1

NONLINEAR ANALYSIS OF SHEAR WALL STRUCTURES BY AN

EFFICIENT REINFORCED CONCRETE MEMBRANE ELEMENT

Leopoldo TESSER1, Michele PALERMO

2 and Filip FILIPPOU

3

ABSTRACT

This paper discusses the parameter calibration of a recently proposed, efficient numerical model for

the inelastic response analysis of reinforced concrete panels under seismic excitations. The concrete

material model is based on damage plasticity with two damage variables. The reinforcing bars are

modelled as smeared steel layers with a uniaxial stress-strain relation in the bar direction following the

Menegotto-Pinto constitutive model with isotropic hardening. For in-plane analysis, a plane stress 2d

membrane model is used with an initial assumption of perfect bond between reinforcing steel and

concrete. After the calibration the model is used to simulate the cyclic response of planar reinforced

concrete shear walls and coupled walls. The specimens are representative of modern mid-rise wall

construction. The evaluation of the model response makes use of global and local response

measurements and compares damage parameter contours of the model against the observed state of the

specimen so as to draw conclusions for the seismic performance assessment of reinforced concrete

shear walls.

INTRODUCTION

The seismic assessment of reinforced concrete shear walls by international standard provisions suffers

from two main sources of uncertainty. The first is the evaluation of the shear demand by linear

analysis methods and the second concerns the determination of the shear wall capacity. In fact, current

design provisions, such as Eurocode 8 (CEN 2004) in Europe and ASCE/SEI 7-10 (2010) in the

United States, might lead to the underestimation of the seismic shear demand in the case of linear

analysis because of the following issues: the flexural over-strength, the higher mode effects for slender

walls in the nonlinear range, and the determination of the force reduction factor for low and mid-rise

walls (Rutenberg and Nsieri 2006, Priestely et al. 2007, Rejec et al. 2012, Pugh 2012). Furthermore,

the formulas for the determination of the shear strength provided by Eurocode 8 (CEN 2004), ACI 318

(2005), ASCE/SEI 43 (2005) do not seem to be effective for all resisting mechanisms and for all wall

configurations (Salonikios 2002, Gulec 2008, Turgeon 2011, Krolicki 2011). General formulas for the

shear strength of shear walls do not account for the presence of barbells, openings, coupling beams,

and special reinforcement detailing.

Consequently, there is a growing consensus that reinforced concrete shear walls should be

designed and assessed with nonlinear analysis methods. However, the use of these methods in

professional practice is still limited because of the lack of physical understanding and numerical

robustness for existing models (Lee and Kuchma 2007, Lee et al. 2008, Gulec and Wittaker 2009,

Hagen 2012). For this reason several studies propose the use of beam-column models with or without

1 Ph.D., Géodynamique & Structure, Bagneux – France, leopoldo.tesser@geodynamique.com

2 Ph.D. candidate, Dept. of Civil Engineering, University of Bologna –Italy.

3 Professor, Dept. of Civil Engineering, University of California, Berkeley – United States of America.

2

shear effect consideration for reinforced concrete shear walls of high to moderate slenderness (Mazars

2006, Saritas 2006, Wallace 2007, Martinelli and Filippou 2009). These models are, however, not

suitable for squat shear walls or for the representation of the multi-axial response in the inelastic zone

of slender walls with the formation of inclined concrete struts and high shear deformations.

Furthermore, the computational complexity of several enhanced beam elements undermines their

advantage relative to membrane and plate finite elements.

In the interest of striking a balance between model accuracy and computational speed while

being able to capture important aspects of the nonlinear inelastic response of reinforced concrete shear

walls an efficient nonlinear reinforced concrete membrane model was recently proposed (Tesser et al.

2011). This paper simply aims at showing the practicality of nonlinear analysis and the reliability of

the developed model for the seismic performance assessment of reinforced concrete structures.

The calibration of the model is presented with the generalization of the relevant steps needed for

every concrete model with softening local response. The procedure to assure mesh objectivity and to

account for transverse confinement and tension stiffening is also clarified.

The model performance is assessed by correlation with the experimental results from the

MUST-SIM project (Lowes et al. 2011) on planar and coupled shear walls. The assessment is done by

comparison of global and local response measures from the nonlinear analysis with test measurements.

MODELLING CONSIDERATIONS

Reinforced concrete shear walls are subjected to a multi-dimensional stress state. Regardless of the

selected finite element, a multi-dimensional concrete material model is critical for capturing the local

material behaviour. Most of the existing macroscopic concrete models fall into two categories. The

models of the first category are directly developed in multi-dimensions starting with a surface limiting

the elastic range and a flow rule governing the nonlinear range (e.g. Darwin and Peknold 1982, Ju

1989). The models of the second category have multiple uniaxial stress-strain constitutive laws acting

in different directions and interacting with each other through empirical formulae (e.g. Vecchio and

Collins 1986, Hsu 1997).

A macroscopic concrete model has to take into account at least the failure modes of cracking in

tension and crushing in compression. The softening response following the peak strength is

fundamental to represent the local failure and the stress redistribution. It is also important to

approximate the shear transfer across cracks in case of non-proportional loadings such as seismic

actions. The model has also to account for the progressive stiffness reduction and for the inelastic

strains to determine the dissipated energy and the residual displacements in cyclic nonlinear analyses.

It is essential to select a method for ensuring the independence of the solution from different mesh

sizes (e.g. crack band theory, nonlocal models).

The reinforcement model should be nonlinear in order to represent the elastic-plastic steel

behaviour. The discrete or the smeared approach for modelling steel bars has to seek an adequate

representation of the complex steel-concrete interaction in the linear and nonlinear range. Other

advanced features are the steel-concrete bond nonlinear behaviour, the steel bar buckling in

compression, the steel fracturing in tension and other secondary shear mechanisms (e.g. dowel action

of steel bars).

Considering the multi-dimensional concrete model, the softening response always slows the

convergence when searching for global equilibrium. This issue can be tackled using an appropriate

nonlinear solution algorithm (e.g. Newton Raphson, quasi-Newton, accelerated Newton, arc-length)

together with the iterative stiffness matrix (i.e. initial tangent, secant stiffness and computational

tangent). The coupling of classical models for stiffness reduction (e.g. damage mechanics) and

residual displacements (e.g. theory of plasticity) strongly affects the convergence rate of the local

material state determination. The most effective algorithms are the explicit ones without internal

iterations. All additional sources of nonlinearity decrease the convergence rate of the model.

3

CONCRETE MODEL

The concrete damage model in this paper belongs to the class of energy-based isotropic continuum

damage models (Tesser et al. 2011). It is a general three-dimensional law for use with any finite

element model. Both tensile and compressive damage modes are taken into account by means of two

scalar damage variables. The two variables represent the material damage by controlling the strain

energy density for tensile stress states (the variable is called tensile damage) and a modified Drucker-

Prager criterion for compressive states (the variable is called compressive damage). Tensile and

compressive stress states are identified by the spectral decomposition of the effective (hyperelastic)

stress tensor. The variables vary from 0 to 1 where 0 means virgin material and 1 meaning that

stiffness and strength are completely lost.

The constitutive model assumes that the damage criteria describe also the plastic surface so that the

development of material damage is simultaneous to the accumulation of irreversible strains for all

stress states. It is to be noted that this assumption does not allow to properly account for dilatancy

effects. A simplified plasticity evolution law represents the residual strains for all stress states. This

feature assures the efficiency of the concrete model since a straightforward predictor-corrector

algorithm is used for the material state determination without internal iterations. The best convergence

rate for the global equilibrium is obtained with the computational tangent for small problems and with

the secant stiffness for large-scale problems. The solution strategy for the global equilibrium is the

Newton method accelerated with the Krylov subspace (Scott and Fenves 2009).

The elastic loading-unloading response is governed by the Young modulus E and the Poisson

ratio . The other model parameters are: the uniaxial compressive strength fcc, the uniaxial tensile

strength fct, the plastic strain ratio , the tensile fracture energy Gft and its equivalent in compression

Gfc. In a uniaxial test, the plastic strain ratio is the ratio between the total strain and its plastic strain

part. It is a constant that replace the consistency parameter of the classical plasticity theory.

For the sake of clarity, the parameters of the model are summarized in Table 1.

Table 1. Parameters for three-dimensional concrete material model

Parameter E fcc fct Gft Gfc

Description elastic

modulus

Poisson’s

ratio

uniaxial

compressive

strength

uniaxial

tensile

strength

plastic

strain

ratio

tensile

fracture

energy

compressive

fracture

energy

REINFORCED CONCRETE MEMBRANE MODEL

For in-plane analysis, a plane stress 2d membrane model has been developed that accounts for the

concrete and reinforcing steel interaction. The general three-dimensional concrete model has been

developed for plane stress states in a plane stress state in agreement with the membrane element

mechanics. The reinforcements are modelled in a discrete or in a smeared way. In the present paper

the embedded approach is preferred and all the parallel rebars are simulated by an additional layer that

comply with the element kinematic and it is characterized by the uniaxial stress and stiffness response

in the direction of the bars. As a result the equivalence of the total strains of the different material

layers is imposed. Perfect bond between concrete and reinforcement is assumed. The smeared model

can be further extended to take into account the bond slip between the two materials at the cost of an

additional computational effort. An important benefit of the membrane model is the ease in handling

any reinforcement configurations.

The Menegotto-Pinto constitutive law with the isotropic hardening is selected for representing

the uniaxial reinforcement response in the direction of a set of steel bars (Filippou et al. 1983).

The main limitations which arise from the current formulation of the model for reinforced

concrete panels are: the concrete dilatancy may be not properly simulated due to the coupling between

damage and plasticity; the perfect bond between concrete and reinforcement bars is assumed; the

failure of reinforcement bars is not simulated.

4

CALIBRATION OF THE CONCRETE CONSTITUTIVE LAW IN COMPRESSION

The model response under cyclic uniaxial compression is represented in Figure 1 and compared with

the experimental results from Sinha et al. 1964. The model parameters for this test are: E=35GPa;

fcc=32.5MPa; Gfc=14.7N/m; =0.55, where fcc is the experimental peak compressive strength. Once the

initial elastic limit, which is assumed to be half of the compressive strength in the model, is exceeded,

the plastic strain and the damage variable increase. The hysteresis of the reloading loop cannot be

simulated by the model because the unloading process is assumed to be elastic.

The three-dimensional concrete constitutive model takes into account the increase of strength

and ductility under biaxial and triaxial compression. In the case of membrane finite elements, the

concrete law needs to be projected in two dimensions and the effect of the transverse stress cannot be

accounted for. Hence the values of fcc and Gfc have to be properly calibrated to represent the

confinement exerted by the transverse reinforcement in the direction normal to the membrane. To this

end, numerical analysis as well as empirical or semi-empirical correlations can be used. In this study

the values of these parameters are selected with the helped of the well-known formulas by Park and

Kent 1971.

For instance, Figure 2a shows the concrete damage constitutive law in compression for the

unconfined case and for two transverse reinforcement ratios T equal to 0.5% and 1.0%. The evolution

of the damage variable dn for the three cases is compared in Figure 2b. The curves displayed in Figure

2 are obtained with a Lagrange four-node quadrilateral with 100 mm side length.

(a) (b)

Figure 1. Cyclic stress-strain response and damage evolution under uniaxial compression.

(a) (b)

Figure 2. (a) Constitutive law in compression for unconfined concrete and confined concrete

(calibrated according to Park and Kent 1971 ); (b) evolution of damage parameter dn.

5

CALIBRATION OF THE CONCRETE CONSTITUTIVE LAW IN TENSION

The numerical response under cyclic uniaxial tension is represented in Figure 3 and compared with the

experimental results from Gopalaratnam and Shah 1985. The model parameters used in this test are:

E=29GPa; fct=3.45GPa; =0.55, Gft=170N/m. The finite element model consists of a Lagrange four-

node quadrilateral with 100 mm side length. For given fracture energy, the local stress-strain response

depends on the finite element size and integration method and differs from the average experimental

response. Nonetheless the example shows the evolution of residual strains and the progressive stiffness

degradation of the unloading-reloading cycles. The evolution of the damage variable, governing the

residual stiffness, and the plastic strain, related to the residual strain upon unloading, provides an

accurate representation of the physical response. Classical plastic flow rules coupled to any damage

model cannot achieve similar accuracy since the consistency parameter is strain dependent. In fact, the

extension of classical plasticity models for steel or soil to the simulation of the cyclic behaviour of

concrete does not seem reasonable.

(a) (b)

Figure 3. Cyclic stress-strain response and damage evolution under uniaxial compression.

If not provided by experiments, the value of the fracture energy can be assumed according to the

expression given by the Model Code 2010:

0.1873F cmG f (1)

where fcm is the mean compressive strength in MPa.

Figure 4a displays the constitutive law in tension for a fracture energy value of 136 N/m

(corresponding to a plain concrete of fcm = 30 MPa according to Eq. 1) and also for two other values of

fracture energy (270 N/m and 540 N/m). Figure 4b displays the evolution damage variable dp for the

three values of fracture energy.

For modelling reinforced concrete specimens, the tensile fracture energy Gft can be modified to

account for the tension stiffening effect. To this end, the model parameters are calibrated against

measurements from uniaxial tension experiments.

6

(a) (b)

Figure 4. (a) Constitutive law in tension for three different values of fracture energy GF; (b) evolution

of damage parameter dp for three different values of fracture energy GF.

SIMULATION OF EXPERIMENTS

In this section the model is used to simulate the cyclic behavior of planar shear walls and coupled

walls which were tested at the NEES facility MUST-SIM (Multi-Axial Full-Scale Sub-Structured

Testing and Simulation) at the University of Illinois in Champaign-Urbana. The selected tests are part

of the research project entitled "Behavior, Analysis, and Design of Complex Wall Systems" which

were aimed at assessing the seismic behavior of modern reinforced concrete wall systems. A total of

four planar shear walls, one coupled wall and three C-shaped (or U-shaped) walls were tested under

lateral reversal loads. A complete description of the test program and test results related to the planar

shear walls and coupled wall is available online at the NEES Project Warehouse website. In the next

section, for the sake of conciseness, only selected experimental data will be briefly presented and

compared with the results of the numerical simulations.

The planar wall specimens are designed in order to be representative of a ten-story prototype

wall of 36.6 m height, 9.1 m length and 46 cm thick. Due to limitation of the facility the specimens are

one-third scale of the first three stories of the prototype. The test parameters are: (i) the lateral load

distribution; (ii) reinforcement layout; (iii) the presence of splice at the base. Two different lateral load

distributions are adopted: the equivalent lateral force distribution according to the ASCE 7-05 (which

is essentially an inverted triangular distribution whose resultant acts at an effective height, Heff equal to

0.71 H, where H is the total height of the prototype building.) and a uniform distribution whose

resultant is applied at 0.50 H. Two reinforcement layouts are adopted, one with longitudinal

reinforcement concentrated in the confined boundary region (referred to as Boundary Element (BE)

layout) and the other with a uniform distribution of longitudinal reinforcement (referred to as uniform

layout). The longitudinal reinforcement ratio in the boundary elements is equal to 3.41%, while it is

equal to 1.57% for the uniform layout. The horizontal reinforcement ratio is equal to 0.27% for both

layouts. Three specimens present spliced longitudinal reinforcement at the base, while one specimen

present continuous bars from the basement to the top of the wall.

The coupled walls test specimen was designed in order to be representative of a typical concrete

core in a 10-story building. The design process was conducted with the following objectives: (i) satisfy

the minimum requirements imposed by the ASCE 318-08 building code; (ii) obtain a prescribed

ductile behaviour under seismic excitation (i.e. performance-based design framework and capacity

design requirements); (iii) be consistent with the characteristics of the planar wall specimens tested

within the Planar Shear Wall Test program (see chapter 2).

During each test reversed lateral loads and a constant axial load were applied. Lateral loads

were applied in displacement control; the imposed drift was increased up to 1.5% for the

7

planar walls and up to 2.0% for the coupled wall. The geometry of the specimen has been modelled using four-node square membrane elements.

The mesh of the planar wall model is composed of 12x14 quadrilateral elements, while the mesh of the

coupled wall is composed of 226 quadrilateral elements. The foundation and the wall cap are included

in the model as linear elastic elements with the same elastic modulus adopted for the non-linear

concrete model. Horizontal loads are applied at the mid height of the wall caps. In order to reproduce

the actual loading protocol and the boundary conditions at the top wall, the effect of the applied top

moment My is simulated by imposing appropriate distribution of the vertical displacements at the node

of the wall cap. A single load path is sufficient to reproduce the applied load protocol, due to the

proportionality between the lateral force Fx and the top moment My.

For the sake of conciseness only selected results of the numerical simulations are here

presented. Firstly, the comparisons between experimental and numerical results in terms of global

response (base shear vs top drift) are provided in Figures 5. In order to better appreciate the model

capabilities selected loading cycles are shown in Figure 6. It can be noted that the model results appear

reasonably accurate at both low and large imposed drifts with the accuracy increasing with the load

amplitude.

Also the capabilities of reproducing local response have been investigated. Figure 7a,b

compares the experimental and numerical strain history of the longitudinal reinforcement at the base

of the wall and at the base of the pier of the coupled wall, respectively. Although perfect bond between

concrete and reinforcement is assumed (i.e. relative slips cannot be explicitly reproduced) the local

strains seem also quite accurate at large drifts.

Figures 8 and 9 show the evolution of the compressive damage variable (dn) for Wall 1 and the

coupled wall, respectively, using contour maps.

The damage variables can be used for the performance assessment of the structure. First, the

local damage variables contours immediately show the cumulative damage distribution in the model

and can be correlated to the physical damage. For instance, a tensile damage greater than 0

corresponds to cracked wall regions and its value can be correlated to the micro-crack width. On the

other hand, a compressive damage of about 0.4-0.5 corresponds to regions with initiation of concrete

crushing following the attainment of the concrete compressive strength.

Secondly, the local damage variable distribution assists with the understanding of the structural

resisting mechanisms. For instance, Figure 7a shows that the compressive damage initially affects the

right side of the specimen and it develops mainly in the vertical direction. This means that the initial

wall resisting mechanism is mainly flexural. Figure 7b shows that the concrete compressive strength is

exceeded in the compression zone corresponding to the state of maximum wall resistance. Figure 7c

and 7d shows that the maximum compressive damage spreads horizontally with increasing lateral

drift, as the wall takes advantage of its shear resistance. The attainment of the unity value of the

compressive damage variable in the element at the bottom left precedes a sudden decrease of the wall

resisting strength. Furthermore, the damage evolution depicted by the contour maps of Figures 8 and 9

agrees very well with the physical damage distribution observed during the experiments (the details

regarding the mechanisms of failure observed during the experimental tests are available in the test

report, Lowes et al. 2011).

Third, the local damage variables can be used to define appropriate global damage indexes that

could be readily correlated with structural limit states so as to define safety margins. Such global

damage indexes were already defined for beam elements (Scotta et al. 2009) showing their suitability

in the seismic performance assessment of reinforced concrete moment resisting frames. The extension

of these global damage indexes to two-dimensional elements is under development.

8

(a) (b)

Figure 5. Comparison between experimental and the numerical results in terms of base shear vs drift:

(a) Wall 1; (b) Coupled Wall.

(a) (b)

(a) (b)

Figure 6. Comparison between the experimental and the numerical results for a single load cycle for

Wall 1: (a) 0.1% drift; (b) 0.5% drift; (c) 0.75% drift; (d) 1.5% drift

9

(a) (b)

Figure 7. Comparison between the experimental and the numerical strain history response for the

longitudinal reinforcement at the base: (a) Wall 1; (b) Coupled Wall.

(a) (b)

(c) (d)

Figure 8. Color contour maps of damage parameter dn for Wall 1: (a) at 0.1% drift; (b) at 0.5% drift;

(c) at 1% drift; (d) at 1.5% drift.

10

(a) (b)

(c) (d)

Figure 9. Color contour maps of damage parameter dn for the coupled wall: (a) at 0.5% drift; (b) at

0.75% drift; (c) at .51% drift; (d) at 2.5% drift.

CONCLUSIONS

The paper highlights the capabilities and limitations of a recently proposed reinforced concrete

membrane model for large-scale seismic analysis. The concrete constitutive law belongs to the class of

energy-based isotropic continuum damage models.

The application of a concrete constitutive law with softening response requires the calibration of

the model parameters for mesh size independence. The effects of confinement and tension stiffening

can be taken into account with parameter calibration as well. After the model calibration with

available material response results, it is used to simulate the hysteretic response of single and coupled

reinforced concrete shear walls with great success.

The damage parameters of the concrete constitutive law permit the identification of the key

resisting mechanisms, the attainment of limit states, and the determination of cumulative physical

damage. These results show that the proposed reinforced concrete membrane model holds significant

promise for the seismic performance assessment of large-scale reinforced concrete structures.

11

REFERENCES

ACI Committee 315, (2005). “Building code requirements for structural concrete (ACI 318-05) and

commentary (318R-05),” American Concrete Institute, Farmington Hills, MI, 1-430

ASCE/SEI 7-10 (2010). “Minimum design loads for buildings and other structures”, American Society

of Civil Engineers, Reston, Virginia.

Bentz EC ( 2005) “Explaining the riddle of tension stiffening models for shear panel experiments”,

ASCE Journal of Structural Engineering, 131(9): 1422–1425.

CEB-FIP, (2010) “Model Code 2010 – First complete draft – Volume 1”, International Federation for

Structural Concrete (fip), Lausanne, Switzerland.

CEN (2004). “Eurocode 8 - Design of structures for earthquake resistance. Part 1: General rules,

seismic actions and rules for buildings” European standard EN 1998-1, December 2004,

European Committee for Standardization, Brussels

Darwin D and Pecknold D.A. (1977) “Nonlinear biaxial stress-strain law for concrete”, Journal of

Engineering Mechanics, 103: 229-241.

Faria, R., Oliver, J., Cervera, M. (1998). “A strain-based plastic viscous-damage model for massive

concrete structures.” International Journal of Solids and Structures, 35, 1533-1558, 1998.

Filippou, F. C., Popov E.P., Bertero V. V. (1983) “Effects of Bond Deterioration on Hysteretic

Behavior of Reinforced Concrete Joints”. Earthquake Engineering Research Center, University

of California, Berkeley, 1983.

Gopalaratnam VS and Shah SP (1985). "Softening response of plain concrete in direct tension" ACI

Journal, 82(3):310-323.

Gulec C.K., Whittaker A.S, (2009). “Performance-based assessment and design of squat reinforced

concrete shear walls”, Technical Report MCEER-09-0010, University of Buffalo

Gulec C.K., Whittaker A.S., Stojadinovic B. (2008). “Shear strength of squat rectangular reinforced

concrete walls”, ACI Structural Journal, 105(4):488-497

Hagen G.R. (2012). “Performance-based analysis of a reinforced concrete shear wall building”, Ph.D.

Thesis, California Polytechnic State University

Hsu TT (1997) “Nonlinear analysis of membrane elements by fixed-angle softened-truss model”, ACI

Structural Journal, 94(5):483-492

Kent DC and Park R (1971) "Flexural members with confined concrete" Journal of the Structural

Division, Proc. of the American Society of Civil Engineers, 97(ST7):1969-1990.

Krolicki, J, Maffei, J., Calvi, G.M. (2011). “Shear strength of reinforced concrete load subjected to

cyclic loadings”, Journal of Earthquake Engineering 15:S1, 30-71, DOI:

10.1080/13632469.2011.562049

Kupfer, H., Hilsdorf, H. K., and Rusch, H. (1969). "Behavior of concrete under biaxial stresses." ACI

J. 66(8), 656-666.

Lee J and Fenves, GL (1998) " Plastic-damage model for cyclic loading of concrete structures”.

Journal of Engineering Mechanics ASCE, 124:892-900

Lee H.J., Kuchma D.A. (2007). “Seismic overstrength of shear walls in parking structures with

flexible diaphragms”, Journal of Earthquake Engineering, 11(1):86-109

Lee H.J., Kuchma D.A., Baker W., Novak L.C., (2008). “Deisgn and analysis of heavily loaded

reinforced concrete link beams for Burj Dubai”, ACI Structural Journal, 105(4):451-459

Lowes, L.N., Lehman, D., C Birely A. C., Kuchma, D., Marley, K., Hart, C.R. (2011), "Behavior,

Analysis, and Design of Complex Wall Systems: Planar Wall Test Program Summary

Document," https://nees.org/resources/3677.

Martinelli, P., Filippou, F.C., (2009). “Simulation of the shaking table test of a seven-story shear wall

building”, Earthquake Engineering and Structural Dynamics, 38:587-607

Mazars, J., Kotronis, P., Ragueneau, F., & Casaux, G. (2006). Using multifiber beams to account for

shear and torsion: Applications to concrete structural elements. Computer Methods in Applied

Mechanics and Engineering, 195(52), 7264-7281.

Palermo M (2014) The effect of two-dimensional squat elements on the seismic behavior of building

structures, Ph.D Thesis, Department DICAM, University of Bologna.

Pramono E and Willam K (1989) "Fracture energy-based plasticity formulation of plain concrete." J.

Engrg. Mech., ASCE, 115(6), 11831203.

12

Priestley, M.J.N., Calvi, G.M., Kowalsky, M.J. (2007). “Displacement-based seismic design of

structures”, IUSS PRESS, Pavia

Pugh J.S. (2012). “Numerical simulation of walls and seismic design recommendations for walled

buildings”, Ph.D. Thesis, University of Washington

Rejec, K., Isakovic, T., Fischinger, M. (2012). “Seismic shear force magnification in reinforced

concrete cantilever structural walls, designed according to Eurocode 8”, Bull Earthquake Eng

10:567–586 DOI 10.1007/s10518-011-9294-y

Rutenberg, A., Nsieri, E. (2006). “The seismic shear demand in ductile cantilever wall system and the

EC8 provisions”, Bull Earthquake Eng 4:1–21 DOI 10.1007/s10518-005-5407-9

Salonikios, T.N. (2002). “Shear strength and deformation patterns of R/C walls with aspectratio 1.0

and 1.5 designed to Eurocode 8 (EC8)”, Engineering Structures 24:39–49

Saritas A. (2006). “Mixed formulation frame element for shear critical steel and reinforced concrete

members”, Ph.D. Thesis, University of California Berkeley

Scott MH and Fenves GL (2009) "Krylov Subspace Accelerated Newton Algorithm: Application to

Dynamic Progressive Collapse Simulation of Frames." Journal of Structural Engineering,

136(5):473-480

Scotta, R., Tesser, L., Vitaliani, R., Saetta, A. (2009). Global damage indexes for the seismic

performance assessement of RC structures. Earthquake Engineering & Structural Dynamics,

38(8), 1027-1049.

Simo, J. C., and Ju, J. W. (1987). "Stress and strain based continuum damage models-I. Formulation."

Int. J. Solids and Struct., 23, 821-840.

Sinha, B.P., Gerstle, K.H., and Tulin, L.G. (1964). “Stress-strain relations for concrete under cyclic

loading”. ACI Journal, 64(2), 195-211.

Turgeon J.A. (2011). “The seismic performance of coupled reinforced concrete walls”, Ph.D. Thesis,

University of Washington

Tesser, L., Filippou, F.C., Talledo, D.A., Scotta, R., Vitaliani, R., (2011). “Nonlinear analysis of R/C

panels by a two parameter concrete damage model”, Compdyn 2011, Corfu Greece

Vecchio F J and Collins MP (1986) “The modified compression-field theory for reinforced concrete

elements subjected to shear”, ACI Journal, 83(2): 219-231

Wallace J.W. (2007). “Modelling issues for tall reinforced concrete care wall building”, The Structural

Design of Tall and Special Building, 6:615-632