Mechanical Engineering Calculator

Simply Supported Beam Stress Calculator

Calculate the maximum bending stress at the extreme fiber of a simply supported beam subjected to a central point load.

Beam Configuration

Simply supported beam with a central point load — maximum bending stress at mid-span extreme fibers

Input Parameters

Enter values using the SI-derived N-mm unit system.

Point Load (P)

Central point load applied to the beam.

Beam Length (L)

Distance between the two supports.

Distance to Extreme Fiber (c)

Maximum distance from neutral axis to outer fiber (usually h/2).

Moment of Inertia (I)

mm⁴

Area moment of inertia about the bending axis.

Allowable Stress (σallow)

MPa

User-defined maximum allowable bending stress.

Engineering Tip

Always verify both strength (stress) and serviceability (deflection). A beam that is strong enough may still deflect excessively under service loads.

Maximum Bending Stress

83.333MPa

Max Moment

2,500,000 N·mm

Section Modulus

30,000.0 mm³

PASS — Stress is within the allowable limit

Calculated stress is 83.333 MPa compared with the user-defined limit of 150 MPa.

Governing Formula

σ = (P L c) / (4 I)

or equivalently σ = M c / I with M = P L / 4

σMPa

Maximum bending stress

Central point load

Lmm

Beam length

cmm

Distance to extreme fiber

Imm⁴

Area moment of inertia

MN·mm

Maximum bending moment

Calculation Assumptions

Simply supported beam
Central point load
Linear elastic behavior
Plane sections remain plane
Constant cross-section
Pure bending (no axial force)

Engineering Code

Reuse the calculation in your own engineering workflow.

Python

def beam_bending_stress(P, L, c, I):
    """
    Calculate maximum bending stress of a simply supported beam
    with a central point load.

    P: Load (N)
    L: Beam length (mm)
    c: Distance from neutral axis to extreme fiber (mm)
    I: Area moment of inertia (mm⁴)

    Returns:
        Maximum bending stress (MPa)
    """

    if P <= 0 or L <= 0 or c <= 0 or I <= 0:
        raise ValueError("All inputs must be greater than zero.")

    M = (P * L) / 4          # N·mm
    sigma = (M * c) / I      # MPa
    return sigma


# Example
P = 5000
L = 2000
c = 50
I = 1500000

sigma = beam_bending_stress(P, L, c, I)

print(f"Maximum Bending Stress: {sigma:.3f} MPa")

MATLAB

function sigma = beam_bending_stress(P, L, c, I)
    % Maximum bending stress of a simply supported beam
    % with a central point load.
    %
    % P = Load (N)
    % L = Beam length (mm)
    % c = Distance to extreme fiber (mm)
    % I = Area moment of inertia (mm^4)

    if P <= 0 || L <= 0 || c <= 0 || I <= 0
        error('All inputs must be greater than zero.');
    end

    M = (P * L) / 4;          % N·mm
    sigma = (M * c) / I;      % MPa
end

% Example
P = 5000;
L = 2000;
c = 50;
I = 1500000;

sigma = beam_bending_stress(P, L, c, I);

fprintf('Maximum Bending Stress: %.3f MPa\n', sigma);

Excel Formula

=(P*L*c)/(4*I)

Example Calculation

For a beam subjected to a 5,000 N central point load, with a length of 2,000 mm, a distance to extreme fiber of 50 mm, and a second moment of area of 1,500,000 mm⁴:

M = (5000 × 2000) / 4 = 2,500,000 N·mm

σ = (2,500,000 × 50) / 1,500,000

σ = 83.333 MPa

Technical Explanation: Beam Bending Stress

Bending stress (flexural stress) is the normal stress induced in a beam due to an applied bending moment. For a simply supported beam carrying a single point load at its center, the maximum bending moment occurs at mid-span, and the maximum tensile/compressive stress occurs at the extreme fibers farthest from the neutral axis.

The magnitude of maximum bending stress depends on the applied load (P), beam length (L), the distance from the neutral axis to the extreme fiber (c), and the area moment of inertia of the cross-section (I).

How to Use This Calculator

Point Load (P): Enter the concentrated load applied to the center of the beam in Newtons (N).
Beam Length (L): Input the total unsupported span between the two pinned/roller supports in millimeters (mm).
Distance to Extreme Fiber (c): Specify the maximum distance from the neutral axis to the outer fiber (typically half the section height) in mm.
Moment of Inertia (I): Provide the area moment of inertia for your chosen cross-section in mm⁴.
Allowable Stress (Optional): Set a maximum allowable bending stress to instantly verify if your design passes safety criteria.

How does beam length affect bending stress?

Maximum bending moment (and therefore stress) is directly proportional to beam length for this loading condition. Doubling the span doubles the peak moment and the resulting extreme-fiber stress.

How does the moment of inertia affect stress?

Stress is inversely proportional to the area moment of inertia. Increasing I (e.g., using a deeper I-beam or adding material farther from the neutral axis) significantly reduces bending stress for the same applied moment.

What is the section modulus?

The elastic section modulus S = I / c is a geometric property that directly relates maximum moment to maximum stress: σ = M / S. A larger section modulus indicates greater resistance to bending stress.

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Real-World Engineering Cases

The Quebec Bridge Collapse (1907)

During construction of the Quebec Bridge, a compression chord failed due to underestimated stresses and inadequate section capacity. The design did not properly account for the actual bending and compressive stresses in the members under self-weight and construction loads.

Engineering Lesson

Always verify that the maximum bending stress remains well below the allowable or critical stress of the material and section. Conservative assumptions on effective length, residual stresses, and load factors are essential for large structures.

Overstressed Crane Girders in Industrial Plants

In several manufacturing facilities, overhead crane runway beams experienced progressive cracking at the extreme fibers after years of cyclic loading. The original design had used static allowable stress without adequate consideration of fatigue and impact factors from the moving crane loads.

Engineering Lesson

Bending stress calculations for cyclic or dynamic applications must incorporate fatigue strength and dynamic amplification. A beam that is safe under static load can still fail under repeated stress cycles if the stress range is too high.

Frequently Asked Questions

What is the formula for maximum bending stress in a simply supported beam?

For a simply supported beam with a central point load, M_max = PL/4 and σ_max = M c / I = (P L c) / (4 I).

Where does maximum bending stress occur?

Maximum bending stress occurs at the mid-span extreme fibers (top and bottom surfaces farthest from the neutral axis).

What units should I use?

This calculator uses N for load, mm for length and distance c, and mm⁴ for the area moment of inertia. The resulting stress is given in MPa (N/mm²).

Can this calculator be used for every beam?

No. This equation applies specifically to a simply supported beam with a central point load under linear-elastic assumptions. Different support conditions, load positions, or load types require different moment equations.

What is the difference between stress and deflection?

Stress (σ) measures the internal force intensity in the material and governs strength/failure. Deflection (δ) measures geometric displacement and governs serviceability. Both must be checked in a complete beam design.