Compute the moment-curvature (M-φ) relationship of a rectangular beam section under bending. Adjust the section size, yield stress and curvature ratio to see the yield moment, plastic moment, shape factor and the through-depth stress distribution update in real time.
Parameters
Section width b
mm
Section depth h
mm
Height of the section in the bending direction
Young's modulus E
GPa
Yield stress f_y
MPa
Curvature ratio φ/φ_y
Current curvature as a multiple of the yield curvature φ_y
Material model
Assumption for the post-yield stress-strain law
Results
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Yield moment My (kN·m)
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Plastic moment Mp (kN·m)
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Shape factor Mp/My
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Yield curvature φ_y (1/mm)
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Current moment M (kN·m)
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Section state
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Section stress distribution and M-φ curve
Left: bending-stress distribution through the depth (elastic triangle, or elastic core sandwiched between yielded blocks). Right: the M-φ curve with a marker at the current point. The yield zones pulse as they grow.
Moment-curvature curve M(φ/φ_y)
Stress distribution through the section σ(position)
Yield moment My and plastic moment Mp. S: elastic section modulus, Z: plastic section modulus, f_y: yield stress. For a rectangle S=bh²/6 and Z=bh²/4, giving a shape factor Z/S=1.5.
Bending moment M beyond the yield curvature φ_y. As the curvature grows, M approaches Mp. For φ ≤ φ_y, M = (φ/φ_y)·My (the whole section is elastic).
$$\phi_y=\frac{2\,f_y}{E\,h}$$
The yield curvature φ_y is the curvature at which the extreme fibre first reaches the yield strain. E: Young's modulus, h: section depth.
What is the Moment-Curvature Relationship?
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"Moment-curvature relationship" came up in my structures class — but what is that graph actually showing?
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In short, it plots how much you bend a cross-section (the curvature φ) against the resistance the section develops (the bending moment M). Think of it as the section-level version of a stress-strain curve. Pull on a material and you get a σ-ε curve; bend a beam section and you get an M-φ curve. So it captures the "bending personality" of a section on a single graph.
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I see. When I raise the curvature ratio on the left, the curve goes straight up at first and then flattens out. What is happening there?
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Good observation. The straight part is the "whole section elastic" state — Hooke's law holds, and the moment grows in proportion to the curvature. But once the very outermost fibre reaches the yield stress f_y, the curve starts to flatten. That point is the yield moment My. Bend it more and the yielded region spreads inward from the surfaces. A yielded fibre can only deliver a constant stress f_y, so adding curvature buys you less and less moment. That is why the curve bends over.
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So how high does the curve go? It can't rise forever, right?
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Right, there is a ceiling — the plastic moment Mp. It is the moment when every fibre of the section has reached the yield stress, that is, when the section is "fully plastified". With an elastic-perfectly-plastic model, no matter how much you increase the curvature, M only approaches Mp asymptotically and never goes higher. Watch the centre canvas: the yielded blocks grow inward from top and bottom, and the elastic core gets thinner and thinner.
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The ratio of My to Mp is called the "shape factor" — what is that good for?
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The shape factor Mp/My tells you how much reserve a section has between first yield and full-section yield. For a rectangle it is always 1.5 — meaning after the extreme fibre yields, the section can still take a 50% larger moment. The practical point: allowable-stress design draws the line at My, but plastic design uses Mp, so the shape factor lets you slim the design down. In a pushover analysis of a building under earthquake, the M-φ curve of each section becomes the nonlinear spring property of the beams and columns. That makes M-φ the very foundation of plastic-hinge analysis.
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If I pick "With strain hardening", the moment slightly exceeds Mp at large curvature. Is that realistic?
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It is realistic. Real steel keeps gaining stress slightly as the strain grows even after it yields — that is strain hardening. So a heavily bent section can edge past Mp. But in design we normally stay on the safe side and treat Mp as the limit. This tool deliberately keeps the hardening modest (2% of the elastic stiffness) so you get the practical feel that Mp dominates and the hardening is a secondary effect.
Frequently Asked Questions
The yield moment My is the bending moment at which the extreme (outermost) fibre of the section first reaches the yield stress f_y, given by My = f_y·S, where S is the elastic section modulus. The plastic moment Mp is the moment at which the entire section has reached the yield stress (full plastification), given by Mp = f_y·Z, where Z is the plastic section modulus. Between My and Mp, yielding spreads inward from the surfaces and the section is in an elasto-plastic state. Mp is the theoretical upper limit of the bending moment the section can carry.
The shape factor is defined as Mp/My = Z/S, a dimensionless number that measures the reserve strength of a section between first yield and full plastification. For a rectangle the elastic section modulus is S = bh²/6 and the plastic section modulus is Z = bh²/4, so the shape factor = Z/S = 1.5. In other words, a rectangular section can carry a moment 50% larger after the extreme fibre yields. The shape factor is a property of the cross-section geometry: about 1.1-1.2 for an I-section and about 1.7 for a solid circular section.
The M-φ curve is the most fundamental description of how a cross-section responds to bending. It rises linearly while the whole section is elastic, bends over once the extreme fibres yield, and asymptotically approaches the plastic moment Mp. The curve acts as the constitutive law that is the starting point of plastic design, plastic-hinge analysis and the nonlinear (pushover) analysis of structures under earthquake loading. To compute the nonlinear behaviour of a beam or column, the M-φ relationship of each section must be known first.
In an elastic-perfectly-plastic model the moment never exceeds the plastic moment Mp; it asymptotically approaches Mp and flattens out as the curvature grows. Real steel, however, shows strain hardening — the stress keeps rising slightly after yield. Selecting "With strain hardening (2%)" in this tool adds a post-yield stiffness equal to 2% of the elastic stiffness, so the moment slightly exceeds Mp at large curvature. This increment is a secondary effect, and design practice normally still treats Mp as the upper limit.
Real-World Applications
Seismic design and pushover analysis: In the nonlinear static (pushover) analysis used to assess the seismic response of buildings, the M-φ curve of each beam and column section is used directly as the nonlinear spring property of a plastic hinge. The yield moment My is the transition from elastic to plastic, the plastic moment Mp is the hinge capacity, and the curvature limit sets the deformation (rotation) capacity of the hinge. Without the M-φ relationship you cannot track when, where and how much a building yields.
Plastic design and collapse-load assessment: In the plastic design of steel structures, a "plastic hinge" is assumed to form wherever a section reaches its plastic moment Mp. Once enough hinges form to turn the structure into a mechanism, the whole frame collapses. Because the shape factor lets you draw out more capacity than allowable-stress design, more economical section sizes become possible. The M-φ curve is the starting point of this approach.
Bending analysis of reinforced-concrete members: In RC beams and columns, the nonlinearity of the concrete in compression combines with yielding of the reinforcement, and the M-φ curve takes an even more complex shape. Even so, tracking the "elastic → cracking → rebar yield → ultimate" stages with an M-φ curve uses exactly the same framework as steel structures. The rectangular, elastic-perfectly-plastic case in this tool serves as the most basic prototype for understanding it.
Assessing section ductility: The "curvature ductility" φ_u/φ_y — the ultimate curvature divided by the yield curvature — indicates how much a section can deform in a ductile manner. For structures subjected to repeated large deformations such as earthquakes, this ductility, not just the strength (Mp), is decisive. The longer the M-φ curve runs flat and horizontal, the more it can be rated as a "tough" section with high energy-absorption capacity.
Common Misconceptions and Pitfalls
The most common mistake is assuming the section fails once it reaches the yield moment My. My is only the point at which the outermost fibre first yields — it is not the end of the section. Everything inside it is still elastic, and the section keeps carrying a larger moment as the yielded region spreads. A rectangular section can take up to 1.5 times My (the plastic moment Mp). Reading My as "failure" throws away the entire reserve capacity the section actually has. Understand correctly that the M-φ curve keeps rising past My.
Next, assuming the shape factor is 1.5 for every section. 1.5 is a value specific to a rectangular (solid rectangular) section. In an I-section, where the material is concentrated at the extreme fibres in the bending direction, the shape factor is only about 1.1-1.2; for a solid circular section it is about 1.7. The shape factor is exactly the ratio of the plastic section modulus Z to the elastic section modulus S, so it must always be recalculated for each section shape. Memorising "steel means 1.5" risks overestimating the reserve capacity of an I-section.
Finally, overconfidence that Mp is a capacity you can always use. The plastic moment Mp assumes the section does not undergo local or lateral-torsional buckling and can deform stably up to a sufficient curvature. Sections with thin flanges or webs (large width-to-thickness ratios) lose capacity to local buckling before reaching Mp, and the M-φ curve turns downward short of Mp. This tool is also a section-level theory that assumes plane sections remain plane, and it does not include shear deformation, residual stresses or strain-rate effects. When you use Mp in plastic design, always check the section's width-to-thickness classification and the lateral-bracing conditions.
How to Use
Enter beam width (b) in mm and height (h) in mm to define rectangular section geometry
Set material properties: Young's modulus E (GPa) and yield strength fy (MPa)
Adjust the applied moment M using the slider to observe real-time changes in curvature, stress distribution, and section state (elastic, yielding, or plastic)
Read outputs: yield moment My, plastic moment Mp, shape factor Mp/My, and yield curvature φ_y
Worked Example
Steel beam section: b=200mm, h=400mm, E=200GPa, fy=250MPa. Calculated yield moment My = 66.7kN·m (stress reaches 250MPa at outer fiber). Plastic moment Mp = 100.0kN·m (entire section at yield stress). Shape factor = 1.5 (typical for rectangular sections). When moment reaches 66.7kN·m, yield curvature φ_y = 0.003125/mm. Increasing moment to 100kN·m causes distributed yielding across the full depth.
Practical Notes
I-sections produce shape factors around 1.1–1.15; rectangular sections give 1.5, requiring less material for elastic design but allowing greater reserve in plastic design
Verify yield curvature matches φ_y = fy/(E·c) where c is distance from neutral axis to extreme fiber
Watch section state transitions: elastic (M < My) shows linear stress gradients; yielding (My < M < Mp) shows nonlinear plasticity; fully plastic (M = Mp) indicates uniform yield stress across depth