Section Modulus Calculator

Q: What units does the section modulus calculator support?

The calculator supports millimeters (mm) , centimeters (cm) , inches (in) , and feet (ft) . You can set one global unit or use the dropdown next to each input field to mix units. All results update to match the unit you pick.

Q: What cross-section shapes can I calculate?

You can choose from 10 shapes : Rectangle, Square, Hollow Rectangle (RHS), I-Section (Wide Flange), Channel (C-Section), Angle (L-Section), Tee (T-Section), Triangle (isosceles or right), Circle, and Pipe / Hollow Circle (CHS).

Q: What is the difference between S and Z in this calculator?

It depends on your notation setting. In American (AISC) mode, S is the elastic section modulus and Z is the plastic section modulus. In British / Eurocode mode, the letters swap — Z is elastic and S is plastic. The actual values do not change. Only the symbols change.

Q: How does the reverse solve feature work?

Reverse solve works backward. You enter a target section modulus you need, pick which single dimension to solve for, and keep all other dimensions fixed. The calculator then finds the value of that dimension that gives you the target modulus. This is useful when you know the required bending strength and need to find the right size.

Q: What is the shape factor shown in the results?

The shape factor is the plastic section modulus divided by the elastic section modulus . It tells you how much extra bending strength a shape has between first yield and full plastification. For example, a solid rectangle has a shape factor of 1.5, meaning it has 50% more strength beyond first yield.

Q: Why do I see two elastic modulus values for some shapes?

This happens when the shape is not symmetric about the bending axis. For example, a Tee or Angle section has its centroid closer to one edge than the other. The elastic section modulus to the top fiber is different from the one to the bottom fiber. The calculator shows both so you can see which side is critical.

Q: What is the difference between moment of inertia and section modulus?

The moment of inertia (I) measures how the area is spread out from the bending axis. The section modulus (S) equals the moment of inertia divided by the distance to the extreme fiber (S = I / y). Section modulus directly tells you bending strength, while moment of inertia is used for stiffness and deflection calculations.

Section Modulus Calculator

Updated Jul 16, 2026 By Jehan Wadia

Introduction

The section modulus tells you how strong a beam or column is when a load tries to bend it. It depends on the shape and size of the cross-section. A higher section modulus means the member can resist more bending before it fails. Engineers use this value every day to pick the right beam size for a structure.

There are two types. The elastic section modulus measures strength up to the point where the material first starts to yield. The plastic section modulus measures the full strength of the cross-section after the entire shape has yielded. Both values matter in structural design codes like AISC and Eurocode.

This Section Modulus Calculator handles 10 common cross-section shapes, including rectangles, I-beams, channels, angles, tees, circles, and hollow sections. Enter your dimensions, and the tool instantly calculates both elastic and plastic section moduli about the x-axis and y-axis. It also gives you the area, centroid location, moment of inertia, radius of gyration, and shape factor. Every result comes with a step-by-step solution so you can follow the math or check it by hand. You can switch between American (AISC) and British (Eurocode) notation, choose your preferred units, and even reverse-solve to find the dimension needed for a target section modulus.

How to Use Our Section Modulus Calculator

This calculator finds the elastic and plastic section modulus of common cross-section shapes. Enter your unit system, pick a shape, type in the dimensions, and get full results with step-by-step math.

Unit System — Pick the unit you want to work in. Choose from millimeters (mm), centimeters (cm), inches (in), or feet (ft). All inputs and results will match the unit you select.

Number Format — Choose how numbers are shown. Select US format (1,234.56) or European format (1.234,56) based on what you are used to.

Notation Convention — Pick American (AISC) or British (Eurocode) notation. In American style, S is elastic and Z is plastic. In British style, Z is elastic and S is plastic. The math is the same — only the letters change.

Cross-Section Shape — Click one of the ten shapes: Rectangle, Square, Hollow Rectangle, I-Section, Channel, Angle (L), Tee (T), Triangle, Circle, or Pipe (CHS). The input fields will update to match the shape you choose.

Dimensions — Type in each measurement for your chosen shape, such as width, height, thickness, or radius. Each field has its own unit dropdown if you need to mix units. All values must be greater than zero.

Calculate Button — Press "Calculate" to run the computation. The tool will display the elastic section modulus, plastic section modulus, shape factors, a scaled diagram, a bar chart, and a full step-by-step solution.

Reverse Solve — Use this feature to work backward. Pick a target property (like elastic modulus about x), choose which dimension to solve for, and enter the section modulus value you need. The calculator will find the required dimension for you.

Download Report — Press "Download Calculation Report (PDF)" to save or print a full summary of your inputs, diagram, and results.

What Is Section Modulus?

Section modulus is a number that tells engineers how strong a beam or column is against bending. The bigger the section modulus, the harder it is to bend the shape. It depends on the size and shape of the cross-section — the flat slice you would see if you cut straight through a beam. Understanding the cross-sectional area is the starting point, but section modulus goes further by accounting for how that area is distributed relative to the bending axis.

Elastic vs. Plastic Section Modulus

There are two types of section modulus. The elastic section modulus measures how much bending a shape can take before any part of it starts to yield (permanently deform). The plastic section modulus measures the full bending strength when the entire cross-section has yielded. The plastic value is always equal to or larger than the elastic value. The ratio between them is called the shape factor, and it shows how much extra strength a shape has beyond first yield.

Why Section Modulus Matters

When engineers design beams for buildings, bridges, or machines, they need to know the section modulus to pick the right size. A beam that is too small will bend too much or break — you can check allowable deflections with a beam deflection calculator. A beam that is too big wastes material and money. Section modulus helps find the right balance. It is used in steel design codes like AISC and Eurocode every day. Whether you are sizing rafters for a roof, selecting members for a truss, or checking a retaining wall stem, the section modulus is central to verifying bending capacity.

How This Calculator Works

This section modulus calculator finds both the elastic and plastic section modulus for 10 common cross-section shapes, including rectangles, I-beams, channels, angles, tees, circles, pipes, and triangles. Enter your dimensions, and the tool instantly computes the section modulus about both the x-axis and y-axis. It also gives you the area, centroid location, moment of inertia, and radius of gyration. A step-by-step solution shows every formula used. You can also use the reverse solve feature to find the dimension needed to reach a target section modulus. For related geometric properties, you may find our triangle area calculator or circle area calculator helpful when working with those specific shapes. If you need to estimate the weight of the structural member you are sizing, try our steel weight calculator or metal weight calculator.

Notation: American vs. British

Different countries use different symbols. In American (AISC) notation, S stands for elastic section modulus and Z stands for plastic section modulus. In British and Eurocode notation, it is the opposite — Z is elastic and S is plastic. This calculator lets you switch between both so the symbols match the design code you are using.

Formulas used

Second Moment of Area (Parallel-Axis Theorem)

I_x = \sum \left( I_{x,i} + A_i \cdot d_i^2 \right), \quad I_{x,i} = \frac{b_i \, h_i^3}{12}

Elastic Section Modulus

S_x = \frac{I_x}{y_{\max}}

Plastic Section Modulus (Rectangle)

Z_x = \frac{b \, d^2}{4}

Plastic Section Modulus (Circle / Hollow Circle)

Z_x = \frac{4}{3} \left( R^3 - r^3 \right)

Shape Factor

f = \frac{Z}{S}

Radius of Gyration

r = \sqrt{\frac{I}{A}}

Frequently asked questions

What units does the section modulus calculator support?

The calculator supports millimeters (mm), centimeters (cm), inches (in), and feet (ft). You can set one global unit or use the dropdown next to each input field to mix units. All results update to match the unit you pick.

What cross-section shapes can I calculate?

You can choose from 10 shapes: Rectangle, Square, Hollow Rectangle (RHS), I-Section (Wide Flange), Channel (C-Section), Angle (L-Section), Tee (T-Section), Triangle (isosceles or right), Circle, and Pipe / Hollow Circle (CHS).

What is the difference between S and Z in this calculator?

It depends on your notation setting. In American (AISC) mode, S is the elastic section modulus and Z is the plastic section modulus. In British / Eurocode mode, the letters swap — Z is elastic and S is plastic. The actual values do not change. Only the symbols change.

What does the governing value mean?

When a shape is not symmetric, the distance from the centroid to the top fiber is different from the distance to the bottom fiber. This gives two elastic section modulus values. The governing value is the smaller one because that side will yield first under bending. This is the value you should use for design.

How does the reverse solve feature work?

Reverse solve works backward. You enter a target section modulus you need, pick which single dimension to solve for, and keep all other dimensions fixed. The calculator then finds the value of that dimension that gives you the target modulus. This is useful when you know the required bending strength and need to find the right size.

What is the shape factor shown in the results?

The shape factor is the plastic section modulus divided by the elastic section modulus. It tells you how much extra bending strength a shape has between first yield and full plastification. For example, a solid rectangle has a shape factor of 1.5, meaning it has 50% more strength beyond first yield.

Why do I see two elastic modulus values for some shapes?

This happens when the shape is not symmetric about the bending axis. For example, a Tee or Angle section has its centroid closer to one edge than the other. The elastic section modulus to the top fiber is different from the one to the bottom fiber. The calculator shows both so you can see which side is critical.

Can I mix different units for different dimensions?

Yes. Each input field has its own unit dropdown. You can enter one dimension in inches and another in millimeters if you want. The calculator converts everything internally before computing the results.

What is the difference between moment of inertia and section modulus?

The moment of inertia (I) measures how the area is spread out from the bending axis. The section modulus (S) equals the moment of inertia divided by the distance to the extreme fiber (S = I / y). Section modulus directly tells you bending strength, while moment of inertia is used for stiffness and deflection calculations.

How accurate is this calculator?

The calculator uses exact closed-form formulas for standard shapes like rectangles, circles, and hollow circles. For composite shapes like I-sections, channels, angles, and tees, it uses the parallel-axis theorem with exact geometry. Plastic section modulus is computed using a numerical integration method with high precision. Results match standard engineering references.

What does the radius of gyration mean?

The radius of gyration (r) is a measure of how spread out the cross-section's area is from the axis. It equals the square root of the moment of inertia divided by the area: r = √(I / A). Engineers use it mainly for column buckling calculations. A larger radius of gyration means the member resists buckling better.

Can I save or print my results?

Yes. Click the "Download Calculation Report (PDF)" button at the bottom. It opens a printable report with your inputs, the cross-section diagram, and all results. Use your browser's print dialog to save it as a PDF or send it to a printer.

Why is the plastic section modulus always larger than the elastic one?

The elastic section modulus only uses the stress at the outermost fiber, assuming stress varies linearly across the section. The plastic section modulus assumes the entire cross-section has yielded, using the full material strength everywhere. This always gives a value equal to or greater than the elastic modulus.

What do the slenderness warnings mean?

These warnings appear when a part of the cross-section, like a flange or web, is very thin compared to its width or height. A slender element may buckle locally before the full section modulus strength is reached. The warnings do not stop the calculation. They remind you to check your design against local buckling rules in your design code.

What is the neutral axis shown in the diagram?

The neutral axis is the horizontal line through the centroid of the cross-section where bending stress is zero. Material above this line is in compression, and material below is in tension (or vice versa). The red dashed line labeled "x̄ (N.A.)" in the diagram marks this axis.

Does this calculator account for rounded corners or fillets?

No. The calculator uses sharp-corner geometry for all shapes. Real steel sections often have small fillets at web-to-flange junctions. These fillets add a small amount of area and slightly increase the section modulus. For precise values of standard rolled sections, refer to your steel manufacturer's tables.

Can I use this for wood or concrete beams?

Yes, for finding geometric section properties. The section modulus depends only on shape and dimensions, not on the material. You can use the results for wood, concrete, steel, aluminum, or any other material. However, the allowable stress and design checks differ by material and code.

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