# Truss Element and its Stiffness Matrix

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## What is Truss Element

Truss elements are oftenly defined as a two-node members which can allow the arbitrary orientation in the coordinate system XYZ. The truss transmits only axial force. In general it is a three degree-of-freedom (DOF) element, such that three global translation components at each end of the member. Trusses are used to model a structures such as bridges, towers and buildings.

Truss element with three-dimensional (3-D) cross-section, is assumed to have a constant cross-sectional area and can be used in linear elastic analysis of a structure. The linear elastic material behavior of the structure is defined only by the modulus of elasticity (E) which is ratio of the stress to the strain. Linear trusses can also be used to simulate transnational and displacement boundary elements.

## Formulation of Truss Element

By its definition, trusses cannot have a rotational degrees of freedoms (DOFs), and even if you released these degrees of freedoms when you applied the boundary conditions to the structure. You can apply translational DOFs as needed, and below figure 1 shows the formulation of a truss element.

## Representation of Spring Element

Spring element is also in many cases used to represents the flexible nature of supports for more complicated systems in automobile and aerospace industry. In a more general application, yet similar characteristics, element is an elastic bar subjected to axial forces. This element, oftently which we simply call a bar (or) truss element, particularly useful in the analysis of both the 2-dimensional and 3-dimensional frame or truss structures.

The formulation of the finite element features of an elastic bar element is completely based on the following assumptions:

• Bar should be geometrically straight in nature.
• The application of the forces should be only at the ends of the bars.
• Structural material should obeys the condition of Hooke’s law.
• The bar can supports only the axial loading; and the torsion, bending, and shear are not transmitted to the element through the nature of its connections to the other elements.

### Stiffness Matrix for Truss Element

First of all, we can acquire an expression for the stiffness matrix “K” for the bar element. As we recollect from elementary strength of materials that the change in length (deflection “δ“) of an elastic bar of length L and the uniform cross-sectional area A when it is subjected to the axial load P, the relation is as follows:

δ = PL/AE

In the above expression, E defines the materials modulus of elasticity. Then the equivalent spring constant “k” defined as

k = P/δ

k = AE/L

Therefore, the stiffness matrix for single element is:

And the equation of equilibrium in matrix form is:

[ke] { } = { }

Substitute the stiffness matrix [ke] in this equation, and we get resultant equilibrium equation in matrix form.

Hence, the above derived expression is Truss Element and its Stiffness Matrix.

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Hello, I am an aircraft structural analyst with industrial experience and a master degree on aerospace structures. Currently working for an aerospace company as a stress

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