| name | yield-line-theory |
| description | Yield line theory for reinforced concrete slabs — upper bound plastic analysis, yield line patterns (sagging/hogging), virtual work equation, isotropic and orthotropic reinforcement, moment ratio (α), concentrated loads, two-way slab yield line analysis, Johansen's work method, affinity transformation for skewed slabs, comparison with elastic analysis, and Eurocode 2/ACI 318 applicability. |
| metadata | {"priority":7,"promptSignals":{"phrases":["yield line theory","yield line analysis","slab yield line","Johansen method","plastic slab analysis","RC slab ultimate load"],"minScore":3}} |
Yield Line Theory — Complete Skill
Fundamentals of Yield Line Analysis
Plastic Analysis Concept
Yield line theory: upper-bound plastic analysis for reinforced concrete slabs
At ultimate load → plastic hinges form in slab at locations of maximum moment
Yield lines: lines of plastic rotation (hinge lines); slab fails by rotation along yield lines
Yield line definition:
Sagging yield line: slab curves downward; bottom reinforcement yields; drawn as solid line
Hogging yield line: slab curves upward; top reinforcement yields; drawn as dashed line
At supported edges: hogging yield line forms along support (if moment continuity)
Upper-bound nature:
Yield line method gives collapse load ≥ true collapse load (unsafe side if wrong yield line pattern)
Must check all plausible yield line patterns; lowest collapse load is governing
Physical insight needed: identify correct pattern from support conditions and loading
Yield Line Rules
Rules for drawing yield line patterns:
- Yield lines are straight (linear)
- Yield lines end at boundaries or at intersection with other yield lines
- Yield lines pass through intersections of axes of rotation
- Axes of rotation lie along lines of support; pass through columns/point supports
- Yield lines separate the slab into rigid regions (between yield lines)
- Each region rotates as a rigid body about its axis of rotation
Johansen's Work Method (Virtual Work)
Principle
Virtual work equation:
External work done by loads = Internal work dissipated at yield lines
W_external = W_internal [at collapse; slab given virtual displacement]
External work:
W_ext = Σ (load × displacement of load centroid)
For UDL w over region area A with centroid displacement δ_centroid:
W_ext = w × A × δ_centroid [integrate over all rigid regions]
Internal work:
W_int = Σ (m_n × θ_n × l_n) [yield moment per unit length × rotation × yield line length for each yield line]
m_n = moment component perpendicular to yield line
θ_n = relative rotation of adjacent rigid regions perpendicular to yield line
Simplified for isotropic slab (moment m same in all directions):
W_int = m × Σ (θ_perp × l) [sum over all yield lines]
Component Method for Internal Work
For orthotropic slab (m_x ≠ m_y):
W_int for one yield line at angle φ to x-axis:
W_int = (m_x × θ_x × l_y + m_y × θ_y × l_x)
l_y = projection of yield line on y-direction = l × cos(φ)
l_x = projection on x-direction = l × sin(φ)
θ_x = rotation component about x-axis; θ_y = rotation component about y-axis
This simplifies:
W_int = Σ (m_x × rotation_x_component × l_y_projection) + Σ (m_y × rotation_y_component × l_x_projection)
Simply Supported Square Slab (Isotropic)
Yield Line Pattern
Two-way simply supported square slab (side L × L):
Positive (sagging) yield lines only: form X-pattern from corner to corner
Four triangular regions; each region rotates about one support edge
Work equation:
Set unit displacement δ = 1 at center of slab (maximum)
Centroid of each triangle is at 1/3 of height → centroid displacement = δ/3 = 1/3
Load on each triangle: w × (L²/4) (area of one triangle = L²/4 for square slab split into 4 triangles)
W_ext = 4 × w × (L²/4) × (1/3) = w × L² / 3
Rotation of each region: θ = (1/0) / (L/2) = 2/L (center displacement 1; distance to support L/2)
Yield line length on each side: half the diagonal = L/√2 × 2 sides...
Simplified using component method:
For isotropic: m = m (positive moment capacity)
W_int = 2m × (L × 2/L) + 2m × (L × 2/L) = 2m×2 + 2m×2 = 8m
Setting W_ext = W_int:
w × L² / 3 = 8m → w_u = 24m / L² [ultimate UDL for square isotropic slab on simple supports]
Note: elastic analysis gives m_max = wL²/47 → w_elastic = 47m/L²
Ratio: yield line gives 40% higher than elastic → redistribution allows this at ultimate
Rectangular Slab (Isotropic)
Simply supported rectangular slab a × b (a < b):
Yield line pattern: diagonal from corners (45° lines) + central longitudinal yield line along a/2
Regions:
End triangles (2): base a; height b/2 ... actually:
For rectangle a × b: yield lines from corners at 45° until intersection; longitudinal central line if b > a
Standard result:
w_u = 24m / (3a² − a³/b²) [for strip along a; assuming a < b]
Or for a = b (square): w_u = 24m/L² (as derived above)
Exact for simple rectangle:
w_u = m × (8/(3L_y)) × (3 − (L_y/L_x)²) for isotropic with L_x ≤ L_y (approximate)
Fixed-Edge Slabs (Hogging Yield Lines)
Fixed (built-in) edges: top reinforcement → hogging yield lines at supports
Let m = positive moment capacity; m' = negative (hogging) capacity at support
Square slab fixed all edges:
W_ext = w × L² / 3 (same as before)
W_int = 4 × (m + m') × L [positive yield lines in middle + hogging at edges]
→ w_u = 24(m + m') / L² [if m' = m → w_u = 48m/L²; double simple support value]
Partial fixity: m' = α × m where α = 0 (simply supported) to ∞ (fully fixed)
For α = 1 (equal hogging and sagging capacity): w_u = 48m/L² = 2 × simply supported
Orthotropic Slabs
Moment Ratios
Orthotropic slab: reinforcement not equal in x and y directions
m_x = positive x-direction moment capacity; m_y = α × m_x [α = moment ratio; orthotropic ratio]
m_x' = β × m_x; m_y' = α × β × m_x (hogging capacities maintaining ratio α)
Yield line for orthotropic slab: positions adjusted for α
Internal work (component method): W_int includes both direction contributions
Pattern changes with α: highly orthotropic slabs (α << 1) may have different yield line geometry
Affinity transformation (Johansen):
Transform orthotropic slab to equivalent isotropic by scaling y-coordinate: y' = y × α^(1/4)
Analyze isotropic slab in transformed coordinates; results back-transform to real coordinates
Simplifies analysis to isotropic case after coordinate scaling
Point Loads
Concentrated load P at center of simply supported square slab:
Circular yield line pattern (for isotropic): yield lines radiate from load point
W_ext = P × δ = P × 1
W_int = m × 2π × 1 = 2πm [total rotation = 2π for all radial yield lines; moment = m; "rotation × length" = 1 for unit displacement at center of unit-radius circle]
→ P_u = 2πm [classic result; per unit width moment → use m per unit width]
For any angle φ yield line fan:
W_int = m × φ (for angle φ); P_u = m × 2π for full circle
Combined: fan + rectangular:
Used for point loads on rectangular slabs; combine fan near load with straight yield lines away from load
Comparison with Elastic Analysis
Yield line vs. elastic design:
Yield line: plastic method; uses ultimate moment capacity; no check of service state stress
Elastic design: satisfies compatibility at all load levels; safe if moment redistribution ability exists
Code applicability:
ACI 318 Commentary: yield line theory acceptable as basis for design if adequate ductility provided
Eurocode 2 Annex: plastic analysis permitted for slabs with adequate redistribution (minimum reinforcement, max steel grade, moment redistribution ≤ 30%)
Ductility requirement:
Yield line depends on rotation capacity at yield lines; concrete must not crush before steel yields
Requirement: x/d ≤ 0.45 (EC2) at yield line locations; maximum reinforcement ratio limited
Practical Design Application
Design procedure:
- Assume yield line pattern (from support conditions and load position)
- Apply virtual work; solve for m from given w or w from given m
- Provide reinforcement for moment m in each direction
- Check all alternative patterns; governing = smallest w_u
- Verify ductility (x/d ≤ 0.45; minimum steel ratio)
Design m for square slab:
Given w_u (factored); simply supported: m_required = w_u × L² / 24
As = m_required / (0.87 × f_y × z) [EC2; z = 0.9d lever arm; f_y = yield strength]
Standards and References
| Standard | Scope |
|---|
| ACI 318-19 | Building code for concrete (yield line acceptable) |
| Eurocode 2 (EN 1992-1-1) | Design of concrete structures |
| Johansen "Yield Line Theory" (1962) | Original comprehensive reference |
| Yield Line Design (The Concrete Centre) | Practical design guide |
| Wood & Armer (1968) | Orthotropic slab design moments |
Output
Provide: slab geometry (shape: square/rectangular/irregular; dimensions a × b [m]; thickness h [mm]; boundary conditions: simply supported/fixed/continuous on each edge), reinforcement arrangement (isotropic: m = m in all directions; orthotropic: m_x, m_y = α×m_x; top/bottom distinction), yield line pattern (describe pattern; sketch description: diagonals from corners/fan around point load/longitudinal central line; axes of rotation at supports), virtual work calculation (virtual displacement δ = 1 at governing point; W_ext = w × Σ(A_i × δ_i) [per unit w]; W_int = Σ(m × θ × l) or component method sum; equate → w_u = f(m, L) or m = f(w_u, L)), ultimate load (w_u [kN/m²] given m [kN·m/m]; or m [kN·m/m] given w_factored; factored load combination per EC2/ACI: 1.35G + 1.5Q), comparison with elastic (elastic m_max from yield line table or FEA; ratio m_yl/m_elastic; redistribution [%]; acceptable if ≤ 30% per EC2), ductility check (x/d at yield line locations ≤ 0.45; steel ratio ρ ≤ ρ_max), reinforcement (As_x = m_x/(0.87×f_y×0.9d); As_y similarly; bar spacing for selected diameter), and applicable standard (EC2 Annex or ACI 318 Commentary; Johansen reference for theory).
