design of non prismatic rc beams using strut and tie models

Journal of Advanced Concrete Technology Vol. 2, No. 2, 249-256, June 2004 / Copyright 2004 Japan Concrete Institute 249

Design of Non-Prismatic RC Beams Using Strut-and-Tie Models Kiang Hwee Tan1

Received 20 October 2003, accepted March 2004

Abstract This study deals with the application of the strut-and-tie models in the analysis and design of non-prismatic reinforced concrete beams. Seven beams were designed, fabricated and tested to failure. Test results showed that the ultimate loads exceeded the design loads for all beams. Non-prismatic beams with a recess through the web performed satisfac-torily, compared to beams with equivalent transverse rectangular openings. For non-prismatic beams with a recess at the bottom, an increase in the recess width resulted in a decrease in the stiffness and an increase in the beam deflection. Non-prismatic beams with a recess in the compression zone performed better with regards to cracking but not deflection, compared to beams with a recess in the tensile zone. Also, beams strengthened with carbon fibrereinforced polymer (FRP) plates performed satisfactory with regard to strength; however, the deflection and crack widths increased rapidly thereafter, leading to a sudden and non-ductile failure of the beam.

1. Introduction

There are many instances where beams can be made non-prismatic in cross-section along its length. For example, in modern buildings where utility ducts and pipes are being accommodated below the floor beams in the space above the false ceiling, the use of a non-prismatic beam with a recess would allow these ducts to pass through the beam, eliminating a significant amount of dead space [Fig. 1(a)]. This would reduce the height of each storey, leading to substantial savings in the material and construction costs.

Similarly, non-prismatic beams could be appropri-ately used as ground beams [Fig. 1(b)] in residential up-grading projects, where existing utility pipes often obstruct the construction of tie beams that connect the newly constructed columns to existing ones. The use of non-prismatic tie beams allows the construction to pro-ceed without the need of relocating these pipes.

A non-prismatic beam in the form of a stepped beam can also be applied to support a split-level floor [Fig. 1(c)]. This application is commonly found in theatres and in private housing for aesthetic reasons.

Finally, in buildings that are being retrofitted, there might be a need to add new service ducts, and this is often a problem due to limited headroom. In this case, a recess can be made in the existing beam to accommo-date the new service ducts. These non-prismatic beams could then be strengthened by externally bonded rein-forcement such as fibre-reinforced polymer (FRP) rein-forcement or steel plates [Fig. 1(d)].

The behaviour of non-prismatic beams is very differ-ent from ordinary prismatic beams, and current codes do

not give specific provisions for the design of such beams. Frequently, actual designs are based on rule of thumb or are empirical in nature and are not adequately backed by research findings. As a result, the design may be too conservative in certain cases while in others, critical issues may be overlooked.

In view of the above, the present study is carried out primarily to evaluate the strut-and-tie method as a de-sign tool for non-prismatic beams. The development and application of strut-and-tie models for design and detailing of structural concrete have been described in detail by Schlaich et al. (1987), Schlaich and Schafer (1991), Marti (1991), and more recently by ACI Sub-Committee 445 (Reineck 2002). For the purpose of the present study, a test programme was carried out on seven non-prismatic beams. As a secondary objec-tive, the behaviour of beams with a recess was com-pared with those with a transverse rectangular opening in the web, and the effect of recess size and location and

1Associate Professor, Department of Civil Engineering, National University of Singapore, Singapore E-mail: [email protected]

Existing structure

Column

G.L.

(c)(a)

(b)

(d)

Extension

Column

Fig. 1 Examples of Recess Beams: (a) Floor Beams; (b) Ground Beam; (c) Stepped Beam; (d) Retrofitted Beam.

250 K. H. Tan / Journal of Advanced Concrete Technology Vol. 2, No. 2, 249-256, 2004

strengthening scheme, on the overall behaviour and strength of non-prismatic beams was investigated. 2. Analytical considerations

A non-prismatic beam may be divided into regions known as the beam regions or the B-regions, in which beam theory, including linear strain distribution across sections, applies, and other regions termed the disconti-nuity regions or D-regions. Based on St. Venants prin-ciple, the D-region extends from the location of discon-tinuity for a distance equal to the member depth.

In general, the design of a non-prismatic beam begins with the isolation of the critical D-regions, followed by the construction of a strut-and-tie model that would transfer the boundary forces through these D-regions. The rest of the beam, that is the B-regions, can be de-signed using the usual procedure for prismatic beams. On the other hand, the beam could also be designed using a strut-and-tie model for the entire beam, as adopted in this study. 2.1 Formulation of strut-and-tie model The formulation of the strut-and-tie models is based on the following assumptions: (1) Strut members are formed by concrete while tie

members are formed by steel or FRP reinforce-ment.

(2) The strut members have a prismatic width. The capacity of the strut members is equal to the effec-tive compressive strength of concrete multiplied by the available concrete area.

(3) The capacity of the tie members is equal to the cross sectional area multiplied by the yield or effec-tive strength of the reinforcement. All tie members are provided with adequate anchorage and there is no bond slip of reinforcement.

(4) The joints (nodal zones), where the strut and tie members meet, do not fail at ultimate.

2.1.1 Design of strut members The effective compressive strength of the concrete struts is taken as:

fce = 12fc' (1) where the product v1v2 is an efficiency factor between 0 and 1. The factor v1 depends on the cracked condition of the strut member, and for the case where the strut is cracked longitudinally and is not confined by transverse reinforcement, v1 is equal to 0.65 (MacGregor 1997).

The factor v2 accounts for the increased brittleness of concrete as the strength increases, and is given by

2 = 0.55 + 1.25/( fc')1/2 (2) where fc' is the cylinder compressive strength in MPa. For the case of fc'= 30 MPa, v2 is equal to 0.778; hence, the value of fce is about 0.5 fc', and was used in the de-sign of all the strut members for the test beams in this

study. The required width of the strut is

a = Fs/tfce (3)

where Fs is the force acting on the strut, and t is the thickness of the member. All struts must fit within the beam, and must also not overlap. Otherwise, additional steel reinforcement would be needed to reinforce the strut so that the required width could be reduced. Such compression reinforcement must be laterally restrained to prevent buckling before failure of the beam occurs. 2.1.2 Design of tie members The required reinforcement for each tie member is cal-culated from:

Asfy Tn (4) where Tn is the calculated force for the tie member, As is the required area of reinforcement and fy is the yield strength of the steel reinforcement or effective strength of the FRP reinforcement. For carbon FRP plates, the plate would debond before the full potential of its ten-sile strength is reached. Based on a previous study (Tan 2001), an effective strength of 0.45 times the rup-ture strength may be used. 2.2 Deflection of beam under service load The deflection under service load, assumed as the ulti-mate load divided by a factor of 1.7, is calculated using the conjugate beam method. To account for cracking, an effective moment of inertia, Ie, is used, where

Ie = Icr +(Ig - Icr)(Mcr/Ma)3 (5)

in which Mcr and Ma are the cracking moment and maximum applied moment, respectively, Ig is the gross moment of inertia, and Icr is the cracked moment of in-ertia of the section. The cracking moment for a section is given by

Mcr = (frIg)/y (6)

where y is the distance from the neutral axis to the ex-treme tensile fibre and fr is the modulus of rupture taken as equal to 0.12(fcu)0.7 (BS 8110, 1997), where fcu is the cube compressive strength of concrete. In general, fcu may be taken as fc divided by a factor of 0.8.

The beam is divided into segments of constant cross-sections. For each prismatic segment, the maxi-mum moment is expressed in terms of applied load, P. This is then substituted for Mcr in Eq. (6) with the re-spective Ig and y values. The lowest value of P thus de-rived is the cracking load for the beam.

The conjugate beam is loaded by distributed loads given by M/EIc where Ic corresponds to the respective prismatic segment. The bending moment evaluated from force equilibrium at any section of the conjugate beam gives the deflection at that section of the actual beam. The bending moments at critical sections, typi-

K. H. Tan / Journal of Advanced Concrete Technology Vol. 2, No. 2, 249-256, 2004 251

cally, at the high-moment end of recess and under the load were thus calculated for the conjugate beam and the higher value gives the maximum deflection.

3. Test programme

Seven beams with dimensions shown in Fig. 2 were designed, fabricated and tested to failure. Beams ST-1, ST-2 and ST-3 each had a recess in the tensile zone (that is, at the bottom) of the beam with widths of 400 mm, 800 mm and 1200 mm respectively, with the center of recess at 1,000 mm from one support. They were de-signed to take a load of 204 kN, 132 kN and 90 kN, respectively, applied at one-third span length from the other support.

Beam ST-2T was designed with a recess in the com-pression zone (that is, at the top) at also 1,000 mm from one support to take a load of 132 kN, applied at one-third span length from the other support. Beam ST-4 had two recesses, one at the top and the other at the bottom and ST-5 was designed as a stepped beam. Both beams were designed for a total load of 150 kN, as a point load at the mid-span for Beam ST-4, and as two point loads, one each at the mid-length of the upper and lower stepped regions, for Beam ST-5.

Beam ST-2R was meant to simulate a beam that was strengthened with externally bonded carbon FRP plates after a recess has been introduced. The geometry of ST-2R was the same as that of beam ST-2. The internal steel reinforcement of ST-2R was first designed assum-ing a solid beam carrying a load of 89 kN. This rein-forcement was curtailed at the faces of introduced recess and was welded to steel plates that lined the recess. The strengthened beam was designed with external car-bon FRP plates to transfer the forces over the recess so that the beam would carry the original design load of 89 kN.

The strut-and-tie-models for the test beams are pre-sented in Figs. 3(a) to (d), with the solid lines indicating the tie members and the dotted lines representing strut members. The reinforcement was designed accord-ingly to resist the forces in the tie members, derived from force equilibrium at the nodes. Typical rein-forcement layout is shown in Fig. 4. Nominal links were provided in accordance with code requirements. 3.1 Materials The concrete mix was designed for a 28-day cylinder compressive strength of 30 MPa. Ordinary Portland cement, natural sand and crushed granite of 10 mm maximum size were mixed in the ratio of 1 : 1.32 : 1.98 by weight. The watercement ratio was 0.45 and the cement content was 495 kg/m3. To increase workability, a superplasticizer was added at a dosage of 0.3 kg per 100 kg of cement to give a slump of about 150 mm.

Mild steel bars designated R6 and R8, and high yield deformed bars designated T10, T13, T16, T20 and T25, were used as internal reinforcement. Tensile tests were

carried out on three specimens of each bar size and the results are presented in Table 1.

The carbon FRP plates had a thickness of 1.2 mm and width of 100 mm. They were stiff in the longitudinal direction, having a high tensile modulus of 150 GPa, but weak in the transverse direction. The plate has a low density of 1.6 g/cm3. Properties of the carbon FRP plates and the epoxy mortar are shown in Table 2. The plates were cut to the required length and width using an ordinary cutting blade. The concrete surface was ground and cleaned of dust and loose particles be-fore the plates were bonded to the beam using the epoxy mortar.

3.2 Preparation of test beams A wooden prismatic formwork was used. The recess was formed by attaching rectangular boxes of the re-quired dimensions into the formwork. The inner surface of the formwork was oiled to facilitate demoulding. Six 100 mm cubes were cast for each beam. The beam

C-C

Sections

A-A

200 mm

B-B

200 mm

200 mm 220mm (140mmfor ST-5) 400mm

(500mm for ST-5)

Fig. 2 Dimensions of Test Beams (all dimensions in mm).

220mm (140mmfor ST-5)

400

150

ST-2T

P 1000 600 600 800

B

B C

C

800 800 1000

150 3000

ST-1 ST-2, 2RST-3

A

A B

B

B

B

ST-4

P 400 600 500 500600 400 B

B

C

CA

A

650 P/2

650 650 650 400

A B

ST-5 P/2

AC

C B

P


and the cubes were stripped off the moulds after one day. Then they were placed under damp hessian for about a week before being left in the laboratory under ambient conditions. The beams were also whitewashed for easy identification of cracks.

For beam ST-2R, holes, 12 mm in diameter and 50 mm deep, were drilled through the beam and the carbon FRP plates after bonding the latter. These holes were vacuum cleaned to remove the dust and metal bolts were inserted and subsequently tightened with nuts, to fasten the carbon FRP plates. Washers were used to

cushion the applied pressure on the plates. 3.3 Test instrumentation and procedure All test beams were simply supported over a span of 3.0 m and tested under a one-point load except Beam ST-5, which was tested under two point loads. The strains in the internal steel bars, external carbon FRP plates and concrete surface were monitored using electrical resis-

(b) ST-5 2 T16

2 T16 2 T20

R8 links at 50 mm

R8 links at 200 mm

R8 links at 50 mm

2 T20 + 1T13

2 T20 + 1 T13

2 T16

2 T16 + 1 T10

2 T13

3 4

21

1,3

2,4

2 T202 T20 100

mm 70 mm 50

mm

R8 links at 70 mm

50 mm 2 T16 50 mm width

Steel Plate

(c) ST-2R

Fig. 4 Typical Reinforcement Details of Test Beams.

2 T10

2 T16

2 T102 T25 +1 T10

2 T10

(a) ST-4

R8 links at 75 mm

2 T20

2 T202 T16 2 T16

2 T25

2 T20

2 T25

160

180

160 180

160 180

140

(a) ST-1 [similar for ST-2, 3, 2R (after introduction of recess)]

P/3 2P/3

P 400 1200 400 1000

(b) ST-2T

P/3 2P/3

P 1000800 600 600

(c) ST-4

400 600 400600 500 500

P/2 P/2

P

300 140

340

(d) ST-2R (Before introduction of recess)

P/2

P/2 40650 650 650 650

P/2

300

P/3 2P/3

P

10002000

(e) ST-5

Fig. 3 Strut-and-Tie Models for Test Beams.

Table 1 Steel Properties.

Type Mild Steel (Round)

High Tensile Steel (Deformed)

Designa-tion R6 R8 T10 T13 T16 T20 T25Actual Diameter (mm)

6.0 8.0 9.7 12.8 15.3 19.6 24.1

Crosssec-tion Area (mm2)

28.3 50.3 78.5 129 183 302 454

Yield strength fy (MPa)

296 346 478 489 532 552 458

Ultimate Strength (MPa)

393 433 571 584 617 684 670

Youngs ModulusE (GPa)

188 160 159 187 165 184 186

Strain at Yield, y ( x 10-3 mm/mm)

1.58 2.16 3.01 2.62 3.22 3.71 2.49

P/2


tance strain gauges. The vertical deflections of the test beams were measured using linear variable displace-ment transducers. Crack widths were measured using a hand-held microscope having a accuracy of 0.02 mm.

The loads on the beam were applied in increments. At every load increment of 1 kN or a deflection increment of 1 mm, strain gauge readings and transducer dis-placements were recorded. After the first crack has occured, the development of cracks was marked and the maximum crack widths were measured at every 20 kN load increment. Finally, the ultimate load and the mode of failure were noted.

4. Test results and discussion

The crack patterns shown in Fig. 5 for six of the beams, corresponded well with the orientation of the tie mem-bers in the strut-and-tie model, shown in Fig. 3. It was also observed that the strains in the longitudinal steel bars were very close to the predicted values, and the measured strains and hence stresses followed closely the general trend predicted by the proposed strut-and-tie model.

A unique crack pattern was observed for beam ST-1 above the recess, indicating the presence of arch action over the recess [Fig. 5(a)]. The first crack appeared as a diagonal crack at the high moment end of the recess for beams ST-3 [Fig. 5(b)], ST-4 [Fig. 5(d)] and ST-2R [Fig. 5 (f)], but as a flexural crack at the recess for the beams ST-2 and ST-2T [Fig. 5(c)]. As for the beam ST5 [Fig. 5(e)], the first crack appeared as a diagonal crack at the corner below the upper stepped portion of the beam.

The cracking loads are indicated in Table 3. As the beam surface of beam ST-2R was covered by carbon FRP plates, the observation of cracks was only possible at the later stages of loading when the cracks had propagated above the plates. Therefore, the recorded cracking load of beam ST-2R was much higher than the other beams. According to BS 8110 (1997), the maxi-mum crack width under service loads should not exceed 0.3 mm. It is clear from Table 3 that this is satisfied in all beams except for beam ST-2, and in particular beam ST-5.

The maximum deflection occurred at the middle of the recess for beams ST-1, ST-2 and ST-3, at the high moment end of the recess for beam ST-2R and ST-4 and

Table 2 Properties of CFRP System. Carbon FRP Plates Epoxy mortar

Colour Black Colour Comp. A: WhiteComp. B: Black Comp. A + B: Light grey

Tensile strength

> 2400 MPa Mix ratio Comp. A : Comp. B = 3 : 1 (by wt. or vol.)

Modulus of elasticity

> 150000 MPa Static E-modulus

12800 MPa

Breaking strain

1.4 % Open time 30 minutes (at 35 C )

Apparent density

1.6 g/cm3 Density 1.77 kg/lit. (A + B)

Temperature resistance

> 500 C Coef. of expansion

9 x 10-5 per C (-10 C to +40 C)

Shelf life Unlimited (no exposure to direct sunlight)

Shelf life 1 year in original packaging at +5 C to 25 C

(a) (c)

ST-2T ST-1

(d)

ST-4

(f)

ST-2R

(e)

ST-5

(b)

ST-3

Fig. 5 Appearance of Test Beams After Failure.

(a) (b) (c)

(d) (e) (f)


at the mid-span for beam ST-5. Table 3 shows that the maximum service load deflections for all the test beams were less than span/350 or 20 mm (BS 8110 1997) ex-cept for beam ST-2R.

Beam ST-2 failed in flexure at the section directly under the applied load. Beams ST-2T, ST-3 and ST-4 failed in flexure with the crushing of concrete at the high moment end of the recess whereas ST-1 failed by concrete crushing at the top of the arch that developed above the recess. ST-2R failed by debonding of the carbon FRP plates from the concrete surface and ST-5 failed by steel yielding at the connection between the upper and the lower stepped regions. 4.1 Comparison of test results with design val-ues The test results are compared with the design or pre-dicted values in Table 3. The predicted cracking loads

are in general less than the test values, except for beam ST-1. This is because the first cracks were diagonal cracks originating from the corners of recess whereas the prediction assumed a flexural crack. The predicted service load deflections are also less than the observed values, indicating that further refinement of the calcula-tion method is necessary.

Excluding ST-5, the ratio of the observed to design ultimate loads varies from 1.07 to 1.69, with an average of 1.29 and standard deviation of 0.21. The main reason for the higher observed ultimate strength is that the strut-and-tie method gives a lower bound solution, al-though the higher concrete strength compared to the design strength of 30 MPa might have some effect.

Beam ST-5 did not satisfy both the ultimate strength and serviceability requirements, due to premature fail-ure. This was the result of a detailing error for the di-agonal bars, which had not been sufficiently anchored.

Table 3 Test Results.

fc (MPa)

Pcr, test (kN)

Pcr, pred (kN)

s,test (mm)

s,pred (mm)

max,s (mm)

Pu,test (kN)

Pu,design (kN)

Pu,test / Pu,design

ST-1 51.9 7.5 12.7 8.2 4.6 0.15 259 204 1.27 ST-2 42.5 28.0 10.2 5.6 5.0 0.35 220 132 1.67

ST-2T 36.5 15.0 7.8 7.5 5.4 0.14 156 132 1.18 ST-2R 39.2 50.0 6.6 7.0 8.4 0.09 95 89 1.07 ST-3 48.0 9.7 8.0 5.7 5.0 0.14 152 90 1.69 ST-4 42.3 17.3 8.5 8.3 3.8 0.14 188 150 1.25 ST-5 37.0 12.6 9.6 17.0 9.1 0.76 133 150 0.89

Note: fc = concrete cylinder compressive strength; Pcr = cracking load; Pu = ultimate load; s = maximum deflection under service load; max,s = maximum crack width under service load.

300

50

100

0

200

10 20 30 40 50

Service Load

R1 ST-1 250

150

Deflection (mm)

Load

(kN

)

Maximum Crack Width

0

50

100

150

200

250

0.2 0.4 0.6 0.8 1.0 1.2 1.4

Service Load

ST-1 R1

Load

(kN

)

02040

6080

100120140

0.2 0.4 0.6 0.8 1.0 1.2 1.4

R5ST-3

Service Load

(a) Deflection Characteristics

0

50

100

150

200

10 20 30 40 50 60 70 80

Service Load

ST-2

R3

250

Deflection (mm)0 10 20 30 40 50 60

Deflection (mm)

406080

120140

100

160

20

Service Load

ST-3

R5

ST-2

0

50

100

200

250

0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

Service Load

R3150

Maximum Crack Width (b) Cracking Characteristics

Maximum Crack Width

Fig. 6 Beams with Recess vs. Beams with Opening.


That is, the diagonal bars indicated by 2-2 and 3-3 in Fig. 4(b) should have been extended to the bottom and top edges of the beam, respectively, so as to effectively control cracking at the re-entrant corners of the beam.

4.2 Comparison of beam performance The beams are first compared with beams with an opening in place of the recess. Next, the effect of re-cess width is investigated using the results of ST1, ST-2 and ST-3. The effect of recess location across the beam depth is examined using the results of ST-2 and ST-2T. Finally, beam ST-2R is compared with beam ST-2 to investigate the effect of strengthening. 4.2.1 Recess vesus web opening Beams ST-1, ST-2 and ST-3, are compared to Beams R1, R3 and R5, respectively, which were tested by Mansur et al. (1985). The latter beams had the same overall cross-section dimensions, beam span and were designed to carry the same ultimate load under the same test set-up as ST-1, ST-2 and ST-3. The only difference between the two groups of beams is that instead of a recess, R1, R3 and R5 each had an opening through the web at mid-depth, having the same dimensions and lo-cation along the beam as the recess in ST-1, 2, and ST-3 respectively.

As shown in Figs. 6(a), the load-deflection character-istics of beams ST-1, ST-2 and ST-3 are similar to those of beam R1, R3 and R5 respectively. The maximum service load deflections are similar for each pair of beams. Figs. 6(b) show the load versus maximum crack width relations for the beams. Beams with a re-

cess had smaller crack widths at service load and hence, more desired cracking characteristics than beams with an opening.

All beams with a recess exhibited a ductile failure. Beams R3 and R5 both failed with the crushing of con-crete on the top and bottom faces of the chord members at the high and low moment ends of the opening respec-tively, while Beam R1 failed at the solid section under the applied load. As beams R1, R3 and R5 were tested using a load-control actuator, the post-peak behaviour could not be obtained.

It is concluded that the provision of recesses offers an alternative solution to openings, and such beams per-form satisfactorily with respect to deflection, cracking and ultimate load behaviour. 4.2.2 Effect of recess width The load-deflection characteristics of Beams ST-1, ST-2 and ST-3, with recess widths of 400 mm, 800 mm and 1200 mm, respectively, are compared in Fig. 7(a). All three beams have recesses at the same location at the bottom of the beam. The maximum deflection oc-curred at the middle of the recess for ST1 and ST-3, and under the load for beam ST-2. The service load deflec-tion decreases with an increase in the recess width.

The load versus maximum crack width curves are compared in Fig 7(b). The maximum crack widths of Beams ST-1, ST-2 and ST-3 under the service load are 0.14 mm, 0.35 mm and 0.15 mm respectively. The larger value for Beam ST-2 is probably due to the failure occurring under the applied load whereas it occurred at the high moment end of the recess in ST-1 and ST-3.

50

100

150

200

250

0 20 40 60 80Deflection (mm)

ST-2R

ST-2(e)

0

50

100

150

200

250

300

10 20 30 40 50 60 70 80

ST-3

ST-1 ST-2

(a)

Deflection (mm)

Load

(kN

) Lo

ad (k

N)

0

50

100

150

200

250

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

ST-2 ST-1

ST-3

(b)

Maximum crack width

0

50

100

150

200

250

0.2 0.4 0.6 0.8 1.0 1.2 1.4

ST-2R

ST-2

Maximum crack width

(f)

0

50

100

150

200

10 20 30 40 50 60 70 80

ST-2

ST-2T

250 (c)

Deflection (mm)

Service Load

0

50

100

150

200

250

0.2 0.4 0.6 0.8 1.0 1.2 1.4

Service Load

ST-2T

ST-2(d)

Maximum crack width

Fig. 7 Effect of Recess Width and Location, and Beam Strengthening.


4.2.3 Effect of recess location Beams ST-2 and ST-2T had the same recess width of 800 mm. The recess is located in the tensile zone for Beam ST-2 but in the compression zone for Beam ST-2T. The deflection characteristics of Beams ST-2 and ST-2T are compared in Fig. 7(c). The deflection at the service load for Beam ST-2T was larger than for ST-2, with values of 7.50 mm and 5.61 mm respectively.

The load versus maximum crack width curves are compared in Fig. 7(d). The maximum crack widths under the service load are 0.14 mm and 0.35 mm for Beams ST-2T and ST-2 respectively. Thus, the provi-sion of the recess in the compression zone offers better cracking characteristics but less desired deflection characteristics, compared to recess in the tensile zone. 4.2.4 Effect of strengthening Beams ST-2 and ST-2R had exactly the same dimen-sions. Beam ST-2R was designed as a truncated beam that was subsequently strengthened with externally bonded carbon FRP plates. Beam ST-2, on the other hand, had been designed from the beginning to accom-modate the recess and was reinforced by internal steel reinforcement bars only. However, the design loads were different, being 132 kN and 89 kN respectively for beams ST-2 and ST-2R. The deflection characteristics are compared in Fig. 7(e), which gives the service load deflections as 7.0 mm and 5.6 mm for ST-2 and ST-2R respectively. The load versus maximum crack width relations shown in Fig. 7(f), give the crack widths at the service loads as 0.35 mm and 0.09 mm, respectively, for beams ST-2 and ST-2R. Beam ST-2R failed in a sudden manner, losing its load carrying capacity once the debonding of carbon FRP plates occurred. On the other hand, beam ST-2 failed in a ductile manner. 5. Conclusion

Strut-and-tie models were presented to design the rein-forcement for non-prismatic reinforced concrete beams. Seven beams designed using these models were fabri-cated and tested. Recess width and location, and strengthening scheme, were considered. The test re-sults were compared with design values and the effects of these parameters on the strength and behaviour of beams were discussed in detail. Also, the performance of the beams was compared to similar beams with web openings.

The following conclusions may be drawn from the investigations carried out:

1. The strut-and-tie method of design was shown to be suitable for application in non- prismatic beams as (a) the crack pattern and measured strains in the rein-forcement agreed with the strut-and-tie model; (b) the strut-and-tie model gives lower bound values for the ultimate load; and (c) the method offers a simple and straightforward solution that is based on established principles to an otherwise complicated problem.

2. Non-prismatic beams with a recess exhibit compara-ble performance to beams with a transverse rectangu-lar opening with respect to deflection and cracking characteristics, and ultimate load behaviour.

3. For non-prismatic beams with a recess in the tensile zone, an increase in the recess width results in smaller ultimate load, higher cracking load and smaller ser-vice load deflection.

4. Beams with a recess introduced and subsequently strengthened with carbon FRP plates performed sat-isfactory with regard to strength, deflection and crack width. However, the failure tends to be non-ductile and sudden.

References British Standards Institution (1997). Structural Use of

Concrete. BS 8110, London. MacGregor, J. G. (1997). Reinforced Concrete

Mechanics and Design. 3rd Ed., Upper Saddle River, N.J.: Prentice Hall.

Mansur, M. A., Tan, K. H. and Lee, S. L. (1985). Design method for reinforced concrete beams with large openings. ACI Journal, USA, 82 (4), 517-524.

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design of non prismatic rc beams using strut and tie models

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