Construction and Building

8/14/2019 Construction and Building

1/17

CFRP strengthened openings in two-way concrete slabs An experimental and numerical study

Ola Enochsson a,*, Joakim Lundqvist a, Bjorn Taljsten a,b,Piotr Rusinowski a,b, Thomas Olofsson a

a Lulea University of Technology, Division of Structural Engineering, 971 87 Lulea, Swedenb Technical University of Denmark, Department of Civil Engineering, Brovej Building 118, 2800 Kgs, Lyngby, Denmark

Received 20 January 2005; received in revised form 24 May 2006; accepted 19 June 2006Available online 1 September 2006

Abstract

Rehabilitation and strengthening of concrete structures with externally bonded fibre reinforced polymers (FRPs) has been a viabletechnique for at least a decade. An interesting and useful application is strengthening of slabs or walls where openings are introduced.In these situations, FRP sheets are very suitable; not only because of their strength, but also due to that they are easy to apply in com-parison to traditional steel girders or other lintel systems. Even though many benefits have been shown by strengthening openings withFRPs not much research have been presented in the literature.

In this paper, laboratory tests on 11 slabs with openings, loaded with a distributed load are presented together with analytical andnumerical evaluations. Six slabs with openings have been strengthened with carbon fibre reinforced polymers (CFRPs) sheets. Theseslabs are compared with traditionally steel reinforced slabs, both with (four slabs) and without openings (one slab). The slabs are qua-dratic with a side length of 2.6 m and a thickness of 100 mm. Two different sizes of openings are used, 0.85 0.85 m and 1.2 1.2 m.

The results from the tests show that slabs with openings can be strengthened with externally bonded CFRP sheets. The performance iseven better than for traditionally steel reinforced slabs. The numerical and analytical evaluations show good agreement with the exper-imental results. 2006 Elsevier Ltd. All rights reserved.

Keywords: Carbon fibre; Concrete; Design; Numerical analysis; Opening; Slab; Strengthening

1. Introduction

Floor and wall structures are some of the most com-

monly existing structural elements in buildings. Nowadays,rebuilding of existing structures has becoming quite com-mon due to structural and/or functional requirements fromthe users. The functional requirements entail often thatstaircases, elevators, escalators, windows, doors and evenelectrical, heating or ventilation systems, have to beinstalled. Thus, there exists a great need to introduce sec-

tional openings in floor as well as in wall structures. Thestructural effect of small openings is often not considereddue to the ability of the structure to redistribute stresses.

However, for larger openings the static system may bealtered when considerable amounts of concrete and rein-forcing steel have to be removed. This leads to a decreasedability of the structure to resist the imposed loads and thestructure needs therefore to be strengthened.

The traditional strengthening methods, such as additionof a girder-column system, or construction of load-bearingwalls along the edges, take up useful space and may not beaesthetically convenient. On the other hand, advancedcomposites as externally bonded reinforcement has beenextensively tested as related to its use for strengthening of

0950-0618/$ - see front matter 2006 Elsevier Ltd. All rights reserved.

doi:10.1016/j.conbuildmat.2006.06.009

* Corresponding author.E-mail address: [email protected] (O. Enochsson).

www.elsevier.com/locate/conbuildmat

Construction and Building Materials 21 (2007) 810826

Constructionand Building

MATERIALS
mailto:[email protected]:[email protected]


2/17

beams and girders in flexure, shear and even for someextent in torsion [112]. This strengthening technique hasbeen successfully used in several repair and strengtheningprojects in Sweden and elsewhere [13].

Today, the use of carbon fibre reinforced polymers(CFRPs) to strengthen existing slabs and walls due to

openings is becoming more popular, partly due to easeof installation and partly due to space saving. In these sit-uations, CFRP sheets are applied to the slab or wallbefore the opening is made, see Fig. 1. The required sec-tional area of CFRP is often calculated by simply convert-ing the area of steel reinforcement according to existingdesign codes, e.g. the Swedish code, BBK 04 [14]. Eventhough CFRP is used for strengthening of openings, veryfew studies on the structural behaviour of slabs with open-ings have been carried out. The flexural behaviour ofCFRP strengthened one-way slabs with cut-outs, subjected to point loads have been studied by Vasques andKarbhari [15]. The purpose was to investigate the effec-

tiveness of externally bonded fibre reinforced polymers(FRPs) composite strips at strengthening of slabs with cut-outs. The failure mechanism and post-debonding responsewere also studied. The outcome of the study was thatexternally bonded FRP strips can be used to restore theoriginal load carrying capacity of slabs weakened by cut-outs. In addition, they observed a more desirable crackpattern for the CFRP strengthened slabs than for thenon-strengthened slabs. However, the used method todecide the anchorage length for the FRP, seems not tobe appropriate in areas of high curvature and initiatedmore or less always peeling followed by debonding before

the final failure. Another study has been carried out byMosallam and Mosalam [16], where the flexural behaviourof FRP strengthened two-way slabs without openings subjected to uniformly distributed loads was investigated.Both repaired and retrofitted specimens were considered.The study shows that FRP systems can successfully beused to strengthen or upgrade the structural capacity of

both two-way reinforced and un-reinforced concrete slabs.In addition, a significant increase in the load carryingcapacity for the CFRP strengthened slabs was observed.However, the loading system with high-pressure waterbags limited the maximum deflection during the test dueto the bags thickness and bedding ability.

It is clear that more studies on the structural behaviourof strengthened slabs are needed, and particularly withopenings. This to obtain a better understanding of the fail-ure mechanism, to verify current design methods and todevelop new efficient strengthening methods. This will leadto a more stringent use of the CFRP strengthening tech-nique and more cost effective design solutions.

In this paper, two-way simply supported reinforced con-crete (RC) slabs subjected to a uniformly distributed loadare studied. Slabs with strengthened and non-strengthenedopenings have been investigated. CFRP sheets have beenused for the strengthening. In addition, the size of theopening is numerically and experimentally analysed and

discussed.

2. Aim and scope

The research work is focused on examining the struc-tural behaviour of two-way RC slabs strengthened withCFRP due to a sawn-up opening, subjected to uniformlydistributed loads. Numerical and experimental methodsare used to analyse the ability of the CFRP reinforcementto take up the additional section forces caused by sawn-upopenings in reinforced concrete slabs. The research workcan be summarized as follows:

The method to calculate the amount of CFRP forstrengthening used today is verified, this to give aload-capacity equivalent to a slab without an opening.

The mechanical behaviour of the strengthened structuralsystem is analysed by numerical methods.

The numerical calculations are verified by experimentaltests.

The outcome of the work can also be used for cast andmade openings in one-way concrete slabs and in concretewalls, i.e. both for additional steel reinforced and CFRPstrengthened slabs or walls due to openings.

3. Experimental program

3.1. General

The experimental program consisted of two-way RCslabs loaded to failure using a uniformly distributed load.The objective was to compare the result between differentslab configurations:

Without an opening (homogeneous slab). With a cast opening, strengthened with additional steel

reinforcement according to Swedish design codes.

Fig. 1. Strengthening with CFRP sheets before making an opening for aventilation duct in an existing slab due to change functional requirements.

Photo: Bjorn Taljsten (2000).

O. Enochsson et al. / Construction and Building Materials 21 (2007) 810826 811


3/17

With sawn-up openings: (a) without additional strength-ening (weakened) and (b) strengthened with CFRP,designed to reach the same load carrying capacity as tra-ditionally steel reinforced slabs without openings.

The slabs are quadratic with a side length of 2.6 m and a

thickness of 100 mm. Two different sizes of openings areused, 0.85 0.85 m and 1.2 1.2 m.

3.2. Test setup

To provide a uniformly distributed load on the slab, anew unique test rig had to be developed, see Fig. 2. Theload is applied using a system of airbags, embedded byan exterior and interior structure, see Fig. 3. The use of air-bags to produce a distributed load is well tested at LuleaUniversity of Technology, and has been used for a longtime to test the load carrying capacity of roof sheeting pro-files. The specimens are simply supported along four edges.

The loading area is 2.4 2.4 m, i.e. somewhat smaller than

the total area of the slabs. For the slabs with openings, theloading area is decreased due to the area of the opening.

The distributed load is calculated from the reactionforces in each corner measured by four load cells. Sincethe slab is loaded upside-down, springs are mounted ineach corner to eliminate the weight of the support struc-

ture. Both deflections and strains are measured accordingto a system of location lines defined over the slab surface.Fig. 4 shows the location of the measuring points at definedlines for the different slab configurations. The deflectionsare measured with linear voltage displacement transducers(LVDTs). The strains are measured with strain gauges(SGs); on the concrete using 50 mm glued SGs, on the rein-forcement using 10 mm welded SGs, and on the CFRPusing 10 mm glued SGs, see Fig. 5. A typical setup of theinstrumentation for a slab with an opening is shown inFig. 6.

3.3. Test specimens

A total number of 11 slabs were manufactured in fourbatches with a designed 28 days characteristic compressivestrength of fcck = 40 MPa. Nine cubes (150 150 150 mm) were cast for each batch to measure the compres-sive strength at 28 days, and the compressive and splittingstrength at the time of testing. The concrete surfaces of theslabs to be strengthened with CFRP were sandblasted andcleaned properly with compressed air before bonding.

All slabs are reinforced with welded steel fabric, Nps 50B 5 s 150, using a concrete cover of 20 mm. Two rein-

forcement bars with the same nominal characteristic yield

Fig. 2. Test setup with slab placed on the bottom structure with airbags.Line support structure placed on top of the slab. The support structure isconnected to the bottom structure through a load cell in each corner.

Fig. 3. System of airbags inside the embedding structure. An ordinary aircompressor is used to fill up one of the airbags. The air can then circulatefreely in the system through valves connecting the airbags (one of the

airbags is removed in the figure).

Displacement

transducers

Strain gauges

on concrete

Strain gauges

on steel bars

Strain gauges

on CFRP

Fig. 4. Instrumentation of the slabs depending on size of openings (no,small or large). The numbers designate the location lines and the letter the

direction of action in the slabs plane.

812 O. Enochsson et al. / Construction and Building Materials 21 (2007) 810826


4/17

strength fyk = 510 MPa, are added in the slabs with castopenings placed in 45 angle as shown in Fig. 7a. A sampleset of three individual steel bars from the welded fabrichave been tested to evaluate the tensile strength at the0.2%-limit f0.2, as is normal for cold worked steel.

The homogeneous reference slab is designed using char-acteristic material properties according to the Swedish con-crete code, BBK 04 (2004), for a uniformly distributed loadof 15 kN/m2.

The required amount of steel reinforcement is calculatedfrom design moments according to the standard method[17], see also next section. The other test specimens aredesigned to be comparable with the homogeneous slab.In case of CFRP strengthened slabs, the amount of appliedCFRP is calculated from the required steel reinforcement

according to a simplified method. Minimum amount of

steel reinforcement for cracking due to shrinkage and tem-perature changes is omitted in the design. The complete test

program is given in Table 1 and Fig. 8.

Fig. 5. Measurement gauges: (a) LVDT, (b) glued 50 mm SG for concrete,(c) glued 10 mm SG for CFRP, and (d) welded 10 mm SG for steelreinforcement.

Fig. 6. A typical setup of the instrumentation for testing a slab with anopening (photo is from an earlier pilot study).

c-90,45c-90c-45

s-90,45s-90s-45

b

a

Fig. 7. Three different arrangements to strengthen a slab with (a)additional steel reinforcement due to a cast opening, or (b) CFRP dueto a sawn-up opening.

Table 1Designation and description of the specimens

Designation Description

H Homogeneous slab: Reference slab traditionally reinforcedSw Slab weakened by a sawn-up small opening (0.85 0.85 m)Ss-90 Slab with a steel strengthened small opening: Cast with a

small opening traditionally steel reinforced along theopening (investigated only numerically)

Ss-45 Slab with a small opening strengthened in corners withsteel reinforcement: Reference slab with a cast smallopening, steel reinforced in corners with 45

Sc-90 Slab with a CFRP strengthened small opening: Slab with asawn-up small opening strengthened with CFRP sheets

along the openingSc-45 Slab with a small opening strengthened in corners with

CFRP: Slab with a sawn-up small opening strengthenedwith CFRP sheets at the openings corners with 45

Sc-45, 90 Slab with a CFRP strengthened small opening: Slab with asawn-up small opening strengthened with CFRP sheetsboth at the openings corners and along its edges

Lw Slab weakened by a sawn-up large opening (1.20 1.20 m)Ls-45 Slab with a large opening strengthened in corners with

steel reinforcement: Reference slab with a cast largeopening, steel reinforced in corners with 45

Lc-90 Slab with a CFRP strengthened large opening: Slab with asawn-up large opening strengthened with CFRP sheetsalong the opening

Lc-45 Slab with a large opening strengthened in corners with

CFRP: Slab with a sawn-up large opening strengthenedwith CFRP sheets at the openings corners with 45

Lc-45, 90 Slab with a CFRP strengthened small opening: Slab with asawn-up large opening strengthened with CFRP sheetsboth at the openings corners and along its edges

H = homogeneous slab, S = small opening (0.85 0.85 m), L = largeopening (1.20 1.20 m), w = weakened, s = steel reinforced, c = CFRPstrengthened, 45 = strengthening applied in corners with 45,90 = strengthening applied in 2-directions orthogonally along the opening,and 45, 90 = a combination of the both former configurations ofstrengthening, see also Figs. 7 and 8.



5/17

Square openings of two different sizes (0.85 0.85 and1.2 1.2 m) were sawn-up in centre of each slab using amobile concrete wet saw. In order to avoid initiation of

cracks in the corners during the tests, a hole (B 70) was firstdrilled out at each corner as a guide prior to the sawing.The sizes of the openings were chosen to be slightly larger

and larger than the limit defined in BBK 04 for a smallopening (1/3 2.4 = 0.8 m < 0.85 and 1.2 m) in order toinvestigate their affect, see next section.

The concrete splitting and compressive strengths, shownin Table 2, are the mean values of three test cubes. The ten-sile strength was evaluated from the splitting strength

SmallWithout Large

Homogeneous (without opening).Designation H

Weakened by a sawn up opening.Designation S/Lw

Strengthened with steel bars in corners with 45.Designation S/Ls-45

Strengthened with CFRP sheets in corners with 45.Designation S/Lc-45

Strengthened with CFRP sheets along edges.Designation S/Lc-90

Strengthened with CFRP sheets along edges and in corners.Designation S/Lc-45,90

S L

S L

S L

S L

S L

Fig. 8. Experimental program and designation of specimens. An opening drawn with solid lines is cast, and one with dashed lines is sawn-up.

Table 2Average concrete strengths from splitting and compressive tests of three cubes and surface shear strength from six torque tests

Slab Cast batch Date forcasting

Date forcube test

Splittingstrength (kN)

Tensile strength(MPa)

Compressivestrength (MPa)

Shear strength(MPa)

H, Sw 1 22/09/2003 20/10/2004 3.95 3.16 46.5 S/Ls-45 2 06/10/2003 15/06/2004 3.90a 3.12a 55.3a 6.2S/Lc-45, S/Lc-90 3 14/10/2003 02/07/2004 4.70a 3.76a 56.3a 5.4Lw 4 24/10/2003 21/11/2003 50.6 7.2S/Lc-45, 90 5 04/11/2003 14/12/2004 4.54a 3.63a 59.0a 8.0

a Performed close to time for experimental test.



6/17

according to BBK 04 (0.8 of the splitting strength). In addi-tion, torque tests were conducted on sawn-out slab partsfrom every batch that included any specimen to bestrengthened with CFRP, this to evaluate the surface shearstrength of the concrete.

The slabs with a sawn-up opening are strengthened

using StoFRP sheet, of two different types: 200 g/m2

and300 g/m2. Table 3 gives the nominal material propertiesof the CFRP sheets, and the length and width of theapplied CFRP are shown in Table 4. A typical setup ofthe CFRP sheets applied along an opening is shown inFig. 9, and the material properties of the used primer andadhesive are shown in Table 5.

3.4. Test procedure

The airbag was carefully arranged to give a well-distrib-uted uniform load during the whole test. A thin protectivelayer of polyester fabric was placed between the airbag andthe concrete surface to protect the airbag at slab failure.Special attention to the contact surfaces to achieve an evendistribution of the reaction forces along the supports wasgiven. An air compressor was used to fill the airbag withair at an approximate loading rate of 40 N/s (417 Pa/min

for the homogeneous reference slab). The load, displace-ment and strains were continuously measured and recordedby a computerized acquisition system until failureoccurred. Both the crack propagation and the crack distri-bution were observed and registered throughout the test.

4. Design and analytical methods

4.1. Background to the Swedish design methods

Classical analytical methods are based on the theory ofelasticity, see [18]. Concrete slabs have a capacity to redis-

tribute high moment concentrations by cracking and bylocal yielding of the reinforcement. This is taken advantageof in the yield line theory, see e.g. [1921]. The yield linetheory gives an upper bound to the load bearing capacityand may over estimate it, if a too simple or optimistic yieldline pattern is assumed. A lower bound to the capacity canbe found with the strip method, see e.g. [22]. In Sweden, astandard method has been developed for slab design [17].The method is originally based on the theory of elasticity,but to get a more economical design of steel reinforcementthe method has been modified to take into account theyield line theory. The method gives the maximum momentm as a simple formula m = aqb2, where q is the distributedload and a is a tabulated coefficient depending on the sup-port conditions and the ratio between the length a and thewidth b of the slab.

4.2. Design method

In Sweden, a floor structure, in case of any opening, isdesigned in two different ways depending on the size ofthe opening in relation to the geometry of the slab. Entryholes for electrical or pipe installations are normally notdefined as an opening. In slabs, subjected to a uniformlydistributed load, a sectional opening with a length of max-

imally 1/3 of the shortest slab span is defined as small in

Table 3Nominal material properties of CFRP sheet

Product Sheetthickness

(mm)

Youngsmodulus

(GPa)

Tensilestrength

(MPa)

Rupturestrain

(%)300Sa 0.17 228 3600 1.5200Sa 0.11 228 3600 1.5

a Number denotes the mass per cross-sectional area (g/m2) a n d Squalifies high strength.

Table 4Location, width and length of the applied CFRP sheets

Slab Length of opening(m)

Typeof sheet

Location inrelation toedges ofopening

Width(mm)

Length(m)

Sc-45 0.85 200S In corner (45) 195 0.85Sc-90 0.85 300S Along (0/90) 185 2.30Sc-45, 90 0.85 200S In corner (45) 195 0.85

300S Along (0/90) 185 2.30

Lc-45 1.20 200S In corner (45) 195 1.20Lc-90 1.20 300S Along (0/90) 240 2.30Lc-45, 90 1.20 200S In corner (45) 195 1.20

300S Along (0/90) 240 2.30

Fig. 9. Slab strengthened with CFRP sheets applied along a sawn-up

small opening.

Table 5Nominal material properties of primer and adhesive

Product Adhesit ivityto concrete(MPa)

Youngsmodulus(GPa)

Tensilestrength(MPa)

Shearstrength(MPa)

Primer 17 Adhesive 2 50 17.6



7/17

BBK 04, otherwise as a large opening. In the latter case, theedges of the large opening are considered as free edges, i.e.the moments acting in the same direction as the edge areredistributed to be more concentrated closer to the open-ing. In the former case, the slab is first designed as a slabwithout an opening, i.e. the moments and shear forces

are calculated as the opening does not exist. The momentand shear forces that would pass each half of the edge ofthe opening is added to existing moment and shear forces,along the closest edge of the opening within a band, that ismaximum 3 times wider than the slab thickness. The rein-forcement is given at least the same length, as it would havehad if the opening had not existed. In Fig. 10, two arrange-ments of additional reinforcement due to a small openingare illustrated, one according to BBK 04 and one accordingto a configuration tested in this study.

4.3. Simplified method

A simplified method to estimate the necessary amount ofCFRP for a structure is to calculate the required steel rein-forcement designed with traditional methods and convert itto CFRP. The calculation is made by accounting for thedifference in the stiffness between the cross-sectional areasof the CFRP and the steel. A more theoretically correctmethod, especially for slabs with openings, is to accountalso for the differences in the lever arms and the effectivewidths of the compression zone, see Fig. 11.

BBK 04 design

Tested design

Introduced opening

in a slab

+

+

Fig. 10. Corresponding methods to reinforce a slab due to a cast smallopening according to BBK 04 (2004) and a tested configuration in thestudy.

Md

bs2

0.8x

fcc

fst

Md

bf

As1

Af

h d

u

x

Ff

Fc0.8x

fcc

ff

h d

u

x

Cross section of a RC slab strengthened with CFRP Afnear an opening.

a

b

Cross section of a RC slab manufactured with additional reinforcement bars As2 near an opening.

NL

cu

f

As1 +As2Opening

cu

s

NLFc

Fs2

0.4x

0.4x

Opening

Fig. 11. Relationships between strain, stresses and internal forces in a cross-section of a RC slab near an opening strengthened with (a) additional steelreinforcement or (b) CFRP. Effective width of compression zone is denoted b, subscripts s2 and f denote additional steel reinforcement and CFRP,

respectively.



8/17

The strengthening effect should be equal between thecross-section with the additional steel reinforcement andthe CFRP strengthened cross-section. Hence, the momentcapacity must also be equal:

Mf Ms 1

where Mf is the moment capacity for the slab with CFRPand Ms is the moment capacity for the slab with additionalsteel reinforcement. To calculate the necessary sectionalarea of CFRP Af, Eq. (1) is expressed as the product ofthe sectional force F and the lever arm:

Fs2d x Ffh x 2

where the subscript s2 is the part of the steel reinforcementbars that will be replaced with the CFRP reinforcement(subscript f). Expressing forces as products of stressesand sectional areas yield:

rs2As2d x rfAfh x 3

Applying Hookes law gives the sectional area of the fibreas

Af Es2esd x

Efefh xAs2 4

The relation between the strain in the steel and CFRP canbe evaluated using Bernoullis hypothesis in Fig. 11, as

ef h x

d xes 5

If perfect bond between the concrete and the reinforcementis assumed, the expression for the sectional area of CFRPbecomes only dependent on the level arms and the elastic

modulus of the steel and the CFRP reinforcement:

Af Es2

Ef

h u x

h x

2As2 6

where d= h u is the effective height and u is the distancefrom the bottom tensile side to the centre of gravity of theadditional steel bars.

The differences in the effective width of the compressionzone b between the CFRP and the substituted additionalsteel reinforcement can be iteratively determined. In thefirst step, the width of the contributing compression zonein the CFRP design bf0 is set equal to bs2 and a new value

ofbf is calculated. The iteration continues until the two val-ues coincide. However, if EfP Es2, bf= bs2 can be useddirectly in the design since it is on the safe side.

5. Numerical analysis

5.1. General

Numerical analysis of the behaviour of two-way con-crete slabs includes several nonlinear considerations, i.e.material, boundary conditions and geometry. Various con-cepts for describing the quasi-brittle mechanisms in rein-

forced concrete have been introduced in the FEM. These

are well known concepts such as discrete crack andsmeared crack approaches, or less used models such asinner softening bands [23]. In this paper, a damaged plas-ticity model is used for the concrete. The steel reinforce-ment is modelled as ideal-elastoplastic and the CFRP aslinear elastic material until failure. Equally important in

the numerical analysis are the boundary conditions. Thisis especially true in this study since the support conditionsfor the concrete slab cannot be prescribed. The analysismust include contact interactions to allow separationbetween the slab and the support.

Numerical analysis of reinforced concrete structures iscustomarily performed by static implicit FE solvers wherethe integration scheme is for example full NewtonRaph-son. The solution is obtained from equilibrium iterationsminimizing the error of the solution. The outcome is a reli-able and stable solution. But this solving technique canhave convergence problems in models that have a largedegree of non-linearity, such as the two-way slabs in this

paper. Apart from the constitutive models that are nonlin-ear, the support conditions must be handled by contactinteractions. This adds complexity to the system that theimplicit scheme is not able to take care of. An optional sol-ver for these kinds of problems is a FE-program with anexplicit time integration scheme. It is normally used fordynamic problems but can also be applied in static prob-lems. Particularly, it is an efficient solver of contactproblems.

Numerical simulations by using the FEM with explicittime integration can be very costly in terms of computertime if not certain adjustments are made. This comprises

of either decreasing the simulation time or increasing thedensity of the material. These two remedies perform essen-tially the same thing; reducing the number of time steps inthe global time integration. The number of time steps in theintegration is set by the inherent critical time step, which isusually governed by element dimensions, the density andthe dilatational wave speed of the material. The criticaltime step is normally very small, inducing a great numberof time steps to be completed. Since the number of timesteps is almost directly proportional to the computer time,it is desirable to decrease this number. Reducing the num-ber of time steps must be done very carefully, otherwise thesimulation becomes unstable and the solution is not reli-able. Care must also be taken in order to ensure that theinertia effects are kept within acceptable limits. For thismodel, an increase in the material density by a factor of100, which means a decrease of computer time by approx-imately 10, does not have an apparent effect on theresponse of the model.

The slab is modelled by eight-node brick elements withreduced integration, the reinforcement in the concrete isrepresented by discrete truss elements, steel support platesare modelled by shell elements with reduced integration,and the CFRP sheets as membrane elements with no stiff-ness perpendicular to the fibres. The test rig is not a part

of the FE model, i.e. the deformations in the test rig are



9/17

not taken into account. The FE model of the slab strength-ened with CFRP along a sawn-up opening is shown inFig. 12. Similar meshes are used for the other models.

Quasi-brittle materials, e.g. concrete, usually experiencea sudden drop in the load carrying capacity during crack-ing and it generally leads to increases in the kinetic energy

content of the response. In the explicit solution of the loaddisplacement response, oscillations will appear due to theseinertia effects after the concrete has cracked significantly.Curve smoothing is used to average the outcome of theFE analysis.

5.2. Material models

5.2.1. Concrete

Typical behaviour of concrete during the growth ofmicro-cracks comprises of strain softening, progressivedeterioration, volumetric dilatancy, and induced anisot-ropy. This complex response must be translated into phe-

nomena, which can be described by continuummechanics. It can be considered as a combination of unre-coverable plastic deformation, degradation of materialstiffness, and the initiation, development, and interactionof defects. The plastic deformation and strain softeningcan be described by using classical plasticity theory. Con-tinuum damage mechanics is used to model the stiffnessdegradation. Essentially, this means that the cracking ofthe concrete reduces the stiffness, e.g. the modulus of elas-ticity. The crack opening and closure is determined by frac-ture mechanics and plasticity.

The constitutive model used in this paper for the con-

crete is the damaged plasticity model included in ABAQUS[24]. It is based on the work by Lubliner et al. [25] and Leeand Fenves [26]. The evolution of the yield surface is con-trolled by two hardening variables, one in tension and onein compression. Non-associated flow is assumed where theflow potential gis the DruckerPrager hyperbolic function.For these functions, a couple of parameters must bedefined; the dilation angle w and the eccentricity e for theflow potential are set to w = 12 and e = 0.1. For the yieldfunction, the ratio of initial equibiaxial compressivestrength to uniaxial compressive strength rb0/rc0, and the

ratio of the second stress invariant on the tensile meridianto that on the compressive meridian at initial yield for anygiven value of the first stress invariant such that the maxi-mum principal stress is negative Kc the default values inABAQUS are used, i.e. 1.16 and 2/3, respectively.

The concrete behaviour in tension is linear elastic until

cracking is initiated and a strain softening response isassumed in the post-failure region. The post-failure behav-iour is specified in terms of the stressdisplacementresponse in order to minimize mesh sensitivity. It definesthe tension softening behaviour and is described here bya bi-linear curve, see Fig. 13. Similarly, the tensile damageis specified by an assumed linear relationship between thetension damage variable dt and the crack opening d. Themaximum value of the damage variable dt0 is set to 0.9,and the maximum crack opening d0 is set to 0.115 mm,see Fig. 13. The facture energy Gf for mode I is the areaunder the softening curve and is estimated to 100 N/m.

In compression, the concrete behaviour is linear elastic

until initial yield stress rc0 is reached. The material entersthe plastic regime with a strain hardening before the ulti-mate compressive stress (strength) rcu, followed by strainsoftening. The initial yield stress for the concrete in thispaper is assumed to be 60% of the ultimate stress and thetypical plastic strain at ultimate stress is 0.2%. In this anal-ysis, the nonlinear part of the constitutive model in com-pression is somewhat unnecessary since the initialcompressive yield stress will not be exceeded. The damageevaluation in compression is omitted since crushing doesnot occur.

5.2.2. SteelThe constitutive model for steel is assumed to be ideal

elasto-plastic. Full bond between steel and concrete isassumed and the tension stiffening effect due to reinforce-ment is not accounted for in the solution.

The steel support plates are considered to behave in alinearly elastic manner.

5.2.3. Carbon fibre reinforced polymer, CFRP

The CFRP-material is considered as linear elastic untilfailure. The interaction between the concrete slab and the

Fig. 12. FE model used to analyse the CFRP strengthened slab, Sc-90.

Fig. 13. Tension softening modelled by a bi-linear relationship andrelationship between the tension damage variable, dt and the crack

opening, d.



10/17

CFRP is modelled without considering debonding. Thedominating failure of the strengthened slabs in the experi-mental tests is rupture of the CFRP, see e.g. Fig. 17, i.e.the bond does not have to be explicitly modelled in thenumerical analysis.

5.3. Boundary conditions

Apparent symmetry in the model has been utilizedresulting in a FE-model using only a quarter of the struc-ture. Although, the behaviour after the first crack in theconcrete dissolves this symmetry, it has been assumed thata quarter is adequate to give the response of the wholestructure with sufficient accuracy.

The supports for the concrete slab cannot be modelledby using simple prescribed boundary conditions. This fol-lows from the uplift of the corners of the slab. The support-ing steel plates are included in the model and boundaryconditions are prescribed for these. Contact interaction is

introduced between the slab and the supports and the for-mulation is penalty based. The contact interaction isdefined by the tangential behaviour, which in this case isset to be rough. This means an infinite coefficient of frictionand no slip between the slab and the support plates.

In establishing correct boundary conditions in thenumerical analysis, the geometrical imperfections of thespecimens and the supporting frame recognised in theexperiments needed to be investigated. Uneven concretesurface was smoothed by means of a plaster layer andremaining gaps were shimmed with thin steel plates, seeFig. 14. However, this solved the problem only partially.

The best contact between a specimen and the supportingstructure was always reached in the corners. Although plas-ter and shims provided some support along the slab edges,it still allowed deformations. It can be assumed that thespecimen was supported stiffly in the corners and elasticallyalong the edges. This can be modelled as a set of discretesprings, proposed by Rusinowski [27], or continually byapplying simply supported deformable plates along slabedges. The latter method is used here and in Fig. 15 theexperimental deflection curves for a slab strengthened with

CFRP along the opening (Sc-90) is compared with thenumerical analyses using stiff and elastic supports. Theanalysis with elastic supports shows good agreement withthe experiment in the elastic region but introduces dynamicinstability in the explicit solution later on. This instabilitycan cause inaccuracies and introduce numerical damagein the concrete model. Therefore, only the model with thestiff line supports is used.

6. Experimental results

The general response during all tests is similar. As the

load is increased, the better contact between the supportand the slab is developed until a load of approximately8 kN/m2 is reached, where after a sudden jump appears,see Fig. 16. This jump is believed to depend on the adap-tion of the bolted joints in the test rig during the loading.After this occurrence, the first crack was observed. Forthe homogeneous slab, the cracking started in the middleof the slab, in contrast to the other slabs where the firstcrack appeared diagonally near a corner due to the open-ings. The load levels at the first observed crack are shownin Table 6. However, the point where the first cracks arenormally observed in the loaddeflection relationships forbeams, does not occur for two-way concrete slabs. Thisdepends probably on the existing membrane effect in two-way RC slabs.

The crack propagation and crack size continued toincrease with the increase of loading. For the CFRPstrengthened slabs, the cracks were narrower and morewidely spread in comparison to the cracks in the steel rein-forced slabs.

In Fig. 16, the loaddeflection relationships until failure,i.e. collapse for all experimentally tested slabs are shown.All slabs strengthened due to openings show, indepen-dently of the size of the opening, a considerably higher loadcarrying capacity than the homogeneous slab (H) especially

for the slabs strengthened with CFRP (Sc and Lc). ThisFig. 14. Example of support conditions indicating an uneven line support.

0 10 20 30 40 50 60

Deflection [mm]

0

10

20

30

40

50

60

Load[kN/m2]

Experiment

Stiff support (FEM)

Elastic support (FEM)

Elastic support

Stiff support

Fig. 15. Loaddeflection relationships from the experiment and the twoanalysed FE models, one with elastic supports and one with stiff supports.The results are compared at the midpoint close to the opening. Slabstrengthened with CFRP along a small opening, Sc-90. The results for theFE-analyses are truncated at 55 mm.



11/17

depends mainly on the existing openings, and accordingly adecreased total available loading area gives a lower total

load. A similar result is seen for the slabs strengthened onlywith additional steel reinforcement in the corners (Ss-45and Ls-45). The slab strengthened with additional steelreinforcement due to a sawn-up large opening (Ls-45) showhigher load carrying capacity and lower deflection at fail-ure than the homogeneous slab (H). A probable reasonfor this is the larger lever arm for the reinforcement placedin the corners.

The slabs weakened by a sawn-up opening (Sw and Lw)show surprisingly a similar loaddeflection behaviour asthe homogeneous slab, except that the deflection at failureis lower.

0 10 20 30 40 50 60 70

Deflection [mm]

0

10

20

30

40

50

60

70

80

0 10 20 30 40 50 60 70

Deflection [mm]

0

10

20

30

40

50

60

70

80

Load[kN/m2]

Sc-45

Sw H

Ss-45

Sc-90

Sc-90,45Lc-45

Lw H

Ls-45

Lc-90

Lc-90,45

a b

Design load 15 kN/m2Design load 15 kN/m2

Fig. 16. Loaddeflection relationships at the midpoint close to the opening, for slabs with (a) a small opening and (b) a large opening. The load capacitiesof the tested slabs are compared with the design load of 15 kN/m2 for a slab without an opening (H).

Table 6Crack and failure load compared to the design load (15 kN/m2)

Slab qcrack (kN/m2) qfailure (kN/m

2) qcrackqdesign

qfailureqdesign

H 7 36.1 0.5 2.41

Sw 20 35.6 1.3 2.37Ss-45 18 41.2 1.2 2.75Sc-45 27 44.2 1.8 2.95Sc-90 34 48.3 2.2 3.22Sc-45, 90 30 51.1 2.0 3.41

Lw 26 34.2 1.7 2.28Ls-45 18 48.0 1.2 3.20Lc-45 32 51.1 2.1 3.41Lc-90 41 57.4 2.7 3.83Lc-45, 90 41 76.8 2.7 5.12

Fig. 17. Crack patterns in a corner of a slab at failure for different configurations of CFRP sheets: (a) along a small opening (Sc-90), (b) in corners of asmall opening (Sc-45), (c) along and in the corners of a small opening (Sc-45, 90), (d) along a large opening (Lc-90), (e) in corners of a large opening (Lc-

45), and (f) along and in the corners of a large opening (Lc-45, 90). The propagation of cracks is governed by the configuration of CFRP sheets.



12/17

All the slabs with openings strengthened with CFRPshows similar loaddeflection behaviour up to failure.However, the failure mode differs in comparison with thetraditional steel reinforced slabs. The steel reinforced slabsshow a more ductile response meanwhile the failure modefor the CFRP strengthened slabs where somewhat more

brittle. The mode of failure is also different, whereas yield-ing followed by large deflections was the dominating failurefor the steel reinforced slabs, the mode of failure for theCFRP strengthened slabs is rupture in the CFRP.However, the rupture is preceded by steel yielding. Theload carrying capacity at failure is considerable larger forthe CFRP strengthened slabs.

The loaddeflection relationship for the CFRP strength-ened slabs needs an explanation. S/Lc-45 are strengthenedwith CFRP sheets only in the corners at 45. S/Lc-90 arestrengthened with CFRP sheets along the edges of theopening. Finally, S/Lc-45, 90 are strengthened both inthe corners and along the edges, see also Fig. 8 and Table

1. Even though all CFRP strengthened slabs failed by fibrerupture, the load carrying capacity and the deflections at

failure varied considerable between the different slabs.The most probable reason for this is the configurationsof the fibre sheets. The highest load carrying capacity isobserved for S/Lc-45, 90 where the crack propagation ishindered by three layers of CFRP sheets. This is illustratedin Fig. 17.

There is a noticeable difference of the load carryingcapacity between the CFRP strengthened small and largeopening where the latter slab configurations carried 2050% higher loads. The main reason for this is most likelythat the behaviour of the slabs with a large opening is clo-ser to a system of beams than a slab.

7. Comparison

The result from the numerical analysis is compared withthe outcome from the experiments in Figs. 1824. The com-parison of deflections in Fig. 18 and the development ofsteel strains show that the global behaviour of the three

analysed slabs in the numeric analysis can reproduce theexperimental results. The main difference between the

0 10 20 30 40 50 60

Deflection [mm]

0

10

20

30

40

50

60

0 10 20 30 40 50 60

Deflection [mm]

0

10

20

30

40

50

60

Load[kN/m2]

SwH

Sc-90

SwH

Ss-90

Sc-90

a b

Fig. 18. Comparison of loaddeflection relationships between results from (a) the experiment and (b) the FE-analyses at the midpoint close to the opening.Slabs with a small opening. The results for the FE-analyses, Sc-90 and the tested, H are truncated at 55 mm.

0 1000 2000 3000 4000

0

10

20

30

40

50

60

Load[kN/m

2]

0 1000 2000 3000 4000

Strain [m/m]Strain [m/m]

0

10

20

30

40

50

60

SwH

Sc-90

Sw

H

Sc-90

a b

In middle

of opening

In corner

of opening

Sw

H

Sc-90

Sc-90

Ss-90

Sw

H

Ss-90

In middle

of opening

In corner

of opening

Fig. 19. Loadstrain relationship in the steel reinforcement along the opening (i.e. in x-direction), for (a) the experiment and (b) the FE-analysis. The levelof strains in the middle and in the corner of a small opening is compared to each other. The final load levels are somewhat to low in figure (a), for the testedslabs H and Sw in the corner (the measuring range at these tests were set too low). The results from the FE-analyses, Sc-90, Ss-90, Sw and H, at the

location in the corner, are truncated to make the comparison clearer.



13/17

observed behaviour and the numerical analysis is found inthe elastic region and at the beginning of the nonlinear

behaviour. This difference is believed to be a consequenceof the boundary conditions and the models inability tomodel the crack propagation in a proper way. Isotropicdamage models are known to give a more brittle behaviourcompared to other softening constitutive models, due tothe stiffness degradation in all directions [23]. The brittle-ness is manifested in the sudden drop directly after theonset of the nonlinear behaviour for the three slabs thatare not strengthened with CFRP, see Fig. 18b. This dropis not present in the modelled slab strengthened withCFRP, or in the reality.

Figs. 19 and 20 show that the strain development in the

CFRP sheet was captured by the numerical model. Still,the failure of the CFRP could not be reproduced in thenumerical analysis. The theoretical strain limit of theCFRP sheet was never reached in the FE analysis. Thiscould be an effect of the biaxial state of stress that theCFRP is exposed to reducing the strain limit of the sheet.

0 1000 2000 3000 4000 5000 6000 7000 8000

Strain [m/m]Strain [m/m]

0

10

20

30

40

50

60

0 1000 2000 3000 4000 5000 6000 7000 8000

0

10

20

30

40

50

60

L

oad[kN/m

2]

a b

Fig. 20. Loadstrain relationships in the CFRP from (a) the experiment and (b) the FE-analysis. The strains are compared along the opening at threedifferent locations; in the middle, at the corner of the opening, and between the middle and the corner of the opening. Slab strengthened with CFRP due toa sawn-up small opening, Sc-90. The results from the FE-analysis are truncated to make the comparison clearer.

0 0.2 0.4 0.6 0.8 1 1.2

Distance, x [m]

0

50

100

150

200

250

300

350

Strain[m/m]

FEM 10 kN/m2

FEM 13.9 kN/m2

FEM 35 kN/m2

Exp 10 kN/m2

Exp 13.9 kN/m2

Exp 33.5 kN/m2

Cracking

Elastic

Plastic

Fig. 21. Strain profiles at three selected stages in the CFRP along theopening at a distance, x from the middle of the opening. The test results(discrete points) are compared to the FE analyses (curves). Slab with asawn-up small opening strengthened with CFRP, Sc-90. The stages areselected at load levels in the elastic area, just at the cracking and when justreaching the plastic area.

-400 -300 -200 -100 0 100 200 300 400

Strain [m/m]

0

20

40

60

80

100

Depth [mm]

Elastic 10 kN/m2

Cracking 12.8 kN/m2

Plastic 27.5 kN/m2

-400 -300 -200 -100 0 100 200 300 400

Strain [m/m]

0

20

40

60

80

100

Depth [mm]

Elastic 10 kN/m2

Cracking 12.8 kN/m2

Plastic 31.5 kN/m2a b

Concrete

Steel

Concrete

Compression

Tension

Fig. 22. Strain distribution ex

through the slab thickness in a corner of the opening for (a) the experiment and (b) the FE analysis of the slab weakened bya small sawn-up opening, Sw. The strains are measured at three selected stages as discrete values in the concrete on the compression side, in the steel and in

the concrete on the tension side, respectively. The experiments show higher strain due to more localised yielding in the steel.



14/17

However, since the strain in the sheet is concentrated towhere the cracks are localised it is difficult to measure theultimate failure strain. The strain gauge must be locatedexactly over the area where the failure occurs.

Fig. 21 shows the development of the strain distributionalong one of the CFRP sheet. The strain distribution indi-rectly shows the development of bond shear stress alongthe sheet. Normally, the rule of thumb states that the bonddevelopment length for CFRP reinforcement needs not tobe greater than 0.2 m [28]. This is clearly the case in thenumerical model and the proof can be found from the factthat the failure is in rupture of the CFRP sheet and not indebonding.

The strain distribution over the cross-section is shown in

Figs. 2224. The Bernoulli hypothesis is clearly valid. The

strains measured in the tests are larger than the strainsfrom the numerical analysis. This is especially apparentin the non-strengthened slab. This is believed to be a con-sequence of the difference in crack localisation betweenthe experiments and the numerical analyses. In the experi-ments, a few localised cracks appear in contrast to thenumerical analyses where a region of cracks is produced.In the CFRP strengthened slab, the cracks are more evenlydistributed resulting in a smoother strain field due to theCFRP. Therefore, a better correspondence is foundbetween the numerical analysis and the experimental result.

This is somewhat reflected in the numerical analysis of thestrain localisation, see Figs. 2527. The figures show thedistribution of the plastic strains at the final load step forthe three analysed slabs, in comparison to the crack pat-

a b

-400 -300 -200 -100 0 100 200 300 400

Strain [m/m]

0

20

40

60

80

100

Depth [mm]

Elastic 10 kN/m2

Cracking 13.9 kN/m2

Plastic 33.5 kN/m2

-400 -300 -200 -100 0 100 200 300 400

Strain [m/m]

0

20

40

60

80

100

Depth [mm]

Elastic 10 kN/m2

Cracking 13.9 kN/m2

Plastic 35 kN/m2

Concrete

Steel

Carbon

Compression

Tension


through the slab thickness in a corner of the opening for (a) the experiment and (b) the FE analysis of the slab strengthenedwith CFRP along a sawn-up small opening, Sc-90. The strains are measured at three selected stages as discrete values in the concrete on the compressionside, in the steel and in the CFRP on the tension side, respectively.

-400 -300 -200 -100 0 100 200 300 400

Strain [m/m]

0

20

40

60

80

100

Depth [mm]

Elastic 10 kN/m2

Cracking 13.6 kN/m2

Plastic 32.5 kN/m2

Concrete

Steel

Concrete

Compression

Tension


through the slab thickness in a corner of theopening, for the FE-analysis of a slab strengthened with additionalreinforcement along a cast small opening, Ss-90. The strains are measuredat three selected stages as discrete values in the concrete on thecompression side, in the steel and in the concrete on the tension side,respectively.

Fig. 25. Comparison between (a) the maximum principal strain achievedin the FE analysis and (b) the final propagation of cracks in theexperiment. Results are shown for a quarter of a slab weakened by a sawn-

up small opening, Sw.



15/17


16/17

model of the reinforcement including the stiffening effect ofthe surrounding concrete will probably give better result.Also, it is always difficult to compare analytical and exper-imental strain readings since the level of strain is highlydependent of the crack localisation in quasi-brittle materi-als like concrete. Apart from these observations, the expli-

cit FE analysis is showing a good agreement with the testand is giving more insight to the strengthening effect ofthe CFRP reinforcement. It is also giving more confidenceto the future utilization of the FE analysis in order to eval-uate larger slabs with different opening configurations.

9. Conclusion

The work presented in this paper shows that CFRPsheets can be used to maintain and even increase the origi-nal load-capacity of two-way concrete slabs with openings.The test results clearly show that the investigated strength-ening system can be used to strengthen existing slabs with

made openings, and even that the load carrying capacitycan be increased when compared to the homogeneous slab(H). For the CFRP strengthened slabs, the load carryingcapacity was increased with 24125% in comparison torespective weakened slab (S/Lw), and with 22110% incomparison to the homogeneous slab.

The general result of the experimental investigation isthat the method to design the required amount of steel rein-forcement due to an opening gives a load carrying capacityon the safe side. The slabs with the larger openings have anoticeable higher load carrying capacity and a stifferloaddeflection response than the slabs with the smaller

openings. This is in contradiction to the proposed designmethod. The reason for this can be that the slabs with thelarge openings behave closer to a system of four beams thana slab. Furthermore, the study shows in particular that:

The traditional way to reinforce additionally with steelbars along an opening gives a higher load capacity thana slab without an opening.

The assumptions made in the simplified method for cal-culating the amount of CFRP are found to be valid.

The numerical analysis showed good agreement with theexperimental result, especially for the CFRP strength-ened slab.

Finally, more advanced design methods should lead to amore efficient use of the CFRP sheets in strengtheningdesign, especially if the goal is to achieve a load carryingcapacity equal to, or slightly larger than the load action.

10. Further research

FE analysis will be used to study different configurationsof openings in larger slabs. This can also be used to inves-tigate how the design of CFRP strengthening should bemade to avoid bond failure. The difference between the

experimental and the numerical strains in the CFRP sheet

needs also to be investigated more thoroughly since it gov-erns the ultimate failure of the slab. Is the difference a resultof inadequate ability to model the crack localisation or dothe biaxial strain field lower the theoretical rupture strainin the CFRP sheets? From these types of analyses, betteranalytical design methods can be derived.

Acknowledgements

This study has been sponsored mainly by SKANSKAand SBUF (The Development Fund of the Swedish Con-struction Industry), and partly by the European Union re-gional funds and Sto Scandinavia AB. Anders Ericsson andTobias Larsson are greatly acknowledged for the initialwork during the pilot study while performing their Mas-ters thesis [29].

The work by Hakan Johansson, Lars Astrom, Hans-Olov Johansson and Georg Danielsson at Testlab, Lulea

University of Technology, in discussion and preparationof the test setup is also appreciated.

References

[1] Taljsten B. Plate bonding, strengthening of existing concrete struc-tures with epoxy bonded plates of steel or fibre reinforced plastics,Doctoral Thesis 1994:152D, Division of Structural Engineering,Lulea University of Technology; 1994, ISSN 0348-8373. p. 308.

[2] Taljsten B. Strengthening of beams by plate bonding. J Mater CivilEng 1997(November):20612.

[3] Triantafillou TC. Shear strengthening of reinforced concrete beamsusing epoxy-bonded FRP composites. ACI Struct J 1998;95(2):10715.

[4] Taljsten B. Forstarkning av betongkonstruktioner med stalplat ochavancerade kompositmaterial utsatta for vridning (Strengthening ofconcrete structures with steel plates and advanced composites fortorsion). Research report 1998:01, Lulea University of Technology,Division of Structural Engineering, Department of Civil & MiningEngineering; 1998, ISSN 1402-1528. p. 56 [in Swedish].

[5] Taljsten B, Elfgren L. Strengthening of concrete beams for shearusing CFRP-materials: evaluation of different application methods.Composites B 2000;31:8796.

[6] Maruyama K, editor. Recommendations for upgrading of concretestructures with use of continuous fibre sheets. Concrete engineeringseries 41. JSCE; 2001, ISBN 4-8106-0355-5. p. 250.

[7] Neale K. Strengthening reinforced concrete structures with externallybonded fibre reinforced polymers, Design Manual No. 4, ISISCanada; September 2001, ISBN 0-9689007-0-4. p. 198.

[8] Teng JG, Chen JF, Smith ST, Lam L. FRP strengthened RCstructures. Chichester: Wiley; 2002, ISBN 0-471-48706-6. p. 254.

[9] Carolin A. Carbon fibre reinforced polymers for strengthening ofstructural elements. Doctoral Thesis 2003:18, Lulea University ofTechnology, Division of Structural Engineering; 2003, ISBN 91-89580-04-4. p. 178.

[10] Carolin A, Taljsten B, Hejll A. Concrete beams exposed to liveloading during CFRP strengthening. J Compos Constr ASCE2003;9(2):17886. MarchApril 2005.

[11] Taljsten B, Carolin A, Nordin H. Concrete structures strengthenedwith near surface mounted reinforcement. Adv Struct Eng2003;6(3):20113.

[12] Taljsten B. Strengthening of concrete structures in torsion with FRP.In: Proceedings of the sixth international symposium on FRPreinforcement for concrete structures (FRPRCS-6), Singapore July

810, 2003, vol. 2, ISBN 981-238-401-4. p. 116776.



17/17

[13] FIB, Bulletin 14. Externally bonded FRP reinforcement for RCstructures, Technical Report, Task Group 9.3 FRP (Fibre ReinforcedPolymer) reinforcement for concrete structures; July 2001, ISBN 2-88394-054-1. p. 130.

[14] BBK 04. Boverkets handbok om betongkonstruktioner (The Swedishbuilding administrations handbook of concrete structures). Stock-holm, Sweden: The Swedish Building Administration, Division ofBuildings; 2004. August 2004 [in Swedish].

[15] Vasques A, Karbhari VM. Fiber-reinforced polymer compositestrengthening of concrete slabs with cutouts. ACI Struct J2003;100(5).

[16] Mosallam AS, Mosalam KM. Strengthening of two-way concreteslabs with FRP composite laminates. Constr Build Mater2003;17:4354.

[17] Hillerborg A. In: Cederwall K, Lorentsen M, Ostlund L, editors.Slabs in Betonghandboken Konstruktion (Concrete handbook Design). Stockholm: Svensk Byggtjanst; 1990, ISBN 91-7332-533-3.p. 614703 [chapter 6.5], [in Swedish].

[18] Timoshenko SP, Woinowsky-Krieger S. Theory of plates and shells.2nd ed. New York: McGraw-Hill; 1959. p. 580.

[19] Johansen KW. Brudlinieteorier (Yield Line Theories). Copenhagen,Denmark: Akademisk Forlag (1963). 1943 [in Danish].

[20] Jones LL, Wood RH. Yield-line analyses of slabs. 1st ed. London:Thames & Hudson and Chatto & Windus; 1967.

[21] Nielsen MP. Limit analysis and concrete plasticity. Englewood Cliffs(NJ): Prentice-Hall; 1984. 420 pp.

[22] Hillerborg Arne. Strip method design handbook. 1st ed. London:Chapman & Hall; 1996, ISBN 0-419-18740-5.

[23] Tano R. Modelling of localized failure with emphasis on band paths,Doctoral Thesis 2001:08, Division of Structural Mechanics, LuleaUniversity of Technology, ISSN 1402-1544.

[24] Hibbitt, Karlsson & Sorensen Inc. ABAQUS theory manual, usermanual and example manual, Version 6.5, Providence, RI; 2005.

[25] Lubliner J, Oliver J, Oller S, Onate E. A plastic-damage model forconcrete. Int J Solids Struct 1989;25(3):229326.

[26] Lee J, Fenves GL. Plastic-damage model for cyclic loading ofconcrete structures. J Eng Mech 1998;124(8):892900.

[27] Rusinowski P. Two-way concrete slabs with openings experiments,finite element analyses and design. Masters Thesis 2005:200 CIV,Division of Structural Engineering, Lulea University of Technology;2005, ISSN 1402-1617. p. 126.

[28] Taljsten B. FRP strengthening of existing concrete structures designguidelines. Lulea, Sweden: Lulea University of Technology; 2004,ISBN 91-89580-03-6. p. 230.

[29] Eriksson A, Larsson T. Kolfiberforstarkning av plattor med hal(CFRP strengthening of slabs with openings), Masters Thesis2003:156 CIV, Division of Structural Engineering, Lulea Universityof Technology; 2003, ISSN 1402-1617 [in Swedish].


Date post:	30-May-2018
Category:	Documents
Upload:	rpatel5509
View:	215 times
Download:	0 times

Construction and Building

Documents