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Queensferry

Module 10 · Lesson 10.1

Geometric stiffness: why load changes stiffness

The second matrix. Where it comes from, why it scales with force rather than with EA, and what that costs you.

Why this matters

Pluck a guitar string and the note rises when you tighten it. Nothing about the string's material or section has changed — only its tension. The stiffness a structure presents depends not just on what it is made of but on what it is already carrying.

For most beams and columns that effect is small enough to ignore, and linear analysis ignores it. For slender frames, tall buildings, cable structures and anything near its buckling load it is not small, and a linear analysis will under-report deflections and moments while giving no sign that it has.

By the end of this lesson you should be able to

  • Explain tension stiffening and compression softening from a free body
  • Write the effective stiffness of a two-bar column with axial load
  • Say why Kg is proportional to P/L and not to EA/L
  • Explain why P-delta results cannot be superposed

What you should already know

  • The elastic stiffness matrix and its assembly (Module 9)
  • Euler buckling of a strut — the classical result, recapped here

The simplest structure that shows it

Two rigid bars, pinned end to end, held at top and bottom, with a lateral spring of stiffness k at the middle joint. Push the joint sideways by δ and the spring resists with kδ. That is the whole of its linear stiffness.

Now add an axial load P down the line of the bars. Once the joint moves sideways the bars are no longer straight, so the axial force has a transverse component at the kink. Work out the geometry and that component is 2Pδ/L — and its direction depends on the sign of P.

In tension it pulls the joint back towards the line: it adds to the spring.

In compression it pushes the joint further out: it subtracts.

So the effective lateral stiffness is:

keff = k − 2P/L, with P positive in compression

That is geometric stiffness in one line. No material property appears in it. It is not a property of the structure at all; it is a property of the structure plus the load it is carrying.

What follows immediately

Kg scales with P/L, not EA/L. Double the section area and the elastic stiffness doubles while the geometric stiffness does not move. Double the axial load and the geometric stiffness doubles while the elastic stiffness does not.

The analysis needs two passes. You cannot build Kg until you know the member forces, and you do not know the member forces until you have solved. So a P-delta analysis runs a linear solve first, forms Kg from the forces it found, then solves again with K + Kg.

Superposition is dead. This is the consequence that catches people. A linear analysis lets you run each load case once and combine the results afterwards with factors. A P-delta analysis does not, because Kg depends on the load — so the structure you are analysing is different in each combination. You must build the factored combination first and analyse that.

Any result that came out of a stiffness that depended on the load may not be scaled, added or enveloped. The only post-processing operation that survives is taking maxima and minima across separately-analysed combinations.

Effective stiffness of the two-bar column

What it calculates: The lateral stiffness that remains once the axial load is accounted for

k
Lateral spring stiffness at the joint (kN/m)
L
Length of each bar (m)
P
Axial load, positive in compression (kN)
Pcrit
Load at which the effective stiffness reaches zero (kN)

This assumes

  • The bars are rigid, so all the flexibility is in the spring
  • Small angles, so the transverse component is 2Pδ/L rather than a trigonometric function of δ

In plain terms: Pcrit contains no reference to the lateral load. That is the first surprising thing about buckling: the critical load is a property of the structure's stiffness and geometry alone, and applying more or less sideways load does not move it.

Try it

Geometric stiffness explorer

Two rigid bars, a lateral spring at mid-height and an axial load. Positive P is compression; drag it negative for tension.

kN/m
m
kN

Positive is compression. Negative is tension, and it stiffens.

Elastic stiffness k
250 kN/m
Geometric contribution −2P/L
-40.0 kN/m

compression softens

Effective stiffness
210.0 kN/m
Critical load Pcrit = kL/2
625 kN
Load factor αcr
6.25
Deflection amplifier α/(α−1)
1.190
Imperfection amplifier 1/(α−1)
0.190

Ignoring second-order effects here costs 19.0 %, which is above the 10 % line. Second-order analysis is required, not optional.

What to notice

  • Change k and Pcrit moves with it. Change L and Pcrit moves with it. Change nothing else and the material never enters.
  • The geometric contribution is proportional to P/L. Doubling the section area would double k and leave it untouched.
  • αcr = 11 is where ignoring second-order effects costs exactly 10 %. The convention quoted in practice is 10, where the cost is 11.1 %.

What this shows: Load changes stiffness. Tension adds 2P/L, compression takes it away, and no material property appears anywhere in it.

Predict first

A cable net is analysed linearly and comes out unrealistically flexible. An engineer suggests running a P-delta analysis. Will that make it stiffer or softer?

Practice

A two-bar column has a lateral spring of 250 kN/m and bars of 5 m. Compute the critical compression load in kN.

Practice

The same column carries 100 kN of compression. What is its remaining effective lateral stiffness, in kN/m?

Worked example

Why P-delta results cannot be added together

Given

  • A frame is analysed for gravity alone and for wind alone, both with P-delta
  • The design combination is 1.35 × gravity + 1.5 × wind

Find

Whether factoring and adding the two results is valid

    Worked example

    Reading an amplification factor

    Given

    • A frame's elastic critical load factor is 5

    Find

    How much the first-order sway is amplified, and what to conclude

      Check yourself

      What does a negative geometric stiffness contribution represent?

      Summary

      • Tension stiffens, compression softens: keff = k − 2P/L
      • Geometric stiffness scales with load over length, not with EA over length
      • It cannot be built until the member forces are known, so the analysis runs twice
      • Pcrit does not depend on the lateral load at all
      • Nothing that came from a load-dependent stiffness may be superposed
      Progress is kept in this browser only.

      This is educational material. It uses simplified examples to teach principles, and must not be relied on for real design or safety-critical decisions. Module overview and checkpoint