CIV5136 Chap.6 Yield Line Patterns and Virtual Work
Yield Line Patterns and Virtual Work
Yield-line analysis divides a slab into rigid regions separated by lines of concentrated plastic rotation. A candidate mechanism must satisfy support compatibility and reinforcement-dependent moment capacity.
External work from applied loads is balanced against internal work along the yield lines, and the lowest result among valid mechanisms governs the upper-bound comparison.
Pattern construction comes before algebra. Yield lines meet boundaries, corners or other lines in a way that allows neighbouring rigid regions to share a compatible displacement along their common edge.
Positive lines represent sagging capacity and negative lines represent hogging capacity under the adopted sign convention. Reinforcement direction decides whether the same capacity can be used in both slab directions or whether directional capacity must be resolved.
Choose one convenient virtual displacement and derive every region rotation from the geometry.
External work sums point loads multiplied by their compatible displacements and integrates distributed load over the displacement surface. Internal work sums plastic moment capacity, relative rotation and yield-line length. Rotation is relative across the line, so opposing region rotations add rather than allowing only one side to enter.
The mechanism method gives an upper bound.
A low numerical answer is useful only when the pattern is kinematically admissible and its work terms use consistent geometry. Compare several plausible patterns using identical load and capacity conventions. The smallest valid result governs among the set examined, while a pattern violating a support condition must be rejected rather than treated as conservative.
What this chapter covers
- 01
Rigid slab regions and rotation axes
- 02
Positive and negative yield lines
- 03
Support and corner compatibility
- 04
Isotropic and directional moment capacity
- 05
Virtual displacement geometry
- 06
External work of point and distributed loads
- 07
Internal work from relative rotations
- 08
Upper-bound comparison of patterns
Worked example · free
Solve a one-way strip mechanism
- +1Give the midpoint a virtual downward displacement delta.
- +1Each 2 m half rotates by delta divided by 2.
- +2Relative rotation at midspan is delta, so internal work is 20 delta.
- +1The triangular displacement area is 2 delta, so external work is 2w delta.
- +1Equating work gives w = 10 kN per metre.
Key terms
- Yield line
- A line where plastic rotation concentrates as adjacent slab regions move as rigid planes.
- Rigid region
- A slab portion assumed not to deform internally during the chosen yield-line mechanism.
- Virtual displacement
- A small compatible movement assigned to the mechanism for calculating external and internal work.
- Relative rotation
- The change in slope between the two rigid regions meeting at a yield line.
- Upper-bound mechanism
- A kinematically admissible collapse pattern whose calculated load is not below the ideal true collapse load.
Yield Line Patterns and Virtual Work FAQ
How is a yield-line pattern chosen?
Use slab geometry, supports, loading and reinforcement directions to draw rigid regions that can move compatibly. Examine more than one plausible pattern because the first admissible sketch may not give the lowest upper bound. Every shared line must carry one compatible displacement from both adjoining regions. Mark each region rotation explicitly. Use the same displacement reference for every region.
Why do rotations on both sides matter?
Internal work uses the relative change in slope across a yield line. When adjacent regions rotate in opposite senses, their rotation magnitudes add rather than using only one side.
What is the role of negative yield lines?
They represent hogging plastic rotation where the relevant reinforcement and support condition provide negative moment capacity. Omitting them can understate internal work and misrepresent the mechanism.
Why does the lowest mechanism load govern?
Each valid kinematic mechanism gives an upper-bound estimate. Among the checked candidates, the smallest value is the first predicted mechanism and the closest upper-bound collapse estimate.
How is distributed-load work calculated?
Integrate load intensity multiplied by the compatible virtual displacement over the loaded slab area. For simple rigid regions this is equivalent to load intensity times the volume beneath the displacement surface. Mark the displacement peak, zero axes and region planes so the centroid or integration used can be checked.
What makes a yield-line pattern impossible?
Adjacent rigid regions must have matching displacement along their common yield line and satisfy support conditions. A line that terminates arbitrarily inside a rigid region, a supported edge that moves incompatibly or inconsistent rotations at an intersection invalidates the pattern before any work equation is considered.
Exam move
Always start with the slab plan, supports, reinforcement directions and a clearly labelled candidate pattern. Mark zero-displacement axes and assign one virtual displacement. Calculate every rigid-region rotation geometrically, then write external work and internal work in separate columns before equating them. Check line lengths and positive or negative capacities directly against the sketch.
Compare several admissible patterns using the same load definition and report why the minimum governs. Add units beside one internal-work term and one external-work term before simplifying; both must reduce to force times length. For distributed loading, mark the displacement volume or centroid used so that the external work can be reconstructed from the diagram.
