CIV5136 Chap.4 Load Vectors and Member Force Recovery
Load Vectors and Member Force Recovery
Member loading enters the stiffness method through consistent or fixed-end actions connected to the chosen coordinates. After global displacements are solved, member end forces combine deformation actions with member-load effects.
Joint equilibrium, whole-structure equilibrium and the correct shear and moment diagram shapes provide independent checks on the recovered result.
A distributed load cannot be replaced by an arbitrary nodal total. Its equivalent vector must do consistent work on the element interpolation and follow the same local end-coordinate order as the stiffness relation.
Fixed-end actions record what the member carries when its end movements are restrained. They remain nonzero even when the solved displacement vector is zero, which is why a fixed beam under load still develops reactions and bending moments.
Recovery combines the stiffness response to end movement with the retained load effects.
The resulting local actions must be transformed once into global joint components when joint equilibrium is checked. Actions on a member and actions on the adjoining joint are equal and opposite, so arrows are safer than isolated signed numbers. At an unloaded joint, all transformed member actions must close.
Internal-force diagrams provide a second layer of evidence.
Between concentrated actions, distributed loading controls the variation of shear and bending moment. Uniform load makes shear linear and moment quadratic; a concentrated force creates a shear jump, while a concentrated couple creates a moment jump. These relationships test whether the original load was carried into post-processing rather than forgotten after the displacement solution.
What this chapter covers
- 01
Equivalent nodal actions for member loads
- 02
Fixed-end action conventions
- 03
Global load-vector assembly
- 04
Member displacement extraction
- 05
End-action recovery formula
- 06
Local and global action reporting
- 07
Joint and structure equilibrium
- 08
Shear and bending diagram relationships
Worked example · free
Recover a fixed-fixed beam response
- +1Both end displacements and rotations are zero, but member-load actions remain.
- +1Each fixed-end moment magnitude is wL squared divided by 12.
- +1The magnitude is 12 times 36 divided by 12 = 36 kN m.
- +1Symmetry gives each vertical reaction as wL divided by 2 = 36 kN.
- +1The reactions sum to the 72 kN total load.
- +1Uniform loading makes shear linear and bending moment quadratic between ends.
Key terms
- Fixed-end action
- The member end action caused by member loading while the member ends are restrained against their coordinate movements.
- Equivalent nodal load
- A nodal action vector that gives work consistent with the original member load for the adopted interpolation.
- Member recovery
- The calculation of local end actions from solved member movements and member-load effects.
- Joint equilibrium
- The requirement that transformed member end actions, nodal loads and reactions balance at a joint.
- Internal force diagram
- A plot of axial force, shear or bending moment along a member under the selected sign convention.
Load Vectors and Member Force Recovery FAQ
Why are fixed-end actions needed?
Distributed and other member loads create end actions even when the member ends do not move. The deformation term alone represents only the actions caused by solved end displacements. Keeping both terms visible prevents a restrained loaded member from being reported as force free. Check the original load resultant afterward. A joint free-body diagram then confirms the reporting convention.
Should equilibrium use the equivalent load or original load?
The final whole-structure check should balance the actual applied loading. Equivalent nodal actions are an analytical representation used inside the displacement formulation, not a replacement for the physical load.
Why can member end signs look opposite at one joint?
Local member face conventions and global nodal conventions describe action on different bodies. Draw arrows and transform to one reporting system before deciding whether equilibrium is violated.
How can a force diagram reveal an error?
Its slope, curvature and jumps must match the applied loading. Constant shear beneath a uniform load or a straight moment line where shear varies indicates that the load was omitted from post-processing.
What is included in a recovered end action?
Combine the action caused by the solved member end movements with the retained action caused directly by member loading. The exact algebraic sign depends on the declared member convention. Write both terms before adding them, because omitting the fixed-end contribution produces zero force for a loaded member whose ends do not move.
Where does maximum bending moment occur?
Within a continuous loaded segment, an interior bending-moment extremum occurs where shear is zero. Solve the shear expression for that location, check that it lies inside the member, then evaluate the moment there. End values alone can miss the governing ordinate beneath a distributed load.
Exam move
Keep three columns in every solution: load conversion, displacement solution and force recovery. Copy the member coordinate order above both its stiffness matrix and fixed-end vector. After solving, extract member displacements in that same order and add the member-load contribution once. Build force diagrams from equilibrium cuts rather than simply joining end ordinates.
Check each unloaded joint, then sum all reactions and compare them with the original loads and moments. For every diagram, predict its qualitative shape before calculating ordinates. Mark where loads, reactions and couples cause jumps or slope changes. If the recovered end values cannot be connected by that expected shape, revisit the fixed-end signs and the member displacement extraction.
