Iranica Journal of Energy & Environment 4 {(3) Geo-hazards and Civil Engineering)}: 243-250, 2013ISSN 2079-2115 IJEE an Official Peer Reviewed Journal of Babol Noshirvani University of TechnologyDOI: 10.5829/idosi.ijee.2013.04.03.10

BUT

Corresponding Author: Sharif Shahbeyk, Faculty of Civil and Environmental Engineering, Tarbiat Modares University, Tehran, Iran Fax: +982182884914, E-mail: shahbeyk@modares.ac.ir

243

Shape Optimization of Hollow Concrete Blocks Using the LatticeDiscrete Particle Model

Fatemeh Javidan, Mohammad Safarnejad and Sharif Shahbeyk

Faculty of Civil and Environmental Engineering, Tarbiat Modares University, Tehran, Iran

(Received: January 2, 2013; Accepted in Revised Form: April 30, 2013)Abstract: Hollow concrete blocks are one of the widely used building elements of masonry structures in whichthey are normally loaded under combined action of shear and compression. Accordingly and due to theirstructural importance, the present study intends to numerically search for an optimum shape of such blocks.The optimality index is selected to be the ratio of block’s failure strength to its weight, a non-dimensionalparameter, which needs to be maximized. The nonlinear analysis has been done using a homemade code writtenbased on the recently developed Lattice Discrete Particle Model (LDPM) for the meso-scale simulation ofconcrete. This numerical approach accounts for the different aspects of concrete’s complex behavior such astensile fracturing, cohesive and frictional shearing and also its nonlinear compressive response. The modelparameters were calibrated against previously reported experimental data. Various two-core configurations forthe hollow blocks are examined, compared and discussed.

Key word: Hollow concrete block Strength Failure mechanism Lattice particle discrete method

INTRODUCTION three dimensional finite element (FE) method [10,11].

Due to their light weight, ease of use and reasonable [12-14] leaded to acceptable estimation of failurecompressive performance, hollow concrete blocks are mechanisms of hollow masonry. Lu et al. [15] have usedwidely used in masonry structures. Hence, its geometrical a nonlinear FE model to analyze slender unreinforceddesign and mechanical properties are of significant masonry hollow walls under eccentric vertical loadsimportance. Accordingly, many experimental and focusing on the cavity depth of the blocks. With thenumerical researches have been conducted in order to purpose of having shape simplicity and ease ofobtain logical relations between the structural manufacture, interlocking mortarless hollow blockcharacteristics of the blocks and their assemblages [1-3]. masonry systems have been developed [16,17].More complicated works examined the effect of mortar Compressive strength correlation between the individualproperties and their interface with the blocks [4,5]. block, prism and basic wall panel for this new type of loadThe responses of hollow concrete blocks under various bearing interlocking blocks has been first obtained bothloading conditions including concentric and eccentric experimentally and numerically [18,19]. A few studies havecompression [6,7], flexure [8] and shear [9] have been also been also carried out on these structural unitsempirically assessed. emphasizing on various loading conditions and effective

Alongside with experimental researches, several geometrical features. Del Coz Díaz et al. [20] haveefforts have been made so far in order to model numerically developed a new type of hollow block andnumerically the complex behavior of blocks, mortar and optimized its compressive behavior, weight and handlinginterface joints. One of the main targets of these studies characteristics.is to link the global behavior of walls and panels to the In this research, the lattice discrete particle modelproperties of individual units and mortar. The most (LDPM) as a powerful numerical tool for the simulation ofwidely-used numerical approaches is found to be the concrete has been employed. This approach let account

Utilizing damage models for the constituent material,

( ) ( )MQ+CQ+P Q F t=

( ) ( ) ( )I Iu x u x-x xI I IA Q = + × =

( )1 1

1 1

1 1

1 0 0 0x 0 1 0 0

0 0 1 0I

z z y yA z z x x

y y x x

− − = − − − −

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244

for the different aspects of concrete nonlinear behaviorsuch as tensile fracturing, cohesive and frictionalshearing and also its nonlinear compressive response.The model is calibrated and validated against availabletest data. At first, a typical two-hole concrete block wasanalyzed. Combined compressive-shear loads of variousratios are applied. An optimality index is defined as theratio of resisted load to block weight. With the aim ofincreasing the optimality index, three more samples werecreated by adding horizontal diaphragm or rearranging theholes. Finally, the blocks were compared from bothoptimality index and failure mechanism points of view. (a)

Lattice Discrete Particle ModelConcrete Meso-structure Modelling: The LDPM is arecently developed method for simulating the failurebehavior of discontinuous and heterogeneous materials,such as concrete. The backbone of the current LDPM isthe innovative meso-scale approach known as theConfinement Shear Lattice (CSL) Model introduced inliteratures [21- 23] and further improved [24]. The LatticeDiscrete Particle Model [25,26] combined CSL model withthe properties of Discrete Particle Models (DPM). In theLDPM, aggregate content is identified using the mixdesign and other input properties such as the cement (b)content, aggregate content per unit volume and thegrading curve. Next, the aggregates of various sizes are Fig. 1: (a) Random aggregate distribution and (b)randomly distributed inside the specimen, which helps three dimensional tetrahedron mesh of a typicalavoid a bias fracture propagation pattern in the concrete dogbone specimen (Mencarelli [26]) (Fig. 1a). Next, a three dimensional lattice structuretessellates the volume and connects the centers of the numerical stability condition is reserved by keeping theadjacent particles resulting in a tetrahedron mesh time step below the permitted limit [23]. Solving the(Fig. 1b). The connection between each aggregate is mentioned equation, the displacement field correspondingcalled a strut. The constitutive laws of the model are to the six degrees of freedom for each aggregate centerformulated on a finite number of points, called the node is determined. Assuming a rigid connection betweencomputational points located between two neighboring nodes and computational points, the displacement jumpsaggregates. of the computational points will be obtained as stated in

Solution Procedure: The basic equation of motion in itsmatrix form is (2)

(1)

Here, M is the mass matrix, P is the vector of internal rotation matrices of each computation point (I),forces, C is the damping matrix (which is neglected here), respectively.F is the given external load history and Q is the vector ofkinematic variables including the translations androtations. The equation is solved by the explicit methodusing the central difference approximation in each of theaggregate centers [23]. During the solution process, the

Eq. (2).

In which A is a matrix presented in Eq. (3) and x,u and are the coordinates, displacement and

(3)

[ ][ ]

[ ]( ) ( ) ( ) ( )( ) ( )

TN N2 2 N1 1

TM M2 2 M1 1

TL N2 2 N1 1

T TNi Li

TMi

n u /1 B Q B Q

M u /1 B Q B Q

n u /1 B Q B Q

1/ n A x , B 1/ l A x ,

B 1/ l m A x ,i i

i=

( )2 2 2 2 2N M L N T

( ) ( ) ( ) ( )0

0

= −

( ) ( )( )

( )( ) ( )

( )( )

( )

( ) ( )

01 00

02 0

1 2 2sin cos

2sin

2sin + 22cos 2 2

2 2 2cos /µ sin

c

t

t ct t

t

=

=+

+ + − + − + =

−

( )

00

0

/ 21/ 2

2 02

0 0

c

t

K

K K

+ − ≤ + = < ≤

≤ ≤

245

are, the slope and the intersection of the( ), asymptote with the

computational points. axis. , and are the mesoscale tensi le , c omp re ssiv e

(4)

Where is the normal strain and and are theN L M

tangential strains. Where n is the unit vector directed

of the strut. Combining these normal and tangentialstrains, the following important stain measures can beintroduced.

(5)

Where and are called the effective andcoupling strains, respectively. The term is the totalT

shear strain. where is a material property used tocontrol the elastic Poisson’s ratio. The constitutiverelation of the model is based on the following elasticboundary.

(6)

)defines the hardening-softening rates for the differentloading paths. where ( ) is the initial effective strength0

function and is defined as below:

(7)

1

N t c s

0

1 2

(8)

c c

compressive and low shear-high compressive responset t

and high shear-low compressive behavior at meso-level.

Calibration and Validation

concrete in the LDPM, one is supposed to de termine thebasic parameters of the model in such a way that the f inalresponses of the numerical models match thecorresponding experimental results. It needs to be notedthat for a successful calibration process, one should knowhow the macroscopic responses of the concrete areconnected to the meso-scale parameters of the LDPM. Inthis study, the calibration procedure proposed byMencarelli [25] has been combined with our exper iencewith the LDPM and a simple reliable calibration procedureis extracted. It requires the results of the three tests ofuniaxial compression, uniaxial tension and hydrosta ticcompression. In the following sections, the experimenta lresults of Barbosa et al. [3] are used to calibrate the mode l.Next, the model is validated in the case of a hollowconcrete block made of the very same concrete .

Calibration:In the first calibration step, N

determined by matching the elastic responses of anunconfined uniaxial compression simulation with those ofthe experiment. It is important to note that, as the fr ic t ioncoefficient between the loading plates and the specimensis not reported [3], this parameter is reasonably assumed

= = −

= = −

= = −

= l l=

ϵϵϵB

ϵ ϵ ϵ

== ++ ϵϵϵϵϵ +ϵϵ ϵ

, expb 0 E

K

≤>

σ σσϵWhere µ is the mesoscale elastic modulus. Also, K(

σσ

tan ω =, αϵ√ϵ

ϵ σσ

σ

Iranica J. Energy & Environ., 4 {(3) Geo-hazards and Civil Engineering)}: 243-250, 2013

The form below is used for K( ) :

perpendicular tangential unit normal. Also, l is the lengthalong the strut and l and m are the two mutually

Parameters K and n control the nonl inear

and K and n govern the nonlinear tensile , shear-tensile

Preliminaries: In order to realist ically s imulate a specif ic

and E are

n t

cn

to the intersection of the curves ( ) and ( ).and shear strength, respectively. The term corresponds

jumps can then be converted to the required strains at the hyperbola, respectively. Also,Constitutive Relations: The quantities of displacement Where µ and

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(a) (b)Fig. 2: Estimating parameters from the normalized hydrostatic curve. (a) and (b) Kc c

(a) (b)Fig. 3: Response of the cylindrical specimen under. (a)Indirect tension and (b) uniaxial compression.

equal to 0.1. The only parameter affecting the Poisson’s of the force-displacement curves. In the last step n and ratio of the model is , thus its accurate value can simply are acquired through a trial and error process with the aimbe estimated. Furthermore, the identification of E requires of reaching the correct peak value and post peak slope ofN

that the initial elastic slope of the stress-strain curve the uniaxial compressive curve. coincide with its experimental counterpart. At the end of Fig. 3b shows a comparison between the numericalthis stage we have the correct values of and E . and experimental results of the uniaxial compression test.N

They will be considered fixed parameters during the It should be noted that the tensile and compression testsremaining steps of calibration. were performed on standard 100×200 mm cylindrical

The next stage is to identify and K . It is found by specimens. The numerical models also correctly predictc c

Mencarelli [25] that these parameters decide the location crack patterns of the cylindrical specimens. A diagonalof slope change on the normalized hydrostatic mode of cracking in addition to a slight crushing at thecompression curve and its initial inelastic slope, middle of the height of the compressive concrete cylinderrespectively. In the absence of appropriate experimental is observed which is the same as the cracks formed in thedata, Mencarelli [25] has suggested to use available experimental test (Fig. 3b) [28]. Also, for both tensileinformation from other experimental work, generally samples, a similar vertical crack is recorded which is duedriven out for the hydrostatic compressive behavior of to the direct loading conditions at the top and bottom ofconcrete material [27]. Figs. 2a and 2b illustrate how the horizontal cylindrical specimen (Fig. 3a). At this stage,these two parameters are estimated in a trial and error all the required parameters to model a desired type ofprocedure. concrete are drawn out which will all be used as inputs for

Next, the tensile strength of meso-structure, ( ), the validation and further numerical analysis. Table 1t

should be found in a way that the tensile strength of the presents concrete meso-scale parameters together withmodel matches the experimental value (Fig. 3a). Here, an their definitions and calibrated values. Table 2 comparesindirect tensile test (splitting test) is performed and the the numerically and empirically obtained properties of thetensile strength is obtained by comparing the peak values concrete.

t s

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(a) (b) (c) (d)Fig. 4: Failure mechanism of the hollow block. (a) Indirect tension (LDPM), (b) indirect tension (experiment), (c) uniaxial

compression (LDPM), and (d) uniaxial compression (experiment).

Table 1: LDPM calibrated parametersCalibrated

Parameter Definition ValueParameter controlling the Poisson’s ratio 0.2

E Normal Young’s modulus of the meso-structure 50 (GPa)N

K Initial hardening modulus 18c

Compressive strength of the meso-structure 85 (MPa)Tensile strength of the meso-structure 3.2 (MPa)t

n Parameter describing the behavior of softening slope 1.0t

Shear strength of the meso-structure 4.8 (MPa)s

Table 2: Final Mechanical properties of the concreteMechanical properties Experimental Numerical (LDPM)Compressive Strength (MPa) 20 20Strain at Failure 2707 2700Modulus of Elasticity (MPa) 19407 19500Tensile Strength (MPa) 2.2 2.2

Validation: In the validation phase, the experimentalresults of Barbosa [28] for the compressive and tensileloading of a specific type of hollow concrete block wereused. The ultimate forces resisted in the numerical modelunder compression and indirect tensions were comparedwith the experimental results. The peak value of thenumerical compressive load-displacement curve was equalto 628.41 KN. Dividing this quantity to the cross-sectionalarea of the block (306 cm ), the strength value of 20.5 MPa2

is obtained which is in good agreement with theexperimental value of 20.1 MPa. Furthermore, the failuremodes of tested specimens were also investigated.Figs. 4a and 4b compare the cracking pattern of the blockunder indirect tension and Fig. 4.c and 4.d show that forthe specimen under compression, both in numerical andexperimental tests, respectively. In the indirect tensiontest, due to the local loading, a vertical crack occursthrough the height of the block. However, because of theconfinement produced by the loading plates at the topand bottom of block in compression test, diagonal crackwith crushing proceeded in the sides of the blocks.

Thus the failure mode is a combination of shear andcompression failures. As demostrated, the experimentalobservations are satisfactorily predicted by the LDPMsimulations.

Block Analysis: The detailed numerical analysis ofthe block units under different loading states to whichare prominently subjected, can lead to a reliabledesign. The types of loads mostly affecting masonrywalls include gravity forces due to the weight ofthe walls and other structural elements standing onthem, as well as lateral loads caused by earthquake orwind. Here, to have a realistic observation of blockresponses in masonry buildings, each block is subjectedto simultaneous gravity and shear loads and isanalyzed using the computer code written based on theLattice Discrete Particle Model. In each loadingstep of the LDPM analysis, a vertical displacement inaddition to a horizontal displacement is applied to thenodes of the top surface of each block while thebottom nodes of the block are constrained according tothe test conditions. In this work, four types of blocks wereanalyzed under combined action of compression andshear. All the blocks considered in this research were140×190×390 mm two-core samples. To optimize theblocks behavior, initially a typical form of two-core blockis taken. In the second and third blocks, a 30 mmhorizontal diaphragm has been inserted in the bottomand middle of the blocks, respectively. An extrafourth sample is also studied in which the holes arerearranged. All four types of blocks have equal crosssectional areas but the weight of each block variesaccording to its geometrical configuration. Table 3presents the cross section, longitudinal section andweight of each block.

Table 3: Analysed blocksBlock type 1 Block type 2 Block type 3 Block type 4

Cross Section

Longitudinal Section

Weight (N) 137 154 154 137

peakPOI

Weight=

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RESULTS AND DISCUSSIONS 4 are similar. Still, block type 1, with its traditional form,

The simulations are carried out by applyingdisplacement-controlled compression while assumingconstant ratios (0, 0.5, 1, 2.5, 5 and 10) of lateral to axialdisplacements. Considering a 2 mm compressivedisplacement, shear displacements equal to 0, 1, 2, 5, 10and 20 are applied to the blocks. Fig. 5a shows the peakcompressive load versus the peak shear load resisted byeach block. It is obvious that block type 2 with thehighest curve and block type 1 with the lowest curve havethe best and worst compressive resistance undercombined effect of shear and compression, respectively.It can also be concluded from the curves in Fig. 5a that,among block types 3 and 4, the former behaves relativelybetter for lower shear loads; however, as the share ofshear increases, block type 3 proves its superiority.

The above comparisons are merely based on the purestrength; however, the blocks used in any load bearingstructure, should have the least possible weight due tostructural and handling requirements. Hence, theoptimum option should be selected according tosimultaneous attention to both the strength and weight.A non-dimensional variable called the optimality index(OI), is calculated as introduced below:

(9)

The curves of Fig. 5b depicts the optimality index ofblocks versus the displacement ratio (shear tocompression) applied to each specimen. According to theresults, block type 4 which no added diaphragm gives thebest response in shear to compression ratios of less than4. This means that in such loading conditions, theincreasing effect of a horizontal diaphragm on the weightis more than that on the strength. For the ratio greaterthan 4 the responses of the three blocks of types 2, 3 and

owns the worst effective strength in any shear-compression combinations.

In order to have a more precise comparison betweenthe responses of blocks, their failure modes should alsobe taken into consideration. Observing the exact crackpatterns, their directions and the amount of damagecaused in the webs, shells or face-shell connections ofeach block, we will come to a complete conclusion for thestrong and weak points. For this purpose, the graphicaloutputs that show the failure patterns of the analyzedblocks are generated by a post-processing code writtenfor the Lattice Discrete Particle Model. In this code, eachaggregate with a specified grout volume surrounding theaggregate is considered as a rigid cell. Each rigid cell isconnected to the adjacent rigid cells of the concretematerial through connection points (Fig. 6). The averagestrain in each of these connection points is calculatedaccording to the effective strains of the computationalpoints. Considering the average strains in eachconnection point, the rotations and displacements of eachparticle and the final deformed and cracked shape of theconcrete specimen are visualized. Fig. 7 shows thesegraphical results.

As shown, a diagonal crack is observed on theface-shell of all blocks. Block type 1 has more damagedand crushed areas than others, especially on the topsurface, which confirm its worst load resistance capacity.The presence of a horizontal diaphragm in block types 2and 3 prevents continuous crack propagations.Suitable block types may be produced by changing thethickness or shape of the diaphragm inserted. It isinteresting that, for block type 4, the presence of twothick internal walls, compared to the three thinnerinternal walls of other samples, increases the effect ofconfinement and thus improves its optimality index.Another important advantage of using block type 4, apartfrom its optimality, is that no additional plate is requiredin its construction.

(a) (b)Fig. 5: Block responses. (a) Peak compressive vs. shear loads and (b) specific strength vs. displacement ratio.

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Fig. 6 : Connection points of a concrete aggregate-mortar cell [24]

Block type 1 Block type 2 Block type 3 Block type 4Fig. 7 : Failure modes of the blocks and effective strain contour of their longitudinal sections

CONCLUSION Furthermore, it is found that, keeping weight constant, the

The application of numerical methods with discrete its both sides behaves better than the block having twonature to the meso-scale simulation of concrete as a multi- complete internal holes. This can be attributed to its morephase material can help us deepen our understanding of thick internal walls in which the effect of confinement isthe underlying mechanisms involved in its failure. more pronounced. Inspired by this fact, the newly developed Lattice DiscreteParticle Model is incorporated in this research. A REFERENCEShomemade graphical code is also utilized to manipulatethe outputs of the models. 1. Maurenbrecher, A., 1986. Axial compression tests on

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