CIV5136 Chap.5 Inclined Members and Coordinate Transformation
Inclined Members and Coordinate Transformation
Coordinate transformation lets a member use its natural local axes while the structure enforces compatibility in shared global axes. Signed direction cosines project global movement into local deformation and transform local member actions back to global joint forces.
Orthogonality, energy preservation and simple unit-vector tests expose sign and orientation errors.
The ordered vector from node i to node j defines the local x-axis. From its signed coordinate differences, calculate length, cosine and sine. Reversing the node order changes the local orientation and the local end-coordinate sequence, even though the physical member is unchanged.
Absolute triangle side lengths are therefore insufficient: a member rising leftwards and one rising rightwards need different signed components.
Displacements travel from the global system to local member coordinates. The local stiffness relation then produces member actions, including any local fixed-end effects. Those actions return to the global axes for joint assembly and equilibrium.
The exact placement of the transpose depends on how the transformation matrix has been defined, so state the mapping rather than memorising an isolated formula.
A pure rotation is orthogonal and preserves length and strain energy. Test a transformation with unit global X and Y movements: their axial projections should be the direction cosines.
After force recovery, the global component resultant should reproduce the local axial magnitude. These small checks expose a swapped sine, incorrect sign or accidental second transformation before it contaminates a large frame model.
What this chapter covers
- 01
Global and local coordinate systems
- 02
Ordered node geometry and signed direction cosines
- 03
Displacement transformation
- 04
Local member stiffness response
- 05
Global member stiffness transformation
- 06
Action transformation back to global axes
- 07
Orthogonality and energy checks
- 08
Node reversal and reporting conventions
Worked example · free
Project a diagonal truss displacement
- +1L = 5 m, so c = 0.6 and s = 0.8.
- +2Project movement: delta = 0.6 times 2 plus 0.8 times minus 1.
- +1The extension is 0.4 mm.
- +1N = 25 times 0.4 = 10 kN tension.
- +1Global component magnitudes are 6 and 8 kN, whose resultant is 10 kN.
Key terms
- Local axis
- A member coordinate direction, commonly with local x aligned from its first node to its second node.
- Global axis
- A structural coordinate system shared by all members and joints for assembly and equilibrium.
- Direction cosine
- A signed projection of a member's local axis onto a global coordinate direction.
- Orthogonal transform
- A pure rotation mapping whose inverse equals its transpose and that preserves vector length and energy.
- Axial projection
- The component of relative global joint movement along an element's local longitudinal axis.
Inclined Members and Coordinate Transformation FAQ
Why are local axes useful?
Member constitutive behaviour is simplest along and across the member. Local axes make axial and bending relations transparent, while transformation lets those relations participate in one global structural system. Shared global axes then allow equilibrium and compatibility to be enforced at connected joints. Label both systems on the sketch. The final global components must still close joint equilibrium.
What changes when node order is reversed?
The local x direction reverses, so signed coordinate differences, direction cosines and end-coordinate order change. A consistent reversal leaves the physical response unchanged but changes local component signs.
How is a transformation matrix checked?
Apply it to unit global movements and compare the expected projections, verify the transpose-inverse relation for a pure rotation, and confirm that strain energy is the same in local and global coordinates.
What does a negative axial deformation mean?
It indicates shortening relative to the chosen positive local axis and therefore compression under the usual convention. It is a result, not a reason to replace signed direction cosines with absolute values.
Why is energy preservation a useful check?
Changing coordinates must not change the physical strain energy of the member. Evaluate one displacement state locally and globally; the scalar energy should agree. A mismatch points to an incorrect transpose, direction cosine or coordinate order even when the transformed matrix still has plausible dimensions.
How do I avoid transforming a force twice?
Label every result as local or global when it is created. Recover the member action in local axes, then apply the action transformation once for joint equilibrium. If software output is already global, do not rotate it again. Compare the component resultant with the local magnitude as a final check.
Exam move
Draw the global axes and the ordered local member axis on every inclined element. Compute delta X, delta Y, length, cosine and sine on one line, preserving signs. Test the mapping with a unit global X movement before transforming a full stiffness matrix. After solution, recover local actions first and draw tension or compression arrows, then transform the nodal actions once for global equilibrium.
Label every vector with its coordinate system to prevent accidental double transformation. Practise the same member with reversed node order and confirm that the local signs change while the global physical action does not. Use horizontal and vertical members as limiting cases, because their expected transformations are simple enough to inspect without matrix multiplication.
