Beam Calculator

What is a Beam Calculator?

A beam calculator is an engineering tool used to determine the structural response of a beam subjected to various loads. It helps engineers calculate key parameters such as deflection (how much the beam bends), bending moment (the internal forces that cause bending), and bending stress (the stress induced in the beam material due to bending).

This specific calculator focuses on a simply supported rectangular beam with a concentrated point load applied at its center. It calculates the maximum deflection and maximum bending stress, which are critical for ensuring the beam's safety and serviceability under load.

Beam Formulas (Simply Supported, Center Point Load, Rectangular Beam)

The following formulas are used for a simply supported rectangular beam with a point load (P) at its center:

Moment of Inertia (I)

For a rectangular cross-section:

I = (b * h³) / 12

Where:

• I = Moment of Inertia (e.g., in m⁴ or mm⁴)
• b = Width of the beam's cross-section (e.g., in m or mm)
• h = Height of the beam's cross-section (e.g., in m or mm)

Maximum Deflection (δ_max)

δmax = (P * L³) / (48 * E * I)

Where:

• δ_max = Maximum deflection at the center of the beam (e.g., in m or mm)
• P = Point load applied at the center (e.g., in N or kN)
• L = Length of the beam between supports (e.g., in m or mm)
• E = Modulus of Elasticity of the beam material (e.g., in Pa or GPa)
• I = Moment of Inertia of the beam's cross-section

Maximum Bending Moment (M_max)

Mmax = (P * L) / 4

Where:

• M_max = Maximum bending moment at the center (e.g., in Nm or kNm)

Maximum Bending Stress (σ_max)

σmax = (Mmax * c) / I

Where 'c' is the distance from the neutral axis to the outermost fiber (c = h / 2 for a rectangular beam):

σmax = ( (P * L) / 4 * (h / 2) ) / I = (P * L * h) / (8 * I)

Alternatively, using Section Modulus (S = I/c = (b*h²)/6 ):

σmax = Mmax / S = ( (P * L) / 4 ) / ( (b * h²) / 6 ) = (3 * P * L) / (2 * b * h²)

Where:

• σ_max = Maximum bending stress (e.g., in Pa or MPa)

Ensure consistent units are used throughout the calculations.

How to Calculate Beam Deflection and Stress: Example

Consider a simply supported wooden beam (Douglas Fir, E ≈ 13 GPa) with a rectangular cross-section.

• Length (L) = 3 m
• Width (b) = 0.05 m (50 mm)
• Height (h) = 0.1 m (100 mm)
• Modulus of Elasticity (E) = 13 GPa = 13 x 10⁹ Pa
• Point Load (P) = 1000 N

1. Calculate Moment of Inertia (I):

   I = (b * h³) / 12 = (0.05 * (0.1)³) / 12 = (0.05 * 0.001) / 12 = 0.00005 / 12 ≈ 4.167 x 10^-6 m⁴

2. Calculate Maximum Deflection (δ_max):

   δ_max = (P * L³) / (48 * E * I)

   δ_max = (1000 N * (3 m)³) / (48 * 13 x 10⁹ Pa * 4.167 x 10^-6 m⁴)

   δ_max = (1000 * 27) / (48 * 13 * 10⁹ * 4.167 * 10^-6) = 27000 / (2600.112 * 10³) ≈ 27000 / 2600112 ≈ 0.01038 m ≈ 10.38 mm

3. Calculate Maximum Bending Stress (σ_max) using (P * L * h) / (8 * I):

   σ_max = (1000 N * 3 m * 0.1 m) / (8 * 4.167 x 10^-6 m⁴)

   σ_max = 300 / (3.3336 x 10^-5) ≈ 8,999,280 Pa ≈ 9.0 MPa

Thus, the maximum deflection is approximately 10.38 mm and the maximum bending stress is approximately 9.0 MPa.

Common Applications

• Structural Design: Ensuring beams in buildings, bridges, and other structures can safely support expected loads without excessive bending or breaking.
• Mechanical Engineering: Designing machine components like shafts and levers that act as beams.
• Material Selection: Comparing the suitability of different materials for beam applications based on their strength (stress limits) and stiffness (deflection limits).
• Furniture Design: Calculating the strength of shelves, table tops, and other load-bearing furniture parts.
• Aerospace Engineering: Analyzing wing spars and fuselage components.