Mechanical Engineering Calculator

Cantilever Beam Calculator

Calculate reactions, shear force, bending moment, deflection and bending stress for a cantilever beam with a point load at the free end.

Beam Configuration

Cantilever beam with point load at the free end — fixed at left, free at right

Input Parameters

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

Point Load (P)

Concentrated load applied at the free end.

Beam Length (L)

Length from fixed support to free end.

Elastic Modulus (E)

MPa

Young's modulus of the beam material.

Moment of Inertia (I)

mm⁴

Area moment of inertia about the bending axis.

Distance to Extreme Fiber (c)

Optional – required only for bending stress.

Allowable Deflection (δallow)

User-defined maximum allowable deflection.

Allowable Stress (σallow)

MPa

User-defined maximum allowable bending stress.

Engineering Tip

Cantilevers develop significantly higher moments and deflections than simply supported beams of the same span. Always verify both the fixed-end moment capacity and the free-end deflection.

Maximum Deflection (Free End)

44.444mm

Deflection / Span = 1 / 45

Reaction

5,000.0 N

Max Shear

5,000.0 N

Max Moment

10,000,000 N·mm

Max Stress

333.333 MPa

FAIL — Deflection exceeds the allowable limit

Calculated deflection is 44.444 mm vs limit of 25 mm.

FAIL — Stress exceeds the allowable limit

Calculated stress is 333.333 MPa vs limit of 250 MPa.

Governing Formulas

R = P

V_max = P

M_max = P L

δ_max = P L³ / (3 E I)

σ_max = M c / I

Calculation Assumptions

Cantilever (fixed-free) beam
Point load at free end only
Linear elastic material
Small deflection theory
Constant cross-section
Negligible self-weight (or included in P)

Engineering Code

Reuse the complete analysis in your own engineering workflow.

Python

def cantilever_beam(P, L, E, I, c=None):
    """
    Complete analysis of a cantilever beam
    with a point load at the free end.

    P: Load (N)
    L: Beam length (mm)
    E: Elastic modulus (MPa)
    I: Area moment of inertia (mm⁴)
    c: Distance to extreme fiber (mm) – optional

    Returns dict with reaction, shear, moment, deflection, stress
    """

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

    reaction = P
    shear = P
    moment = P * L
    deflection = (P * L**3) / (3 * E * I)
    stress = (moment * c) / I if c and c > 0 else None

    return {
        "reaction_N": reaction,
        "shear_N": shear,
        "moment_Nmm": moment,
        "deflection_mm": deflection,
        "stress_MPa": stress
    }


# Example
P = 5000
L = 2000
E = 200000
I = 1500000
c = 50

results = cantilever_beam(P, L, E, I, c)
for k, v in results.items():
    print(f"{k}: {v}")

MATLAB

function results = cantilever_beam(P, L, E, I, c)
    % Complete analysis of a cantilever beam
    % with a point load at the free end.
    %
    % P = Load (N)
    % L = Beam length (mm)
    % E = Elastic modulus (MPa)
    % I = Area moment of inertia (mm^4)
    % c = Distance to extreme fiber (mm) – optional

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

    results.reaction = P;
    results.shear = P;
    results.moment = P * L;
    results.deflection = (P * L^3) / (3 * E * I);
    if nargin >= 5 && c > 0
        results.stress = (results.moment * c) / I;
    else
        results.stress = NaN;
    end
end

% Example
P = 5000;
L = 2000;
E = 200000;
I = 1500000;
c = 50;

r = cantilever_beam(P, L, E, I, c);
disp(r);

Excel Formulas

Reaction: =P
Shear: =P
Moment: =P*L
Deflection: =(P*L^3)/(3*E*I)
Stress: =(M*c)/I

Example Calculation

Cantilever beam with P = 5,000 N at the free end, L = 2,000 mm, E = 200,000 MPa, I = 1,500,000 mm⁴, c = 50 mm:

R = 5,000 N

V_max = 5,000 N

M_max = 10,000,000 N·mm

δ_max = 22.222 mm

σ_max = 333.333 MPa

Technical Explanation: Cantilever Beam Analysis with Point Load

1. Technical Introduction

A cantilever beam is a fundamental structural element fixed rigidly at one end and entirely free at the other. Unlike simply supported beams, the fixed support must resist both the vertical shear force and the rotational bending moment induced by external loads. Engineers calculate these reactions, internal forces, and deflections to ensure that the beam will not fail structurally (yielding or fracture) and will not deflect beyond serviceability limits. This specific calculator addresses the most common baseline scenario: a single concentrated point load applied precisely at the free end of the beam.

2. Governing Formulas

The analysis of a cantilever beam with an end point load relies on static equilibrium and Euler-Bernoulli beam theory. The key equations computed by this tool are:

Vertical Reaction: R = P
Maximum Shear Force: V_max = P (constant along the entire length)
Maximum Bending Moment: M_max = P × L (occurs at the fixed support)
Maximum Deflection: δ_max = (P × L³) / (3 × E × I) (occurs at the free end)
Maximum Bending Stress: σ_max = (M_max × c) / I

3. Calculation Method and Derivation

The derivation begins with static equilibrium. Summing vertical forces yields the support reaction R = P. Summing moments about the fixed support yields the reaction moment M = P × L. Because there are no other loads between the support and the free end, the internal shear force remains constant at V = P throughout the beam. The internal bending moment varies linearly, starting at zero at the free end and reaching its absolute maximum of P × L at the fixed support. The deflection formula is derived by integrating the beam curvature equation, E × I × (d²y/dx²) = M(x), twice and applying the boundary conditions that both deflection (y) and slope (dy/dx) are zero at the fixed end (x = 0).

4. Input Parameters Explained

Point Load (P): The concentrated transverse force applied at the free tip, measured in Newtons (N). Common mistakes include forgetting to convert kilonewtons (kN) to Newtons or neglecting to include dynamic impact factors if the load is applied suddenly.
Beam Length (L): The distance from the fixed support face to the point of load application, in millimeters (mm). Using the clear span rather than the center-to-center distance is critical for accurate moment calculation.
Elastic Modulus (E): Young's modulus, representing the material's stiffness in Megapascals (MPa). For example, structural steel is typically 200,000 MPa, while aluminum is approximately 69,000 MPa.
Moment of Inertia (I): The second moment of area about the neutral axis, in mm⁴. This is a purely geometric property of the cross-section that dictates its resistance to bending.
Distance to Extreme Fiber (c): The perpendicular distance from the neutral axis to the outermost fiber of the cross-section, in mm. For a symmetric section, this is exactly half the total height.

5. Engineering Interpretation

The calculated maximum bending moment dictates the required section modulus (S = I / c) to prevent yielding. The maximum bending stress (σ_max) must be compared against the material's allowable stress, which incorporates an appropriate factor of safety. Furthermore, the maximum deflection (δ_max) must be checked against serviceability limits (e.g., L/180 or L/240) to prevent aesthetic issues, cracking of attached finishes, or user discomfort. A high deflection result does not necessarily mean the beam will break, but it may render the structure unfit for its intended purpose.

6. Units and Unit Consistency

This calculator enforces a consistent metric unit system to prevent catastrophic calculation errors. Forces must be in Newtons (N), lengths in millimeters (mm), modulus in Megapascals (MPa), and inertia in mm⁴. Note that 1 MPa = 1 N/mm². Therefore, the resulting bending stress naturally outputs in MPa, and deflection outputs in mm. If your load is given in kilonewtons (kN), multiply by 1,000 before entering it.

7. Alternative and General Cases

This tool is specifically optimized for a single point load at the free end. In real-world scenarios, loads may be distributed. For a uniformly distributed load (UDL) of intensity w (N/mm) across the entire length, the maximum moment becomes M_max = w × L² / 2, and the maximum deflection becomes δ_max = (w × L⁴) / (8 × E × I). If the point load is applied at an intermediate distance a from the fixed support (rather than the free end), the maximum moment remains P × a, but the deflection formula changes significantly, as the segment beyond the load remains straight but rotates.

8. Assumptions and Limitations

The formulas used in this calculator rely on several standard engineering assumptions:

Linear Elasticity: The material obeys Hooke's Law, and stresses remain below the proportional limit.
Small Deflections: The beam's deflection is small compared to its length, meaning the original geometry is used for equilibrium equations (no P-Delta effects).
Prismatic Beam: The cross-sectional area and moment of inertia (I) are constant along the entire length.
Negligible Self-Weight: The weight of the beam itself is not included in the point load P. If self-weight is significant, it must be calculated separately as a UDL and superimposed.
Rigid Support: The fixed end is assumed to be perfectly rigid, with zero rotation or settlement.

9. Practical Engineering Applications

Cantilever beam analysis is ubiquitous across multiple engineering disciplines. In civil and structural engineering, it models balconies, building canopies, retaining walls, and cantilevered roof trusses. In mechanical engineering, it represents tool holders on CNC machines, aircraft wing spars (simplified), crane jibs, and mounting brackets for heavy equipment. Accurate calculation of the fixed-end moment is especially critical in these applications, as it is the primary location for potential fatigue cracking or connection failure.

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

Balcony and Canopy Failures

Several residential balcony collapses have been traced to underestimation of the fixed-end moment and inadequate reinforcement or connection detailing at the wall or beam support. The high moment concentration at the root was not properly designed for.

Engineering Lesson

The fixed end of a cantilever is a critical section. Both the moment capacity and the connection (or embedment) must be verified with appropriate safety factors and construction quality control.

Overhanging Sign and Billboard Structures

Large cantilevered advertising structures have failed under wind load when the design used only static gravity loads. The combination of self-weight moment and wind-induced moment at the fixed base exceeded capacity.

Engineering Lesson

Cantilevers are sensitive to both gravity and lateral loads. Always include environmental loads (wind, snow, ice) and dynamic effects when analysing outdoor cantilever structures.

Frequently Asked Questions

What does this calculator compute?

It computes the fixed-end reaction, maximum shear force, maximum bending moment, maximum deflection and (optionally) maximum bending stress for a cantilever beam with a point load at the free end.

What are the governing formulas?

R = P, V_max = P, M_max = PL, δ_max = PL³/(3EI), σ_max = Mc/I.

Where do the maximum values occur?

Maximum shear and moment occur at the fixed end. Maximum deflection and slope occur at the free end.

What units should I use?

N for load, mm for length and c, MPa for elastic modulus, mm⁴ for moment of inertia. Results are given in N, N·mm, mm and MPa.

Can I use this for uniform distributed load?

No. This calculator is limited to a single concentrated load at the free end. For uniform load the formulas change (M_max = wL²/2, δ_max = wL⁴/8EI).