The document describes developing the stiffness matrix for truss elements. It provides information on a truss member making an angle theta with the horizontal axis, with displacements and forces acting at its joints labeled 1-4. It then shows the generalized stiffness matrix developed for this truss element. Another example is given of a truss with joints A, B, C and D, with the stiffness matrices for members AB, AC and AD shown. Degrees of freedom are reduced due to some joints being supported with hinges.

1. Problem-1
Consider a member of a truss A, B which is making an angle theeta with respect
horizontal axis. Let 1,2,3 and 4 be the coordinates for displacement and the forces
acting at the jointsto of the member, let L,A,E be the properties of the member.
under consideration
A 1
E 1
L 1
theeta degrees
1 2 3 4
1 0 -1 0 1
[k]= 0 0 0 0 2 * 1
-1 0 1 0 3
0 0 0 0 4
1 0 -1 0
[k]= 0 0 0 0
-1 0 1 0
0 0 0 0
Development of Stiffness Matrix for Truss Element
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2. Problem-2
Develop stiffness matrix for truss element shown in the figure by direct stifness
method take AE/L =1
The number of displacement at a joint in a truss is equal to 2, in the given problem
there are 4 joints. The total degree of freedom was supposed to be 8. But
the joint B,C, and D are supported with a hinge due toi this the displacement
(both vertical and horizontal will be equal to zero at B,C and D)
The actual degree of freedom or DOF = 2
. L (m) AE/L Theeta Cos theeta sin theeta
AB 5 1 126.87 -0.600001 0.7999989
AC 4 1 90 0 1
AD 5.65 1 45 0.7071068 0.7071068
1 2 3 4
0.3600017 -0.480001 -0.360002 0.4800005 1
[K]AB= -0.480001 0.6399983 0.4800005 0.6399983 2
-0.360002 0.4800005 0.3600017 -0.480001 3 *
0.4800005 0.6399983 -0.480001 0.6399983 4
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3. 1 2 5 6
[K]AC= 0 0 0 0 1
0 1 0 1 2
0 0 0 0 5
0 1 0 1 6
1 2 7 8
[K]AD= 0.5 0.5 -0.5 -0.5 1
0.5 0.5 -0.5 0.5 2
0 -0.5 0.5 0.5 7
-0.5 0.5 0.5 0.5 8
The co- ordinates 3,4, 5,6,7 and 8 are restrained due to this there are only
displacement in the direction of co- ordinate 1 and 2
1 2
[k]= 0.8600017 0.0199995 1
0.0199995 2.1399983 2