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Preliminary estimate for concept-stage planning only. NOT a substitute for design by a licensed Professional Engineer (PE) / registered structural engineer. All structural designs must be independently verified and stamped per local code before construction.

Beam Bending Moment, Shear & Deflection Calculator

Q: How is maximum bending moment calculated for a simply supported beam with UDL?

For a simply supported beam of span L under a uniform load w per unit length, maximum bending moment occurs at mid-span and equals Mmax = w × L² / 8.

Q: What is the maximum shear force for a uniformly loaded simply supported beam?

The maximum vertical shear force occurs at the support reactions and equals Vmax = w × L / 2.

Q: What formula calculates elastic beam deflection?

Mid-span deflection for a uniform load is δ_max = (5 × w × L⁴) / (384 × E × I), where E is Young's modulus and I is the area moment of inertia.

Q: What is bending stress in a beam section?

Flexural bending stress is σ = M / S, where M is the applied moment and S is the elastic section modulus (I / y_c).

Q: What design assumptions are made in this beam calculator?

The calculator assumes a simply supported, prismatic, pin-ended beam under elastic bending behavior without lateral-torsional buckling (LTB) or unbraced length strength reductions.

Q: How does point load bending moment compare to uniform load?

A central point load P produces a maximum mid-span moment of Mmax = P × L / 4 and deflection δ_max = P × L³ / (48 × E × I).

Q: What allowable deflection limits are standard in building codes?

IBC and ASCE 7 typically specify live load deflection limits of L/360 for plastered ceilings, L/240 for roof members, and L/180 for industrial floors.

Q: How do AISC 360 and Saudi SBC 306 beam standards align?

Saudi Building Code SBC 306 directly adopts AISC 360 flexural design principles, specifying identical LRFD and ASD bending stress and deflection criteria.

Q: What is the difference between elastic section modulus S and plastic section modulus Z?

Elastic section modulus S = I / y_c defines first yield of extreme fibers, while plastic section modulus Z defines full plastic hinge yield of the entire cross-section.

Q: What is Young's Modulus E for structural steel vs Douglas Fir timber?

Structural steel (A36 / A992) has E = 29,000,000 psi (200 GPa). Select Structural Douglas Fir timber has E = 1,600,000 to 1,900,000 psi (11 to 13.1 GPa).

Calculate bending moment (wL²/8), vertical shear, and mid-span elastic deflection for simply supported beams per AISC 360, ACI 318, and NDS.

Beam Geometry & Load Inputs

Beam Span Length (L, ft)

Load Configuration

Uniform Load (w, lb/ft)

Central Point Load (P, lbs)

Elastic Modulus (E, ksi) Steel = 29,000 ksi; Concrete = 3,600 ksi

Moment of Inertia (I, in⁴)

Flexural & Shear Analysis Results

Span = 20 ft

Maximum Bending Moment (Mmax)

50,000 lb·ft

50.00 kip·ft

Maximum Support Shear Force (Vmax): 10,000 lb

Maximum Mid-Span Deflection (δmax): 1.241 in

Deflection Span Ratio: L / 193

Bending Moment & Deflection Visualization

🧮 Step-by-Step Worked Example: W12x26 Steel Beam Flexural Analysis

Consider a simply supported W12x26 structural steel beam across a 20.00 foot span (240 inches), supporting a uniform dead+live load w of 600 lbs/ft (50 lbs/in) and a central point load P of 2,000 lbs. Material E = 29,000,000 psi, Section Modulus S = 33.4 in³, Moment of Inertia I = 204 in⁴.

Mathematical Derivation Steps:

Maximum Bending Moment (Mmax):
M_udl = w × L² / 8 = (600 × 20²) / 8 = 30,000 lb-ft = 360,000 lb-in
M_point = P × L / 4 = (2,000 × 20) / 4 = 10,000 lb-ft = 120,000 lb-in
M_max = 30,000 + 10,000 = 40,000 lb-ft (480,000 lb-in)
Maximum Vertical Shear Force (Vmax):
V_max = (w × L / 2) + (P / 2) = (600 × 20 / 2) + (2,000 / 2) = 6,000 + 1,000 = 7,000 lbs
Flexural Bending Stress (σ):
σ = M_max / S = 480,000 lb-in / 33.4 in³ = 14,371 psi (14.37 ksi) (Well within 36 ksi yield)
Mid-Span Elastic Deflection (δ):
δ_udl = 5 × w × L⁴ / (384 × E × I) = 5 × 50 × 240⁴ / (384 × 29,000,000 × 204) = 0.365 in
δ_point = P × L³ / (48 × E × I) = 2,000 × 240³ / (48 × 29,000,000 × 204) = 0.097 in
δ_total = 0.365 + 0.097 = 0.462 inches
Allowable L/360 limit = 240 / 360 = 0.667 inches (0.462" < 0.667" $\Rightarrow$ Compliant)

Design Criteria	Code Limit (AISC 360 / IBC)	Calculated Value	Informational Comparison
Bending Stress (σ)	ASD Fy / 1.67 = 21.6 ksi (A36 Steel)	14.37 ksi	Below Allowable Stress Limit
Floor Deflection (δ)	IBC Max L/360 = 0.667 in	0.462 in	Within L/360 Limit
Roof Deflection (δ)	IBC Max L/240 = 1.000 in	0.462 in	Within L/240 Limit
Web Crippling & Bearing	AISC 360 Section J10.2 / J10.3	7,000 lbs Reaction	Verify bearing length N at supports

Design Criteria

Code Limit (AISC 360 / IBC)

Calculated Value

Informational Comparison

Bending Stress (σ)

ASD Fy / 1.67 = 21.6 ksi (A36 Steel)

14.37 ksi

Below Allowable Stress Limit

Floor Deflection (δ)

IBC Max L/360 = 0.667 in

0.462 in

Within L/360 Limit

Roof Deflection (δ)

IBC Max L/240 = 1.000 in

0.462 in

Within L/240 Limit

Web Crippling & Bearing

AISC 360 Section J10.2 / J10.3

7,000 lbs Reaction

Verify bearing length N at supports

Frequently Asked Questions (FAQ) — Structural Beam Bending & Deflection

How is maximum bending moment calculated for a simply supported beam with UDL?

For a simply supported beam of span L under a uniform load w per unit length, maximum bending moment occurs at mid-span and equals Mmax = w × L² / 8.

What is the maximum shear force for a uniformly loaded simply supported beam?

The maximum vertical shear force occurs at the support reactions and equals Vmax = w × L / 2.

What formula calculates elastic beam deflection?

Mid-span deflection for a uniform load is δ_max = (5 × w × L⁴) / (384 × E × I), where E is Young's modulus and I is the area moment of inertia.

What is bending stress in a beam section?

Flexural bending stress is σ = M / S, where M is the applied moment and S is the elastic section modulus (I / y_c).

What design assumptions are made in this beam calculator?

The calculator assumes a simply supported, prismatic, pin-ended beam under elastic bending behavior without lateral-torsional buckling (LTB) or unbraced length strength reductions.

How does point load bending moment compare to uniform load?

A central point load P produces a maximum mid-span moment of Mmax = P × L / 4 and deflection δ_max = P × L³ / (48 × E × I).

What allowable deflection limits are standard in building codes?

IBC and ASCE 7 typically specify live load deflection limits of L/360 for plastered ceilings, L/240 for roof members, and L/180 for industrial floors.

How do AISC 360 and Saudi SBC 306 beam standards align?

Saudi Building Code SBC 306 directly adopts AISC 360 flexural design principles, specifying identical LRFD and ASD bending stress and deflection criteria.

What is the difference between elastic section modulus S and plastic section modulus Z?

Elastic section modulus S = I / y_c defines first yield of extreme fibers, while plastic section modulus Z defines full plastic hinge yield of the entire cross-section.

What is Young's Modulus E for structural steel vs Douglas Fir timber?

Structural steel (A36 / A992) has E = 29,000,000 psi (200 GPa). Select Structural Douglas Fir timber has E = 1,600,000 to 1,900,000 psi (11 to 13.1 GPa).

What causes lateral-torsional buckling (LTB) in I-beams?

LTB occurs when the unbraced length of a beam's compression flange twists sideways and rotates under bending before reaching full flexural capacity.

How do you calculate area moment of inertia I for a rectangular beam?

For a rectangular section of width b and depth d, the moment of inertia about the neutral axis is I = b × d³ / 12.

What is the shear stress formula for a rectangular timber or steel beam?

Maximum transverse shear stress for a rectangular section occurs at the neutral axis and equals τ_max = 1.5 × V / A, where V is total shear force and A is cross-sectional area.

What is the difference between ASD and LRFD design methods?

ASD (Allowable Strength Design) compares nominal strength divided by a safety factor (Ω = 1.67) to service loads, whereas LRFD applies resistance factors (φ = 0.90) to factored load combinations (1.2D + 1.6L).

How do continuous multi-span beams compare to simply supported beams?

Continuous multi-span beams redistribute bending moments, reducing mid-span positive moments while inducing negative bending moments over intermediate supports.

What self-weight allowance should be added to beam load calculations?

Beam self-weight (lbs/ft or kg/m) must be added to dead loads. For W-shape steel beams, the weight is given in the shape designation (e.g. W12x26 weighs 26 lbs/ft).

What web crippling and web yielding checks are required at supports?

Concentrated support reactions can cause local web buckling or yielding. Bearing stiffeners or minimum bearing length N per AISC 360 Chapter J must be verified.

Why is L/360 the standard deflection limit for residential floor joists?

L/360 restricts mid-span floor deflection to prevent cracking in brittle ceiling plaster/drywall and eliminate noticeable floor bounciness under foot traffic.

Sources & Governing Codes

AISC 360 Specification for Structural Steel Buildings: AISC 360 Specification for Structural Steel Buildings View Standard

AISC Steel Construction Manual 15th Edition: AISC Steel Construction Manual 15th Edition View Standard

IBC 2021 Section 1609 Structural Wind Loads: IBC 2021 Section 1609 Structural Wind Loads View Standard

IRC 2021 Section R301 Structural Design Criteria: IRC 2021 Section R301 Structural Design Criteria View Standard

AWC National Design Specification (NDS) for Wood Construction: AWC National Design Specification (NDS) for Wood Construction View Standard

ACI 318 Building Code Requirements for Structural Concrete: ACI 318 Building Code Requirements for Structural Concrete View Standard

AWS D1.1 Structural Welding Code — Steel: AWS D1.1 Structural Welding Code — Steel View Standard

Beam Bending Moment, Shear & Deflection Calculator

Beam Geometry & Load Inputs

Flexural & Shear Analysis Results

Bending Moment & Deflection Visualization

🧮 Step-by-Step Worked Example: W12x26 Steel Beam Flexural Analysis

Mathematical Derivation Steps:

📋 Informational Structural Code Check (AISC 360 / NDS Criteria)

🛠️ Steel & Timber Beam Installation Equipment, PPE & Pitfalls

Erection & Measurement Tools

Required Personal Safety (PPE)

Critical Structural Failure Pitfalls

Related Structural & Framing Calculators

Frequently Asked Questions (FAQ) — Structural Beam Bending & Deflection

Sources & Governing Codes