Civil & Structural 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 resists the curvature of a beam subjected to transverse loading. In a simply supported beam carrying a single concentrated load at mid-span, the bending moment is zero at both supports and increases linearly to a maximum value at the center of the span. This peak value, known as the maximum bending moment, is the governing internal action for strength and serviceability checks in the elastic range.

This calculator computes the maximum bending moment for a simply supported beam with a central point load. It is intended for preliminary sizing, quick verification of hand calculations, and educational use in structural and mechanical engineering.

Governing Formula

The calculator uses the closed-form expression for the peak moment:

M_max = P L / 4

where P is the concentrated load applied at mid-span (N), and L is the clear span between supports (mm). The result M_max is expressed in N·mm. The expression is dimensionally consistent: force multiplied by length yields a moment.

The formula is valid only for a prismatic, simply supported beam with a single point load acting at the exact mid-span, with negligible self-weight and linear-elastic material behaviour.

Derivation

Consider a simply supported beam of span L with a central point load P. By symmetry, each support reaction equals P/2. Taking a section at a distance x from the left support, for 0 ≤ x ≤ L/2, the internal bending moment is:

M(x) = (P/2) · x

At mid-span, x = L/2, so M = (P/2)(L/2) = P L / 4. The bending moment diagram is triangular, with zero moment at the supports and a peak at the load point. The shear force diagram is constant at ±P/2 and changes sign at mid-span, which is consistent with V = dM/dx and confirms that the maximum moment occurs where shear crosses zero.

Input Parameters

Point Load (P): The concentrated force applied at the center of the span, in newtons. Doubling P doubles M_max. Common mistake: entering a mass in kilograms instead of a force in newtons. Multiply mass by 9.81 m/s² to obtain the weight force.
Beam Length (L): The clear unsupported span between the two supports, in millimeters. Doubling L doubles M_max. Common mistake: using the total member length instead of the span between supports.
Allowable Moment (optional): A user-defined limiting moment used to compare against the calculated M_max. It is a screening aid, not a substitute for a full design check.

Engineering Interpretation

The maximum bending moment alone does not determine whether a beam is safe. It is the input to the bending stress equation:

σ = M c / I = M / S

where c is the distance from the neutral axis to the extreme fibre, I is the second moment of area, and S = I/c is the section modulus. A designer compares the resulting stress against the allowable stress (or factored resistance) and also checks deflection, shear, buckling, and connection capacity. The calculated moment is therefore a necessary but not sufficient condition for design acceptance.

Units and Conversion

This calculator uses N for force and mm for length, giving moment in N·mm. Useful conversions:

1 N·m = 1,000 N·mm
1 kN·m = 1,000,000 N·mm
1 kgf ≈ 9.81 N

Alternative and General Cases

The formula M = P L / 4 is a special case. For an off-center point load at distances a and b from the supports (a + b = L), the maximum moment occurs under the load and equals M_max = P a b / L. For a uniformly distributed load w over the full span, M_max = w L² / 8 at mid-span. For cantilevers, multiple loads, or continuous beams, a shear and moment diagram must be constructed. This calculator does not handle those cases.

Assumptions and Limitations

Simply supported beam with one pinned and one roller support.
Prismatic, straight member with constant cross-section.
Single static point load applied exactly at mid-span.
Negligible self-weight of the beam.
Linear-elastic, homogeneous, isotropic material.
Small deformations and validity of Euler–Bernoulli beam theory.
No axial loads, torsion, or lateral-torsional buckling considered.
No dynamic, fatigue, or impact amplification included.

Practical Engineering Use

The simply supported beam with a central load is a standard idealisation used for machine frames, temporary support beams, lifting spreader bars, laboratory test rigs, floor joists under a concentrated appliance load, and many educational problems. In real structures, the actual support conditions, load distribution, and connection stiffness often differ from the ideal model, so the calculated M_max should be treated as a first-order estimate and verified with a complete structural analysis.

Continue Calculating

calculators

Calculators

Quickly calculate and analyze parameters for your calculators requirements.

Open

calculators

Civil

Quickly calculate and analyze parameters for your civil requirements.

Open

calculators

Beam Bending Moment

Quickly calculate and analyze parameters for your beam bending moment requirements.

Open

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.