Coordinate Transformation and Global Element Stiffness

Each structural element is conveniently described in a local coordinate system, while the equations of the complete structure must use one common global system. A coordinate-transformation matrix connects the two descriptions.

Local and global displacements

Let $\{d_e\}$ be the element nodal-displacement vector in global coordinates and $\{u_e\}$ the corresponding local vector. With a consistent definition of $[T]$:

$$\{u_e\}=[T]\{d_e\}.$$

For planar structures, $[T]$ contains the direction cosines $c=\cos\alpha$ and $s=\sin\alpha$ of the member axis.

Element stiffness in global coordinates

Combining the displacement transformation with the consistent force transformation gives

$$[k_e^{(g)}]=[T]^T[k_e][T].$$

The matrix $[k_e^{(g)}]$ represents the same physical element, but its rows and columns correspond to global nodal displacement components.

Why transformation is necessary

Two identical members with the same $E$, $A$, $I$, and $L$ but different orientations have the same local matrix and different global matrices. The transformation accounts for their orientation within the structure.

Force recovery

After solving the global problem, the end displacements of a particular member are extracted from the global vector and transformed to local coordinates. Axial force, shear, and end moments can then be evaluated in the member’s natural coordinate system.

Checks

A coordinate transformation must not alter the physical strain energy. For an orthogonal rotational transformation it also preserves stiffness symmetry and the consistency of nodal-force work.