What is yield line theory?

Yield line theory is a plastic analysis method for RC slabs, systematized by the Danish engineer K. W. Johansen in the 1940s. Just before failure, plastic hinges are assumed to form along straight lines called yield lines, and the ultimate load is obtained from the principle of virtual work. It predicts a larger capacity than elastic analysis and is used for ultimate limit state design.

How does elastic design differ from yield line (plastic) design?

Elastic design defines capacity at the first point that yields (the maximum stress point), while plastic design considers the reserve until the structure as a whole forms a collapse mechanism. Because RC slabs behave nearly as a ductile material, yield line analysis typically predicts 20 to 40 percent more capacity than elastic design, enabling more rational designs.

How do upper bound and lower bound methods differ?

The lower bound method (lower bound theorem) satisfies only the stress field and gives a result below the true ultimate load (safe side). The upper bound method (upper bound theorem) satisfies only the displacement field (collapse mechanism) and gives a result above the true ultimate load (unsafe side). Yield line theory is an upper bound method, so accuracy improves as the assumed yield line pattern approaches the true collapse mechanism. In design, the minimum among several patterns is adopted.

How is it related to plastic design under seismic loads?

In modern seismic design, members are allowed to undergo plastic deformation under large earthquakes to absorb energy. By evaluating the plastic capacity of floor slabs with yield line analysis, the collapse mechanism and energy absorption of the whole frame can be rationally assessed. For flat slab construction in particular it is, alongside punching shear, an essential check.

Yield Line Theory Simulator — Free Online Calculator

Parameters

Short side L_x

Long side L_y

Yield moment m_p

kN·m/m

Safety factor γ

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If L_y < L_x, the sides are swapped automatically to keep β = L_y / L_x ≥ 1. Isotropic yield moment (same m_p in both orthogonal directions) is assumed.

Results

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Ultimate distributed load q_u

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Design distributed load q_d = q_u/γ

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Equivalent point load P_u=q_u·A

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Aspect ratio β=L_y/L_x

Yield Line Pattern (Center-Diverging Orthogonal Lines)

Red lines = yield lines (plastic hinges) / divides into triangles ①② and trapezoids ③④ / blue triangles = four-edge simple supports / arrows = dimensions

Theory & Key Formulas

For a four-edge simply supported rectangular slab under uniformly distributed load q, assuming the typical pattern with yield lines diverging from the centre to the four edge midpoints (X plus I shape), the upper bound method (Johansen) gives the ultimate load.

With aspect ratio β = L_y / L_x (≥1) and isotropic yield moment m_p (per unit width), the ultimate distributed load q_u is expressed with the coefficient α(β):

$$q_u = \alpha(\beta)\,\frac{m_p}{L_x^2}$$

Textbook values (linearly interpolated): β=1 → α=24 (square), β=1.5 → α=18, β=2 → α=16, β=3 → α=12, β=∞ → α=8 (one-way slab).

Design distributed load q_d and equivalent point load P_u (A=L_x·L_y):

$$q_d = \frac{q_u}{\gamma}, \qquad P_u = q_u \cdot A$$

For a square slab (β=1), q_u = 24 m_p / L_x². The larger β becomes, the more bending in the short direction dominates, and α decreases toward the one-way slab value of 8.

What is the Yield Line Theory Simulator

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RC slab design is usually done with elastic analysis, right? What is different about "yield line theory"?

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Roughly speaking, yield line theory thinks about "the moment the slab actually fails". Elastic analysis defines the limit at the first point to yield, but a real RC slab does not collapse immediately when one point yields. Plastic hinges spread along straight "yield lines" and the true limit comes only when a collapse mechanism is formed. Johansen of Denmark systematized this in the 1940s. In the simulator above, change "short side L_x" and you can see the ultimate load q_u scales with 1/L_x².

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When I move the "aspect ratio" slider, the length of the central line in the yield line pattern changes.

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Yes, that is the interesting part of this theory. A square (β=1) has an "X pattern" of yield lines from one central point to the four corners. A rectangle adds an "I" of central yield line, producing triangles at the short ends and trapezoids at the long sides. In the simulator, ①② are triangles and ③④ are trapezoids. As β grows, bending in the short direction dominates, and the coefficient α(β) decreases from 24 toward 8 (the one-way slab value).

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Is the design distributed load q_d just q_u divided by the safety factor?

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Yes. q_u is the theoretical load at which the slab collapses — the "cross this and you're out" line. Real design divides it by a safety factor γ to account for concrete strength scatter, construction tolerance and long-term effects, giving q_d = q_u/γ. In Japan γ ≈ 1.5 is standard, and Eurocode is similar. Set γ to 1.0 in the simulator and you see q_d = q_u — no safety margin at all.

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What is the "equivalent point load P_u"? The area load gathered into one point?

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Yes — simply q_u times floor area A = L_x·L_y, the total load. It is a handy figure for estimating loads transferred to columns and foundations, often used in preliminary calculations. With the defaults (L_x=4, L_y=6, m_p=50, γ=1.5), β=1.5, α=18, q_u=18·50/16=56.25 kN/m², P_u=56.25·24=1350 kN. That's like supporting 1000 cars on a single floor.

Frequently Asked Questions

Elastic analysis is appropriate for serviceability limit states (deflection, cracking), and yield line analysis is appropriate for the ultimate limit state (collapse capacity). They look at different limit states, so modern design codes (Eurocode 2, ACI 318, etc.) generally use both. Relying on yield line analysis alone may overlook excessive deformation or cracking, so evaluating with both is essential.

Yield line theory is an upper bound (upper limit theorem) method. When the assumed collapse mechanism matches the true mechanism, the true ultimate load is obtained; otherwise a result larger than the true value (unsafe side) is obtained. The representative lower bound method is the strip method (Hillerborg). In design practice, adopting the lower bound is the principle, but yield line theory is simple and often accurate enough to be widely used. Taking the minimum among several patterns brings the result back toward the safe side.

A slab is a thin two-dimensional structure across its thickness, so it is conventional to use bending resistance per unit width [kN·m/m] rather than a beam-like sectional moment [kN·m]. It is computed from the slab section (thickness h, effective depth d) and the tensile reinforcement A_s as m_p ≈ A_s·f_y·(d − a/2). The simulator takes m_p as direct input, but in practice the reinforcement is decided first and m_p is back-calculated from it.

Modern seismic design is based on allowing members to deform plastically under large earthquakes to absorb energy ("ultimate horizontal strength design", "ductility design"). Evaluating the plastic capacity of floor slabs with yield line analysis allows the collapse mechanism and energy absorption of the whole frame to be assessed rationally. Especially in flat slab construction (no beams, directly supported by columns), yield line analysis is indispensable, alongside punching shear capacity.

Real-World Applications

Floor slabs of RC buildings: Yield line theory is used in the floor slab design of almost all reinforced concrete buildings — office buildings, apartments, schools and more. For long-span four-edge supported slabs and flat-slab construction in particular, elastic analysis alone makes rational reinforcement design difficult, and plastic capacity evaluation by yield line analysis is standard.

Bridge deck design: Deck slabs of highway and railway bridges are evaluated for their plastic capacity under concentrated wheel loads of vehicles and trains. Real bridge design includes many corrections — different yield line patterns for various wheel-load positions, composite girder effects, prestress influence and so on. Japan's road bridge specifications also incorporate yield line theory into ultimate limit state checks.

Seismic assessment and retrofitting: Seismic assessment of existing RC buildings evaluates the plastic capacity of floor slabs and walls, and computes the collapse mechanism and ultimate horizontal strength under earthquakes. For buildings designed before the 1981 (new seismic code) in Japan, yield line analysis sometimes reveals that the actual capacity is larger than the nominal value, helping to avoid unnecessary retrofitting.

Nuclear facilities and protective engineering: In special structures such as nuclear reactor buildings, nuclear fuel storage facilities and missile-protection shelters, evaluating the plastic capacity of slabs under impact and blast loads is important. Yield line analysis is effective for predicting the collapse mechanism under instantaneous large loads and is applied to evaluating energy absorption and peak displacement.

Common Misconceptions and Cautions

The most common misconception is to think that "the result of yield line theory is the true ultimate load". Yield line theory is an upper bound method, and the true value is obtained only when the assumed pattern matches the true collapse mechanism. In reality, several patterns (X, I, diagonal lines, etc.) must be tried and the smallest q_u adopted. The simulator implements only the simplified pattern (centre-diverging orthogonal lines), but in practice more complex patterns must be examined (local yield lines around columns, eccentric patterns around openings, etc.). The textbook α values are values for "the most basic pattern".

The next most common error is to think that "using yield line theory lets us reduce reinforcement". It is true that the plastic capacity looks 20 to 40 percent larger than the elastic capacity, so designing at ultimate capacity seems to allow less reinforcement. But reducing reinforcement tends to cause problems at the serviceability limit state (deflection, cracking), and in many cases serviceability ends up governing the reinforcement quantity anyway. Yield line analysis is used to "confirm ultimate capacity", and the standard approach is to size reinforcement by elastic analysis plus serviceability checks. Doing both in parallel is essential.

Finally, note that this simulator gives results under the idealized conditions of "isotropic yield moment, four-edge simply supported, uniformly distributed load". Real slabs have many situations that need correction factors: anisotropic slabs with different reinforcement in X and Y, fixed or continuous supports, complex floor plans with openings or steps, non-uniform loads such as vehicle or crowd loads. The simulator values are reference values under the most basic conditions, and design must always apply the modification factors and safety factors of the relevant national code.

Yield Line Theory Simulator — Ultimate Load of a Rectangular Slab

What is the Yield Line Theory Simulator

Frequently Asked Questions

Real-World Applications

Common Misconceptions and Cautions

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