Institute of Structural Engineering Page 1

Method of Finite Elements I

Chapter 2b

2nd order Effects & Structural Dynamics:Modal Analysis with the DSM

Method of Finite Elements I

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Method of Finite Elements I

Goals of this Chapter• 2nd Order Effects• Review of structural dynamics• Dynamic analysis with the DSM• DSM software workflow for …

• Modal analysis

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Method of Finite Elements I

2nd Order Effectsor the influence of the axial normal force

Normal forces change the stiffness of the structure !

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Method of Finite Elements I

Geometrical Stiffness Matrix

kG = geometrical stiffness matrix of a truss element

p = ( k + kG ) u

Very small element rotation

=> Member end forces (=nodal forces p )perpendicular to axis due to initial N

Truss

NOTE:It’s only a

approximation

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Method of Finite Elements I

Beams: Geometrical Stiffness

kG = geometrical stiffness matrix of a beam element

kG =

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Method of Finite Elements I

Linear Static Analysis (2nd order)

Global system of equations

( K + KG ) U = F U = ( K + KG ) -1 F

Inversion possible only if K + KG is non-singular, i.e.- the structure is sufficiently supported (= stable)- initial normal forces are not too big

What are the 2nd order nodal displacements fora given structure due to a given load ?

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Method of Finite Elements I

Linear Static Analysis (2nd order)

Workflow of computer program

1. Perform 1st order analysis2. Calculate resulting axial forces in elements (=Ne)3. Build element geometrical stiffness matrices due to Ne

4. Add geometrical stiffness to global stiffness matrix5. Solve global system of equations (=> displacements)6. Calculate element results

NOTE: Only approximate solution !

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Method of Finite Elements I

Stability AnalysisHow much can a given load be increased until a given structure becomes unstable ?

(K + λmax KG0) U = F

Nmax = λmax N0

KG = f(Nmax)KG(Nmax) = λmax KG(N0) = λmax KG0

2nd order analysis No additional load possible(K + λmax KG0) ΔU = ΔF = 0

linear algebra(A - λ B) x = 0 Eigenvalue problem

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Method of Finite Elements I

Stability AnalysisEigenvalue problem

(A - λ B) x = 0

λ = eigenvaluex = eigenvector

(K - λ KG0) x = 0

λ = critical load factorx = buckling mode

e.g. Buckling of a column

λ N0

λ F

x

Solution

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Method of Finite Elements I

Stability Analysis

Workflow of computer program

1. Perform 1st order analysis2. Calculate resulting axial forces in elements (=N0)3. Build element geometrical stiffness matrices due to N0

4. Add geometrical stiffness to global stiffness matrix5. Solve eigenvalue problem

NOTE: Only approximate solution !

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Method of Finite Elements I

Newton’s law: force = mass x acceleration

Common cyclic or periodic loads• people rhythmically dancing (0.5- 3 Hz)• Marching soldiers (1 Hz)• Rotation machinery (0.2 – 50 Hz)• wind gusts (0.3 – 2 Hz)• earthquakes (0.4 – 6 Hz)

Structural Dynamics

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Method of Finite Elements I

Dynamic ResonanceTruss element under cyclic load

load frequency Ω = 1T

UDYN dynamic responseUSTA static response

Ω1 resonance frequency= eigenfrequency

Ω1 =1

2π𝐸𝐸 𝐴𝐴L M

load independent

elastic materialno damping

M

𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠mass

proportional to

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Method of Finite Elements I

EigenmodesFrame structureSupported beam

deformed shape = eigenform i

Ωi = eigenfrequency ieigenmode i

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Method of Finite Elements I

Eigenmodes

• physical structure: unlimited number• numerical model: number of dofs

How many eigenmodes do exist for a certain structure ?

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Method of Finite Elements I

Modal AnalysisGoal of structural design for dynamic effects:

load frequencies ≠ eigenfrequencies

Find the dynamic eigenmodes (frequency/form)this process is known as

modal analysis

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Method of Finite Elements I

DSM: Dynamic Nodal Forces

P1 P1

p = k u p = m u(t)

Statics Dynamics

Inertia nodal forces:

Nodal displacements Nodal accelerations

p = k u(t) + m u(t)Equilibrum equation:Nodal forces of a ‘vibrating’ element:

m : (element) mass matrix

Nodal forces:

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Method of Finite Elements I

UXE=1

FYS

S

E..

FXS =

FYS =

MZS =

FXE =

FYE =

MZE =

UXS UYS UZS UXE UYE UZE

m14 m15 m16

m24 m25 m26

m34 m35 m36

m44 m45 m46

m55 m56

m66

m11 m12 m13

m22 m23

m33

symm.

e.g. m24 =

reactionin global direction Yat start node S

due to a

unit acceleration in global direction Xat end node E

Element: Mass Matrix

p = m u(t) Element mass matrix in global orientation

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Method of Finite Elements I

Beam: Mass Matrix

m m

Lumped mass Distributed mass

NOTE:It’s only a

approximation

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Method of Finite Elements I

K = global stiffness matrix = Assembly of all ke

F(t) = K U(t) + M U(t)

Global System of Equations

Equilibrium at every node of the structure:

F(t) = global load vector = Assembly of all fe

U(t) = global displacement vector

M = global mass matrix = Assembly of all me

U(t) = global acceleration vector

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Method of Finite Elements I

Modal AnalysisWhat are the eigenmodes of a given structure ?

Global system of equations K U(t) + M U(t) = F(t)

Harmonic displacementsfor eigenmode i (Ei ,Ωi) Ui(t) = Ei cos(2π Ωi t)

( K – (2π Ωi t)2 M ) Ei cos(2π Ωi t) = 0

valid at any time ( K – (2π Ωi t)2 M ) Ei = 0

Solution of eigenvalue problem: Ωi = (dynamic) eigenfrequencyEi = (dynamic) eigenform

load independent!

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Method of Finite Elements I

Eigenmodes and DeformationsNumerical structural model: Deformations of a structure

U = u1, u2, u3, u4, u5, u6

u1u2

u3u4

u5 u6

φ1 φ2 φ3 φ4 φ5 φ6

Every deformed configuration can be described as…

Nodal dof and corresponding amplitudes Eigenmodes and corresponding amplitudes

U= φ1 q1 + φ2 q2 +...+ φ6 q6= Φ q

or

ui nodal displacement

ui φi

φi modal coordinateequivalent

Every mode is like a independent structure !

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Method of Finite Elements I

Types of Modal AnalysisResponse spectra analysis Time history analysis

load-timefunction

T1 Ti Tj

response spectrum

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Method of Finite Elements I

Modal AnalysisWorkflow of computer program

1. System identification: Elements, nodes, support and loads2. Build element stiffness and mass matrices3. Assemble global stiffness and mass matrices4. Solve eigenvalue problem for a number of eigenmodes5. Perform further analysis (time-history or response spectra)

NOTE: Only approximate solution !