This chapter extends the formulation and solution methods of beam theory to two-dimensional (planar) framed structures. As was the case with the truss, frame structures can have multiple elements oriented in different directions sharing load at the places where they connect. Like the truss, we will establish balance of linear momentum at the nodes. Unlike the truss, angular momentum must also balance at the nodes to recognize that transmission of bending moment at the intersection points of elements is a fundamental feature of continuous frames. This final chapter of the book gets us to a practical endpoint in our study of structural dynamics, providing a context to tie together all of the main themes of the book and yielding a computational framework in which many practical problems can be explored.
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The term continuous frame means that all elements share the same rotation at the joint where they meet. It is, of course, possible to have a structure where truss elements frame into a joint shared by frame elements. We will not pursue that more general modeling option in this book. Hence, the term frame will be used hereafter to imply a continuous frame.
Note that the letter m gets a pretty hard workout in this chapter. We use it for the nodal mass, usually with a subscript i, as in m i. We use it for the applied nodal moment, but always with a hat, as in \(\hat m_i\). And we use it for the applied distributed moment with a subscript e, as in m e. The context should usually make the usage clear.
There are also global degrees of freedom associated with the bubble functions in each element, but these degrees of freedom do not play a role in establishing continuity of the structure because they are zero at the element ends.
Any method can be used here, including the generalized trapezoidal rule.
There is a little bit more than notational convenience operating here. This model is what is usually called rate independent , which means that while the constitutive equations are rate equations, the outcome of the model is not sensitive to rate. One manifestation of this is that the time step size h is always associated with the consistency parameter and we solve for the product of the two. Hence the size of h does not play a role in the result.
Note that we are using W i for nodal displacements and W e for element transverse displacement parameters. The context will generally distinguish the usage.
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Hjelmstad, K.D. (2022). Dynamic Analysis of Planar Frames. In: Fundamentals of Structural Dynamics. Springer, Cham. https://doi.org/10.1007/978-3-030-89944-8_14
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-89943-1
Online ISBN: 978-3-030-89944-8