Beam Laboratory

The beam is discretised into two-node Hermitian Euler-Bernoulli elements with a node placed at every load discontinuity. The global stiffness matrix is assembled, boundary conditions are applied by elimination, and the banded system is solved directly. Reactions are recovered as R = Kd − F at the restrained degrees of freedom.

Shear and bending moment are then obtained by statics integration from the left using those reactions, which is exact for determinate and indeterminate beams alike. Deflection comes from the nodal degrees of freedom interpolated with the element shape functions.

V(x) = Σ upward forces left of the cut   ·   M(x) sagging positive   ·   deflection plotted downwards positive

Assumptions. Prismatic member, linear-elastic material, small displacements, plane sections remain plane.

What this does not do. These are real limitations, not a disclaimer:

• Shear deformation is neglected. For spans shorter than roughly ten times the depth, a Timoshenko formulation would give larger deflections.
• No axial force and therefore no second-order or P-delta effects.
• No torsion and no lateral-torsional buckling check.
• Self-weight is not added automatically. Include it as a distributed load if you want it.
• Gross uncracked section properties. For reinforced concrete at serviceability, a cracked section would be more appropriate.
• This is an analysis tool. It performs no code check and is not a design verification to SANS, Eurocode or any other standard.

The solver was checked against standard results before this interface was built. Every case below agrees to within 0.6 per cent, and most to within 0.001 per cent.

The equilibrium residual shown above the results is recomputed live on every solve, so the tool reports its own balance rather than asking you to trust it.