#2 Ranked Tool · 20,000–40,000 searches/mo
Bending Moment Calculator
Calculate bending moments and shear forces for beams with interactive diagrams.
Structural Analysis Structural engineers, students Commercial Intent: HIGH
Engineering Formulas
Simply Supported — UDL
Mmax = wL²/8
Vmax = wL/2
M_max: Maximum bending moment (kNm)
w: UDL (kN/m)
L: Span (m)
Simply Supported — Point Load at Center
Mmax = PL/4
Vmax = P/2
P: Point load (kN)
L: Span (m)
Cantilever — UDL
Mmax = wL²/2
Vmax = wL
M_max: Max moment at fixed end (kNm)
w: UDL (kN/m)
L: Length (m)
Cantilever — Point Load at Free End
Mmax = PL
Vmax = P
P: Point load (kN)
Fixed — UDL
Mmax = wL²/12
Vmax = wL/2
M_max: Max moment at mid-span (kNm)
Worked Example
Simply Supported Beam with UDL
beamType: simply-supportedloadType: udllength: 6loadValue: 10
Reactions
Ra = Rb = 10 × 6 / 2 = 30 kN
Max Moment
M_max = 10 × 6² / 8 = 45 kNm
Max Shear
V_max = 30 kN
Result: M_max = 45 kNm at mid-span, V_max = 30 kN at supports
Engineering Notes
For continuous beams (multi-span), use moment distribution method or software.
Deflection criteria: L/250 for floors, L/175 for roofs typically.
Check both moment and shear — one often governs the design.
Assumptions
• Beam is prismatic (constant cross-section)
• Material is homogeneous and isotropic
• Small deflections (linear elastic analysis)
• Self-weight may need to be added separately
Common Mistakes
✕ Using UDL formula for point loads
✕ Forgetting sign conventions in diagrams
✕ Not considering self-weight of beam
Frequently Asked Questions
What is a bending moment?
A bending moment is the internal moment that causes a beam to bend. It is calculated as the sum of moments about a point on the beam.
What is the difference between UDL and point load?
A UDL (Uniformly Distributed Load) is spread evenly across a beam length, measured in kN/m. A point load acts at a single location, measured in kN.
When is a beam fixed vs simply supported?
A simply supported beam rests on supports at both ends and can rotate. A fixed beam has restrained ends that prevent rotation, creating negative moments at supports.
References & Standards
Timoshenko Beam TheoryACI 318BS 5950
Timoshenko & Gere
Mechanics of Materials — definitive beam theory reference
ACI 318
Building Code Requirements for Structural Concrete
BS 5950
Structural Use of Steelwork in Building