Beam-Column Elements in OpenSees
August 22, 2001
Michael H. Scott
PEER, University of California, BerkeleyThis document provides a brief description of the interaction between a beam-column element and the SectionForceDeformation and CoordTransformation classes in OpenSees. Material and geometric nonlinearities are abstracted, or separated, from the element formulation by using the SectionForceDeformation and CoordTransformation classes. As a result, an element can be programmed in the “basic system” to account for material nonlinearities, then use one of many transformation types to pick up geometric nonlinearities. A displacement based, distributed plasticity formulation is presented as an example of how a beam-column element is formulated in the basic system.
Geometric Nonlinearity
In general, the transformation of nodal displacements, u, in the global system to deformations, v, in the basic system is described by a nonlinear function,
\[ \label{eq:v=v(u)} {\bf v} = {\bf v}({\bf u}). \]
In a similar manner, the transformation of basic forces, q, to forces p in the global frame of reference is given by
\[\label{eq:p=p(q,u)} {\bf p} = {\bf p}({\bf q}({\bf u}), {\bf u}), \]
where p is implicitly a function of u via the basic forces, q, as well as an explicit function of u. The explicit dependence on u takes into account approximate geometric nonlinearities such as P-\(\Delta\). These transformations are shown schematically in figure [fig:Transformation].
As seen in figure [fig:BeamClass], a beam-column element acquires geometric nonlinearity from the CoordTransformation class, and material nonlinearity from the SectionForceDeformation class.
Material Nonlinearity
At every cross-section along the element length, a force-deformation relationship holds, providing section stress resultants, s, as a function of section deformations, e,
\[\label{eq:s=s(e)} {\bf s}(x) = {\bf s}({\bf e}(x)).\]
Linearizing the force-deformation relationship with respect to deformations reveals the section tangent stiffness, \({\bf k}_s\),
$$ \[\begin{aligned} \Delta{\bf s} &= \frac{\partial{\bf s}}{\partial{\bf e}} \Delta{\bf e} \ \Delta{\bf s} &= {\bf k}_s \Delta{\bf e}, \end{aligned}\]$$
where \({\bf k}_s = \frac{\partial{\bf s}}{\partial{\bf e}}\), the partial derivative of the section stress resultants with respect to the section deformations.
A beam-column element obtains material nonlinearity through use of the SectionForceDeformation class, as seen in figure fig:BeamClass.
Class Hierarchy
Figure fig:BeamClass shows the class interaction between a beam-column element and the CoordTransformation and SectionForceDeformation classes. An element can use any one of Linear, PDelta, or Corotational transformations; and any one of ElasticSection or FiberSection constitutive models. When a new transformation or section class is added to the framework, the element can use the new class without modification.
Displacement Based Element Formulation
This section describes the formulation of a displacement based, distributed plasticity beam-column element. The governing compatibility and equilibrium equations are covered along with the consistent element stiffness. Bending deformations are assumed to be small, and shear deformations are neglected.
Compatibility
For displacement based elements, there is a strong form of compatibility between basic displacements, v, and section deformations e, satisfied pointwise along the element length,
\[\label{eq:e=av} {\bf e}(x) = \left[ \begin{array}{c} \varepsilon(x) \\ \kappa(x) \end{array} \right] = {\bf a}(x) {\bf v}, \]
where a is the strain-displacement matrix. The section deformations are the axial strain, \(\varepsilon\), and curvature, \(\kappa\). Assuming linear axial displacement and transverse displacement based on cubic Hermitian polynomials, the shape functions in the basic system are
$$ {}(x) = = .$$
The strain-displacement matrix contains the shape function derivatives. Axial strain is the first derivative of the axial displacement, and curvature is the second derivative of the transverse displacement,
$${}(x) = .$$
Using the shape functions defined in equation [eq:N], the strain-displacement matrix is then,
$${}(x) = .$$
The basic displacements, v, can be obtained by invoking the method getBasicTrialDisp(). After computing section deformations from basic displacements via equation [eq:e=av], the method setTrialSectionDeformation() may be invoked with the updated deformations, e.
Equilibrium
Using the principle of virtual displacements (virtual work), equilibrium between element end forces, q, and section stress resultants, s, is satisfied weakly, or in an average sense, along the element length,
\[\label{eq:q} {\bf q} = \int_0^L {\bf a}(x)^T {\bf s}(x) \: dx,\]
where the section stress resultants are the axial force, \(P\), and bending moment, \(M\),
\[{\bf s}(x) = \left[ \begin{array}{c} P(x) \\ M(x) \end{array} \right].\]
To obtain the current value of section stress resultants, s, the method getStressResultant() must be invoked. To perform the transformation from basic to global resisting force (equation [eq:p=p(q,u)]), the method getGlobalResistingForce() should be invoked.
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=p(q,u)] with respect to displacements, u.
$$ \[\begin{aligned} {\bf k} &= \frac{\partial{\bf p}}{\partial{\bf q}}\frac{\partial{\bf q}}{\partial{\bf u}} + \left.\frac{\partial{\bf p}}{\partial{\bf u}}\right|_{\bf q} \ &= \frac{\partial{\bf p}}{\partial{\bf q}} \frac{\partial{\bf q}}{\partial{\bf v}} \frac{\partial{\bf v}}{\partial{\bf u}} + \left.\frac{\partial{\bf p}}{\partial{\bf u}}\right|_{\bf q} \ {\bf k} &= \label{eq:stiff} \frac{\partial{\bf p}}{\partial{\bf q}} {\bf k}_b \frac{\partial{\bf v}}{\partial{\bf u}} + \left.\frac{\partial{\bf p}}{\partial{\bf u}}\right|_{\bf q}\end{aligned}\]$$
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,
$$ \[\begin{aligned} {\bf k}_b &= \frac{\partial{\bf q}}{\partial{\bf v}} \\ &= \int_0^L {\bf a}(x)^T \frac{\partial{\bf s}}{\partial{\bf v}} \: dx \\ &= \int_0^L {\bf a}(x)^T \frac{\partial{\bf s}}{\partial{\bf e}} \frac{\partial{\bf e}}{\partial{\bf v}} \: dx \\ {\bf k}_b &= \label{eq:kb} \int_0^L {\bf a}(x)^T {\bf k}_s(x) {\bf a}(x) \: dx \end{aligned}\]$$
The section tangent stiffness matrix, \({\bf k}_s\), is returned upon invoking the method getSectionTangent(). After computing the basic stiffness, \({\bf k}_b\), the method getGlobalStiffMatrix() should be invoked to perform the transformation in equation [eq:stiff]. The remaining partial derivatives in equation [eq:stiff] are computed by the getGlobalStiffMatrix() method.
Numerical Quadrature
In general, the element integrals, equations [eq:q] and [eq:kb], cannot be evaluated in closed form due to nonlinearities in the section constitutive model. These integrals must be approximately evaluated by numerical quadrature,
$$ \[\begin{aligned} %\label{eq:qapprox} {\bf q} &\approx \sum_{i=1}^{N_s} {\bf a}(x_i)^T {\bf s}(x_i) \: W_i \\ %\label{eq:kbapprox} {\bf k}_b &\approx \sum_{i=1}^{N_s} {\bf a}(x_i)^T {\bf k}_s(x_i) {\bf a}(x_i) \: W_i ,\end{aligned}\]$$
where \(N_s\) is the number of integration points, i.e., the number of section sample points along the element length.
Integration points, \(\xi_i\), and weights, \(\omega_i\), are typically defined over a fixed domain such as \(\left[-1,1\right]\) or \(\left[0,1\right]\), then mapped to the element domain \(\left[0,L\right]\), where \(L\) is the element length. Assuming points and weights defined on \(\left[-1,1\right]\), the following relationships hold,
$$ \[\begin{aligned} x_i &= \frac{L}{2} \left( \xi_i+1 \right) \\ W_i &= \frac{L}{2} \: \omega_i . \end{aligned}\]$$
After mapping the points and weights to the element domain, equations [eq:qapprox] and [eq:kbapprox] can be evaluated.