Remark
If there is an applied distributed moment m(x) per unit of beam length, the external energy (13.8) must be augmented with a fQ m(x)O(x) dx term. This is further elaborated in Exercises 13.4 and 13.5. Such kind of distributed loading is uncommon in practice although in framework analysis occassionally the need arises for treating a concentrated moment C between nodes.
Figure 13.7. The twonode plane beam element with four degrees of freedom.
Figure 13.7. The twonode plane beam element with four degrees of freedom.
§13.5. BEAM FINITE ELEMENTS
Beam finite elements are obtained by subdividing beam members longitudinally. The simplest BernoulliEuler plane beam element, depicted in Figure 13.7, has two end nodes, i and j, and four degrees of freedom:
IdjA
§ 13.5.1. Finite Element Trial Functions
The freedoms (13.9) must be used to define uniquely the variation of the transverse displacement v(e)(x) over the element. The C1 continuity requirement stated at the end of the previous Section says that both w and the slope 9 = v' must be continuous over the entire beam member, and in particular between beam elements.
C1 continuity can be trivially satisfied within each element by choosing polynomial interpolation functions as shown below. Matching the nodal displacements and rotations with adjacent beam elements enforces the necessary interelement continuity.
§13.5.2. Shape Functions
The simplest shape functions that meet the C1 continuity requirements for the nodal freedoms (13.9) are called the Hermitian cubic shape functions. The interpolation formula based on these functions may be written as v(e) = [ N(e)
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