yield-line-theory

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:

1. Yield lines are straight (linear)
2. Yield lines end at boundaries or at intersection with other yield lines
3. Yield lines pass through intersections of axes of rotation
4. Axes of rotation lie along lines of support; pass through columns/point supports
5. Yield lines separate the slab into rigid regions (between yield lines)
6. 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:

1. Assume yield line pattern (from support conditions and load position)
2. Apply virtual work; solve for m from given w or w from given m
3. Provide reinforcement for moment m in each direction
4. Check all alternative patterns; governing = smallest w_u
5. 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).