Mechanical Engineering Calculator

Simply Supported Beam Calculator

Calculate reactions, shear force, bending moment, deflection and bending stress for a simply supported beam with a central point load — all in one place.

Beam Configuration

Simply supported beam with a central point load — complete static and deflection analysis

P = 5000 NDeflected shapeL = 2000 mmPinnedRoller

Input Parameters

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

N

Central point load applied to the beam.

mm

Distance between the two supports.

MPa

Young's modulus of the beam material.

mm⁴

Area moment of inertia about the bending axis.

mm

Optional – required only for bending stress.

mm

User-defined maximum allowable deflection.

MPa

User-defined maximum allowable bending stress.

Engineering Tip

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

Maximum Deflection

2.778mm

Deflection / Span = 1 / 720

Reactions

2,500.0 N

Max Shear

2,500.0 N

Max Moment

2,500,000 N·mm

Max Stress

83.333 MPa

PASS — Deflection is within the allowable limit

Calculated deflection is 2.778 mm vs limit of 5 mm.

PASS — Stress is within the allowable limit

Calculated stress is 83.333 MPa vs limit of 150 MPa.

Governing Formulas

RA = RB = P / 2
Vmax = P / 2
Mmax = P L / 4
δmax = P L³ / (48 E I)
σmax = M c / I

Calculation Assumptions

  • Simply supported beam
  • Central point load 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 simply_supported_beam(P, L, E, I, c=None):
    """
    Complete analysis of a simply supported beam
    with a central point load.

    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 / 2
    shear = P / 2
    moment = (P * L) / 4
    deflection = (P * L**3) / (48 * 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 = simply_supported_beam(P, L, E, I, c)
for k, v in results.items():
    print(f"{k}: {v}")
MATLAB
function results = simply_supported_beam(P, L, E, I, c)
    % Complete analysis of a simply supported beam
    % with a central point load.
    %
    % 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 / 2;
    results.shear = P / 2;
    results.moment = (P * L) / 4;
    results.deflection = (P * L^3) / (48 * 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 = simply_supported_beam(P, L, E, I, c);
disp(r);
Excel Formulas
Reactions: =P/2
Shear: =P/2
Moment: =(P*L)/4
Deflection: =(P*L^3)/(48*E*I)
Stress: =(M*c)/I

Example Calculation

Beam with P = 5,000 N, L = 2,000 mm, E = 200,000 MPa, I = 1,500,000 mm⁴, c = 50 mm:

RA = RB = 2,500 N
Vmax = 2,500 N
Mmax = 2,500,000 N·mm
δmax = 1.389 mm
σmax = 83.333 MPa

Technical Explanation: Simply Supported Beam Analysis

A simply supported beam is a fundamental structural element supported by a pin (or hinge) at one end and a roller at the other. This configuration permits rotation at both supports and horizontal translation at the roller, rendering the system statically determinate and analytically tractable. When a single concentrated point load acts at mid-span, the internal shear force, bending moment, deflection, and bending stress can be computed using closed-form equations derived from static equilibrium and Euler-Bernoulli beam theory.

Governing Formulas and Derivation

For a simply supported beam of span L subjected to a central point load P, the support reactions are obtained from vertical force equilibrium and symmetry:

ΣFy = 0 ⇒ RA + RB = P By symmetry: RA = RB = P / 2

The shear force diagram consists of two constant segments: +P/2 from the left support to mid-span, and −P/2 from mid-span to the right support. The maximum absolute shear force is therefore:

Vmax = P / 2

The bending moment at any section x from the left support (0 ≤ x ≤ L/2) is M(x) = RA · x = (P/2) · x. The maximum moment occurs at mid-span (x = L/2):

Mmax = (P / 2) × (L / 2) = P L / 4

Under the assumptions of linear elasticity, small deformations, and prismatic cross-section, the maximum deflection at mid-span is given by the standard Euler-Bernoulli beam formula:

δmax = P L³ / (48 E I)

where E is the elastic (Young's) modulus and I is the second moment of area (area moment of inertia) about the neutral axis. The maximum bending stress at the extreme fiber is:

σmax = Mmax · c / I = (P L / 4) × c / I

where c is the perpendicular distance from the neutral axis to the outermost fiber (e.g., half the beam depth for a symmetric section).

Input Parameters and Physical Meaning

Each input to this calculator corresponds to a measurable mechanical or geometric property:

  • Point load (P): The concentrated force applied at mid-span, in Newtons (N). Represents a machine load, equipment weight, or localized live load.
  • Beam length (L): The span between supports, in millimeters (mm). Defines the lever arm for moment and the cubic term in deflection.
  • Elastic modulus (E): Material stiffness, in MPa (N/mm²). Typical values: 200,000 MPa for structural steel, 70,000 MPa for aluminum, 10,000–30,000 MPa for timber.
  • Moment of inertia (I): Geometric property of the cross-section, in mm⁴. For a rectangle: I = b h³ / 12; for a circle: I = π d⁴ / 64.
  • Distance to extreme fiber (c): Half the section depth for symmetric sections, in mm. Required only for bending stress calculation.

Common mistakes include entering length in meters instead of millimeters (causing deflection errors by a factor of 10⁹), using inconsistent units for E and I, or omitting c when stress output is expected.

Engineering Interpretation of Results

The computed quantities serve distinct roles in beam design and verification:

  • Support reactions (RA, RB): Used to design bearings, columns, or foundation connections that must resist the transferred load.
  • Maximum shear force (Vmax): Checked against allowable shear stress or shear capacity of the section, particularly for short, deep beams or materials weak in shear (e.g., timber).
  • Maximum bending moment (Mmax): The primary driver of bending stress and section selection. Compared against the section's moment capacity (Mn / Ω or φ Mn per LRFD/ASD).
  • Maximum deflection (δmax): A serviceability limit state. Excessive deflection can cause cracking of finishes, vibration discomfort, or functional impairment. Typical limits: L/360 for floors, L/240 for roofs (varies by code).
  • Maximum bending stress (σmax): Compared to allowable bending stress (Fb / Ω) or yield strength (Fy) with appropriate safety factors. Does not alone constitute a complete design check.

A beam may satisfy strength criteria (stress and moment) yet fail serviceability (deflection or vibration). Both limit states must be evaluated per applicable codes (AISC 360, Eurocode 3, ASCE 7, etc.).

Units and Unit Consistency

This calculator operates in a consistent SI-derived unit system:

  • Force: Newtons (N)
  • Length: millimeters (mm)
  • Elastic modulus: MPa = N/mm²
  • Moment of inertia: mm⁴
  • Moment: N·mm
  • Deflection: mm
  • Stress: MPa = N/mm²

Key conversions: 1 kN = 1,000 N; 1 m = 1,000 mm; 1 N·m = 1,000 N·mm; 1 GPa = 1,000 MPa. Mixing units (e.g., L in meters, I in mm⁴) will produce erroneous results by orders of magnitude.

Assumptions and Limitations

The closed-form solutions implemented in this tool rely on several explicit assumptions:

  • Support conditions: Ideal pin and roller with no rotational restraint. Real connections may provide partial fixity, altering moments and deflections.
  • Loading: Single concentrated load at exact mid-span. Off-center loads, multiple point loads, or distributed loads require different formulas.
  • Material behavior: Linear elastic, homogeneous, isotropic material obeying Hooke's law. Plasticity, creep, and nonlinear stress-strain are not considered.
  • Beam theory: Euler-Bernoulli assumptions: plane sections remain plane, small deflections (δ ≪ L), prismatic cross-section, and negligible shear deformation. Short, deep beams may require Timoshenko beam theory.
  • Self-weight: Beam self-weight is not included. For heavy sections, add distributed load due to self-weight separately.
  • Lateral-torsional buckling: Not considered. Slender beams in bending may fail by lateral buckling before reaching yield stress.

This calculator is intended for preliminary sizing, educational demonstration, and quick verification of idealized cases. Final design must account for load combinations, safety factors, connection details, stability, fatigue, and code compliance.

Alternative and General Cases

Real-world beams often deviate from the idealized mid-span point load case:

  • Off-center point load: Maximum moment becomes P a b / L, where a and b are distances from the load to the supports (a + b = L).
  • Uniformly distributed load (w): Mmax = w L² / 8; δmax = 5 w L⁴ / (384 E I).
  • Multiple point loads: Superposition of individual load effects or use of shear/moment diagrams.
  • Fixed or continuous supports: Indeterminate structures requiring moment distribution, slope-deflection, or matrix methods.
  • Non-prismatic beams: Variable I(x) requires integration or numerical methods (e.g., finite element analysis).

This calculator addresses only the special case of a simply supported beam with a single central point load. For other configurations, consult structural analysis references or use general-purpose beam analysis software.

Practical Engineering Applications

Typical applications of this simply supported beam calculator include:

  • Machine frames: Sizing support beams for presses, conveyors, or lifting equipment with concentrated loads.
  • Structural members: Preliminary design of floor beams, bridge girders, or roof purlins under point loads from secondary members.
  • Test rigs and fixtures: Verifying deflection and stress in laboratory loading frames with calibrated point loads.
  • Educational examples: Demonstrating shear force, bending moment, and deflection concepts in mechanics of materials courses.

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Real-World Engineering Cases

The Hyatt Regency Walkway Collapse (1981)

A mid-construction change doubled the load on the connections supporting the walkways. The altered load path and resulting moments and shears were never fully re-analyzed, leading to progressive collapse and 114 fatalities.

Engineering Lesson

Any modification to supports, connections or load paths requires a complete re-evaluation of reactions, shear, moment and deflection. A ‘minor’ detail change can invalidate the entire structural model.

Serviceability Failures in Long-Span Floor Beams

Several office buildings experienced excessive floor vibrations and visible sagging under normal occupancy. Strength checks (stress and moment) had passed, but deflection and natural frequency limits had been ignored during design.

Engineering Lesson

A complete beam check must include both strength (shear, moment, stress) and serviceability (deflection, vibration). Passing one does not guarantee the other.

Frequently Asked Questions

What does this calculator compute?

It computes support reactions, maximum shear force, maximum bending moment, maximum deflection and (optionally) maximum bending stress for a simply supported beam with a central point load.

What are the governing formulas?

R = P/2, V_max = P/2, M_max = PL/4, δ_max = PL³/(48EI), σ_max = Mc/I.

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 distributed loads or off-center point loads?

No. This calculator is limited to a single concentrated load acting exactly at mid-span. Other load cases require different equations.

Why are both strength and deflection results shown?

A beam can be strong enough (stress and moment within limits) yet still deflect excessively. Both strength and serviceability must be checked.

Engineering calculations provided by this tool are for educational and preliminary design purposes. Always verify calculations, loading conditions, material properties, applicable standards, safety factors, and design requirements before using results in a final engineering design.