Beam Types, Support Types, and Load Formulas
| Beam Type | Support Type | Load Type | Bending Moment (M) | Shear Force (V) | Reaction (R) | Deflection (δ) |
|---|---|---|---|---|---|---|
| Simply Supported Beam | Pin & Roller Support | Point Load at Center | M = P × L / 4 | V = P / 2 | R = P / 2 | δ = (P × L³) / (48 × E × I) |
| Simply Supported Beam | Pin & Roller Support | Uniformly Distributed Load (UDL) | M = w × L² / 8 | V = w × L / 2 | R = w × L / 2 | δ = (5 × w × L⁴) / (384 × E × I) |
| Cantilever Beam | Fixed Support | Point Load at Free End | M = P × L | V = P | R = P | δ = (P × L³) / (3 × E × I) |
| Cantilever Beam | Fixed Support | Uniformly Distributed Load (UDL) | M = w × L² / 2 | V = w × L | R = w × L | δ = (w × L⁴) / (8 × E × I) |
| Fixed Beam | Fixed at Both Ends | Point Load at Center | M = P × L / 8 | V = P / 2 | R = P / 2 | δ = (P × L³) / (192 × E × I) |
| Fixed Beam | Fixed at Both Ends | Uniformly Distributed Load (UDL) | M = w × L² / 12 | V = w × L / 2 | R = w × L / 2 | δ = (w × L⁴) / (384 × E × I) |
| Propped Cantilever Beam | Fixed Support & Roller | Point Load at Free End | M = P × L / 4 | V = P | R = P / 2 | δ = (P × L³) / (24 × E × I) |
| Propped Cantilever Beam | Fixed Support & Roller | Uniformly Distributed Load (UDL) | M = w × L² / 12 | V = w × L | R = w × L / 2 | δ = (5 × w × L⁴) / (384 × E × I) |
| Overhanging Beam | Pin & Roller Support | Point Load at Overhang | M = P × a / L | V = P × (L - a) / L | R = P | δ = (P × a² × L) / (9 × E × I) |
| Overhanging Beam | Pin & Roller Support | Uniformly Distributed Load (UDL) | M = w × L² / 8 | V = w × L / 2 | R = w × L | δ = (5 × w × L⁴) / (384 × E × I) |
Explanation of Symbols
P = Point Load
w = Uniformly Distributed Load (UDL)
L = Length of the Beam
a = Overhang Length
E = Modulus of Elasticity
I = Moment of Inertia
M = Bending Moment
V = Shear Force
R = Reaction Force
δ = Deflection
Additional Formulas
Moment of Inertia for Rectangular Section:
I = b × h³ / 12
Moment of Inertia for I-Beam:
I = (b₁ × h₁³ / 12) + (b₂ × h₂³ / 12)
Moment of Inertia for Circular Section:
I = Ï€ × d⁴ / 64
Torsion and Shear Stress in Beams
Torsional Moment:
T = Ï„ × J / r
Torsional Shear Stress:
Ï„ = T × r / J
Polar Moment of Inertia (J):
J = Ï€ × d⁴ / 32
Shear Stress in Beams:
Ï„ = V × Q / (I × b)
Additional Support and Load Types
Propped Cantilever Beam: A beam with one fixed support and one roller support. It is partially restrained.
Overhanging Beam: A beam that extends beyond one or both supports, creating a cantilever at one end.
Fixed Beam: A beam that is restrained at both ends, preventing rotation and translation.
Simply Supported Beam: A beam supported at both ends by pin and roller supports, allowing rotation but preventing vertical translation.
Key Concepts
Moment of Inertia (I): The resistance of a cross-section to bending and deflection. It depends on the geometry of the section (b = base width, h = height).
Modulus of Elasticity (E): The material property that measures the stiffness of the material and its resistance to deformation.
Bending Moment (M): The internal moment that causes a beam to bend due to external forces.
Shear Force (V): The force that acts perpendicular to the cross-section of the beam, leading to shear deformation.
Deflection (δ): The displacement of the beam due to applied loads. It depends on the load type, length, modulus of elasticity, and moment of inertia.
Useful Formulas for Beam Design
Shear Force Calculation:
V = (w × L / 2) for a uniformly distributed load (UDL) on a simply supported beam
Bending Moment Calculation:
M = (w × L² / 8) for a uniformly distributed load on a simply supported beam
Deflection Calculation:
δ = (5 × w × L⁴) / (384 × E × I) for a simply supported beam under UDL
General Torsion and Shear Stress Formulas
Torsion Formula:
T = Ï„ × r / J
Shear Stress in a Rectangular Beam:
Ï„ = V × Q / (I × b)
Tips for Practical Applications
When designing beams for construction, always consider the load types, support types, and deflection limits. The moment of inertia plays a crucial role in minimizing deflection. Make sure to calculate both bending and shear stresses and compare them to the material's yield strength to ensure safety.
For more complex beam configurations, like overhanging or propped cantilever beams, additional moment and reaction calculations are required to account for the additional load cases and restraint conditions.
