2025-08-21 14 min

Simply Supported Beam Calculator

Calculate deflection, slope, moment, and stress for simply supported beams with point load or UDL.

Calculator

Unit System

Load Type

Point Load (P)(N)

Beam Span (L)(mm)

Young Modulus (E)(MPa)

MPa

Moment of Inertia (I)(mm^4)

mm^4

Section Width (b)(mm)

Section Height (h)(mm)

Decimal Precision

Results update live as you type

Results

Normal

Max Deflection (midspan)

0.42mm

Max Slope (supports)

0.00rad

Max Moment (midspan)

5000000.00N*mm

Max Stress

3.13MPa

Support Reaction

2500.00N

Visualizations

Results Overview

0.4mmMax Deflection

5,000,000mmMax Moment

3.1mmMax Stress

2,500mmSupport Reaction

Formula

Simply supported beam formulas for central point load and UDL. Maximum values at midspan.

Point load: delta = PL^3/(48EI), M = PL/4

Alternative Form

UDL: delta = 5wL^4/(384EI), M = wL^2/8

Variables

Symbol	Name	Unit	Description
delta	Maximum Deflection	mm	Deflection at midspan
theta	Maximum Slope	radians	Slope at supports
M	Maximum Moment	N*mm	Moment at midspan
sigma	Maximum Stress	MPa	Bending stress at midspan
P	Point Load	N	Concentrated force at center
w	UDL Intensity	N/mm	Distributed load per unit length
L	Beam Span	mm	Distance between supports
E	Young Modulus	MPa	Material stiffness
I	Moment of Inertia	mm^4	Section property

Explanation

A simply supported beam has pin and roller supports. Max deflection and moment at midspan. Max slope at supports.

Derivation

From EI*y = M(x). For central point load M = Px/2. Integrating with boundary conditions y(0) = y(L) = 0.

Step-by-Step Calculation

Follow each step to understand how the result is calculated from your inputs.

Step 1

Identify loading type (point load or UDL)

Result: Loading determined

Step 2

Calculate section properties I and c

Result: Section properties known

Step 3

Compute deflection at midspan

Result: Deflection calculated

Step 4

Compute slope at supports

Result: Slope calculated

Step 5

Compute moment and stress at midspan

Result: Stress and moment calculated

Step 6

Verify against allowable limits

Result: Design check complete

Final Answer

0.42

Worked Examples

See how the calculator works with different scenarios.

Scenario

4 m timber joist (E=12 GPa, I=80E6 mm^4) with 5 kN center load.

Inputs

Load:5000 N

Span:4000 mm

Calculation

delta = 5000*4000^3/(48*12000*80E6) = 5.21 mm.

Result

delta = 5.21 mm

Interpretation

Within L/360 limit.

Understanding Your Result

Your Result: Max Deflection (midspan): 0.42 mm

Typical range: Floors: L/240-L/360. Roofs: L/180-L/240. Bridges: L/600-L/800.

What This Means

Simply supported beams are the most common structural element, supported at both ends.

Why It Matters

Fundamental to structural design of buildings, bridges, and mechanical systems.

Professional Interpretation

Check both stress and deflection. Deflection often governs for long spans.

Recommendations

•Increase depth for deflection reduction
•Check midspan values
•Verify support reactions

Educational Content

Introduction

Simply supported beams are the most fundamental structural configuration.

Concept Explanation

Pin and roller supports allow rotation. Max moment and deflection at midspan, max slope at supports.

Applications

▸Floor framing

▸Bridge spans

▸Machine beds

▸Conveyor frames

▸Stadium seating

Benefits

•Simple to construct and analyze
•Accommodates thermal expansion
•Most common beam type

Limitations

•Requires two supports per span
•Higher moments than continuous beams
•Deflection can govern

Best Practices

•Check stress and deflection
•Verify bearing reactions
•Consider self-weight

Common Misconceptions

Myth: Zero slope everywhere

Truth: Slope is zero only at midspan for symmetric loading.

Practical Tips

•Midspan has max deflection and moment
•Supports have max slope
•Doubling depth reduces deflection by 8x

Common Mistakes

Ignoring self-weight

Add self-weight UDL to applied loads.

Wrong moment formula

Use PL/4 for point, wL^2/8 for UDL.

Assumptions

This calculator makes the following assumptions to provide estimates:

Elastic Behavior: Linear elastic range.

Small Deflections: Small relative to span.

Limitations

Be aware of these limitations when using this calculator:

Single Span: Formulas for single-span beams.

Frequently Asked Questions

Answers to common questions about this calculator.

What is a simply supported beam?

A simply supported beam has a pin support at one end and a roller at the other. It is free to rotate at both supports but cannot translate vertically.

What is the deflection formula for a simply supported beam?

For central point load: delta = PL^3/(48EI). For UDL: delta = 5wL^4/(384EI). Maximum deflection at midspan.

What is the maximum moment for a simply supported beam?

For central point load: M = PL/4. For UDL: M = wL^2/8. Both occur at midspan.

What is the slope formula for a simply supported beam?

For central point load: theta = PL^2/(16EI) at supports. For UDL: theta = wL^3/(24EI) at supports.

What are the support reactions?

For symmetric loading: each support carries half the total load. For central point load: R = P/2. For UDL: R = wL/2.

How does span affect deflection?

Deflection scales with L^3 (point load) or L^4 (UDL). Doubling span increases deflection by 8x or 16x.

How does depth affect deflection?

I scales with h^3 for rectangular sections. Doubling depth reduces deflection by 8x.

What is the allowable deflection?

L/240 for floors, L/360 for roofs, L/480 for finishes. Serviceability often governs design.

What is the difference between point load and UDL?

Point load concentrates force at one location. UDL distributes load uniformly. UDL creates higher total load for the same intensity.

What is the moment diagram shape?

Point load: triangular with peak at center. UDL: parabolic with maximum at center.

What is the shear force distribution?

Point load: constant shear in each half, jumping at the load. UDL: linear shear from +wL/2 at left to -wL/2 at right.

How do you calculate deflection for eccentric loads?

For load at distance a from left: delta_max near load point. Use standard formulas from beam tables.

What is the slope at midspan for symmetric loading?

Zero by symmetry. The slope is maximum at the supports and zero at midspan for symmetric loading.

What is the maximum shear force?

For central point load: V = P/2. For UDL: V = wL/2. Maximum shear is at the supports.

How do multiple loads affect deflection?

Use superposition: calculate deflection for each load separately and sum the results.

What is a real-world example of a simply supported beam?

Bridge spans, floor joists, roof rafters, machine tool beds, and conveyor frames commonly use simply supported configurations.

What is the advantage of simply supported beams?

Simple to construct, allow thermal expansion, easy to analyze, and the most common beam configuration in construction.

What is the deflection at the supports?

Zero by definition. The supports prevent vertical translation but allow rotation.

How does material choice affect a simply supported beam?

Material affects E (deflection) and yield strength (allowable stress). Steel deflects less than wood for the same section due to higher E.

What is the relationship between shear and moment?

V = dM/dx. Shear is the slope of the moment diagram. Where shear is zero, moment is maximum.

What is the deflection for a two-point symmetric load?

For two equal loads P at distance a from each support: delta = Pa(3L^2-4a^2)/(24EI) at midspan.

How do you size a simply supported beam?

Calculate required S = M/allowable_stress. Select section with S >= required. Verify deflection.

What is the effect of self-weight on deflection?

Self-weight acts as UDL. For rectangular beam: w = rho*g*b*h. Add self-weight deflection to applied load deflection.

What is a continuous beam vs simply supported?

Continuous beams span multiple supports. They have lower moments and deflections than simply supported beams of the same span.

How do you find the maximum deflection location?

For symmetric loading, maximum deflection is at midspan. For asymmetric loads, it is near but not exactly at midspan.

What is the effect of axial load on a simply supported beam?

Axial compression amplifies bending effects (P-delta). Axial tension reduces deflection. Both modify the deflection formula.

What is the deflection of a simply supported beam with end moments?

For equal end moments M: delta = ML^2/(16EI) at midspan. The deflected shape is a circular arc.

How does the support type affect the beam?

Pin support resists horizontal and vertical forces. Roller resists only vertical force. Both allow rotation. Thermal expansion is accommodated.

What is the effective span for a simply supported beam?

Effective span is the center-to-center distance between supports. For masonry, it may include partial bearing length.

What is the deflection due to temperature?

Uniform temperature causes axial expansion, not deflection. Temperature gradient across the depth causes curvature and deflection.

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Glossary

Technical terms used in mortgage calculations explained in plain language.

Simply Supported: Pin and roller supports. Free to rotate.
Reaction: Force at each support maintaining equilibrium.
Midspan: Center of beam span.

Authoritative References

Trusted sources and standards referenced by this calculator.

Shigleys Mechanical Engineering Design AISC Steel Construction Manual Eurocode 3

Applicable Standards

AISC 360Eurocode 3ACI 318

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Page Information

Last Updated:

2025-08-21

Version:

2.0.0

References:

License:

Free to use for educational and professional purposes.