Theory: analysing frame structures

This section explains how the frame calculator works and what lies behind its inputs – written like lecture notes, with derivations, formulas and worked examples. Every topic ends with a matching structure that you can open in the calculator and check for yourself.

Topics

• Direct stiffness method

  The method behind the calculator: the element stiffness matrix of a frame member, transformation into the global system, assembly, support conditions and recovery of the internal forces – with a fully worked two-span beam.

• Releases and degrees of freedom

  Moment, shear and axial releases at member ends, their treatment by static condensation and what they mean for the degrees of freedom – with a fully worked Gerber beam.

• Support types

  Roller, pinned, fixed and spring supports as restrained or elastically held degrees of freedom, support reactions, support settlement and the proper arrangement of supports – with two fully worked examples.

• Static and kinematic determinacy

  The counting criterion for frame structures, counting releases correctly, critical forms and kinematic indeterminacy as the measure of the unknowns in the stiffness method – with a fully worked T-frame.

Prerequisites and scope

These pages are written for civil engineering students. They assume the basics of structural analysis: equilibrium, internal forces and the deflection curve. Like the calculator itself, all derivations apply to plane frame structures under first-order theory, with linear-elastic material and slender members without shear deformation (Euler–Bernoulli).

Signs and notation are the same on every page as in the calculator: $x$ to the right, $y$ upwards, rotations positive counter-clockwise; axial force positive in tension, bending moment positive when the bottom side of the member is in tension.