Mechanical Engineering Calculator

Simply Supported Beam Bending Moment Calculator

Calculate the maximum bending moment of a simply supported beam subjected to a central point load.

Beam Configuration

Simply supported beam with a central point load — bending moment diagram shown schematically

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.

Allowable Moment (Mallow)

N·mm

User-defined maximum allowable bending moment.

Engineering Tip

Once you have M_max, the required section modulus is simply S_req = M_max / σ_allow. Always check both strength (moment/stress) and serviceability (deflection).

Maximum Bending Moment

2,500,000N·mm

Location

Mid-span

Allowable Limit

3000000 N·mm

PASS — Moment is within the allowable limit

Calculated moment is 2,500,000 N·mm compared with the user-defined limit of 3000000 N·mm.

Governing Formula

M_max = P L / 4

MN·mm

Maximum bending moment

Central point load

Lmm

Beam length

Calculation Assumptions

Simply supported beam
Central point load
Static equilibrium
Negligible self-weight (or included in P)
Prismatic beam
No horizontal forces or moments at supports

Engineering Code

Reuse the calculation in your own engineering workflow.

Python

def beam_bending_moment(P, L):
    """
    Calculate maximum bending moment of a simply supported beam
    with a central point load.

    P: Load (N)
    L: Beam length (mm)

    Returns:
        Maximum bending moment (N·mm)
    """

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

    return (P * L) / 4


# Example
P = 5000
L = 2000

M = beam_bending_moment(P, L)

print(f"Maximum Bending Moment: {M:.0f} N·mm")

MATLAB

function M = beam_bending_moment(P, L)
    % Maximum bending moment of a simply supported beam
    % with a central point load.
    %
    % P = Load (N)
    % L = Beam length (mm)

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

    M = (P * L) / 4;
end

% Example
P = 5000;
L = 2000;

M = beam_bending_moment(P, L);

fprintf('Maximum Bending Moment: %.0f N·mm\n', M);

Excel Formula

=(P*L)/4

Example Calculation

For a beam subjected to a 5,000 N central point load with a length of 2,000 mm:

M_max = (5000 × 2000) / 4

M_max = 2,500,000 N·mm

(Maximum moment occurs at mid-span)

Technical Explanation: Beam Bending Moment

Bending moment is the internal moment that causes a beam to bend. For a simply supported beam carrying a single point load at its center, the bending moment is zero at the supports and reaches its maximum value at mid-span.

The magnitude of the maximum bending moment depends only on the applied load (P) and the beam length (L). This peak moment is the primary input for calculating bending stress (σ = M c / I) and for selecting an adequate section modulus.

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).
Allowable Moment (Optional): Set a maximum allowable bending moment to instantly verify if your design passes safety criteria.

Where does maximum bending moment occur?

For a simply supported beam with a central point load, the bending moment diagram is triangular. Moment is zero at both supports and increases linearly to the peak value M_max = PL/4 at the center of the span.

How is bending moment related to shear force?

Shear force is the derivative of bending moment (V = dM/dx). Therefore the maximum moment occurs exactly where the shear force changes sign (at mid-span for this symmetric loading).

Why is maximum moment important?

The maximum moment governs the required section strength. Once M_max is known, the required section modulus is simply S_req = M_max / σ_allow. Both stress and deflection checks ultimately depend on this peak moment value.

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Real-World Engineering Cases

The Hyatt Regency Walkway Collapse (1981)

A critical design change doubled the load on the box-beam connections supporting the walkways. The increased moment and connection forces were never recalculated, leading to catastrophic failure under the actual occupancy load.

Engineering Lesson

Any change to load path or support conditions requires a complete re-evaluation of the bending moment diagram. Even a seemingly minor connection detail can invalidate the original moment calculations.

Overstressed Crane Runway Beams

In several industrial facilities, overhead crane runway girders developed progressive fatigue cracks at mid-span. The original design used static moment values without adequate dynamic amplification factors for the moving crane loads.

Engineering Lesson

Peak bending moment under dynamic or moving loads is higher than the static PL/4 value. Always apply appropriate impact and fatigue factors when the load is not truly static.

Frequently Asked Questions

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

For a simply supported beam with a central point load, the maximum bending moment is M_max = P L / 4. It occurs at mid-span.

Where does maximum bending moment occur?

The maximum bending moment occurs at the center of the span, directly under the point load.

What units should I use?

This calculator uses N for load and mm for length. The resulting bending moment is given in N·mm.

Does the location of the load affect the maximum moment?

Yes. The formula M = PL/4 is valid only when the point load is exactly at mid-span. An off-center load produces a smaller maximum moment given by M_max = P a b / L, where a and b are the distances from the load to each support.

How is bending moment used in design?

The maximum moment is divided by the allowable stress (or multiplied by a safety factor) to obtain the required section modulus: S_req = M_max / σ_allow. The chosen cross-section must provide at least this section modulus.