Plane and Spatial Frame Structures
Within this chapter the procedure for the analysis of a load-bearing structure will be introduced. Structures will be considered, which consist of multiple elements and are connected with each other on coupling points. The structure is supported properly
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Plane and Spatial Frame Structures
Abstract Within this chapter the procedure for the analysis of a load-bearing structure will be introduced. Structures will be considered, which consist of multiple elements and are connected with each other on coupling points. The structure is supported properly and subjected with loads. Unknown are the deformations of the structure and the reaction forces on the supports. Furthermore, the internal reactions of the single element are of interest. The stiffness relation of the single elements are already known from the previous chapters. A total stiffness relation forms on the basis of these single stiffness relations. From a mathematical point of view the evaluation of the total stiffness relation equals the solving of a linear system of equations. As examples plane and general three-dimensional structures of bars and beams will be introduced.
7.1 Assembly of the Total Stiffness Relation It is the goal of this section to formulate the stiffness relation for an entire structure. It is assumed that the stiffness relations for each element are known and can be set up. Each element is connected with the neighboring elements through nodes. One obtains the total stiffness relation by setting up the equilibrium of forces on each node. The structure of the total stiffness relation is therefore predetermined: F = Ku.
(7.1)
The dimension of the column matrices F and u equals the sum of the degrees of freedom on all nodes. The assembly of the total matrix K can be illustrated graphically by sorting all submatrices ke in the total stiffness matrix. Formally this can be noted as follows: ke . (7.2) K= e
A. Öchsner and M. Merkel, One-Dimensional Finite Elements, DOI: 10.1007/978-3-642-31797-2_7, © Springer-Verlag Berlin Heidelberg 2013
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7 Plane and Spatial Frame Structures
The setup of the total stiffness relation occurs in multiple steps: 1. The single stiffness matrix ke is known for each element. 2. It is known which nodes are attached to each element. The single stiffness relation can therefore be formulated for each element in local coordinates: F e = k e up . 3. The single stiffness relation, formulated in local coordinates has to be formulated in global coordinates. 4. The dimension of the total stiffness matrix is defined via the sum of the degrees of freedom on all nodes. 5. A numeration of the nodes and the degrees of freedom on each node has to be defined. 6. The entries from the single stiffness matrix have to be sorted in the corresponding positions in the total stiffness matrix. This can be shown with the help of a simple example. Given is a bar-similar structure with length 2L and constant cross-section A. The structure is divided into two parts with length L with differing material (this means different moduli of elasticity). The structure has a fixed support on one side and is loaded with a point load F on the other side (Fig. 7.1).
Fig. 7.1 Bar-shaped structure with the length 2L
For further analysis the structure will be divided into two parts each of le
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