earthquake engineering by the beachthe beach - reluis engineering by the beachthe beach ......

Earthquake Engineering by the beachthe beachCapri, July 2-4 2009

Nonlinear modelling of i f d t t treinforced concrete structures

Jesús Miguel Bairán GarcíaLecturer of Civil Engineering

Universitat Politecnica de CatalunyaDepartment of Construction Engineering

Main research areas

• Nonlinear sectional analysis accounting for biaxial y gshear-bending-torsion and axial forces

• Seismic designSeismic design

• Prestressed concrete structures

• High-rise precast concrete and hybrid wind-towers

D t i ti d d bilit f t t t• Deterioration and durability of concrete structures

• D-regions

2J. M. Bairán2009

Contents of the presentation

Sectional analysisPart 1: Beam regions

• Sectional analysis • Objectives for further steps• Problematic• TINSA: “Total interaction nonlinear sectional analysis”• Implementation• Applications• Further topics

Part 2: Disturbed regions• Problematic• Automatic generation of Strut & Tie models• Nonlinear assessment of D regions• Further topics

3J. M. Bairán2009

Further topics

PART 1

Beam regionsBeam regions

4J. M. Bairán2009

Sectional analysisLevels of structural modeling

Solid Bars+Sections system

5J. M. Bairán2009

Sectional analysis

• Easy and quick model construction

Advantages of frame modeling• Easy and quick model construction.

• Result interpretation in terms of generalized forces and deformations (directly used for ULS design and verification).

• Reduced degrees of freedom system.

• Computational cost.

• Excellent results in “B” regions governed by normal stresses• Excellent results in B regions governed by normal stresses.

• Possibilities of force-based or displacement based elements.

• Versatile geometric definition through fiber sectional discretization.

6J. M. Bairán2009

Sectional analysis

Nonlinear analysis for normal stressesTraditional fiber element

( )y z z yε ε φ φ= + −• Define strain in each fiber.

• Evaluate material response

Traditional fiber element

0( , )x y zy z z yε ε φ φ= +• Evaluate material response.

• Integrate internal forces

“well solved”...

“versatile and aplicable to most load cases”...

7J. M. Bairán2009

Sectional analysis

What does Navier-Bernoulli hypothesis imply?

• Bars are large.

• What happens in short directions is not important.

• Fibers only respond to normal stresses.

• Only axial force and bending can be considered.

• Only applicable to B regions

Can it be improved?

• Only applicable to B regions.

8J. M. Bairán2009

What can be improved?

Main limitations of traditional frame element modelling

• Uniaxial σ-ε laws.

• Limited confinement modeling.

• Tangential forces (shear and torsion) usually neglected or considered in a• Tangential forces (shear and torsion) usually neglected or considered in a simplified manner.

• Interaction between normal and tangential forces is not considered

T

xN dAσ= ∫∫x

yM z dA

M dA

σ= −

∫∫∫∫

∫∫

Accuracy for tangential forces

Accuracy for normal forces<

φxzM y dAσ= ∫∫

9J. M. Bairán2009

Should we try to improve sectional analysis?Non linear behavior of RC structures

• All engineering structures are subjected to a combination of normal and tangential forces. The matter is which is dominant.

• Most earthquake collapses of modern structures are related to shear forces.

10J. M. Bairán2009

Objectives

• A sectional model for arbitrary geometry capable of reproducing the

non-linear response of reinforced concrete under fully 3D loading (6

internal forces) ( Nx, Vy, Vz, Tx, My, Mz )

• Reproduce other 3D phenomena taking place in RC sections.

(i.e. confinement, etc.)

• To extend the concept of fiber discretization to tangential forces (shear

and torsion) and achieved balanced accuracy in all 6 internal forcesand torsion) and achieved balanced accuracy in all 6 internal forces.

11J. M. Bairán2009

Cracked concreteProblematic (1)B regions under combined normal and tangential forces:

• 6 internal forces

• Big difference between tensile and compression strength

• Inclined cracking

12J. M. Bairán2009

Cracked concreteProblematic (1)B regions under combined normal and tangential forces:

• Cracked induced anisotropy

• Coupling of previously uncoupled forces i e V M• Coupling of previously uncoupled forces, i.e. V-M

⎟⎞

⎜⎛ ++ )(t)(( t)(t θηθVMT u ⎟

⎠⎜⎝

+−+= )(cot)(·(cot2

)(cot· αθηθ gggVz

T uu

u

13Not the only coupling produced...J. M. Bairán2009

Cracked concrete

• In general with an inclined crack pattern all internal forces may be coupled

Problematic (1)

In general, with an inclined crack pattern all internal forces may be coupled.

Traditional sectional stiffness matrix (only vertical cracks):Sectional stiffness matrix after inclined cracking:

xNV

⎡ ⎤⎢ ⎥⎢ ⎥

0εγ

⎡ ⎤⎢ ⎥⎢ ⎥

11 15 160 0 00 0 0 0 0

K K KK

⎡ ⎤⎢ ⎥⎢ ⎥

11 12 13 14 15 16K K K K K KK K K K K K

⎡ ⎤⎢ ⎥⎢ ⎥

Traditional sectional stiffness matrix (only vertical cracks):Sectional stiffness matrix after inclined cracking:

y

z

VVT

⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

=

y

z

γγφ

⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

22

33

0 0 0 0 00 0 0 0 00 0 0 0 0

KK

K

⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

21 22 23 24 25 26

31 32 33 34 35 36

K K K K K KK K K K K KK K K K K K

⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥x

y

TMM

⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

x

y

φφφ

⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

44

51 55 56

61 65 66

0 0 0 0 00 0 00 0 0

KK K KK K K

⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

41 42 43 44 45 46

51 52 53 54 55 56

61 62 63 64 65 66

K K K K K KK K K K K KK K K K K K

⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦zM⎢ ⎥⎣ ⎦ zφ⎢ ⎥⎣ ⎦61 65 660 0 0K K K⎢ ⎥⎣ ⎦

Shear and torsion are uncoupled

61 62 63 64 65 66K K K K K K⎢ ⎥⎣ ⎦

Totally coupled

14sectional response

J. M. Bairán2009

Cracked concrete

Experimental evidence:Problematic (1)

15J. M. Bairán2009

Stress-strain distribution

Patterns of shear stress and strain are not constant along loading (state dependent):

Problematic (2)

•1D equilibrium among layers.

P d fi d h fl di i P l di i iStatically determined

problem

Aproaches:

• Predefined shear flow direction. Panel discretization. problem

Actual patterns depend on geometry, reinforcement, and concrete state.

3D problem is statically undetermined

Cracked induced anisotropy → 3D ilib i d tibilit i h fib

Jourawski like approachBiaxial bending and shear:

equilibrium and compatibility in each fiber

Shear flow predefined

and shear: Lateral reinforcement arrangement:

162D shear flowJ. M. Bairán

2009


• The problem of fixed strain distribution patterns• Extensively used.

γ τ

y• Not good results in cracked RC sections...

γUncracked Cracked- small shear Cracked-large shear

ConstantConstant pattern

Parabolic pattern

17J. M. Bairán2009


• The problem of fixed strain distribution patterns

Vecchio y

18

yCollins, 1988

J. M. Bairán2009

TOTAL INTERACTION SECTIONAL MODEL

Ideas:

TINSA: “Total Interaction Nonlinear Sectional Analysis”

• Any kinematical hypothesis implies an additional constraint in the solution space. Hence,

the model produces the “best” possible considering the new artificial constraint..

Sometimes Plain-Section kinematics is not good enough.

• PS solution can be improved “as much as necessary” by means of more

d d d f ti d i di t ti19

advanced deformation modes: warping + distortion.

J. M. Bairán2009

Section distortionWhy consider distortion?

In reinforced concrete:

•Stretching of lateral reinforcement.Stretching of lateral reinforcement.

• 2D shear flows.

• Wide sections.

• Shear-torsion resistance mechanisms.

• Confinement.

In composite laminates:

L l f il• Local failure

• Delamination

20J. M. Bairán2009


Hypotheses

1. Displacement decomposition

Plane Section (PS) Warping Distortion (w)

2. Small strains

Plane Section (PS) Warping-Distortion (w)

Strain decomposition

3. Stress decompositionp

21J. M. Bairán2009


• Full 3D equilibrium in fibers

Special weak for full 3D sectional analysis:

22J. M. Bairán2009


Dual system of equilibrium:

1. Structural level: Traditional beam theories

Solved by traditional frame elements (1D domain)

2. Sectional level: warping-distortion.

To be solved internally in the cross-section (2D domain)( ) ( )

23J. M. Bairán2009






To be solved internally in the cross-section (2D domain)

( )

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To be solved internally in the cross-section (2D domain)( ) ( )

25J. M. Bairán2009

Implementation

1. 2D FEM of the cross-sectionTwo approaches to warping-distortion definition:

2. Generalized coordinates

26γ γ1 γ2 γ3

J. M. Bairán2009

Implementation

2D FEM Generalized coordinates

• More accuracy for 3D effects (i.e. spalling, etc.)• Better distribution of stresses

• Less degrees of freedom.• More suitable for full structural level implementationAdvantages • Better distribution of stresses

and strains.• More versatile for arbitrary geometries.

implementation.• Accuracy may be improved by addition of more shape functions.

Disadvantages • Computational cost • Local errors may exist.

27J. M. Bairán2009

Comparison with other methods for tangential forces

Description Characteristics

• Extended usedV i th

• Easy to implementFib ilib i t li itl id dFixed patterns • Various authors • Fiber equilibrium not explicitly considered.

• In general not correct distribution of strains or stresses.

Panel discretization• Extended used• Various authors

• Easy to implement• Fiber equilibrium not explicitly considered• In general not correct distribution of strains or stresses.

• Sectional program based on MCFT.• Requires analyzing two simultaneous sections under normal forces

• Fiber equilibrium considered through finite differences approach• Distortion not explicitly considered

RESPONSE –Dual sectional analysis

sections under normal forces• Constitutive model 2D

Distortion not explicitly considered• Only 2D in plane loading and vertically symmetric sections (N-V-M).• Solution depends on the distance between analysedsections.• Requires a specific frame element• Requires a specific frame element.

RESPONSE 2000 –Longitudinal stiffeness

th d

• Sectional program based on MCFT. • Analyzes a single section using differential equilibrium equation.•Constitutive model 2D

• Fiber equilibrium considered in a differential approach.• Distortion implicit• Only 2D in plane loading and vertically symmetric sections (N-V-M)method Constitutive model 2D sections (N V M).• Does not requires a specific frame element.

• Various constitutive models (3D)• Analyzes a single section using

• Fiber equilibrium considered in a differential approach.• Distortion explicitly considered

28J. M. Bairán2009

TINSA

y g gdifferential equilibrium equation.•3D equilibrium equation• Any geometry and type of load

p y• Any type of load 3D• Reproduces confinement• Does not requires a specific frame element.

MATERIALSConcrete: main aspects3D effects

• Current strength is the projection from the current stress state to a 3D failure surface, Willam & Warnke surface is used.

•Concrete in compression behaves non-linear with residual strains

Compression behaviour

• Collins & Porasz backbone σ-ε curve.Considers influence of concrete strength in curve’s shape.

Tension behaviour• Concrete in tension behaves non-linear with degrading modulusg g

• Damage is only active in tension and varies independently in each principal direction• Cervenka backbone σ-ε curve

29J. M. Bairán2009

MATERIALSConcrete: cyclic response

Cyclic compression Cyclic tension

30J. M. Bairán2009

Cyclic shear

MATERIALSReinforcing steelBilinear elasto-plastic with kinematic hardening

31J. M. Bairán2009

APLICATIONS

K i (1977)

Shear strengthPure shear. V-γ diagrams

Kani (1977)

32J. M. Bairán2009

APLICATIONS

Moderate to high strength concrete with different reinforcing arrangementesCl d (2002)

Shear strength

Cladera (2002)• fc = 50 MPa

33J. M. Bairán2009

APLICATIONS

Moderate to high strength concrete with different reinforcing arrangementesShear strength

34J. M. Bairán2009

APLICATIONS

Location of strain gages

Crack patterns in H502 specimen

35V - εtransV - εlong

J. M. Bairán2009

APLICATIONSBending-shear interaction

M-ϕ and V-γ diagrams for different M/V ratios

36J. M. Bairán2009

APLICATIONS

M-εlong and V- εtrans diagrams for different M/V ratios

Bending-shear interaction

ε o g a d εt a s d ag a s o d e e t / at os

M-εlong V- εtrans

• Effect of shear in longitudinal reinforcement • Effect of bending in transversal reinforcement

37J. M. Bairán2009

APLICATIONSSlender cantilever

RC cantilever pier (L/h=4.92)

38J. M. Bairán2009


engt

h

Curvature distribution

plas

tic le

nsm

atio

n of

nd

rota

tion

Shear strain distributionstic

est

im an

Shear strain distribution

Rea

li

39J. M. Bairán2009


σz in stirrups at z=h/2 along length σx in concrete in critical section

Componente horizontal bielas

τxz in concrete in critical sectionσz distribution in stirrups in critical section

40J. M. Bairán2009

APLICATIONSConfinement in RC sections

Centered loading

• Rectangular section L = 120 mm• Stirrups: 61 5 mm2 / 100 mm• Stirrups: 61.5 mm2 / 100 mm• Cover: 10 mm

41J. M. Bairán2009


Centered loading

42J. M. Bairán2009


In-plane and biaxial bending and compression

In-plane bending and compressionN=980 kN

Biaxial (45º) bending and compressionN=980 kN

Concrete: fc=38 Mpa

Steel: fy=480 Mpafs=648 MPa

43J. M. Bairán2009

APLICATIONS

Momento vs. CurvaturaAxil= 980kN

Confinement in RC sectionsIn-plane and biaxial bending and compression

0.015; 231.4

0.023; 240.1

0.0927; 208.80.0296; 221.6 0.1616; 187.2200.0

250.0

300.0

Axil 980kN

M-ϕ diagrams

In-plane bending and 0.0125; 186.1

50 0

100.0

150.0

M kN‐m

TINSA

Fibras

p gcompressionN=980 kN

0.0

50.0

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

φ 1/mMomento vs. CurvaturaAxil= 980kN

0.008; 204.70.0125; 221.0

0.035; 199.20.0122; 202.2

0.0714; 181.1

200.0

250.0

Biaxial (45º) bending and compression

0.0069; 165.3

100.0

150.0

M kN‐m

TINSA MC45

Flexion esviada 45º

pN=980 kN

440.0

50.0

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

φ 1/m

J. M. Bairán2009


Momento vs. CurvaturaAxil= 980kN

Experimental investigation on non-linear cyclic “P i ” t i

0.015; 231.4

0.023; 240.1

0.0927; 208.80.0296; 221.6 0.1616; 187.2200.0

250.0

300.0

Axil 980kN“Poisson” strains

0.0125; 186.1

50 0

100.0

150.0

M kN‐m

TINSA

Fibras

0.0

50.0

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

φ 1/mMomento vs. CurvaturaAxil= 980kN

0.008; 204.70.0125; 221.0

0.035; 199.20.0122; 202.2

0.0714; 181.1

200.0

250.0

0.0069; 165.3

100.0

150.0

M kN‐m

TINSA MC45

Flexion esviada 45º

45J. M. Bairán2009

0.0

50.0

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

φ 1/m

APLICATIONSConfinement in RC sectionsExperimental investigation on non-linear cyclic “Poisson” strains

60

ensi

ón (M

Pa)

σ(M

Pa)

25 H60 - Muestra 8H60-Simulación

40

Teσ

15

20

- ε3 (

*10-3

)

2010

15

form

ació

n la

tera

l -tra

ns (1

E-3

)

0

5

Def εt

25 20 15 10 5 0Deformación lateral - ε3 (*10-3)

0 5Deformación axial - ε1(*10-3)

εtrans (1E-3) εlong(1E-3) 0 1 2 3 4 5Deformación axial - ε1(*10-3)

0

εlong(1E-3)

46J. M. Bairán2009

APLICATIONSBending and shear cyclic loading

fc=37 MPa

f 414 MPfy=414 MPa

47J. M. Bairán2009

APLICATIONS

Shear force history

25000

Shear historyBending moment history

12000

Bending moments history

5000

0

5000

10000

15000

20000

Vz [k

N] 1

2

3

4

6000

8000

10000

Mz

[kN

m]

1

2

3 6

7

8ed lo

ad

-25000

-20000

-15000

-10000

-5000

0 5000 10000 15000 20000

V

5

6

7

0

2000

4000

-15000 -10000 -5000 0 5000 10000 15000

M

4 5App

lie

3x 10-3 Shear strain history

7

8x 10-5 Curvature history

d ϕ

Vy [kN] My [kNm]

1

2

5

6

pons

e γ

an

2

-1

0γ z

2

3

4φ z

Res

p

0 1 2 3

x 10-3

-3

-2

γy

-4 -2 0 2 4

x 10-5

0

1

φy

48J. M. Bairán2009

APLICATIONS

20

Vy - strain

20

Vz - strain

Bending and shear cyclic loading

10

15

y [kN]

5

10

15

20

V-γ diagrams

-5

0

5Vy

-10

-5

0

Vz [kN]

γ g

01

23

x 10-3-5

0

5

x 10-3

5

γy

γz0

12

3

x 10-3

-50

5

x 10-3

-20

-15

γyγz

15My - curvature

12Mz - curvature

5

10

8

10

M-ϕ diagrams

-5

0

My [kN-m

]

2

4

6

Mz [kN-m

]

-4 -2 0 2 4

x 10-5

-15

-10

φy

0 2 4 6 8

x 10-5

-2

0

φz

49J. M. Bairán2009

APLICATIONS

Distribution of de σx

Distribution of σs in stirrups

50J. M. Bairán2009

APLICATIONS

Di ib i f d iDistribution of damage in concrete

Distribution of damage h tshear component

51J. M. Bairán2009

APLICATIONSPure torsion

Torsion stiffnessSh t flShear stress flow

φ

φ

φ

φφ

52J. M. Bairán2009

APLICATIONS

Principal compression

Pure torsionTorsion stiffness

Principal compression

53J. M. Bairán2009

APLICATIONS

Cracked stiffeness evaluated by Collins and Lampert (1972)

Torsion-axial strain coupling Torsion-bending couplingLongitudinal reinforcement stressTransversal reinforcement stress

Uncracked sectional stiffness matrix:

Cracked sectional stiffness matrix:

54J. M. Bairán2009

APLICATIONSBending-torsion interaction

Bending-torsion interaction diagramsO (1978)Onsongo (1978)

55J. M. Bairán2009

APLICATIONSBending-torsion interaction

Bending-torsion interaction diagrams

56J. M. Bairán2009

APLICATIONSInfluence in the response of complete structures

Effect of tangential forces in the non-linear response of concrete structures

57J. M. Bairán2009

APLICATIONS

Influence in the response of complete structuresEffects of tangential forces in the non linear response of concrete structuresEffects of tangential forces in the non-linear response of concrete structures

S1S1•Neglecting shear forces.•Navier-Bernoulli

l t

S2•Timoshenko elementelement.

• Traditional fiber discretization

element.• TINSA –Generalized coordinates

S3•Timoshenko element.• TINSA –Generalized

58

coordinates

J. M. Bairán2009

APLICATIONS

Influence in the response of complete structuresEffects of tangential forces in the non linear response of concrete structures

Load-displacement

Effects of tangential forces in the non-linear response of concrete structures

59J. M. Bairán2009

APLICATIONS


Moment - Load


InternalInternal Support

Mid-spanp

60J. M. Bairán2009

APLICATIONS


Stresses in longitudinal reinforcements


61J. M. Bairán2009

APLICATIONS


Stresses in transversal reinforcements


62J. M. Bairán2009

APLICATIONS


S1


Crack patterns

S2

S3

63q=43.8 kN/mJ. M. Bairán2009

APLICATIONS


S1Crack patterns


S2

S3

64q=127 kN/mJ. M. Bairán2009

APLICATIONS


Left support

Distribution of stresses and strains in some sections q=127 kN/m

Mid-Span 1 Interior support


Left support p

65J. M. Bairán2009

Further topics

E t di li d di t f l ti

Cross-section modelling

• Extending generalized coordinates formulation.

• Dynamic and seismic response of structures sensible to shear forces and torsionand torsion.

• Stage construction.

• Evaluation of repaired and retrofitted structures.

• Computational cost .Material modelling

• Modelling lateral strains under cyclic loading.

66J. M. Bairán2009

PART 2

Disturbed regionsDisturbed regions

67J. M. Bairán2009

Disturbed regions

Geometric and load discontinuity

When a rod is not good eno ghWhen a rod is not good enough….

68J. M. Bairán2009

Disturbed regions

Problematic

Design methodology: Strut and Tie

•Not uniqueness of Strut and Tie models for design.

• Sometimes it is difficult to find a plausible Strut and Tie scheme for a new elementSometimes it is difficult to find a plausible Strut and Tie scheme for a new element.

• Constructability of the resulting reinforcement arrangement.

• Strut and Tie models are a representation of the ultimate limit state.

• Lack of explicit rules for damage control and serviceability.

Assessment

• Real load carrying capacity.

• Damage assessment in different load levels.

69J. M. Bairán2009

AUTOMATIC GENERATION OF STRUT & TIE MODELS

Idea

T l i l ti i ti th h d i i t ki it i b d th l i f• Topological optimization through decision taking criteria based on the analysis of energy density distribution in linear elastic analysis.

• Importance or efficiency of the each element is defined by means of a efficiency p y y yfactor.

• Element stiffness is modified according to its efficiency. Less important elements collaborate lesscollaborate less.

Implementation and applicability

• Several truss-and-tie schemes are generated according to the criteria used.

• Constructability can be considered through decision criteria and by orthotropic elastic behaviour

70J. M. Bairán2009


Deep wall Linear elastic response

71J. M. Bairán2009


Strut & Tie model from criterion 1 : no special constructability considerationsDeep wall

72J. M. Bairán2009


Strut & Tie model from criterion 2 : constructability considerationsDeep wall

73J. M. Bairán2009

EVALUATION OF “D” REGIONS

A t l f

Goal

• Actual performance

• Damage assessment

• Load carrying capacity

Approach

• Concrete: non-linear biaxial behaviour

• Softening induced by transversal tensile strains (according to MCFT)

• Smeared crack approach• Smeared crack approach

74J. M. Bairán2009


Crack patterns for strut & tie model 1Deep wall

75J. M. Bairán2009


Crack patterns for strut & tie model 2Deep wall

76J. M. Bairán2009


Force-displacement curves for models 1 and 2Deep wall

6000

7000

5000

6000

Criterion 1

3000

4000 Criterion 2P (kN)Pd=3000 kN

1000

2000

0 10 20 30 40 50 600

1000

77J. M. Bairán2009

d (mm)

Further topics

• Performance based design – damage control

• Optimization - design

Possibilit of local fail re modes imperfect bond anchorage• Possibility of local failure modes: imperfect bond, anchorage failure

78J. M. Bairán2009

Acknowlegments

Some of the works here presented have been conducted under the support of:

Spanish Ministry of Education and Science through the research programs:•SARCS: “Seismic Assesment of Reinforced Concrete Structures” (BIA-2006-05614)•SEDUREC: “Security and durability in Construction Structures” (CSD-2006-00060)

Institute for the Promotion of Certified Reinforcements (IPAC) Through the Research Agreement for the Study of Structural Advantages of Using Very High Ductility Reinforcement.

Spanish Lamination Company (CELSA) Research project to study effects of straightening of reinforcing steel coils in their mechanics and ductility characteristics.

The author wishes to acknowledge SARCS research team In particular to

y

The author wishes to acknowledge SARCS research team. In particular to Prof. Antonio MaríPh.D. students Steffen Mohr and Edison Osorio.

79J. M. Bairán2009

Earthquake Engineering by the beachthe beachCapri, July 2-4 2009

Nonlinear modelling of i f d t t treinforced concrete structures

Jesús Miguel Bairán GarcíaLecturer of Civil Engineering

Universitat Politecnica de CatalunyaDepartment of Construction Engineering

81J. M. Bairán2009

earthquake engineering by the beachthe beach - reluis engineering by the beachthe beach ......

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