Fem global source vector 2d4/30/2023 Recall that the truss element has only one degree of freedom at each node (axial deformation), and the beam element has two degrees of freedom at each node (transverse deformation and rotation). Therefore, the element matrices for a frame element can be simply formulated by combining element matrices for truss and beam elements, without going through the detailed process of formulating shape functions and using the constitutive equations for a frame.įigure 6.2. As mentioned, a frame element contains both the properties of the truss element and the beam element. Equations in Local Coordinate SystemĬonsidering the frame element shown in Figure 6.2 with nodes labelled 1 and 2 at each end of the element, it can be seen that the local x-axis is taken as the axial direction of the element with its origin at the middle of the element. Therefore, each element with two nodes will have a total of six DOFs. They are the axial deformation in the x direction, u deflection in the y direction, v and the rotation in the x-y plane and with respect to the z-axis, θζ. In a planar frame element, there are three degrees of freedom (DOFs) at one node in its local coordinate system, as shown in Figure 6.2. The elements and nodes are numbered separately in a convenient manner. Each element is of length le = 2a, and has two nodes at its two ends. FEM Equations For Planar FramesĬonsider a frame structure whereby the structure is divided into frame elements connected by nodes. Of course, if the variation in cross-section is too severe for accurate approximation, then the equations for a varying cross-sectional area can also be formulated without much difficulty using the same concepts and procedure given in this topic.įigure 6.1. If a structure of varying cross-section is to be modelled using the formulation in this topic, then it is advised that the structure is to be divided into smaller elements of different constant cross-sectional area so as to simulate the varying cross-section. In this topic, it is assumed that the frame elements have a uniform crosssectional area. Frame members in a frame structure are joined together by welding so that both forces and moments can be transmitted between members. A typical three-dimensional frame structure is shown in Figure 6.1. Hence, it can be considered to be the most general form of element with a one-dimensional geometry.įrame elements are applicable for the analysis of skeletal type systems of both planar frames (two-dimensional frames) and space frames (three-dimensional frames). A three-dimensional spatial frame structure can practically take forces and moments of all directions. Commercial software packages usually offer both pure beam and frame elements, but frame structures are more often used in actual engineering applications. The frame element developed is also known in many commercial software packages as the general beam element, or even simply the beam element. The development of FEM equations for beam elements facilitates the development of FEM equations for frame structures in this topic. In fact, the frame structure can be found in most of our real world structural problems, for there are not many structures that deform and carry loadings purely in axial directions nor purely in transverse directions. Therefore, a frame element is seen to possess the properties of both truss and beam elements. The bar is capable of carrying both axial and transverse forces, as well as moments. A frame element is formulated to model a straight bar of an arbitrary cross-section, which can deform not only in the axial direction but also in the directions perpendicular to the axis of the bar.
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