Method of Finite Elements I of Structural Engineering Page 1 Method of Finite Elements I Chapter 2b The Direct Stiffness Method: 2nd Order and Stability Analysis Method of Finite ...

Institute of Structural Engineering Page 1

Method of Finite Elements I

Chapter 2b

The Direct Stiffness Method:

2nd Order and Stability Analysis




2nd Order Effectsor the influence of the axial normal force

Normal forces change the stiffness of the structure !



Types of Linear Analysis

Analysis type 1st Order 2nd Order 3rd Order

Deformations small small large

Strains small small small

Equilibrum

formulation

undeformed

shape

deformed

shape

deformed

shape

Usage Floor slab Column

Cable bridge

Form finding

(Cable-nets)



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



Beams: Geometrical Stiffness

kG = geometrical stiffness matrix of a beam element

kG =



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 for

a given structure due to a given load ?



Linear Static Analysis (2nd order)

Workflow of computer program

1. Perform 1st order analysis

2. Calculate resulting axial forces in elements (=Ne)

3. Build element geometrical stiffness matrices due to Ne

4. Add geometrical stiffness to global stiffness matrix

5. Solve global system of equations (=> displacements)

6. Calculate element results

NOTE: Only approximate solution !



Stability Analysis

How much can a given load be increased until a

given structure becomes unstable ?

(K + λmax KG0) U = F

Nmax = λmax N0

KGmax = f(Nmax)KGmax(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



Stability Analysis

Eigenvalue problem

(A - λ B) x = 0

λ = eigenvalue

x = eigenvector

(K - λ KG0) x = 0

λ = critical load factor

x = buckling mode

e.g. Buckling of a column

λ N0

λ F

x

Solution



Stability Analysis


1. Perform 1st order analysis

2. Calculate resulting axial forces in elements (=N0)

3. Build element geometrical stiffness matrices due to N0

4. Add geometrical stiffness to global stiffness matrix

5. Solve eigenvalue problem






Chapter 2c

Structural Dynamics:

Modal Analysis with the DSM




Goals of this Chapter

• Review of structural dynamics

• Dynamic analysis with the DSM

• DSM software workflow for …

• Modal analysis



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



Dynamic Resonance

Truss element under cyclic load

load frequency W =1

T

UDYN dynamic response

USTA static response

W1 resonance frequency

= eigenfrequencyW1 =

1

2p

𝐸 𝐴

L M

load independent

elastic material

no damping

M

𝑠𝑡𝑖𝑓𝑓𝑛𝑒𝑠𝑠

massproportional to



Eigenmodes

Frame structureSupported beam

deformed shape = eigenform i

Wi = eigenfrequency ieigenmode i



Eigenmodes

• physical structure: unlimited number

• numerical model: number of dofs

How many eigenmodes do exist for a certain structure ?



Modal Analysis

Goal of structural design for dynamic effects:

load frequencies ≠ eigenfrequencies

Find the dynamic eigenmodes (frequency/form)

this process is known as

modal analysis



DSM: Dynamic Nodal Forces

P1P1

p = k u p = m ሷu(t)

Statics Dynamics

Inertia nodal

forces:

Nodal displacements Nodal accelerations

p = k u(t) + m ሷu(t)

Equilibrum equation:

Nodal forces of a ‘vibrating’ element:

m : (element) mass matrix

Nodal forces:



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 =

reaction

in global direction Y

at start node S

due to a

unit acceleration in

global direction X

at end node E

Element: Mass Matrix

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



Beam: Mass Matrix

m m

Lumped mass Distributed mass

NOTE:

It’s only an

approximation



K = global stiffness matrix = Assembly of all ke

F(t) = K U(t) + M ሷU(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

ሷU(t) = global acceleration vector



Modal AnalysisWhat are the eigenmodes of a given structure ?

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

Harmonic displacements

for eigenmode i (Ei ,Wi)Ui(t) = Ei cos(2p Wi t)

( K – (2p Wi t)2 M ) Ei cos(2p Wi t) = 0

valid at any time ( K – (2p Wi)2 M ) Ei = 0

Solution of eigenvalue problem:Wi = (dynamic) eigenfrequency

Ei = (dynamic) eigenform

load independent!



Eigenmodes and DeformationsNumerical structural model: Deformations of a structure

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

u1u2

u3

u4

u5u6

1 2 3 4 5 6

Every deformed configuration can be described as…

Nodal dof and corresponding amplitudes Eigenmodes and corresponding amplitudes

U= { 1 q1 + 2 q2 +...+ 6 q6}= q

or

ui nodal displacement

uii

i modal coordinateequivalent

Every mode is like a independent structure !

qi



Types of Modal Analysis

Response spectra analysis Time history analysis

load-time

function

T1 Ti Tj

response spectrum



Modal Analysis


1. System identification: Elements, nodes, support and loads

2. Build element stiffness and mass matrices

3. Assemble global stiffness and mass matrices

4. Solve eigenvalue problem for a number of eigenmodes

5. Perform further analysis (time-history or response spectra)




Difference betweenDSM and FEM

How to find the local matrices k , kG and m

and local load vectors f?for a specific finite element ?

Chapters 3 and 4

Method of Finite Elements I of Structural Engineering Page 1 Method of Finite Elements I Chapter 2b The Direct Stiffness Method: 2nd Order and Stability Analysis Method of Finite ...

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