Beam Load Calculator
Calculate reactions, bending moment, shear force, and deflection for simply supported beams.
📖 What is a Beam Load Calculator?
A beam load calculator determines the internal forces and deformations in a beam subjected to external loads. It is a fundamental tool in structural and civil engineering, used in the design of floors, bridges, roof structures, industrial frames, and any structure where beams carry loads.
This calculator focuses on the most common scenario: a simply supported beam - one that rests on two supports and is free to rotate at each end. Two load types are supported: a point load (a concentrated force at a specific location, such as a column bearing onto a beam) and a uniformly distributed load or UDL (load spread evenly across the whole span, such as the self-weight of a slab or wind pressure).
For each load case, the calculator finds: support reactions at both ends, the maximum bending moment (and its location), the maximum shear force, and the maximum midspan deflection. These four outputs are the key quantities needed for beam selection and structural adequacy checks.
Note that this calculator provides idealised elastic results for a single span, single load. Real structures often have multiple loads, continuous spans, or non-uniform sections - consult a structural engineer and applicable design codes for complex or safety-critical designs.
📝 Beam Load Formulas
Simply Supported Beam - Point Load P at position a from left (b = L − a):
R_A = P × b / L R_B = P × a / L
M_max = P × a × b / L (at load position)
δ_max = P × a × b × (L² − a² − b²)^(3/2) / (9√3 × EI × L) [at x = √((L² − b²)/3)]
For central load (a = b = L/2): δ_max = PL³ / (48EI)
Simply Supported Beam - Uniformly Distributed Load w (kN/m):
R_A = R_B = wL / 2
M_max = wL² / 8 (at midspan)
δ_max = 5wL⁴ / (384EI)
Where: P = point load (kN) | w = distributed load (kN/m) | L = span (m) | a, b = distances from supports | EI = flexural rigidity (kN·m²)
R_A = P × b / L R_B = P × a / L
M_max = P × a × b / L (at load position)
δ_max = P × a × b × (L² − a² − b²)^(3/2) / (9√3 × EI × L) [at x = √((L² − b²)/3)]
For central load (a = b = L/2): δ_max = PL³ / (48EI)
Simply Supported Beam - Uniformly Distributed Load w (kN/m):
R_A = R_B = wL / 2
M_max = wL² / 8 (at midspan)
δ_max = 5wL⁴ / (384EI)
Where: P = point load (kN) | w = distributed load (kN/m) | L = span (m) | a, b = distances from supports | EI = flexural rigidity (kN·m²)
✍️ How to Use This Calculator
- Select Point Load or Distributed Load (UDL) using the tabs above.
- Enter the beam span L in metres.
- For a point load: enter the load P in kN and its distance from the left support (a).
- For a UDL: enter the load intensity w in kN/m (total load = w × L).
- Enter EI (flexural rigidity in kN·m²) for deflection. For a standard steel I-beam, EI is typically 5,000–100,000 kN·m².
- Click Calculate to see reactions, maximum moment, shear, and deflection.
📄 Example Calculations
Example 1 - Central point load:
A 6 m simply supported beam carries a 50 kN point load at midspan. EI = 10,000 kN·m².
R_A = R_B = 50/2 = 25 kN
M_max = 50 × 3 / 4 = 37.5 kN·m (at midspan)
δ_max = (50 × 6³) / (48 × 10,000) = 10,800/480,000 = 0.0225 m = 22.5 mm
L/267 - check if this is within the allowable deflection limit (typically L/250 = 24 mm ✓)
Example 2 - Full-span UDL:
A 5 m beam carries a UDL of 15 kN/m (e.g., floor slab + finishes). EI = 8,000 kN·m².
Total load = 15 × 5 = 75 kN
R_A = R_B = 75/2 = 37.5 kN
M_max = 15 × 25 / 8 = 46.9 kN·m (at midspan)
δ_max = (5 × 15 × 5⁴) / (384 × 8,000) = 46,875/3,072,000 = 0.0153 m = 15.3 mm Try this example →
A 6 m simply supported beam carries a 50 kN point load at midspan. EI = 10,000 kN·m².
R_A = R_B = 50/2 = 25 kN
M_max = 50 × 3 / 4 = 37.5 kN·m (at midspan)
δ_max = (50 × 6³) / (48 × 10,000) = 10,800/480,000 = 0.0225 m = 22.5 mm
L/267 - check if this is within the allowable deflection limit (typically L/250 = 24 mm ✓)
Example 2 - Full-span UDL:
A 5 m beam carries a UDL of 15 kN/m (e.g., floor slab + finishes). EI = 8,000 kN·m².
Total load = 15 × 5 = 75 kN
R_A = R_B = 75/2 = 37.5 kN
M_max = 15 × 25 / 8 = 46.9 kN·m (at midspan)
δ_max = (5 × 15 × 5⁴) / (384 × 8,000) = 46,875/3,072,000 = 0.0153 m = 15.3 mm Try this example →
Frequently Asked Questions
What is a simply supported beam?
A simply supported beam is supported at both ends with pin connections that allow rotation but prevent vertical movement. It is the most common beam configuration in structural engineering. The supports provide vertical reactions but no moment resistance, so the beam is statically determinate - reactions can be calculated from statics alone.
What is bending moment and why does it matter?
Bending moment at a cross-section is the net moment of all forces acting on one side of that section, measured in kN·m or N·m. It determines the internal stress in a beam. The maximum bending moment governs the required beam size - higher moments need deeper or wider beams. For a simply supported beam with a central point load P, the maximum moment is PL/4 at midspan.
What is shear force in a beam?
Shear force at a section is the net vertical force on one side of that section. It creates shear stresses in the beam cross-section. Shear force is highest at the supports and zero at the midspan for symmetric loading. Beams must be checked for both bending and shear failure.
How do I calculate beam deflection?
Deflection depends on the load, beam length, material stiffness (E, Young's modulus), and cross-section moment of inertia (I). For a simply supported beam with a central point load: δ_max = PL³ / (48EI). For a UDL: δ_max = 5wL⁴ / (384EI). This calculator computes deflection for steel beams - adjust E for other materials.
What is EI (flexural rigidity)?
EI is the product of Young's modulus (E) and second moment of area (I). It represents the beam's resistance to bending. A higher EI means a stiffer beam with less deflection. For steel, E = 200 GPa. The moment of inertia I depends on the cross-section shape - a deeper I-section has a much higher I than a square bar of the same area.
How do point loads and distributed loads affect beam design differently?
A point load concentrates force at a single location, creating a sharp peak in the shear force diagram and a triangular bending moment diagram with maximum moment directly under the load. A uniformly distributed load (UDL) spreads force over the span, producing a trapezoidal shear diagram and a parabolic bending moment diagram with maximum moment at midspan. For equal total load, a UDL produces half the maximum bending moment of a central point load (M = wL²/8 vs PL/4), making distributed loading less demanding on the beam section.
What is the difference between a simply supported beam and a fixed beam?
A simply supported beam rests on two supports that allow rotation but prevent vertical displacement. It develops reactions at the supports but no bending moments at the supports. A fixed (built-in) beam has both ends clamped, preventing both rotation and displacement. Fixed beams develop bending moments at the supports (called fixed-end moments) and have lower midspan deflection and moment than simply supported beams for the same load. Most practical structures use combinations of simply supported and fixed conditions depending on connection details.
When should I use a simply supported beam vs a cantilever beam model?
Use a simply supported beam model when both ends rest on supports that allow rotation (e.g. a floor joist between two walls). Use a cantilever model when one end is rigidly fixed and the other is free (e.g. a balcony slab or a diving board). The support conditions dramatically affect bending moments and deflections - a cantilever with a point load at the tip has a maximum moment at the fixed end equal to W x L, while a simply supported beam has maximum moment at midspan of W x L / 4. Always match the model to the actual physical support conditions.
What is the difference between a simply supported and a cantilever beam?
A simply supported beam rests on two supports at its ends with no moment resistance - free to rotate. A cantilever beam is fixed at one end and free at the other; the fixed support resists both shear and moment. Cantilevers experience maximum moment at the fixed end; simply supported beams experience maximum moment at midspan for a uniformly distributed load.
What units should I use in beam calculations?
Use consistent units throughout. Common SI sets: length in metres (m), force in kilonewtons (kN), resulting moments in kN.m and deflection in mm. If you use mm for length, use N for force, and results are in N.mm and mm. Mixing units (e.g. kN with mm) will give wrong answers. This calculator accepts the most common engineering unit combinations and converts internally.