Introduction
A beam is another simple but commonly used structural component. It is also geometrically a straight bar of an arbitrary cross-section, but it deforms only in directions perpendicular to its axis. Note that the main difference between the beam and the truss is the type of load they carry. Beams are subjected to transverse loading, including transverse forces and moments that result in transverse deformation. Finite element equations for beams will be developed in this topic, and the element developed is known as the beam element.
In beam structures, the beams are joined together by welding (not by pins or hinges, as in the case of truss elements), so that both forces and moments can be transmitted between the beams. In this topic, the cross-section of the beam structure is assumed uniform. If a beam has a varying cross-section, it is advised that the beam should be divided into shorter beams, where each can be treated as beam(s) with a uniform cross-section. Nevertheless, the FE matrices for varying cross-sectional area can also be developed with ease using the same concepts that are introduced. The beam element developed in this topic is based on the Euler-Bernoulli beam theory that is applicable for thin beams.
FEM Equations
In planar beam elements there are two degrees of freedom (DOFs) at a node in its local coordinate system. They are deflection in the y direction, v, and rotation in the x-y plane, θζ with respect to the z-axis (see Section 2.5). Therefore, each beam element has a total of four DOFs.
Shape Function Construction
Consider a beam element of length I = 2a with nodes 1 and 2 at each end of the element, as shown in Figure 5.1. The local x-axis is taken in the axial direction of the element with its origin at the middle section of the beam. Similar to all other structures, to develop the FEM equations, shape functions for the interpolation of the variables from the nodal variables
Figure 5.1. Beam element and its local coordinate systems: physical coordinates x, and natural coordinates ξ.
would first have to be developed. As there are four DOFs for a beam element, there should be four shape functions. It is often more convenient if the shape functions are derived from a special set of local coordinates, which is commonly known as the natural coordinate system. This natural coordinate system has its origin at the centre of the element, and the element is defined from —1 to +1, as shown in Figure 5.1.
The relationship between the natural coordinate system and the local coordinate system can be simply given as
To derive the four shape functions in the natural coordinates, the displacement in an element is first assumed in the form of a third order polynomial of ξ that contains four unknown constants:
where a0 to α3 are the four unknown constants. The third order polynomial is chosen because there are four unknowns in the polynomial, which can be related to the four nodal DOFs in the beam element. The above equation can have the following matrix form:
or
where p is the vector of basis functions and α is the vector of coefficients.The rotation θ can be obtained from the differential of Eq. (5.2) with the use of Eq. (5.1):
The four unknown constants α0 to α3 can be determined by utilizing the following four conditions:
The application of the above four conditions gives
or
Solving the above equation for α gives
where
Hence, substituting Eq. (5.10) into Eq. (5.4) will give
where N is a matrix of shape functions given by
in which the shape functions are found to be
It can be easily confirmed that the two translational shape functions Ni and N3 satisfy conditions defined by Eqs. (3.34) and (3.41). However, the two rotational shape functions
N2 and N4 do not satisfy the conditions of Eqs. (3.34) and (3.41). This is because these two shape functions relate to rotational degrees of freedom, which are derived from the deflection functions. Satisfaction of N1 and N3 to Eq. (3.34) has already ensured the correct representation of the rigid body movement of the beam element.
Strain Matrix
Having now obtained the shape functions, the next step would be to obtain the element strain matrix. Substituting Eq. (5.12) into Eq. (2.47), which gives the relationship between the strain and the deflection, we have
where the strain matrix B is given by
In deriving the above equation, Eqs. (2.48) and (5.1) have been used. From Eq. (5.14), we have
where
Element Matrices
Having obtained the strain matrix, we are now ready to obtain the element stiffness and mass matrices. By substituting Eq. (5.16) into Eq. (3.71), the stiffness matrix can be obtained as
where
is the second moment of area (or moment of inertia) of the cross section of the beam with respect to the z axis. Substituting Eq. (5.17) into (5.19), we obtain
Evaluating the integrals in the above equation leads to
To obtain the mass matrix, we substitute Eq. (5.13) into Eq. (3.75):
where A is the area of the cross-section of the beam. Evaluating the integral in the above equation leads to
The other element matrix would be the force vector. The nodal force vector for beam elements can be obtained using Eqs. (3.78), (3.79) and (3.81). Suppose the element is loaded by an external distributed force fy along the x-axis, two concentrated forces fs1 and fs2, and concentrated moments ms1 and ms2, respectively, at nodes 1 and 2; the total nodal force vector becomes
The final FEM equation for beams has the form of Eq. (3.89), but the element matrices are defined by Eqs. (5.21), (5.23) and (5.24).
Remarks
Theoretically, coordinate transformation can also be used to transform the beam element matrices from the local coordinate system into the global coordinate system. However, the transformation is necessary only if there is more than one beam element in the beam structure, and of which there are at least two beam elements of different orientations. A beam structure with at least two beam elements of different orientations is commonly termed a frame or framework. To analyse frames, frame elements, which carry both axial and bending forces, have to be used, and coordinate transformation is generally required. Detailed formulation for frames is discussed in the next topic.
