(13) Maheri

Engineering Structures 25 (2003) 1639–1654www.elsevier.com/locate/engstruct

Static and seismic design of one-way and two-way jack archmasonry slabs

Mahmoud R. Maheri∗, Hamid RahmaniShiraz University, Department of Civil Engineering, Shiraz, Iran

Received 7 June 2001; received in revised form 26 May 2003; accepted 26 May 2003

Abstract

Steel I-beam, brick jack arch slabs have long been used to floor and roof industrial and residential buildings in many parts ofthe world. Collapse of a large number of these non-homogeneous one-way slabs during past earthquakes has highlighted their poorseismic performance. However, due to their easy construction together with low cost, the jack-arch slab is still widely used in manycountries. In this article, the weaknesses inherent in the traditional one-way jack-arch slabs are explored. To overcome these short-comings, a new two-way system is proposed. Results of static and dynamic tests on full scale two-way and one-way jack-archslabs and finite element numerical analyses, aimed at investigating the effectiveness of the proposed two-way system, are presentedwith favourable conclusions. Following these investigations the static and seismic design of jack arch slabs are discussed. Theproposed, allowable stress design method is based on designing for the steel grid and controlling the stresses in brick arches.Parameters necessary for an equivalent static seismic load calculation are first determined. Finite element numerical analyses arethen conducted to investigate the effects of a number of parameters on the design of the slab and the necessary design factors areevaluated. In addition, appropriate tables and figures are presented to facilitate the design of the one-way and two-way jack archslabs. It is concluded that the jack arch slab system, designed and constructed as presented in this article, provides a viable, lowcost alternative to other forms of flooring in seismic zones and elsewhere. 2003 Elsevier Ltd. All rights reserved.

Keywords: Jack arch slab; Masonry slab; Design of jack arch slab; Seismic response; Dynamic testing

1. Introduction

The traditional steel I-beam, jack arch flooring systemwas developed in Britain towards the end of nineteenthcentury and was used extensively to cover large floorareas in factories and other industrial buildings. Thetechnique spread eastwards and, by the middle of thetwentieth century, it became a popular flooring systemin parts of East Europe, the Middle East and the Indiansubcontinent. Due to its technical simplicity, speed inconstruction and low cost, traditional jack arch slabs arestill very popular in the Middle East, where, not onlyindustrial buildings and ordinary dwellings but alsomany high-rise steel and concrete framed buildings arefloored by this method.

∗ Corresponding author. Tel:+98-711-832-1353; fax:+98-711-2352-725.

E-mail address: [email protected] (M.R. Maheri).

0141-0296/$ - see front matter 2003 Elsevier Ltd. All rights reserved.doi:10.1016/S0141-0296(03)00143-3

The floor slabs constructed using the steel I-beam jackarch system are stable under normal static conditions asthe brick arches transfer the gravity loads, mainly incompression, along the arch to the supporting beams(Fig. 1). The load is then transferred along the parallelsteel beams to the supporting walls or beams. The geo-metric form of the steel I-beam jack arch system and theload path through the steel beams, make the slab act asa one-way system.

Despite the wide spread use of the jack arch slabs andtheir advantages, there are no particular procedures fortheir engineered design and there is no mention of thesystem in codes of practice. Indeed, a search of theliterature reveals no reference to any particular scientificresearch directed at studying this slab system or anyattempts to provide an engineering basis for its designand construction. Design engineers, using the jack archslab in framed buildings, consider the brick arches asmerely dead loads, carried by the steel beams, and aresufficient in designing the steel beams. This assumption

1640 M.R. Maheri, H. Rahmani / Engineering Structures 25 (2003) 1639–1654

Nomenclature

A Design base accelerationa Length of slabB Slab dynamic response coefficientb Width of slabC Earthquake coefficientCs Slab design coefficientE Earthquake loadEeff Effective modulus of elasticity of slabEm Elastic modulus of brickworkfball Allowable tensile stress of masonryfmall Allowable compressive stress of masonryfsall Allowable stress of steelfy Yield stress of steelf’m Compressive strength of masonry prismh Thickness of slabI Importance factorIreq Required moments of inertia for slab beams (Ixx)In Minimum normalized moments of inertia of slab beams (Ixx) given in design tableIxx, Iyy Moments of inertia of slab beams around axis parallel to and perpendicular to the slab surface,

respectivelyRw Slab working performance factorRm Ductility reduction factorS Section modulus of steel beamsSn Minimum normalized section modulus of slab beams given in design tableT Fundamental period of vibrationTo Type of soil parameterV In-plane shear force on the slabW Design loadWe Weight of slab for calculating the earthquake loadWs Weight of steel grid per unit area of slabWst Weight of transverse beams per unit area of slabY Allowable stress factor�max Maximum allowable deflection� Over-strength factor.a Correction factor for support conditionb Correction factor for width of brick archg Slab deflection parameter ( = αβ�λ)gmin Deflection control parameter� Correction factor for elastic modulus of brickworkλ Correction factor for allowable stress of brickworkl1 Dimensionless parameter depending on geometry and boundary conditionsnm Poisson’s ratio of brickworkr Mass per unit area of slabw Correction factor for design load

ignores the role of brick arches in transferring slab loadsand the resulting large stresses developed in them.Despite the lack of a proper design basis and the poorperformance of the jack arch system under earthquakeloading, this type of flooring is still used extensively inmany countries. The reason for this being a number ofpractical and economical advantages including simple

construction technique, speed in construction, low costand the ability to alter the slab after construction, whencompared to conventional reinforced concrete or con-crete beam-block slabs.

The performance of the traditional one-way jack archslab in a number of recent earthquakes in Eastern Europeand the Middle East, particularly in Iran, has generally

1641M.R. Maheri, H. Rahmani / Engineering Structures 25 (2003) 1639–1654

Fig. 1. Construction detail of a traditional, one-way jack arch slab.

been poor. Collapse of a large number of jack arch slabsand damage to many more was reported from theRomanian earthquake of 1990. The Manjil, Iran earth-quake of 1990 [3,4] is of particular interest in this regard.It provided a real testing ground for different forms ofthe one-way jack arch system. Typical weaknesses andmodes of failure of the traditional one-way slab include(i) movement of simply supported steel beams from theirposition under earthquake shaking, causing the collapseof brick arches; (ii) inability of the brick arches to trans-fer in-plane loads perpendicular to the steel beams (Fig.2a); (iii) concentration of stresses in the stiff brick archesdue to out-of-plane vibration of the slab (Fig. 2b); (iv)weakness of the slab system in transferring in-planeshear (Fig. 2c); (v) dynamic interaction between the stiffbrick arches and the more flexible steel beams under ver-tical vibration; and (vi) inability of the slab to act as adiaphragm as is required for good seismic performance

Fig. 2. Different modes of failure of traditional one-way jack arch slab. (a) inability to transfer in-plane axial loads; (b) concentration of stressesin brick arch under out-of-plane bending; (c) inability to transfer in-plane shear force; (d) inability to act as a diaphragm.

(Fig. 2d). Restraining the ends of the I-beams with trans-verse steel beams, or fixing the ends of the beams to theconcrete ring beam and using transverse or diagonal steeltie bars over the span are two earlier recommendations,presented by Moinfar [8], for improving the seismic per-formance of the traditional one-way system.

2. The proposed two-way, engineered, jack archsystem

To overcome the shortcomings of the one-way jackarch slab, the senior author has recently proposed usinga number of transverse steel beams spanned between themain I-beams to form a steel grid [5]. In this way theunconnected parallel steel beams will become part of aninter-connected steel grid, allowing the vertical loads tobe transferred in two directions, also enabling the trans-fer of the in-plane forces (Fig. 3). In fact, by using asteel grid, the grid will act as the main load-carryingelement in the slab, while the brick arches act mainly asin-fill panels. The proposed two-way, steel grid, jackarch system therefore addresses all the weaknesses ofthe traditional one-way slab as discussed below:

(i) The steel grid will confine the brick arches in small(steel framed) panels where no differential move-ments of the supporting steel occur during an earth-quake and as a result the arches remain in place.


Fig. 3. Details of a typical configuration of the proposed two-waysystem.

(ii) The inability of the one-way slab in transferring thetransverse in-plane loads through the brick arches isovercome by inclusion of transverse steel beams.

(iii) By creating a stiff steel grid, the in-plane shearforces will be directed towards the steel grid andaway from the brick arches, hence the build up ofstresses in brick arches is avoided.

(iv) In a two-way grid system, the brick arches are madediscontinuous at the location of transverse beamsand become divided into a number of smaller panelsundergoing much smaller out-of-plane bendingstresses.

(v) Discontinuities in brick arches at the locations oftransverse beams also act as plastic hinges, which inturn reduces the overall stiffness of brick arches. Onthe other hand, a steel grid will be stiffer than indi-vidual I-beams. Therefore, the two systems will bedynamically more compatible and the effects ofdynamic interaction will be reduced.

(vi) The steel grid creates a homogeneous diaphragmaction for the slab in such a way that if some localfailure in supporting walls or support beams occurs,the load from the unsupported part of the slab willbe carried by the continuous grid. This will stop thepartial collapse of the slab.

(vii) Finally, the use of a steel grid as opposed to indi-vidual parallel steel beams does not necessarilyincrease the amount of steel required for the slab.In many instances, it may in fact reduce the amountof necessary steel and therefore reduce the cost ofthe slab. The reason for this being that the steelgrid creates a two-way action for the slab and, assome of the load is transferred to the supports bythe transverse beams, the load on the main longi-tudinal beams is reduced and smaller sections willtherefore be required.

3. Experimental investigations

To investigate the behaviour of the jack arch systemin detail, numerical (finite element) models were neces-sary to represent the real system. However, the presenceof many unknown parameters regarding the mechanical,dynamic and strength properties of the slab materials andelements, as well as the boundary conditions requiredsome prior experimental investigations. Static tests werefirst conducted to determine the mechanical and strengthproperties of slab elements, particularly those of thebrick arches. Dynamic tests on full-scale prototypes ofthe one-way and two-way slabs were then carried out todetermine the dynamic properties and boundary con-ditions. The results from dynamic tests also provided abasis for adjusting, correcting and, in general, improvingthe finite element models of the slab.

3.1. Material properties of brick arches

The proposed brick arches are lightweight perforatedbricks joined together by a clay-gypsum mortar. No dataon the strength and mechanical properties of this typeof masonry were available. For this reason, standardstatic tests were carried out on brick units, mortar andbrickwork. Details of the tests are presented elsewhere[6]. Typical results include the minimum compressivestrength of brickwork, f�m = 4.8 MPa and average elasticmodulus of brickwork, Em = 2.5 GPa. The stress–strainrelations for a number of brickwork test samples areshown in Fig. 4. A near linear relationship, up to theultimate load, can be seen in all the tests.

3.2. Dynamic properties

The four-storey, steel-framed, main building of a flourfactory was used to test the idea of the two-way jackarch system in a full-scale building. The detailed planfor the position of equipment, machinery, belts, cranes,piping etc were not available at the time of construction.It was, therefore, decided to use the jack arch flooringsystem, whereby the desired holes and openings couldeasily be made in the floors at a later date. For the pur-pose of our comparative investigations, the first and thirdfloors of the building were covered using the two-waysteel grid system and the second and fourth floors of thebuilding were covered using the one-way system.Dimensions and details of a 5.0 by 6.0 m floor panel,used for experiments is given in Fig. 5. Non-destructivedynamic tests were carried out on the second and thirdfloors of the building. These two floors have identicaldetails and boundary conditions, except that in the firstfloor two, equally spaced, transverse beams were usedto create a two-way jack-arch system. Dynamic impulsetests were carried out on the floors to determine the natu-ral frequencies, mode shapes and damping of their out-


Fig. 4. The stress–strain curves of four lightweight brick lime-claymortar prisms.

of-plane bending vibration. The test procedure anddetails of processing and extraction of data are reportedelsewhere [6]. A typical time history and the relatedpower spectrum function extracted from one of the testsare shown in Fig. 6. The natural frequencies of each slab,appearing as strong peaks in power spectra of theirrespective records, were extracted from the spectra. Thedamping ratio associated with each frequency was alsocalculated from the spectrum, using the half-powermethod.

Due to the nature of the tests, it was not possible todirectly determine the mode shapes associated with eachfrequency. Therefore, a mapping technique was adoptedin which, for any given frequency, the peak amplitudesrelated to that frequency at different locations of the slabwere compared and in this way the associated mode wasdetermined. The main natural frequencies of the out-of-plane vibration of the two slabs extracted from the tests,together with their associated mode shapes and dampingratios, are give in Table 1.

A review of the frequencies given in Table 1 indicatesan increase in the stiffness of the floor, due to transversebeams. The natural frequencies of the two-way systemare all higher than the corresponding frequencies of the

Fig. 5. Construction details of the tested two-way jack arch panel ofthe flour factory.

Fig. 6. Typical (a) response time-history and (b) auto-power fre-quency spectrum of the two-way jack arch slab of the flour factory.

one-way system. A marked increase in damping ratiocan also be noted for the two-way system.

4. Loading

Loading to be considered in the design of jack archslabs, are gravity and earthquake loads. The gravity, ser-vice dead and live loads may be determined using appro-


Table 1Experimental natural frequencies and damping ratios of the one-way and two-way slabs

Mode number (1) Two-way system One-way system

Frequency (Hz) (2) Damping % (3) Frequency (Hz) (4) Damping % (5)

Mode 1 (1st bending) 11.6 4.2 10.9 2.8Mode 2 (2nd bending) 19.1 5.0 14.4 2.9Mode 3 (3rd bending) 30.5 4.8 24.2 2.8Mode 4 (4th bending) 37.9 4.3 39.4 3.0

priate codes of practice. The earthquake loads, however,warrant further discussion. The earthquake loads, actingon a jack arch slab are in-plane horizontal loads and out-of-plane vertical loads.

4.1. Horizontal earthquake loads

The earthquake-induced horizontal forces acting onthe slab may be either in-plane axial or in-plane shearforces. The in-plane axial forces are caused by the inertiaof the slab alone. These forces are generally small andthe resulting stresses are minimal. The in-plane axialforces, therefore, are neglected for design purposes. Thein-plane shear forces, however, may be large and shouldbe considered in design. These forces are the result ofdifferential movements of the parallel slab supportscaused by the torsion in the building. The horizontalshear force acting on the slab may be determined byanalysing the building for earthquake loads, using anequivalent-static, a pseudo-dynamic or a dynamicmethod.

In a two-way jack arch system, in-plane shear forceis carried mainly by the frame action of the steel grid,whereas, in a one-way system, the brick arches are themain load-carrying elements. The brick arches are weakin transferring in-plane stresses, therefore, the steel gridshould carry the majority of the load. In other words, aone-way system will not be suitable when large in-planeshears are present. For the two-way system, the shear-induced stresses in both the steel grid and the brickarches should be checked against the allowable stresses.

4.2. Vertical earthquake loads

The majority of stresses developed in the slab are dueto the combined effects of gravity and vertical earth-quake loads, both causing out-of-plane bending andshear in the slab. Design of the slab will, therefore, begoverned by the out-of-plane bending of the steel beams.Out-of-plane shear stresses in both the steel beams andthe brick arches should also be checked against theallowable stresses.

To determine the vertical earthquake loads, it is pro-posed that the simple, equivalent-static, method be used.

In this method, the earthquake load, E, is related to theweight of the system, We, in the following form:

E � CWe (1)

We, is usually considered as the total dead load plus apercentage of the live load, the recommendation forwhich differs in seismic codes for different structures.The earthquake coefficient, C, may be determined fromthe relation:

C �ABIRw

(2)

in which, A, B, I and Rw are the design base acceleration,the dynamic response coefficient, the importance factorand the performance factor, respectively. The designbase acceleration, A, is given by seismic codes for differ-ent localities. This coefficient is, however, determinedfor the horizontal earthquake loading and should bemodified for the vertical loading. A 33% reduction isoften used for this conversion. For design of jack archslabs, the importance factor, I, may be taken as that forthe whole building. The other two coefficients needfurther considerations as follows.

4.2.1. Dynamic response coefficient: BThis coefficient is a function of the soil type, To, and

the fundamental period of vibration of the system, T, andrepresents the dynamic magnification of the system’sresponse. Seismic codes provide information regardingthe soil type, To. Designer must determine the fundamen-tal period of vibration of the system, T. For jack archslab, the fundamental mode of vibration will be the firstout-of-plane bending mode. A simple method ofdetermining T, for the jack arch slabs, is discussed here.Assuming the slab as a linear, elastic rectangular plate-bending element, the classic solution for the fundamentalbending mode of vibration is given in the followingform [1]:

T �2pa2

l21� Eeffh3

12r(1�J2m)�1/2

(sec.) (3)

In the above equation, a, h, Eeff, r and nm are length,


thickness, effective elastic modulus, mass per unit areaand Poisson’s ratio of the plate, respectively and l1 is adimensionless parameter depending on the geometry ofplate and its boundary conditions. To be able to applythis equation to a non-homogenous composite plate, suchas a jack arch slab, equivalent effective parametersshould be used. The effective thickness, h, may safely beconsidered as the full thickness of the flat slab, includingflooring and the effective density, r, should be determ-ined considering slab materials. The effect of Poisson’sratio will be small. Therefore, as a safe choice, the Pois-son’s ratio may be considered to be the same as thatfor brickwork.

The effective elastic modulus, Eeff, of the slab has anappreciable effect on the fundamental period of vibrationand should be evaluated more accurately. The para-meters affecting the Eeff of the slab are the dimensionsand boundary conditions of the slab, the elastic modulusof brick arches and the number, and size and configur-ation of the steel beams. A number of numerical dynamicanalyses were carried out on jack arch slabs with differ-ent dimensions, boundary conditions, steel configur-ations and types of brickwork and from the results ofeach analysis, an equivalent elastic modulus was calcu-lated. Details of these analyses are reported elsewhere[10]. It was found that the effective elastic modulus ofslab, Eeff, is linearly proportional to the elastic modulusof brickwork, Em. As for the slab boundary conditions,the results from the worse case scenario, i.e. the simplysupported situation, are used. The best parameter to rep-resent the other two variables, i.e. dimensions of the slaband steel grid configuration, is the weight of steel perunit area of slab, Ws (kg/m2). The results of these analy-ses are plotted in Fig. 7. In this figure, the ratio of effec-

Fig. 7. The effective modulus of elasticity (Eeff) of the jack arch slabs.

tive elastic modulus of slab to elastic modulus of brick-work (Eeff/Em) is plotted against Ws.

After determining Eeff, it is used in Eqn. (3) to deter-mine the fundamental period of vibration of the slab. Tocheck the validity of Eqn. (3) for jack arch slabs, thissimple method was used to calculate the fundamentalperiods of the one-way and two-way jack arch slabs ofthe flour factory discussed earlier. The calculated valueswere compared with the periods determined from theexperiments. The fundamental periods of vibration, T,calculated using Eqn. (3), are 0.0989 s for the one-wayslab and 0.0912 s for the two-way slab. These valuescompare well with the experimental results of 0.0913 sand 0.0858 s, respectively. The higher calculated valuesare due to the simple-support assumption made for theslab, as the worst case scenario.

To determine the dynamic response coefficient, B, dif-ferent seismic codes provide different relations and dia-grams, the majority of which are based on To and T.These relations are derived for the building as a wholeand, in most codes, the region in the diagram coveringsmall periods of vibration is considered a constantmaximum. This can be seen in Fig. 8 in which, as anexample, the diagram given by the Iranian Seismic Code[2] is shown. The dotted line for periods between 0 and0.4 s is recommended by the Iranian code for buildings.The fundamental periods of vibration of jack-arch slabsare, however, well below those of ordinary buildings andsometimes fall within this range. It is, therefore, prudentto determine the actual values of B in this period range.The actual values of B for the four different soil typesare calculated and plotted, in the same figure (Fig. 8),as full lines.

4.2.2. Performance factor of jack arch slab, Rw

The jack arch slab system discussed here may be con-sidered as a confined masonry construction. For confinedmasonry walls, a value of 4 is often used for the per-

Fig. 8. The proposed dynamic response coefficient (B) for the jackarch slabs.


formance factor Rw [9]. However, the jack arch slab sys-tem is expected to behave differently to a conventionalmasonry building system. A preliminary discussion onRw factor for jack arch system will follow.

The system performance factor, Rw, corresponding tothe UBC allowable stress design format, is given by:

Rw � Rm�Y (4)

In which, Rm, � and Y are ductility reduction factor,over-strength factor and allowable stress factor, respect-ively [11]. Design of jack arch slab will be based ondesigning the steel grid and controlling the brick archesfor allowable stresses. Ductility of the slab, therefore,will depend on the level of stresses in the brick archeswhen the steel grid is at its allowable stress limit. Thenumber and size of transverse beams in a two-way sys-tem will govern the ductility of the slab. Assuming noductility for the one-way system (i.e. Rm = 1), a numberof finite element stress analyses were conducted on slabsof different sizes and transverse beam configurationsand, in each analysis, the level of stresses in brick archeswere compared with those of the one-way system. Thiscomparison was used as a basis for determining theavailable ductility of the slab. Results of these analysesare plotted in Fig. 9. In this figure, the ductility factorof the slab, Rm, is plotted against the amount of trans-verse beams per unit area of the slab. In the expectedrange of transverse beam density, ductility factor variesbetween 1.0 and 2.0. For the one-way system, Rm, maybe assumed as unity.

A safe value of � = 1.5 may be proposed for theoverstrength of jack-arch slab. The allowable stress fac-tor, Y, on the other hand may be determined by consider-

Fig. 9. Ductility reduction factor (Rm) for the jack arch slabs.

ing the critical element of the slab, i.e. the steel grid.For steel construction, Y, varies between 1.4 to 1.5. Theperformance factor, Rw, may therefore be considered as2.0 (Rm = 1.0, � = 1.5, Y = 1.4) for the one-way systemand 2.1Rm for the two-way system. Design tables andfigures presented in the following sections have beenprepared in such a way that allowable stresses and load-ing parameters, other than those suggested in this article,can be easily accommodated.

5. Design method

The jack arch slab is primarily a masonry construc-tion. In the one-way system, brick arches undergo largestresses as they transfer the static and seismic loads tothe steel beams. In the two-way system, although therole of brick arches in the transfer of loads is reduced,they are still required to transfer their own loads. Themasonry buildings and steel buildings may be designedusing the allowable (working) stress design method(ASD). It is, therefore, plausible to base the design ofsteel-beam jack arch slabs on the same method.

The design procedure to be adopted will be based ondesigning the steel grid, as the main load-bearingelement, and controlling the brick arches for allowablestresses. The failure criterion for the slab is assumed tobe bending failure of the steel beams or compressivefailure of the brick arches. In steel beams, shear failureis unlikely to occur prior to bending failure. The tensilefailure in brick arches is governed by the bond tensilestrength. The bond tensile strength in masonry construc-tion is low. For the lime-clay mortar brick prism, thiswas determined as 0.2 MPa. Tensile bond failurebetween brick units and mortar will be allowed underhigh frequency cyclic loads since this failure does notaffect the load-carrying capability of brick arches incompression.

6. Design parameters and procedure

Similar to other slab systems, in the jack arch slab,there are a number of parameters that affect the design.The most important parameters in this respect are recog-nized as (i) dimensions of the slab; (ii) layout of the steelgrid; (iii) boundary conditions and connections of slabelements; (iv) width and rise of brick arches; (v) loading,W; (vi) elastic modulus of brickwork, Em; and (vii)allowable stress of brickwork (fmall). In the followingsections the allowable stresses of the slab materials arefirst explored. Typical slab dimensions and steel gridconfigurations are also discussed. Then a large numberof slabs having different dimensions and steel configur-ations are selected and designed according to the designmethod outlined above. In designing these ‘ reference’


slabs, constant ‘ reference’ values are considered for thedesign parameters, (iii) to (vi). Finally, for design casesin which one or more of the design parameters, (iii) to(vi) have values other than the reference values, appro-priate penalty or conversion factors are evaluated usinga series of parametric analyses.

6.1. Allowable stresses

The allowable stresses for steel and masonry aredetermined by codes of practice. For this study, in linewith the conventional codes of practice, allowable flex-ural tensile or compressive stress for steel will beassumed to be; fsall = 0.6 fy. Since no references existconcerning the allowable stresses used for jack archmasonry made with lime-clay mortar, the followingare proposed:

Allowable compressive stress for masonry:

fmal � 0.2f�m;

Allowable tensile stress for masonry:

fball � 100 kPa.

The proposed allowable tensile stress is in line withthe UBC codes of practice [12] and the test results dis-cussed earlier. For allowable compressive strength, fmall,the UBC gives a value of 0.16 f’m for masonry walls.The brick arches in a jack arch system are, however,less slender than masonry walls and due to the loadingsituation and the geometry of the arch, an increase incompressive stress (increase in load) will cause anincrease in the stability of the arch. For this reason,fmall = 0.2 f’m is proposed for jack-arch construction.Further work will be necessary to establish a more rep-resentative relation between fmall and f’m.

6.2. Dimensions and layout of slabs

Geometry and dimensions of the slab are governed bythe architectural layout of the building. It is, however, auseful common practice to limit the span of the mainbeams to 6.0 m. For larger floor areas, deep or stiff sup-port beams will be necessary to reduce the effective spanof the longitudinal dimension of the slab. In the proposedtwo-way system, if the transverse beams are to be effec-tive in carrying the load, i.e. the slab to remain as a two-way system, it will be necessary to limit the transversedimension of the slab to twice the span of the mainbeam. This can be achieved by providing support beams.To validate the above recommendations, stress analyseswere made on slabs of larger dimensions. It was foundthat, to limit the stresses in brick arches and the deflec-tion of the slabs to the allowable limits, steel beams of

uncommon sizes are required, rendering a very expens-ive slab.

The number of main beams required for a jack archslab will be determined by considering the transversedimension of the slab and the width of brick arches.Within the limits discussed later, a designer may selectan appropriate width of arch and determine the numberof required main beams.

The span of main beams and the level of loading dic-tate the number of required transverse beams. One of theobjectives of providing transverse beams is to divide thelong brick arches into a number of smaller brick panels.The maximum distance between the transverse beamsshould preferably be fixed as twice the width of the arch.This would give brick panels of a maximum length towidth ratio of 2.0. The number of required transversebeams and their section modulus, as design parameters,will depend on the level of horizontal and vertical load-ing.

6.3. Design of reference slabs (design table)

Over 350 individual rectangular slabs of differentdimensions, ranging from 3.0 × 3.2 m to 6.0 × 12.0 mand different steel configurations were selected as refer-ence slabs. Each slab was then designed with otherdesign parameters ((iii) to (vi)) having reference valuesor conditions. These include a non-yielding support forthe main beams (a = 1.0), a width of brick arch equalto 80 cm with a rise of 5 cm (b = 1.0), an equivalentvertical load of W = 10.0 kN/m2 (w = 1.0), a brickworkelastic modulus of Em = 2.5 MPa (� = 1.0) and an allow-able compressive stress for brickwork of fmall = 1.0MPa (l = 1.0). In the above, a, b, w, � and l are penaltyfactors, values of which are unity for design parameterswith reference values. For design parameters havingvalues other than those specified above, these factorsassume different values as will be discussed later. Alsoin all analyses and designs, the allowable stress for steelis considered as fsall = 144.0 MPa and the brickworkPoisson’s ratio is assumed as n = 0.25.

To design the slabs, finite element linear stress analy-ses of the slabs were carried out using the general-pur-pose, SAP-90 program. Beam elements were used tomodel slab beams and shell elements were utilized tomodel the brick arches. The number of beam and shellelements used in each model varied according to the slabsize and steel configuration. In some models in excessof 1000 shell elements were used to model the brickarches. The stress contours plotted for typical one-wayand two-way slabs are shown in Fig. 10.

For each slab, the required section modulus, Sn andmoment of inertia, In of the main and transverse beamswere thus determined using the allowable stress limitsfor steel and brick arches and deflection limits (discussedlater) as controlling parameters. For an optimum design


Fig. 10. Typical stress results for (a) a one-way and (b) a two-wayjack arch slab showing the transverse bending stresses around axisparallel to main beams in the brick arches.

of the main steel beams and in line with design pro-cedures for RC slabs, the jack arch slab is divided intothree strips: a central strip and two side strips. As in any,simply supported, plate-bending system, the maximumstresses will normally be in the central part of the slab.By dividing the main beams into ‘central beams’ and‘side beams’ , it will be possible to use smaller steel sec-tions for the side beams. Unlike the RC slab, thisdivision cannot be made on a fixed length and, therefore,the division is made on the number of main beams. Itshould be noted that, if slab beams span over two ormore slab panels, the negative moments in side beamsbecome critical. In this case, all main beams should beof the same size as the central beams.

The results of the design of each reference slab in theform of the evaluated section modulus, Sn and momentof inertia, In of the steel beams are compiled in a tablereferred to as a ‘design table’ . A typical section of thedesign table is shown in Table 2. The complete designtable is presented elsewhere [6]. For every slab size, dif-ferent steel grid configurations are presented in the tablewhich include both the one-way and two-way systems.All the given configurations will be suitable for a designcoefficient of Cs = 1.0, based on vertical gravity andseismic loads only. The required moments of inertia ofthe main and transverse beams, Ireq, will be determinedfrom the following equation:

Ireq � CsIn (5)

where

Cs � abwel (6)

In Eq. (5), In, is the minimum moments of inertia of themain and transverse beams, given in the design table anda, b, w, � and l are the factors discussed in the nextsection. The actual design for every steel configurationis carried out by designer using Ireq. Design coefficient,Cs, will be applied to both the main and transversebeams.

In earthquake-prone areas, the slab may undergo hori-zontal in-plane shear forces for which it should be con-trolled. The in-plane shear force may act parallel to themain beams or perpendicular to the main beams. Theweakest direction of the slab for this loading is the direc-tion parallel to the main beams. In another set of analy-ses, for each of the slabs given in the design table, thefinite element model was fixed at one side (parallel to themain beams), and shear force was applied, distributedbetween the beam nodes, at the other side. Themaximum stress developed in the brick arches and theflexural stress in weaker, transverse beams, were usedas parameters for determining the in-plane, shearcapacity of the slab. The in-plane shear strengths of allsteel grid configurations for every slab size were thuscalculated and are presented in the design table. In thisway, the choice of a particular configuration for a givenslab size will depend on two parameters, namely thelevel of in-plane shear force on the slab and the economyof steel configuration. A designer will choose the cheap-est steel configuration, which is capable of resisting thegiven in-plane shear force. It should be noted that themaximum in-plane shear capacity of the steel configur-ation is calculated using the Iyy for the transverse beams.

6.4. Parametric evaluation of design parameterfactors

To design a jack arch slab with design parameters(items iii to vii) having different values to their referencevalues, conversion factors are necessary to account forthe effects of the change in the values on the design.For this purpose a series of parametric analyses wereconducted on 15 different jack arch slabs, covering awide range of slab sizes and steel grid configurations inwhich the effect of each design parameter on the out-come of the design was evaluated. The slab dimensionsconsidered included 3.0 by 3.2, 3.0 by 5.6, 4.5 by 4.8,4.5 by 8.8, 6.0 by 6.4 and 6.0 by 12.0 m and steel gridconfigurations included (a) no transverse beams (one-way system), (b) one transverse beam and (c) two trans-verse beams. Other parameters were kept constantthroughout the parametric study when not under study,with reference values.


Table 2Typical format of the design table

Slab dimensions Minimum normalized moments of Inertia (In) and the section modulus (Sn) In-plane shear(m) capacity of slab

(kN)

Main beams Transverse beams

Central beams End beams

No. of Sn (cm3) In (cm4) No. of Sn (cm3) In (cm4) No. of Sn (cm3) In (cm4)beams beams beams

3.0 × 3.2 1 146 1320 2 78 541 – – –1 109 869 2 53 318 1 53 318 301 78 541 2 53 318 1 109 8693 78 541 – – – 1 53 3181 109 869 2 53 318 2 53 318 403 78 541 – – – 2 53 3181 78 541 2 53 318 2 109 869

3.0 × 4.0 2 146 1320 2 78 541 – – – –2 109 869 2 53 318 1 53 318 404 78 541 – - – 1 53 3182 78 541 2 53 318 1 146 13202 84 600 2 53 318 2 53 318 554 78 541 – - – 2 53 3182 78 541 2 53 318 2 78 541

6.4.1. Slab support condition (factor, a)The jack arch slab may be supported on walls or deep

beams, in which case, the deflection of the slab on theline of its support will be nil or negligible. The flexiblebeams of a framed building may also support the slab.In this case, the slab will deflect on the support lineswith the supporting beams. This changes the stress levelin the steel beams and the brick arches. All the analysesleading to the design table were made assuming the for-mer, unyielding support condition for the main beamsand the latter, flexible support condition for the trans-verse beams. A factor, a, will therefore be necessary toaccount for other forms of support conditions. This fac-tor will be applied to the required moments of inertia ofthe steel beams. To determine the influence of the flexi-bility of the support beams on the amount of steelrequired for the slab, numerical analyses were carriedout on the selected slabs with different support con-ditions. In each case, the slab was analysed and thestresses in brick arches and steel beams were extractedand compared with those of the slabs with reference sup-port condition. The change in the moments of inertia ofslab beams, required to bring the stresses in steel beamsand brick arches to the level of the reference model, wasthen calculated in each case, using trial and error analy-ses. In Fig. 11, the correction factor, a, indicating therequired change in the moment of inertia of steel sec-tions, is plotted for slabs of different sizes, steel gridconfigurations and support conditions. It can be seen in

this figure that as the size of slab increases, the effectof support condition on the slab decreases.

6.4.2. Width of brick arches (factor, b)In the construction of traditional jack arch slabs, the

width of arch varies from 80 cm to 1.0 m. Wider archesrequire a higher rise and, therefore, a thicker slab andshorter arches mean extra steel beams. The rise of archis another important parameter affecting the performanceof the slab. To determine the optimum rise of arch,numerical models of one-way jack arch slabs were ana-lysed. They had standard 80 cm wide arches, with risesof 3 cm, 5 cm, 7 cm, and 10 cm and were analysedunder a unit vertical load and the maximum compressivestresses in the brick arches were determined. It wasfound that the stresses in brick arches increase rapidlyfor width to rise ratios higher than 16, corresponding toa rise of 5 cm for an 80 cm wide arch. Numerical analy-ses carried out on slabs having different arch widthsshowed similar optimum width to rise ratio of 16. It is,therefore, recommended that when geometry of the slabdictates using shorter arches, the rise of arch to be takenas 5 cm and when a wider arch (up to 100 cm) becomesnecessary, the rise of arch should correspond to thewidth to a rise ratio of 16.

The reference width of arch used for design tableanalyses is 80 cm. When a wider arch is used, a factor,b, should be applied to the supporting steel beams totake into account the increased stresses in the brick


Fig. 11. Factor for slab support condition (a). 1, rigid support for main beams and flexible support for transverse beams; 2, rigid support formain beams and transverse beams; 3, flexible support for main beams and transverse beams; 4, flexible support for main beams and rigid supportfor transverse beams.

arches. The width of arch factor, b, is determined fromthe curves shown in Fig. 12. In this figure, b, is a factorwhich indicates the increase in the moment of inertia ofthe main and transverse beam sections required to keepthe stresses in brick arches and steel grid at the samelevel as the 80 cm wide arch.

6.4.3. Loading, W (factor, w)Loading is a design parameter calculated by the

designer. The design table is based on an equivalentstatic, vertical load of 10.0 kN/m2. A factor is, therefore,necessary to account for variations in the load intensity.The selected slabs were analysed with different loads ofbetween 5.0 and 20.0 kN/m2. For every slab size andsteel configuration and for each loading, the steel sec-tions required to bring the level of stresses in steel andbrick arches to the level of stresses for the 10.0 kN/m2

load, were calculated. These sections were then normal-ised to the steel section required for the load of 10.0kN/m2. The normalised results, corresponding to a loadfactor, w, are plotted against the load intensity, W, inFig. 13. As is evident in this figure, a near linear relationexists between loading and the required steel sections.This figure also indicates that the relation between theload factor, w, and the load intensity, W, is effectivelythe same for all slab sizes and steel-grid configurations.

6.4.4. Elastic modulus of brickwork, Em (factor, �)Similar stress analyses to those mentioned above were

carried out on the selected slabs, this time changing thevariable parameter to elastic modulus of brickwork.Values between 2.5 and 20.0 GPa were used for Em indifferent analyses. The value of elastic modulus used asa basis for comparison was taken as Em = 2.5 GPa whichcorresponds to the elastic modulus of the lightweight,perforated brick-lime mortar prism previously tested.The steel sections required to bring the maximum stressin brick arches to the level of the reference slabs (Em

= 2.5 GPa) were then normalised to the steel sections ofthe reference slabs. The normalized values are equivalentto a factor, �, as shown plotted against the Em values inFig. 14. This figure shows that the factor for elasticmodulus, �, is also independent of slab size and steelconfiguration and is directly proportional to Em.

6.4.5. Allowable stress of brickwork, fmall (factor, l)The allowable compressive stress of brickwork used

to evaluate the design table was assumed as 1.0 MPa(fmall = 0.2 f’m). For allowable stresses, fmal � 1.0MPa, another factor, l, will be used. The values of thiscorrection factor are shown in Fig. 15. Fig. 15 has simi-larly been extracted from analyses in which steel sec-tions for different values of fmall were calculated.


Fig. 12. Factor for width of brick arch (b).

Fig. 13. Factor for design load (w).


Fig. 14. Factor for elastic modulus of brickwork (�).

Fig. 15. Factor for allowable stress of brickwork (l).


7. Control of deflection

Control of deflection is an important aspect of a struc-tural design procedure. This is particularly true for thejack arch slabs as they are generally more flexible thanthe conventional concrete slabs. Some codes of practicesuggest a maximum allowable out-of-plane deflection of�max = l / 360. This limit has been used for control ofdeflections in jack arch slab design. For every slab sizeand steel configuration presented in the design table, themaximum out-of-plane deflection of the slab wasdetermined from the numerical analyses. This deflectionwas in turn compared with the maximum allowabledeflection and the necessary changes in the design para-meters; a, b, � and l, required to reduce the deflectionto the allowable limit and keep the stresses in brickarches and steel beams within their specified allowablestresses, were calculated. A deflection parameter, g, wasthen calculated as:

g � abel�gmin (7)

This process was repeated for all the slabs considered.It was noted that the variation in deflection control para-meter for different slab sizes and steel configurationswas small, but it changed markedly with the allowablestress for steel and the width of brick arches. Themaximum values of deflection control parameter evalu-ated for different slab types were, therefore, selected asthe controlling limits (gmin). These are presented inTable 3.

8. Conclusions

A type of engineered jack arch slab, in the form of atwo-way action, steel-grid system was introduced in thispaper. Experimental and numerical studies of full size,one way and two-way jack arch systems show the effec-tiveness of the proposed system. Conclusions drawnfrom the material presented in this paper may be sum-marised as follows:

Table 3Minimum values of g (= ab�l) for control of deflection in jack archslabs

Allowable stress of steel, fsall, Deflection control parameter,(MPa) (1) gmin

Width of arch (cm)

80 (2) 90 (3) 100 (4)

100 1.2 1.46 1.73144 0.95 1.16 1.37200–240 0.90 0.9 0.9

1. The proposed steel-grid, two-way jack arch systemhas none of the shortcomings of the traditional one-way system. It provides a more homogeneous slabcapable of diaphragm action and is able to transfergravity and seismic loads. In the proposed system, thein-plane and out-of-plane loads are both carried,mainly, by the steel grid. This reduces the role of thelow strength, brittle brick arches to small in-fillpanels.

2. The use of lightweight perforated brick units togetherwith clay-gypsum mortar greatly reduces the weightof the slab and enhances the ductility and flexibility ofthe brickwork. All these factors improve the seismicperformance of the slab.

3. The static tests showed that the brickwork made oflightweight bricks and clay-gypsum mortar has almosta linear behaviour up to failure. Considering the lin-ear-elastic behaviour of steel beams and the hingedconnection between the brick arches and steel beams,linear, elastic analysis of the slab system is justified.This also indicates the applicability of the ASDmethod for designing this type of masonry slab.

4. Dynamic tests carried out on full scale one-way andtwo-way systems showed the improved dynamicbehaviour of the two-way system. An increased stiff-ness and damping ratio was observed in the two-wayslab. Dynamic test results were also used to createrepresenting numerical models for the slabs.

5. The advantages of the proposed two-way system,regarding performance and cost, compared with thoseof the one-way system, can be deduced from theirrespective design. It is also concluded that an engine-ered design of the jack arch slab, as presented in thisarticle, will pave the way for this popular method offlooring to be considered as an engineered construc-tion and a viable alternative to other forms of flooring.

Acknowledgements

The author wishes to acknowledge the financial sup-port provided by the International Institute of EarthquakeEngineering and Seismology (IIEES), under the grantNo. ZL-4744 and the Iranian National Research Councilfor the grant No. NRCI-ZL-479 of National ResearchCouncil Projects. Thanks are also due to Mr. Jafar Fal-lahi, Managing Director of the Naghshe-Rostam flourfactory. Mr. A. Khoddam-Mohammadi and Mr. A.Imanipour for their useful contributions to thisresearch program.

References

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[4] Maheri MR. Manjil, Iran earthquake of June 1990, some aspectsof structural response. J of Structural Engineering Review1992;4(1):1–16.

[5] Maheri MR, Imanipour A. Seismic evaluation of a proposed two-way jack arch slab. In: Proceedings of the 3rd International Con-ference on Earthquake Engineering and Seismology, Vol. III,Tehran, Iran. 1999.

[6] Maheri MR. The gravity and seismic design of jack arch slabs,Iranian National Research Council Report No. NRCI-ZL-479,2001, (in Farsi).

[8] Moinfar AA. Seismic activities and conditions of rural houses inthe countries of the region. In: Proc CENTO Conf on EarthqHazard Mitigation, Turkey. 1968.

[9] Moroni MO, Astroza M, Gomez J, Guzman R. Establishing Rw

and Cd factors for confined masonry buildings. J Struct Engineer-ing, ASCE 1996;122(10):1208–15.

[10] Rahmani H. The basis for static and seismic design of two-wayjack arch slabs. MSc thesis, Shiraz University, Iran, 2001 (inFarsi).

[11] Uang C. Establishing R (or Rw) and Cd factors for building seis-mic provisions. J of Struct Engineering, ASCE1991;117(1):19–28.

[12] Uniform Building Code. International Conference of BuildingOfficials, California, 1997.

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