Module 10
Multi-degree-of-freedom systems
Assembling the matrices of a real model — lumped against consistent mass, rotational inertia, rigid diaphragms — and the entries that get left at zero without anyone noticing.
What this module covers
- Assemble mass and stiffness matrices for a multi-storey shear building
- Explain the difference between lumped and consistent mass and when each is right
- Derive the consistent mass matrix of an axial element from its shape functions
- Compute the mass moment of inertia of a floor diaphragm and say why it matters
- Explain how mass eccentricity couples translation and rotation
- Recognise the modelling faults that leave a matrix entry silently at zero
Lessons
The two-storey pattern, extended — and the reason every entry of a shear-building stiffness matrix can still be checked by inspection at fifty storeys.
Start lesson →Two ways to describe where the mass is, both approximations, missing the exact answer from opposite sides — which makes the gap between them a mesh check that needs no exact solution.
Start lesson →A floor has three degrees of freedom, not two — and the third one is the entry most often left at zero, which does not make the torsional mode wrong but removes it.
Start lesson →
Module checkpoint
Check what you have taken in
5 questions
Question 1
A six-storey shear building has all storey stiffnesses of 550 MN/m. What is the sum of the diagonal entries of its stiffness matrix, in MN/m?
Question 2
A bar element has ρAL = 1 200 kg. What is the off-diagonal entry of its consistent mass matrix, in kg?
Question 3
A square floor is 20 m by 20 m with a mass of 600 tonnes. What is its mass moment of inertia, in kg·m²?
Question 4
A modal analysis of an eight-storey rigid-diaphragm model returns 26 modes. What does this indicate?
Question 5
Why do lumped and consistent mass models bracket the exact natural frequency?