Explore how much reserve a beam section keeps between "the instant it first yields" and "the instant the whole section yields into a plastic hinge". Change the dimensions of a rectangular, solid circular or I-beam section and see the elastic section modulus S, plastic section modulus Z, shape factor f, yield moment My and fully plastic moment Mp update in real time.
Parameters
Section shape
Bending about the strong (horizontal) axis
Width b (diameter for circle)
mm
Section width for rectangle/I-beam, diameter d for the circle
Height h
mm
Overall section depth (not used for the circle)
Flange thickness tf (I-beam only)
mm
Web thickness tw (I-beam only)
mm
Yield stress σy
MPa
About 235 for S235/SS400, about 325 for S355/SM490
Results
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Elastic section modulus S (cm³)
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Plastic section modulus Z (cm³)
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Shape factor f = Z/S
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Yield moment My (kN·m)
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Fully plastic moment Mp (kN·m)
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Reserve ratio Mp/My
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Section & bending stress — elastic to fully plastic
Left is the chosen cross-section, right is the bending stress distribution. It sweeps from the elastic state (triangular profile, σy only at the extreme fibre) to the fully plastic state (rectangular ±σy block over the whole depth). Shaded bands mark the yielded regions.
Elastic and plastic section moduli of a rectangle. Their ratio is the shape factor f, which is exactly 1.5 for a rectangle. b: width, h: height.
$$M_y=S\,\sigma_y,\qquad M_p=Z\,\sigma_y$$
Yield moment My (the moment at which the extreme fibre first yields) and fully plastic moment Mp (the moment at which the whole section yields). σy: yield stress.
The shape factor f measures how much extra moment a section can carry between first yield and a full plastic hinge.
What is the Plastic Section Modulus and the Shape Factor?
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The "section modulus" is the value that describes a beam's bending strength, right? But there are two of them — "elastic" and "plastic". How do they differ?
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Good question. As you bend a beam, the extreme fibre — the outermost layer of the section — is the first to reach the yield stress. The elastic section modulus S governs that instant. It is S = I/c, where I is the second moment of area and c is the distance from the neutral axis to the edge. If you keep bending, the yielded region spreads inward from the surface until eventually the whole section has yielded. The plastic section modulus Z governs that "fully plastic" instant.
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I see. So what does the ratio of those two — the "shape factor f" — actually mean?
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Roughly speaking, it is the "headroom after first yield". With f = Z/S, it tells you how many times the first-yield moment My the fully plastic moment Mp can climb to. Set the section shape on the left to "Rectangular" — f should read exactly 1.5. That is a mathematically exact value: 3/2 for any rectangle. For a solid circle it is 16/(3π) ≈ 1.698.
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Wait — when I switch the shape to "I-beam" the factor drops to just over 1.1. Why does the section shape change it so much?
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That is the most interesting thing about the shape factor. The reason is "where the material sits". In an I-beam most of the area is in the top and bottom flanges — that is, close to the extreme fibre. So by the time the extreme fibre first yields, almost all of the material is already on the verge of yielding. There is little reserve left, so f is small. A rectangle, by contrast, has plenty of material right next to the neutral axis that has not yielded at all. That un-yielded material is the "headroom", and it is large.
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So how do engineers actually use My and Mp in design?
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My = S·σy is the moment at which the extreme fibre first yields — the limit used in allowable-stress (elastic) design. Mp = Z·σy is the moment at which the section is fully plasticised, loses its bending stiffness and becomes a plastic hinge. The ultimate-strength check of steel structures and plastic design use Mp as the basis. People often say "elastic design throws away the reserve worth f". Look at the moment-curvature chart below: past My the curve flattens out and approaches a plateau just below Mp.
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One last thing — for the I-beam, won't the calculation break if I make the flanges too thick?
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Sharp point. For an I-beam, the web clear height is hw = h − 2tf, the total height minus the two flange thicknesses. If you make the flanges so thick that hw drops to zero or below, the section no longer exists. This tool guards hw > 0 and shows a warning for an invalid input. In real H-section steel the flange thickness is only about 10% of the total height. Probing how a tool behaves at extreme dimensions is one good way to use it.
Frequently Asked Questions
The elastic section modulus S governs the moment at which the extreme fibre of the section just reaches the yield stress (first yield). It is S = I/c, where I is the second moment of area and c is the distance from the neutral axis to the extreme fibre. The plastic section modulus Z governs the state where the whole section has yielded and a fully plastic hinge has formed. Z is the sum of the first moments of the two half-areas about the neutral axis. S gives the yield moment My = S·σy, and Z gives the fully plastic moment Mp = Z·σy.
The shape factor f = Z/S is the ratio of extra moment a section can carry between first yield and a full plastic hinge. For a rectangle f = 1.5 (exactly 3/2), for a solid circle f = 16/(3π) ≈ 1.698, and for typical I-beams f ≈ 1.10-1.18. The difference comes from where the material sits. In an I-beam most of the area is already near the extreme fibre, so when the extreme fibre first yields almost all the material is on the verge of yielding too, leaving little reserve. In a rectangle or circle a lot of un-yielded material remains near the neutral axis, giving a larger reserve.
The yield moment My = S·σy is the bending moment at which the extreme fibre first yields, and it is the limit used in allowable-stress (elastic) design. The fully plastic moment Mp = Z·σy is the moment at which the section is fully plasticised, loses its bending stiffness and forms a plastic hinge. Plastic (limit-state) design and the ultimate-strength check of steel structures use Mp as the basis. The ratio of Mp to My is exactly the shape factor f, so switching from elastic to plastic design raises the apparent capacity by that factor f.
For a symmetric I-section (flange width b, flange thickness tf, total height h, web thickness tw), the plastic section modulus about the strong axis is Z = b·tf·(h−tf) + tw·hw²/4, where hw = h − 2tf is the web clear height. The first term is the contribution of the two flanges, the second term is the web contribution, each taken as a first moment of area about the neutral axis. For thick flanges, check that hw > 0; an input that makes hw negative is not physically valid.
Real-World Applications
Plastic and ultimate-strength design of steel structures: For the beams and columns of steel moment frames, the section capacity is evaluated with the fully plastic moment Mp. In a "weak-beam" design, where beam ends form plastic hinges in an earthquake to absorb energy and protect the building, the Mp of each member is the starting point of design. The "plastic section modulus Zx" listed in H-section steel tables is exactly the Z this tool computes.
Comparing allowable-stress and plastic design: For the same beam, allowable-stress (elastic) design uses My as the capacity basis while plastic design uses Mp. The difference between them is the shape factor f. Switching design method changes the apparent capacity by about 1.5× for a rectangular section and about 1.1× for an I-beam. This tool lets you quantify which design method to adopt and how the choice of section shape affects capacity.
Machine parts, shafts and arms: For crane booms, the arms of construction machinery and rotating shafts, how much plastic deformation the part can sustain under overload is key to safe design. A solid round bar (circular section) has a large shape factor of about 1.7, giving it "toughness" — it does not fail abruptly once first yield is exceeded. The tool helps compare this toughness across section shapes.
Pre-study and sanity check for non-linear FEM: Before running an elasto-plastic FEM analysis, estimating the section Mp with this tool gives a check on the plausibility of the results. If the ultimate bending capacity from FEM differs greatly from Z·σy, suspect a mistake in the material model or section definition. If they agree, it is evidence that the analysis is correctly capturing plasticity.
Common Misconceptions and Pitfalls
The biggest misconception is assuming you can safely use a section right up to the fully plastic moment Mp. Mp is the moment at which the section is completely plasticised into a plastic hinge — it is not a value to be used routinely in design. A section that has reached Mp has almost no bending stiffness left and deflects greatly for a tiny increase in load. Real design applies a safety factor to Mp, or, when a plastic hinge is allowed, separately checks the deformation magnitude and the location and number of plastic hinges. Read Mp not as "the safe upper limit you may use" but as "the theoretical maximum bending capacity the section can possibly carry".
Next, thinking a section with a larger shape factor f is always better. A large f (1.7 for a circle, 1.5 for a rectangle) means a large "reserve after first yield", and that does add safety margin against fracture. But a section with a large f delivers a smaller elastic section modulus S for the same amount of material, so it is worse for deflection and first yield. The I-beam has a small f (about 1.1), yet by concentrating material at the extreme fibre it achieves a large S and Z, and the highest bending efficiency per unit weight. Choosing a section by f alone is a mistake — judge it by the balance of required stiffness, capacity and weight.
Finally, the misconception that this calculation accounts for local buckling or lateral-torsional buckling. The Mp = Z·σy of this tool is a theoretical value that assumes the section reaches full plasticity smoothly without buckling. In real thin-walled sections, the flange or web often buckles locally before reaching Mp, or the whole beam tips sideways (lateral-torsional buckling), so the capacity is governed by a moment lower than Mp. Steel design classifies sections by width-to-thickness ratio (compact or not), and Mp cannot be used for sections whose capacity is governed by buckling. This tool deals only with the plastic capacity of the section itself; a separate buckling check is still required.
How to Use
Enter flange width (b) and height (h) in mm for your I-beam or H-section geometry.
Input flange thickness (tf) and web thickness (tw) in mm.
The simulator calculates elastic section modulus S and plastic section modulus Z automatically.
Review the shape factor (f = Z/S) to quantify reserve capacity between first yield and full plasticity.
Compare yield moment My against fully plastic moment Mp to assess ductility margin in your design.
Worked Example
Consider an IPE 240 steel section: b=120mm, h=240mm, tf=9.8mm, tw=6.2mm. The simulator yields S≈354cm³ and Z≈484cm³, giving shape factor f=1.37. For steel with Fy=355MPa, My=12.6kN·m and Mp=17.2kN·m. The reserve ratio Mp/My=1.37 shows the section can sustain 37% additional load beyond first fiber yield before complete section collapse, critical for seismic and impact design scenarios.
Practical Notes
Shape factors for I-beams typically range 1.10–1.20; rectangular sections reach 1.50. Higher factors indicate greater post-yield deformation capacity before fracture.
In plastic analysis and limit-state design, use Mp instead of My·safety factor to exploit material reserve and optimize steel tonnage in continuous beams and frames.
Verify plastic hinge formation: tf and tw must be Class 1 (compact) per Eurocode or AISC to prevent local buckling before reaching full plasticity.
Web slenderness h/tw affects which limit state governs; thick webs push failures toward global buckling rather than section yield.