By Jacob Fish, Ted Belytschko

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12) gives " ð1Þ # " # ! F1 kð1Þ Àkð1Þ u3 ¼ : ð2:14Þ ð1Þ u2 Àkð1Þ kð1Þ F2 Notice that we have replaced the nodal displacements by the global nodal displacements. This enforces compatibility as it ensures that the displacements of elements at common nodes are identical. 12) gives " ð2Þ F1 ð2Þ F2 # " ¼ kð2Þ Àkð2Þ Àkð2Þ kð2Þ # ! 13) because the matrices are not of the same size. 15) by adding zeros; we similarly augment the displacement matrices. 15) are rearranged into larger augmented element stiffness matrices and zeros are added where these elements have no effect.

A linear law relating the flux to the potential; 3. e. a compatible potential). Two examples are described in the following: steady-state electrical flow in a circuit and fluid flow in a hydraulic piping system. In an electrical system, the potential is the voltage and the flux is the current. 10. By Ohm’s law, the current from node 1 to node 2 is given by ie2 ¼ ee2 À ee1 ; Re ð2:34Þ where ee2 and ee1 are the voltages (potentials) at the nodes and Re is the resistance of the wire. This is the linear flux–potential law.

Structures, it is necessary to develop an element stiffness matrix for a bar element aligned arbitrarily in two- or three-dimensional space. 2(b). Trusses differ from networks such as electrical systems in that the nodal displacements in multidimensional problems are vectors. The unknowns of the system are then the components of the vector, so the number of unknowns per node is 2 and 3 in two and three dimensions, respectively. We begin by developing the element stiffness matrix for a bar element in two dimensions.