# Plane Trusses

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## Introduction on Plane Trusses :

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.

## Concepts and Formulas of Plane Trusses:

A plane truss element is an axial deformation element oriented arbitrarily in two-dimensional space.

Definitions:

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:

$u(s)=(\frac{L-s}{L}-\frac{s}{L})\begin{pmatrix} d_1\\ d_2 \end{pmatrix};\; \; 0\leq s\leq L$

and the element equations in the local coordinate system are as follows:

$\frac{EA}{L}\begin{bmatrix} 1 & -1\\ 1 & -1 \end{bmatrix}\begin{bmatrix} d_1\\ d_2 \end{bmatrix}=\begin{bmatrix} P_1\\ P_2 \end{bmatrix}\Rightarrow k_1d_1=r_1$

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:

$\frac{EA}{L}\begin{bmatrix} l_s^2 & l_sm_s & -l_s^2 & -l_sm_s\\ l_sm_s & m_s^2 & -l_sm_s & -m_s^2\\ -l_s^2 & -l_sm_s & l_s^2 & l_sm_s\\ -l_sm_s & -m_s^2 & l_sm_s & m_s^2 \end{bmatrix}\begin{bmatrix} u_1\\ v_1\\ u_2\\ v_2 \end{bmatrix} = \begin{bmatrix} F_{1x}\\ F_{1y}\\ F_{2x}\\ F_{2y} \end{bmatrix}\Rightarrow KU=F$

where

$l_s=cos\alpha=\frac{x_2-x_1}{L}$

$m_s=sin\alpha=\frac{y_2-y_1}{L}$

$L=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$

for solving the above-mentioned equation

$U = K^{-1}F$

after solving it, the axial displacements can be obtained from:

$\begin{bmatrix} d_1\\ d_2 \end{bmatrix} =\begin{bmatrix} l_s & m_s & 0 &0 \\ 0 & 0 & l_s & m_s \end{bmatrix}\begin{bmatrix} u_1\\ v_1\\ u_2\\ v_2 \end{bmatrix}$

and from that, the axial strain is simply the first derivative of the axial displacement, giving constant strain over the element as

$\epsilon=\frac{du}{ds}=\frac{1}{L}(-d_1+d_2)$

and from the axial strain, one can get to the axial stress of each member using the Hook's law, where

$\sigma=E\epsilon$

and consequently, the force of each member can be obtained from

$F = \sigma A$

Equations given above are only for one member, for a plane truss structure containing more than member, stiffness, displacement, and force matrices need to be assembled before solving. On the other hand, sufficient number of boundary conditions are required to get a unique solution. Using the files under download section may help the readers understand the concepts better.

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## Solved sample problems of Plane Trusses:

Group 1:

Nodes file for plane truss analysis
Element files for plane truss analysis

put these three files in the same folder.

Group2:

Nodes file for plane truss analysis
Element files for plane truss analysis

put these three files in the same folder.

Group3:

Nodes for plane truss analysis including initial strains and temperature loads
Elements for plane truss analysis including initial strains and temperature loads
put these three files in the same folder.