Construction and Building Materials 83 (2015) 159–173
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Construction and Building Materials journal homepage: www.elsevier.com/locate/conbuildmat
Simplified micro modeling of partially grouted masonry assemblages Mohammad Bolhassani a,⇑, Ahmad A. Hamid a, Alan C.W. Lau b, Franklin Moon a a b
Department of Civil, Architectural and Environmental Engineering, Drexel University, Philadelphia, PA, United States Mechanical Engineering and Mechanics, Drexel University, Philadelphia, PA, United States
h i g h l i g h t s A simplified micro model for masonry assemblages is proposed. The model is used for hollow and grouted prisms, shear and diagonal tension specimens. Mortar joints and units are smeared into one homogeneous material. The model relies on CDP and surface-based cohesive behavior for units and interface elements. The proposed model can be successfully used to model grouted, ungrouted and partially grouted masonry walls.
a r t i c l e
i n f o
Article history: Received 2 October 2014 Received in revised form 23 February 2015 Accepted 4 March 2015 Available online 16 March 2015 Keywords: Masonry assemblages Cohesive surface-based behavior Micro modeling Concrete damage plasticity Partially grouted masonry
a b s t r a c t Masonry is an anisotropy structure due to the presence of different components within the assembly. Although, the concepts of concrete modeling are applicable to fully grouted masonry there are difficulties for modeling hollow and partially grouted masonry. Cohesive surface-based behavior (interface elements) has been used in this study as a discontinuity for hollow and grouted masonry. The mortar joints and concrete masonry units were smeared into one homogeneous material using concrete damage plasticity model (CDP). The traction–separation behavior of the cohesive element was employed to model the mortar joints. Damage initiation was considered based on compressive strength of mortar and grout in the hollow and grouted masonry, respectively. A set of tests were conducted on masonry assemblages and properties were used as input in the model. It is evident from results that the responses predicted by the analysis are generally in good agreement with the behavior observed in the experiments for the hollow and grouted prisms, shear and diagonal tension assemblages. The proposed model can be successfully used to model hollow and partially grouted masonry walls. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction Masonry is the oldest structural material still in use for a vast variety of construction. It has been continually employed for different construction purposes. Old masonry structures were designed and constructed without any complicated mathematics, but through the expert masons with trial and error methods. In the last few decades engineers’ attention are being devoted to the new methods of analysis of masonry structures [18,29,19,9,22,23,24, 25,31,26]. These methods are mostly based on developing numerical methods calibrated with experimental test results. The Finite Element Method (FEM) is one of the powerful tools for modeling a continuous structure with an infinite number of complex elements. This is done by converting the structure into simple ⇑ Corresponding author. E-mail addresses:
[email protected] (M. Bolhassani),
[email protected] (A.A. Hamid),
[email protected] (A.C.W. Lau), fl
[email protected] (F. Moon). http://dx.doi.org/10.1016/j.conbuildmat.2015.03.021 0950-0618/Ó 2015 Elsevier Ltd. All rights reserved.
finite elements with estimated mechanical properties such as modulus of elasticity compressive and tensile strength. There are two main categories of FEM approaches for masonry structures modeling called heterogeneous and homogeneous. Mortar joints and units are considered separately in the heterogeneous approach while they are assumed to be smeared into a uniform composite material with average properties in the homogeneous approach. Though homogeneous models are traditionally used for masonry modeling, heterogonous models are more representative [7]. Considering masonry as a heterogeneous material can improve the results compared to the homogenous modeling. Minaie [26] developed a three dimensional FEM for a partially grouted wall using the CDP model for grout and units in Abaqus. The CDP could not represent cracks in a realistic fashion especially in presenting the fracture behavior of the mortar joints. Even though the model could duplicate the cyclic load displacement response fitting to experimental results, the failure mechanism of the wall was not properly captured.
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Nomenclature E0
rt rc rc0 rt0 rtu rcu w fbo fco Em epl eel ec
un-damaged modulus of elasticity, MPa (psi) tensile strength of masonry, MPa (psi) compressive strength of masonry, MPa (psi) initial compressive strength of masonry, MPa (psi) initial tensile strength of masonry, MPa (psi) ultimate tensile strength of masonry, MPa (psi) ultimate compressive strength of masonry, MPa (psi) dilation angle, degrees bidirectional compressive strength of masonry, MPa (psi) unidirectional compressive strength of masonry, MPa (psi) modulus of elasticity for masonry, MPa (psi) plastic strain elastic strain compressive strain
A more detailed model combining smeared and discrete crack was successfully used by Koutromanos et al. [15] to simulate the inelastic behavior of masonry infilled reinforced concrete frames and capture the failure mechanism of a partially grouted wall under cyclic reversed quasi static and earthquake loads. The model used cohesive crack interface elements to model flexural and shear cracks in reinforced concrete columns and mixed mode fracture of masonry mortar joints in the infill. The computational model closely captured the failure mechanism and load displacement hysteresis curves of an infilled frame tested on a shake table [31]. In overall to capture the response of masonry in a more realistically fashion a numerical model requires interface elements. One possible option is to use interface elements for the joints and suitable model for masonry units, grout and mortars. In this case, most of the important factors affecting the wall behavior are taking into account and the model can be solved with general purpose finite element software. Therefore, a simplified 3D micro model considering units and grout is employed to build the model in this study. Proposed model includes material parameters such as elastic and inelastic property, stress–strain, failure and yield criteria of hollow and grouted masonry. For simplicity, masonry units are modeled as a solid block with thickness equal to the sum of the thicknesses of the two face shells. Grout is also represented by solid concrete, mortar joints and blocks are combined into a homogeneous unit material by applying CDP model. Contact elements were used as a discontinuity in the hollow portion using cohesive surface-based behavior.
tensile strain compressive damage factor tensile damage factor sc critical shear stress, MPa (psi) l friction coefficient factor 0 fd diagonal tensile strength, MPa (psi) A area, mm2 (in2) 0 fv bed joint shear stress, MPa (psi) dn ; ds and dt initial separation in normal, in-plane and out of plane shear, mm (in) t n ; t s and t t traction stress vector in normal and two shear direction, MPa (psi) dnf ; dsf and dtf ultimate separation in normal, in-plane and out of plane shear, mm (in) Knn, Kss and Ktt stiffness coefficients, N/m (lb/in) et dc dt
1.1. Macro and micro modeling Masonry is heterogeneous and anisotropic material due to different component constituent materials such as mortar, unit, grout and presence of mortar joints in the two directions. Considering the masonry as a homogeneous or heterogeneous material is a key decision when modeling masonry walls. Macro and micro models have already been developed for masonry as sub-branches of aforementioned approaches. Different strategies of masonry modeling are summarized in Fig. 1 and they will be briefly discussed in this section. Macro modeling: The macro model is based on the homogeneous material and it can provide an approximate response only for the basic design. In the macro approach, masonry is considered as a composite material and this type of model is used to study the overall response of the structure. One of the methodologies to model a system such as shear wall using macro element is to adopt different type of springs instead of structural elements [11]. To simplify the modeling, some researchers considered the homogeneous approach to present the mortar and units with average mechanical properties. This method was used for the large scale models in such a way that mortar joints and units are smeared into one isotropic or anisotropic material. However, since masonry is not homogeneous, this type of model will not be able to properly predict the local behavior of the wall assembly. Some other studies have been conducted to consider masonry homogeneous by defining the smeared crack which called micro/macro modeling. This
Fig. 1. Modeling strategies of masonry.
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method is based on the same approach used in reinforced concrete. Reducing the modulus of elasticity by increasing the load in the model represents the propagation of cracks in the elements. The method has been conducted to evaluate the lateral resistance of hollow and fully grouted masonry shear walls. Lotfi and Shing [18] used the smeared crack in finite element model analysis to investigate the capability of the model in capturing the failure mechanism and ultimate strength of fully grouted concrete shear walls. Their model has shown excellent agreement with respect to flexural behavior, but it was not capable to predict the brittle shear behavior of the walls. This was found to be caused by unrealistic kinematic constraint on the crack opening. The cracks were modeled by introducing displacement in the continuum field. Using the smeared concept is appropriate for fully grouted masonry walls because of their similarity to concrete structures. For hollow masonry, smeared method will be less appropriate. In this category of modeling (micro/macro), Zhuge et al. [32] developed a comprehensive analytical model using two dimensional plane stress concrete behavior of unreinforced masonry. The model was affected by in-plane dynamic load and has been solved in a nonlinear finite element program. The failure envelopes were employed for modeling a homogeneous material. The model was calibrated with experimental tests and results showed reasonable agreement between the analytical and experimental outputs. Micro modeling: To reach more accurate response of partially grouted masonry, different types of masonry components need to be considered. Therefore, the detailed micro modeling is the most appropriate method to predict the actual behavior of masonry. In this method, units, mortar and unit/mortar interface are represented by continuum and discontinuous elements respectively. Different properties of both units and mortar, such as Poisson’s ratio, Young’s modulus and inelastic characteristics, were taken into account in this approach. Although this approach is more realistic and predicts the local behavior of masonry, modeling becomes a complicated question by considering all behaviors of the masonry constituent and this makes the heterogeneous approach uneconomic and inefficient in terms of time. To overcome this problem, simplified micro modeling has been established and most of the studies on finite element modeling of grouted and ungrouted masonry, such as Shing et al. [29], Berto et al. [9], Milani [22], Stavridis and Shing [31], and La Mendola et al. [16] have been conducted on this type of modeling. In this approach, mortar joints are clamped into the unit/mortar interface as a discontinuous element. Expanded units up to half of the mortar thickness in vertical and horizontal directions were simulated to continuum elements. Shing et al. [30] have used discrete crack approach to model the brittle shear behavior of masonry shear walls as a simplified model. Interface elements were used for mortar joints as the primary crack source. Secondary source of cracks were modeled using smeared crack approach in the units. This model successfully predicted the shear behavior of brittle reinforced shear wall when compared to the experimental results. Lourenço and Rots [19] used an interface model for analyzing unreinforced masonry structures. Different failure modes were considered in the model and a new strategy for modeling masonry was developed (details of model are shown in Fig. 2). In this approach, the mortar joints are considered as the weakest elements and modeled by an elastic–plastic interface behavior. Results showed that the model was able to reproduce the experimental response without any numerical difficulties. Additionally, the model can estimate cracks inside the units. Capturing this type of failure was a new achievement in masonry finite element modeling. Shing and Cao [28] employed two different types of elements to model the behavior of partially grouted masonry walls; smeared and interface elements. Smeared elements were used to model
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Fig. 2. Suggested modeling strategies [19].
the behavior of masonry units and plasticity interface elements were used for the tensile and shear behavior of the mortar joints. Results showed that the walls lateral strength predicted numerically are higher than defined experimental outputs. However, the failure mode of the walls was well captured in this model. Moreover, the results showed that the behavior of partially grouted masonry is similar to infill reinforced concrete frame. Homogenization approach is another method in masonry modeling which intended as intermediate between macro and micro modeling. This approach which is studied mainly by Milani et al. [21], Casolo and Milani [10], and Milani [24], Milani [25], is an appropriate method for a heterogeneous structure consist of a periodic cells. In this approach the user only requires a reduced number of material parameter only for an assigned cell by avoiding independent modeling of all joints and units existing through the structure as can be seen in Fig. 3. Implementing this strategy leads to a simplification in numerical modeling and computational cost of masonry. 2. Constitutive models 2.1. Concrete damage plasticity To simulate the nonlinear response of the masonry units and grout, the CDP model available in Abaqus was used. CDP model has been developed to predict the behavior of concrete and other quasi-brittle materials such as rock and mortar under cyclic loading. Cracks in tension or crushing in compression are the main failure modes of this model. The model is based on primary models proposed by Lubliner et al. [20] and Lee and Fenves [17]. The tension and compression damage from micro to macro cracking can be tracked separately in this model. CDP model assumes that the uniaxial compressive and tensile response of concrete is characterized by damaged plasticity (see Fig. 4). 2.2. Cohesive surface-based element Generally, cohesive interactions are a function of displacement separation between the edges of potential cracks. The concept of cohesive zone was employed by Dugdale [12] for the first time. Barenblatt [8] adopted the concept of cohesive stress zone into account for the finite strength of brittle materials in the fracture modeling. Needleman [27] recognized that cohesive elements are partially useful when interface strength are relatively weak
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Fig. 3. Homogenization [21].
(a) Tension
(b) Compression
Fig. 4. Response of concrete to uniaxial loading Abaqus theory manual [1].
compared to the adjoining materials. Composite parts and laminates bonded with adhesives–cohesive behavior are examples for modeling with the adhesive bonded interfaces. Adhesive bonded interface is appropriate to model the separation between two initially bonded surfaces. Cohesive zone model has been extensively employed to study the failure of different materials [13]. It can be used where the interface strength and integrity of structure may be of interest. The mechanical constitutive behavior of cohesive elements can be defined in three methods: (1) uniaxial stress-based, (2) continuum based and (3) traction–separation constitutive model. Where two bodies are connected by a third part material like glue, the continuum based modeling is appropriate for the adhesive. In this case, glue should be considered with a finite thickness. The mechanical properties of adhesive material were employed directly in the model from the experimental results. In general, the adhesive material has more impact than the surrounding material in real structures. Therefore, the damage initiation and propagation dominate the ultimate behavior of composite material. The traction–separation constitutive models can also be used when the glue is very thin and for the practical purpose may be considered as a zero thickness material. The cohesive
(a) Traction-separation response
elements can be applied in situations where cracks are expected to propagate. In this model, cracks are restricted to develop along the layers at the head and bed joints. Prior to damage, the cohesive behavior follows a linear traction–separation law and progressive degradation of the bond stiffness leads to the bond failure. Once a damage initiation in the interface element is met, damage will take place based on the user defined damage factor. A typical traction–separation response is presented in Fig. 5. In the elastic part, the traction stress vector consists of normal, tn and two shear traction components ts ; t t . These components represent mode I, II and III of fracture modes shown in Fig. 5. Also in this model dn ; ds and dt represent the corresponding initial separation caused by pure normal, in plane and out-of-plane shear stresses, respectively. These values can be calculated using the stiffness and strength of each fracture modes. The second part of traction–separation response shows the damage propagation of bond which can be determined in different ways. The maximum nominal stress (MAXS available in Abaqus library) for damage propagation was selected here. The damage initiates when the maximum nominal stress ratio reaches a value of one. Damage evolution in this model describes the degradation
(b) Fracture modes
Fig. 5. Typical traction–separation behavior and fracture modes.
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(the shear stresses are below lp). And surfaces moves infinitely when the critical shear is reached. Basically, there is a finite sliding between contacts even by imposing a small amount of shear stress. The dotted line shows the realistic behavior of Coulomb model. 3. Finite element idealization 3.1. Summary of the experiment
Fig. 6. Frictional behavior (Abaqus theory manual).
of the cohesive stiffness. For this purpose, the maximum separation at the end of graph can be specified either by using the total fracture energy or the post damage-initiation effective separation at failure (i.e. dnf ; dsf and dtf . Post damage-initiation separation was used to consider the damage evolution in the present study. Cohesive elements are used to bond two bodies and they degrade after applying load due to the tensile or shear deformation. Subsequently, the two bonded component come into a contact after debonding. Therefore, Coulomb frictional contact behavior is also defined in the current model. Coulomb friction describes the interaction of contacting surfaces and the model characterizes the frictional behavior using a coefficient of friction, l. It is important to avoid components penetration after forming the contact, especially for the normal behavior of contacts. This allows the assemblages to take apart in presence of the critical force. Abaqus provides two options for the standard contact including surface-to-surface and self-contact. For the presented problem, surface-to-surface contact was used and contacting properties for the tangential and normal behavior were specified. This contact is applicable for modeling two surfaces that are deformable. The coefficient of friction can be defined based upon slip-rate data. In this study, contact-pressure dependent behavior was used based on the results from the shear test assemblages. The tangential motion is zero until the surface traction reaches a critical shear stress value which is dependent upon the normal contact pressure, according to the Eq. (1).
sc ¼ lp
ð1Þ
where l is the coefficient of friction and p is contact pressure between the two surfaces. This equation introduces the limiting frictional shear stress for the contacting surfaces. The contacting surfaces do not slip until the shear stress across their interface reaches the limiting frictional shear stress. The solid line in Fig. 6 represents the ideal behavior of Coulomb friction model; there is zero relative motion (slip) of the surfaces when they are sticking
Regular hollow concrete blocks with nominal thickness of 200 mm (8 in), meeting ASTM C90 [6] provisions, were used in this research as shown in Fig. 7. Different types of hollow and fully grouted concrete masonry assemblages were constructed and tested to determine the physical and mechanical properties of masonry under axial compression, bed joint shear and diagonal tension. Three samples of each specimen were tested to account for variability. The test results demonstrate the distinct difference in behavior (failure mode, strength and deformation capacity) between ungrouted and fully grouted concrete masonry. Grouting provides continuity and uniformity and results in higher strength and deformation capacity compared to ungrouted masonry. The average properties of the assemblages, described in the following sections, were used in the numerical modeling. Type S Portland cement–lime mortar was used in construction of the test specimens. Proportions by volume of Portland, lime and masonry sand were 1:0.5:4.5 following ASTM C270 [2]. Two inch mortar cubes were tested under axial compression to determine compressive strength. The average compressive strength of mortar was 13.1 MPa (1.9 ksi). The model masonry grout consisted of 1.0:2.78:0.74 by weight of cement, sand and water, respectively. The water to cement ratio of the grout was chosen to achieve sufficient workability for good flow of grout into the cells, without any segregation while pouring. Also, during assemblage construction, a steel rod was used to agitate the grout and achieve compaction and good bond between the grout and the blocks. To determine compressive strength of grout, block-molded specimens (prepared as per ASTM C1019 [5] were tested under axial compression, the average grout compressive strength was 23.4 MPa (3.4 ksi). 3.1.1. Axial compression tests Three ungrouted full block wide by three courses high prisms were constructed in stack bond and tested under axial compression following ASTM C1314 [3] to determine the compressive strength of ungrouted masonry (see Fig. 8a). Same setup was used for all other tests. Two grouted half blocks wide by three courses high were tested under axial compression to determine the compressive strength of grouted masonry. Vertical strain was measured using LVDT strain gauges. Load was applied using MTS actuator under force control. Failure mode of the ungrouted was characterized by vertical tensile splitting cracks initiated at the middle web and spreading to the top and bottom units (Fig. 8b).
Fig. 7. Unit configuration.
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Fig. 8. Axial compression test.
Table 1 Compressive strength of ungrouted and grouted prisms. Results specimen
ID
Strength Individual MPa (ksi)
Average MPa (ksi)
COV%
Ungrouted
PU1 PU2 PU3
17.9 (2.5 ksi) 20.7 (3.0 ksi) 20.7 (3.0 ksi)
19.8 (2.8 ksi)
11
Grouted
PG1 PG2
21.6 (3.1 ksi) 29.1 (4.2 ksi)
25.4 (3.7 ksi)
–
For the grouted specimen, failure mode was characterized by diagonal crack as shown in Fig. 8c. Table 1 contains test results of the ungrouted and grouted prisms. For the ungrouted (hollow) prisms compressive strength is presented based on mortar net area. The compressive strength of the grouted prism is higher than that of the ungrouted prisms (based on net area) because the compressive strength of the grout, occupying 51% of the gross area, is much higher than the compressive strength of the outer shell (hollow prisms).
Fig. 9. Shear test specimens.
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mortar joints and resulted in shear slip failure at the block–mortar interfaces. 0 Bed joint shear strength f v is calculated as,
Table 2 Shear strength of ungrouted and grouted specimens. Results specimen
ID
Individual MPa (psi)
Average MPa (psi)
COV%
Ungrouted
SU1 SU2 SU3 SU4
0.22 0.18 0.27 0.16
(31 psi) (25 psi) (39 psi) (24 psi)
0.20 (30 psi)
23
Shear strength
SG1 SG2 SG3
0.58 (84 psi) 0.57 (83 psi) 0.61 (89 psi)
0.60 (85 psi)
0
Grouted
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3.5
3.1.2. Bed joint shear tests The assemblage shown in Fig. 9, originally developed by Hamid et al. [14], was chosen to determine joint shear slip resistance. Three ungrouted (hollow) and three grouted model bed joint shear assemblages with two units height were constructed flat-wise using two full blocks at the middle and one full model block at the top and bottom. Vertical load was applied at the top of the middle block as shown in Fig. 9. This load created pure shear at the
fv ¼
P A
ð2Þ
where P is the applied ultimate load and A is the net and gross contact area between one of the central blocks and the two end blocks for ungrouted and grouted specimens, respectively. Table 2 presents the test results for the ungrouted and grouted specimens. Shear strength of grouted masonry is four times that of ungrouted masonry. This is attributed to the high shear strength of the grout column for grouted prisms compared to the limited mortar bond strength at the block–mortar interface for ungrouted prisms. 3.1.3. Diagonal tension tests Hollow (ungrouted) and fully grouted diagonal tension (DT) assemblages with six units height and three units long were constructed with a running bond and tested diagonally (Fig. 10a) following ASTM E519 [4] Standard. The specimens were constructed by a qualified mason and were filled with grout 24 h after construction. The load was applied uniformly in constant intervals
Fig. 10. Diagonal tension specimens.
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Table 3 Diagonal tension strength of ungrouted and grouted DT specimens. Results specimen
ID
Ungrouted
Grouted
DT strength Individual MPa (psi)
Average MPa (psi)
COV%
DTU1 DTU2 DTU3 DTU4
0.63 0.45 0.43 0.55
0.51 (75 psi)
17.8
DTG1 DTG2 DTG3
1.50 (217 psi) 1.00 (146 psi) 0.85 (124 psi)
1.12 (162 psi)
14.3
(93 psi) (65 psi) (63 psi) (80 psi)
using a vertical MTS actuator under force control. The failure mode of the ungrouted specimens was characterized as step-wise crack at the block–mortar interfaces as shown in Fig. 10b. For the grouted specimens, however, the failure plane followed a straight line through a combination of head joints and masonry units as shown in Fig. 10c. Grout-filled cells tend to reinforce the mortar joints at those locations and force the crack through the units. 0 Horizontal diagonal tensile strength f d at the center of the specimen is calculated as, 0
fd ¼
0:707P A
ð3Þ
Fig. 11. Stress–strain curves of prism, shear and diagonal tension specimens.
Table 4 Mechanical properties of ungrouted and grouted masonry assemblages. Sample
Ungrouted Grouted
Mass
Elasticity
Density kg/m3 (lbf. s2/in4)
Young’s modulus, E0 GPa (ksi)
Poisson’s ratio t
Dilation angle w
Plasticity Eccentricity
fbo/fco
K
Viscosity parameter
26,717 (0.0025)
26.2 (3804) 33.7 (4890)
0.2
32 34
0.1
1.16
0.67
0.001
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M. Bolhassani et al. / Construction and Building Materials 83 (2015) 159–173 Table 5 Compressive and tensile behavior of the model. Concrete damage plasticity Hollow
Grouted
Compressive behavior
Tensile behavior
Compressive behavior
Tensile behavior
Yield stress MPa (psi)
Inelastic strain
Yield stress MPa (psi)
Cracking strain
Yield stress MPa (psi)
Inelastic strain
Yield stress MPa (psi)
Cracking strain
13.8 (2000) 17.2 (2500) 17.9 (2596) 13.8 (2000) 7.6 (1100) 4.4 (640) 2.2 (320) 1 (140)
0 0.00012 0.00032 0.00057 0.00131 0.00191 0.00245 0.00295
0.64 0.62 0.34 0.21 0.12 0.05 0.03 0.02
0 0.00006 0.00028 0.00045 0.00079 0.00139 0.00299 0.00349
17.4 (2523) 20.7 (3000) 21.6 (3135) 20.9 (3038) 19.6 (2851) 15.1 (2191) 10.3 (1490) 2.0 (295)
0 0.0003 0.0005 0.0010 0.0015 0.0029 0.0045 0.0099
1.5 (217) 1.15 (167) 0.83 (120) 0.3 (43) 0.1 (15) 0.06 (9) 0.05 (7) –
0 0.00029 0.00044 0.00092 0.00300 0.00450 0.00530 –
(93) (90) (50) (30) (17) (7) (5) (3)
Fig. 12. Compressive and tensile behavior of ungrouted and grouted model.
Table 6 Cohesive behavior of joints. Sample
Contact Tangential behavior
Normal behavior
Friction coefficient Ungrouted Grouted
0.78
Hard contact
Cohesive behavior Traction–separation behavior Stiffness coefficients MN/m (Kipf/in)
Damage
Knn
Kss
Ktt
Normal
Shear I
Shear II
Plastic displacement mm (in)
8.7 (50) 14 (80)
8.7 (50) 14 (80)
(0) (0)
12.6 (1825) 23.7 (3434)
0.21 (30) 0.60 (85)
0 0
2.0 (0.08) 2.3 (0.09)
Initiation MPa (psi)
Fig. 13. Tension–separation behavior of masonry models.
Evolution
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Fig. 14. Modeled units, finite element mesh and boundary condition.
Fig. 15. Geometry of assemblages and surface-based interaction of units.
where P is the applied ultimate vertical load and A is the net and gross area of the vertical diagonal section for ungrouted and grouted specimens, respectively. The net area is calculated as the gross area times the average percent solid of the block which is taken equal to 51%. Table 3 presents test results of the ungrouted and grouted specimens. As shown, grouting significantly increased the diagonal tensile strength. The strengthening of the bed joints due to the continuity of grout resulted in higher and more uniform strength. 3.2. Tests outputs Stress–strain results of grouted and ungrouted prism specimens were used in the model as inputs. These include elastic and inelastic parameters of both prisms. For this purpose PU1 and PG1 specimens with 17.9 MPa (2.5 ksi) and 21.6 (3.1 ksi) compressive
strength were selected. In elastic and inelastic portions of each stress–strain curve, modulus of elasticity and compressive behavior of concrete damage plasticity were calculated modified and implemented in the subsequent finite element model (Fig. 11a and b). The same procedure in diagonal tension specimens was also followed and stress–strain curve of DTU1 and DTG1 specimens with 0.63 MPa (93 psi) and 1.5 MPa (217 psi) diagonal tensile strength was used in the model. In this case results of horizontal diagonal LVDT were picked up for simulating the tensile behavior of concrete damage plasticity as it shown in Fig. 11e and f. Mechanical parameters of SU1 and SG1 specimens with 0.22 and 0.58 MPa (31 and 84 psi) nominal shear strengths were also employed for modeling mode I of shear in cohesive behavior of mortar (Fig. 11c and d). Additionally, elongation of horizontal diagonal in DT test, reaching to 10% of its maximum strength, was used as plastic displacement in defining the damage
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parameter of contact cohesive behavior. By considering 1.5 m (60 in) gauge length, plastic displacements in damage evolution of ungrouted and grouted specimens measured 2 and 2.3 mm (0.08 and 0.09 in), respectively. Moreover, type S Portland cement–lime mortar was tested under axial compression to determine compressive strength. The average compressive strength of mortar was 12.6 MPa (1.8 ksi). This value was used as damage initiation in the normal direction of ungrouted specimens. However, in grouted specimens grout is the major source of masonry strength against shear. Therefore, in this case average compressive strength of masonry grout, 23.7 MPa (3.4 ksi), was
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employed in order to model the normal damage initiation. All the test outputs which were used in the model will be discussed in the following section. 4. FE model, results and discussion 4.1. Model inputs 4.1.1. Concrete damage plasticity parameters Table 4 shows the material properties which were used for modeling the ungrouted and grouted masonry assemblages. The
Fig. 16. Prism stress–strain curves, principal and Mises stress and strain contours.
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Fig. 17. Shear bond stress–strain curves, principal and Mises stress and strain contours.
plasticity characteristics of material need different types of experimental tests which are beyond the scope of this research. In the absence of such data, the plasticity parameters were determined indirectly by trial and error in the calibration process, and by use of common values recommended in the literature. Modulus of elasticity and the compressive behavior of CDP model were extracted from stress–strain curves of PU1 and PG1 prisms. Elasticity modulus of ungrouted and grouted specimens are 26.2 and 33.7 GPa (3804, 4890 ksi), respectively (Fig. 11a and b). Data collected in the horizontal direction of DTU1 and DTG1 (diagonal tension, Fig. 11e and f) specimens were used for the tensile behavior of the model. Table 5 shows the yield stress versus the inelastic strain and cracking strain calculated from the mentioned tests. For a better presentation, smooth inelastic and crack strain versus stress curves are illustrated in Fig. 12.
4.1.2. Joints cohesive behavior parameters Cohesive behavior of mortar was defined based on information presented in Table 6. Mortar is the only source of bond resistance against shear forces along the bed joints. Therefore, the compressive strength of S type mortar was used for the cohesive behavior of ungrouted specimens in mode I (normal). However, for grouted specimens the compressive strength of grout was used as a normal mode of masonry fracture (see Table 6). As mentioned in Section 3.2, normal strength of tested mortar and grout are 12.6, 23.7 MPa (1.8 and 3.4 ksi), respectively. Shear strength of SU1 and SG1 specimens were also used for mode II which called shear I in the Table 6. For simplicity, in the traction–separation model the same stiffness in the normal and shear directions were assumed. Since there is no out-of-plane shear in force applied to the tested specimens, to simulate the mode III (shear II), shear value of
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Fig. 18. DT load–displacement curves, principal and Mises stress and strain contours.
masonry was taken equal to zero in this mode. Plastic displacement values, which employed in the strength degradation of mortar, were calculated based on the maximum separation of tip displacement in the diagonal tension for the ungrouted masonry assemblages as discussed earlier. Fig. 13 shows the employed traction–separation behavior of ungrouted and grouted specimens in the model. In this graph the maximum stresses are the strength of mortar and grout [12.6 and 23.7 MPa (1825 and 3434 ksi)] for ungrouted and grouted specimens, respectively. In addition, the maximum separation considered the same as plastic displacement of the diagonal specimen, calculated based on displacement related to 10% of strength of smooth curve of DTU1 and DTG1 specimens. Maximum separation of ungrouted and grouted specimens considered 2 and 2.3 mm (0.08 and 0.09 in), respectively. Since the traction–separation graph is linear the slope of each line which called stiffness coefficient are 8.7 and 14 MN/m (50 and 80 kipf/in) for ungrouted and grouted specimens only based on
mortar and grout strengths and also maximum separation. Contact assumed to be zero thickness, therefore hard contact was assigned for normal behavior of contact. It is supposed that ‘‘Hard’’ contact refers to an interaction without any softening, in other words, no penetration of the surfaces can occur in the model. Also, the most common friction coefficient of concrete masonry is in the range of 0.6–0.8, which the best fit was captured using 0.78 in this study. The finite element mesh and boundary conditions of assemblages are shown in Fig. 14. Units are modeled using 50 mm cubical mesh (2 in). All models were tested under displacement control by applying displacement at the top of the specimen using Abaqus implicit. Because there are a large number of elements in the masonry micro-modeling the use of higher order elements often results in extensive computation time without adding too much accuracy into the overall analysis outcomes. Therefore, an eightnode 3D stress linear brick, reduced integration elements
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(C3D8R) were used for modeling the masonry units. These elements are based on linear interpolation and a 4 4 Gauss integration scheme. The model adopts a linear varying normal strain and a constant shear strain over the elements area. The geometry of assemblages and defined interaction surfaces between units are shown in Fig. 15. The size of the grouted unit is 406 203 203 mm (16 8 8 in) and the thickness of mortar is considered zero. However for simplicity, the hollow portions of ungrouted units were ignored, as shown in Fig. 14a and b. The masonry block is modeled as a solid unit with equivalent thickness equal to the thicknesses of unit face-shells. Although omitting the hollow portions of ungrouted units in the modeling resulted in a simplified model, one may say the accuracy of outputs and behavior of masonry assemblages will be affected due to different mechanical behavior of real and simplified hollow units. The major source of strength in hollow prism is provided by the concrete flesh of unit so considering the solid part of unit instead of detailed model will be reasonable in this case. In the shear specimen mortar resists against the load and due to relatively high strength of concrete to mortar, the unit just transfers the stresses. Therefore, in the shear case this assumption will not cause a huge difference between the real and simplified model and using simplified unit will be numerically cost effective. Lastly, though mechanical behavior and fracture mechanism of DT is complex than the other two specimens, it mostly takes advantage of unit compressive strength in the vertical direction and mortar strength in the horizontal direction. Vertical core of DT has the same performance of prism with different joint alignment which is horizontal in the prisms but has 45 degree in the DT. Therefore, the simplified concept will be good enough to represent the behavior of such structure. Additionally, for defining the relation between different surfaces and applying the path of load between units in a logical way, master and slave surfaces were defined in the bed and head joints between the units and the traction–separation behavior was assigned to the surface-based cohesive of mortars as shown in Fig. 15. 4.2. Model outputs Fig. 16 shows the load–displacement relationship of the ungrouted and grouted prisms from the numerical analysis and experimental tests by reaching to peak load. The numerical results showed good agreement with the experimental tests. Despite similar strength at ultimate load in the experiments, axial deformation at peak load of the ungrouted and grouted prisms are similar. As described before, the failure mode of the ungrouted specimen was characterized by vertical tensile splitting cracks initiated at the middle web and spreading to the top and bottom units (Fig. 8b) and for the grouted specimen, the failure mode was characterized by diagonal crack as shown in Fig. 8c. The final Mises stresses, maximum principal stress and strain in ungrouted and grouted specimens are presented in Fig. 16. As can be seen, the maximum stress in the ungrouted specimen occurred at the bottom edges of prism and the maximum stress shear of the grouted prisms is located at the bottom of sample, which coincides with the experimental results. Fig. 17 shows the experimental and numerical stress–strain curves of the ungrouted and grouted shear specimens, respectively. The deformation at the ultimate load of the grouted specimens is much higher than that of the ungrouted specimens. Adhesion mortar bond at the block–mortar interfaces has very small deformation, indicating high degree of brittleness for this mode of failure. It is to be noted that the axial stress normal to bed joints, induced by gravity and axial loads, maintains a large amount of slip deformation after failure of adhesion bond. Thereby, this creates a ductile mode of failure and large energy absorption capacity in the
axially loaded prisms. The stiffness of specimens obtained from the numerical analysis is the same as obtained from the tests however, the maximum shear strength in both models are a little bit greater than experimental results. In addition as shown in Fig. 17, the maximum stress occurred at the neighbor of surfaces and gap. Fig. 18 shows load–deflection curves of the vertical diagonal in the compression path of diagonal tension specimens. Grouted specimens showed much higher deformation capacity compared to the ungrouted specimens. The model was able to predict the behavior of ungrouted and grouted diagonal tension with good agreement. The stress distributions in the ungrouted and grouted specimens are almost the same and it was shown that the diagonal portion of the specimen carried the most of load. As expected, the left and right sides of model take zero load and deformation. 5. Summary and conclusions For having a better understanding of the behavior of masonry assemblages a simplified micro model of assemblages has been developed. Failure and yield criteria, elastic and inelastic property of masonry and stress–strain were employed in this model based on different experimental test results. Cohesive surface-based behavior (interface elements) was used in this study as a discontinuity. Mortar joints and units were smeared into one homogeneous material, and they were modeled using concrete damage plasticity model. The traction–separation behavior of cohesive element was employed for modeling the mortar joints. The initiation of damage was considered based on strength of mortar and grout in the ungrouted and grouted masonry, respectively. Damage evolution, maximum crack opening when the load becomes zero, was defined as a plastic displacement. The mechanical properties of the ungrouted and grouted masonry were extracted from the prism and diagonal tension tests, respectively. It is evident from the numerical results that the responses predicted by the analysis are generally in agreement with the behavior and strength of the test specimens. In the experimental part of the study, test results showed that there is a distinct difference in behavior (failure mode, strength and deformation capacity) between the ungrouted and fully grouted concrete masonry assemblages. Grout-filled cells tend to reinforce the week mortar bed joints resulting in more continuity and uniformity. Therefore, fully grouting the cells of concrete masonry units increased the compressive strength, shear strength and diagonal tensile strength of assemblages. The model also was able to capture these differences with the minimum error. The proposed model can be successfully used to model ungrouted and partially grouted masonry walls. Acknowledgments This project is supported by a Grant from National Science Foundation (NSF) Grant No. 1208208. The support of Delaware Valley Masonry Institute and Sabia Mason Contractors in providing the mason to build the test specimens is acknowledged. The results, opinions, and conclusions expressed in this paper are solely those of the authors and do not necessarily reflect those of the sponsoring organizations. References [1] Abaqus analysis user’s manual, 6.13-3. RI, USA: Dassault Systems Providence; 2013. [2] ASTM C270. Specification for mortar for unit masonry. West Conshohocken, PA; 2012. [3] ASTM C1314. Standard test method for compressive strength of masonry prisms. West Conshohocken, PA; 2014.
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