Global Assembly

The fixing force contributions of each element k have to be assembled into a global system column vector of fixing forces.

(1)\[\boldsymbol{r} = \bigcup_{k=1}^{n_{el}} \boldsymbol{r}_k\]

The same also applies for the tangent stiffness contributions k of each element into a global system stiffness matrix.

(2)\[\boldsymbol{K}_{T} = \bigcup_{k=1}^{n_{el}} \boldsymbol{K}_{T,k}\]

Example

The contributions of a truss k=1 with it’s connectivities (A,E) = (1,3) and a truss k=2 with (A,E) = (3,2) to a global system with a total number of 3 nodes is assembled as follows: The system fixing force column vector \(\boldsymbol{r}\) has a total length of (nnodes*ndim)=(9) (assuming ndim=3). The contributions of the trusses k=1 and k=2 are then assembled by their connectivity information A and E.

(3)\[\begin{split}\boldsymbol{r} = \left[\begin{array}{c} \boldsymbol{r}_1 \\ \boldsymbol{r}_2 \\ \boldsymbol{r}_3 \\ \end{array}\right]\end{split}\]

(4)\[\begin{split}\boldsymbol{r} &= \quad \boldsymbol{r}_{k=1} &+ \quad \boldsymbol{r}_{k=2} \\ \boldsymbol{r} &= \left[\begin{array}{c} \boldsymbol{r}_{A=1} \\ \boldsymbol{0} \\ \boldsymbol{r}_{E=3} \\ \end{array}\right] &+ \left[\begin{array}{c} \boldsymbol{0} \\ \boldsymbol{r}_{E=2} \\ \boldsymbol{r}_{A=3} \\ \end{array}\right]\end{split}\]

The total system fixing force column vector of all elements k is finally obtained with equation (1). For the stiffness contribution the same method applies to A end E. For example, the submatrix (AE) of element k is assembled at system position (13).

(5)\[\begin{split}\boldsymbol{K}_{T~(9,9)} = \begin{bmatrix} \boldsymbol{K}_{11} & \boldsymbol{K}_{12} & \boldsymbol{K}_{13}\\ \boldsymbol{K}_{21} & \boldsymbol{K}_{22} & \boldsymbol{K}_{23}\\ \boldsymbol{K}_{31} & \boldsymbol{K}_{32} & \boldsymbol{K}_{33} \end{bmatrix}\end{split}\]

The total system stiffness matrix of all elements k is again obtained with equation (2).

(6)\[\begin{split}\boldsymbol{K}_T &= \quad \quad \quad \boldsymbol{K}_{T,k=1} &+ \quad \quad \quad \boldsymbol{K}_{T,k=2} \\ \boldsymbol{K}_T &= \begin{bmatrix} \boldsymbol{K}_{k=1,AA} & \boldsymbol{0} & \boldsymbol{K}_{k=1,AE}\\ \boldsymbol{0} & \boldsymbol{0} & \boldsymbol{0}\\ \boldsymbol{K}_{k=1,EA} & \boldsymbol{0} & \boldsymbol{K}_{k=1,AA} \end{bmatrix} &+ \begin{bmatrix} \boldsymbol{0} & \boldsymbol{0} & \boldsymbol{0}\\ \boldsymbol{0} & \boldsymbol{K}_{k=2,EE} & \boldsymbol{K}_{k=2,EA}\\ \boldsymbol{0} & \boldsymbol{K}_{k=2,AE} & \boldsymbol{K}_{k=2,AA} \end{bmatrix}\end{split}\]