STRUCTURAL EXPLORER

Periodic Beam · Support Configuration

Adjust the geometry to explore internal forces and deflection in real time

Maximum absolute moment |M|max

—

Maximum absolute deflection |y|max

—

Maximum absolute shear |V|max

—

Structural layout

A, C: internal hinges / B, D: vertical supports

Bending moment diagram

Positive moment upward · kN·m

Move over the diagram to inspect values at any position.

Shear force diagram

V = dM/dx · kN

Move over the diagram to inspect values; left and right shear values are shown at supports.

Deflection diagram

Positive downward · mm (vertical scale adjusted automatically)

Move over the diagram to inspect values at any position.

Reactions and extrema

Item	Value	Position x · m

Support reactions are positive upward and negative downward. Deflection is measured relative to the support elevation; displacement at A and C may be nonzero.

Model assumptions, equations and scope

The periodic unit uses constant EI, a uniform load over its full length, and small-deflection Euler–Bernoulli beam theory. A (x=0) and C (x=L) are internal hinges at fixed positions in this model: M(0)=M(L)=0 and y(0)=y(L). Rotation may be discontinuous at the hinges; equal end rotations are not imposed. This tool does not relocate inflection points of a continuous beam and does not represent an arbitrary continuous beam without internal hinges.

Supports restrain vertical displacement and allow rotation. Both positive and negative reactions are permitted. A negative reaction requires hold-down or bidirectional restraint. Compression-only supports require a separate contact model.

Taking upward hinge-end force H_A and support reactions Rᵢ as positive:
V(x)=H_A − wx + ΣRᵢ H(x−aᵢ), where H is the unit step function
M(x)=H_A x − wx²/2 + ΣRᵢ⟨x−aᵢ⟩
EI y(x)=−H_A x³/6 + wx⁴/24 − ΣRᵢ⟨x−aᵢ⟩³/6 + C₁x + C₀
Deflection is positive downward, so EI y″=−M. The solution enforces periodic equilibrium ΣRᵢ=wL, zero end moments, zero displacement at each support, and equal end displacements. The end force at C is H_C=−H_A.

Shear extrema are checked at both ends and on both sides of each support; the shear jump equals the support reaction. Moment extrema are checked where V=0 and at segment boundaries. Deflection extrema are checked where y′=0 and at the ends. Extrema are located using piecewise polynomials and root finding, independently of the plotting grid. Diagrams use 600 sampling intervals; CSV output also includes support and extremum positions.

Units: L and x in m; w in kN/m; E in GPa; I in mm⁴; EI in kN·m²; M in kN·m; shear and reactions in kN; displayed deflection in mm. EI (kN·m²) = E (GPa) × I (mm⁴) × 10⁻⁶.