FEM Nonlinear

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BAUHAUS--UNIVERSITÄTWEIMAR

FAKULTÄT FÜR BAUINGENIEURWESEN

INSTITUT FÜR STRUKTURMECHANIK PROFESSUR FÜR BAUSTATIK

UNIV.--PROF. DR.--ING. HABIL. C. KÖNKE

HANDOUTS

ADVANCED FINITE ELEMENTMETHODS



Preface 1

PREFACE

The intention of this scriptum is to offer a handout which can be used by interested students par-

ticipatinginthelecturesandexercisesofthemastercourseon AdvancedFiniteElementMethods.

This script is not intended to replace textbooks on the Finite Element Method applied to geomet-

rical and physical nonlinear problems. A script presentation is by intention less systematically

and less carefully worded than a textbook. The major difference between a handout (useful only

when attending the lectures and exercises) and a textbook is the incomplete text in the scriptum,

which has to be completed by the student, while attending the course. Therewith the scriptum

should be of value for all students who have been attending lecturesand exercises and have been

completing the text with their own notes. A script is never intended for use as an exclusive read-

ing.This course is following the idea of generic problem descriptions and of generic solution

methods. Starting from a very broad and universal point of view we will focus on specificdetails

in a second stepand therewith willreduce complexity of theproblems andits solutions.Proceed-

ing in such a way requires the ability of abstract thinking. The Finite Element Method is based

on abstract mechanical and mathematical formulations and therewith requires to deal with these

types of formulations. This already could be seen when studying linear problems and will be re-

peatedly observed when adressing nonlinear problems.

The intensive study of the governing equations, the mechanical, mathematical and algorith-

mic basics of the FEM is fundamental in order to become a responsible user of this numerical

discretization technique and to overcome the status of using such programs only as black boxes.Critical assessment of computed results will be only possible for someone who has obtained a

deeper insight into the basic principles of the method.

It is recommended to study additional literature on this topic. The following page givessome

recommendations for additional textbooks on FEM.



Preface2

List of textbooks

Thefollowing textbooks canbe used as additionalreading forthe courseon “AdvancedFinite

Element Methods” (March 2004):

[ 1 ] Bathe, Klaus--Jürgen: Finite Element Procedures, Prentice Hall 1996

One of the two bibles for the FEM. Covers nearly all different aspects of the method

and refers to additional special literature. Very comprehensible presentation.

[ 2 ] Zienkiewicz, Oleg, C.; Taylor, Robert L.: The Finite Element Method, Volumes 1+2,

Mc Graw--Hill 1989, 4th. Edition

Second bible of FEM community. Written by engineers in an engineering language.

Additionally there are some textbooks which are focussing on special topics, such as varia-

tional formulations or non--linear problems:

[ 3 ] Oden, J.T.: Finite Elements of Nonlinear Continua, McGraw--Hill 1972

Covers FEM for nonlinear problems in structural mechanics.

[ 4 ] Washizu, K.: Variational Methods in Elasticity and Plasticity, Pergamon Press 1975

Specializing on energy and variational formulations, which are the mechanical basic

for the FEM and other numerical discretization techniques.

The following lecture notes from colleagues have been used to develop this scriptum:

[ 5 ] Krätzig, W.: Nichtlineare Finite Element Methoden, Lehrstuhl für Statik und Dyna-

mik, Ruhr--Universität Bochum

[ 6 ] Zahlten: Finite Element Methoden

Lehrstuhl für Baumechanik, Universität Gesamthochschule Wuppertal



1 Introduction into nonlinear strcutural analysis 3

1 INTRODUCTION INTO NONLINEAR STRUCTURAL

ANALYSIS

1.1 Linear and nonlinear structural analysis

Up till now we have restricted ourselves to linear structural behavior, defining the structural re-

sponse (displacements, strains and stresses) as proportional to the applied external loads. This

restriction of our mechanical model has led to the following stiffness relation:

(1.1)

The stiffness term in equation (1.1)

(1.2)

is for linear problems independent of:

S the current strain and stress status of the material

S the current size of displacements.

Asaresultofthisconditiontheglobalequationsystem(linearsystemofequations)couldbequite

easily solved and additionaly we had been able to benefit from the superposition principle:

Right hand side of equation (1.1) could be divided into abitrary load cases F i , which then have

been used each to solve for the associate displacements ui . Finally all displacement results from

partial load cases could be superimposed to the displacements for the total load.

K u1 = F1

+ K u2 = F2

+ K ui = Fi

+ K un = Fn

K ( u1 + u2 + + ui + un ) = ( F1 + F2 + + Fi + Fn )

(1.3)



1 Introduction into nonlinear structural analysis4

The load--displacement curve for any linear problems is a straight line (Fig 1.1).

F

u

u1 ui un

F1

Fi

Fn K

Figure 1.1: Load--displacement curve for a linear problem (single degree of freedom problem)

Thefollowing requirementsmust be fulfilled to classifya problem as a linear structural problem:

S Infinitesimal small deformations.

Thiscondidtion allows to formulate the equilibriumequations using the undeformedconfi-

guration. The directions of all external load vectors are constant (conservative loads).

S Linear elastic material behavior, given by a linear dependency between σ and

σ = C

(1.4)

S Linear kinematical equations,

= Dk u

(1.5)

Observing structures in the real world by experiments and comparing external loads and measu-

red displacements always results in nonlinear functions between these two measurements.




F

wl

F

w

linearstructural behavior

nonlinearstructural behavior

Figure 1.2: Experimentally obtained load--displacement curve for a cantilever beam

→ Assumptionof linear strcutural behavior can be formulated for a large number of structures

under service loads. The error introduced by the assumption of linear structural behavior

remains neglectable in these cases.

Nverthelessthere area number of problems for which theconditionsof linear structuralbehavior

are no longer fulfilled. We will investigate typical engineering examples of nonlinear strcuturalbehavior, for which the assumption of linearity would leed to non--tolerable errors in the re-

sponse. Examples:

S Slender structures can show large deformations already under service loads → geometri-

cal nonlinear behavior

lw

F2

F2

F1

linear theory:

nonlinear theory:

Figure 1.3: geometrical nonlinear behavior for slender structures




In case of geometrical nonlinear problems the equilibrium equations will be formulated for the

deformed structure. Even for theses cases strains can (but must not) remain small.

S A lot of engineeringmaterials, e.g. concrete or reinforced concrete, show nonlinear stress--

strain function already under small stresses.

→ cracking under tension

→ nonlinear σ− function in compression regime

→ physical nonlinear behavior

F

w

σy

σy

ε

F

full plasticity

onset of plasticity

elastic branch

w

onset of plasticity

full plasticity

σ y

σ yσ y

σ y

material law

stress function over thecroos section of a beam

Figure 1.4: physical nonlinear behavior

Investigation of ultimate limit states for structures always requires to account for nonlinear ef-

fects, because structures will behave profoundly nonlinear (geometrical and physical) when co-

ming close to their collapse loads.

Which patterns of nonlinear structural behavior can we distinguish in principle ?




F

u

k2

k1

increasing stiffness(stiffening)k2 > k1 linear

(constant stiffness)

declining stiffness(softening)k2 < k1

Figuer 1.5: nonlinear structural behavior

The strcutural stiffness of a system is variable for nonlinear problems.

(1.6)

→ New methods are required to solve nonlinear systems of equations

For this reason the superposition principle holds no longer. All possible different load combina-

tions have to be studied seperately.

The above mentioned patterns in structural behavior can appear in arbitrary combinations, as itis shown in Figure 1.6. Sometimes this can lead to situations where the system stability is lost.

F

u

Figure 1.6: nonlinear response paths

S All paths in Figure 1.6 describe admissible situations of equilibrium (equilibrium between

external forces and internal stresses is always guaranteed).

S Regionsofstructuralinstabilityarecharacterizedbyadecreaseinexternalforceandincrea-

sing dispplacements at the same time.




S The structure will branch into a buckling path whenever an infinitesimal small pertubation

can initiate a second equilibrium situation. Reaching the branching point in the response

path the structure can decide which of several available equilibrium paths it will follow.

→ Asa result theuniqienessof our solution is lost. We will obtain severalload situatuionswhich

will lead to the same deformations situation.

1.2 Geometrical nonlinear structural behavior of a truss system

Example: Geometrical nonlinear behavior of a simple truss system

Let us study the two truss system in Figure 1.7. The system shall be symmetrically to the middle

plane and shallbe also staticallydeterminate.A single nodal forcein verticaldirection is applied

in the center.

H

P

L L

α0

H = 50 cm

L = 100 cm

P = 8000 kN

E = 21000 kN/cm2

A = 10 cm

2l0

Figure 1.7: truss system

We will solve the system for displacements and internal forces. Due to the statically determinatesituation, we can directly apply equilibriumprinciplesto obtain internal forces.We willcompare

the influence of two different kinematical assumptions onto the results.




Equilibrium:

Figure 1.8: Equilibrium conditions

Material law:

In order to take into account the deformations of the structure we will formulate euqilibrium

conditions with respectto the deformed system. Because the vertical displacement of the middel

node is unknown, we have to solve the system iteratively. We will see in the following that theassumptions about the kinematics of oursystem is considerably influencing the obtained results.

We will investigate theinfluence of kinematics withrespect to response quality andconvergence

rate.

kinematical model A:

Figure 1.9: kinematical model A




kinematical model B B:

Figure 1.10: kinematical model B

The equilibrium condition is requesting equilibrium between external loads and internal forces

(stresses), which are activated by strains, which again are activated by external displacements.

Therefore we can only come to an equilibrium condition if we allow structral defomations and

therewith strains to take place. Otherwise no internal forces will be activated. Therewith we ob-

tainthe following iterative procedure to solve for the deformed equilibrated status of a strcuture:




Equilibrium is reached, whenever results -- e.g. normal forces -- from one iterative step to next

one remain nearly constant. Comparing the results for the two kinematical models, we obtain:




Table 1.1: Convergence behavior for kinematical model A

kinematical model A: exact

i αi [˚] Ni [kN] ∆ li [cm] wi [cm]

1 26,57 8944 4,7619 11,8178

2 20,90 11214 5,9702 15,3493

3 19,11 12217 6,5044 17,0169

4 18,25 12770 6,7987 17,9691

5 17,76 13113 6,9813 18,5727

6 17,45 13342 7,1030 18,9810

7 17,23 13501 7,1881 19,2698

...

20 16,63 13976 7,4409 20,1415

21 16,62 13981 7,4434 20,1503

22 16,62 13985 7,4454 20,1573

Table 1.2: Convergence behavior for kinematical model B

kinematical model B: approximated

i αi [˚] Ni [kN] ∆ li [cm] wi [cm]

1 26,57 8944 4,7619 10,6479

2 21,48 10923 5,8156 13,0040

3 20,30 11528 6,1376 13,7240

4 19,94 11730 6,2449 13,9639

5 19,82 11799 6,2816 14,04616 19,78 11823 6,2943 14,0745

7 19,76 11831 6,2987 14,0844

8 19,76 11834 6,3003 14,0878

9 19,75 11835 6,3008 14,0890

10 19,75 11835 6,3010 14,0894

The following fundamental questions arise:




S automation ?

-- generic formulation of problem

-- FE formulation

S choice of the kinematical model ?

-- validity limits

-- computational effort

S is there always a convergent unique solution ?

-- stability of process

S influence of the iteration method ?

-- convergence rate-- stability of solution

-- criteria for stopping the iteration



2 Nonlinear structural behavior14

2 NONLINEAR STRCUTURAL BEHAVIOR

2.1 Introduction

What are the characteristics of nonlinear strcutural response ?

S no linear stiffness relation.

S superposition principle is no longer valid.

The equilibrium equation -- which always has to be fulfilled -- can be formulated as:

G ( V) = P stiffness relation

internal forces (nonlinearfunction of displacements)

external loads

(2.1)

= (2.2)

We will see that an analytical solution can be only obtained for very simple systems with nonli-

near structural responses. In general the solution has to be obtained by numerical approximation

methods and within the frame of an incrementaly--iterative procedure.

In order to develop the necessary fundamental equations to describe a nonlinear structural re-

sponse, we imagine to follow the response path in increments and progressively obtain an un-

known neighboring state NS from an already known initial state IS. Let us assume that all state

variables -- displacements forces, strains, stresses and so on -- of the initial state IS are known.



2 nonlinear strcutural behavior 15

V

P

unknown

response path

IS

NS

residual forces whichare not in equilibrium

Figure 2.1: Initial state and unknown neigbouring state

How can we obtain the necessary stiffness relation in order to step from the initial state IS into

the new (unknown) neighboring state NS ?

1. Increment all variables

→ Therewith we obtain for the new neighouring state NS

2. We can now develop a taylor series expansion.




Taylor series expansion for function f(x)

x

f(x)

Figure 2.2: taylor series expansion of an arbitratry function f(x)

S let us assume that the function value f(x) in x should be known

S we want to calculate the function value in x + ∆ x : f (x + ∆ x)

f (x + ∆ x) = f (x) + ∂f ∂ x

| x · ∆ x + 12

∂2 f ∂ x 2

| x · ∆ x 2 +

infinite series

(2.3)

Often the taylor series expansion is truncated with the linear part.

G ( V + V +) = G ( V ) + ∂G∂ V

| V = V

· V +

tangential stiffness in theinitial state K T

(2.4)

3. Linearized stiffness relation (tangential stiffness)

G ( V ) + K T ( V ) · V +

= P + P+ (2.5)

We can now distinguish two situations:

a. The initial state is in equilibrium and we want to step forward from the IS to the NS by ap-

plying the next load increment P+ .

G ( V ) = Pthis means

⇒ K T ( V ) · V + = P+ load step

⇒ solve for V

+

(2.6)




b. The new neighboring state NS, which will become our next initial state IS, is not in equiili-

brium.Wewillnowiterateaslongasequilibriumbetweeninternalforcesandexternalloads

is ibtained. While this iteration is taking place no new load increment will be applied.

P+ = 0this means

⇒ K T ( V ) · V + = P − G ( V )

⇒ solve for V +

∆P = residual forces whichare not in equilibrium

iteration

(2.7)

2.2 Tangential stiffness matrices from principle of virtual works (vir-

tual displacements principle)

We start fromthe equilibrium formulationusingtheprinciple of virtualwork (principle of virtual

displacements):

− δ Tσ dV + δ uT p dV = 0 (2.8)

Question: Where is the difference in compraison to the linear theory?

S kinematic equation

linear: = DK u DK = linear differential operator

geometrical nonlinear = DKnl(u) u DKnl = nonlinear differential operator

constitutive law

linear elasticity σ

=E

We cannow investigate thestep from theinitial state IS (which shallbe in equilibrium and which

also shallbe well known in allhistory variables) tothe neighboringstate NS. In order to step from

IS to NS we increment the external loads as well as the external displacements.




load

displacement

strains

IS NS

stresses

We can now insert all variables in their incremental form into the principle of virtual works and

by regarding the variational forms for external and internal displacements:

u = u + u+ δ u = δ u+

the initial state is fixedand therewith can not be varied

⇒ variation can onlyaffect the incrementalcomponent

(2.9)

= + + +

++ δ = δ + + δ++ (2.10)

we finally obtain:

P. V. V. (it should be pointed out that boundary loads will be neglected for the sake of simplicity)




− V

δ Tσ dV +

V

δ uT p dV = 0

− V

(δ +T + δ++T) (σ+ σ

+ + σ++) dV +

V

δ u+T (p + p+) dV = 0 (2.11)

Let us now study the different parts in eq. (2.11) in detail. We negelct terms of higher order than

++

.

virtual work of external loads in initial

state

virtual work of incremental loads when stepping from initial state toneighboring state

virtual work of internal forces (stres-ses) in the initial state, which are al-ready in equilibrium with the appro-priate external loads of the ISP

initial stress matrix orgeometrical stiffness K g

This matrix can be splitted into

a linear initial stress matrix

and a

nonlinear initial stress matrix

elastic stiffness matrix K el

This matrix can be splitted into:

linear stiffness matrix(independent of )

linear initial displacement matrix(linear dependency of )

nonlinear initial displacement matrix(nonlinear dependency of )

u

u

u




How can we obtain theses matrices for a specific structural element ?

Similar to our procedure in linear finite element methods we will interpolate the state variables

by chosen functions. These functions are supported by appropriate state variables in the element

nodal points.

u = N v (2.12)

N istheshapefunctionmatrix,whichisdefiningthefunctionalbehaviorofthedisplacementfield

u with supports in the element nodal points v.

Let us define the nonlinear kinematical equation in the following form.

= DKnl(u) · u

= DKL +12 DKN(u) · u

(2.13)

→ Insert the incremental form of the displacement field:

u = u + u+

= DKL + 12

DKN (u + u+) · (u + u+)

(2.14)




This transformation is requiring the symmetry of the nonlinear differential operator DKN which

is fulfilled:

DKN(u) · u+ = DKN(u+) · u(2.15)

Finally we introduce the discretized field variables (discretized form of the displacement field)

into the kinematical equation and obtain

= DKL + 12

DKN(N · v) N · v

+ = DKL + DKN(N · v) N · v+

++ = 1

2DKN(N · v+) N · v+

(2.16)

Variation of these displacement fields is resulting in:

δ u+ = N δ v+

δ + = BL · δ v+ + BN( v) · δ v+

δ ++ = BN ( v+) · δ v+

(2.17)

Everything is now inserted into the principle of virtual works, leading to following equation:




δ v+T V

NT p dV + V

NT p+ dV

− V

BTL + BN( v) T

· E · BL + 12

BN( v) · v dV

− V B

T

N( v+

) · E · BL +12 BN( v) · v dV

− V

BTL + BN( v) T

· E · BL + BN( v) · v+ dV = 0

(2.18)

Let us resort this equation:

Thefirst twointegrals obviously describe external loads in theinitial state IS andtheneighboring

state NS.

The third integral describes internal forces in the initial state IS whichare already in equilibrium

with the associated external loads in the IS.

− V

BTL + BN( v) T

· E · BL + 12

BN( v) · v dV = Fi (2.19)

We will now study the fourth and fifth integral in detail:




(2.20)

Therewith we can write the already known stiffness equation

K T · v+ = P − Fi(2.21)

also in the following form

( K e + K uL + K uN + K σL + K σN) · v+ = P − Fi (2.22)

2.3 Example: truss element

We will now specialize the generic matricesfrom above for a specific element: 2D truss element

Nonlinear kinematical equation

A differential small 2D truss elementwill be studiedin Figure2.3in theinitial undeformedconfi-

guration and in a deformed configuration. Take into account that neither u nor w will be assumed

as infinitesimal small:




z

x

dx

wu

u + du

w + dw

dl

Figure 2.3: Nonlinear kinematic of truss element

The length of deformed truss element can be calculated by Pythagoras’ theorem

dl = (dx + du)2 + (dw)2 = 1 + 2 u + u2 + w2 dx (2.23)

When the strain is defined as the ratio of elongation to the original length we obtain:

= dl − dx dx

= 1 + 2 u + u2 + w2 − 1 (2.24)

Develop the Taylor series expansion

1 + x = 1 + 12

x − 18

x 2 + 124

x 3 − (2.25)

and abort after 1. term results in the approximation:

= 1 + 12

2 u + u2 + w2 − 1

= u + 12

u2 + w2 (2.26)

The longitudinal displacement u is significant less than the transversal displacement w becauseof the high extensional stiffness. So approximatively we can disregard the term u 2 in relation

to w2 and the simplified kinematic equation is:

= u + 12

w2

= DKL u + 12

DKN(u) · u

(2.27)

Incremental formulation

According to section 2.1 the displacement state can be divided into a known basic state and an

unknown increment:




u = u + u+

w = w + w+(2.28)

Therewith the total strain results in:

= + + + ++

= ( u + u+ ) + 12

( w + w+ )2

= u + 12

w2 + u+ + w w+ + 12

(w+)2 (2.29)

with the single terms

=

+ =

++ =

(2.30)

According to eq. (2.9) and eq. (2.10) we obtain the variation of displacement variables

δ u = δ u+

δ = δ + + δ++

(2.31)

with the variations of strain increments

δ + = δu+ + w δ w+

δ ++ = w+ δ w+

(2.32)

Discretization

Now we have to find approximations for all displacement components in the kinematical equa-

tions. Additionally to the approximative function of longitudinal displacement u in the linear

theory we also choose in the nonlinear theory a trial function for the transversal displacement w.

z

x

l1

2w1

u1s,x

w2

u2

Figure 2.4: Displacement variables in nodes according to the nonlinear truss element.




The nonlinear approximations within the normalized coordinate system s are obtaint equivalent

to the linear case:

u = Nu · v u+ = Nu · v+

w = N w · v w+ = N w · v+(2.33)

with the shape functions

Nu = 1 − s 0 s 0 Nu = 1l

− 1 0 1 0

N w = 0 1 − s 0 s N w = 1l

0 − 1 0 1

ddx

= 1l

dds

x = s l

(2.34)

and the vectors of local element degrees of freedom

vT = u1 w1 u2 w2

v+T = u+1 w+

1 u+2 w+

2

(2.35)

With theknown discretedisplacements v of the basic state all displacement variables of thebasic

state can be calculated in every point of the element using shape functions N. Therefor we need

the partial derivatives that are constant over the element domain because of linear shape func-tions.Inserttheshapefunctionsintheincrementalprincipleofworkleadstotheelementmatrices

(dx = l ds):

Linear stiffness matrix:

K e =

BTL · E · BL dx = EA

1

0

Nu · Nu ds (2.36)

Initial deformation matrix, linear in v :

K uL =

BTL · E · BN( v) + BT

N( v) · E · BL dx

= E A w 1

0

NT w · Nu + NT

u · N w ds

(2.37)




Initial deformation matrix, quadratic in v :

K uN = BT

N( v) · E · BN( v) dx = E A w2

1

0

NT

w · N w ds (2.38)

Initial stress matrix, linear in v :

K σL =

BTN( v) · E · BL dx = N

1

0

NT w · N w ds (2.39)

Initial stress matrix, quadratic in v :

K σN = 0 (2.40)

Vector of internal forces:

Fi =

vT · BL + BN( v) T· E · BL + 1

2BN( v) dx

= N 1

0

NTu ds + N w

1

0

NT w ds

(2.41)

K σ causes stiffness terms according to displacement variables w1 andw2 , thatact perpendicular

to the longitudinal axis.

For compression in truss K σ will become negative → stiffness decreases.

For tension in truss K σ will become positive → stiffness increases

The evaluation of the above integrals results in:




K e = E A

1

0

− 1

0

0

0

0

0

− 1

0

1

0

0

0

0

0

K uL = E A w

0

1

0

− 1

1

0

− 1

0

0

− 1

0

1

− 1

0

1

0

K uN = E A w2

0

0

0

0

0

1

0

− 1

0

0

0

0

0

− 1

0

1

(2.42)

K σ = N

0

00

0

0

10

− 1

0

00

0

0

− 10

1

Fi = N

− 1

0

1

0

+ N w

0

− 1

0

1

(2.43)

The linear stiffness matrix depending only on the original element stiffness is constant. The in-

itial deformation matrix includes the influence of the current deformation state on the stiffness.

And the initial stress matrix includes the current state of internal forces. The vector of internal

forces represents the nodal forces that are energetically equivalent to the state of internal forces

within the element. All matrices given above are described in local displacement components.

So finally we have to do a transformation into global nodal degrees of freedom. Doing this the

incremental element stiffness relation can be written as:

K e + K uL + K uN + K σ · v+ = K T · v+ = P − Fi (2.44)




The development of the global equation system on the system level

K T · V + = P − Fi (2.45)

can be done as known for the linear case, but for solution we now need iterative solution algo-

rithms. That is the topic of the following sections.

2.4 Tangential matrices of the beam element

Atnextwewanttodevelopthetangentialmatricesofanevenbeamelementwithoutsheardeflec-

tion. Therefore at first we want to write the basic equations of mechanic:

Kinematik

external displacement variables: u = u w

internal displacement variables: =

dx

u

w

u + du

w + dw

(1 + ) dxε

d (1 + u’ ) dxl ≈

⋅

dw = w dx

Figure 2.5: Nonlinear kinematic of beam element, truss part

The length of the deformed element can be obtained by theorem of Pythagoras (analog eq. 2.23)

dl = (dx + du)2 + (dw)2 = 1 + 2 u + u2 + w2 dx (2.46)

If we define the strain as the ratio of elongation to the original length we obtain:

= dl − dx dx

= 1 + 2 u + u2 + w2 − 1 (2.47)

Then we apply the Taylor series expansion

1+

x =

1+

1

2x

−1

8x 2

+1

24x 3

− (2.48)




and consideration of only the 1. Term results in the approximation

= 1 + 12

2 u + u2 + w2 − 1

= u + 12

u2 + w2 (2.49)

The longitudinal displacement u is significant less than the transversal displacement w because

of high extensional stiffness. So approximatively we can disregard the term u2 inrelationto w2

and the simplified kinematic equation of the truss part is:

= u + 12

w2

= DKL u + 12

DKN(u) · u

(2.50)

Using a beam element we must also consider the distortion.

R

x

z w

ds (1 + ) dx≈

ε

+ φ

− φ ≈ dwdx

δ φ

Figure 2.6: Nonlinear kinematic of beam element, bending part

Arc length:

ds ≈ (1 + ) dx (1 + ) dx = R dφ (2.51)

Curvature:

= 1

R

= 1

1 +

dφ

dx

≈ φ (2.52)




dw

− φ

d ≈ (1 + u) dx

Figure 2.7: Nonlinear kinematic of beam element

− φ = dw(1 + u) dx

φ = − 1(1 + u)

dwdx

≈ − w(2.53)

Therewith we obtain:

=

d x

0

0

− d2 x + 1

20

0

w dx

0

· u w

= ( DKL + 12

DKN ) · u

(2.54)

Advice: For the kinematic equation of bending part we use the relation, which is known from

the linear theory.

Constitutive law:

Internal forces: σ = N

M

Stress--strain relation:

N

M =

E A

0

0

E I ·

σ = E ·

(2.55)

Equilibrium:




According to the procedure for the truss element we also develop shape functions for the beam

element, which describe the displacements within an element depending on the nodal values.

Then we insert the discretized form of basic mechanical equations (including the shape func-

tions) in the principle of virtual displacements. By integration of the resulting terms we obtain

the nonlinear stiffness matrices of the beam element. Even though we can analytically integrate

the stiffness matrices, the matrices are very complex, so that we can only handle them numeri-

cally. Thus the single matrices are not illustrated here.



3 Solution methods for nonlinear equation systems 33

3 SOLUTION METHODS FOR NONLINEAR EQUA-

TION SYSTEMS / PATH FOLLOWING ALGORITHMS

3.1 Introduction

We are not able to solve the nonlinear equation system of chapter 2 in one single step as it could

be done for the linear equation system. Due to the applied nonlinear stiffness operator the solu-

tion must be applying iterative algorithms.

When we additionally apply the loading as a number of increments we call it an incremental--ite-

rativ solution method. Such methods are also characterised as path following algorithms, be-

cause the structural behaviour is analysed for all load steps of the load history and therewith forthe complete load path.

In the following part we want to consider laod histories with a total load vector

P = λ1 P1 + λ2 P2 + + λn Pn (3.1)

depending on n scalar magnification factors, the loadfactors λi. The single loadvectors Pi repre-

sent the nodal force values caused by arbitrary load collectives consisting of point, line and sur-

face or volume loads.

In order to simplify the following considerations we will restrict to an one--parameter load sy-

stem; all loadfactors are constant and only one will be incremented. Because of simplicity we

don’t need to index the load factor and so the remaining load factor is called λ.

P = λ P0 (3.2)

P0 is a reference load. Figure 3.1 shows qualitative some possible nonlinear response paths wit-

hin the load--deformation--diagram.

Maximum

Umkehrpunkte

MinimumVerzweigungspunkt

primärer Pfad

sekundärer Pfad

tertiärer PfadVerzweigungspunkt

displacement V

loadfactor λ

Figure 3.1: Qualitative illustration of nonlinear response paths.



3 Solution methods for nonlinear equation systems34

A nonlinear response path canbe very complex. Ascan be seen in figure 3.1 such a response path

can include snap through points and rebound points,the primary path can branch into secondary

paths which then can also branch to tertiary paths.

Algorithms that allowsus to followarbitrary nonlinear paths in the load--displacement--space are

called path following algorithms. We can differ between three types of such algorithms:

S Newton Raphson method: The load axis is incremented. The equilibrium state is iterati-

vely searched for a fixed load level.

S Arc length method: The response path is incremented so that the load level has to change

iteratively.

S Modified Newton method: Instead of formulating the exact tangential stiffness matrix a

secant stiffness matrix will be developped by using so called update methods. Therewith

the secant stiffness willbe updatedduringthe iteration. A well--establishedalgorithm is theBFGS--update algorithm according to its inventors Broyden, Fletcher, Goldfarb and

Shanno.

Criteria for chossing solution algorithms for nonlinear equation systems are:

S Accuracy: How is the accuracy of the approximative solution (using an acceptable calcula-

tion time) ?

S Robustness:Howsensitiveisthecalculationprocessduetochangesininputparametersand

the current problem? E.g.: Is it possible to use arbitrary large load steps or does the algo-

rithm show divergence for large load steps ?

S Computional efficiency: How much computing time is required to solve the complete pro-

blem? The time depends on the number of load steps, the number of iterations per load step

and the calculating time per iteration.

We will have a closer look at different iateration algorithms in the following sections.

3.2 Newton--Raphson method

The Newton--Raphson method, which is one of the most frequently used iteration schemes, is

based on the incremental solution concept. For a fixed current load level the iterative corrections

of displacement arecalculated fromthe out of balance forces. Therefore respectively the tangen-

tial stiffness matrix is used. The method is characterised by the following attributes:

S It ispureforce controlledmethod. The loadlevelis fixedand thealgorithmwilltryto iterati-

vely find an equilibrium state for that given load level. Therewith it is unpossible for this

method to overcome extremal points, such as snap--through points.

S Thealgorithmshowsaquadraticconvergenceintheiterationwhenweusetheexacttangen-

tialstiffnessmatrix.Fromthisitfollowsthattheerrordecreaseswithquadraticalorderfrom

one iteration to the next. (Example: error in thefirst iteration: 10--1,errorinthe2nd,3rdund

4th iteration: 10--2, 10 --4, 10 --8 and so on.)




λ

∆λ

0V+

1V+

2V+

3V+

4V+

V

initial statein equilibrium

neighbour statein equilibrium

iterative temporary configurations(not in equilibrium)

t+∆t K 0 t+∆t K 1

t+∆t K 2

t+∆tF

1i

Figure 3.3: Standard Newton--Raphson method

Taking the incremental--iterative equilibriumequation from chapter 2, we have to finda solution

for the displacement state V, which is representing an internal force situation, described by the

vector Fi (V) , in equilibrium with all appplied external forces Ft+∆t F − t+∆t Fi(V) = 0 . (3.3)

Applying a Taylor series expansion we obtain the the increment in the displacement by

t+∆t K ( i−1) ∆V( i ) = t+∆t F − t+∆t F( i−1 )i

(3.4)

where t+∆t K ( i−1) is the current tangent stiffness matrix.

The improved displacement at the end of the actual iteration step is given by

t+∆t V(i) = t+∆t V(i−1) + ∆ V( i ) (3.5)

3.2 Modified Newton--Raphson method

In the Modified Newton--Raphson method we use the same tangent stiffness matrix for all itera-

tions; the tangent stiffness is formulated once in every load increment at the beginning of theite-

rative process.

The method is characterised by the following attributes:

S It ispureforce controlledmethod. The loadlevelis fixedand thealgorithmwilltryto iterati-

vely find an equilibrium state for that given load level. Therewith it is unpossible for this

method to overcome extremal points, such as snap--through points.




S The algorithm shows slower convergence thanthe StandardNewton--Raphson method,but

is requiring fewer computing timeto reestablishthe stiffness matrix thanthe StandardNew-

ton--Raphson method.

λ

∆λ

0V+

1V+

2V+

iV+

V



iterative temporary configurations(not in equilibrium)

t+∆t K 0 t+∆t K 0

Figure 3.4: Modified Newton--Raphson method

The governing equation is given as follows

t+∆t K 0 ∆V( i ) = t+∆t F − t+∆t F( i−1 )i

(3.6)

where t+∆t K 0 istheinitialtangentialstiffnessmatrixinthecurrentloadincrement.Thistangent

stiffness is not updated in every iteration cycle but kept constant.

3.4 Arc--length method

In order to overcome extremumpoints in the nonlinear response path(e.g. snap--throughpoints), we have to find an iterative method which can increase and decrease the load level. Arc--length

methods are incrementing the response path itself, therewith they allow an iteratively changing

load level.

The method is characterised by the following attributes:

S It is combined method, which is controlling the displacement and the forces at the same

time. The load level is changing iteratively. Therewith it becomes possible to overcome ex-

tremal points, such as snap--through points.

S The algorithm is converging also for nonlinear response paths in regions where an instable

behaviour can be observed (e.g. incresing displacements fro decreasing loads).




0V

+

V



∆

t λ F

( t+∆t ) λ F

F

Figure 3.5: Arc--length method

The governing equilibrium equation at time t + ∆t is given by

t+∆t λ F0 − t+∆t Fi(V) = 0 , (3.7)

withanunknownscalarfactor (t+∆t) λ whichhastobedetermined,and F0 asreferenceloadvec-

tor.Thescalarloadfactor(t+∆t)

λ canincreaseordecreaseineachstep.Applyingthesametaylorseries expansion as already done in the Newton Raphson iteration schemes, we obtain

t+∆t K ( i−1 ) ∆V( i ) = t+∆t λ( i−1 ) + ∆ λ ( i ) F0 − t+∆t F( i−1 )i

(3.8)

Equation (3.8), which consists of n equations, has (n +1) unknowns. These are the n unknown

displacement increments in the vector ∆V( i ) andtheunkonwn scalar load factor ∆ λ ( i ).Inorder

tosolveequation(3.8)wehavetodefineanadditionalequation,linkingthetheloadfactor ∆ λ ( i )

and the displacement increments ∆V( i ).

Lets define the total diplacement increment for the current step as

V( i ) = t+∆t V( i ) − t V (3.9)

V( i ) is definingthe sumover all displacement increments ∆V( i ) up to the currentiteration cycle

(i). Similar we can define the total load factor increment for the current step as

λ( i ) = t+∆t λ( i ) − t λ (3.10)

defining the he sum over all load factor increments ∆ λ ( i ) up to the current iteration cycle (i).

Onepossibleconstraintequation betweenthese twomeasures is given bythe constant arc--length

criterium

λ( i ) 2

+

V( i )T V( i )

β =( ∆

)2

(3.11)




β is a normalisation termmaking thesecondtermdimensionless.Solving thissystem of equation

is done by rewriting eq. (3.8)

t+∆t

K

( i−1 )

∆V

( i )

=t+∆t

λ

( i−1 )

F0 −t+∆t

F

( i−1 )

i (3.12)

t+∆t K ( i−1 ) ∆V = F0 (3.13)

Therewith the displacement increment is obtained as

∆V( i ) = ∆ V( i )

+ ∆λ( i ) ∆V (3.14)

The total displacement increment and the total load factor increment are given by

V( i ) = V( i−1 ) + ∆V( i ) = V( i−1 ) + ∆V( i )

+ ∆ λ ( i ) ∆V (3.15)

λ( i ) = λ( i−1 ) + ∆ λ ( i ) (3.16)

Inserting eqs. (3.16) and(3.15) intoeq. (3.11) is leadingto a quadratic equation for ∆ λ ( i ),which

can be solved.

3.5 Convergence criteria

An optimal algorithm is obtaining very accurate results calculated in little time and with high

robustness. Often the requirements of high computational efficiency and highaccuracy are con-

tradictory. In order to increase the accuracy of the obtained result the number of iteration will

normally increase and therewith more computing time has to be spend.

All iterative soluton strategies have to define convergence criteria allowing to decide upon the

end of the iterative process. Therefor we define the ratio of an iterative norm Zit to a reference

norm Zref :

=Zit

Zref (3.3)

If the value is smaller as a given tolerance tol the iteration stops. As tolerance norm we can

use arbitrary vector or energy norms. E.g. the euklidian vector norm is defined as the length of

a vector (square root of the sum over the squares of the vector components):

Z = V 2 = n

i=1

( V i )2 (3.4)

A norm which is combining force and displacement measures is given by the energy norm,defi-

ned as the scalar product of force and displacement variables (external or internal force and dis-

placement variables):

Z = PT · V (3.5)

For simple vector norms we have the two possibilities to use force or displacement norms. But

in both cases of the simple vector norm there exist the problem that the single components have

different units. E.g. the force vector includes forces and moments thathave should be considered

in the vector norm calculation with different weightings. Furthermore the following figure 3.6




shows that force and displacement norms show different informations. For a softening system

a small error in the equilibrium state (force vector norm) can be correlated to a large error in the

displacement variables. For a hardening system we have the opposite effect. Therefor a small

error in displacement state (displacement vector norm) can be correlated to a large error in theequilibrium of force values.

PIteration i+1

Iteration i

P -- Fuii

V

iV+

P

P -- FuiiIteration i+1

Iteration i

V

iV

+

Figure 3.6: Differences in accuracy for force and displacement values

In the case of a vector norm the iterativ norm Zit includes the displacement or force values of the

last iteration step and in the case of an energy norm it includes the product of both vectors. The

reference norm Zref refers either to the current total state or to the current load increment.



4 Stability problems40

4 LOSS OF STABILITY

4.1 Plane arc

geometrical nonlinear stress problem S snap--through problem S braching, buckling problems S

geometrical imperfections

w

RR

L

P0

L = 100 m

Figure 4.1: Plane circular arc

In the following we will study a plane circular arc, with a concentrated force in the symmetry

axis and varying radius R.S R large → shallow circular arcs

S R small → steep circular arcs

4.1.1 Geometrical linear analysis

We will vary the radius R of the circular arc and perform a geometrical linear analysis for each

arc. If we plot the middle displacement w with respect to the changing radius R, we obtain figure

4.2.

min R ≈ 85 mR = L 2

displacement w

radius R

Figure 42: Geometrical linear analysis of the plane circular arc



4 Stability problems 41

4.1.2 Complete geometrical nonlinear analysis

We will now fix the radius to R = 800 m and will analyze the response behaviour by applying

a full geometrical nonlinear (incrememtal--itearative analysis) analysis. Therefore we apply the

load in increments P = λ P0 stepwise.

λ

w

Figure4.3: Complete geometricalnonlinear analysis of the plane circulararc with radius R = 800

m

Figure 4.3 is showing a monotonously increasing function. Each load level λ corresponds to ex-

actly onedisplacement situation for w. All points along thisreponse pathrepresent stableequili-

brium situations. The final collapse of thesystem is governed by a stress criterion; if thecritical

stresses in one cross section of the arc are reached, the structure will collapse.

The load bearing capacity of the system is governed by the maximum acceptable stress

→ nonlinear stress problem

If the acceptable stress for the used material would be infinite, the load bearing capacity would

be infinite hight.

Wecanthereforeconcludethatanonlinearstressproblemisleadingtoalocalfailureinstructural

systems (e.g. plastic hinge in one cross section of one structural member).

4.1.3 Snap--trough problem

The gradient ofthe reponse curve in figure 4.3 is showing a minimum in itsmiddle sectionbefore

it is showing an increasing gradient towards the end. The structure has therefore regions where

it is showing softening behaviour (middle section of the response curve) and finally a stiffening

behaviour in the final section. The distinctness of the minimum is depending on the radius of the

arc.

Lets study an arc of radius R = 400 m. We will obtain the following load--displacement curce.




λ

w A

B

C

D

Figure4.4: Complete geometricalnonlinear analysis of the plane circulararc with radius R = 400

m

In this case the load--bearing capacity of the structure is limited, even in the case of an infinite

high acceptable stress, by the local maximum in point B of the response curve. This point is des-

cribing the snap--trough limit load.

Thestructuralfailureisa global failure, without cross sections reaching thecritical stress condi-

tion:

→ loss of stability problem

How can we realize such a behaviour in an experiment?

load controlled experiments

the load will incrementally increase until point B is reached, from there a dynamic snap--

through process will lead to point D. We will observe a damped dynamic vibration in the

vicinity of the equilibrium point D.

displacement controlled experiments

thedisplacementwwillincrementallyincreaseuntilpointBisreached.Thenecessaryforce

to produce this displacement can be measured. Using displacement controlled experiments

enables us to follow the fullresponse path, even in unstable sections (e.g. postbuckling sec-

tions).

The section BC of the response curve is representing instable equilibrium situations.

Onecriterion to decide on stable or unstable equilibriumsituations is based on the maindiagonal

coefficients of the Choleski factorized tangent stiffness matrix. We can distinguish three cases:

all main diagonal elements are positiv

→ stable equilibrium situation

one or more main diagonal elements are zero

→ we observe a critical point (e.g. point B in response curve figure 4.4)

one or more main diagonal elements are negative

→ unstable equilibrium situation

the number of negative main diagonal elements is describing the degree on instability.




In the observed example section BCof theresponse curve is characterized by one negative main

diagonal element.

→ degree of instability i = 1.

Snap--trough problems often occur suddenly, without showing large displacements before the

critical pointis reached. The loss of stabilityis therefore a very dangereous failure type for struc-

tures.

4.1.4 Branching, buckling

We will now construct the circular arc steeper, by decreasing its radius to R = 150 m.

λ

w A

B

C

D F

E

V

W

Figure 4.5: Complete geometrical nonlinear analysis for the plane circular arc with R = 150 m

We can now observe sections along the reponse path which are representing instable situations

between B and E. Their degree of instability is i = 1 and i = 2.Up to now wehave beeninvestiga-

ting the plane circular arc with a perfect geometry and without any temporary geoemetrical per-

tubations.If wenowadd small geometricalperturbationswhile we followthe nonlinear response

path (in each increment we add a small geometrical perturbation before searching for the new

equilibrium), we can observe a new phenomenon.

In point V before we reach the snap--through load in point B, the structure buckles and the re-

sponse path is branchingoff into a secondary path VW. Afterreaching point W thecurve is following again the primary response path.

This behaviour is called buckling or branching:

S In a branching point (e.g. V) the structure can decide to follow the primary resposne path

or to branch off into a secondary path. Both pathes describe admissible equilibrium situa-

tions.

S A geometrical perfect structure would always follwo the primary path, if no external per-

tubations would be applied.

S If an external geometrical pertubations is applied, which can be arbitrary small, the

response path would branch off into the secondary path.




S The primary and the secondary path can describe instable situations, but not necessarily.

The symmetry which has been observed in the circular arc system is now lost. In order to find

the branching point V, the total structure has to be investigated.

4.1.5 Geometrical imperfect structures

All previous investigations in chapter 4 have been focusing on geometrical perfect structures.

The geometrical pertubation in oint V, which has been leading to a buckling or branching pro-

blem has been only temporarily applied. The pertubation has been applied in each incremental

load step in order to investigate if the structure will buckle (if the response path will branch off

into a secondary path).

Real structures always show imperfect geometries. We will e.g. never find perfect straight co-

lumns which are perfectly oriented in the vertical plane. Geometrical imperfections which are

affin to the eigenforms of a structure are the most unfavourable shapes (called eigenform--affinimperfections). Lets study the circular arc with radius R = 150 m as an example. An eigenform--

affin imperfection representing the first eigenform and sacled with a factor λ is applied to the

structure in the beginning.

λ

w A

B

f = 0.0V

f = 0.1

f = 0.5

Figure4.6: Complete geometricalnonlinearanalysis of theplanecircularand geometricalimper-

fect arcwith radius R = 150 m, f is describing thescalar factor for theappliedgeometrical imper-fection

The geometricalimperfections are resulting in an unsymmetrical structural response frombegin

on. Therewith the branching point V is disappearing and the branching problem of the perfect

structure is becoming a snap--through problem for the imperfect structure. The snap--trough

loads are always smaller than the load level in the branching point V.

4.2 Stability characteristics of nonlinear response pathes

The governing equation for geometrical nonlinear problems has been formulated in chapter 2 in

the form of a tangential stiffness equation.

K T ( V ) · V + = P+ incremental step




K T ( V ) · V + = P − Fi iterative step

We will now use this equation to investigate the stability behaviour of a point IS (initial state)

and restrict the investigation to one--parametric load systems. The load will be increased by a

scalar factor λ:

P = λ P0(4.1)

stability criterion:

We investigate a point in the initial state IS. The first linear step of a Newton--Raphson Iteration

is obtained by:

K T · V + = P+

= λ+ · P0

λ+ = λ− λ

(4.2)

In order to solve this linear equation system, we transform the tangential stiffnessK

T into a dia-gonal matrix (coefficients only and only on the main diagonal of the matrix) K *T :

K *T · V +* = λ+ · P0*

= λ+ ·

(4.3)

Therewith we can obtain the solution for each line independently by subdivision:

V+*i =

λ+ · P 0*i

K *Tii

(4.4)

We now define the invetigated initial state IS as stable, if an increase in the load λ+ is leading

to an increase in the displacements V+* .

definition:

S each point of arbitrary nonlinear resposne path ist stable (instable), if an increase in the load

+λ+ is leading to an increase (decrease) of the displacement increments V+*

ifor all de-

grees of freedom (at least for one degree of freedom).

S In this case every (at least one) element of the diagonal form K *Tii becomes positive (nega-

tive) and the appropriate increment in the energy

V+*i · P 0*

i

becomes positive (negative).

How can we transform the tangential stiffness matrix K T into the diagonal form ?

While solving the linearized equation system (e.g. using the Choleski scheme), we have to trans-

form the tangential stiffness matrix K T into the follwing form




K T = LTD L

=K T

(4.5)

This leads to: det K T = det D

We nowobserve, thatwe canspare the transformationof K T into K *T and usethe diagonal matrix

D, which we have to establish in the course of the solution process.Therewith we can summarize:

S stable equilibrium: det K T > 0 ∀ Dii > 0 (4.6)

S instable equillibrium: det K T > 0 ∃ Dii < 0< (4.7)

It should be mentioned that the instability condition which often can be found in literature:

det K T < 0 is not correct. This is due to the fact that with an even number of negative coeffi-

cients Dii the determinant det K T > 0.

We will now check if the initial state IS is showing ambigous solutions. If mor than one solutionis possible, the stiffness equations should have more than one solution for the same load level.

This eqaution only shows a nontrivial solution, if and only if

det K T = 0 ⇒ V +e (4.8)

Now we can conclude:

Ambigous points (indifferent or critical equilibrium) are requiring:

det K T = 0 ; ∃ Dii = 0 (4.9)

We therefore have to solve an eigenvalue problemin order to find these critical points along

the response path.




The appropriate eigenvector V +e is describing the alternative solution, which can be found

instead of the primary solution.

This alternative solution is known as buckling shape

Summary:

K T = LT · D · L equilibrium

∀ Dii > 0 stable

∀ Dii ≥ 0

∃ Dii = 0indifferent

∃ Dii < 0 instable

(4.10)

This investigation has to be performed in each equilibrium pointin thecourse of a complete geo-

metrical nonlinear analysis. This procedure is requiring some computation time. In order to re-

duce thenecessarycomputationaleffort wewill searchfor anapproximativemethodto formulate

the eigenvalue problem.

We substract the linear stiffness K e from the tangential stiffness matrix K T andobtain the nonli-

near matrix part of the full tangential stiffness matrix

K N = K T − K e . (4.11)

The equilibrium condition in the indifferent point is then obtained by K e + K N

· V +e = 0 . (4.12)

The nonlinear stiffness matrix is developped into a Taylor series expansion with respect to the

current load level P = λ P0

K N = Λ K NL + Λ2K NQ +Λ

3K NK +

(4.13)

All matrices of eqs. (4.12) and (4.13) are well known. As a result the eigenvalue problem can

be fomulated as:

K e + Λ K NL + Λ2K NQ +Λ

3K NK + · V +e = 0 (4.14)

This equation is describing a general eigenvalue problem. Depending on the highest power of

Λ the problem is a linear, quadratic, cubic ... eigenvalue problem.

The simplest case is given by the linear eigenvalue problem:




K e + Λ · K NL · V +e = 0 (4.15)

S The obtained eigenvalues Λi are describing the distance between critical load levels i and

the current load level:

Pcritical = Λ · P (4.16)

S The approriate eigenvectors V +are representing the buckling shapes (deformation pattern

wchich is characterizing the alternative response pathes). These eigenvectors are giving a

qualitativeinformationaboutthebucklingshapebutnoquantitativeinformation(noab-

solutedisplacements). This restrictionto a qualitativeinformationis a fundamental charac-

teristic of eigenvalue problems.

S The Taylor series exapnsion and the truncation of the infinite series after the linear term is

leading to an appoximation error. The magnitude of this error is depending on the degreeof nonlinearity for the specific problem before the critical point is reached.

Let us study this effect by investigating an example.

TheplanecirculararcwithradiusR=250hasbeeninvestigatedbyacompletegeometricalnonli-

near analysis. Additionaly we have been solving the linear eigenvalue problem of eq. (4.15) for

each incrementally applied load level. For each load level we established the nonlinear part K Nof the full tangential stiffness matrix K T andsolvedthelinearized eigenvalueproblem. Thecriti-

cal load factor is obtained by

λ

critical =Λ · λ (4.16)

The results are described in figure 4.7

λ = load factor

w

λ = 3,24

Figure 4.7: stability analysis for the plane circular arc with radius R = 250 m

The prediction of the critical load from the load level A will lead to λcritical = 4,58 which is

overestimating the load bearing capacity of the structure λcritical = 3,24 by 41 %.

4.3 Classical Stability



5 Material nonlinear behavior 49

5 MATERIAL NONLINEAR BEHAVIOR

5.1 Nonlinear stress--strain diagram

Up to now we have assumed a linear stress--strain diagram as material law.

σ

E

Figure 5.1: linear--elastic material law

σ = E · E = const. (5.1)

For a linear--elastic material law the Youngs modulus is constant, independent of the current

strain state.

In order to obatin internal forces (internal bending moments), we have to integrate stresses overthe cross section (assuming a truss or beam element). Assuming constant strains and therewith

constant stresses over the cross section we obtain

(5.2)

Assuming a linear stress and strain distribution over the beam height z, we obtain



5 Material nonlinear behavior50

(5.3)

Summarizing, we can conclude the following material law. This constitutive law is defining a

relationbetween stresses andstrainsfor eachmaterialpoint, or a relation between internal forces

and internal kinematical variables, such as the axial strain or the curvature .

σ = N

M =

EA

0

0

EI

·

σ = E ·

(5.4)

As long as the components in thematerial matrix E remain indpendent of the currentstrain/stress

situation, the material law is characterized as linear.

Definition: Material nonlinear behavior canbe attributedto a nonlinear stress--strainrelation

on the material point level.

We can distinguish different nonlinear material behavior.

a. Nonlinear--elastic material behavior

σ

Figure 5.2: Nonlinear --elastic material behavior




b. Elastic--plastic material behavior

σ

yieldstress σF

Figure 5.3: Elastic--plastic material behavior

c. steel

σ

yieldstress σF

Figure 5.4: steel

The stress--strain diagram in figure 5.4 is describing the dependency between technical strains

and technical stresses for an axial tensile experiment.

technical strains

= ∆0

technical stresses,1. Piola--Kirchhoff--stresses

nominal stresses

σ = N

A 0

(5.5)




Alternative strain and stress measures are

Hencky strains,logarithmic strains

true strains Hencky = 0+∆

0

d = ln |0+∆0

Cauchy stresses,true stresses

σ = N A

= ln (0 + ∆) − ln 0

= ln0 + ∆

0= ln 1 + ∆

0

= ln (1 + )

(5.6)

Assuming a constant volume we can obtain the relation between technical and true stresses.

(5.7)

σ, σW

, W

yield

stress σF

Figure 5.5: Relation between technical and true strains and stresses

Restriction: Strainsremain so small thatwe do nothave to distinguish betweentechnical andtrue

strains.




d. concrete

σ

f t

≈ 0,3 · f c

1,0 · f c

c1

Ec,m

Figure 5.6: concrete

EUROCODE 2 is describing the nonlinear material behavior of concrete in the compression re-

gime by

σc = f c

1,1 Ec,m

f c−

c12

1 + 1,1 Ec,mc1

f c− 2

c1

(5.7)

We can also observe a time dependent behavior for certain materials. All above described mate-

rial laws have been assumed to show a constant behavior over the time (independent of time).




t

t

σ

t

σ

t

creep relaxation

Figure 5.7: time dependent matrial behavior, viscos behavior, creep, relaxation

5.2 system behavior, example




Incremental--iterative procedure

P = 32

NPinitial state: w = 34

NP

EA = 3

4

σF

E

increment: P+ = 1 · NP

(5.8)

EA + EA

2 + EA 2

· w+ = P+ = NPequilibrium:

⇒ w+ = 12

NP

EA = 1

2

σF

E

w = w + w+ =54

σF

E

displacement in the

neighbouring stateNS:

(5.9)

We therewith have obtained a first (linearized) approximation of the new neighbouring state. Si-

milar to thealgorithm used for geometricalnonlinear problems, we nowhave to check theequili-

brium condition.Thereforethe internalforces, resulting from thecurrentdisplacement state, and

external applied loads have to be compared.

N1 = σF A internal forces:

Fi = N1 + N2 + N2 = 18

8

NP = 9

4

NP ≠ 10

4

NP

N2 = EA 2 = EA w2 = 5

8σF A

external load

(5.10)

We observe that the equilibrium condition is not fulfilled after the first iterative step. The next

iterative step should improve the equilibrium situation.

0 + EA 2 + EA

2 · w+ = 10

4NP − 9

4NP = 1

4NP

⇒ w+ = 14

NP

EA = 1

4

σF

E

w = w + w+ = 54σF

E + 1

4σF

E = 3

2σF

E

(5.11)

N1 = σF A internal forces:

Fi = N1 + N2 + N2 = 104

NP = 104

NP

N2 = EA 2 = EA w2 = 3

4σF A

external load

(5.12)

The equilibrium condition is fulfilled after the first iterative step ! No furtheriterations arerequi-

red.




Let us check the tangential stiffness for the system after the yield stress in bar no 1 has been re-

ached. For a displacement w ≥ wF we obtain:

0 + EA 2 + EA

2

tangential stiffness of element no. 1

tangential stiffness of element no. 2(5.13)

The algorithm for geometrical nonlinear problems can be easily adapted to material nonlinear

problems:

S the material matrix id depending arbitrarily on the displacement / strain situation

S the tangential stiffness matrices of all elementsnow have to takeintoaccount thetangential

material stiffness matrices

σ = E

σ

until now:

now: σ = σ ()

E = const.

ET ≠ const.

dσd

= ET()

Figure 5.9: tangential material stiffness

The ultimate limit load is obtained by:

instable system

Pultimate = 3 NP

Pultimate

Figure 5.10: Ultimate limit load




What is happening if the load is relieved after the system has been loaded ?

N

w wF

32

wF

NP

2 NP

3 NP

2,5 NP

N1 N2

14

wF

Figure 5.11: loading followed by a relieve of load and resulting residual stress situation

After a full relieve of the applied load, the following residual stress situation is obtained:

NR1 = NP − 5

4wF

EA = σF A − 5

4σF

E EA

= − 14σF A = − 1

4NP

NR2 = 1

4wf

EA 2 = 1

8NP

(5.14)

The residual stresses are in equilibrium; no external resulting force is generated. The residual

forces of all three bars are in equilibrium.

If the system will be again loaded (after the first loading and unloading cycle has been finished)

the system will react linear elastic up to a load of P = 2,5 NP . From thereon the system is

“remembering” the first load cycle and will follow the initial response path.




5.3 Elasto--Plastizität auf Tragelementebene (Biegebalken)

5.3.1 Reine Biegung

Wir betrachten einen Einfeldträger unter reiner Biegung, ohne Normalkraft und vernachlässig-

barer Querkraft.

System:

Moment:

Krümmung:

P

Verschiebung:

Bild 5.12: Biegeträger im teil-- und vollplastischen Zustand

Betrachten wir nun zusätzlich die einzelnen Dehnungs-- und Spannungzustände im Querschnitt

in der Mitte, so können wir drei ausgezeichnete Zustände unterscheiden




z

b

h

Grenzeelastisch--plast.

teilplastisch vollplastisch

Dehnung

Es gelte weiterhin die Hypothese vom Ebenbleiben der Querschnitte

Bild 5.13: Biegeträger im teil-- und vollplastischen Zustand

Im nächsten Bild ist der Verlauf des Biegemoments über der Krümmung dargestellt. Das Mo-

ment verläuft linear bis die erste Faser die Fließspannung erreicht.




M = P 4

MF

32

MF

F 2 F

Bild5.14:BelastungmitanschließenderEntlastungundresultierendemEigenspannungszustand

Die Grenztragfähigkeit eines Biegebalkens (elastisch--ideal plastisches Materialverhalten) liegt

um einen Faktor α über demjenigen Biegemoment, bei dem die erste Randfaser plastiziert.

MMF

F

2

1I--Profile 1,12 -- 1,18

Rohrquerschnitt 1,27Rechteckquerschnitt 1,5

quadratischerQuerschnitt 2,0

Bild 5.15: Grenztragfähigkeit unterschiedlicher Profile

Eine relativ einfache Näherung zur Beschreibung des Versagensverhaltens bei elastisch--ideal

plastischem Materialverhalten liefert die Fließgelenktheorie.




5.3.2 Biegung mit Längskraft

Elastischer Bereich

z

h

M

N

Bild 5.16: Biegung und Längskraft im elastischen Zustand

σ = N

M =

EA

0

0

EI

·

σ = E ·

(5.15)

N = 0

M = 0

(5.16)




Interaktionsdiagramm

M

MF

NNF

1

1

-- 1

-- 1

elastischerBereich

Bild 5.17: Interaktionsdiagramm für Normalkraft und Biegemoment

(5.17)

Teilplastischer Bereich

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