Simply Supported Beam Calculator (UDL)
Enter the properties of your simply supported beam with a uniformly distributed load (UDL) to calculate its maximum bending moment, shear force, and deflection.
Calculation Results:
Maximum Bending Moment (Mmax): —
Maximum Shear Force (Vmax): —
Maximum Deflection (δmax): —
Understanding Simply Supported Beams and Their Properties
Beams are fundamental structural elements designed to carry loads primarily by resisting bending. They are ubiquitous in construction, from the floors and roofs of buildings to bridges and industrial frameworks. Understanding how beams behave under load is crucial for ensuring structural integrity and safety.
What is a Simply Supported Beam?
A simply supported beam is one of the most basic and commonly analyzed beam types. It is supported at both ends, typically by a pin support at one end (allowing rotation but preventing translation) and a roller support at the other (allowing rotation and horizontal translation, but preventing vertical translation). This configuration ensures that the beam is statically determinate, meaning its reactions and internal forces can be calculated using only the equations of static equilibrium.
What is a Uniformly Distributed Load (UDL)?
A uniformly distributed load (UDL) is a load that is spread evenly across a length of the beam. Examples include the weight of a concrete slab, the pressure from water or snow, or the weight of a wall resting on a beam. For calculation purposes, a UDL is typically expressed in units of force per unit length (e.g., kilonewtons per meter, kN/m).
Key Beam Properties Calculated:
1. Maximum Bending Moment (Mmax)
The bending moment is a measure of the internal forces that cause a beam to bend. It is highest where the beam experiences the greatest tendency to curve. For a simply supported beam with a UDL, the maximum bending moment occurs at the exact center of the beam. A high bending moment indicates significant stress in the beam's fibers, with tension on one side and compression on the other. The formula for maximum bending moment (Mmax) under a UDL is:
Mmax = (w × L2) / 8
Where:
w= Uniformly Distributed LoadL= Beam Length (Span)
2. Maximum Shear Force (Vmax)
Shear force is the internal force acting perpendicular to the beam's longitudinal axis, tending to cause one part of the beam to slide past the adjacent part. For a simply supported beam with a UDL, the maximum shear force occurs at the supports. Shear forces are critical for designing connections and ensuring the beam doesn't fail by shearing. The formula for maximum shear force (Vmax) under a UDL is:
Vmax = (w × L) / 2
Where:
w= Uniformly Distributed LoadL= Beam Length (Span)
3. Maximum Deflection (δmax)
Deflection refers to the displacement or deformation of the beam under load. While a beam might be strong enough to resist bending and shear stresses, excessive deflection can lead to aesthetic issues, damage to non-structural elements (like plaster or finishes), or even affect the functionality of machinery. For a simply supported beam with a UDL, the maximum deflection occurs at the center of the beam. The formula for maximum deflection (δmax) under a UDL is:
δmax = (5 × w × L4) / (384 × E × I)
Where:
w= Uniformly Distributed LoadL= Beam Length (Span)E= Modulus of Elasticity (a material property indicating its stiffness)I= Moment of Inertia (a cross-sectional property indicating resistance to bending)
Understanding Modulus of Elasticity (E) and Moment of Inertia (I)
- Modulus of Elasticity (E): Also known as Young's Modulus, this is a fundamental material property that measures its stiffness or resistance to elastic deformation under stress. Materials with a higher 'E' (like steel) are stiffer than materials with a lower 'E' (like wood). It's typically expressed in Pascals (Pa) or Gigapascals (GPa).
- Moment of Inertia (I): This is a geometric property of a beam's cross-section that quantifies its resistance to bending. A larger moment of inertia means the beam is more resistant to bending and will deflect less under a given load. It depends on the shape and dimensions of the cross-section (e.g., I-beams have a high moment of inertia for their weight). It's typically expressed in units of length to the fourth power (e.g., m4 or mm4).
How to Use the Calculator:
Simply input the required values for your beam:
- Beam Length (L): The total span of your beam in meters.
- Distributed Load (w): The total uniformly distributed load acting on the beam in kilonewtons per meter (kN/m).
- Modulus of Elasticity (E): The material's stiffness in Gigapascals (GPa). Common values are ~200 GPa for steel, ~30 GPa for concrete, and ~10-15 GPa for wood.
- Moment of Inertia (I): The cross-sectional resistance to bending in meters4 (m4). This value can be found in structural handbooks for standard beam sections or calculated for custom shapes.
Click "Calculate Beam Properties" to see the maximum bending moment, shear force, and deflection.
Example Calculation:
Let's consider a steel beam with the following properties:
- Beam Length (L): 6 meters
- Distributed Load (w): 10 kN/m
- Modulus of Elasticity (E): 200 GPa (for steel)
- Moment of Inertia (I): 0.0000455 m4 (typical for a W200x46.1 steel I-beam)
Using the calculator, you would find:
- Maximum Bending Moment (Mmax): 45.00 kN.m
- Maximum Shear Force (Vmax): 30.00 kN
- Maximum Deflection (δmax): 18.544 mm
Disclaimer: This calculator provides theoretical values for a simply supported beam with a uniformly distributed load. It is intended for educational and preliminary estimation purposes only. Actual structural design requires consideration of many other factors, including safety factors, load combinations, support conditions, local buckling, fatigue, and specific building codes. Always consult with a qualified structural engineer for professional design and analysis.