Beam grid (grillage)
A grid of beams in two directions, connected where they cross and loaded normal to the plan: the deflection and rotations of every joint, the moment, shear and torque along every beam, and the support reactions. Use it for floor and roof framing where beams of both directions share loads (a two-way grid of girders, a mezzanine, a beam grid under equipment, a framed opening), and to see how much a crossing beam relieves another.
Inputs of the beam grid #
- Spans along x and y: the lengths of the beams of each direction, end to end. The plan is rectangular and the beams orthogonal.
- Beams along x and along y: up to 12 each, at their y (or x) positions, with their moment of inertia I, torsional constant J (AISC Manual Part 1) and a uniform load w along each. Every beam is connected to every beam it crosses.
- E and G: steel 29,000 and 11,200 ksi (AISC 360-22); your material's values otherwise.
- Include torsion: with it, a beam twists where a crossing beam bends, and its St. Venant stiffness GJ spreads load; without it the beams only cross and share loads by bending. Warping torsion of open sections is never counted.
- Supports: the ends of each family: pin (deflection held), fixed (deflection and the bending rotation held) or free (the beam frames into the beams it crosses, for example edge beams). Whether a pinned end also holds the beam's twist is your judgment; fixed ends hold it.
- Point loads: downward, at a point of any beam (its x and y); a joint is added there.
The beam grid has no auto-design: size the members in their material modules with these forces.
Checks performed #
| Result | Basis |
|---|---|
| Joint deflections and rotations | Grid stiffness method: three unknowns per joint, Hermite beam elements with consistent loads, GJ/L torsion |
| Moment, shear and torque along each beam, largest values | Element end forces and statics inside each element |
| Deflection along each beam | End values interpolated, plus the deflection of the element's own uniform load |
| Support reactions and equilibrium | K u − F at the supports |
Deflection is positive downward, moment positive sagging, torque by the right-hand rule along the beam.
Limitations of the beam grid #
- Orthogonal beams spanning the full grid, a rectangular plan, supports at the beam ends only; no skew, columns inside the grid, springs or partial beams.
- Linear elastic, first order; shear deformation and warping torsion neglected; the beams are connected rigidly at every crossing (bending of one to twist of the other).
- One load set, as entered; no member strength is checked.
Verification examples #
Verification status: 11 values compared in 3 examples (0 published, 3 hand calculations from the cited equations): 11 within tolerance, 0 recorded deviations. No published example: every value is a hand calculation. That checks the program against this reading of the code, not the reading itself. Code basis: written without the code document on hand, and confirmed only in part through published examples.
| Example | Source | Values |
|---|---|---|
| Two simple beams crossing at midspan | Hand calculation (load shared by 48EI/L³) | Both moments, deflection |
| A cantilever stub twisting a fixed beam | Hand calculation (PL³/192EI, TL/4GJ, Pl³/3EI) | Tip and midspan deflection, torque, moments |
| One simple beam, uniform load | Hand calculation | wL²/8, wL/2, 5wL⁴/384EI |
No published grillage example is on hand; the cases are hand calculations that each test one action (bending share, torsion, element load).