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Finite element analysis of interlocking mortarless hollow block masonry prism
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Computers and Structures 86 (2008) 520–528 www.elsevier.com/locate/compstruc
Finite element analysis of interlocking mortarless hollow block masonry prism Waleed A.M. Thanoon a,*, Ahmed H. Alwathaf b, Jamaloddin Noorzaei c, Mohd. Saleh Jaafar c, Mohd. Razali Abdulkadir c a
Civil Engineering Department, Faculty of Engineering, University Technology Petronas, 31750 Tronoh, Perak, Malaysia b Civil Engineering Department, Faculty of Engineering, Sana’a University, Sana’a, Yemen c Civil Engineering Department, Faculty of Engineering, University Putra Malaysia, 43400 UPM-Serdang, Malaysia Received 5 October 2005; accepted 30 April 2007 Available online 22 June 2007
Abstract Interlocking mortarless masonry system has been developed as an alternative system for the conventional bonded masonry. This paper covers the analysis of interlocking mortarless hollow concrete block system subjected to axial compression loads using FEM. An incremental-iterative finite element code is written to analyze the masonry system till failure. The stress–strain relation obtained from test is employed and equivalent uniaxial strain concept is used to account for the material nonlinearity in the compression stress field. The developed program is also capable of simulating the nonlinear progressive contact behaviour (seating effect) of dry joint taking into account the block bed imperfection. The comparison shows a good agreement between the developed FE program and the experimental test results. Ó 2007 Civil-Comp Ltd and Elsevier Ltd. All rights reserved. Keywords: Finite element method; Nonlinear analysis; Constitutive relations; Masonry; Interlocking block; Dry-stacked masonry; Seating effect
1. Introduction Interlocking mortarless load bearing hollow block system is different from conventional mortared masonry systems in which the mortar layers are eliminated and instead the block units are interconnected through interlocking protrusions and grooves [1,2]. The behaviour of this system is affected highly by the behaviour of the mortarless (dry) joint in both elastic and inelastic stages of loading. The geometric imperfection of the bed blocks is an important factor influences the structural behaviour of the mortarless masonry system [3,4]. Hence, modelling of mortarless joint plays a significant role in the overall simulation of the system. Also, an accurate estimation for the
*
Corresponding author. Tel.: +60 3 89466370; fax: +60 3 86567129. E-mail address: [email protected] (W.A.M. Thanoon).
masonry strength and failure mechanism requires an adequate simulation for the material stress–strain behaviour. FEM has been extensively used in analyzing of masonry structures and numerous models have been developed to simulate the behaviour of different types of the conventional mortared masonry systems using FE technique. However, analytical studies on the mortarless block systems are limited and depend mainly on the type of block used to assemble the walls. A finite element model was proposed by Oh [4] to simulate the behaviour of interlocking mortarless block developed in Drexel University. The procedure implemented to simulate the contact behaviour of mortarless joint including geometric imperfection of the dry joint is suitable only for modelling of small masonry assemblages. Material nonlinearity is not considered to account for the behaviour of the masonry near the ultimate load and to predict failure mechanism. Alpa et al. [5] suggested a macro-model based on homogenization techniques to model the joint and the block as a homogenous material.
0045-7949/$ - see front matter Ó 2007 Civil-Comp Ltd and Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruc.2007.05.022
W.A.M. Thanoon et al. / Computers and Structures 86 (2008) 520–528
That model focuses on the joints movement mechanism assuming perfect joint. Using the model in the structural application without modification is difficult because it ignores significant issues such as, progressive material failure, material nonlinearity and joint imperfection. In this study, a finite element model is proposed and an incremental-iterative program is developed to predict the behaviour and failure mechanism of the system under compression. The nonlinear progressive contact behaviour of dry joint that takes into account the geometric imperfection of the block bed interfaces is included based on experiment testing. The developed contact relations for dry joint within specified bounds can be used for any mortarless masonry system efficiently with less computational effort. Furthermore, the best fit equation for the experimental test data is used in the program to describe the stress–strain behaviour of the masonry block unit under compression for the uniaxial and biaxial stress state. Material nonlinearity in the compressive stress field is considered for the masonry block in the orthogonal directions and the effect of microcracking confinement and softening on the stress–strain relationship under biaxial stresses are included employing the equivalent uniaxial strain concept. The program allows for the progressive local failure of the masonry block (crushing and cracking). After cracking, a smeared crack model is adopted and the compressive strength reduction in the cracked block is considered. Previous studies on hollow block masonry under compression showed that the differences in ultimate strength using 2D (plane stress and plane strain) and 3D FE analysis were not more than 5% [6,7]. The accuracy can be increased if the FE mesh is used in the critical direction where the failure is initiated. Also using 3D discretization requires high computational effort. Therefore, plane stress 2D continuum is adopted in this study. Eight-nodded isoparametric element is used to model block unit and three-nodded isoparametric interface element of zero thickness located between two material elements is employed to
521
model the interface characteristics of the dry joint, as shown in Fig. 1. For numerical integration and stress calculation, nine and three Gaussian points are used in the plane stress element and interface element, respectively. 2. Masonry material modelling 2.1. Stress–strain relation In this study, the best fit equation of the experimental data of masonry block [8] under uniaxial compression test for both ascending and descending parts is adopted. It can be expressed as [9] r¼
pðe=e0 Þr0 p p 1 þ ðe=e0 Þ
where r, e
ð1Þ
instantaneous values of the stress and the strain, respectively the ultimate stress (peak) and the corresponding strain, respectively a constant called material parameter depends on the shape of the stress–strain diagrams
r0, e0 p
An equation was suggested in Ref. [9] to calculate the material parameter (p), which depends on the initial tangent modulus of the stress–strain curve. To avoid the inaccuracy in evaluating the initial tangent modulus, nonlinear regression analysis has been used in this study to determine the material parameter (p), which yields more accurate value for the parameter [8]. Eq. (1) is capable of simulating the stress–strain relation for different masonry materials and can be incorporated efficiently in the biaxial stress model. Fig. 2 shows the experimental test data of the block and the best fit curve drawn by Eq. (1). A comparison with the formula suggested by Saenz [10], which is frequently used for simulation of compressive stress–strain curves of concrete and masonry under biaxial stress state [11,12], is also shown in Fig. 2.
1
2
1
Compressive stress N/mm 2
25
20
15
10 Saenz's Eq [10] 5
Eq.1 Test results [8]
1 Masonry element 2 Joint element (zero thickness) Fig. 1. Masonry and joint element.
0 0
0.0005
0.001
0.0015
0.002
0.0025
0.003
0.0035
Strain mm/mm Fig. 2. Comparison of test data and the best fit relation.
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To include the biaxial stress effect on the uniaxial stress– strain relation given by Eq. (1), the following procedure has been suggested Rewriting Eq. (1) in terms of equivalent uniaxial strain, we obtain (for i = 1, 2): ri ¼
pðeiu =eip Þrip p p 1 þ ðeiu =eip Þ
where rip eip eiu
ð2Þ
maximum (peak) compressive principal stress in direction i equivalent uniaxial strain corresponding to maximum (peak) compressive principle stress rip the equivalent uniaxial strain
The equivalent uniaxial strain eiu essentially removes Poisson’s effect; whereas the strengthening due to the microcracking confinement in biaxial compression stress and softening in compression–tension stress fields are incorporated in rip and eip, respectively [13–15]. Thus a single relation (Eq. (2)) can represent the infinity variety of monotonic biaxial loading curves (Fig. 3). The tangent moduli E1t and E2t for a given principal stress directions are found as the slopes of the r1 versus e1u and the r2 versus e2u curves for the current e1u and e2u as follows: p
Eit ¼
p
pEs ½p 1 þ ðeiu =eip Þ pðeiu =eip Þ
ð3Þ
½p 1 þ ðeiu =eip Þp 2
where Es is the secant modulus at the peak (maximum stress) rip/eip. Fig. 3 depicts the equivalent uniaxial stress–strain curves for masonry element loaded with different biaxial stress ratio, a (a = r1/r2) in the compression and tension fields. In the tension field, the relation is linear and the slope is equal to the initial tangent modulus (E0) at the origin. The strength is reduced when a < 0.0 whereas for a > 0.0,
the strength is enhanced due to microcracking confinement. The maximum stress (peak), rip, and the corresponding strain eip will be found from the biaxial failure criteria illustrated in the next section. 2.2. Failure criteria and constitutive laws The proposed model uses the biaxial compression strength envelope proposed by Veccho [16]. Fig. 4 describes the failure under biaxial compression. The principal stresses in two orthogonal directions are denoted by r1 and r2 with jr1j 6 jr2j (using negative sign (ve) for compressive stress and strain). The following equations define the failure surface: 2
ð4aÞ
0:76ðr1 =fc0 Þ2
ð4bÞ
K c1 ¼ 1 þ 0:92ðr2 =fc0 Þ 0:76ðr2 =fc0 Þ K c2 ¼ 1 þ
0:92ðr1 =fc0 Þ
r1p ¼ K c1 fc0
ð5aÞ
K c2 fc0
ð5bÞ
r2p ¼
fc0
where is the uniaxial compressive strength of the block unit which will be taken here equal to r0. The strains corresponding to the ultimate strengths in the orthogonal directions are e1p ¼ K c1 e0 e2p ¼ K c2 e0
ð6aÞ ð6bÞ
where e0 is uniaxial strain corresponding to the ultimate strength, fc0 . It follows that readily the above relation result in a symmetric state of stress and strain under equal biaxial compression. For tension–compression region; Cerioni and Doinda [12] proposed a smooth curve for masonry material instead of the straight line developed by Kupfer and Gerstle [17]. The envelope relation that is used in this region can be written as
-σi
σip σo=
fc ' α > 0.0 α = 0.0 α < 0.0 εcr
εo
εip
feq Fig. 3. Stress–strain curves of masonry block for different biaxial stress ratio.
- εiu
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based on Eq. (3) and in the tensile field from linear relation (Fig. 3). For undamaged block, the envelope curve in Fig. 4 based on the formulations presented in Eqs. (4)–(10) for different stress states is adopted. After cracking, the tangential elasticity modulus and the Poisson’s ratio are reduced to zero in the direction perpendicular to the crack direction. Instead of zeros, very small values are substituted in the program to avoid singularity in stiffens matrices. The masonry along cracks is still resisting compressive stress after cracking. It was found [18,19] that the compressive strength of concrete after cracking can be significantly reduced by the tensile strain in the transverse direction. To account for this effect, the following formula is adopted to obtain r2p after cracking:
f't /f'c σ 1/ f ' c
1.0
1.0 Proposed envelope Kupfer envelope [17]
σ2 /f'c
r2p 1 6 1:0 ¼ 0:8 þ 0:34ðe1 =e0 Þ fc0
Fig. 4. Masonry envelope for different stress states.
r1p ¼ feq ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 ðr2 =fc0 Þ ft0 0:65fc0
r2p ¼ r1p =a 6
ð7aÞ ð7bÞ
3. Modelling of dry joint
ð8aÞ 3
2
e2p ¼ e0 ½1:6q þ 2:25q þ 0:35q
ð8bÞ
where ft0 feq E0
tensile strength of block unit equivalent tensile strength the initial tangent modulus of elasticity (at the origin) q ¼ r2p =fc0 For tension–tension region:
r1p ¼ ft0 P r2p e1p ¼
ft0 =Ec
ð9Þ
P e2p ¼ r2p =E0
ð10Þ
The failures envelop is shown in Fig. 4 for all stress states. An incrementally relationship is assumed between strains and stresses, which in differential form and in the principal directions can be written for undamaged masonry: fdrg ¼ ½Dc fdeg 8 9 2 pffiffiffiffiffiffiffiffiffiffi E1 m E1 E2 > < dr1 > = 1 6 pffiffiffiffiffiffiffiffiffiffi dr2 ¼ E2 4 m E1 E2 2 > > : ; 1m ds12 0 0 8 9 > < de1 > = de2 > > : ; dc12
where pffiffiffiffiffiffiffiffi v ¼ v1 v2 and
v1 ¼ v2 ¼ 0:2
ð14Þ
where e0 is the compressive strain relative to the uniaxial compressive strength, fc0 , and e1 is the tensile strain normal to the crack direction.
in which r1p tensile stress and r2p compressive stress and the maximum strain can be evaluated as e1p ¼ r1p =E0
523
ð11Þ 3
0 7 0 5 pffiffiffiffiffiffiffiffiffiffiffiffi 0:25ðE1 þ E2 2m E1 E2 Þ ð12Þ
ð13Þ
The tangent moduli of elasticity, E1 and E2, along the principle stress directions are evaluated in the compressive field from a nonlinear equivalent uniaxial stress–strain relation
The characteristics of dry joint under compression and shear loading has been investigated experimentally [3,20] and load–deformations relations that are required for the FE modelling is formulated based on the experimental results. Contact test of block beds was carried out to investigate the behaviour of mortarless joint under compression. The test was carried out on small wall panels at different joint locations to take into consideration the different sources of block bed imperfection [3]. Modified triplet tests on mortarless interlocking panels with different levels of axial compression were carried out to ascertain the deformation and shear strength characteristics of the system at the joint interface. Failure criterion and stiffness of the interlocked bed joints under combined normal–shear load was proposed [20]. The nonlinear behaviour of the mortarless bed joint under compressive load is simulated taking into consideration the block bed imperfection. The proposed mathematical model that can describe the nonlinear compressive stress in terms of the joint closure is rn ¼ ad bn þ cd n where rn dn a, b, c
ð15Þ
compressive stress (N/mm2) close-up deformation (mm) constants determined from data analysis of the test results
Eq. (15) yields an average compressive stress on the contacted area for family curves that consider the variety of geometric imperfection of block beds. Fig. 5 shows the upper and lower bounds of Eq. (15). Although high variation in close-up deformations was measured at different dry joints, the average deformation of all specified joints in a
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W.A.M. Thanoon et al. / Computers and Structures 86 (2008) 520–528
Compressive stress N/mm2
10
8 Lower bound model Average model Upper bound model Lower bound -Test data [3] Average - Test data [3] Upper bound - Test data [3]
6
4
Lower Average Upper a 3272.02 93.75 132.02 b 4.28 3.01 7.46 c 5.00 5.00 5.00
2
0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Close-up deformation mm Fig. 5. Close-up deformation versus the compressive stress of dry joint.
wall yields the average deformation [3]. The normal tangent stiffness of the joint, kn, developed by differentiating of Eq. (15) as follows: k n ¼ Ad Bn þ c
- σn f 'c
ð16Þ
where A = ab and B = b 1 (Table 1). At zero deformation (at the origin) kn = kni = c, where kni is the initial normal stiffness and Eq. (16) can be written as k n ¼ k ni þ Ad Bn
ð17Þ
The relative displacement between the corresponding two points lying on the opposite faces of the continuum elements (Fig. 1) can be expressed as [8] fdg ¼ ½Bj fdg ð18Þ where {d}
[B]j {d}
relative displacement vector at a point (tangential (ds) and normal (dn) displacement, i.e. shear slip and close up displacement) strain–displacement matrix of the zero thickness interface element nodal displacement of the top and bottom nodes of the interface
The stress–deformation constitutive relation for the dry joint is expressed as in the standard form: frg ¼ ½Dj fdg or
ss rn
where kn ks
¼
ks 0
0 kn
ds dn
Opening
dno
-dn Closure
Fig. 6. Closure and opening criteria of dry joint.
ds, dn
relative shear and normal movement of the contacted interfaces
The dry joint is assumed to lose all its stiffness when rn reaches the masonry block compressive strength ðfc0 Þ. Also, when the normal stress, rn, is tension, joint opening occurs and the joint will lose all its stiffness at that point. In both cases, the normal and shear stiffness, kn and ks, are assumed to instantaneously drop to almost zero value. The previous provisions can be shown in Fig. 6. 4. Finite element analysis
ð19Þ
normal tangent stiffness of the joint determined from Eq. (17) shear stiffness (estimated from shear load–slip curves of the bed joints from the shear test [20], 103 N/mm3)
Incremental-iterative finite element program has been developed to implement the proposed mortarless masonry model. Nonlinear analysis is started from the beginning of the load application because of the nonlinearity characteristic of the joint contact behaviour. Fig. 7 shows the adopted calculation to predict the complete nonlinear response of the masonry system for ith iteration for any incremental load. In order to validate the FE program,
W.A.M. Thanoon et al. / Computers and Structures 86 (2008) 520–528
525
Fig. 7. Nonlinear solution procedure.
compression tests results of interlocking mortarless hollow prisms [3] are compared with the developed program results. Fig. 8a shows a half of the three courses height prism that used in the analysis. 2D finite element discretization of the prism in x–y plane is adopted as shown in Fig. 8b. The web and face-shell elements are modelled with different thicknesses. There is no element in the webs between joint elements because the webs of the hollow units are not aligned vertically in successive courses as shown in Fig. 8a. Properties of joints and blocks that used in the analysis are shown in Tables 1 and 2, respectively. All the symbols shown in the tables are identified above. The compressive load was applied incrementally and the boundary
condition at the top and bottom of the prism were simulated as reported in the experimental test [3]. Fig. 9a shows the deformed mesh of the prism and Fig. 9b shows the average axial deformation of the gauge length between the Demec Points located on both prism sides. The results obtained from the program and the test results [3] are shown together in this figure for comparison purposes. A good agreement between the program results and the experimental work is distinguished. It can be shown that the dry joints affect predominantly the hollow prism deformation especially at the lower and middle load levels (45% of the ultimate load). This behaviour is due to the nonlinear gradual closure of the contacted interfaces (or seating) of the dry bed joints. Beyond this load, the
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Fig. 8. Hollow prism and FE mesh.
Table 1 Dry joint element properties (JE) [8] Joint
kni, N/mm3
ks, N/mm3
Coeff. A
Coeff. B
1 and 4 2 and 3
5.0 5.0
103 103
14004.20 984.90
3.28 6.46
Table 2 Masonry block properties [8] Type of material
E0, N/mm2
fc0 , N/mm2
e0
Material parameter, p
ft0 , N/mm2
m
Concrete block
10,932
22.0
0.0023
8
1.98
0.2
joints will be mostly in full contact and the prism deformation affected by the masonry block shortening only till the
a
b
sudden failure. The compressive strength provided by the FE model is 13.2 N/mm2 (11.2 N/mm2 from the test) which is 15.0% higher than the average strength of the tested prisms. Maximum and minimum principal stresses (r1, r2) obtained by the FE analysis at a compressive load of 48 kN (average compressive stress is 2.0 N/mm2) are shown in Fig. 10a and b, respectively. The little unsymmetrical distribution of the stresses in the prism is due to the block bed imperfection. As predicted by the program, high tensile stress is induced in the webs as shown in Fig. 10a and compressive stress is higher at the face-shell as shown in Fig. 10b. Cracks initiate when the principal tensile stress reaches the equivalent tensile strength (Eqs. (7a) and (9)). Fig. 11a shows the predicted cracks pattern near the failure
350
Compressive load kN
300 Demec Points
250 200 150
FE model 100
- - - - Test Results[3]
50 0 0
Deformed mesh
0.4
0.8
1.2
Axial deformation mm Axial deformation versus compressive load Fig. 9. Mode of deformation of hollow prism.
1.6
W.A.M. Thanoon et al. / Computers and Structures 86 (2008) 520–528
Face-shell
Web
Face-shell
Face-shell
a
Web
527
Face-shell
b 550
550
S1
N/mm2
500
S2 N/mm2
500 1.5
450 1.4 1.2
400
0.2
450
0 400
-0.4
1 350 0.8 0.6
300
-0.8
350
-1.2 300 -1.6
0.4 250 0.2 0
200
250
-2 -2.4
200
-2.8
-0.2
150
-0.4 -0.6
100 50
150
-3.2 -3.6
100 50
50
100
Maximum principal stress, σ1
50
100
Minimum principal stress, σ2
Fig. 10. Principal stresses distribution of hollow prism.
and Fig. 11b shows the observed cracks in the experimental test [3].. As can be seen, a relatively good agreement between the FE modelling results and the experimental test. The pattern of cracking here is also affected by the dry joints imperfection in which different joint properties yields unsymmetrical mode of failure. 5. Conclusion
Fig. 11. Cracks pattern in hollow prism.
1. A detailed micromodel for the mortarless block masonry system has been presented. The constitutive relationships proposed in the model concerning the masonry materials and mortarless joint. 2. The presented stress–strain relationship of masonry block based on fitting experimental data is capable of simulating the material nonlinear stress–strain behaviour effectively. Furthermore, it can be incorporated successfully into the model to include the biaxial state of stress and the material nonlinearity in the orthogonal directions using the equivalent uniaxial strain concept. The effect of microcracking confinement and softening on the masonry behaviour under biaxial stresses can be included efficiently also by the presented stress–strain relation. 3. The contact behaviour of mortarless joint taking into consideration the geometric imperfection of the block bed interfaces is simulated depending on the results
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obtained experimentally. The developed contact relations for dry joint within specified bounds describe accurately the variation of the contact properties and can be used for any mortarless masonry system efficiently in the nonlinear solution procedure of the system. 4. The FE model is capable of predicting accurately the deformation hollow prism under compression. The dry joint influences the prism deformation predominantly from the initial loading stage up to 45% of the ultimate loads reflecting the seating effect of dry joint due to the block bed imperfection. In this range of loading, the stiffness of ungrouted prism increases due to increasing of the contacted particles in the dry joint interface. 5. The developed FE model can predict the cracking pattern of hollow prism. The compressive strength predicted by the FE model is 13.2 which is 15% higher than the experimental value.
References [1] Harris H, Oh K, Hamid A. Development of new interlocking and mortarless block masonry units for efficient building systems. In: Proceeding of the sixth Canadian masonry symposium, Saskatoon, Canada, June 15–17, 1992. [2] Thanoon WA, Jaafar MS, Abdul Kadir MR, Ali AA, Trikha DN, Najm AM. Development of an innovative interlocking load bearing hollow block system in Malaysia. Construct Build Mater 2004;18:445–54. [3] Jaafar MS, Alwathaf AH, Thanoon WAM, Noorzaei J, Abdulkadir MR. Characteristic of interlocking mortarless block masonry under compression. In: Proceedings of ICE, construction materials, UK, vol. 159 (3), 2006. p. 111–7. [4] Oh K. Development and investigation of failure mechanism of interlocking mortarless block masonry system, Ph.D. Thesis, Drexel University, Philadelphia, 1994.
[5] Alpa G, Gambarotta L, Monetto I. Dry block assembly continuum modelling for the in-plane analysis of shear walls. In: Proceeding of the fourth international symposium on computer methods in structural Masonry, E&FN Spon, Italy, 1998. p. 111–8. [6] Guo P. Investigation and modelling of the mechanical properties of masonry, Ph.D. Thesis, McMaster University, 1991. [7] Suwalski P, Drysdale R. Influence of slenderness on the capacity of concrete block walls. In: Proceeding of fourth Canadian masonry symposium, fredericton, New Brunswick, 1986. p. 122–35. [8] Alwathaf AH. Development of finite element code for non-linear analysis of interlocking mortarless masonry system, Ph.D. Thesis, Civil Engineering Department, Universiti of Putra Malaysia, Malaysia, 2006. [9] Carreira DJ, Chu KC. Stress–strain relationship for plain concrete in compression. ACI J 1985;82:797–804. [10] Saenz LP. Discussion of ‘‘Equation for the stress–strain curve of concrete’’, by Desayi and Krishnan. ACI J 1964;61:1229–35. [11] Chen WF. Plasticity in reinforced concrete. USA: McGraw-Hill Book Company; 1982. [12] Cerioni R, Doinda G. A finite element model for the nonlinear analysis of reinforced and prestressed masonry wall. Comput Struct 1994;53:1291–306. [13] Darwin D, Pecknold D. Nonlinear biaxial stress–strain law for concrete. J Eng Mech Div, ASCE 1977;103:229–41. [14] Foster S, Gilbert R. Rotating crack finite element model for reinforced concrete structures. Comput Struct 1996;58:43–50. [15] Ayoub A, Filippou F. Nonlinear finite element analysis of RC shear panels and walls. J Struct Eng, ASCE 1998;124:298–308. [16] Vecchio F. Finite element modelling of concrete expansion and confinement. J Struct Eng, ASCE 1992;118:2390–405. [17] Kupfer H, Gerstle K. Behaviour of concrete under biaxial stresses. J Eng Mech Div, ASCE 1973;99:853–67. [18] Vecchio F, Collins M. The modified compression-field theory for reinforced concrete element subjected to shear. ACI J 1986;83: 219–31. [19] Vecchio F, Collins M. Compressive response of cracked reinforced concrete. J Struct Eng, ASCE 1993;119:3590–610. [20] Alwathaf AH, Thanoon WAM, Noorzaei J, Jaafar MS, Abdulkadir MR. Shear characteristic of interlocking mortarless block masonry joints masonry international. J Br Masonry Soc 2005;18(3):139–46.
Computers and Structures 86 (2008) 520–528 www.elsevier.com/locate/compstruc
Finite element analysis of interlocking mortarless hollow block masonry prism Waleed A.M. Thanoon a,*, Ahmed H. Alwathaf b, Jamaloddin Noorzaei c, Mohd. Saleh Jaafar c, Mohd. Razali Abdulkadir c a
Civil Engineering Department, Faculty of Engineering, University Technology Petronas, 31750 Tronoh, Perak, Malaysia b Civil Engineering Department, Faculty of Engineering, Sana’a University, Sana’a, Yemen c Civil Engineering Department, Faculty of Engineering, University Putra Malaysia, 43400 UPM-Serdang, Malaysia Received 5 October 2005; accepted 30 April 2007 Available online 22 June 2007
Abstract Interlocking mortarless masonry system has been developed as an alternative system for the conventional bonded masonry. This paper covers the analysis of interlocking mortarless hollow concrete block system subjected to axial compression loads using FEM. An incremental-iterative finite element code is written to analyze the masonry system till failure. The stress–strain relation obtained from test is employed and equivalent uniaxial strain concept is used to account for the material nonlinearity in the compression stress field. The developed program is also capable of simulating the nonlinear progressive contact behaviour (seating effect) of dry joint taking into account the block bed imperfection. The comparison shows a good agreement between the developed FE program and the experimental test results. Ó 2007 Civil-Comp Ltd and Elsevier Ltd. All rights reserved. Keywords: Finite element method; Nonlinear analysis; Constitutive relations; Masonry; Interlocking block; Dry-stacked masonry; Seating effect
1. Introduction Interlocking mortarless load bearing hollow block system is different from conventional mortared masonry systems in which the mortar layers are eliminated and instead the block units are interconnected through interlocking protrusions and grooves [1,2]. The behaviour of this system is affected highly by the behaviour of the mortarless (dry) joint in both elastic and inelastic stages of loading. The geometric imperfection of the bed blocks is an important factor influences the structural behaviour of the mortarless masonry system [3,4]. Hence, modelling of mortarless joint plays a significant role in the overall simulation of the system. Also, an accurate estimation for the
*
Corresponding author. Tel.: +60 3 89466370; fax: +60 3 86567129. E-mail address: [email protected] (W.A.M. Thanoon).
masonry strength and failure mechanism requires an adequate simulation for the material stress–strain behaviour. FEM has been extensively used in analyzing of masonry structures and numerous models have been developed to simulate the behaviour of different types of the conventional mortared masonry systems using FE technique. However, analytical studies on the mortarless block systems are limited and depend mainly on the type of block used to assemble the walls. A finite element model was proposed by Oh [4] to simulate the behaviour of interlocking mortarless block developed in Drexel University. The procedure implemented to simulate the contact behaviour of mortarless joint including geometric imperfection of the dry joint is suitable only for modelling of small masonry assemblages. Material nonlinearity is not considered to account for the behaviour of the masonry near the ultimate load and to predict failure mechanism. Alpa et al. [5] suggested a macro-model based on homogenization techniques to model the joint and the block as a homogenous material.
0045-7949/$ - see front matter Ó 2007 Civil-Comp Ltd and Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruc.2007.05.022
W.A.M. Thanoon et al. / Computers and Structures 86 (2008) 520–528
That model focuses on the joints movement mechanism assuming perfect joint. Using the model in the structural application without modification is difficult because it ignores significant issues such as, progressive material failure, material nonlinearity and joint imperfection. In this study, a finite element model is proposed and an incremental-iterative program is developed to predict the behaviour and failure mechanism of the system under compression. The nonlinear progressive contact behaviour of dry joint that takes into account the geometric imperfection of the block bed interfaces is included based on experiment testing. The developed contact relations for dry joint within specified bounds can be used for any mortarless masonry system efficiently with less computational effort. Furthermore, the best fit equation for the experimental test data is used in the program to describe the stress–strain behaviour of the masonry block unit under compression for the uniaxial and biaxial stress state. Material nonlinearity in the compressive stress field is considered for the masonry block in the orthogonal directions and the effect of microcracking confinement and softening on the stress–strain relationship under biaxial stresses are included employing the equivalent uniaxial strain concept. The program allows for the progressive local failure of the masonry block (crushing and cracking). After cracking, a smeared crack model is adopted and the compressive strength reduction in the cracked block is considered. Previous studies on hollow block masonry under compression showed that the differences in ultimate strength using 2D (plane stress and plane strain) and 3D FE analysis were not more than 5% [6,7]. The accuracy can be increased if the FE mesh is used in the critical direction where the failure is initiated. Also using 3D discretization requires high computational effort. Therefore, plane stress 2D continuum is adopted in this study. Eight-nodded isoparametric element is used to model block unit and three-nodded isoparametric interface element of zero thickness located between two material elements is employed to
521
model the interface characteristics of the dry joint, as shown in Fig. 1. For numerical integration and stress calculation, nine and three Gaussian points are used in the plane stress element and interface element, respectively. 2. Masonry material modelling 2.1. Stress–strain relation In this study, the best fit equation of the experimental data of masonry block [8] under uniaxial compression test for both ascending and descending parts is adopted. It can be expressed as [9] r¼
pðe=e0 Þr0 p p 1 þ ðe=e0 Þ
where r, e
ð1Þ
instantaneous values of the stress and the strain, respectively the ultimate stress (peak) and the corresponding strain, respectively a constant called material parameter depends on the shape of the stress–strain diagrams
r0, e0 p
An equation was suggested in Ref. [9] to calculate the material parameter (p), which depends on the initial tangent modulus of the stress–strain curve. To avoid the inaccuracy in evaluating the initial tangent modulus, nonlinear regression analysis has been used in this study to determine the material parameter (p), which yields more accurate value for the parameter [8]. Eq. (1) is capable of simulating the stress–strain relation for different masonry materials and can be incorporated efficiently in the biaxial stress model. Fig. 2 shows the experimental test data of the block and the best fit curve drawn by Eq. (1). A comparison with the formula suggested by Saenz [10], which is frequently used for simulation of compressive stress–strain curves of concrete and masonry under biaxial stress state [11,12], is also shown in Fig. 2.
1
2
1
Compressive stress N/mm 2
25
20
15
10 Saenz's Eq [10] 5
Eq.1 Test results [8]
1 Masonry element 2 Joint element (zero thickness) Fig. 1. Masonry and joint element.
0 0
0.0005
0.001
0.0015
0.002
0.0025
0.003
0.0035
Strain mm/mm Fig. 2. Comparison of test data and the best fit relation.
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To include the biaxial stress effect on the uniaxial stress– strain relation given by Eq. (1), the following procedure has been suggested Rewriting Eq. (1) in terms of equivalent uniaxial strain, we obtain (for i = 1, 2): ri ¼
pðeiu =eip Þrip p p 1 þ ðeiu =eip Þ
where rip eip eiu
ð2Þ
maximum (peak) compressive principal stress in direction i equivalent uniaxial strain corresponding to maximum (peak) compressive principle stress rip the equivalent uniaxial strain
The equivalent uniaxial strain eiu essentially removes Poisson’s effect; whereas the strengthening due to the microcracking confinement in biaxial compression stress and softening in compression–tension stress fields are incorporated in rip and eip, respectively [13–15]. Thus a single relation (Eq. (2)) can represent the infinity variety of monotonic biaxial loading curves (Fig. 3). The tangent moduli E1t and E2t for a given principal stress directions are found as the slopes of the r1 versus e1u and the r2 versus e2u curves for the current e1u and e2u as follows: p
Eit ¼
p
pEs ½p 1 þ ðeiu =eip Þ pðeiu =eip Þ
ð3Þ
½p 1 þ ðeiu =eip Þp 2
where Es is the secant modulus at the peak (maximum stress) rip/eip. Fig. 3 depicts the equivalent uniaxial stress–strain curves for masonry element loaded with different biaxial stress ratio, a (a = r1/r2) in the compression and tension fields. In the tension field, the relation is linear and the slope is equal to the initial tangent modulus (E0) at the origin. The strength is reduced when a < 0.0 whereas for a > 0.0,
the strength is enhanced due to microcracking confinement. The maximum stress (peak), rip, and the corresponding strain eip will be found from the biaxial failure criteria illustrated in the next section. 2.2. Failure criteria and constitutive laws The proposed model uses the biaxial compression strength envelope proposed by Veccho [16]. Fig. 4 describes the failure under biaxial compression. The principal stresses in two orthogonal directions are denoted by r1 and r2 with jr1j 6 jr2j (using negative sign (ve) for compressive stress and strain). The following equations define the failure surface: 2
ð4aÞ
0:76ðr1 =fc0 Þ2
ð4bÞ
K c1 ¼ 1 þ 0:92ðr2 =fc0 Þ 0:76ðr2 =fc0 Þ K c2 ¼ 1 þ
0:92ðr1 =fc0 Þ
r1p ¼ K c1 fc0
ð5aÞ
K c2 fc0
ð5bÞ
r2p ¼
fc0
where is the uniaxial compressive strength of the block unit which will be taken here equal to r0. The strains corresponding to the ultimate strengths in the orthogonal directions are e1p ¼ K c1 e0 e2p ¼ K c2 e0
ð6aÞ ð6bÞ
where e0 is uniaxial strain corresponding to the ultimate strength, fc0 . It follows that readily the above relation result in a symmetric state of stress and strain under equal biaxial compression. For tension–compression region; Cerioni and Doinda [12] proposed a smooth curve for masonry material instead of the straight line developed by Kupfer and Gerstle [17]. The envelope relation that is used in this region can be written as
-σi
σip σo=
fc ' α > 0.0 α = 0.0 α < 0.0 εcr
εo
εip
feq Fig. 3. Stress–strain curves of masonry block for different biaxial stress ratio.
- εiu
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based on Eq. (3) and in the tensile field from linear relation (Fig. 3). For undamaged block, the envelope curve in Fig. 4 based on the formulations presented in Eqs. (4)–(10) for different stress states is adopted. After cracking, the tangential elasticity modulus and the Poisson’s ratio are reduced to zero in the direction perpendicular to the crack direction. Instead of zeros, very small values are substituted in the program to avoid singularity in stiffens matrices. The masonry along cracks is still resisting compressive stress after cracking. It was found [18,19] that the compressive strength of concrete after cracking can be significantly reduced by the tensile strain in the transverse direction. To account for this effect, the following formula is adopted to obtain r2p after cracking:
f't /f'c σ 1/ f ' c
1.0
1.0 Proposed envelope Kupfer envelope [17]
σ2 /f'c
r2p 1 6 1:0 ¼ 0:8 þ 0:34ðe1 =e0 Þ fc0
Fig. 4. Masonry envelope for different stress states.
r1p ¼ feq ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 ðr2 =fc0 Þ ft0 0:65fc0
r2p ¼ r1p =a 6
ð7aÞ ð7bÞ
3. Modelling of dry joint
ð8aÞ 3
2
e2p ¼ e0 ½1:6q þ 2:25q þ 0:35q
ð8bÞ
where ft0 feq E0
tensile strength of block unit equivalent tensile strength the initial tangent modulus of elasticity (at the origin) q ¼ r2p =fc0 For tension–tension region:
r1p ¼ ft0 P r2p e1p ¼
ft0 =Ec
ð9Þ
P e2p ¼ r2p =E0
ð10Þ
The failures envelop is shown in Fig. 4 for all stress states. An incrementally relationship is assumed between strains and stresses, which in differential form and in the principal directions can be written for undamaged masonry: fdrg ¼ ½Dc fdeg 8 9 2 pffiffiffiffiffiffiffiffiffiffi E1 m E1 E2 > < dr1 > = 1 6 pffiffiffiffiffiffiffiffiffiffi dr2 ¼ E2 4 m E1 E2 2 > > : ; 1m ds12 0 0 8 9 > < de1 > = de2 > > : ; dc12
where pffiffiffiffiffiffiffiffi v ¼ v1 v2 and
v1 ¼ v2 ¼ 0:2
ð14Þ
where e0 is the compressive strain relative to the uniaxial compressive strength, fc0 , and e1 is the tensile strain normal to the crack direction.
in which r1p tensile stress and r2p compressive stress and the maximum strain can be evaluated as e1p ¼ r1p =E0
523
ð11Þ 3
0 7 0 5 pffiffiffiffiffiffiffiffiffiffiffiffi 0:25ðE1 þ E2 2m E1 E2 Þ ð12Þ
ð13Þ
The tangent moduli of elasticity, E1 and E2, along the principle stress directions are evaluated in the compressive field from a nonlinear equivalent uniaxial stress–strain relation
The characteristics of dry joint under compression and shear loading has been investigated experimentally [3,20] and load–deformations relations that are required for the FE modelling is formulated based on the experimental results. Contact test of block beds was carried out to investigate the behaviour of mortarless joint under compression. The test was carried out on small wall panels at different joint locations to take into consideration the different sources of block bed imperfection [3]. Modified triplet tests on mortarless interlocking panels with different levels of axial compression were carried out to ascertain the deformation and shear strength characteristics of the system at the joint interface. Failure criterion and stiffness of the interlocked bed joints under combined normal–shear load was proposed [20]. The nonlinear behaviour of the mortarless bed joint under compressive load is simulated taking into consideration the block bed imperfection. The proposed mathematical model that can describe the nonlinear compressive stress in terms of the joint closure is rn ¼ ad bn þ cd n where rn dn a, b, c
ð15Þ
compressive stress (N/mm2) close-up deformation (mm) constants determined from data analysis of the test results
Eq. (15) yields an average compressive stress on the contacted area for family curves that consider the variety of geometric imperfection of block beds. Fig. 5 shows the upper and lower bounds of Eq. (15). Although high variation in close-up deformations was measured at different dry joints, the average deformation of all specified joints in a
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Compressive stress N/mm2
10
8 Lower bound model Average model Upper bound model Lower bound -Test data [3] Average - Test data [3] Upper bound - Test data [3]
6
4
Lower Average Upper a 3272.02 93.75 132.02 b 4.28 3.01 7.46 c 5.00 5.00 5.00
2
0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Close-up deformation mm Fig. 5. Close-up deformation versus the compressive stress of dry joint.
wall yields the average deformation [3]. The normal tangent stiffness of the joint, kn, developed by differentiating of Eq. (15) as follows: k n ¼ Ad Bn þ c
- σn f 'c
ð16Þ
where A = ab and B = b 1 (Table 1). At zero deformation (at the origin) kn = kni = c, where kni is the initial normal stiffness and Eq. (16) can be written as k n ¼ k ni þ Ad Bn
ð17Þ
The relative displacement between the corresponding two points lying on the opposite faces of the continuum elements (Fig. 1) can be expressed as [8] fdg ¼ ½Bj fdg ð18Þ where {d}
[B]j {d}
relative displacement vector at a point (tangential (ds) and normal (dn) displacement, i.e. shear slip and close up displacement) strain–displacement matrix of the zero thickness interface element nodal displacement of the top and bottom nodes of the interface
The stress–deformation constitutive relation for the dry joint is expressed as in the standard form: frg ¼ ½Dj fdg or
ss rn
where kn ks
¼
ks 0
0 kn
ds dn
Opening
dno
-dn Closure
Fig. 6. Closure and opening criteria of dry joint.
ds, dn
relative shear and normal movement of the contacted interfaces
The dry joint is assumed to lose all its stiffness when rn reaches the masonry block compressive strength ðfc0 Þ. Also, when the normal stress, rn, is tension, joint opening occurs and the joint will lose all its stiffness at that point. In both cases, the normal and shear stiffness, kn and ks, are assumed to instantaneously drop to almost zero value. The previous provisions can be shown in Fig. 6. 4. Finite element analysis
ð19Þ
normal tangent stiffness of the joint determined from Eq. (17) shear stiffness (estimated from shear load–slip curves of the bed joints from the shear test [20], 103 N/mm3)
Incremental-iterative finite element program has been developed to implement the proposed mortarless masonry model. Nonlinear analysis is started from the beginning of the load application because of the nonlinearity characteristic of the joint contact behaviour. Fig. 7 shows the adopted calculation to predict the complete nonlinear response of the masonry system for ith iteration for any incremental load. In order to validate the FE program,
W.A.M. Thanoon et al. / Computers and Structures 86 (2008) 520–528
525
Fig. 7. Nonlinear solution procedure.
compression tests results of interlocking mortarless hollow prisms [3] are compared with the developed program results. Fig. 8a shows a half of the three courses height prism that used in the analysis. 2D finite element discretization of the prism in x–y plane is adopted as shown in Fig. 8b. The web and face-shell elements are modelled with different thicknesses. There is no element in the webs between joint elements because the webs of the hollow units are not aligned vertically in successive courses as shown in Fig. 8a. Properties of joints and blocks that used in the analysis are shown in Tables 1 and 2, respectively. All the symbols shown in the tables are identified above. The compressive load was applied incrementally and the boundary
condition at the top and bottom of the prism were simulated as reported in the experimental test [3]. Fig. 9a shows the deformed mesh of the prism and Fig. 9b shows the average axial deformation of the gauge length between the Demec Points located on both prism sides. The results obtained from the program and the test results [3] are shown together in this figure for comparison purposes. A good agreement between the program results and the experimental work is distinguished. It can be shown that the dry joints affect predominantly the hollow prism deformation especially at the lower and middle load levels (45% of the ultimate load). This behaviour is due to the nonlinear gradual closure of the contacted interfaces (or seating) of the dry bed joints. Beyond this load, the
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Fig. 8. Hollow prism and FE mesh.
Table 1 Dry joint element properties (JE) [8] Joint
kni, N/mm3
ks, N/mm3
Coeff. A
Coeff. B
1 and 4 2 and 3
5.0 5.0
103 103
14004.20 984.90
3.28 6.46
Table 2 Masonry block properties [8] Type of material
E0, N/mm2
fc0 , N/mm2
e0
Material parameter, p
ft0 , N/mm2
m
Concrete block
10,932
22.0
0.0023
8
1.98
0.2
joints will be mostly in full contact and the prism deformation affected by the masonry block shortening only till the
a
b
sudden failure. The compressive strength provided by the FE model is 13.2 N/mm2 (11.2 N/mm2 from the test) which is 15.0% higher than the average strength of the tested prisms. Maximum and minimum principal stresses (r1, r2) obtained by the FE analysis at a compressive load of 48 kN (average compressive stress is 2.0 N/mm2) are shown in Fig. 10a and b, respectively. The little unsymmetrical distribution of the stresses in the prism is due to the block bed imperfection. As predicted by the program, high tensile stress is induced in the webs as shown in Fig. 10a and compressive stress is higher at the face-shell as shown in Fig. 10b. Cracks initiate when the principal tensile stress reaches the equivalent tensile strength (Eqs. (7a) and (9)). Fig. 11a shows the predicted cracks pattern near the failure
350
Compressive load kN
300 Demec Points
250 200 150
FE model 100
- - - - Test Results[3]
50 0 0
Deformed mesh
0.4
0.8
1.2
Axial deformation mm Axial deformation versus compressive load Fig. 9. Mode of deformation of hollow prism.
1.6
W.A.M. Thanoon et al. / Computers and Structures 86 (2008) 520–528
Face-shell
Web
Face-shell
Face-shell
a
Web
527
Face-shell
b 550
550
S1
N/mm2
500
S2 N/mm2
500 1.5
450 1.4 1.2
400
0.2
450
0 400
-0.4
1 350 0.8 0.6
300
-0.8
350
-1.2 300 -1.6
0.4 250 0.2 0
200
250
-2 -2.4
200
-2.8
-0.2
150
-0.4 -0.6
100 50
150
-3.2 -3.6
100 50
50
100
Maximum principal stress, σ1
50
100
Minimum principal stress, σ2
Fig. 10. Principal stresses distribution of hollow prism.
and Fig. 11b shows the observed cracks in the experimental test [3].. As can be seen, a relatively good agreement between the FE modelling results and the experimental test. The pattern of cracking here is also affected by the dry joints imperfection in which different joint properties yields unsymmetrical mode of failure. 5. Conclusion
Fig. 11. Cracks pattern in hollow prism.
1. A detailed micromodel for the mortarless block masonry system has been presented. The constitutive relationships proposed in the model concerning the masonry materials and mortarless joint. 2. The presented stress–strain relationship of masonry block based on fitting experimental data is capable of simulating the material nonlinear stress–strain behaviour effectively. Furthermore, it can be incorporated successfully into the model to include the biaxial state of stress and the material nonlinearity in the orthogonal directions using the equivalent uniaxial strain concept. The effect of microcracking confinement and softening on the masonry behaviour under biaxial stresses can be included efficiently also by the presented stress–strain relation. 3. The contact behaviour of mortarless joint taking into consideration the geometric imperfection of the block bed interfaces is simulated depending on the results
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obtained experimentally. The developed contact relations for dry joint within specified bounds describe accurately the variation of the contact properties and can be used for any mortarless masonry system efficiently in the nonlinear solution procedure of the system. 4. The FE model is capable of predicting accurately the deformation hollow prism under compression. The dry joint influences the prism deformation predominantly from the initial loading stage up to 45% of the ultimate loads reflecting the seating effect of dry joint due to the block bed imperfection. In this range of loading, the stiffness of ungrouted prism increases due to increasing of the contacted particles in the dry joint interface. 5. The developed FE model can predict the cracking pattern of hollow prism. The compressive strength predicted by the FE model is 13.2 which is 15% higher than the experimental value.
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