Beam Deflection Calculator
Professional structural engineering calculator for beam deflection, stress, and moment analysis. Support for multiple beam types, materials, and loading conditions with safety verification.
Beam Deflection Calculator
Professional structural engineering calculator for beam deflection, stress, and moment analysis
Beam Configuration
Simply Supported Beam
Beam supported at both ends, free to rotatePoint Load
Concentrated load at a specific pointStructural Steel S275
E = 200 GPa, fy = 275 MPaRectangular Section
Simple rectangular cross-sectionDimensions
m
Overall beam length
mm
Cross-section width
mm
Cross-section height
Loading Conditions
kN
Concentrated load magnitude
m
Distance from left support (default: center)
Structural Formulas
Essential beam deflection formulas for different support conditions and loading types.
Simply Supported Beam - Point Load at Center
Maximum deflection for point load at beam center
δ = (P × L³) / (48 × E × I)
Variables:
δ = Maximum deflection (mm)
P = Point load (N)
L = Beam length (mm)
E = Elastic modulus (MPa)
I = Moment of inertia (mm⁴)
Applications:
Simply Supported Beam - Distributed Load
Maximum deflection for uniformly distributed load
δ = (5 × w × L⁴) / (384 × E × I)
Variables:
δ = Maximum deflection (mm)
w = Distributed load (N/mm)
L = Beam length (mm)
E = Elastic modulus (MPa)
I = Moment of inertia (mm⁴)
Applications:
Cantilever Beam - Point Load at Free End
Maximum deflection for cantilever with end load
δ = (P × L³) / (3 × E × I)
Variables:
δ = Maximum deflection (mm)
P = Point load (N)
L = Beam length (mm)
E = Elastic modulus (MPa)
I = Moment of inertia (mm⁴)
Applications:
Fixed-Fixed Beam - Point Load at Center
Maximum deflection for fixed ends beam
δ = (P × L³) / (192 × E × I)
Variables:
δ = Maximum deflection (mm)
P = Point load (N)
L = Beam length (mm)
E = Elastic modulus (MPa)
I = Moment of inertia (mm⁴)
Applications:
Beam Types
Different beam support conditions and their structural characteristics.
Simply Supported Beam
Beam resting on two supports with freedom to rotate
Characteristics:
Pin support at one end
Roller support at other end
No moment restraint at supports
Statically determinate
Applications:
Key Formulas:
Moment: M_max = P×L/4 (point load at center)
Deflection: δ_max = P×L³/(48×E×I) (point load at center)
Cantilever Beam
Beam fixed at one end and free at the other
Characteristics:
Fixed support at one end
Free end
Moment restraint at fixed end
Statically determinate
Applications:
Key Formulas:
Moment: M_max = P×L (point load at free end)
Deflection: δ_max = P×L³/(3×E×I) (point load at free end)
Fixed-Fixed Beam
Beam with both ends fully restrained against rotation and translation
Characteristics:
Fixed supports at both ends
No rotation at supports
Moment restraint at both ends
Statically indeterminate
Applications:
Key Formulas:
Moment: M_max = P×L/8 (point load at center)
Deflection: δ_max = P×L³/(192×E×I) (point load at center)
Propped Cantilever
Cantilever beam with additional support at the free end
Characteristics:
Fixed support at one end
Pin/roller support at other end
Statically indeterminate
One degree of redundancy
Applications:
Key Formulas:
Moment: Variable (depends on support conditions)
Deflection: Reduced compared to simple cantilever
Continuous Beam
Beam extending over multiple supports with continuous connection
Characteristics:
Multiple supports
Continuous over intermediate supports
Highly indeterminate
Moment redistribution
Applications:
Key Formulas:
Moment: Negative moments at supports, positive in spans
Deflection: Significantly reduced due to continuity
Material Properties
Mechanical properties of common structural materials used in beam design.
Structural Steel
S275 (Mild Steel)
| E | 200 GPa |
| fy | 275 MPa |
| fu | 430 MPa |
| ρ | 7850 kg/m³ |
Applications:
Characteristics:
Good weldability
Moderate strength
Cost effective
S355 (High Strength Steel)
| E | 200 GPa |
| fy | 355 MPa |
| fu | 510 MPa |
| ρ | 7850 kg/m³ |
Applications:
Characteristics:
High strength
Good toughness
Excellent weldability
S420 (Very High Strength)
| E | 200 GPa |
| fy | 420 MPa |
| fu | 550 MPa |
| ρ | 7850 kg/m³ |
Applications:
Characteristics:
Very high strength
Special welding requirements
Expensive
Concrete
C25/30 (Standard Grade)
| E | 31 GPa |
| fy | 25 MPa (compression) |
| fu | 30 MPa |
| ρ | 2400 kg/m³ |
Applications:
Characteristics:
Standard grade
Good workability
Cost effective
C40/50 (High Strength)
| E | 35 GPa |
| fy | 40 MPa (compression) |
| fu | 50 MPa |
| ρ | 2400 kg/m³ |
Applications:
Characteristics:
High strength
Good durability
Requires quality control
C60/75 (Very High Strength)
| E | 39 GPa |
| fy | 60 MPa (compression) |
| fu | 75 MPa |
| ρ | 2500 kg/m³ |
Applications:
Characteristics:
Very high strength
Special mix design
Expensive
Timber
C24 (Softwood)
| E | 11 GPa |
| fy | 24 MPa |
| fu | 40 MPa |
| ρ | 420 kg/m³ |
Applications:
Characteristics:
Sustainable
Good strength-to-weight
Easy to work
GL32h (Glulam)
| E | 13.6 GPa |
| fy | 32 MPa |
| fu | 32 MPa |
| ρ | 410 kg/m³ |
Applications:
Characteristics:
Engineered timber
Consistent properties
Large sizes available
LVL (Laminated Veneer)
| E | 14 GPa |
| fy | 35 MPa |
| fu | 44 MPa |
| ρ | 500 kg/m³ |
Applications:
Characteristics:
High strength
Dimensional stability
Consistent quality
Deflection Limits
Standard deflection limits for different structural applications and building types.
| Application | Limit | Description |
|---|---|---|
General Building Beams | L/250 | Standard deflection limit for building beams under service loads Reasoning: Prevents damage to finishes, partitions, and provides acceptable visual appearance |
Floors Supporting Partitions | L/350 | Stricter limit for floors supporting non-structural partitions Reasoning: Prevents cracking of partitions and finishing materials |
Roofs (General) | L/200 | Deflection limit for roof beams under service loads Reasoning: Prevents ponding, drainage issues, and damage to roofing materials |
Cantilevers | L/125 | More relaxed limit for cantilever structures Reasoning: Cantilevers naturally have higher deflections, limit prevents excessive sag |
Industrial Floors | L/300 to L/500 | Stringent limits for industrial applications Reasoning: Precision requirements for machinery, equipment operation |
Bridges | L/300 to L/1000 | Variable limits based on bridge type and usage Reasoning: User comfort, dynamic effects, fatigue considerations |
Load Types
Different types of loads considered in structural beam design and analysis.
Dead Loads
Permanent loads that remain constant throughout the life of the structure
Examples:
Self-weight of structure
Fixed equipment
Architectural finishes
MEP systems
Typical Values:
Concrete slab (150mm): 3.6 kN/m²
Steel beam (UB 305×165×40): 0.39 kN/m
Masonry wall (200mm): 4.0 kN/m²
Roofing materials: 0.5-1.5 kN/m²
Design Considerations:
Calculate based on actual dimensions
Include all permanent fixtures
Account for construction tolerances
Live Loads
Variable loads due to occupancy and use of the structure
Examples:
People
Furniture
Equipment
Storage materials
Typical Values:
Residential floors: 1.5-2.0 kN/m²
Office floors: 2.5-3.0 kN/m²
Retail floors: 4.0 kN/m²
Industrial floors: 5.0-15.0 kN/m²
Design Considerations:
Follow local building codes
Consider actual usage patterns
Account for dynamic effects
Wind Loads
Loads caused by wind pressure and suction on the structure
Examples:
Lateral pressure on walls
Uplift on roofs
Dynamic effects
Typical Values:
Basic wind speed: 26-50 m/s
Wind pressure: 0.6-2.0 kN/m²
Exposure categories: Urban, Rural, Coastal
Design Considerations:
Site wind conditions
Building height and shape
Dynamic amplification factors
Snow Loads
Loads due to accumulation of snow on roof surfaces
Examples:
Uniform snow load
Drift loads
Sliding snow
Ice dams
Typical Values:
Ground snow load: 0.5-4.0 kN/m²
Roof snow load: 0.7×ground load
Drift surcharge: Variable
Design Considerations:
Local climate conditions
Roof geometry and slope
Heat loss effects
Seismic Loads
Loads due to earthquake ground motion and structural response
Examples:
Base shear
Story forces
Overturning moments
Typical Values:
Seismic zone factors: 0.1-0.4
Structural response factors: 1.5-8.0
Site coefficients: 1.0-2.0
Design Considerations:
Local seismic hazard
Soil conditions
Structural system type
Design Considerations
Key engineering considerations for safe and efficient beam design.
Serviceability Limit States
Limits related to normal use and occupant comfort
Key Criteria:
Deflection limits (visual and functional)
Vibration limits (human comfort)
Crack width control (reinforced concrete)
Durability requirements
Ultimate Limit States
Limits related to structural safety and collapse prevention
Key Criteria:
Flexural strength (bending capacity)
Shear strength (shear failure)
Lateral-torsional buckling (steel beams)
Overall stability
Load Combinations
Various combinations of loads for design verification
Key Criteria:
1.35×Dead + 1.5×Live (Ultimate)
1.0×Dead + 1.0×Live (Serviceability)
Wind and seismic combinations
Construction stage loading
Material Safety Factors
Factors accounting for material variability and uncertainties
Key Criteria:
Steel: γm = 1.0-1.1 (yield), 1.25 (ultimate)
Concrete: γm = 1.5 (compression), 1.15 (reinforcement)
Timber: γm = 1.3 (strength), 1.0 (stiffness)
Connection factors: 1.25-2.0
Calculator Features
Deflection Analysis
Accurate deflection calculations with serviceability limit verification and deflection ratios.
Stress Analysis
Complete stress analysis with safety factors and allowable stress verification.
Moment Analysis
Maximum moment and shear calculations for different loading and support conditions.
Material Database
Comprehensive material properties for steel, concrete, and timber structures.
Loading Conditions
Support for point loads, distributed loads, and various loading configurations.
Safety Analysis
Comprehensive safety factor analysis and structural integrity assessment.
Beam Types
Multiple beam configurations including simply supported, cantilever, and fixed beams.
Educational Content
Comprehensive structural engineering education with formulas, examples, and design guides.
Frequently Asked Questions
What is beam deflection and why does it matter?
Beam deflection is the displacement of a beam from its original position under load. Limiting deflection prevents cracking of finishes, misalignment of doors/windows, and vibration issues. Common serviceability limits: L/360 for floor beams (live load only) and L/240 for total load per most building codes.
What is the formula for a simply supported beam with a central point load?
Maximum deflection at center: δ = PL³ / (48EI). Slope at supports: θ = PL² / (16EI). Where P is applied load (N), L is span (m), E is Young’s modulus (Pa), and I is the second moment of area (m⁴). Selecting a larger I — a deeper beam or I-section — is the most efficient way to reduce deflection.
What is the second moment of area (moment of inertia)?
The second moment of area I measures a cross-section’s resistance to bending. For a rectangle: I = bh³/12 (about the neutral axis). For a standard I-beam, I is listed in steel section tables. Doubling beam depth increases I by 8× and halves deflection — depth is far more effective than width for reducing deflection.
What is bending stress and how is it calculated?
Maximum bending stress: σ = M × y / I, where M is the bending moment (N·m), y is the distance from the neutral axis to the extreme fiber (m), and I is the second moment of area. The section modulus Z = I/y simplifies this to σ = M/Z. Steel A36 has a yield strength of 250 MPa; design stress is typically limited to 60–65% of yield.
How do different support conditions affect beam deflection?
Fixed (cantilever) beams deflect the most at the free end: δ = PL³/3EI for a point load. Simply supported beams deflect half that for midspan loading. Continuous beams with multiple supports have the least deflection. Changing from simply supported to fixed-fixed reduces midspan deflection by 80%.