Truss Element

This section describes the Kinematics and Constitution of a Truss element trusspy.elements.element_definition.truss. A Truss k is connected to a system by it’s begin (A) and end (E) nodes. The cross section remains constant over the element length. It has three coordinates and three degrees of freedom.

Arbitrary deformed (stretched and rotated) truss k and it’s contribution to the equilibrium on nodes A and E.

Kinematics

For a truss element the stretch may be calculated as

(1)\[\Lambda = \frac{l}{L} = \sqrt{1 + 2 \left(\frac{\boldsymbol{\Delta X}}{L}\right)^T \left(\frac{\boldsymbol{\Delta U}}{L}\right) + \left(\frac{\boldsymbol{\Delta U}}{L}\right)^T \left(\frac{\boldsymbol{\Delta U}}{L}\right)}\]

which follows from

(2)\[\begin{split}l^2 &= \boldsymbol{\Delta x}^T \boldsymbol{\Delta x} \\ L^2 &= \boldsymbol{\Delta X}^T \boldsymbol{\Delta X}\end{split}\]

and enables the Biot strain measure:

(3)\[E_{11} = \Lambda - 1\]

Constitution

The normal force of a truss depends directly on the geometric exactly defined strain measure \(E_{11}\). For the general case of a nonlinear material behaviour the normal force is defined as

(4)\[N = S_{11}(E_{11})~A + N_0\]

and the derivative

(5)\[\frac{\partial N}{\partial E_{11}} = \frac{\partial S_{11}(E_{11})}{\partial E_{11}}~A\]

For the case of a linear elastic material this reduces to

(6)\[\begin{split}S_{11}(E_{11}) &= E~E_{11} \\ N &= EA~E_{11} + N_0 \\ \frac{\partial N}{\partial E_{11}} &= EA\end{split}\]

Kinetics

The (nonlinear) fixing force column vector with dimension (ndim) may be expressed as a function of the elemental force \(N_k\) and the deformed unit vector \(\boldsymbol{n}_k\).

(7)\[\begin{split}\boldsymbol{r}_k = \begin{bmatrix} \boldsymbol{r}_A \\ \boldsymbol{r}_E \end{bmatrix} = N_k \begin{pmatrix} -\boldsymbol{n}_k \\ \phantom{-}\boldsymbol{n}_k \end{pmatrix}\end{split}\]

Stiffness Matrix

The elemental stiffness matrix of a truss has dimensions (2*ndim,2*ndim) and contains partial derivatives of the element fixing forces w.r.t to the displacements. The matrix components for the case of ndim=3 results in

(8)\[\begin{split}\boldsymbol{K}_{k~(6,6)} = \left[\begin{array}{ccc:ccc} \frac{\partial r_{A,x}}{\partial U_{A,x}} & \frac{\partial r_{A,y}}{\partial U_{A,x}} & \frac{\partial r_{A,z}}{\partial U_{A,x}} & \frac{\partial r_{E,x}}{\partial U_{A,x}} & \frac{\partial r_{E,y}}{\partial U_{A,x}} & \frac{\partial r_{E,z}}{\partial U_{A,x}}\\ \frac{\partial r_{A,x}}{\partial U_{A,y}} & \frac{\partial r_{A,y}}{\partial U_{A,y}} & \frac{\partial r_{A,z}}{\partial U_{A,y}} & \frac{\partial r_{E,x}}{\partial U_{A,y}} & \frac{\partial r_{E,y}}{\partial U_{A,y}} & \frac{\partial r_{E,z}}{\partial U_{A,y}}\\ \frac{\partial r_{A,x}}{\partial U_{A,z}} & \frac{\partial r_{A,z}}{\partial U_{A,z}} & \frac{\partial r_{A,z}}{\partial U_{A,z}} & \frac{\partial r_{E,x}}{\partial U_{A,z}} & \frac{\partial r_{E,y}}{\partial U_{A,z}} & \frac{\partial r_{E,z}}{\partial U_{A,z}}\\ \hdashline \frac{\partial r_{A,x}}{\partial U_{E,x}} & \frac{\partial r_{A,y}}{\partial U_{E,x}} & \frac{\partial r_{A,z}}{\partial U_{E,x}} & \frac{\partial r_{E,x}}{\partial U_{E,x}} & \frac{\partial r_{E,y}}{\partial U_{E,x}} & \frac{\partial r_{E,z}}{\partial U_{E,x}}\\ \frac{\partial r_{A,x}}{\partial U_{E,y}} & \frac{\partial r_{A,y}}{\partial U_{E,y}} & \frac{\partial r_{A,z}}{\partial U_{E,y}} & \frac{\partial r_{E,x}}{\partial U_{E,y}} & \frac{\partial r_{E,y}}{\partial U_{E,y}} & \frac{\partial r_{E,z}}{\partial U_{E,y}}\\ \frac{\partial r_{A,x}}{\partial U_{E,z}} & \frac{\partial r_{A,z}}{\partial U_{E,z}} & \frac{\partial r_{A,z}}{\partial U_{E,z}} & \frac{\partial r_{E,x}}{\partial U_{E,z}} & \frac{\partial r_{E,y}}{\partial U_{E,z}} & \frac{\partial r_{E,z}}{\partial U_{E,z}} \end{array}\right]\end{split}\]

For a truss the stiffness matrix may be divided into four block matrices of the same components but with different signs.

(9)\[\begin{split}\boldsymbol{K}_{k~(6,6)} = \begin{bmatrix} \boldsymbol{K}_{AA} & \boldsymbol{K}_{AE}\\ \boldsymbol{K}_{EA} & \boldsymbol{K}_{EE} \end{bmatrix} = \begin{pmatrix} \phantom{-}\boldsymbol{K}_{EE} & -\boldsymbol{K}_{EE}\\ -\boldsymbol{K}_{EE} & \phantom{-}\boldsymbol{K}_{EE} \end{pmatrix}\end{split}\]

Whereas a change in the fixing force vector at the end node E w.r.t. a small change of the displacements at node E is defined as the tangent stiffnes EE.

(10)\[\begin{split}\boldsymbol{K}_{EE} &= \frac{\partial \boldsymbol{r}_E}{\partial \boldsymbol{U}_E} \\ \boldsymbol{K}_{EE} &= \frac{A}{L} ~ \frac{\partial S_{11}(E_{11})}{\partial E_{11}} \boldsymbol{n} \otimes \boldsymbol{n} + \frac{N}{l} \left( \boldsymbol{1} - \boldsymbol{n} \otimes \boldsymbol{n} \right)\end{split}\]

with the identity matrix \(\boldsymbol{1}\)

(11)\[\begin{split}\begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1\\ \end{bmatrix}\end{split}\]

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