Civil Engineering Calculator

Simply Supported Beam Deflection Calculator

Calculate the maximum vertical deflection at the center of a simply supported beam subjected to a central point load.

Beam Configuration

Simply supported beam with a central point load

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.

Elastic Modulus (E)

MPa

Young's modulus of the beam material.

Moment of Inertia (I)

mm⁴

Area moment of inertia about the bending axis.

Allowable Deflection (δallow)

User-defined maximum allowable deflection.

Engineering Tip

Always verify your theoretical calculations with physical testing and standard safety factors before real-world application.

Maximum Deflection

2.778mm

Deflection / Span

1 / 720

Allowable Limit

5 mm

PASS — Deflection is within the allowable limit

Calculated deflection is 2.778 mm compared with the user-defined limit of 5 mm.

Governing Formula

δ = P L³ / (48 E I)

δmm

Maximum beam deflection

Central point load

Lmm

Beam length

EMPa

Elastic modulus

Imm⁴

Area moment of inertia

Calculation Assumptions

Simply supported beam
Central point load
Linear elastic behavior
Small deflection assumption
Constant elastic modulus
Constant cross-section

Engineering Code

Reuse the calculation in your own engineering workflow.

Python

def beam_deflection(P, L, E, I):
    """
    Calculate maximum deflection of a simply supported beam
    with a central point load.

    P: Load (N)
    L: Beam length (mm)
    E: Elastic modulus (MPa = N/mm²)
    I: Area moment of inertia (mm⁴)

    Returns:
        Maximum deflection (mm)
    """

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

    return (P * L**3) / (48 * E * I)


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

delta = beam_deflection(P, L, E, I)

print(f"Maximum Deflection: {delta:.3f} mm")

MATLAB

function delta = beam_deflection(P, L, E, I)
    % Maximum deflection 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)

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

    delta = (P * L^3) / (48 * E * I);
end

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

delta = beam_deflection(P, L, E, I);

fprintf('Maximum Deflection: %.3f mm\n', delta);

Excel Formula

=(P*L^3)/(48*E*I)

Example Calculation

For a beam subjected to a 5,000 N central point load, with a length of 2,000 mm, an elastic modulus of 200,000 MPa, and a second moment of area of 1,500,000 mm⁴:

δ = (5000 × 2000³) / (48 × 200000 × 1500000)

δ = 1.389 mm

Technical Guide: Simply Supported Beam Deflection

Beam deflection represents the transverse displacement of a structural or mechanical member under transverse loading. In structural engineering and mechanical design, understanding and calculating maximum deflection is essential for serviceability design. While stress calculations verify that a beam will not yield or fracture under load, deflection analysis guarantees that the member satisfies functional stiffness limits, preventing excessive sagging, mechanical binding, vibration susceptibility, and damage to adjacent structural elements.

Governing Equation and Elastic Curve

For an idealized, prismatic simply supported beam carrying a single concentrated point load at mid-span, the maximum vertical deflection occurs directly at the center of the span ($x = L/2$). Based on classical Euler-Bernoulli beam theory, this maximum central displacement is defined by:

δ_max = (P · L³) / (48 · E · I)

Where:

P: Concentrated point load applied at the midpoint (N).
L: Unsupported beam span between support centerlines (mm).
E: Modulus of elasticity (Young's modulus) of the beam material (MPa or N/mm²).
I: Second moment of area (area moment of inertia) about the bending axis (mm⁴).
δ_max: Maximum vertical downward deflection at mid-span (mm).

Derivation via Double Integration

The governing differential equation for the elastic curve of a beam in pure bending is given by:

E · I · (d²v / dx²) = M(x)

For a symmetric simply supported beam of span L loaded with a central point load P, static equilibrium yields reaction forces at both end supports equal to R_A = R_B = P / 2. Considering the beam segment from the left support to the center ($0 \le x \le L/2$), the internal bending moment is M(x) = (P / 2) · x.

Integrating the moment equation once yields the slope equation:

E · I · (dv / dx) = (P · x² / 4) + C₁

Because of geometric and loading symmetry, the slope at the beam midpoint is zero ($dv/dx = 0$ at $x = L/2$). Substituting this boundary condition solves the first integration constant: C₁ = - (P · L² / 16).

Integrating a second time gives the deflection equation:

E · I · v(x) = (P · x³ / 12) - (P · L² · x / 16) + C₂

At the pinned support ($x = 0$), deflection is constrained to zero ($v(0) = 0$), which sets C₂ = 0. Evaluating the deflection curve at mid-span ($x = L/2$) yields v(L/2) = - (P · L³ / 48 · E · I). The negative sign designates downward displacement, matching the standard scalar formula implemented in this calculator.

Input Parameters and Sensitivity Analysis

A critical insight revealed by the formula is parameter sensitivity:

Span Length Sensitivity (L³): Deflection is proportional to the third power of unsupported span. Doubling the span length increases mid-span deflection by an factor of eight ($2³ = 8$), assuming load and cross-section remain constant. Span measurement errors or span expansions produce drastic stiffness penalties.
Flexural Rigidity (E · I): Deflection varies inversely with the product of material stiffness ($E$) and cross-sectional geometry ($I$). Selecting an I-beam or rectangular profile oriented along its strong axis exponentially increases $I$ without adding proportional material weight ($I = b · h³ / 12$ for rectangles).
Common Input Pitfalls: Entering span length in meters rather than millimeters or entering modulus in GPa instead of MPa will produce output errors of several orders of magnitude. Ensure all inputs are unified into consistent N and mm base units.

Engineering Interpretation and Serviceability Limits

Calculating deflection is only one step within a comprehensive structural check. Engineers distinguish between:

Ultimate Limit State (Strength): Verifies that the maximum bending stress ($\sigma = M / Z$) and shear stress do not exceed the yield strength or allowable working stress of the material, incorporating an adequate factor of safety.
Serviceability Limit State (Stiffness): Verifies that deformation under working (unfactored) loads remains within functional tolerances. Standard structural codes define acceptable deflection thresholds based on span fraction:
- L / 250: General beams and machinery baseplates to avoid noticeable visual sag.
- L / 360: Commercial floors supporting plaster ceilings to prevent cracking.
- L / 500 to L / 1000: Precision machine ways, crane runways, and shafting to prevent gear misalignment and bearing race binding.

Assumptions and Boundary Conditions

The validity of this calculator relies strictly on the assumptions of Euler-Bernoulli beam theory:

Small Deflections: Displacements and rotations are small enough that the original geometry and coordinate systems remain valid ($dv/dx \ll 1$).
Linear Elastic Material: The material obeys Hooke's Law, remains isotropic and homogeneous, and does not exceed the proportional limit.
Slender Beam Geometry: The length-to-depth ratio exceeds approximately 10:1 ($L/h \ge 10$), meaning shear deformation (Timoshenko effect) is negligible.
Ideal Pin/Roller Supports: Supports restrict vertical displacement completely while permitting free longitudinal translation and rotational slope.
Symmetric Loading: The concentrated point load acts vertically through the shear center of the cross-section, inducing bending without torsion.
Self-Weight Excluded: The calculated value accounts solely for the concentrated external load. For heavy spans, the deflection due to distributed beam self-weight ($5 w L⁴ / 384 E I$) must be superimposed.

Alternative and General Loading Conditions

In real engineering applications, beams frequently experience more complex boundary configurations than a single central load:

Off-Center Point Load: If load $P$ is placed at distance $a$ from the left support and $b$ from the right, the deflection under the load becomes $\delta = P a² b² / (3 E I L)$, and maximum deflection shifts away from mid-span.
Uniformly Distributed Load (UDL): A beam carrying load $w$ per unit length exhibits maximum central deflection of $\delta = 5 w L⁴ / (384 E I)$.
Fixed-End Supports: Clamping the ends against rotation decreases maximum central deflection by 75% for a central point load ($\delta = P L³ / 192 E I$).

This calculator addresses specifically the simply supported, centrally loaded case. Always verify that actual physical boundary conditions match the theoretical model before applying these results to production designs.

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)

During a dance competition, two suspended walkways collapsed, killing 114 people. A subtle, unapproved mid-construction design change doubled the stress on the box beams and hanger rod connections, fundamentally altering the load path and deflection characteristics.

Engineering Lesson

Never alter a beam's loading or support configuration without completely recalculating the shear, moment, and maximum allowable deflection. Even a 'minor' connection change can invalidate the entire structural model.

Excessive Deflection in Heavy Machinery Frames

In a prototype electric tow tractor design, engineers initially underestimated the sustained central point load from a high-capacity battery bank. The resulting mid-span beam deflection caused the drive axles to misalign under dynamic loading.

Engineering Lesson

Always account for dynamic amplification factors. Calculating the theoretical static deflection is only the first step; you must ensure the structural stiffness (I) is robust enough to prevent mechanical binding under operational stress.

Frequently Asked Questions

What is the formula for simply supported beam deflection?

For a simply supported beam with a central point load, the maximum deflection is δ = PL³/(48EI).

Where does maximum deflection occur?

For a symmetric simply supported beam with a central point load, the maximum deflection occurs at the center of the span.

What units should I use?

This calculator uses N for load, mm for length, MPa for elastic modulus, and mm⁴ for the area moment of inertia. The resulting deflection is given in mm.

Can this calculator be used for every beam?

No. This equation applies specifically to a simply supported beam with a central point load under the stated linear-elastic assumptions. Different support conditions or loading configurations require different beam equations.