A truss is a structure in which members are arranged in such a way that they are subjected to axial loads only. The joints in trusses are considered to be pinned. In plane trusses, all members are assumed to be in the x-y plane.
A plane truss element is an axial deformation element oriented arbitrarily in two-dimensional space.
s and t = local coordinates along the member's axis and perpendicular to it, respectively.
x and y = global coordinates
d1 and d2 = nodal displacements along s axis (local coordinates)
u1 and u2 = nodal displacements along x axis (global coordinate)
v1 and v2 = nodal displacements along y axis (global coordinate)
L = length of each member
Using the following interpolation we can find the deformation at any point on the member based on the known values of nodal displacements:
and the element equations in the local coordinate system are as follows:
where k1= the local stiffness matrix of element; d1 = local degrees of freedom; r1 = local applied forces; E = elastic modulus of the material; and A = cross-sectional area.
For the sake of brevity, transformation matrices are not described in here.
Final form of element equations are as follows:
for solving the above-mentioned equation
after solving it, the axial displacements can be obtained from:
and from that, the axial strain is simply the first derivative of the axial displacement, giving constant strain over the element as
and from the axial strain, one can get to the axial stress of each member using the Hook's law, where
and consequently, the force of each member can be obtained from