Section Modulus Calculator

Calculate elastic and plastic section modulus for various cross-sections

Cross-Section Configuration

Rectangle: Rectangular solid cross-section

Dimensions

Output Units

Section Properties

Section Modulus

Elastic X-axis (Sx)
666666.67
mm³
Elastic Y-axis (Sy)
333333.33
mm³
Plastic X-axis (Zx)
1000000.00
mm³
Plastic Y-axis (Zy)
500000.00
mm³

Moment of Inertia

X-axis (Ix)
66666666.67
mm⁴
Y-axis (Iy)
16666666.67
mm⁴

Additional Properties

Cross-sectional Area
20000.00
mm²
Neutral Axis X
100.00
mm
Neutral Axis Y
50.00
mm

Section Modulus Formulas

Elastic X: S_x = b × d² / 6

Elastic Y: S_y = d × b² / 6

Plastic X: Z_x = b × d² / 4

Plastic Y: Z_y = d × b² / 4

S = Elastic Section Modulus, Z = Plastic Section Modulus, I = Moment of Inertia

Example: Rectangular Section

Given Values

Width (b): 100 mm

Height (d): 200 mm

Calculation Steps

S_x = b × d² / 6 = 100 × 200² / 6

S_x = 100 × 40,000 / 6 = 666,667 mm³

I_x = b × d³ / 12 = 100 × 200³ / 12

I_x = 100 × 8,000,000 / 12 = 66,666,667 mm⁴

Results

The elastic section modulus about the X-axis is 666,667 mm³ and the moment of inertia is 66,666,667 mm⁴.

Cross-Section Types

Basic Shapes
Rectangle, Square, Circle
Simple geometric shapes
Hollow Shapes
Tubes, Pipes, Hollow rectangles
Empty interior sections
Structural Shapes
I-beams, Channels, T-sections
Standard structural sections

Section Modulus Types

Elastic (S)S = I/c
Plastic (Z)Z > S
I = Moment of inertia
c = Distance to extreme fiber
Plastic modulus is always larger

Engineering Applications

Beam Design:σ = M/S for bending stress
Structural Analysis:Optimizing member sizes
Material Selection:Comparing efficiency
Failure Analysis:Plastic vs elastic behavior

Design Tips

Higher section modulus means lower bending stress

I-beams are efficient for bending applications

Plastic modulus is used for ultimate load design

Consider both axes for biaxial bending

Wall thickness affects hollow section efficiency

Understanding Section Modulus

What is Section Modulus?

Section modulus is a geometric property that relates a cross-section's resistance to bending. It's defined as the ratio of the moment of inertia to the distance from the neutral axis to the extreme fiber. Engineers use it to quickly calculate maximum bending stress in beams.

Key Relationships

S = I / c

σ_max = M / S

M_plastic = Z × σ_yield

S = Section modulus, I = Moment of inertia, c = Distance to extreme fiber, M = Bending moment, Z = Plastic section modulus

Elastic vs Plastic Modulus

Elastic Section Modulus (S)

Used for elastic behavior, assumes linear stress distribution

Based on extreme fiber stress

Plastic Section Modulus (Z)

Used for plastic analysis, assumes uniform stress at yield

Always greater than elastic modulus

Shape Factor

Z/S ratio indicates plastic reserve capacity

Rectangular: 1.5, I-beam: ~1.1-1.2

Design Applications

Beam Sizing

Select sections with adequate section modulus

Stress Analysis

Calculate maximum bending stress quickly

Optimization

Compare section efficiency for weight savings

Common Section Formulas

Rectangle

S = bd²/6

Circle

S = πD³/32

I-Beam

S = I/c (complex I)