Frame elements

For frame elements in a plane the ${\mathbf{n}}_1$-direction is always (0.0, 0.0, −1.0); that is, normal to the plane in which the motion occurs. Therefore, planar frame elements can bend only about the first axis direction.

For frame elements in space the approximate direction of ${\mathbf{n}}_1$ must be defined directly as part of the element section definition or by specifying an additional node off the element's axis. This additional node is included in the element's connectivity list (see Element definition).

• If an additional node is specified, the approximate direction of ${\mathbf{n}}_1$ is defined by the vector extending from the first node of the element to the additional node.

• If both input methods are used, the direction calculated by using the additional node will take precedence.

• If the approximate direction is not defined by either of the above methods, the default value is (0.0, 0.0, −1.0).

The ${\mathbf{n}}_1$-direction is then the normal to the element's axis that lies in the plane defined by the element's axis and this approximate ${\mathbf{n}}_1$-direction. The ${\mathbf{n}}_2$-direction is defined as $\mathbf{t}\times {\mathbf{n}}_1$.

The elastic response of a frame element is governed by Euler-Bernoulli beam theory. The displacement interpolations for the deflections transverse to the frame element's axis (the local 1- and 2-directions in three dimensions; the local 2-direction in two dimensions) are fourth-order polynomials, allowing quadratic variation of the curvature along the element's axis. Thus, each single frame element exactly models the static, elastic solution to force and moment loading at its ends and constant distributed loading along its axis (such as gravity loading). The displacement interpolation along an element's axis is a second-order polynomial, allowing linear variation of the axial strain. In three dimensions the twist rotation interpolation along an element's axis is linear, allowing constant twist strain. The elastic stiffness matrix is integrated numerically and used to calculate 15 nodal forces and moments in three dimensions: an axial force, two shear forces, two bending moments, and a twist moment at each end node, and an axial force and two shear forces at the midpoint node. In two dimensions 8 nodal forces and moments exist: an axial force, a shear force, and a moment at each end, and an axial force and a shear force at the midpoint. The forces and moments are illustrated in Figure 1.

Figure 1. Forces and moments on a frame element in space.

The plastic response of the element is treated with a “lumped” plasticity model such that plastic deformations can develop only at the element's ends through plastic rotations (hinges) and plastic axial displacement. The growth of the plastic zone through the element's cross-section from initial yield to a fully yielded plastic hinge is modeled with nonlinear kinematic hardening. It is assumed that the plastic deformation at an end node is influenced by the moments and axial force at that node only. Hence, the yield function at each node, also called the plastic interaction surface, is assumed to be a function of that node's axial force and three moment components only. No length is associated with the plastic hinge. In reality, the plastic hinge will have a finite size determined by the element's length and the specific loading that causes yielding; the hinge size will influence the hardening rate but not the ultimate load. Hence, if the rate of hardening and, thus, the plastic deformation for a given load are important, the lumped plasticity model should be calibrated with the element's length and the loading situation taken into account. For details on the elastic-plastic element formulation, see Frame elements with lumped plasticity.

You can obtain a frame element's response to uniaxial force only, based on linear elasticity, buckling strut response, and tensile yield. In that case all transverse forces and moments in the element are zero. For linear elastic response the element behaves like an axial spring with constant stiffness. For buckling strut response if the tensile axial force in the element does not exceed the yield force, the axial force in the element is constrained to remain inside a buckling envelope. See Frame section behavior for a description of this envelope. Inside the envelope the force is related to strain by a damaged elastic modulus. The cyclic, hysteretic response of this model is phenomenological and approximates the response of thin-walled, pipe-like members. When the element is loaded in tension beyond the yield force, the force response is governed by isotropic hardening plasticity. In reverse loading the response is governed by the buckling envelope translated along the strain axis by an amount equal to the axial plastic strain. For details of the buckling strut formulation, see Buckling strut response for frame elements.

The frame element uses a lumped mass formulation for both dynamic analysis and gravity loading. The mass matrix for the translational degrees of freedom is derived from a quadratic interpolation of the axial and transverse displacement components. The rotary inertia for the element is isotropic and concentrated at the two ends.

For buckling strut response a lumped mass scheme is used, where the element's mass is concentrated at the two ends; no rotary inertia is included.