a one hour course on nonlinear modeling of...

1

A one hour course onNonlinear Modeling of Structures

Filip C. FilippouProfessor of Structural Engineering

University of California, Berkeley

Computational framework for earthquake simulation

• OpenSees (OPEN Software for Earthquake Engineering Simulation)http://opensees.berkeley.edu (last release 2.2.0, August 2010)

• FEDEASMatLab for teaching and concept developmenthttp://fedeaslab.berkeley.edu (last release 3.1 July 2010)

at Berkeley

2

Element Selection in EQ Engineering Practice

• Criteria– Economy in model development and result interpretation considering parameter

sensitivity and multiple ground motions– Knowledge and experience of analysis team– Detail of response (global, regional or local) and accuracy

• Selection (in decreasing popularity and increasing cost and expectations)– Linear elastic elements of any type (1d, 2d, 3d)– Nonlinear beam and column elements with “plastic hinges”– Nonlinear beam and column elements with material response integration

(fiber, fiber-hinge, inelastic beam-column finite elements)– 2d and 3d finite elements (few robust constitutive models, few advanced

features when compared with expectations: e.g. bond-slip, buckling of reinforcement, large discrete cracks, shear sliding, local buckling, fatigue, tearing of steel, etc)

Beam-Column Models

• Concentrated plasticity models = one rotational spring at each end + elastic element– Advantages: relatively simple, good for interface effects (e.g. bar pull-out) – Disadvantages: force-deformation of rotational spring depends on geometry and

moment distribution; relation to strains requires “plastic hinge length”; interaction of axial force, moment good for “metallic” elements; more complex interaction questionable; numerical robustness difficult

• Distributed inelasticity models (FE model) = consistent integration of section response at specific “control” or monitoring points– Advantages: versatile and consistent; material response can be incorporated in

section response; interaction of axial force and moment (and shear and torsion) can be rationally developed, thus numerical robustness is possible

– Disadvantages: can be expensive (not clear), require good understanding of integration to determine validity of strains and local response

3

Structural Beam-Column Models

M

N

Distributed inelasticity models

• 2 monitoring points usually at element ends = inelastic zone model- Good for columns and girders with low gravity loads- Consistent location of integration point?- Value of fixed length of inelastic zone?- Good for softening response (Fenves/Scott, ASCE 2006)- Variable inelastic zone element (CLLee/FCF, ASCE 2009); good for hardening

response- >2 monitoring points

- Good for girders with significant gravity loads- 4-5 integration points are advisable (“good plastic hinge length value with

corresponding integration weights)- One element per girder -> avoid very high local deformation values

4

Beam-Column Models: Concentrated Inelasticity

• Concentrated plasticity models =one or more rotational springs at each end + elastic element– Advantages:

relatively simple, good(?) for interface effects(e.g. shear sliding, rotation due to bar pull-out)

– Disadvantages:properties of rotational spring depend on geometry and moment distribution; relation to strains requires “plastic hinge length”;interaction of axial force, moment and shear ????;generality??? numerical robustness???M

N

Beam-Column Models: Distributed Inelasticity

• Distributed inelasticity models (1d FE model) = consistent integration of section response at specific “control” or monitoring points– Advantages:

versatile and consistent;section response from integration of material response thus N-My-Mz interaction (Bernoulli)(shear and torsion?? Timoshenko, …)thus numerical robustness is possible

– Disadvantages:can be expensive (what is the price $$$$) with “wasted sections” for localized inelasticityinaccuracy of local response (localization)thus better understanding of theory for interpretation of local response and damage is necessary

x

y

z

5

“Economic” Distributed(?) inelasticity models for Columns

• 2 monitoring points at element ends =inelastic zone model- Good for columns and girders with low gravity loads- N-My-Mz interaction straightforward- shear and torsion ???- Consistent location of integration point?- Value of fixed length of inelastic zone?- Good for softening response

(Fenves/Scott, ASCE 2006)- Hardening response Spreading inelastic zone

element SIZE (CL Lee/FCF, ASCE 2009 to appear)without N-M interaction; is generalization possible??

x

y

z

Good Distributed inelasticity models for Girders

- For girders with significant gravity loads- 4-5 integration points are advisable (“good plastic hinge length value with

corresponding integration weights)- One element per girder -> avoid very high local deformation values

6

Section Response for Distributed Inelasticity Models

• Section resultant formulation based on plasticity theory, damage, etc, etc– Relatively economical, effects can be “lumped” in a composite section response– Generalization and extension to other than the calibrated cases may not be

straightforward; hardening is very difficult to incorporate let alone softening– Section geometry must always be accounted for (are limit capacities sufficient?)

• Integration of 1d, 1 ½ d, 2d, and 3d material response– For midpoint integration the name fiber model is used; but other integration

methods are possible– For many fibers it can be expensive; how many fibers should be used?

Section Response for “Distributed” Inelasticity Models

• Section resultant formulation based on plasticity theory, damage, etc, etc– Relatively economical, effects can be

“lumped” in a composite section response

– Generalization and extension to other than the calibrated cases may not be straightforward; hardening is difficult to incorporate, let alone softening

– Section geometry must always be accounted for (are limit capacities sufficient?)

– Hardening ???– RC section, softening ???

-1 -0.5 0 0.5 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1numerical solutionanalytical solutionpolynomial approximation

2 2 2 23.5 1.2 1x x y y+ + =

7

Section Response for “Distributed” Inelasticity Models

• Integration of 1d, 1 ½ d, 2d, and 3d material response– For midpoint integration the name

fiber model is used; are other integration methods better??

– how many fibers should be used and for what purpose?

z

ya) yb)

z

MID25y

z

y

z

8

Two key ideas for beam-column elements

• How to incorporate nonlinear geometry under large displacements?– Formulate the element in the basic reference system without rigid body modes– Use corotational formulation to transform basic variables to global system– Keep element force-deformation relation in basic system simple: use linear

geometry under small deformations– Use one element per structural member (see point below)– If second-order effects within structural member are significant, break structural

member into 2 elements by inserting middle node• How to formulate a robust frame element?

– Use Hu-Washizu functional with independent interpolation of forces (exact under certain conditions), displacements and section deformations(Taylor/Filippou/Saritas Comp Mech, 2003)

– No need for mesh refinement; keep integration points to 4, or 2 with variable integration weight (5 respectively 3 points for girders under large element loads)

Advantages of Mixed Formulation in Corotational Framework

• Nonlinear geometry is uncoupled from basic element response; thus, geometric transformation classes (e.g. large displacements, P-Δ, linear) can be implemented once for all user frame elements (OpenSees, FEDEASLab)

• Force interpolation functions are exact under certain conditions

• Element deformations arise in weak form and represent well the inelastic strain distribution (see next point)

• Without need for mesh and integration point refinement there is no danger of localization; local strains are relatively accurate as long as the “plastic hinge length” is accurate ( = integration weight of end points) – fiber hinge is a good choice for columns under plastic or softening response

• The effect of distributed element loads can be accounted for exactly with the force interpolation functions

9

PEER 2004 Annual Meeting

4u

5u

1u

2u

X

Y

xy

L

L

4 1xΔ = −u u u

i

j

5 2yΔ = −u u u

nL

3u

6u

β

Corotational formulation for large displacement geometry

2 3

3 6

ββ

= −= −

v uv u

1 nL Lv = −

Tgu=p a q

gu∂

=∂v au

Te g gu t gu= +k k a k a


Nonlinear geometry with large displacements

-40 -20 0 20 40 60 80 100 120 140 160-40

-20

0

20

40

60

80

100

120

x

y

Lee`s Frame

E

E EH

y

=

=

=

70608

01

1020

MPa

MPa

.

σ

120

cm

P,w

120 cm

96 cm24 cm2 cm

3 cm

Frame by Lee et al. (1968)

10


Lee Frame (Lee et al. 1968)

0 10 20 30 40 50 60 70 80 90 100-10

-5

0

5

10

15

20

linear elas tic

elas to-plas ticwith k inem atichardening

Lee`s Fram e

Displacem ent v (cm )

App

lied

Load

P (

kN)

proposed flex . form ul. - 3 elm tsCic hon (1983) - 10 elm ts


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.5

1

1.5

2

2.5x 104

End Rotation

Axi

al F

orce

2 E lements 4 E lements 8 E lements16 E lements32 E lements

Force-displacement of eccentric column (basic element)

Linear element

P

P

11


Brace buckling (Black and Popov 1980)

-2 .5 -2 -1 .5 -1 -0 .5 0 0 .5 1 1 .5 2-1 0 0

-5 0

0

5 0

1 0 0

1 5 0

V e rt ic a l D is p la c e m e n t

Ver

tical

For

ce

B u c k lin g B ra c e w ith P -δ


Brace buckling (Black and Popov 1980)

12


P. Uriz and S. MahinUC Berkeley


Lateral Buckling: CST and Quad element in 3d space

General multi-step analysis Analysis with scripts Script files NR and its Variants Results

Newton-Raphson algorithm and its variants

The Newton-Raphson algorithm requires that a new tangent stiffness matrixbe assembled at every iteration of every load step. This means that the linearsystem of equations for the displacement correction needs to be solved fromscratch at every iteration of every load step. For very large structural modelsthis is a very expensive proposition.

In the modified Newton-Raphson method the tangent matrix is not updatedat every iteration, but only once at the beginning of each load step. In theinitial stiffness method, the initial stiffness is used throughout the incremen-tal analysis. Alternative strategies that update the stiffness matrix every sooften are also possible. If the stiffness matrix is not updated, the last de-composition of the stiffness matrix is used for the solution of the linearizedequilibrium equations and only the load changes at each iteration.

Finally, quasi-Newton methods do not use the tangent stiffness matrix ofthe structure but obtain secant stiffness approximations of the inverse ofthe stiffness matrix from the displacement vectors of previous iterations.Among the best known quasi-Newton methods is the BFGS method, whichwas originally developed for nonlinear optimization problems. For a briefdescription of the method consult Bathe’s 1982 book pp. 759-761.

c©Filip C. Filippou - UC Berkeley

Lecture 11 / page 14


Response of 2-dof truss under 2 load increments with Δλ = 0.5

8

1

8

EA=25000k=5,000

a

b1

2

3

no s t ep = 2 ;Dlam0 = 0 . 5 ;t o l = 1 . e−16;max i t e r = 10 ;S I n i t i a l i z e S t a t ePf = ze ro s ( nf , 1 ) ;S S i m p I n i t i a l i z ef o r k=1: no s t ep

S t i fUpd t = ’ ye s ’ ;S S impIncrementS t i fUpd t = ’ ye s ’ ;S S imp I t e r a t ei f ( ConvFlag )

S Update Sta teend

end

0 0.05 0.1 0.15 0.2 0.25 0.30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Vertical translation (downward)

Load

fact

or λ

numerically exact solutionNR stepsfinal result for algorithm

Figure: Newton-Raphson method





8

1

8

EA=25000k=5,000

a

b1

2

3


S t i fUpd t = ’ ye s ’ ;S S impIncrementS t i fUpd t = ’ no ’ ;S S imp I t e r a t ei f ( ConvFlag )

S Update Sta teend

end

0 0.05 0.1 0.15 0.2 0.25 0.30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Load

fact

or λ

numerically exact solutionmodified NR stepsfinal result for algorithm

Figure: Modified Newton-Raphson method





8

1

8

EA=25000k=5,000

a

b1

2

3

no s t ep = 2 ;Dlam0 = 0 . 5 ;t o l = 1 . e−16;max i t e r = 10 ;S I n i t i a l i z e S t a t ePf = ze ro s ( nf , 1 ) ;S S i m p I n i t i a l i z eS t i fUpd t = ’ ye s ’ ;f o r k=1: no s t ep

S SimpIncrementS t i fUpd t = ’ no ’ ;S S imp I t e r a t ei f ( ConvFlag )

S Update Sta teend

end

0 0.05 0.1 0.15 0.2 0.25 0.30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Load

fact

or λ

numerically exact solutionmodified NR stepsfinal result for algorithm

Figure: Initial stiffness method





8

1

8

EA=25000k=5,000

a

b1

2

3


S t i fUpd t = ’ ye s ’ ;S S impIncrementS Update Sta te

end

0 0.05 0.1 0.15 0.2 0.25 0.30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Load

fact

or λ

numerically exact solutionstepsfinal result for algorithm

Figure: Incrementation without correction





8

1

8

EA=25000k=5,000

a

b1

2

3



end

0 0.05 0.1 0.15 0.2 0.25 0.30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Load

fact

or λ







8

1

8

EA=25000k=5,000

a

b1

2

3



end

0 0.05 0.1 0.15 0.2 0.25 0.30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Load

fact

or λ





LF control in incrementation Stiffness parameter Algorithm with LF Control LF control in iteration Examples


8

1

8

EA=25000k=5,000

a

b1

2

3

no s t ep = 23 ;Dlam0 = 0 . 1 0 ;t o l = 1 . e−16;max i t e r = 10 ;S I n i t i a l i z e S t a t e ;Pf = ze ro s ( nf , 1 ) ;S I n i t i a l i z ef o r k=1: no s t ep

S t i fUpd t = ’ ye s ’ ;LoadCt r l = ’ ye s ’ ;S Inc r ementS t i fUpd t = ’ ye s ’ ;S S imp I t e r a t ei f ( ConvFlag )

S Update Sta teend

end

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.450

0.2

0.4

0.6

0.8

1

Vertical displacement (downward)

Load

fact

or λ

Figure: Load factor control during incrementation; only21 steps are able to complete; the last step ”flip-flops”



LF control in incrementation Stiffness parameter Algorithm with LF Control LF control in iteration Examples


8

1

8

EA=25000k=5,000

a

b1

2

3

no s t ep = 120 ;Dlam0 = 0 . 1 0 ;t o l = 1 . e−16;max i t e r = 10 ;S I n i t i a l i z e S t a t e ;Pf = ze ro s ( nf , 1 ) ;S I n i t i a l i z ef o r k=1: no s t ep

S t i fUpd t = ’ ye s ’ ;LoadCt r l = ’ ye s ’ ;S Inc r ementS t i fUpd t = ’ ye s ’ ;LoadCt r l = ’ ye s ’ ;S I t e r a t ei f ( ConvFlag )

S Update Sta teend

end

0 0.5 1 1.5 2 2.5-1.5

-1

-0.5

0

0.5

1

1.5

Vertical displacement (downward)

Load

fact

or λ

Figure: Response of 2-dof truss with load factor control



13

Correlation studies of analysis with experiment: important but …

• Many tests have been conducted and more are under way– Before understanding the behavior of assemblies one should understand the

behavior of the constituent parts; not always possible or available– Reduced scale models require attention to scaling laws (e.g. weld fractures,

bond-slip)– “Older” tests are not complete either for lack of enough channels of

measurement or for lack of reporting (lost data); it is hard to obtain funding to “repeat” old tests

– Tests may have experimental errors (these are not reported always)– Success or failure can be decided by looking at all experimental data, not a

suitable subset of them– We can learn from failure as much as we learn from success, even though this

is not accepted practice in research publications; better paradigm is necessary

A simple start …

RC columns with biaxial bending and variable axial force

14

Low-Moehle Specimen 5: Load-Displacement Response in y

-3 -2 -1 0 1 2 3-40

-30

-20

-10

0

10

20

30

40

Tip Displacement y (cm)

Load

y (k

N)

y

z

x P

51.44 cmpz

x =44.48 kNp

y

experimentanalysis

Low-Moehle Specimen 5: Load-Displacement Response in z

-3 -2 -1 0 1 2 3-40

-30

-20

-10

0

10

20

30

40

Tip Displacement z (cm)

Load

z (k

N)

y

z

x P

51.44 cmpz

x =44.48 kNp

y

experimentanalysis

15

Low-Moehle Specimen 5: Reinforcing Steel Strain History

-80 -60 -40 -20 0 20 40 60 80-2000

-1500

-1000

-500

0

500

1000

1500

2000

Fiber Strain (mil-cm/cm)M

omen

t Mz

(kN

-cm

) z

y

-2000

-1500

-1000

-500

0

500

1000

1500

2000

-40 -30 -20 -10 0 10 20 30 40Steel Strain (mil-cm/cm)

Mom

ent M

z (k

N-c

m)

z

y

Effect of reinforcing bar pull-out from base

Low-Moehle Specimen 5: Reinforcing Steel Strain History

-80 -60 -40 -20 0 20 40 60 80-2000

-1500

-1000

-500

0

500

1000

1500

2000

Fiber Strain (mil-cm/cm)

Mom

ent M

z (k

N-c

m)

z

y

-2000

-1500

-1000

-500

0

500

1000

1500

2000

-40 -30 -20 -10 0 10 20 30 40Steel Strain (mil-cm/cm)

Mom

ent M

z (k

N-c

m)

z

y

16


Shear Link

Eccentrically braced steel frames

Eccentrically Braced Frame


L=28 in

δ

fb 5.985 in=

d 17.88 in=

f

w

t 0.521 int 0.314 in==

W18 40×

Loading History

0 2 4 6 8 10-4

-2

0

2

4

Pseudo time

Dis

plac

emen

t, in

Imposed vertical displacement at the right end of the element.

Equal moments at member ends

Shear Link Experiment (Hjelmstad/Popov 1983)

17


Shear Link Experiment (Hjelmstad/Popov 1983)

-4 -3 -2 -1 0 1 2 3 4-250

-200

-150

-100

-50

0

50

100

150

200

250

Imposed Displacement δ (in)

Shea

r For

ce (k

ips)

ExperimentAnalysis


b

d

Monitoring sectionv vA f

2σ1σ

s

Monitoring point

Concrete shear model

18


Vecchio/Shim (2004) Beams A1 and A2 (shear-compresion)P

L=3660 mm (Beam A1)L=4570 mm (Beam A2)L=6400 mm (Beam A3)

b=307mm

D5 ties

M10

Beam A2

M30 M25

b=307mm

h=561mm

Beam A3

D4 ties

64mmeach

M10

M30M25

b=307mm

M30

D5 ties

M10

64mmeach

Beam A1

M25

0 5 10 15 20 25 300

100

200

300

400

500

600

Midspan Deflection (mm)

Load

(kN

)

Point10

Point25

Point45

Point113

Experiment2D-FEM ModelProposed Model

0 5 10 15 20 25 30 35 40 450

100

200

300

400

500

600

Midspan deflection (mm)

Experiment2D-FEM ModelProposed ModelA1 A2


Vecchio/Shim (2004) Beam A1 (shear-compresion failure)

Point 10

Point 25

Point 45

Point 113

0 5 10 15 20 25 300

100

200

300

400

500

600

Midspan Deflection (mm)

Load

(kN

)

Point10

Point25

Point45

Point113


19


Vecchio/Shim (2004) Beam A3 (compression failure)

0 10 20 30 40 50 60 70 80 900

100

200

300

400

500

600

Midspan deflection (mm)

Load

(kN

)

Point10

Point25

Point40 Point65


Point 10

Point 25

Point 40

Point 65

One element per half span with 5 Lobatto integration points is usedThe cross-section is divided into 10 midpoint layersBasic concrete material parameters are taken from Vecchio-Shim


Thomsen-Wallace (2004) Slender Shear Walls

-100 -50 0 50 100-200

-150

-100

-50

0

50

100

150

200

Top Displacement (mm)

144"

48"

A A

48"

4"

#2 bars @7.5"8- #3 bars

(Concrete cover 0.75")

Section A-A-100 -50 0 50 100

-200

-150

-100

-50

0

50

100

150

200

Top Displacement (mm)

She

ar F

orce

(kN

)

20


Lefas-Kotsovos-Ambraseys (1990) Shear Walls

•Shear walls SW21 and SW22 have aspect ratio L/h=2.0

Loading Direction

d

L

0 5 10 15 20 250

20

40

60

80

100

120

140

160

180

Top Displacement,mm

She

ar F

orce

,kN

Experiment ν=0Analysis ν=0Experiment ν=0.1Analysis ν=0.1


Point35

Column depth-0.2 0 0.2

0.2

0.4

0.6

0.8

1

1.2Point40


0.2

0.4

0.6

0.8

1

1.2Point60


0.2

0.4

0.6

0.8

1

1.2Point97

Column depth

Col

umn

axis

-0.2 0 0.2

0.2

0.4

0.6

0.8

1

1.2

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9 Point30


0.2

0.4

0.6

0.8

1

1.2

Evolution of shear wall cracking – Tensile damage

a) Point 30 b) Point 35 c) Point 97

21

Paolo Martinelii, Politecnico di Milano, ItalyFCF, University of California, Berkeley

Analytical Model of

Reinforced Concrete Walls

Height ≈ 20 m

Weight ≈ 226 tons

Web Wall

Flange Wall

Table motion direction

Gravity Column Precast Column

Full scale 7 story wall building (Panagiotou, Restrepo, Conte, UCSD)


Analytical Model of


Confined and unconfined zones in the web wall

22


Analytical Model of


Interstory drift and displacement envelopes

• Excellent agreement in terms of interstory drift and displacement

• Increasing damage due to increasing intensity motions is visible


Analytical Model of


• Good agreement in terms of residual displacement

• Satisfactory results in terms of floor acceleration

Residual displacement and floor acceleration envelopes

23


Analytical Model of


• Good agreement also in terms of internal forces

• Maximum discrepancy during last motion at shear wall base

Story shear and overturning moment envelopes


Analytical Model of


• Significant lengthening of fundamental period of specimen (> than 2.5 x)

• Damage evolution tracked with remarkable accuracy

f1 = 1.83 Hz (0.55 s)Beginning of EQ1

f1 = 0.67 Hz (1.43 s)End of EQ4

Experimental and analytical frequency spectrum

24


Analytical Model of


Top displacement time history

Conclusions

• Nonlinear Analysis is gradually going to become a designer’s tool for the evaluation of existing structures and the design of new important structures

• It can offer significant insights into the global, but particularly into the local response of structures and, thus serve for the identification of local failure mechanisms

• The current state of the art permits the simulation of the hysteretic behavior of structural elements with limited success; deeper understanding of material behavior under cyclic loading is, however, indispensable in order to arrive at failure mode prediction

• Thorough correlations between numerical models and specimens of increasing complexity are indispensable; limited progress has been made to date, particularly regarding 3d response

25

Continuing Challenges

• Effect for shear, torsion and interaction with axial force and bending moment (3d and not just 2d analysis for shear)

• Effect of bond-slip, pull-out of reinforcing steel

• 3d beam-column joint model that is robust and efficient

• 3d constitutive model for concrete under large inelastic strains (damage, dilatation, …)

• Buckling of reinforcing steel (global and not local)

• Low cycle fatigue of structural steel; fracture

• Simulation of structural subassemblies and full-scale structures

• Many more: partitions, slab-wall-column interactions, cladding, infills …

Future outlook

• Much work needs to be done before nonlinear analysis can have widespread use, because of the complexity of nonlinear solution algorithms and the lack of training of modern engineers; thus researchers and educators need to redouble their efforts in clarifying the concepts

• I hope that with the contributions of all those present in an open forum (open platform) we will be able to experience significant progress in the nonlinear simulation of concrete structures in the future

a one hour course on nonlinear modeling of...

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