', 8, 4),
       ('f', 8, 0),
       ]

Member Incidences

This section defines each member in the truss by specifying the ids of the joints at either end of each member. Each member is connected to exactly two joints.

mi = [ ('a', 'b',),   # there is a member from joint a to joint b, etc.
       ('b', 'c',),
       ('c', 'd',), 
       ('d', 'e',),
       ('e', 'f',),
       ('f', 'a',),
       ('b', 'e',),
       ('c', 'f',),
       ('a', 'd',),
       ]

Reactions

This section specifies the reaction components. Each line specifies one independent reaction by naming the joint on which it impinges and the direction of the reaction. Directions are given as relative direction components. For example, a direction component of ‘0,1’ means a vertical reaction (in the $y$-direction). A direction component of ‘3,4’ would specify an angle of 53.13 degrees to the $x$-axis ($\tan^{-1}{4\over3}$ - relative components of 3 in $x$ and 4 in $y$).

rf = [ ('a', 1, 0),             # reaction force dirns - horizontal at a
       ('a', 0, 1),             # vertical at a
       ('f', 0, 1),
       ]

Joint Loads

Joint loads arfe specified by naming the joint, providing the magnitude, and direction components in which it acts.

jl = [ ('c', 1, 1, 0),      # joint loads (j, p, dx, dy) - this is a unit load at c, in x-direction
       ]

Analysis

Now we analyze by providing the above data to the library function.

try:
    sdtruss( jc, mi, rf, jl )
except SDTError as err:
    print('**** Error: {0}'.format(err))

**** Error: 'C' matrix is rank deficient.  Truss is unstable.

We see above the the rank-check of the matrix failed. The rank of the coefficient matrix was less than the number of columns and thus no solution was possible.

Change the Geometry

Here we see a very small change made to the geometry, by moving joint c up 1mm.

jc = [ ('a', 0, 0),
       ('b', 0, 4),
       ('c', 2, 8.001),
       ('d', 6, 8),
       ('e', 8, 4),
       ('f', 8, 0),
       ]

try:
    sdtruss( jc, mi, rf, jl )
except SDTError as err:
    print('**** Error: {0}'.format(err))

unknown  joint-i  joint-j        force
-------  -------  -------        -----
ab       a        b               1779 T
bc       b        c               1988 T
cd       c        d               2222 T
de       d        e               1988 T
ef       e        f               1778 T
fa       f        a               1334 T
be       b        e              889.1 C
cf       c        f               2224 C
ad       a        d               2222 C
Rax      a                          -1  @ 1,0
Ray      a                          -1  @ 0,1
Rfy      f                           1  @ 0,1

Mathematically, the truss is no longer unstable, but notice the very large member forces (compared to the small applied load). Forces were amplified by about a factor of 2000. This demonstrates that trusses that are unstable or nearly so will have very high member forces, leading to large distortions and geometry changes. This is very undesireable in a civil engineering structure.