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mm⁴
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mm³
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mm⁴
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mm³
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mm³
The Section Modulus Calculator determines the elastic section modulus \(S\) and plastic section modulus \(Z\) of common cross-sectional shapes used in structural engineering. These properties are essential for evaluating a beam's ability to resist bending and are used daily by structural, mechanical, and civil engineers worldwide.
The section modulus links the internal bending moment in a beam to the maximum stress at the extreme fiber. For elastic analysis, the fundamental relationship is:
$$\sigma_{max} = \frac{M}{S} = \frac{Mc}{I}$$
where \(M\) is the bending moment, \(I\) is the second moment of area (moment of inertia), \(c\) is the distance from the neutral axis to the extreme fiber, and \(S = I/c\) is the elastic section modulus. A larger section modulus means the beam can resist a greater bending moment before yielding.
The plastic section modulus \(Z\) quantifies the capacity of a cross-section when the entire section has yielded. The plastic moment is:
$$M_p = f_y \cdot Z$$
where \(f_y\) is the yield stress. The ratio \(Z/S\) is called the shape factor, indicating how much additional moment capacity exists beyond first yield. For a rectangle, this factor is 1.5; for a solid circle, it is approximately 1.70; and for a typical I-beam, it ranges from 1.10 to 1.18.
This calculator supports three common shapes: rectangular, solid circular, and I-beam (doubly symmetric). For rectangular sections, the inputs are width \(b\) and depth \(h\). For circular sections, the diameter \(d\) is used. For I-beams, the overall width \(b\), total depth \(h\), flange thickness \(t_f\), and web thickness \(t_w\) define the geometry.
Engineers use section modulus values when selecting steel beams from catalogs, checking timber joists for code compliance, or designing concrete cross-sections. The elastic modulus governs the allowable stress design (ASD) method, while the plastic modulus underpins load and resistance factor design (LRFD) methods used in modern steel codes such as AISC 360 and Eurocode 3.
Understanding both moduli gives a complete picture of a section's bending performance, from first yield through full plastification, enabling safe and efficient structural design.
The calculator computes the moment of inertia \(I\), then divides by the extreme fiber distance \(c\) to obtain the elastic section modulus \(S\). The plastic section modulus \(Z\) is calculated directly from geometry.
Rectangular section (b × h):
$$I = \frac{bh^3}{12}, \quad c = \frac{h}{2}, \quad S = \frac{bh^2}{6}, \quad Z = \frac{bh^2}{4}$$
Solid circular section (diameter d):
$$I = \frac{\pi d^4}{64}, \quad c = \frac{d}{2}, \quad S = \frac{\pi d^3}{32}, \quad Z = \frac{d^3}{6}$$
Doubly symmetric I-beam (b, h, t_f, t_w):
$$I = \frac{bh^3 - (b - t_w)(h - 2t_f)^3}{12}$$
$$S = \frac{I}{h/2}$$
$$Z = b \cdot t_f (h - t_f) + \frac{t_w (h - 2t_f)^2}{4}$$
All dimensions are in millimeters, giving \(I\) in mm⁴, \(S\) and \(Z\) in mm³.
A higher elastic section modulus \(S\) means the section can resist a larger bending moment within the elastic range. The plastic section modulus \(Z\) is always greater than or equal to \(S\) and represents the full plastic moment capacity. I-beams are efficient because they concentrate material away from the neutral axis, maximizing \(I\) and \(S\) per unit area. Compare your computed values with required section moduli from structural design codes to verify adequacy.
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Results
A 100 mm × 300 mm timber joist has I = 225 × 10⁶ mm⁴, elastic section modulus S = 1.5 × 10⁶ mm³, and plastic section modulus Z = 2.25 × 10⁶ mm³. The shape factor Z/S = 1.50, characteristic of all rectangles.
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Results
An I-beam with 200 mm flanges, 400 mm total depth, 20 mm flanges, and 12 mm web. S ≈ 1.93 × 10⁶ mm³ and Z ≈ 2.12 × 10⁶ mm³. The shape factor is about 1.10, typical for I-sections — material is efficiently placed far from the neutral axis.
Section modulus is a geometric property of a cross-section that measures its resistance to bending. The elastic section modulus \(S = I/c\) relates bending moment to maximum elastic stress. The plastic section modulus \(Z\) relates bending moment to full plastic capacity. Both are expressed in units of length cubed (mm³ or in³).
The elastic section modulus \(S\) applies when the stress distribution is linear (Hooke's law region). The plastic section modulus \(Z\) applies when the entire cross-section has yielded, giving the maximum possible moment capacity. The ratio \(Z/S\) is the shape factor — for rectangles it is 1.5, for I-beams typically 1.10–1.18.
I-beams place most of the material in the flanges, far from the neutral axis, which maximizes the moment of inertia \(I\) per unit area of material. A rectangular section of the same weight would have a much smaller \(I\) and \(S\). This efficiency makes I-beams the preferred shape for steel construction.
Calculate the maximum bending moment \(M\) in your beam (from loading and support conditions). Then check that \(\sigma = M/S \leq f_{allowable}\) for allowable stress design, or that \(M \leq \phi f_y Z\) for LRFD design. If the check fails, select a larger section with greater \(S\) or \(Z\).
In SI, section modulus is commonly in mm³ or cm³. In US customary units, in³ is standard. Ensure consistency: if bending moment is in N·mm, use \(S\) in mm³ to get stress in MPa (N/mm²). If moment is in kip·in, use \(S\) in in³ to get stress in ksi.
This calculator covers solid rectangular, solid circular, and doubly symmetric I-beam shapes. For hollow sections (HSS, pipes), channel sections, or asymmetric shapes, the general approach is the same — compute \(I\) about the centroidal axis and divide by \(c\) — but the formulas differ. Refer to engineering handbooks or use numerical integration for complex profiles.
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