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Shear Force Diagram Calculator
Calculate shear forces and bending moments for statically determinate beams with point loads, UDL, triangular loads, and applied moments.
Structural Analysis Structural engineers, civil engineering students Commercial Intent: MEDIUM
Engineering Formulas
Simply Supported — Point Load
RA = P × b / L
RB = P × a / L
V(x) = RA for x < a, RA - P for x > a
M(x) = RA × x for x < a, RA × x - P(x-a) for x > a
R_A, R_B: Reactions at supports A and B (kN)
P: Point load (kN)
a: Distance from A to load (m)
b: Distance from B to load (m)
Simply Supported — UDL
RA = RB = wL/2
V(x) = w(L/2 - x)
M(x) = wx(L-x)/2
Mmax = wL²/8 at midspan
w: UDL (kN/m)
L: Span (m)
Cantilever — Point Load
V = P (constant)
M(x) = -P × (L - x)
Mmax = -P × L at support
P: Point load at free end (kN)
Cantilever — UDL
V(x) = w × x
M(x) = -w × x²/2
Mmax = -wL²/2 at support
w: UDL (kN/m)
Worked Example
Simply Supported Beam with Point Load
beamType: simply-supportedspanLength: 6pointLoads: [{"position":2,"value":20}]udlValue: 0momentLoad: 0selfWeight: 0
Reaction Ra
Ra = 20 × 4 / 6 = 13.33 kN
Reaction Rb
Rb = 20 - 13.33 = 6.67 kN
Max Shear
V_max = 13.33 kN (left of load)
Max Moment
M_max = 13.33 × 2 = 26.67 kN·m
Result: Ra = 13.33 kN, Rb = 6.67 kN, M_max = 26.67 kN·m at x=2m
Engineering Notes
Shear and moment diagrams are essential for determining critical design sections.
For steel design, check both shear yielding and buckling.
For concrete design, shear reinforcement (stirrups) is designed based on shear diagram.
The area under the shear diagram equals the change in moment between two points.
Assumptions
• Linear elastic behavior
• Small deflections
• Beam is prismatic
• Sections remain plane after bending
Common Mistakes
✕ Forgetting sign convention for shear and moment
✕ Not including distributed load effects in moment calculation
✕ Incorrect reaction calculation for overhang beams
Frequently Asked Questions
What is a shear force diagram?
A shear force diagram (SFD) shows the variation of internal shear force along a beam. It is essential for beam design as it identifies critical shear locations.
How do you calculate reactions for a simply supported beam?
Use ΣM = 0 about one support to find the reaction at the other, then ΣV = 0 to find the remaining reaction.
What is the relationship between shear and moment?
The slope of the shear diagram equals the negative of the distributed load. The slope of the moment diagram equals the shear. dM/dx = V, dV/dx = -w.
Where does maximum shear occur?
Maximum shear typically occurs at supports for simply supported beams and at the fixed end for cantilevers.
References & Standards
IS 456AISCBS 5950
Timoshenko & Gere
Mechanics of Materials
Hibbeler
Structural Analysis
IS 456
Plain and Reinforced Concrete — Code of Practice
AISC
Steel Construction Manual