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Let us use lower numbers to denote unknown degrees of freedom (unconstrained degrees of freedom) and higher numbers to denote known (constrained) degrees of freedom. The possible displacement degrees of freedom of the beam are also shown in the figure. Hence, there are four nodes and eight degrees of freedom. The beam ABCD is divided into three beam members.
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27.1b, numbers enclosed in a circle represents beam numbers. If the axial deformations are neglected then each node of the beam will have two degrees of freedom: a vertical displacement (corresponding to shear) and a rotation (corresponding to bending moment). This is done whenever the cross sectional area changes suddenly or if it is required to calculate vertical or rotational displacements at an intermediate point. However sometimes it is required to consider a node between support points. Thus each span is treated as an individual beam. It is noticed that, in this case, nodes are located at the supports. The given continuous beam is divided into three beam elements as shown in Fig. For example, consider a continuous beam ABCD as shown in Fig. The stiffness matrix of each individual beam element can be written very easily. Towards this end, we break the given beam into a number of beam elements. In a similar way, one could obtain the global stiffness matrix of a continuous beam from assembling member stiffness matrix of individual beam elements. In the case of truss, the stiffness matrix of the entire truss was obtained by assembling the member stiffness matrices of individual members.
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However this can be much simplified provided we follow the procedure adopted for trusses. This procedure runs into trouble when the structure is large and complex. In fact the load displacement relation for the entire structure was derived from fundamentals. The procedure adopted therein is not suitable for computer implementation. In previous chapter, a few problems were solved using stiffness method from fundamentals. Write the global load-displacement relation for the beam Write down global load vector for the beam problem.Ĥ. Assemble member stiffness matrices to obtain the global stiffness matrix for a beam.ģ. Derive member stiffness matrix of a beam element.Ģ. After reading this chapter the student will be able toġ.