the finite element method for the analysis of non-linear and

The Finite Element Method for the Analysis ofNon-Linear and Dynamic Systems

Prof. Dr. Eleni Chatzi

Lecture 10 - 4 December, 2012

Institute of Structural Engineering Method of Finite Elements II 1

Nonlinear Dynamic Analysis

Example CaseDynamic Shake Table Test of a Reinforced Concrete Column

250

1285

1535Specimen

Servo hydraulic test cylinderPiston: + / - 125 mmPiston force: + / - 100 kN

ETH ShakeTable

DamperAnchorage


Example Case

Elevation Drawings


Example Case

Plan Drawings


Expected Behavior

RC Response in Cyclic Loading

C. Greifenhagen, P. Lestuzzi / Engineering Structures 27 (2005) 1703–1712 1709

Fig. 6. Force–displacement relationships observed in static cyclic tests (RF—failure of vertical reinforcement, DC—diagonal cracking, CF—concretecompression failure).

displacement level of 9 mm was almost equal to thebase shear at nominal flexural strength. The shear capacitydecreased in the second cycle at this displacement levelby 13% while in previous cycles up to 7.5% loss of shearcapacity due to repeated loading was observed. Furtherincrease in displacement led to the failure of both thevertical rebars in tension and concrete in compression at thewall edges. The maximum top lateral displacement yielded12.5 mm.

5.4. Discussion of test results

The strength, top lateral displacements and achievedductility are shown inTable 4. Failure of the specimen isassumed to occur when the shear capacity falls below 80%of the shear capacity that was observed in the second cycle atthe top lateral displacement corresponding to the maximumobserved base shear. The ductility is then computed from theultimate displacement and the extrapolated displacement atfirst yield.

The drift capacity of all specimens is greater than orequal to 0.8%. The test results for specimens M2, M3,and M4 indicate that the higher the axial force ratio, the

lesser the drift capacity. For specimen M1, the smallest driftcapacity was observed due to concentration of deformationsin the base crack. This concentration depends on concretequality, the axial force ratio and the detailing of thehorizontal reinforcement. Although significant degradationof shear capacity was observed,the vertical load bearingcapacity of the wall was maintained until the end of thetest.

The concentration of deformation in the base crackreduces the effect of aggregate interlocking because ofsignificant crack widths. Thus, it is almost solely thecompressed part of the wall that contributes to the transferof shear forces to the footing. Taking into account the lowratio of vertical reinforcement, inclined tensile strains thatcan cause inclined cracking are less important for lower axialforce ratios and diagonal tensile failure is prevented. This isevident when comparing the behavior of specimens M3 andM4.

Due to sliding movements in regions of the intercon-nected cracks at the base of the wall, residual displacementsincrease as the axial force ratio decreases. The tested speci-mens had no construction joint between the footing and thewall. A construction joint can result in a reduction of the

Experimental DataAccelerometer MeasurementsDisplacement Measurements

RC Response Characteristics

Strength Deterioration

Stiffness Degradation

Pinching Behavior


Modeling of Reinforced Concrete Behavior

Concrete Material Models

σcu E0

peak compressive stress

-σ

strain at maximum stress

Compression

σtu = maximum tensile strength of concrete Tension

+ε εo εcu

-ε

+σ

softening

Figure 2.5: Typical uniaxial compressive and tensile stress-strain curve for concrete (Bangash 1989)

In compression, the stress-strain curve for concrete is linearly elastic up to about 30 percent of the maximum compressive strength. Above this point, the stress increases gradually up to the maximum compressive strength. After it reaches the maximum compressive strength σ cu , the curve descends into a softening region, and eventually crushing failure occurs at an ultimate strain ε cu . In tension, the stress-strain curve for concrete is approximately linearly elastic up to the maximum tensile strength. After this point, the concrete cracks and the strength decreases gradually to zero (Bangash 1989).

2.3.1.1 FEM Input Data

For concrete, ANSYS requires input data for material properties as follows:

Elastic modulus (Ec).Ultimate uniaxial compressive strength (f’c).Ultimate uniaxial tensile strength (modulus of rupture, fr).Poisson’s ratio (ν).Shear transfer coefficient (βt).Compressive uniaxial stress-strain relationship for concrete.

9

Typical uniaxial compressive and tensile

stress-strain curve for concrete



Failure Criteria for Concrete

The determination of failure criteria is very important for the propersimulation of the degrading behavior of concrete structures.A. Discrete CrackingThe discrete crack approach to concrete fracture is intuitively appealing: a crackis introduced as a geometric entity. Initially, this was implemented by letting acrack grow when the nodal force at the node ahead of the crack tip exceeded atensile strength criterion. Then, the node is split into two nodes and the tip ofthe crack is assumed to propagate to the next node. When the tensile strengthcriterion is violated at this node, it is split and the procedure is repeated.

We begin by giving a concise overview of some historical developments of the discrete and thesmeared crack approaches. Mathematical deficiencies are noted which have surfaced forsmeared crack models and a now widely used solution}gradient enhancement}is brieflydiscussed. A recent development in computational mechanics, namely the use of the partition-of-unity property of finite element shape functions, turns out to be key for modelling thecoalescence of distributed cracks into one or more dominant cracks. This is first shown for agradient enhanced continuum damage model and then for the cohesive segments model. Sincewater and chloride ion transport play an important role in the deterioration and failure ofconcrete structures, we conclude with a formulation incorporating models for these phenomenainto a cohesive framework.

2. DISCRETE VS SMEARED CRACK MODELS

2.1. The discrete crack approach

The discrete crack approach to concrete fracture is intuitively appealing: a crack is introduced asa geometric entity. Initially, this was implemented by letting a crack grow when the nodal forceat the node ahead of the crack tip exceeded a tensile strength criterion. Then, the node is splitinto two nodes and the tip of the crack is assumed to propagate to the next node. When thetensile strength criterion is violated at this node, it is split and the procedure is repeated, assketched in Figure 1 [1].

The discrete crack approach in its original form has several disadvantages. Cracks are forcedto propagate along element boundaries, so that a mesh bias is introduced. Automatic remeshingallows the mesh bias to be reduced, if not eliminated, and sophisticated computer codes withremeshing were developed by Ingraffea and co-workers [3]. Nevertheless, a computationaldifficulty, namely, the continuous change in topology, is inherent in the discrete crack approachand is to a certain extent even aggravated by remeshing procedures.

The change in topology was to a large extent alleviated by the advent of meshless methods,such as the element-free Galerkin method [4]. Indeed, successful analyses have been carried outusing these methods, but disadvantages including difficulties with robust three-dimensionalimplementations, the large computational demand compared with finite element methods, thesomewhat ad hoc manner in which the support of a node is changed in the presence of a crack [5]

Figure 1. Early discrete crack modelling.

Copyright # 2004 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech. 2004; 28:583–607

RENE DE BORST ET AL.584



B. Smeared Cracking

In a smeared crack approach, the nucleation of one or more cracks inthe volume is translated into a deterioration of the current stiffnessand strength.

Generally, when the combination of stresses satisfies a specifiedcriterion, e.g. the major principal stress reaching the tensile strengthft; a crack is initiated.

This implies that at the integration point where the stress, strain and

history variables are monitored, the isotropic stress - strain relation is

replaced by an orthotropic elasticity-type relation with the n,s-axes

being axes of orthotropy; n is the direction normal to the crack and s

is the direction tangential to the crack.



One such criterion is utilized by ANSYS accounting for both crushing & cracking.

surface for the concrete. Consequently, a criterion for failure of the concrete due to a multiaxial stress state can be calculated (William and Warnke 1975).

A three-dimensional failure surface for concrete is shown in Figure 2.7. The most significant nonzero principal stresses are in the x and y directions, represented by σxp and σyp, respectively. Three failure surfaces are shown as projections on the σxp-σyp plane. The mode of failure is a function of the sign of σzp (principal stress in the z direction). For example, if σxp and σyp are both negative (compressive) and σzp is slightly positive (tensile), cracking would be predicted in a direction perpendicular to σzp. However, if σzp is zero or slightly negative, the material is assumed to crush (ANSYS 1998).

fc ’ fr

fc ’

fr

Figure 2.7: 3-D failure surface for concrete (ANSYS 1998)

In a concrete element, cracking occurs when the principal tensile stress in any direction lies outside the failure surface. After cracking, the elastic modulus of the concrete element is set to zero in the direction parallel to the principal tensile stress direction. Crushing occurs when all principal stresses are compressive and lie outside the failure surface; subsequently, the elastic modulus is set to zero in all directions (ANSYS 1998), and the element effectively disappears.

During this study, it was found that if the crushing capability of the concrete is turned on, the finite element beam models fail prematurely. Crushing of the concrete started to develop in elements located directly under the loads. Subsequently, adjacent concrete

13

For σxp, σyp ≤ 0 (compressive) and σzp > 0(tensile), cracking would be predicted in a directionperpendicular to σzp. However, if σzp ≤ 0, thematerial is assumed to crush.

After cracking, the elastic modulus of the concrete element is set to zero in thedirection parallel to the principal tensile stress direction.Crushing occurs when compressive stresses exceed the compressive failurestrength.

In practice, a pure compression failure of concrete is unlikely. Therefore, crushing

is ignored and cracking controls failure.



Steel Material Models

An aspect of utmost importance, for a non linear analysis, is the hysteretic rule needed to model the cyclic response of the structure. Over the last twenty years, significant development has occurred in the so-called phenomenological approach of hysteresis. Beginning with Bouc’s original formulation (1967, 1969, 1971) of the single degree degrading hysteresis model with pinching, many modifications have been subsequently introduced, such as the Bouc-Wen model (1976, 1980), the Baber-Noori model (1985, 1986) and the Reinhorn model (1996). These hysteresis models –also known as smooth hysteretic models- are capable of simulating a number of different types of loops using a single smooth hysteretic function affected by a set of user-defined parameters. In doing so, one can easily model the three main phenomena describing the cyclic response of R/C elements namely; stiffness degradation, strength deterioration and pinching behaviour due to bond-slip effects.

Following these rules, many computer programs have been developed, capable to perform a non-linear structural analysis such as DRAIN-2D (Kanaan and Powell,1973), SARCF (Chung et al.,1998; Gomez et al.,1990), IDARC (Park et al., 1978;Kunnath et al., 1992) and ANSR (Oughourlian and Powell,1982). The “Plastique” code presented herein, although maintains the elastoplastic behaviour within the 2D plane frames, works with a 3D stiffness of the entire structure based on diaphragmatic action.

2 Material Properties

Material properties are defined through certain conventional stress-strain curves both for unconfined concrete and reinforcing steel. In the former a parabolic stress-strain relationship with a softening branch is used, while in the latter a bilinear stress-strain diagram with hardening is implemented. The aforementioned stress – strain curves are depicted in Figure 1.

Figure 1. Stress –Strain diagrams a) for unconfined concrete and b) for reinforcing steel

3 Element Modeling

Three different types of two dimensional structural elements namely; beams, columns and shear walls are adequate to model the response of multi-storey buildings. By combining such elements one assembles multi-storey plane frames, which are linked together through diaphragms at the floor levels to create a 3D model of the structure.

3.1 Beam Element

Beam elements are considered as flexural elements with shear deformations with no axial deformations as beams belong to inextensional diaphragms and rigid zones at the ends to account for the stiffness increase at the joint, if needed. The element stiffness matrix varies throughout the analysis due to plasticity effects. In order to simulate such effects a hysteretic law and a spread plasticity model are introduced. The hysteretic model is formulated based on an initial moment-curvature relationship which represents the backbone skeleton curve. Such skeleton curves must be defined for each edge section of all elements. These curves can be either user defined or can be computed using a fiber

V.K. Koumousis et al.368

Bilinear Model with Hardening

0 0.002 0.004 0.006 0.0080

20

40

60

80

STRAIN [in/in]ST

RESS

[ksi

]

R=20

R=5

EpE

f y

Figure 15. Material Parameters of Monotonic Envelope of Steel_2 Model

-80

-60

-40

-20

0

20

40

60

80

100

-0.010 0.000 0.010 0.020 0.030 0.040 0.050 0.060Strain [in/in]

Stre

ss [k

si]

Figure 16. Hysteretic Behavior of Steel_2 Model w/o Isotropic Hardening

Giuffre-Menegotto-Pinto Model

R controls the transition from elastic to

inelastic branch.



Steel Cyclic Model

06734

ε

Confined

Unconfined

σ

εcoεcm

fco

fcm

εts

fo+

= fto

Figure 1. 1D behaviour for confined and Figure 2. Steel cyclic model.unconfined concrete.

Denoting by cof and coε the compressive peak stress and strain obtained from standard 1D unconfined tests, the

effect of confinement may lead to the following increments on concrete strength and peak strain

cocm fkf = (16)

cocm k ε=ε 2 (17)

As for parameter Z, an estimation of the form

[ ]cmcvcoco shffZ ε−ρ+−+= 43)1000145()29.03(5.0 ( cof in MPa) (18)

was assumed.

STEEL CYCLIC MODEL

The explicit formulation proposed by Giuffré and Pinto and implemented in reference [Menegotto et al. 1973]was chosen to model reinforcement cyclic behaviour. As illustrated in Figure 2, a family of transition curvesbetween two asymptotes intersecting at point ),( oo σε and with slopes E and hE (the elastic and the hardening

modulus) is defined according to equation

( ) RRbb 1**** ])(1[1 ε+ε−+ε=σ (19)

where

)()( o*

rr σ−σσ−σ=σ )()( o*

rr ε−εε−ε=ε (20)

EEb h= )( 21o ξ+ξ−= aaRR (21)

)()( oomax rr ε−εε−ε=ξ (22)

As for ),( rr σε , they represent the co-ordinates of the last reversal point ( maxrε is the maximum rε ever reached)

and R is the parameter which tunes Bauschinger’s effect. Parameters 1a , 2a and oR should be established on the

basis of experimental results.

VALIDATION: PSEUDODYNAMIC TESTS OF BRIDGE PIERS

The application that follows documents the numerical simulation of a complex experimental, which concerns toa quasi-static cyclic test of a reduced scale bridge pier, reported in Guedes (1997). Pier’s height is 8.4 m, and thecross section is a 0.8×1.6m2 hollowed rectangle, with walls 0.16m thick, as depicted in Figure 3. Steel


Modeling of Reinforced Concrete Behavior - 3D SolidApproach

Concrete Modeling using Solid Elements

A solid (3D) finite element can be used to model the concrete. Forexample, ANSYS uses an eight node element (Solid 65) with three degreesof freedom at each node translations in the nodal x, y, and z directions.The element is capable of plastic deformation, cracking in three orthogonaldirections, and crushing.

This chapter discusses model development for the full-size beams. Element types used in the models are covered in Section 2.2 and the constitutive equations, assumptions, and parameters for the various materials are discussed in Section 2.3. Geometry of the models is presented in Section 2.4, and Section 2.5 discusses a preliminary convergence study for the beam models. Loading and boundary conditions are discussed in Section 2.6. Nonlinear analysis procedures and convergence criteria are in explained in Section 2.7. The reader can refer to a wide variety of finite element analysis textbooks for a more formal and complete introduction to basic concepts if needed.

2.2 ELEMENT TYPES

2.2.1 Reinforced Concrete

An eight-node solid element, Solid65, was used to model the concrete. The solid element has eight nodes with three degrees of freedom at each node – translations in the nodal x, y, and z directions. The element is capable of plastic deformation, cracking in three orthogonal directions, and crushing. The geometry and node locations for this element type are shown in Figure 2.1.

Figure 2.1: Solid65 – 3-D reinforced concrete solid (ANSYS 1998)

A Link8 element was used to model the steel reinforcement. Two nodes are required for this element. Each node has three degrees of freedom, – translations in the nodal x, y, and z directions. The element is also capable of plastic deformation. The geometry and node locations for this element type are shown in Figure 2.2.

6


Modeling of Reinforced Concrete Behavior - 3D SolidApproach

Reinforcing Steel Modeling using Solid Elements

A truss element can be used to model the steel reinforcement. Two nodesare required for this element. Each node has three degrees of freedom,translations in the nodal x, y, and z directions.For example, ANSYS uses the LINK8, a uniaxial tension-compressionelement, which is also capable of plastic deformation.

LINK8 is a spar which may be used in a variety of engineering applications. Depending upon theapplication, the element may be thought of as a truss element, a cable element, a link element, a springelement, etc. The three-dimensional spar element is a uniaxial tension-compression element with threedegrees of freedom at each node: translations in the nodal x, y, and z directions. As in a pin-jointedstructure, no bending of the element is considered. Plasticity, creep, swelling, stress stiffening, and largedeflection capabilities are included. See Section 14.8 in the ANSYS Theory Reference for more detailsabout this element. A tension only compression-only element is defined as LINK10 and is described inSection 4.10.

Figure 4.8-1 LINK8 3-D Spar

The geometry, node locations, and the coordinate system for this element are shown in Figure 4.8-1. Theelement is defined by two nodes, the cross-sectional area, an initial strain, and the material properties. Theelement x-axis is oriented along the length of the element from node I toward node J. Properties not inputdefault as described in Section 2.4. The initial strain in the element (ISTRN) is given by /L, where is thedifference between the element length, L, (as defined by the I and J node locations) and the zero strainlength.

Element loads are described in Section 2.7. Temperatures and fluences may be input as element bodyloads at the nodes. The node I temperature T(I) defaults to TUNIF. The node J temperature T(J) defaultsto T(I). Similar defaults occurs for fluence except that zero is used instead of TUNIF.

A summary of the element input is given in Table 4.8-1. A general description of element input is given inSection 2.1.

Table 4.8-1 LINK8 Input Summary

Element Name LINK8

4.8 LINK8 3-D Spar (UP19980821 ) http://mechanika2.fs.cvut.cz/old/pme/examples/ansys55/html/elem_55/...

1 of 4 25.11.2010 11:38


Modeling of Reinforced Concrete Behavior - BeamApproach

Alternative View to the simulation of Degrading HystereticBehavior

The well known 1D beam element can be used as a simplified toolfor the simulation of the reinforced column behavior in place of the3D solid element formulation.

Beam Elements - Galerkin

Finally, the weight functions and trial solutions are

Finally, the weight functions and trial solutions

Where the shape functions are


This element has two degrees of freedom per node, one translational(perpendicular to the beam axis) and one rotational.



The shape functions utilized from this element are the HermitePolynomials (see Lecture 6)

Beam Elements - Shape Functions

Hermite PolynomialsHermite Polynomials




Then, as we saw in Lecture 6, the elastic force deformation relationship, fora prismatic beam without shearing deformations, is

Fi

Mi

Fj

Mj

=EI

L3

12 6L −12 6L6L 4L2 −6L −2L2

−12 −6L 12 −6L−6L −2L2 −6L 4L2

viφivjφj

or FE = KEv

Whilst , from Lecture 8, we saw that in case P-Delta effects are taken intoaccount, the geometric (nonlinear) stiffness is:

Fi

Mi

Fj

Mj

=T

30L

36 3L −36 3L3L 4L2 −3L −L2

−36 −3L 36 −3L3L −L2 −3L 4L2

viφivjφj

or FG = KGv

Therefore, the total forces acting on the beam element will be:

FT = FE + FG = [KE +KG]v = KT v


Moment Curvature Envelope

In order to simulate the effects of varying stiffness due to plasticityappropriate plasticity model and a hysteretic law will beintroduced.

The hysteretic model is formulated based on an initialmoment-curvature relationship otherwise known as the backboneskeleton curve.

Such skeleton curves must be defined for each different section type.For instance, the bottom sections are more heavily reinforced thanthe top. These curves can be either user defined or can be computedusing a fiber model.



Reinforced Concrete Design Calculations normally assume a simple materialmodel for the concrete and reinforcement so as to determine the momentcapacity of a section. The Whitney stress block for concrete along with an elastoplastic reinforcing steel behavior is one widely used material model.



However, the actual material behavior is nonlinear and can be described byidealized stress-strain (material) models, as the ones introduced earlier.



Moment Curvature Analysis

is a method to accurately determine the load - deformation behavior of aconcrete section using nonlinear material stress-strain relationships.

For a given axial load there exists an extreme compression fiber strainand a section curvature φ at which the nonlinear stress distribution isin equilibrium with the applied axial load. Dividing the section intofibers at distance z from the CG axis the strain distribution will be:

ε(z) = ε0 + zφ

A unique bending moment can be calculated at this section curvaturefrom the stress distribution.

The extreme concrete compression strain and section curvature can beiterated until a range of moment curvature values is obtained.



Software packages are available for generating Moment - Curvaturerelationships by inputing the, geometry, reinforcement characteristics,material properties and axial load for a given section.

Material properties for concrete can be obtained as a result of labcompression tests on the utilized concrete mix.Material properties from Steel can be directly obtained from the quality ofthe reinforcing Steel

Software packages that can be used for the generation of MomentCurvature Envelopes are:

SAP section designer (for those that have access to SAP2000)

Response 2000: http://www.ecf.utoronto.ca/ bentz/r2k.htm

MyBiaxial:http://users.ntua.gr/vkoum/links-prog/MyBiaxial/mybiaxial.htm


Plasticity Model

There are different approaches for the modeling of inelastic behavior.

Concentrate Plasticity (plastic hinge approach)

Distributed Plasticity

Spread Plasticity


Plasticity Model

Concentrated Plasticity Model


Plasticity Model

Distributed & Spread Plasticity Models


Plasticity Model

Distributed & Spread Plasticity Models

source: Hwasung Roh, Andrei M. Reinhorn, Jong Seh Lee, Power spread plasticity model for inelastic analysis of

reinforced concrete structures, Engineering Structures, Volume 39, June 2012, Pages 148-16


Bouc - Wen Hysteretic Model

The smooth hysteretic model presented herein is a variation of themodel originally proposed by Bouc (1967) and modified by severalothers (Wen 1976; Baber Noori 1985).

The use of such a hysteretic constitutive law is necessary for theeffective simulation of the behavior of R/C structures under cyclicloading, since often structures that undergo inelastic deformationsand cyclic behavior weaken and lose some of their stiffness andstrength. Moreover, gaps tend to develop due to cracking causingthe material to become discontinuous.

The Bouc-Wen Hysteretic Model is capable of simulating stiffnessdegradation, strength deterioration and progressive pinching effects.

(see: V. Koumousis, E. Chatzi and S. Triantafillou: Plastique “A Computer Program For 3D Inelastic Analysis Of

Multi-Storey Buildings, Advances in Engineering Structures, Mechanics & Construction, Solid Mechanics and Its

Applications, 2006, Volume 140, Part 3, 367-378)



The model can be visualized as a linear and a nonlinear element in parallel:

referred to as “moment” (M) and the strain variable as “curvature” ( ). In the case of shear-walls the hysteretic loop can be described in terms of a shear force-shear deformation relationship.

The use of such a hysteretic constitutive law is necessary for the effective simulation of the behavior of R/C structures under cyclic loading, since often structures that undergo inelastic deformations and cyclic behavior weaken and lose some of their stiffness and strength. Moreover, gaps tend to develop due to cracking causing the material to become discontinuous. The Bouc-Wen Hysteretic Model is capable of simulating stiffness degradation, strength deterioration and progressive pinching effects.

The model can be visualized as a linear and a nonlinear element in parallel, as shown in Figure 3. The relation between generalized moments and curvatures is given by:

( )( )( ) 1 ( )= + −yy

tM t M z tφα αφ

(1)

where My is the yield moment; y is the yield curvature; is the ratio of the post-yield to the initial elastic stiffness and z(t) is the hysteretic component defined below.

Figure 3. Bouc-Wen Hysteretic Model

The nondimensional hysteretic function z(t) is the solution of the following non-linear differential equation:

1 1 1 ( ) | ( ) | ( ) or alternatively where2 2

1 ( ) | ( ) | ( ) 1 ( ) | ( ) | ( ) 1 ( ) | ( ) | ( )2 2 2 2 2 2

B

C D E

n

z zy y

n n n

dz sign d z t z tK K A Bd

sign d z t z t sign d z t z t sign d z t z tC D E

φφ φ φ

φ φ φ

+ += − −

+ − − + − −− −

(2)

In the above expression A, B, C, D & E are constants which control the shape of the hysteretic loop for each direction of loading, while the exponents nB, nC, nD & nE govern the transition from the elastic to the plastic state. Small values of ni lead to a smooth transition, however as ni increases the transition becomes sharper tending to a perfectly bilinear behavior in the limit (n ∞).The program defaults are:

1 11, =0 & , where , =1 and −

+

−= = = =

B E

yB En n

y

MA C = D B E e b n = n = n

Mb e (3)

The parameters C, D control the gradient of the hysteretic loop after unloading occurs. The assignment of null values for both, results to unloading stiffness equal to that of the elastic branch. Also, the model is capable of simulating non symmetrical yielding, so if the positive yield moment is regarded as a reference point, the resulting values for B and E are those presented in equation (3). The hysteretic parameter Kz is then limited in the range of 0 to 1, while the hysteretic function z varies from

- +y y- M / M to 1.

z t( ) = =f (φ(t), z t( )). .


The relation between generalized moments and curvatures is given by:

M(t) =My

[αφ(t)

φy+ (1− α)z(t)

]where My is the yield moment; φy is the yield curvature; α is the ratio ofthe post-yield to the initial elastic stiffness and z(t) is the hystereticcomponent defined as:

referred to as “moment” (M) and the strain variable as “curvature” ( ). In the case of shear-walls the hysteretic loop can be described in terms of a shear force-shear deformation relationship.

The use of such a hysteretic constitutive law is necessary for the effective simulation of the behavior of R/C structures under cyclic loading, since often structures that undergo inelastic deformations and cyclic behavior weaken and lose some of their stiffness and strength. Moreover, gaps tend to develop due to cracking causing the material to become discontinuous. The Bouc-Wen Hysteretic Model is capable of simulating stiffness degradation, strength deterioration and progressive pinching effects.

The model can be visualized as a linear and a nonlinear element in parallel, as shown in Figure 3. The relation between generalized moments and curvatures is given by:

( )( )( ) 1 ( )= + −yy

tM t M z tφα αφ

(1)

where My is the yield moment; y is the yield curvature; is the ratio of the post-yield to the initial elastic stiffness and z(t) is the hysteretic component defined below.

Figure 3. Bouc-Wen Hysteretic Model

The nondimensional hysteretic function z(t) is the solution of the following non-linear differential equation:

1 1 1 ( ) | ( ) | ( ) or alternatively where2 2

1 ( ) | ( ) | ( ) 1 ( ) | ( ) | ( ) 1 ( ) | ( ) | ( )2 2 2 2 2 2

B

C D E

n

z zy y

n n n

dz sign d z t z tK K A Bd

sign d z t z t sign d z t z t sign d z t z tC D E

φφ φ φ

φ φ φ

+ += − −

+ − − + − −− −

(2)

In the above expression A, B, C, D & E are constants which control the shape of the hysteretic loop for each direction of loading, while the exponents nB, nC, nD & nE govern the transition from the elastic to the plastic state. Small values of ni lead to a smooth transition, however as ni increases the transition becomes sharper tending to a perfectly bilinear behavior in the limit (n ∞).The program defaults are:

1 11, =0 & , where , =1 and −

+

−= = = =

B E

yB En n

y

MA C = D B E e b n = n = n

Mb e (3)

The parameters C, D control the gradient of the hysteretic loop after unloading occurs. The assignment of null values for both, results to unloading stiffness equal to that of the elastic branch. Also, the model is capable of simulating non symmetrical yielding, so if the positive yield moment is regarded as a reference point, the resulting values for B and E are those presented in equation (3). The hysteretic parameter Kz is then limited in the range of 0 to 1, while the hysteretic function z varies from

- +y y- M / M to 1.

z t( ) = =f (φ(t), z t( )). .




In the above expression A, B, C, D & E are constants which control theshape of the hysteretic loop for each direction of loading, while theexponents nB , nC , nD & nE govern the transition from the elastic to theplastic state. Small values of ni lead to a smooth transition, however as niincreases the transition becomes sharper tending to a perfectly bilinearbehavior in the limit (ni →∞).Finally, the flexural stiffness can be expressed as:

Finally, the flexural stiffness can be expressed as:

( ) ( ) ( )01 1 11 1 1= = = + − = + − = + −y y z zy y y

dM dzK EI M M K EI Kd d

α α α α α αφ φ φ φ φ

(4)

6.1 Hysteretic behavior Variations

a) Stiffness Degradation

The stiffness degradation that occurs due to cyclic loading is taken into account by introducing the parameter into the differential equation:

( ) max0

1 1 1.02

+= → = + − = +z z

ky

K Kdz K EI where Sd

µ µα α ηφ η φ η

(5)

The parameter depends on the current, / yµ φ φ= , and maximum achieved plasticity, max max / yµ φ φ= .Sk is a constant which controls the rate of stiffness decay. Common values for Sk are 0.1 and 0.05.

b) Strength Deterioration

The strength deterioration is simulated by multiplying the yield moment My with a degrading parameter S :

( )( )( ) 1 ( )= + −yy

tM t S M z tβφα αφ

(6)

The parameter S depends on the damage of the section which is quantified by the Damage Index DI:

2

max

1

1 11 where 1

14

−= − =

−−

pd Sc p diss

mon

S S DI DIS dE

E

βµµ

(7)

In the above expression Sd, Sp1, Sp2 are constants controlling the amount of strength deterioration; µc is the maximum plasticity that can be reached, /=c u yµ φ φ ; dissdE is the energy dissipated before

unloading occurs and finally Emon is the amount of energy absorbed during a monotonic loading until failure as shown in Figure 4.

Figure 4. Dissipated Energy (Ediss) and Monotonic Energy (Emon).

“Plastique” – A Computer Program for Analysis of Multi-Storey Buildings 371


Modeling of Degradation

Stiffness Degradation



Modeling of Degradation


Finally, the flexural stiffness can be expressed as:

( ) ( ) ( )01 1 11 1 1= = = + − = + − = + −y y z zy y y

dM dzK EI M M K EI Kd d

α α α α α αφ φ φ φ φ

(4)

6.1 Hysteretic behavior Variations

a) Stiffness Degradation

The stiffness degradation that occurs due to cyclic loading is taken into account by introducing the parameter into the differential equation:

( ) max0

1 1 1.02

+= → = + − = +z z

ky

K Kdz K EI where Sd

µ µα α ηφ η φ η

(5)

The parameter depends on the current, / yµ φ φ= , and maximum achieved plasticity, max max / yµ φ φ= .Sk is a constant which controls the rate of stiffness decay. Common values for Sk are 0.1 and 0.05.

b) Strength Deterioration

The strength deterioration is simulated by multiplying the yield moment My with a degrading parameter S :

( )( )( ) 1 ( )= + −yy

tM t S M z tβφα αφ

(6)

The parameter S depends on the damage of the section which is quantified by the Damage Index DI:

2

max

1

1 11 where 1

14

−= − =

−−

pd Sc p diss

mon

S S DI DIS dE

E

βµµ

(7)

In the above expression Sd, Sp1, Sp2 are constants controlling the amount of strength deterioration; µc is the maximum plasticity that can be reached, /=c u yµ φ φ ; dissdE is the energy dissipated before

unloading occurs and finally Emon is the amount of energy absorbed during a monotonic loading until failure as shown in Figure 4.

Figure 4. Dissipated Energy (Ediss) and Monotonic Energy (Emon).

“Plastique” – A Computer Program for Analysis of Multi-Storey Buildings 371

The model can also be appropriately modified to simulate pinching.

Note: The Matlab code for the Bouc Wen Model will be provided!


Bouc Wen Model

Resulting Hysteretic Loops

Stiffness & Strength Degradation Pinching


Bouc Wen Model

Dynamic Equation of Motion

This is a dynamic problem (input: base excitation Ug). The generalequation of motion is therefore written as:

MX(t) +CX(t) +KX(t) = −MUg(t)

The Newmark Constant acceleration method outlined in Lecture 8 canbe used for the direct integration of the above equation.(You can neglect the effect of damping for this project)

In order to achieve equilibrium within each time step, it is necessary toimplement a Newton - Raphson iterative scheme as outlined in Lecture 3.


the finite element method for the analysis of non-linear and

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