Linear static analysis

The workhorse analysis — what K u = f actually assumes, what it returns, and where linearity stops being a reasonable model.

Analysis types 2 min read

Linear static analysis solves

K u = f

for the displacement vector $u$ , given the assembled stiffness matrix $K$ and the load vector $f$ . Internal forces, stresses and reactions are all recovered from $u$ .

What "linear" is promising

Three separate assumptions travel under that one word:

Assumption	Means	Fails when
Material linearity	Stress proportional to strain, no yield	Steel yields, concrete cracks, timber creeps
Geometric linearity	Equilibrium written on the undeformed shape	Slender frames sway, cables change geometry
Boundary linearity	Restraints do not change with load	Foundations lift off, gaps close, friction slips

All three hold in most everyday design. Know which one you are relying on.

What you get back

Displacements — translations and rotations at nodes, interpolated along members.
Internal forces — $N$ , $V_{y}$ , $V_{z}$ , $T$ , $M_{y}$ , $M_{z}$ along beams and bars; moments and membrane forces in surfaces.
Reactions — at every restrained degree of freedom.
Stresses — derived from internal forces and section properties.

Superposition

Because the system is linear, the response to a combination equals the factored sum of the responses to its cases:

u_{comb} = i \sum γ_{i} u_{i}

In practice:

Each load case is solved once, all of them in one pass.
Every combination is a factored sum of the load-case results; nothing is solved again.
Fifty combinations cost barely more than one.

Reading the result

Deflected shape, then equilibrium, then one hand check, then the governing numbers with their combinations. The order matters, and it is written out once in Troubleshooting.

Drag the load in the demo below and watch reactions, moment and deflection follow: the same quantities, in the same order, you read off a real model.

Max. moment: 57.6 kNm
Max. deflection: 9.7 mm · L/617
Reactions: 24.0 / 16.0 kN
Bending · §6.2.5: η = 0.39 OK

Deflection limits

Deflection is checked at SLS, not ULS, and against the correct combination for what is being protected — see Automatic combinations. Common limits:

Situation	Typical limit
Beams, total deflection	span / 250
Beams supporting brittle finishes	span / 500 after finishes
Cantilevers	length / 180
Inter-storey drift, no brittle cladding	height / 300

These are indicative. The applicable limit comes from the code, the national annex and the project specification, each overriding the one before. Members can be checked against the tabulated limits automatically, each on its own combination, with a serviceability check; slabs are read off the contour.

Where linear static is not enough

Situation	What to do
Slender sway frame	Check $α_{cr}$ with linear buckling, then run a second-order analysis
Foundation uplift	Look at spring forces; interpret by hand where tension appears
Cracked concrete deflection	Use the code's cracked-section method; the linear result is uncracked
Vibration and dynamics	Modal analysis
Post-yield behaviour	Not in this release; material nonlinearity is on the roadmap

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