The OpenSees Truss Element

August 22, 2001

Michael H. Scott

PEER, University of California, Berkeley

This document provides a brief description of the interaction between a truss element and the UniaxialMaterial class in OpenSees. Material nonlinearity is abstracted, or separated, from the element formulation by using the UniaxialMaterial class. Figure [fig:TrussClass] shows the class interaction between a truss element and the UniaxialMaterial class. A truss element can use any one of ElasticMaterial or HardeningMaterial models. When a new uniaxial material class is added to the framework, the truss can use the new class without modification.

 

The formulation of a linear geometry, material nonlinear truss element is covered in the remainder of this document. First, the linear transformation is described, followed by the truss element formulation.

Geometric Transformation

A linear transformation of displacements and forces between the global and basic frames of reference is assumed. The transformation of global displacements,u, to a single axial displacement, \(v_1\), is given by the linear relation,

\[\label{eq:v=Tu} {\bf v} = \left[ \begin{array}{c} v_1 \end{array} \right] = {\bf T}{\bf u}.\]

Based on force equilibrium, the transformation of axial force in the basic system to global forces is also linear,

\[\label{eq:p=Tq} {\bf p} = {\bf T}^T{\bf q} = {\bf T}^T \left[ \begin{array}{c} q_1 \end{array} \right].\]

The transformation between global and basic systems is shown schematically in figure [fig:TrussTransf]. The transformation matrix, T, is given in terms of the element orientation, \(\theta\), in the global system,

\[{\bf T} = \left[ \begin{array}{cccc} -\cos\theta & -\sin\theta & \cos\theta & \sin\theta \end{array} \right].\]

 

Truss Element Formulation

This section describes the formulation of a displacement based truss element. The governing compatibility and equilibrium equations are covered along with the consistent element stiffness. Axial deformations are assumed to be small.

Compatibility

For the truss element, there is a strong form of compatibility between basic displacements, v, and section deformations e, satisfied pointwise along the element length,

\[\label{eq:e=av} {\bf e}(x) = \left[ \begin{array}{c} \varepsilon(x) \end{array} \right] = {\bf a}(x) \: v_1,\]

where a is the strain-displacement matrix and \(v_1\) is computed from equation [eq:v=Tu]. There is one section deformation, the axial strain, \(\varepsilon\). Assuming linear axial displacement the shape function in the basic system is

\[\label{eq:N} {\bf N}(x) = \left[ \begin{array}{c} N_1(x) \end{array} \right] = \left[ \begin{array}{c} \frac{x}{L} \end{array} \right].\]

The strain-displacement matrix contains the shape function derivative. Axial strain is the first derivative of the axial displacement,

\[{\bf a}(x) = \left[ \begin{array}{c} N_{1,x} \end{array} \right].\]

Using the shape function defined in equation [eq:N], the strain-displacement matrix is then,

\[{\bf a}(x) = \left[ \begin{array}{c} \frac{1}{L} \end{array} \right].\]

Thus, the axial strain, \(\varepsilon\), is constant along the element length and equation [eq:e=av] reduces to

\[\varepsilon(x) = \frac{v_1}{L}.\]

After computing the axial strain, the method setTrialStrain() should be invoked with the updated strain.

Equilibrium

Using the principle of virtual displacements (virtual work), equilibrium between element end force, q, and section stress resultant, s, is satisfied weakly, or in an average sense, along the element length,

\[\label{eq:q} {\bf q} = \left[ \begin{array}{c} q_1 \end{array} \right] = \int_0^L {\bf a}(x)^T {\bf s}(x) \: dx,\]

where the section stress resultant is the axial force, \(P\). For the truss element, the stress resultant is computed by integrating constant material stress, \(\sigma\), over the cross-section area, \(A\),

\[{\bf s}(x) = \left[ \begin{array}{c} P(x) \end{array} \right] = \left[ \begin{array}{c} \sigma A \end{array} \right],\]

where \(\sigma\) is constant since the axial strain does not vary along the element length. The integral in equation [eq:q] reduces to

\[q_1 = \sigma A,\]

since \({\bf a}(x) = \frac{1}{L}\). To obtain the current value of material stress, \(\sigma\), the method getStress() must be invoked. Then the truss force can be transformed to the global system via equation [eq:p=Tq], i.e.,

\[{\bf p} = {\bf T}^T q_1.\]

Element Stiffness

To solve the structural system of equations, the element stiffness must be assembled along with the resisting force. The element stiffness is obtained by taking the partial derivative of equation [eq:p=Tq] with respect to displacements, u.

$$ \[\begin{aligned} {\bf k} &= \frac{\partial{\bf p}}{\partial{\bf u}}\ &= \frac{\partial{\bf p}}{\partial{\bf q}} \frac{\partial{\bf q}}{\partial{\bf v}} \frac{\partial{\bf v}}{\partial{\bf u}} \ {\bf k} &= \label{eq:k} {\bf T}^T {\bf k}_b {\bf T}\end{aligned}\]

$$

The basic element stiffness, \({\bf k}_b\), is the partial derivative of the basic forces, q, with respect to the basic displacements, \({\bf v}\). Differentiating equation [eq:q] gives,

$$ \[\begin{aligned} {\bf k}_b &= \frac{\partial{\bf q}}{\partial{\bf v}} \ &= \int_0^L {\bf a}(x)^T \frac{\partial{\bf s}}{\partial{\bf v}} \: dx \ &= \int_0^L {\bf a}(x)^T \frac{\partial{\bf s}}{\partial{\bf e}} \frac{\partial{\bf e}}{\partial{\bf v}} \: dx \ {\bf k}_b &= \int_0^L {\bf a}(x)^T {\bf k}_s(x) {\bf a}(x) \: dx \\\end{aligned}\]

$$

Recalling that \({\bf a}(x) = \frac{1}{L}\), integration along the element length gives,

\[{\bf k}_b = \label{eq:kb} \frac{{\bf k}_s}{L},\]

for a prismatic element where \({\bf k}_s\) is constant.

The section tangent stiffness matrix can be manipulated further and put in terms of the material tangent. Recalling that \({\bf s} = \sigma A\) and \({\bf e} = \varepsilon\),

$$ \[\begin{aligned} {\bf k}_s &= \frac{\partial{\bf s}}{\partial{\bf e}} \ &= \frac{\partial{\bf s}}{\partial{\varepsilon}}\frac{\partial{\varepsilon}}{\partial{\bf e}} \ &= \frac{\partial{\sigma}}{\partial{\varepsilon}} A \ {\bf k}_s &= \label{eq:ks} D_t A,\end{aligned}\]

$$

where \(D_t\) is the material tangent, which is returned upon invoking the method getTangent(). Combining equations [eq:k], [eq:kb], and [eq:ks], the familiar truss stiffness equation is recovered,

\[{\bf k} = {\bf T}^T \frac{D_t A}{L} {\bf T}.\]