Mechanical Engineering Calculator

Simply Supported Beam Shear Force Calculator

Calculate the maximum shear force and support reactions of a simply supported beam subjected to a central point load.

Beam Configuration

Simply supported beam with a central point load — shear force 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 (does not affect V_max).

Cross-Section Area (optional) (A)

mm²

Used only to compute average shear stress τ = V/A.

Allowable Shear Force (Vallow)

User-defined maximum allowable shear force.

Engineering Tip

Always check both shear and bending. A beam that is safe in flexure can still fail in shear near the supports, especially with thin webs or rectangular timber sections.

Maximum Shear Force

2,500.0N

Support Reactions

2,500.0 N each

Avg. Shear Stress

—

PASS — Shear force is within the allowable limit

Calculated shear force is 2,500.0 N compared with the user-defined limit of 3000 N.

Governing Formula

V_max = P / 2

R_A = R_B = P / 2

Maximum shear force

Central point load

R_A / R_BN

Support reactions

τ_avgMPa

Average shear stress (optional)

Amm²

Cross-section area (optional)

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

Engineering Code

Reuse the calculation in your own engineering workflow.

Python

def beam_shear_force(P):
    """
    Calculate maximum shear force of a simply supported beam
    with a central point load.

    P: Load (N)

    Returns:
        Maximum shear force (N)
    """

    if P <= 0:
        raise ValueError("Load must be greater than zero.")

    return P / 2


# Example
P = 5000

V = beam_shear_force(P)

print(f"Maximum Shear Force: {V:.1f} N")
print(f"Support Reactions: {V:.1f} N each")

MATLAB

function V = beam_shear_force(P)
    % Maximum shear force of a simply supported beam
    % with a central point load.
    %
    % P = Load (N)

    if P <= 0
        error('Load must be greater than zero.');
    end

    V = P / 2;
end

% Example
P = 5000;

V = beam_shear_force(P);

fprintf('Maximum Shear Force: %.1f N\n', V);
fprintf('Support Reactions: %.1f N each\n', V);

Excel Formula

=P/2

Example Calculation

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

V_max = 5000 / 2

V_max = 2,500 N

(Left reaction R_A = Right reaction R_B = 2,500 N)

Technical Explanation: Beam Shear Force

Shear force is the internal force that acts parallel to the cross-section of a beam. For a simply supported beam carrying a single point load at its center, the shear force is constant in magnitude between each support and the load, and reverses direction at the mid-span.

The maximum shear force occurs at the supports and is equal to half the applied point load. This value is critical for checking shear capacity of the cross-section and for designing connections or web stiffeners in steel beams.

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). Length does not affect the magnitude of maximum shear for this load case, but is required for the schematic and consistency.
Cross-Section Area (A) – Optional: Provide the cross-sectional area in mm² to also compute the average shear stress τ = V / A.
Allowable Shear Force (Optional): Set a maximum allowable shear force to instantly verify if your design passes safety criteria.

Where does maximum shear force occur?

For a simply supported beam with a central point load, the shear force diagram is a step function: +P/2 from the left support to the center, and –P/2 from the center to the right support. The maximum absolute value therefore occurs along the entire length between supports and the load (i.e., at the supports and throughout each half-span).

How is shear force related to bending moment?

The shear force is the derivative of the bending moment (V = dM/dx). Consequently, the maximum moment occurs where the shear force changes sign (at mid-span for this symmetric case).

Average vs. maximum shear stress

The simple formula τ_avg = V / A gives the average shear stress across the section. The true maximum shear stress for a rectangular section is 1.5 × τ_avg and occurs at the neutral axis. For I-beams the maximum is higher still and is concentrated in the web.

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Real-World Engineering Cases

Web Crippling and Shear Buckling in Steel Beams

In several warehouse projects, slender-web steel beams experienced local web buckling near the supports under concentrated reactions. The design had adequately checked bending stress but underestimated the high shear force and the need for web stiffeners or thicker webs at the reaction points.

Engineering Lesson

Always check shear capacity at the supports in addition to bending. High shear combined with thin webs can lead to buckling or crippling even when flexural stress is well within limits.

Timber Beam Shear Failures in Residential Construction

A number of large-span timber floor beams developed horizontal shear cracks near the neutral axis close to the supports after years of service under heavy point loads from partitions. The original design used average shear stress without applying the 1.5 factor required for rectangular sections.

Engineering Lesson

For rectangular timber or concrete sections, the maximum shear stress is 50 % higher than the average V/A value. Ignoring this can lead to unexpected horizontal shear failures.

Frequently Asked Questions

What is the formula for maximum shear force in a simply supported beam?

For a simply supported beam with a central point load, the maximum shear force is V_max = P / 2. Each support reaction is also equal to P / 2.

Where does maximum shear force occur?

The shear force has constant magnitude P/2 between each support and the central load. The maximum absolute value therefore occurs throughout both half-spans (including at the supports).

What units should I use?

This calculator uses N for load and shear force, and mm for length. If cross-section area is provided, average shear stress is returned in MPa (N/mm²).

Does beam length affect the maximum shear force?

No. For a central point load on a simply supported beam, V_max = P/2 is independent of span length. Length does affect the bending moment and deflection, but not the peak shear force for this load case.

What is the difference between shear force and shear stress?

Shear force (V) is the internal force (in Newtons). Shear stress (τ) is the force intensity distributed over the cross-sectional area (τ ≈ V/A). Design codes typically limit shear stress, not the force itself.