CVE 805:

ADVANCED
STRUCTURAL
ANALYSIS

Multiple degree of freedom systems

OUTLINE
 MDOF spring-mass systems
 definition, free vibration, modes of vibration.
 MDOF structures
 stiffness matrix, mass matrix
 modes of vibration, orthogonality, modal mass and stiffness
 general damping, Rayleigh damping
 Problem reduction, static condensation, Rayleigh-Ritz
reduction.
 Forced vibration
 vibration absorbers, modal decomposition methods.
 Earthquake loading of buildings
 general ground motion, response spectrum method, modal
combination, shear buildings, base shear and overturning
moments, torsion, Eurocode 8.
MDOF SPRING-MASS SYSTEMS
 Consider the following simple spring-mass system:

 The equilibrium equation is:

 or in terms of the stiffness and mass matrices as:

FREE VIBRATION
 Trying solutions as:

 leads to:

 For non-trivial solutions to exist

 The solutions 𝜔1 and 𝜔2 of this equation are the natural

frequencies of vibration of the system and the
corresponding vectors v1and v2 are the modes of vibration.
For instance if k1=k2=k3=k and m1=m2=m:
MDOF STRUCTURES - STIFFNESS
 The stiffness of any structure can be represented
by the stiffness matrix which in statics gives:

 where u is a displacement vector containing n

degrees of freedom and F is vector containing all
the external nodal forces (or moments). For
instance, for trusses or frames:
MDOF STRUCTURES - MASS
 The distributed mass of a structure can be
lumped or condensed at the degree of freedom
nodes:
MDOF STRUCTURES – FREE VIBRATION
 The dynamic equilibrium equation is:

 with general solution:

where C𝛼 and 𝜑𝛼 are functions of the initial conditions.

 The n natural frequencies of vibration 𝜔𝛼 and

corresponding modes of vibration vectors v𝜶 are the
solutions of the eigenvalue problem:

 Note that the vectors v𝜶 can be arbitrarily scaled.

MODES OF VIBRATION – TYPICAL TYPES
MODES OF VIBRATION - ORTHOGONALITY
 The modes of vibration satisfy the following orthogonality
conditions:

 For the case where, 𝛼 = 𝛽, the above products define the

modal mass and modal stiffness

 The ratio between modal stiffness and mass is the Rayleigh

quotient and gives the frequency squared:
FREE VIBRATION - INITIAL CONDITIONS
 Given the free vibration equation:

 the orthogonality conditions enable the constants C𝛼 and 𝜑𝛼

to be evaluated from the initial conditions as:

 where u0 is the vector of initial displacements and is the

vector of initial velocities.
FREE VIBRATION -DAMPING
 By analogy with SDOF, a modal damping ratio 𝜉𝛼 can be
defined so that the resulting free vibration is:

 This is equivalent to introducing viscous forces in the

equilibrium equation given by a damping matrix C so that
the dynamic equilibrium equation becomes:

where C satisfies

 Other commonly used damping definition is Rayleigh

damping given by:
PROBLEM REDUCTION
 Solving the eigenvalue problem for a large
structure is a very costly and complex process.
Often only a few modes are necessary for the
dynamic analysis. The size of the problem can be
reduced by:
 Structural simplifications: for the purpose of the
dynamic analysis it is often common to assume that
many parts of the structure are rigid:

 Static condensation
 Rayleigh-Ritz reduction
 Vector iteration methods
STATIC CONDENSATION
 The mass of the structure can be lumped on a few nodes

 The original eigenvalue problem becomes:

 where s denotes static d.o.f (no mass) and d dynamic d.o.f.

The static d.o.f.’s are obtained as:

and the reduced problem is:

RAYLEIGH-RITZ REDUCTION
 It is often possible to have a good idea of the first few modal
shapes. Let r1, r2,…, rm approximate v1,…, vm.

 The eigenvalue problem is then reduced to:

 from which the first m modes of vibration can be obtained

as:
 Typically the Ritz vectors are obtained as the
displacements that result from a set of load patterns p1,
p2,…, pm:
MASS DAMPER – VIBRATION ABSORBER
 The vibration of a SDOF system near resonance can be
damped by attaching a mass damper: a small mass-spring
system with about the same natural frequency.

 The maximum displacement of the main spring is now:

GENERAL FORCED VIBRATION
 Given a general structure with applied forces:

 Newton’s second law applied at each mass gives:

 This is the dynamic equilibrium equation for a general

structure.
MODAL DECOMPOSITION
 The vibration can be expressed in terms of the modal
components as:

 Substituting into the dynamic equation and using the

orthogonality conditions gives:

 where the k𝛼 , c𝛼 and m𝛼 are the modal stiffness, damping

and mass and the modal force F𝛼 (t) is:

 Each component is given by a Duhamel integral:

EARTHQUAKE LOADING
 Consider a structure under a general earthquake defined
by a direction vector d:

 The total acceleration is:

 so the equilibrium equation becomes:

EARTHQUAKE LOADING – MODAL
DECOMPOSITION
 Using modal decomposition:
gives

where the participating factors l𝛼 are:

 If the response spectrum is known, then the maximum

modal displacements components are:

and the corresponding maximum modal displacement vectors

and equivalent forces are:
EFFECTIVE MODAL MASS
 The equation for the maximum force vector can be
expressed as:

 where the effective modal mass and modal force

distribution vector are:

 Given that the sum of the modal masses is the total mass of
the structure:

 Eurocode 8 requires that the number of modes N<n

considered must be such that:
PEAK MODAL COMBINATION
 Given a particular effect E (displacement,…), the peaks of
each modal component take place at different times. Hence
they cannot be added. Instead a square root sum of squares
(SRSS) is used:

 This assumes that frequencies are different from each

other. Otherwise the complete quadratic combination
(CQC) must be used:
SIMPLIFIED MODELS FOR BUILDINGS
 For common building designs, a simple rigid
slab/columns structural model (shear building)
can be used:

 Eurocode 8 establishes the following rules:

SHEAR BUILDING – STRUCTURAL MODEL
 One degree of freedom per storey at a time:

 Eurocode 8 allows the first mode to be approximated as

SHEAR BUILDINGS – SIMPLIFIED METHOD
 Only the fundamental mode of vibration needs to be
considered, but with an effective mass equal to the total
mass of the building. The distribution of shear forces is:

 where the total base shear Vb and overturning moment Mb

are:
DESIGN STRATEGIES
 Symmetry, regularity and ductility

 Base isolation: placing either

rollers or laminated rubber
bearings

 Cross-bracing: specially used

to reinforce older buildings

 Mass dampers