Calculator · Structural Mechanics
Beam Deflection Calculator
Slope and deflection by direct integration of the bending moment the beam engine already produced — not by a table of standard cases. Any determinate arrangement of point loads, couples and uniform loads is handled the same way.
Beam
Scroll the schematic sideways to see the whole beam; every result is also listed in the tables below.
Beam and loads
The beam runs from x = 0 to x = L.
Supports
Point loads
Point moments
Uniform distributed loads
Downward loads are negative (q < 0, Fy < 0); an applied couple is positive counter-clockwise. Internally, M > 0 is sagging, v > 0 is upward and θ > 0 is counter-clockwise.
Section and material (optional)
Leave a field empty to leave it unspecified: reactions, shear and bending moment are reported without any section data. A value that is zero or negative is not unspecified — it is reported as an invalid entry. Bending stress additionally needs I and an extreme-fibre distance c; slope and deflection need both E and I.
Evaluation station
Measured from the left end of the beam.
Results
| Deflection v(x) | -28.125 mm (downward) |
|---|---|
| Slope θ(x) | 0 mrad |
| Largest deflection in magnitude | -28.125 mm at 3 m |
| Largest downward deflection | -28.125 mm at 3 m |
| Largest slope in magnitude | 14.063 mrad at 6 m |
| Flexural rigidity EI | 1.6000e+6 N·m² |
Assumptions
- · Statically determinate planar beam; reactions come from the shared equilibrium solver.
- · Small displacements, linear elasticity, plane sections remain plane (Euler-Bernoulli); shear deformation is neglected.
- · Uniform E and I along the beam when slope and deflection are reported.
Open in the Lab
The Lab draws the deflected shape against the shear and moment diagrams.
Technical reference
EI v''(x) = M(x)
θ(x) = v'(x)
v and θ continuous at every interior station
- v
- transverse deflection, positive upward[m]
- θ
- slope of the deflected axis, positive counter-clockwise[rad]
- E
- Young's modulus[Pa]
- I
- second moment of area about the bending axis[m⁴]
M is integrated twice over each interval between structural events. The two constants per interval are fixed by requiring v and θ to be continuous across every interior station and by the support conditions: v = 0 at a pin or roller, and v = θ = 0 at a fixed support.
Euler–Bernoulli theory: shear deformation is neglected, so results are accurate for slender beams. For a span-to-depth ratio below about ten, real deflections exceed these values.
Small-displacement, linear-elastic response with constant EI along the span. A concentrated force or couple never makes v or θ jump — only the curvature changes abruptly.