479464263-Finite-element-interface-modelling-and-experimental-verification-of-masonry-infilled-rc-frame-Al-Chaar-2008

Prévia do material em texto

See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/284415010
Finite element interface modeling and experimental verification of masonry-
infilled R/C frames
Article · January 2008
CITATIONS
23
READS
1,946
3 authors, including:
Some of the authors of this publication are also working on these related projects:
Emerging materials for lightweight ferro-cement structural systems View project
Study on silicon nanoparticles View project
Ghassan Al-Chaar
US Army Corps of Engineers
39 PUBLICATIONS   688 CITATIONS   
SEE PROFILE
All content following this page was uploaded by Ghassan Al-Chaar on 26 July 2017.
The user has requested enhancement of the downloaded file.
https://www.researchgate.net/publication/284415010_Finite_element_interface_modeling_and_experimental_verification_of_masonry-infilled_RC_frames?enrichId=rgreq-2df088577c0adf96ee721125627f99a2-XXX&enrichSource=Y292ZXJQYWdlOzI4NDQxNTAxMDtBUzo1MjAyMjc4NjkzNDM3NDRAMTUwMTA0MzM5NzM2Mg%3D%3D&el=1_x_2&_esc=publicationCoverPdf
https://www.researchgate.net/publication/284415010_Finite_element_interface_modeling_and_experimental_verification_of_masonry-infilled_RC_frames?enrichId=rgreq-2df088577c0adf96ee721125627f99a2-XXX&enrichSource=Y292ZXJQYWdlOzI4NDQxNTAxMDtBUzo1MjAyMjc4NjkzNDM3NDRAMTUwMTA0MzM5NzM2Mg%3D%3D&el=1_x_3&_esc=publicationCoverPdf
https://www.researchgate.net/project/Emerging-materials-for-lightweight-ferro-cement-structural-systems?enrichId=rgreq-2df088577c0adf96ee721125627f99a2-XXX&enrichSource=Y292ZXJQYWdlOzI4NDQxNTAxMDtBUzo1MjAyMjc4NjkzNDM3NDRAMTUwMTA0MzM5NzM2Mg%3D%3D&el=1_x_9&_esc=publicationCoverPdf
https://www.researchgate.net/project/Study-on-silicon-nanoparticles?enrichId=rgreq-2df088577c0adf96ee721125627f99a2-XXX&enrichSource=Y292ZXJQYWdlOzI4NDQxNTAxMDtBUzo1MjAyMjc4NjkzNDM3NDRAMTUwMTA0MzM5NzM2Mg%3D%3D&el=1_x_9&_esc=publicationCoverPdf
https://www.researchgate.net/?enrichId=rgreq-2df088577c0adf96ee721125627f99a2-XXX&enrichSource=Y292ZXJQYWdlOzI4NDQxNTAxMDtBUzo1MjAyMjc4NjkzNDM3NDRAMTUwMTA0MzM5NzM2Mg%3D%3D&el=1_x_1&_esc=publicationCoverPdf
https://www.researchgate.net/profile/Ghassan_Al-Chaar?enrichId=rgreq-2df088577c0adf96ee721125627f99a2-XXX&enrichSource=Y292ZXJQYWdlOzI4NDQxNTAxMDtBUzo1MjAyMjc4NjkzNDM3NDRAMTUwMTA0MzM5NzM2Mg%3D%3D&el=1_x_4&_esc=publicationCoverPdf
https://www.researchgate.net/profile/Ghassan_Al-Chaar?enrichId=rgreq-2df088577c0adf96ee721125627f99a2-XXX&enrichSource=Y292ZXJQYWdlOzI4NDQxNTAxMDtBUzo1MjAyMjc4NjkzNDM3NDRAMTUwMTA0MzM5NzM2Mg%3D%3D&el=1_x_5&_esc=publicationCoverPdf
https://www.researchgate.net/institution/US_Army_Corps_of_Engineers?enrichId=rgreq-2df088577c0adf96ee721125627f99a2-XXX&enrichSource=Y292ZXJQYWdlOzI4NDQxNTAxMDtBUzo1MjAyMjc4NjkzNDM3NDRAMTUwMTA0MzM5NzM2Mg%3D%3D&el=1_x_6&_esc=publicationCoverPdf
https://www.researchgate.net/profile/Ghassan_Al-Chaar?enrichId=rgreq-2df088577c0adf96ee721125627f99a2-XXX&enrichSource=Y292ZXJQYWdlOzI4NDQxNTAxMDtBUzo1MjAyMjc4NjkzNDM3NDRAMTUwMTA0MzM5NzM2Mg%3D%3D&el=1_x_7&_esc=publicationCoverPdf
https://www.researchgate.net/profile/Ghassan_Al-Chaar?enrichId=rgreq-2df088577c0adf96ee721125627f99a2-XXX&enrichSource=Y292ZXJQYWdlOzI4NDQxNTAxMDtBUzo1MjAyMjc4NjkzNDM3NDRAMTUwMTA0MzM5NzM2Mg%3D%3D&el=1_x_10&_esc=publicationCoverPdf
TMS Journal July 2008 9
1 Structural Engineer, U.S. Army Engineer Research and 
Development Center – Construction Engineering Labo-
ratory, 2902 Newmark Dr., Champaign, IL 61822.
2 President, Bridge Engineering Solutions, Inc., 1706 
Aralia Dr., Mt. Prospect, IL 60056.
3 President, Trilogy Consultants, Inc., 440 Huehl Rd., 
Northbrook, IL 60062.
Finite Element Interface Modeling and 
Experimental Verificaion of Masonry-infilled R/C Frames
Ghassan Al-Chaar1, Armin B. Mehrabi2, and Teymour Manzouri3
Masonry walls have been used as both load-bearing 
structural elements and architectural non-structural ele-
ments in single- and multi-story buildings. Both reinforced 
and unreinforced masonry partitions have been used, 
sometimes filling the space within structural frames and 
other times not bounded by any confining structure. In 
load-bearing walls of the latter type, ties and columns 
are generally used to increase the structural integrity. In 
cases where the masonry infill interacts with the bounding 
frame, that interaction must be accounted for in design 
and evaluation of structures subjected to lateral loading 
such as an earthquake. 
Although there has been much previous work to 
develop analytical models that realistically capture 
the behavior characteristics of experimentally tested 
masonry-infilled R/C frame prototypes, many technical 
difficulties in analytical modeling remain unresolved. 
A simplified analytical model that captures the salient 
characteristics of infilled masonry structures has not yet 
been developed. Success toward that end will require 
understanding masonry-infilled frame behavior in much 
more detail than offered by a strut or beam model. The 
required understanding can be accomplished only through 
experimental investigation followed by numerical simu-
lation and parametric studies. The correct approach will 
facilitate the introduction and calibration of a simple yet 
accurate model for infill walls. 
Recent research has shed new light on the behavior 
of infill frames and has produced advanced analytic tools. 
Classical diagonal strut models have been subjected to 
more thorough evaluations with new experimental data, 
and various limit analysis methods have been developed 
to account for the different load-resisting mechanisms of 
infilled frames. Sophisticated finite element models have 
been developed to analyze the nonlinear behavior of infilled 
frames in a detailed manner. This paper summarizes some 
of those findings and developments, identifies numerical 
models, and demonstrates their capabilities. Also, to facili-
tate the use of these models by researchers and designers, a 
commercial finite element program is identified with similar 
capabilities and is tested for its capabilities. 
The finite element models identified in this study for 
modeling of unreinforced concrete masonry infill walls 
and R/C frames are (1) a cohesive interface model to 
simulate the behavior of mortar joints between masonry 
units and the behavior of the frame/panel interface, and 
(2) a smeared crack finite element formulation to model 
concrete in R/C frames and masonry units. The interface 
model is used to analyze a simple combination of concrete 
blocks and mortar joints, and it also can account for the 
shearing, residual shear strength, and opening and closing 
of joints under cyclic shear loads. 
The behavior of infilled frames is briefly discussed 
to determine important aspects that need to be considered 
and accommodated by the FE modeling approaches. Then, 
a brief review of the modeling approaches and models 
used specifically for masonry-infilled frames is presented. 
From the successful experiences of other investigators, as-
semblies and infilled frame samples for which laboratory 
test and numerical results are available are selected. A 
summary of the results of verification studies conducted at 
the constitutive model level using the selected commercial 
software program is presented. The models in the com-
mercial program were first used to analyze masonry prism 
subassemblies for validation and possible calibration. Then 
the program was used to develop a sophisticated model and 
analysis of the behavior of two types of infill frames, with 
two distinctive failure mechanisms, subjected to in-plane 
lateral loading. The numerical results are compared with 
available experimental and other numerical results. 
CuRREnT STATE oF KnowlEdGE
Behavior of Masonry-infilled Frames 
Subjected to In-plane loads
The behavior of masonry-infilled steel and reinforced 
concrete frames subjected to in-plane lateral loads has 
been investigated by a number of researchers. Fiorato et 
al. (1970) tested 1/8-scale non-ductile R/C frames infilledstats
https://www.researchgate.net/publication/284415010with brick masonry under both monotonically increasing 
and cyclic lateral loads. That work was followed by the 
studies of Klingner and Bertero (1976), Bertero and Brok-
ken (1983), Zarnic and Tomazevic (1985), and Schmidt 
(1989). More recently, single-story reinforced concrete 
frames with masonry infills were studied by Mehrabi et al. 
10 TMS Journal July 2008
(1994), Angel et al. (1994), Buonopane and White (1997), 
and Al-Chaar et al. (1998, 2002)
All studies have shown that the behavior of an infilled 
frame is heavily influenced by the interaction of the infill 
with its bounding frame. At a low lateral load level, an 
infilled frame acts as a monolithic load-resisting system. 
As the load increases, the infill tends to partially separate 
from the bounding frame and form a compression strut 
mechanism as observed in many early studies (e.g., Stafford 
Smith (1962)). However, the compression strut may or may 
not evolve into a primary load-resistance mechanism for the 
structure depending on the strength and stiffness properties 
of the infill with respect to those of the bounding frame.
FE Modeling Studies of Masonry-infilled 
Frames 
Dhanasekar and Page (1986) and Liauw and Lo (1988) 
have used linear and nonlinear beam elements to model the 
behavior of steel frames, and interface elements to model 
the interaction between the infill and the frame. Dhanasekar 
and Page used a nonlinear orthotropic model to simulate the 
behavior of brick infills, and Liauw and Lo used a simple 
smeared crack model to simulate the behavior of micro-
concrete infills. Schmidt (1989) used smeared crack ele-
ments to model both reinforced concrete frames and brick 
infills. In these three analyses, the infill panels have been 
modeled as a homogenous material before fracture and the 
effects of mortar joints have been smeared out.
Mehrabi et al. (1994) demonstrated through experi-
mental and analytical studies that diagonal and horizontal 
cracking within the infill and slip at cracked joints is the 
dominant failure mechanism for R/C frames infilled with 
unreinforced masonry. Although cracking, crushing, and 
accumulation of damage occur also in the masonry units, 
it is the degradation of shear resistance in cracked masonry 
joints that defines loss of lateral stiffness and resistance of 
a masonry-infilled frame. It can be seen that smeared crack 
models have a deficiency in modeling unreinforced ma-
sonry infills because they alone cannot realistically capture 
diagonal shear cracking and the shear sliding of cracked 
concrete or masonry mortar joints. Those deficiencies are 
inherent in the kinematic constraints related to trying to 
account for cracking in a continuum (i.e., a homogeneous 
material). Consequently, the use of smeared crack ele-
ments alone will lead to non-conservative design results 
for unreinforced masonry infill. In order to realistically 
account for the natural planes of weakness in unreinforced 
masonry infill, interface elements must be incorporated 
into the model. 
A number of plasticity-based continuous interface 
models have been developed to model the tension and 
shear behavior of masonry mortar joints [Rots (1991); 
Lotfi and Shing (1994); Lourenco (1996)]. These models 
account for the interaction between normal compression 
and shear as well as the shear dilatation often observed in 
experiments. Mehrabi and Shing (1997) have developed 
an interface model that accounts for the increase of contact 
stress due to joint closing, the geometric shear dilatation, 
and the plastic compaction of a mortar joint for analyzing 
masonry infills. The failure surface of the model is based 
on a hyperbolic function proposed by Lotfi and Shing 
(1994), and is capable of modeling damage accumulation 
at mortar joints under increasing displacement and cyclic 
loading. This is reflected by shear strength reduction and 
mortar compaction (i.e., loss of material) at interfaces. The 
model has been used to analyze the infilled frames tested 
by Mehrabi et al. (1994).
The literature review strongly suggests that the most 
realistic approach to modeling masonry-infilled R/C frames 
would combine both continuum and interface constitutive 
models: the continuum model captures the behavior of the 
reinforced concrete in frame and masonry units in infill, and 
the interface model captures the behavior of mortar joints 
between individual masonry units and between the infill and 
frame. Mehrabi and Shing (1997) have tested the validity 
of this approach and the capability of related constitutive 
models. The use of a plasticity-based total strain, rotating 
smeared crack model with tension softening and shear re-
tention for modeling the concrete continuum in concert with 
a combined Coulomb friction/tension cutoff/compression 
cap interface model for masonry joints was recognized to 
be in good agreement with the requirements established by 
the modeling approach [Lofti and Shing (1994); Mehrabi 
and Shing (1997)]. In general, the models were shown to 
be able to represent the important behavioral aspects of the 
materials and elements used in infilled-frame structures.
Review of Commercial FE Programs
The approach and constitutive models described above 
have thus far been implemented mostly by researchers in 
an academic setting. Those implementations have been 
developed using analytical software programs that are 
not necessarily available in the public domain and typi-
cally have limited scope and applicability. For effective 
modeling and simulation of infilled frames in the design 
and engineering community, a commercial finite element 
program must be employed. The advantages of commercial 
modeling programs include ease of model construction, 
large element and model libraries, user-friendly input and 
output formats, and integrated graphics capabilities. Four 
available FE programs capable of modeling concrete and 
interfaces were selected for review. These were Ansys, 
AdinA, AbAqus, and diAnA. The programs were reviewed 
for their capabilities in modeling structural discontinuities, 
such as mortar joints, in otherwise heterogeneous materi-
als such as concrete and masonry. For the type of infilled 
frames considered in this study, i.e., R/C frames infilled 
with unreinforced concrete masonry (UCM) blocks, the 
TMS Journal July 2008 11
defining parameters are (1) separation of mortar joints and 
infill-to-frame interfaces, and (2) cracking and crushing of 
the infill material. Degradation of shear properties, espe-
cially at the interfaces and joints, was of specific interest. 
Based on stated selection criteria, the diAnA finite ele-
ment program was identified as the best fit for the purpose 
of this study. 
ModElInG BRITTlE MATERIAlS In 
Diana
General Capabilities
diAnA offers a broad range of element types for model-
ing structures made of brittle and quasi-brittle materials, 
including concrete. The constitutive behavior of quasi-
brittle material is characterized by tensile cracking and 
compressive crushing, and by long-term effects such as 
shrinkage and creep. 
Cracking can be modeled in diAnA using plasticity-
based total strain models with multidirectional fixed or 
rotating smeared crack features, and tension softening and 
shear retention. Brittle cracking, linear tension softening, 
multilinear softening, and nonlinear softening are avail-
able. The combination of tensile and compressive stresses 
also can be modeled with a multi-surface plasticity model, 
which is available for biaxial stress states.
Reinforcement in a concrete structure can be modeled 
with the embedded reinforcement types available in diAnA. 
The constitutive behavior of the reinforcement can be mod-
eled by an elasto-plastic material model with hardening. 
Specific Masonry Modeling Capabilities
Masonry structures are analyzed on two different levels: 
one where the global behavior is simulated and the other 
where the behavior is analyzed in more detail. At the global 
level,diAnA offers smeared crack, plasticity-based models 
to simulate cracking. At the detailed level, Diana can model 
the bricks using continuum elements and the joints using 
interface elements. Various models for concrete in the R/C 
frame and in masonry units are available, as described previ-
ously. Other models are available to describe the interface 
behavior: a discrete crack model, a Coulomb friction model, 
and a combined Coulomb friction/tension cutoff/compres-
sion cap model. The latter model, which seems more ap-
propriate for the purpose of modeling masonry joints, was 
formulated by Lourenço and Rots (1996) and enhanced by 
Van Zijl (2000). It is based on multi-surface plasticity, com-
prising a Coulomb friction model combined with a tension 
cutoff and an elliptical compression cap. The model acts in 
softening in all three modes, and is preceded by hardening 
in the case of the cap mode.
The models described above are not the only ones that 
can be applied to masonry. In some cases, inclusion of the 
elastic orthotropy of the masonry may not be essential and 
a standard smeared or total strain crack model with isotro-
pic elasticity may be applied as well. Also, the combined 
friction/tension/compression interface model is not always 
required, and one may choose to use a standard discrete 
crack or Coulomb friction model. 
In summary, diAnA offers a broad range of constitu-
tive material models for concrete and masonry. Modeling 
and analysis at a detailed level is recommended, as noted 
previously. Specifically, masonry units and R/C concrete 
are modeled using continuum elements with a smeared-
crack-type constitutive model capable of cracking and 
crushing; and mortar joints are modeled using structural 
interfaces capable of modeling cohesion, separation, shear 
degradation, cyclic behavior, and closing.
VERIFICATIon oF Diana ModElS And 
CAlIBRATIon PRoCESS
The experimental work by Mehrabi et al. (1994) was 
selected as a baseline for this verification study. That work 
provides detailed test results for masonry subassemblies 
and for masonry-infilled R/C frames as well as numerical 
results following the analytical approach selected here 
in this study. Mehrabi et al. used half-scale specimens of 
conventional concrete masonry blocks, both hollow and 
solid, to construct half-scale R/C infilled frames. Material 
properties and parameters are also well defined in their 
work. Two types of tests are selected: one using masonry 
prisms under axial loading, the other using one-bay, one-
story masonry-infilled R/C frames under constant vertical 
loading and increasing lateral loading. The masonry prisms 
represented the material used to construct the masonry 
infills for the R/C frames. Therefore, the modeling of 
prisms followed by simulation of the behavior of the infilled 
frames provides a consistent and effective verification and 
calibration process.
As discussed previously, the most suitable models for 
the current study were determined to be the smeared crack 
and interface models developed and used by Lotfi and Shing 
(1991) and Mehrabi and Shing (1997). Those models are 
hereinafter referred to as the comparative models. The 
comparative models have been tested and validated through 
several investigations by those researchers, and their results 
are considered to be a useful basis for comparison, along 
with available experimental results. 
In order to evaluate the performance of the finite 
element models used for simulating the behavior of unre-
inforced masonry, it is necessary to observe the behavior 
of the models under different basic modes of stress and 
deformation. The performance of these models in simu-
12 TMS Journal July 2008
lating the behavior of a physical specimen under loading 
is also tested against other numerical investigations and 
experimental results (if available). 
Investigating the Interface Model
The following basic modes of behavior were generated 
for the interface model: (1) separation of interface under 
pure tension and tension softening, (2) shearing and loss of 
cohesion of the joint and residual shear strength under zero 
normal stress, (3) shearing and loss of cohesion of the joint 
and residual shear strength under nominal normal stress, 
and (4) shear behavior under cyclic shear displacement. It is 
believed that an interface model able to realistically capture 
these four modes of behavior has the basic required capa-
bilities for modeling masonry joints. The diAnA interface 
model selected for this evaluation is the combined Coulomb 
friction/tension cut-off/compression cap model.
Prior to performing such evaluation, the parameters 
required for defining the diAnA interface model and the 
comparative interface model is introduced. The diAnA 
model is based on multi-surface plasticity, where the yield 
criterion comprises a Coulomb friction model combined 
with a tension cutoff and an elliptical compression cap, 
as shown in Figure 1. The comparative interface model 
is an elastic/plastic softening, dilatant constitutive model 
whose hyperbolic yield criterion is shown in Figure 2. The 
parameters for these two models are listed and defined in 
Table 1. 
A comparison between the major features and capa-
bilities of the diAnA interface model and the comparative 
interface model is shown in Table 2. The diAnA interface 
model possesses a compression cap that can represent 
compressive damage associated with the failure in joints. 
The comparative interface model does not have this feature. 
The diAnA interface model is not capable of representing 
accumulative damage to the joints in the form of loss of 
material, while the comparative model has this feature. This 
feature affects the active dilatancy and can be of importance 
once the joints are placed under constraints such as in an 
infilled frame. Nevertheless, the two interface models share 
a wide basis for addressing frictional behavior, frictional 
and tensile degradation, and the progression of damage, and 
for the most part they are expected to behave similarly. 
Figure 1—Diana Interface Model Yield Criterion
Coulomb Friction Mode
Cap Mode
Intermediate Yield 
surface
Initial Yield Surface
Residual Yield Surface
Tension Mode
 
Figure 2. Comparative Interface Model Yield 
Criterion [Mehrabi and Shing (1997)].
 
TMS Journal July 2008 13
Table 1. Parameters defining Interface Models (1 psi = 0.00689 MPa, 1 in. = 25.4 mm)
Diana Interface Model Comparative Interface Model Value for Prism
Analysis
Knn = normal numerical elastic stiffness 
parameter, psi/in
Knn = normal numerical initial elastic 
stiffness parameter, psi/in 74E+4 (psi/in)
Kss = tangential numerical elastic 
stiffness parameter, psi/in
Dtt = tangential numerical elastic stiffness 
parameter, psi/in 90E+4 (psi/in)
ft = joint tensile strength, psi so = joint tensile strength, psi 40 (psi)
Gf 
I = first mode fracture energy, psi-in Gf 
I = first mode fracture energy, psi-in 1.61 (psi-in)
co = initial cohesion, psi η = parameter controlling loss of material 
/ plastic flow direction 40 (psi)
φ = internal friction angle, radian ξo = initial asperity slope 0.9
ψ = dilatancy angle, radian ξr = residual asperity slope 0.005
φi = initial internal friction angle, 
radian
µO = initial friction coefficient
0.75
φr = residual internal friction angle, 
radian
µr = residual friction coefficient
150 (psi)
σu = confining stress over which the 
dilatancy will be zero, psi
δ = interface closing distance, in.
2.3
δ = dilatancy shear slip degradation 
coefficient
γ = asperity angle degradation coefficient
16.1 (psi-in)
Gf
II = shear mode fracture energy, psi-in Gf
II = shear mode fracture energy, psi-in 2100 (psi)
cf ′ = compressive strength, psi ro = initial radius of curvature for vertex 
of yield hyperbola 1
Cs = parameter controlling the shear 
stress contribution to failure
rr = residual radius of curvature for 
vertex of yield hyperbola 55 (psi-in)
Gfc = fracture energy in compression,interface, psi-in 
α = parameter controlling the rate of 
friction reduction/softening 0.006
 κp = Norm of plastic strain associated 
with peak compressive strength
β = parameter controlling the rate of 
curvature reduction/softening 74E+4 (psi/in)
Other Notations:
ν = Poisson’s ratio, diAnA smeared crack model
E = Modulus of elasticity, psi, diAnA smeared crack model
cmf ′ = compressive strength of masonry mortar cubes, psi
cuf ′ = compressive strength of masonry units, psi (with respect to net cross-sectional area)
mf ′ = compressive strength of masonry prisms, psi (with respect to net cross-sectional area)
so = joint tensile strength, psi, comparative interface model
εu = strain at maximum strength for axial testing of masonry prisms (in./in.) 
Table 2. Comparison between the Diana and Comparative Interface Models, Features, and Relevant Parameters.
Model Feature
Diana Model Comparative Interface Model
Shape Parameters Shape Parameters
Y
ie
ld
 
Su
rf
ac
e
Shear slipping Coulomb φ, co Hyperbola µ,s,r 
Tension softening Exponential Ψ, δ Exponential s,r
Tension Cut-Off Cut-Off Line ft Tip of Hyperbola s,r
Compression Cap Elliptic Cs , fc NA NA
Flow Rule Exponential Ψ, δ, σu Elliptic a,η
Dilatancy Exponential Ψ, δ,σu Exponential ξ,γ
Loss of Material NA NA Exponential ξ,γ
14 TMS Journal July 2008
Investigating Diana Interface Model Performance
The diAnA interface model was investigated in terms 
of some important behavioral modes. The model for this 
study is a single joint, 8 in. (203 mm) long, 3/8 in. (10 mm) 
thick, and unit depth. This joint is placed between two 
elastic masonry blocks and is analyzed under tension and 
direct shear displacement loading one at a time. The case 
of shear loading is analyzed first without any normal stress 
acting on the interface, then with normal pressure. A com-
bination of the Modified Newton-Raphson and the BFGS4 
secant solution methods was used for analysis at the mate-
rial and structural levels, respectively. The direct solver is 
employed, and the convergence criterion is set for a small 
value of the norm of the residual energy at each iteration. 
The step size is varied for each run in order to find and fit 
the parameters to their best performance. Table 3 shows 
the parameters used for the diAnA interface model for these 
analyses. These parameters are adopted representing a bed 
joint for clay brick masonry [Mehrabi et al. (1994)].
The first analysis was performed to check the tensile ca-
pacity and after-peak behavior of the interface element. The 
interface was subjected to incremental normal displacement. 
The interface element reached a maximum tensile stress of 
38.2 psi (263 kPa) (compared with 40 psi (276 kPa) defined 
by the input parameters), after which significant joint open-
ing and tension softening occurred. The second analysis was 
carried out on the same element, this time subjected to shear 
displacement loading without normal pressure. The maxi-
mum shear strength was approximately 40 psi (276 kPa), 
the same as defined for the model by cohesion factor. The 
residual shear became zero in absence of normal pressure. 
The third analysis was also carried out under shear 
displacement loading, but with an applied constant normal 
pressure of 100 psi (689 kPa). As expected, the normal 
compressive pressure increased the shear capacity, yielding 
a maximum shear stress of 123.8 psi (854 kPa), which is 
comparable by the cohesion resistance (40 psi (276 kPa)) 
plus friction that resulted from 100 psi (689 kPa) normal 
pressure (0.79 × 100 = 79 psi (544 kPa)). The model suc-
cessfully followed the sharp drop after loss of cohesion 
and approached the residual shear resistance defined by 
model parameters. 
The last verification analysis was carried out on the 
same interface element subjected to shear displacement 
loading and under 100 psi (689 kPa) constant normal 
pressure. A full shear loading cycle was applied with 1.2 
in. (30 mm) shear displacement in both directions, a total 
of 2.4 in. (61 mm) shear displacement. As for the shear 
stress, the joint reached a maximum stress of 123.8 psi 
(854 kPa), after which the shear resistance dropped with 
softening toward residual shear strength. The model was 
able to trace the full cycle, and pass unloading and loading 
successfully. In return, after 1.7 in. (43 mm) cumulative 
shear displacement, the shear strength reached the residual 
shear strength defined by the corresponding material pa-
rameter (φr = 0.65).
Verification and Comparison
To validate the performance of the diAnA interface 
model, the model was used to simulate two direct shear 
laboratory tests reported by Mehrabi et al. (1994). For those 
tests, numerical results using the comparative interface 
model were also available. The tests were performed on 
two mortar bed joints—one between two concrete hollow 
blocks and the other between two solid concrete blocks. 
Each joint was subjected to 100 psi (689 kPa) normal pres-
sure in Test 1, and 150 psi (1,034 kPa) normal pressure in 
Test 2. Cyclic shear displacement loading was used.
 
For numerical modeling of these tests, Mehrabi et al 
(1994) used the comparative interface model with parameters 
reported in Table 4 for Tests 1 and 2. The diAnA interface 
model parameters were selected to match those of the com-
parative interface model as much as possible. Both tests were 
modeled using the same material parameters because in the 
comparative models, those parameters defined differently be-
tween Tests 1 and 2 do not have equivalents in the diAnA model 
and/or they have no influence in the analysis performed here 
(e.g., initial asperity angle). The parameters used for analysis 
with the diAnA interface model are shown in Table 5. Some 
of the parameters had to be specified independently because 
there was no equivalent for them in the comparative interface 
model. Parameters Ψi , σu , δ, cf ′ , Cs , Gfc , and κp were taken 
either in the range recommended by the diAnA manual or 
assigned values within a reasonable range. For example, the 
compressive strength, cf ′ , which defines the limit of the com-
pression cap on the yield surface for diAnA interface model, is 
not reflected or used in the comparative interface model but 
is assumed to be 1,500 psi 10.3 MPa, which falls within the 
customary range of masonry strength. The cohesion factor, c, 
was selected to be 40 psi (276 kPa)to approximate the cohesion 
factor calculated for the comparative interface model, based 
on the other input material parameters. 
Table 3. Material Parameters for Diana Interface Model Verification Analysis (1 psi = 0.00689 MPa and 
1 in. = 25.4 mm)
Knn
(psi)
Kss
(psi)
ft
(psi)
Gf
I
(psi-in)
co
(psi) tgφi tgΨ tgφr
σu
(psi) δ Gf
II
(psi-in)
′cf
(psi)
Cs Gfc κp
2.8E+4 3.5E+4 40 1.61 40 0.79 0.005 0.65 150 2.3 16.1 1,500 1.0 2 0.006
4 BFGS: Broyden-Fletcher-Goldfarb-Shanno algorithm.
TMS Journal July 2008 15
Figure 3 shows the shear stress/shear displacement 
curve generated by the simulation of Test 1 using the di-
AnA interface model. Figure 4 shows the shear stress/shear 
displacement curves generated by the laboratory test, and 
the numerical simulation of Test 1 using the comparative 
interface model. These simulation results clearly show 
the capability of the diAnA interface model to accurately 
simulate the test and other numerical results. 
Investigating the Smeared Crack Model
diAnA finite element program has a wide range of 
continuum models for concrete and masonry materials. All 
of these models follow well established and verified formu-
lations, so a full validation investigation is not necessary. 
The purpose of this section is to demonstrate the capability 
of the selected model, which is a plasticity-based, total-
strain, rotating smeared crack model intended to provide 
an acceptable form of stress/strain relationship for concrete 
used in R/C frames and masonry blocks. Without loss of 
generality, a plane stress conditionwas considered in this 
evaluation to match the two-dimensional interface model 
described previously because the two models will be used 
together to model masonry assemblages. Therefore, the 
selected diAnA smeared crack model was tested for its 
overall behavior.
A single block was tested under compression and 
tension. The material parameters, type of tension and 
compression curves, and assumed for them are shown in 
Table 6. Values for parameters such as E, ν, ft, and cf ′ were 
selected to represent a normal weight concrete material. 
These parameters usually define only the strength limits 
and slope of the initial ascending branch of the stress/strain 
curves. Parameters Gf
I, and Gfc, on the other hand, define the 
curvature and descending (softening) branch of the stress/
strain curve. The initial values reflected in Table 6 for these 
parameters were estimated by the area under the expected 
stress/strain curves. By varying the latter parameters, a 
limited calibration of the stress/strain curves was performed 
to provide numerical robustness to the model and also to 
generate desirable shapes for stress-strain curves. The first 
objective is especially important for implementation of the 
constitutive model in a large-scale model for analysis.
 
Table 4. Material parameters for the comparative interface model. (1 psi = 0.00689 MPa, 1 in. = 25.4 mm)
Test Knn 
(psi)
 Dtt 
(psi)
so 
(psi)
Gf
I 
(psi-in) µo µr α δ Gf
II 
(psi-in)
β ro 
(psi)
rr 
(psi) ςο ςr γ η
1 2.8E+4 3.5E+4 40 1.61 0.9 0.75 4 0.2 16.1 2 10 5 0.45 3E-4 3 15.7
2 2.8E+4 3.5E+4 40 1.61 0.9 0.75 2 0.2 16.1 2 10 5 0.15 3E-4 3 55
Table 5. Material Parameters for Diana Interface Model in Comparison Analyses (1 psi = 0.00689 MPa, 
1 in. = 25.4 mm)
Knn
(psi)
Kss
(psi)
ft
(psi)
Gf
I
(psi-in)
co
(psi) tgφi tgΨ tgφr
σu
(psi) δ Gf
II
(psi-in)
′cf
(psi)
Cs Gfc κp
2.8E+4 3.5E+4 40 1.61 40 0.9 0.005 0.75 150 2.3 16.1 1500 1.0 2 0.006
 
Shear Displacement (in.)
S
he
ar
 S
tre
ss
 (p
si
)
Figure 3. Test 1 Simulated using Diana Interface 
Model (1 psi = 0.00689 MPa, 1 in. = 25.4 mm)
 
Shear Displacement (in.)
S
he
ar
 S
tre
ss
 (p
si
)
Figure 4. laboratory Result for Test 1 [Mehrabi et al. 
(1994)]. (1 psi = 0.00689 MPa, 1 in. = 25.4 mm)
16 TMS Journal July 2008
A four-node finite element representing a concrete 
block was considered for these analyses. The two-
dimensional analysis focused on the fracture energy in 
compression, Gfc, and its effect on the parabolic curve of 
the compressive failure. The model was subjected to in-
creasing uniaxial compression in a series of analyses with 
Gfc as the variable. 
ModElInG oF MASonRY PRISMS
numerical Analysis
Two masonry prisms, one with hollow blocks and 
the other with solid blocks, were modeled using diAnA. 
Two-dimensional modeling and analysis were performed. 
Dimensions and geometry were as described in the previ-
ous section. For the prisms with hollow blocks, equiva-
lent thickness of the blocks was considered to be 1.8 in. 
(45.7 mm) and the thickness of the mortar joints was 
1.25 in. (32 mm). For the prisms with solid blocks, the 
thickness of blocks was considered to be 3.625 in. (92 mm) 
and width of mortar bed joints to be 3.5 in. 88.9 mm). 
Masonry units were modeled using rotating smeared 
crack model, and mortar joints were modeled using Diana 
interface models. Two-dimensional, plane-stress, four-node 
elements with four integration points were used for masonry 
units. Two-dimensional four-node interface elements with 
two integration points were used for mortar joints. Mate-
rial parameters used for modeling the concrete in hollow 
and solid blocks are shown in Table 6. The values for the 
parameters were determined by the material test results 
and the infilled frame analysis calibration process reported 
by Mehrabi et al. (1994), and also by values defined by 
verification studies in the first phase of the current study. 
For simplicity, interface normal and shear stiffnesses for 
solid and hollow blocks are assumed to be the same, and 
minor differences introduced by Mehrabi et al. in their 
calibration process were ignored. The tensile strength of 
concrete for which no test result was reported is assumed 
to be 10% of the compressive strength. The fracture energy 
in compression, Gfc , was increased to 22 psi-in. (3.85 MPa-
mm) to provide strain at maximum strength and a shape of 
descending branch (softening) in the normal stress/normal 
strain curve similar to that obtained from the test results.
Material parameters and their definitions for interface 
elements are shown in Table 1. The values for the param-
eters were assigned mostly according to parameters used 
by Mehrabi et al. (1994) for bed joints in modeling of the 
infill walls, and by values defined in verification studies 
in the first phase of this investigation. Mehrabi et al. have 
used shear stiffness values for bed joints in analysis of infill 
walls that are considerably larger than those used in their 
verification studies for individual mortar joints. It was their 
conclusion that lower values calibrated according to single 
joint shear testing do not correspond to practical values 
due to the fact that deformation of the test machine distorts 
the joint deformation measurements. The higher stiffness 
values were the result of their calibration efforts in model-
ing of infilled frames. The compression cap value, cf ′ , was 
defined by the compressive strength of masonry mortar 
reported by Mehrabi et al. (1994). The fracture energy in 
compression, Gfc , was increased to 55 psi-in. (9.63 MPa-
mm) to provide strain at maximum strength and a gradual 
descending branch (softening) in the normal stress/normal 
strain curve similar to that obtained from test results. 
All nodes at the bottom of the lower masonry unit were 
restrained in every direction and a uniform vertical displace-
ment was applied at upper nodes in the top masonry unit. 
The loaded nodes were restrained from horizontal movement 
assuming a perfect bond between loading cap and masonry 
Table 6. Material Parameters for Concrete in Masonry units, Prism Analysis (1 psi = 0.00689 MPa and 1 in. = 25.4 mm)
Parameter Parameter definition Value for Hollow 
Blocks Value for Single unit CMu Value for Solid 
Blocks
E Modulus of elasticity 2 E+6 (psi) 2.0E+6 (psi) 2 E+6 (psi)
ν Poisson’s ratio 0.16 0.16 0.16
ft Tensile strength 240 (psi) 200 (psi) 230 (psi)
Gf
I First mode fracture 
energy 0.09 (psi-in) 0.09 (psi-in) 0.09 (psi-in)
β Shear retention factor NA NA NA
cf ′ Compressive strength 2400 (psi) 3,000 (psi) 2,300 (psi)
Gfc
Fracture energy in 
compression 22 (psi-in) 2 (psi-in) 22 (psi-in)
Tension Curve Shape of tensile stress-
strain curve Exponential Exponential- defined by other 
parameters Exponential
Compression 
Curve
Shape of compressive 
stress-strain curve Parabolic Parabolic- Defined by other 
parameters Parabolic
TMS Journal July 2008 17
units in the prism test. A sample finite element mesh and 
stress distribution is shown in Figure 5. The stress distribu-
tion reflects the formation of shear cones in the prisms that 
agree well with those in the standard prism tests. Stress rise 
at the upper bed joint shows initiation of the failure at that 
joint. Figure 6 shows normal stress/normal strain curves 
for hollow and solid block prisms. The maximum strength, 
strain at maximum strength, and softening curves agree well 
with the test results. The normal stress is calculated based on 
equivalent thickness of 1.8 in. (46 mm) and 3.625 in. (92 mm) 
for hollow and solid block prisms, respectively. It should be 
noted, however, that the failure of the prisms in the analysis 
was initiated and dominated by the failure of mortar joints. 
This is not precisely consistent with the laboratory results, 
in which the failure is dominated by conic shear failure of 
masonry units. The two-dimensional model used for the 
analysis is not capable of generating the confinementin the 
mortar required for transferring the failure to the masonry 
blocks. This represents a limitation of the two-dimensional 
model used in this study. 
Experimental Results
The three-unit, one-half-scale masonry prisms tested 
by Mehrabi et al. (1994) consisted of concrete hollow 
or solid blocks with nominal dimensions of 4 x 4 x 8 in. 
(3.625 x 3.625 x 7.625 in. (92 x 92 x 194 mm) actual) and 
3/8 in. (10 mm), Type S masonry mortar joints. The mortar 
was applied to face shells only for the hollow block prisms 
and to the entire bed joints for the solid block prisms. The 
thickness of each face shell in hollow blocks was 0.625 
in. (16 mm) Therefore, the equivalent width of mortar 
joint for hollow blocks was assumed to be twice the shell 
thickness, i.e., 1.25 in. (32 mm). An equivalent thickness of 
1.8 in. (46 mm) for hollow blocks, including face shells and 
webs, was calculated based on proportion of net concrete 
cross-section with respect to the gross cross-section of the 
block. Table 7 summarizes the average material proper-
ties and test results for hollow and solid masonry prisms 
calculated based on the results reported by Mehrabi et al. 
Figure 5—A Sample Finite Element Mesh and Stress 
Contour for Masonry Prism Analyses
 
 
Figure 6—normal Stress/normal Strain Curves for Hollow and Solid Block Masonry Prisms (1 psi = 0.00689 
MPa, 1 in. = 25.4 mm)
 
0
500
1000
1500
2000
2500
0.000 0.001 0.002 0.003 0.004 0.005 0.006
Normal Strain (in/in)
N
or
m
al
 S
tr
es
s 
(p
si
)
Solid block prism
Hollow block prism
18 TMS Journal July 2008
This table includes the average compressive strength of 
masonry units with respect to the net cross-sectional area 
( cuf ′ ), average compressive strength of masonry mortar 
tested on cubes ( cmf ′ ), compressive strength of masonry 
prisms with respect to the net cross-sectional area ( mf ′ ), 
and strain at maximum strength for masonry prisms (εu). 
A sample stress contour for one of the prism tests is shown 
in Figure 5. The high stress concentration is evident near 
the mortar joint between the top and the middle masonry 
units. The stresses at the ends of the prisms are also high 
but attributed to the loading applied directly at both ends 
of the prism.
ModElInG And AnAlYSIS oF 
MASonRY-InFIllEd R/C FRAMES
The purpose of this section is to examine the capa-
bilities and to identify the limitations of the diAnA finite 
element program in simulating the behavior of masonry-
infilled structures subjected to lateral loading and to estab-
lish a framework for future modeling and analysis of these 
highly nonlinear structures. Two specimens among those 
tested by Mehrabi et al. (1994) are considered with distinc-
tively different load carrying and failure mechanisms. The 
two specimens both have frames that are not designed for 
high seismicity regions (referred to as weak frames) and 
therefore are susceptible to development of shear failure 
in the frame columns—a failure that is of the most concern 
in R/C infilled frame structures. 
diAnA is here to model and analyze one frame with 
a relatively weak infill to further calibrate the models for 
obtaining agreement between analytical and experimental 
results. The failure of such specimen is expected to be gov-
erned by shearing and slip along masonry bed joints. Then a 
second frame with a relatively strong infill will be analyzed 
with the calibrated models. The failure mechanism for that 
frame is expected to be governed by diagonal cracking of 
the infill and shear failure of columns. The goal is to verify 
that the models calibrated with the results of one test could 
be applied to another frame. 
Experimental Results of Masonry-infilled 
R/C Frames
The specimens tested by Mehrabi et al. (1994), as 
modeled in the current study, were half-scale frame physical 
models representing the interior bay at the bottom story of a 
prototype frame. The prototype frame was a six-story, three-
bay moment resisting R/C frame with a 45 x 15 ft tributary 
floor area at each story. Two types of frames were designed 
for the prototype structure with respect to the lateral loadings. 
One was a “weak” frame, designed only for a strong wind 
load, and the other was a “strong” frame, designed to resist 
the equivalent static forces of strong seismic loading. For 
infill panels, 4 x 4 x 8 in. (100 x 100 x 200 mm) (nominal) 
Table 7. Average Material Properties and Axial Test Results for Masonry Prisms [Mehrabi et al. (1994)] 
(1 psi = 0.00689 MPa and 1 in. = 25.4 mm)
Type of Blocks cuf ′ (psi) cmf ′ (psi) mf ′ (psi) εu (in./in.)
Hollow 2,400 2,100 1,550 0.0031
Solid 2,300 2,100 1,930 0.0027
Figure 7—A Sample Stress/Strain Curve for Axial Testing of Masonry Prisms 
(1 psi = 0.00689 MPa and 1 in. = 25.4 mm)
 
TMS Journal July 2008 19
hollow and solid concrete masonry blocks were used to 
represent weak and strong infill panels, respectively. 
The accuracy and reliability of an analytical model 
for simulating the behavior of an infilled frame strongly 
depend on the capability of the model to predict the load-
carrying and failure mechanisms in addition to estimate 
strength and deformations. The specimens considered for 
this study are one with a weak frame and weak infill, and 
one with a weak frame and strong infill. These are common 
failures for frames with unreinforced masonry infill that are 
not designed for high seismicity in accordance with recent 
design code revisions.
Geometry and details of the selected specimens are 
shown in Figure 8. Each test specimen was subjected to con-
stant vertical loading and monotonically increasing lateral 
loading. In this figure, P2 is 22 kips (97.9 kN), P3 is 11 kips 
(48.8 kN), and d is equal to 16.5 in. (419 mm) Figure 9 shows 
damage to the two specimens mapped after completion of 
the tests for weak and strong infill specimens.
Mehrabi et al. (1994) used finite element modeling to 
analyze their test specimen. They used their experimental 
results to calibrate the model and determine parameter 
values. A smeared-crack finite element formulation was 
used to model concrete in the R/C frame and masonry 
units; a nonlinear constitutive model was used for bond-
slip behavior between steel reinforcement and concrete, 
and an interface model was used for the mortar joints. 
For the specimen with strong infill, Mehrabi et al. used 
interface elements at column ends to allow shear failure 
of the columns that otherwise could not be modeled with 
their smeared crack model for concrete. 
Figure 8—Geometry and loading details of Test Specimen [Mehrabi et al. (1994)] (1 in. = 25.4 mm)
 
 
P1 
P2 P3 P3 
P2 
d d 
Figure 9—Failure Pattern from laboratory Tests on weak Infill (left) and Strong Infill (Right)
 
20 TMS Journal July 2008
Modeling of Masonry-infilled R/C Frames
The infilled frame with geometry and details shown 
in Figure 8 was modeled with diAnA. As before, two-
dimensional, plain-stress, four-node elements with four 
integration points were used to model the concrete in R/C 
frame and masonry units. Two-dimensional, four-node 
interface elements with two integration points were used to 
model the mortar bed joints, head joints, and joints between 
the infill and the frame. The diAnA rotating smeared crack 
and interface element models were utilized as described 
earlier in this paper. Reinforcement bars for the frame were 
modeled using diAnA elastic-hardening plastic, two-node 
discrete bar elements. The loading plates were modeled 
with linear elastic, four-node plane-stress elements. The 
loading top beam was modeled with two-node beam ele-
ments using linear-elastic material with steel properties 
while the frame footing was modeled with four-node plane-
stress elements using linear-elastic material with concrete 
properties. The finite element mesh is shown in Figure 10 
at initial stages of lateral loading. Each masonry unit was 
discretized into two elements with an aspect ratio of 1:1. 
Theloading scheme followed the one illustrated in Fig-
ure 20. The vertical load was first applied in increments to 
its maximum and kept constant while lateral displacement 
loading was applied gradually. 
Model Parameter Setting and Analysis
When analyzing infilled frames, Mehrabi et al. (1994) 
adjusted the normal and shear stiffnesses of mortar joints 
to about 30 times those calibrated by their laboratory direct 
shear test results. The need for adjustment was attributed to 
the inaccuracy of interface normal and shear displacements 
in the elastic region of their laboratory test responses, which 
was caused by deflections of the test fixture. They had 
differentiated among mortar bed joints, head joints, joints 
between wall and frame by introducing different interface 
material parameters and thicknesses. 
Parameter Setting for Frame with Weak Infill
For the first trial using diAnA, material parameters 
were selected in agreement with those used by Mehrabi et 
al. (1994). The parameters for the bed joints were the same 
as those used for the prism analysis described previously 
and reflected in Table 1. The use of the normal and shear 
stiffnesses shown in Table 1 resulted in divergence and lack 
of solution at initial stages of the analysis. This divergence 
occurred with initiation of shear cracking and slipping 
at interfaces. The numerical process for determining the 
stress on the yield surface (i.e., return mapping) failed in 
tension-shear or compression-shear corner zones. Persistent 
efforts to prevent the divergence of the algorithm, both with 
reduction of step sizes and tolerances in practical range and 
the application of various available solution methods, did 
not resolve the issue. It was concluded that sharp corners, 
especially at tension cutoff zone, must be repaired in the 
yield surface of the interface model if this problem is to 
be avoided. Mehrabi et al. (1994) used a hyperbolic yield 
surface that avoided corners in the shear-tension zone, and 
their model did not include a compression cap. 
Figure 10—Finite Element Mesh for Frame with weak Infill, Analysis using Diana.
 
TMS Journal July 2008 21
It should be noted that during the earlier verification 
of the constitutive models with the parameter set calibrated 
based on laboratory material test results, the analysis did 
not encounter any problem with convergence. That result 
was due to much lower normal and shear stiffness values 
than those utilized by Mehrabi et al. (1994) for their infilled 
frame analyses. Therefore, it was decided to carry out the 
analysis with lower stiffness values according to those used 
in the material-level investigation. Another trial analysis 
using diAnA confirmed the conclusion by Mehrabi et al. 
(1994) that use of these low values for stiffness does not 
yield an agreement between experimental and analytical 
responses. This trial analysis clearly showed that low nor-
mal stiffness for interfaces resulted in transferring much of 
the vertical load to the frame columns and less to the infill 
wall. Because the shear resistance of the infill wall strongly 
depends on normal stresses, low normal stresses on infill 
resulted in significantly lower lateral resistance for the 
infilled frame. Also, lower shear stiffness for mortar joints 
of the infill resulted in a significantly lower initial stiffness 
for the analytical response curve than that of experimental 
results. To find an agreement between analytical and ex-
perimental results, higher stiffnesses had to be simulated 
while avoiding the numerical convergence problem. It was 
decided to use the lower stiffness values for all interfaces 
as the base stiffness parameters (for which good numerical 
convergence is guaranteed at material level) while increas-
ing the assumed width of joints to provide the higher overall 
stiffness required to obtain agreement with the experimen-
tal results. In order to avoid an unwanted and unrealistic 
increase in the overall interface strength values due to the 
width increase, the parameters affecting the strength (i.e., 
compressive strength, tensile strength, and cohesion) had to 
be reduced by the same proportion to which the thicknesses 
are increased. The confining stress, σu, is related only to 
evolution of dilatancy and is not reduced. 
After a series of trial runs with various width increases, 
a tenfold increase in the stiffness of bed joints from those 
of base values (estimated originally by calibration using 
material test results) was found to provide a good agreement 
between the analytical and experimental responses for the 
frame with weak infill. The stiffnesses of the head joints 
and the joints between the frame and the infill were adjusted 
accordingly, retaining the same proportion considered by 
Mehrabi et al. (1994) in their modeling effort with respect 
to the bed joints. Material parameters for the concrete 
material in the frame and masonry units were calculated 
according to the test results provided by Mehrabi et al. 
(1994) and the verification process discussed previously. 
Table 8 shows material parameters and the definition of 
each for the concrete in the frame and the masonry units. 
Tables 9, 10, and 11 show target material parameters and 
the parameters actually used for bed joints, head joints, and 
joints between frame and wall, respectively. 
Analysis Results for Frame with Weak Infill
With the geometry and material properties described 
above, the analysis on the weak-infill R/C frame was car-
ried out for a maximum lateral displacement of 0.8 in. 
(20 mm) Figure 10 shows the principal stress contour at 
0.07 in. (1.5 mm) lateral displacement. This figure shows 
signs of forming diagonal struts that serve as the load-
carrying mechanism of infilled frames at initial loading 
stages. Figure 11 shows the deformed shape and principal 
stress contour at 0.31 in. (7.9 mm) lateral deflection, be-
fore the maximum strength is reached. This figure shows 
separation and slip at the bed and head joints, which is 
the governing failure mechanism for this infilled frame. 
It also shows that stress at loaded corners of the infill is 
approaching the compressive strength of the masonry, 
Table 8. Material Parameters for Concrete in Frame and Masonry units (1 psi = 0.00689 and MPa, 1 in. = 25.4 mm)
Parameter description Concrete in 
Frame
Concrete in Masonry 
Hollow units
Concrete in Masonry 
Solid units
E Modulus of elasticity 3.55 E+6 (psi) 2.0 E+6 (psi) 2.0 E+6 (psi)
ν Poisson’s ratio 0.16 0.16 0.16
ft Tensile strength 390 (psi) 240 (psi) 230 (psi)
Gf
I First mode fracture 
energy 0.09 (psi-in) 0.09 (psi-in) 0.09 (psi-in)
β Shear retention factor NA NA NA
cf ′ Compressive strength 3,900 (psi) 2,400 (psi) 2,300 (psi)
Gfc
Fracture energy in 
compression 22 (psi-in) 22 (psi-in) 22 (psi-in)
Tension Curve Shape of tensile 
stress-strain curve Exponential Exponential Exponential
Compression 
Curve
Shape of compressive 
stress-strain curve Parabolic Parabolic Parabolic
22 TMS Journal July 2008
Table 9. Material Parameters for Bed Joints, Hollow Block Infill, Diana Interface Model (1 psi = 0.00689 MPa and 
1 in. = 25.4 mm)
Parameter
Set
width
(in.)
Knn
(psi)
Kss
(psi)
ft
(psi)
Gf
I
(psi-in)
co
(psi) tgφi tgΨ tgφr
σu
(psi) δ Gf
II
(psi-in)
′cf
(psi)
Cs Gfc κp
Target 
values 1.25 280,000 350,000 40 1.61 40 0.9 0.005 0.75 150 2.3 16.1 1,500 1.0 55 0.006
Actual 
values used 12.5 28,000 35,000 4 1.61 4 0.9 0.005 0.75 150 2.3 16.1 150 1.0 55 0.006
Table 10. Material Parameters for Head Joints, Hollow Block Infill, Diana Interface Model (1 psi = 0.00689 MPa 
and 1 in. = 25.4 mm)
Parameter
Set
width
(in.)
Knn
(psi)
Kss
(psi)
ft
(psi)
Gf
I
(psi-in)
co
(psi) tgφi tgΨ tgφr
σu
(psi) δ Gf
II
(psi-in)
′cf
(psi)
Cs Gfc κp
Target 
values 1.25 215,300 269,200 10 1.61 10 0.8 0.005 0.7 150 2.3 16.1 1,500 1.0 55 0.006
Actual 
values used 12.5 21,530 26,920 1 1.61 1 0.8 0.005 0.7 150 2.3 16.1 150 1.0 55 0.006
Table 11. MaterialParameters for Joints between Frame and wall, Hollow Block Infill, Diana Interface Model 
(1 psi = 0.00689 MPa and 1 in. = 25.4 mm)
Parameter
Set
width
(in.)
Knn
(psi)
Kss
(psi)
ft
(psi)
Gf
I
(psi-in)
co
(psi) tgφi tgΨ tgφr
σu
(psi) δ Gf
II
(psi-in)
′cf
(psi)
Cs Gfc κp
Target 
values 1.4 215,300 269,200 20 1.61 20 0.8 0.005 0.7 150 2.3 16.1 1,500 1.0 55 0.006
Actual 
values used 14 21,530 26,920 2 1.61 2 0.8 0.005 0.7 150 2.3 16.1 150 1.0 55 0.006
Figure 11—deformed Shape and Stress Contour for Frame with weak Infill, Analysis using Diana
 
TMS Journal July 2008 23
signaling crushing at loaded corners under higher lateral 
displacements. These results agree well with the experi-
mental and analytical results by Mehrabi et al. (1994) and 
illustrate the capability of the models to predict the load-
carrying and failure mechanisms of the R/C frame with 
weak masonry infill. Figure 12 shows response curves 
obtained from this analysis compared with those from 
the laboratory test. Stiffness, response trend (including 
initiation of nonlinear behavior due to mortar joint shear 
cracking and separation), and lateral resistance obtained 
numerically agree well with the experimental results. 
Parameter Setting for Frame with Strong Infill
To examine the capability of the calibrated models 
to predict the behavior of another infilled frame with dif-
ferent characteristics, the diAnA constructed and material 
parameters calibrated for the weak-infill frame were used 
to analyze the frame with strong infill. The material param-
eters for masonry units were changed to those of the solid 
blocks (see Table 8). The only parameter in the interface 
model needing adjustment was the compressive strength 
(or the compression cap), f ’c, to reflect the higher strength 
of the masonry assembly made of solid blocks. The width 
of the mortar joints had to be also adjusted to higher val-
ues. These changes are shown in Table 12. A thickness of 
3.625 in. (92 mm) was used for masonry units in the infill. 
The model was then analyzed with the same loading scheme 
used for the weak-infill frame.
Analysis Results for Frame with Strong Infill
The analysis results indicated that the predicted behav-
ior of the frame with strong infill using the abovementioned 
models agreed well with the experimental behavior for 
the initial portion of the response curve, i.e., for lateral 
displacement smaller than 0.2 in. (5 mm). However, the 
numerical response did not flatten and the lateral resis-
tance continued to increase with increasing displacement 
far beyond the strength obtained in the experimental test 
(see Figure 13). Reviewing the numerical results indicated 
that the shear failure at the top end of windward column, 
expected to govern the failure of the infilled frame based 
on the experimental results, did not occur. Furthermore, 
with stronger mortar joints in the strong infill, sliding of 
bed joints did not occur as it did in the weak-infill frame. 
Table 12. Material Parameters for Mortar Joints in Solid Block Infill that are different from Hollow Block Infill, 
Diana Interface (1 psi = 0.00689 MPa and 1 in. = 25.4 mm)
Parameter
Set
width of Bed Joints
(in.)
width of Head Joints 
(in.)
width of Frame to 
wall Joints 
(in.)
′cf
(psi)
Target values 3.5 3 3.5 2,000
Actual values used 35 30 35 200
Figure 12—Experimental and numerical (Diana) lateral load/lateral displacement Curves for Frame with weak 
Infill (1 kip = 4.45 kn and 1 in. = 25.4 mm)
 
0
10
20
30
40
50
60
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Lateral Displacement (in)
La
te
ra
l L
oa
d 
(k
ip
s)
Experimental (Mehrabi et all, 1994)
Numerical using DIANA program
24 TMS Journal July 2008
Only some diagonal cracking and mortar joint separation 
were observed. This behavior resulted in the development 
of confined a diagonal compression strut in the infill, pro-
viding a much higher lateral strength than expected. This 
problem was attributed to the inefficiency of the smeared 
crack model for concrete in the R/C frame for modeling 
the shear failure. 
To overcome this modeling problem, a discontinuity 
in the form of an interface model was introduced at the 
column ends. The purpose was to allow shear failure of the 
columns along these interfaces, as was anticipated for the 
strong-infill frame. The material properties assigned to the 
interface element at the column ends were similar to those 
of the interfaces in the masonry. However, some param-
eters, such as tensile strength, cohesion, and compressive 
strength, were adjusted to reflect properties of concrete in 
the frame. To avoid problems with convergence, the inter-
face thicknesses were increased here by the same method 
as used for the interfaces in infill. 
With the above modification, the model was analyzed 
again. As can be seen in Figure 14, the response agreed 
well with the experimental results. Figure 14 shows the 
deformed shape and displacement field contour at 0.795 in. 
(20 mm) lateral displacement. It also shows the shear fail-
ure of the top end of the windward column. A large part 
of the lateral displacement has been absorbed by the shear 
failure of the column and displacement of the upper right-
portion of the frame and infill. Load-carrying and failure 
mechanisms of the modified model agree well with the 
experimental results.
In general, the diAnA program performed well for 
modeling of masonry-infilled R/C frames of various 
characteristics. However, the capabilities and limitations 
of the models should be recognized, and modifications or 
improvements be applied for better performance. 
 ConCluSIonS
This study identified a modeling approach that treats 
the discontinuities in masonry by introducing interface 
elements for the masonry joints. Furthermore, the study 
identified constitutive material models through an exten-
sive literature review, and demonstrated their capabilities 
through verification analyses. It was concluded that an 
approach using both continuum and interface constitutive 
models is most suitable for R/C frames infilled with UCM 
blocks. To facilitate the use of these models by researchers 
and designers, a commercial finite element program having 
similar capabilities was identified. The constitutive mod-
els used by this program, diAnA, were examined through 
analysis of data available from earlier experimental physi-
cal modeling tests to verify their strengths, weaknesses, 
capabilities and limitations. 
diAnA was first used to analyze masonry prisms made 
of hollow and solid blocks. Next, one frame with weak infill 
(hollow blocks) was modeled and analyzed, and further 
calibration was performed to obtain agreement between 
the analytical and experimental results. The failure of the 
weak-infill frame was governed by shear cracking and slip 
along masonry bed joints. A second frame, with strong infill 
(solid blocks), was analyzed with the calibrated models. 
Figure 13—Experimental and numerical (Diana) lateral load/lateral displacement Curves for Frame with Strong 
Infill (1 kip = 4.45 kn and 1 in. = 25.4 mm)
 
0.0
10.0
20.0
30.0
40.0
50.0
60.0
70.0
80.0
90.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Lateral Displacement (in)
La
te
ra
l L
oa
d 
(k
ip
s)
Experimental (Mehrabi et al. 1994)
DIANA Model W Column Interface
DIANA Model W/O Column Interface
TMS Journal July 2008 25
The anticipated failure mechanism for this frame was di-
agonal cracking in the infill and shear failure of windward 
column at the top end. The goal was to investigate whether 
the models calibrated with the results of one test could be 
applied to another test as well. In order to obtain agreement 
in this second test, interface elements had to be introduced 
into the columns to allow the shear expected failure. That 
adjustment was necessary because the shear failure could 
not be obtained using the smeared crack model applied to 
the concrete material in the frame. 
The models in diAnA showedgood capabilities for 
predicting load-carrying and failure mechanisms for such 
complex structures as infilled frames. This capability is 
recognized to be the most crucial for reliable strength and 
ductility predictions. Among limitations of the models in 
diAnA, the convergence problem with the numerical scheme 
in return mapping on the yield surface for higher stiff-
nesses for interface elements is a significant impediment 
to a robust and reliable solution. A short-term solution was 
implemented to overcome this problem, but is believed that 
this numerical inefficiency in the models can be overcome 
by improvements to the failure surface and return mapping 
schemes of the interface constitutive model. For the case 
of potential shear failure in columns, it was shown that the 
use of interface elements in the column ends is required to 
compensate for inability of the smeared crack formulation 
to model of shear failure. In any case, it is understood that 
for successful application of any FE program to infilled 
frames, the model parameters must be calibrated using ap-
propriate material-level and structural-level experimental 
results. Once calibrated, the models can be used reliably 
for parametric study of the behavior of infilled frames 
subjected to lateral loading.
REFEREnCES
Al-Chaar, G. K., “Evaluating Strength and Stiffness of 
Unreinforced Masonry Infill Structures,” ERDC/CERL 
TR-02-1/ADA407072, Champaign, IL:Engineer Research 
and Development Center–Construction Engineering Re-
search Laboratory, January, 2002.
Al-Chaar, G. K., “Non-Ductile Behavior of Reinforced 
Concrete Frame with Masonry Infill Panels Subjected to In-
Plane Loading,” USACERL Technical Manuscript 88/18. 
Champaign, IL: U. S. Army Construction Engineering 
Research Laboratory, 1988.
Bertero, V. V., and Brokken, S., “Infills in Seismic Resis-
tant Building,” Journal of Structural Engineering, ASCE, 
U.S.A.: 109(6): pp. 1337-1361, 1983.
Buonopane, S. G., and White, R. N., “Pseudodynamic 
Testing of Masonry Infilled Reinforced Concrete Frame,” 
Journal of Structural Engineering, ASCE, U.S.A.: 125(6): 
pp. 578-589, 1999.
Dhanasekar, M., and Page, A. W., “The Influence of Brick 
Masonry Infill Properties on the Behavior of Infilled 
Frames,” Proceedings of the Institute of Civil Engineers, 
UK: 81(2): pp. 593-606, 1986.
Figure 14—deformed Shape and displacement Field Contour for Frame with Strong Infill at 0.795 in. (20 mm) 
lateral displacement, Analysis using Diana
 
26 TMS Journal July 2008
diAnA 8.1 Finite Element Program, Users’ Manual, 
FEMSYS, TNO Building and Construction Research, 
Department of Computational Mechanics, Delft, The 
Netherlands, 2002.
Fiorato, A.E., Sozen, M.A., and Gamble, W.L., “An 
Investigation of the Interaction of Reinforced Concrete 
Frames with Masonry Filler Walls,” Report No. UILU-
ENG-70-100, Dept. of Civil Engineering, University of 
Illinois, Urbana-Champaign, IL, U.S.A., 1970.
Klingner, R.E., and Bertero, V.V., “Infilled Frames in Earth-
quake-Resistant Construction,” Report No. EERC/76-32, 
Earthquake Engineering Research Center, University of 
California, Berkeley, CA, U.S.A., 1976.
Liauw, T.C., and Lo, C.Q., “Multibay Infilled Frames 
without Shear Connectors,” ACI Structural Journal, ACI, 
U.S.A., pp. 423-428, July-August 1988.
Lotfi, H.R., and Shing, P.B., “An Appraisal of Smeared 
Crack Models for Masonry Shear Wall Analysis,” Comput-
ers and Structures, 41(3), pp. 413-425, 1991.
Lotfi, H.R., and Shing, P.B., “An Interface Model Applied 
to Fracture of Masonry Structures,” Journal of Structural 
Engineering, ASCE, U.S.A., 120(1), pp. 63-80, 1994.
Lourenco, P.B., “Computational Strategies for Masonry 
Structures,” Doctoral Thesis, Civil Engineering Depart-
ment, Delft University, The Netherlands, 1996. 
Mehrabi, A.B., and Shing, P.B., “Finite Element Modeling 
of Masonry-infilled RC Frames,” Journal of Structural 
Engineering, ASCE, U.S.A., 123( 5), pp. 604-613, 1997.
Mehrabi, A.B., Shing, P.B., Schuller, M.P., and Noland, 
J.L., “Performance of Masonry-infilled R/C Frames under 
In-plane Lateral Loads,” Report No. CU/SR-94-6, Dept. 
of Civil, Environmental, and Architectural Engineering, 
University of Colorado, Boulder, CO, U.S.A., 1994.
Rots, J.G., “Numerical Simulation of Cracking in Structural 
Masonry,” HERON, Netherlands School for Advanced 
Studies in Construction, The Netherlands, 36(2), pp. 
49-63, 1991.
Rots, J.G., and Borst, “Analysis of Mixed-mode Fracture 
in Concrete,” Journal of Engineering Mechanics, ASCE, 
113(11), pp. 1739-1758, R, 1987. 
Schmidt, T., “An Approach of Modeling Masonry Infilled 
Frames by the F. E. Method and a Modified Equivalent 
Strut Method,” Darmstadt Concrete, Annual Journal on 
Concrete and Concrete Structures, Darmstadt University, 
Darmstadt, Germany, 1989.
Stafford Smith, B., “Lateral Stiffness of Infilled Frames,” 
Journal of the Structural Division, ASCE, U.S.A., 88(6), 
pp. 183-199, 1962.
Van Zijl, G. P. A. G., “Computational Modeling of Masonry 
Creep and Shrinkage,” Ph.D. Thesis, Delft University of 
Technology, 2000.
Zarnic, R., and Tomazevic, M., “Study of the Behav-
ior of Masonry Infilled Reinforced Concrete Frames 
Subjected to Seismic Loading,” Proceedings of the 7th 
International Conference on Brick Masonry, Australia, 
pp. 1315-1325, 1985.
noTATIonS
co = initial cohesion, psi.
Cs = parameter controlling the shear stress contri-
bution to failure.
Dtt = tangential numerical elastic stiffness param-
eter, psi/in, MPa/mm.
E = modulus of elasticity, psi, MPa, diAnA 
smeared crack model.
FE = finite element.
cf ′ = compressive strength, psi, MPa.
cmf ′ = compressive strength of masonry mor-
tar cubes.
cmf ′ = average compressive strength of masonry 
mortar tested on cubes. 
cuf ′ = compressive strength of masonry units (with 
respect to net cross-sectional area).
cuf ′ = average compressive strength of masonry units 
with respect to the net cross-sectional area. 
mf ′ = compressive strength of masonry prisms (with 
respect to net cross-sectional area).
mf ′ = compressive strength of masonry prisms with 
respect to the net cross-sectional area. 
ft = joint tensile strength, psi, MPa.
Gf
I = first mode fracture energy, psi-in, MPa-mm.
Gf
II
 = shear mode fracture energy, psi-in, MPa-mm.
Gfc = fracture energy in compression, interface, 
psi-in, MPa-mm. 
Knn = normal numerical elastic stiffness parameter, 
psi/in, MPa/mm.
Knn = normal numerical initial elastic stiffness 
parameter, psi/in, MPa/mm.
Kss = tangential numerical elastic stiffness param-
eter, psi/in, MPa/mm.
R/C = reinforced concrete.
r = radius of curvature for vertex of yield hy-
perbola.
ro = initial radius of curvature for vertex of yield 
hyperbola.
rr = residual radius of curvature for vertex of yield 
hyperbola.
TMS Journal July 2008 27
s = joint tensile strength, psi, MPa.
so = joint tensile strength, psi, MPa.
so = joint tensile strength, psi, comparative inter-
face model.
UCM = unreinforced concrete masonry.
α = parameter controlling the rate of friction 
reduction/softening.
β = parameter controlling the rate of curvature 
reduction/softening.
µ = friction coefficient.
δ = dilatancy shear slip degradation coefficient.
δ = interface closing distance, in., mm.
εu = strain at maximum strength for masonry 
prisms in/in(mm/mm).
φ = internal friction angle, radian.
φi = initial internal friction angle, radian.
φr = residual internal friction angle, radian.
γ = asperity angle degradation coefficient.
η = parameter controlling loss of material/plastic 
flow direction.
κp = norm of plastic strain associated with peak 
compressive strength.
µO = initial friction coefficient.
µr = residual friction coefficient.
ν = Poisson’s ratio, diAnA smeared crack model.
σu = confining stress over which the dilatancy will 
be zero, psi, MPa.
ξo = initial asperity slope.
ξr = residual asperity slope.
Ψi = initial dilatancy angle, radian.
ψ = dilatancy angle, radian.
View publication statsView publication