Structure and Infrastructure Engineering

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Flexural strengthening of RC beams using distributed prestressed high strength steel wire rope: theoretical analysis a

a

a

b

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Gang Wu , Zhishen Wu , Yang Wei , Jianbiao Jiang & Yi Cui a

Key Laboratory of Concrete and Prestressed Concrete Structures of the Ministry of Education, Southeast University, Nanjing, China b

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Beijing Texida Technology R & D Co., Ltd., Beijing, China Published online: 23 Aug 2012.

To cite this article: Gang Wu, Zhishen Wu, Yang Wei, Jianbiao Jiang & Yi Cui (2014) Flexural strengthening of RC beams using distributed prestressed high strength steel wire rope: theoretical analysis, Structure and Infrastructure Engineering: Maintenance, Management, Life-Cycle Design and Performance, 10:2, 160-174, DOI: 10.1080/15732479.2012.715174 To link to this article: http://dx.doi.org/10.1080/15732479.2012.715174

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Structure and Infrastructure Engineering, 2014 Vol. 10, No. 2, 160–174, http://dx.doi.org/10.1080/15732479.2012.715174

Flexural strengthening of RC beams using distributed prestressed high strength steel wire rope: theoretical analysis Gang Wua*, Zhishen Wua, Yang Weia, Jianbiao Jiangb and Yi Cuib a

Key Laboratory of Concrete and Prestressed Concrete Structures of the Ministry of Education, Southeast University, Nanjing, China; bBeijing Texida Technology R & D Co., Ltd., Beijing, China

Downloaded by [Southeast University] at 06:25 27 January 2015

(Received 23 March 2012; final version received 24 June 2012; accepted 20 July 2012; published online 23 August 2012) Various strengthening techniques for structural elements using different materials have been investigated. Recently, a new, reliable and cost-effective strengthening technique with distributed prestressed high strength steel wire rope (P-SWR technique) was proposed. This paper mainly focuses on theoretical analysis of the flexural behaviour of reinforced concrete (RC) beams strengthened with the P-SWR strengthening technique. First, mechanical properties of steel wire rope such as ultimate strength, ultimate tensile strain and relaxation were tested. Second, an evaluation method, including the prediction of cracking load and flexural capacity of RC beams strengthened with P-SWR, was proposed. Third, prestressed level of P-SWR, ratio of reinforcement, and bond strength of P-SWR and concrete responsible for short-term cross-sectional stiffness were studied and associated calculation equations are suggested. Finally, according to parametric studies, an entire evaluation system, including a modified Rao & Dilger code calculation method and hypothetical tension method, as well as a simplified method for predicting the maximum crack width, is proposed. All of these analytical procedures are based on experimental studies. A great similarity between the experimental and analytic results suggests that the proposed methods are highly accurate. Keywords: prestressing; high strength steel wire rope; flexural strengthening; concrete beams; theoretical analysis

1. Introduction Strengthening of existing structures is an accepted and guaranteed means of improving both load-carrying capacity and serviceability of structures. As a result, different strengthening techniques of structure elements using various materials have been thoroughly investigated for flexural strengthening, including section enlargement (Thanoon, Jaafar, Kadir, & Noorzaei, 2005), external bonding steel plates (Jones, Swamy, & Charif, 1988), external bonding fibre reinforced polymer (FRP) (Choi, West, & Soudki, 2008; Ebead, 2011; Kim, Wight, & Green, 2008; Martin & Lamanna, 2008; Teng et al., 2006; Wu et al., 2005) and near-surface mounted (NSM) FRP strips or bars (Hajihashemi, Mostofinejad, & Azhari, 2011), external posttensioning tendons (Lorenc & Kubica, 2006), rehabilitation with steel wire mesh (Kubaisy & Jumaat 2000) have all been used to strengthen and repair of structures. The structural behaviour of cracked reinforced concrete (RC) one-way slab repaired with different techniques was investigated by Thanoon et al. (2005). The specimen repaired by section enlargement exhibits higher stiffness and the load–deflection curve shows much stiffer behaviour. However, enlarging section

*Corresponding author. Email: [email protected] © 2012 Taylor & Francis

brings structures additional remarkable deadweight, and need to build form and pour concrete in site. Bonding steel plates (Jones et al., 1988) as a type of traditional technique has been investigated and used for strengthening reinforced concrete beams extensively. But bonding steel plates has the disadvantages of corrosion over the long term, difficult handling due to weight and easy to occur interface failure. Studies have shown that externally bonded FRP increase the bearing capacity of flexural members significantly. Fibre reinforced polymer has a high strength to weight ratio, excellent resistance to chemical corrosion and ease of handling. Test results show that when FRP are used as externally bonded reinforcement, the flexural stiffness has very little improvement, peeling failure is often occurred without warning and the ductility is reduced extremely than unstrengthened flexural members (Choi et al., 2008; Teng et al., 2006; Wu et al., 2005). To avoid premature failure due to debonding and improve the failure ductility, some techniques have been developed, such as the NSM strengthening technique (Hajihashemi et al., 2011) and mechanically fastened anchorage systems (Ebead, 2011; Martin & Lamanna, 2008). Adding prestressing by external tendons can significantly increase the yield load and the ultimate resistance of the beams, the deflection at


Structure and Infrastructure Engineering the serviceability state is also reduced, the behaviour of composite beams prestressed with external tendons was investigated (Lorenc & Kubica, 2006). However, a large jack must be provided for drawing of the prestressing tendons, and need a big drawing space during construction. Kubaisy and Jumaat (2000) studied the flexural behaviour of reinforced concrete slabs strengthened with ferrocement tension zone cover. The results indicate that the use of ferrocement cover slightly increases the ultimate flexural load and increases in the first crack load. The first crack load increased with the increase in the percentage of mesh reinforcement and the ferrocement layer thickness. Recently, a new strengthening technique with distributed prestressed high strength steel wire rope (P-SWR technique) was proposed by Wu, Wu, Jiang, Tian, and Zhang (2010). This new strengthening technique for RC beams, which was proposed to overcome the shortcomings of existing strengthening methods, utilises the advantages of traditional materials and achieves an active strengthening of RC beams with less influence on the original structure and better comprehensive performance. Laboratory experiments have illustrated the effectiveness of using P-SWR in repairs. There is no analytical procedure for evaluating the flexural behaviour of RC beams strengthened with P-SWR technique. This paper focuses on a systemic study concerning with theoretical analysis model and design method of parameters such as bearing capacity, deflection and the maximum crack width of RC beams strengthened with the P-SWR technique. 2.

Figure 1.

The cross-section of 1 6 19 steel wire rope.

Figure 2.

Failure mode of specimen.

161

Mechanical properties of P-SWR

2.1. Stress–strain curves High strength, low relaxation steel wire ropes with high tensile strain and good corrosion resistance were used in this test. The high strength and low relaxation ensured effective prestress load and a comparatively high bearing capacity, while high tensile strain guaranteed good ductility. In addition, steel wire ropes with a moderate diameter were used to avoid construction difficulties caused by too much stretching force. Considering all of the above factors, high strength steel wire rope with a nominal diameter of 3 mm and nominal section area of 5.37 mm2 were chosen and twisted with 19 steel wires with diameters of 0.6 mm. The cross-section is shown in Figure 1. A test was applied to determinate the mechanical properties of the steel wire ropes with lengths of 85 mm, all of which had a middle fracture, as shown in Figure 2. Stress–strain curves of some specimens are shown in Figure 3, in which strain was recorded by strain gauges fixed 50 mm away from the middle of the

Figure 3. Stress–strain curves of some steel wire rope specimens.

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gauge length. Before the proportional limit point, the curves indicate a fine linear relationship between stress and strain and there is no obvious yield point at the non-linear stage. The average ultimate tensile strength was measured to be 1240.7 MPa and the average elastic modulus was found to be 143.7 GPa. The average ultimate tensile strain was 3.17%, with each specimen over 3%. The stress when residual strain was 0.2% is defined to be the nominal yield stress, which is about 85% of its ultimate tensile strength. 2.2. Relaxation property The relaxation property of steel wire ropes played an important role in effective load application and final strengthening results. Many factors, such as time, grade of steel and initial time, might contribute to stress relaxation. The stress relaxation test was conducted with two groups of specimens with lengths of 3150 mm. Considering the fact that stress relaxation of tensile steel in the first 24 h might account for over 50% of the whole relaxation, the stress relaxation test of two specimens was first conducted over 24 h: the initial stress of Specimen 1 was 1098.7 MPa, which became 1080.1 MPa after 24 h, for a relaxation rate of 1.7%; the initial stress of Specimen 2 was 894 MPa, which became 857 MPa after 24 h, for a relaxation rate of 4.1%. A relaxation test lasting 192 h was conducted with a specimen in the second group: its initial stress was 806.3 MPa, which became 797.4 MPa after 192 h, for a relaxation rate of 1.1%. Later, the specimens were divided into three groups. A relaxation test containing three specimens of steel wire ropes in each group with the same specifications but different

Figure 4.

production batches was conducted. The initial stress of these specimens was 70% of their ultimate tensile strength, with relaxation rates of 2.94, 2.65, and 3.088%, respectively, after 300 h. Thus, the steel wire ropes have a comparatively low relaxation rate. Furthermore, anchoring and protection from mortar used commonly in practical engineering could further reduce relaxation, thus making steel wire rope a prominent material in construction. 3. Experimental research Nine reinforced concrete beams 150 6 300 mm in cross-section and 2000 mm in length were cast (Figure 4). The reinforcement consisted of three 14 mm diameter steel bars at the bottom and two 6 mm diameter bars on the top. The web reinforcement consisted of 8 mm diameter closed stirrups spaced 80 mm centre-to-centre throughout the span of the beams. The yield stress of the steel was 382.4 MPa. Ready-mixed concrete with an average 150 6 150 6 150 mm cubic compressive strength of 43.5 MPa, approximately equivalent to 150 6 300 mm cylinder compressive strength fco0 of 34.3 MPa, was used for all beams. Thepelastic modulus of concrete is approxiffiffiffiffiffi mately 3950 fco0 (Ahmad & Shah, 1982). The thickness of the concrete cover, which is the distance from the outside edge of the cross-section to the lateral side of stirrup, was 20 mm. Among these beams, one was not strengthened to provide a baseline. Five beams were strengthened with P-SWRs, and the others were strengthened by carbon fibre reinforce plastics (CFRP) laminates and steel plates. This article is mainly concerned with beams strengthened with P-SWRs, as

Main dimensions and schematic illustration of strengthened beams.

Structure and Infrastructure Engineering Table 1. Beam

163

Strengthening schemes and failure modes. Strengthening mode

Effective stress (MPa)

Failure characteristics

L1 L4

Unstrengthened beam One-layer 12 steel wire ropes without mortar

– 761

L5

One-layer 12 steel wire ropes with mortar

741

L6

One-layer steel wire rope cut down at anchorage support only by mortar One-layer 12 steel wire rope with mortar after preloading Two-layer 23 steel wire ropes with mortar

566

Steel yield, concrete compression failure Steel yield, concrete compression failure and steel wire rope rupture Steel yield, compression failure, midspan steel wire rope rupture Bond failure

L7 L9

717 665/592

Steel yield, concrete compression failure and steel wire rope rupture Steel yield, concrete compression failure and steel wire rope rupture


Note: Beams L2, L3 and L8 are strengthened with CFRP and steel plate nor listed here, which can be found in Wu et al. (2010).

shown in Table 1. Cross-section dimensions and the test setup are shown in Figure 4. More details about the experiment are described by Wu et al. (2010). As shown in Figure 5, the main procedures for this new P-SWR strengthening technique for RC beams (Figure 5a) included: (1) notching the flexural member (Figure 5b); (2) fixing the anchorage (Figure 5c); (3) placing mortar into notches (Figure 5d); (4) setting reacting points (Figure 5d); (5) baiting steel wire ropes and producing extruding anchor heads (Figure 5e); (6) tensioning and anchoring steel wire ropes (Figure 5f) and (7) protection by mortar (Figure 5h). Figure 5(h) also shows the beam bottom after the beam was strengthened. Two major innovations of this new technology are that the previously centralised prestress is now well distributed and that there is reliable anchorage at the end to ensure convenient and efficient strengthening. Two strengthening modes, direct strengthening and strengthening after preloading were applied in this test. In the former mode, the contrastive beam and strengthened beams were first strengthened and then loaded gradually until failure; in the latter, preloading was first executed until the main steel yielded, which is defined as when the deflection of the midspan was approximately 1/360 the length of the span and the maximum crack width about 0.2 mm, and then the beams were strengthened after unloading and consequently loaded gradually again until failure. Table 1 shows the different strengthening methods of beams, of which Beam L1 is the unstrengthened beam. Beams L4–L7 were strengthened by one-layer steel wire ropes with different parameters, as shown in Figure 5. After tensioning and anchoring, the mean strain and corresponding stress of Beam L4 were 5016 me and 761 MPa, respectively, 61.3% of the ultimate strength, and those of Beam L5 were 4846 me and 741 MPa, respectively, 60% of the ultimate strength. After being strengthened by steel wire ropes, Beam L5 was coated with mortar with a thickness of 15 mm. The strain of steel wire ropes on Beam L6 was 3494 me and

its corresponding stress was 566 MPa. In addition, Beam L6 was also protected and anchored by mortar with a thickness of 15 mm. To identify the effect of the anchorage, the steel wire ropes on Beam L6 were cut down after the mortar achieved certain strength so that the steel wire ropes were solely anchored by mortar. After being loaded to the yield of its main steel, Beam L7 was unloaded and strengthened by one-layer steel wire ropes whose effective stress was 717 MPa, while the other parameters were the same as those for Beam L5. Beam L9 was first strengthened by two-layer steel wire ropes and consequently anchored and protected by mortar. The mean strain and corresponding stress of the first layer were 4236 me and 665 MPa, respectively, and those of the second layer 3682 me and 592 MPa, respectively. Load–deflection curves of these beams are shown in Figure 6. The test showed that strengthening with prestressed steel wire ropes could achieve better results in improving crack load, yield load and ultimate bearing capacity. Compared with the unstrengthened Beam L1, the crack loads of Beams L9 and L5 increased by 111 and 178%, respectively, the yield load by 49 and 121%, respectively, and the ultimate bearing capacity by 53 and 106%, respectively. Since the failure of Beam L6 anchored solely by mortar was caused by a bond failure, the test results were not satisfying. In practical construction, this should be avoided, and will not be discussed further in this paper. All other beams succumbed to ductile failure, consisting of steel yielding, crushing of concrete and fractures in the steel wire ropes. 4. 4.1.

Flexural capacity Crack load

Since prestress was applied to beam soffit (the bottom of the beam), a force the beam must counteract before cracking, the influence of an external load can be analysed by examining the material mechanics. The

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Figure 6.

Load mid-span deflection curves of beams.

cracking moment of beams strengthened with P-SWR can be calculated by Equation (1).


Mcr ¼

Figure 5. Technological process of P-SWR strengthening method. (a) Member to be strengthened; (b) end getting notched; (c) fixing anchorage; (d) impregnating mortar and setting reacting points; (e) producing steel wire rope and extruded anchor head; (f) tensioning and anchoring of steel wire rope; (g) beam bottom after strengthening; (h) external anchoring and protective mortar.

Pe Pe e fct Ig þ yb þ Ag yb Ig

ð1Þ

where Pe is the force of effective prestress; e is eccentricity of resultant force of prestress; yb is distance from extreme tensile fibre to the centroid of the uncracked section; Ig is the moment of inertia of gross (uncracked) concrete section; Ag is the gross area of section and fct is the average splitting tensile strength of concrete. A comparison between test values and calculated values of the crack load is shown in Table 2. The crack value of the unstrengthened beam was the average value of that of Beam L1 and those of the other two beams on which cracks first appeared. As shown in Table 2, crack loads of beams strengthened with PSWR were clearly improved, and more strengthening materials could bring better results. Test values go along well with calculated values were shown in spite of some errors. These errors are primarily because the calculated value of concrete tensile strength was determined by the test value of the cube block, which results in certain discreteness; the load in the test was applied with a grade of 5 kN, while the crack load was observed with the naked eye, which might cause some discreteness; the initial effective tensile stress was determined by tensile strain obtained by strain gauges on the steel wire ropes which might have some errors. For example, the test values of the pre-compressive stress of Beams L4 and L9 were around 5.3 and 9.2 MPa, respectively (Wu et al., 2010), which are both greater than their calculated values of 4.2 and 7.0 MPa. In fact, the calculated values in Table 2 are all much smaller than their corresponding test values. 4.2. Ultimate bearing capacity The ultimate strength of each beam can be found in Table 3 and Figure 7. The ultimate load of Beam L1 was


Structure and Infrastructure Engineering

4643 me , which was much smaller than that of Beam L5. Even if the ultimate load was applied, the ultimate strain was only 10,488 me . That is to say, the stress in the external unbonded tendons depends on the deformation of the whole structure and is assumed uniform at all sections. Many researchers have suggested various methods to predict the ultimate stress in the unbonded tendons by introducing the concept of a bond reduction coefficient. While considering design convenience, the ultimate bearing capacity of unbonded beams strengthened with high strength steel wire rope in this paper is evaluated through its nominal yield strength. The nominal yield strength measures several properties, including similarity to the favourable influence of non-prestressed steel bars to the stress of the unbonded tendon in PRC structure. Non-prestressed steel rebar usually makes up a large part of RC beams, which may have a positive influence to P-SWR. For PRC beams, the prestressed steel rebar is on the internal side of non-prestressed rebar, while for beams strengthened with high strength steel wire rope, the steel wire ropes are on the outermost edge to their full use. And test has proved that when the ultimate strain of Beam L4 was 10,488 me , its corresponding stress was 1155 MPa, which already surpasses the nominal yield stress 1055 MPa and is close to its ultimate strength 1241 MPa. There are some differences in the stress–strain relationship between steel wire ropes and prestressed steel strands. For example, the yield platform of the former is flatter than the latter. Finally, in practical engineering, considering the durability of steel wire ropes, protective measures are taken after strengthening is complete, which allows protective materials to be used to enable the steel wire ropes to make their full potential. Figure 7 shows the stress relationship of beams at the ultimate stage. From the equilibrium condition (ACI318–02, 2001):

159.4 kN, while the ultimate bearing capacities of Beams L4 and L5 strengthened by one-layer steel wire ropes were 244.1 and 243.8 kN, respectively. Thus, the bearing capacity increased by approximately 53% compared with that of the unstrengthened Beam L1. The ultimate strength of Beam L9 strengthened by two-layer steel wire ropes reached 328 kN, increasing by 106 and 34.5%, respectively, compared to those of the unstrengthened Beams L1 and L5, which were strengthened by one-layer steel wire ropes. Beams L4 and L5 sustained ductile failure, consisting of steel yielding, crushed concrete and fractured steel wire ropes which made possible further increase of the ultimate bearing capacity using more strengthening materials (Wu et al., 2010). The test results also showed that there was concurrent deformation between steel wire ropes and existing steels (Wu et al., 2010), allowing the strain in the cross-section to agree with the strain calculated by assuming a plane section. Hence, classical theories of reinforced concrete or prestressed RC beams are applicable to calculate the ultimate strength. Bonded Beams L5 and L9 experienced steel wire rope fracture at midspan. Therefore, the ultimate bearing capacity could be obtained from the ultimate tensile strength of the steel wire ropes. The mechanical properties of Beam L4 with no mortar coat were similar to those of common unbonded prestressed beams. The strain of steel wire ropes was uniform along the length. The stress was smaller than that of bonded steel wire ropes and often could not reach its ultimate strength. The bearing strength was also a little smaller than that of the bonded steel wire ropes. For example, when the load reached 234.3 kN, the strain of bonded Beam L5 was 12,622 me, which kept increasing as the load increased. No accurate data were recorded because the strain exceeded the measurement range of the strain gauge. When the load reached 234.7 kN, the strain of unbonded Beam L4 was Table 2.

Comparison between test results and theoretical values of cracking load.

Test value (kN) Calculated value (kN) Error %

Table 3.

165

Unstrengthened beam

Beam L4

Beam L5

Beam L6

Beam L9

41.7 37 11

95 76 20

90 75 17

85 68 20

125 98 21

Comparison between test results and theoretical values of maximum flexural capacity.

Test value (kN) Calculated value (kN) Error %

Beam L1

Beam L4

Beam L5

Beam L7

Beam L9

159.4 150.0 5.9

244.1 224.3 8.1

243.8 224.3 8.0

248.7 224.3 9.8

328.0 294.0 10

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G. Wu et al.


Figure 7. Stress profile at failure of cross-section. (a) Reinforcement at cross-sections; (b) stress relation.

0:85f0c ba ¼ fy As þ fps Aps

ð2Þ

Mn ¼ fy As ðds a=2Þ þ fps Aps dps

ð3Þ

and ecu is the ultimate compressive strain for the concrete. Comparison between calculated values obtained from above equations and test values can be seen in Table 3, where the ultimate tensile strength of steel wire ropes is assumed to be their strength under the ultimate load. As shown in the table, the calculated values go along well with test values, with a discreteness ratio within 10%, which ensures safety. Beam L7 was strengthened after failure, based on the relevant test results and research (Wu et al., 2010). Although initial damage may affect stiffness and cracks, it has little influence on the ultimate bearing capacity. Thus, the calculation methods suggested above are still appropriate. 5. 5.1.

where a is the depth of the equivalent rectangular concrete stress block; b is the width of the crosssection; ds is the distance from extreme compression fibre to centroid of the reinforcing steel; As is the area of non-prestressed reinforcing steel; Aps is the area of prestressed steel wire ropes; ds is the distance from extreme compression fibre to centroid of the reinforcing steel; dps is the distance from extreme compression fibre to centroid of prestressed steel wire ropes; f c0 is the specified compressive strength of concrete; fy is the yield strength of the reinforcing steel; fps is the stress in the steel wire rope; and Mn is the nominal moment capacity for the section. The strengthened beams should not be overreinforced to ensure their ductility. In other words, the relative depth of the compression zone x should be smaller than its balanced relative depth xb. Since the strengthened beams consist of prestressed steel wire ropes and non-prestressed steel rebar, their balanced relative depth of relative compression zone should be calculated separately, and the smaller value is taken. This is described in Equations (4) and (5). xb1 ¼

xb2 ¼

b1 f

1 þ Es eycu b1

1 þ 0:002 ecu þ

fps fse Eps ecu

ð4Þ

ð5Þ

where b1 is the factor relating depth of equivalent rectangular compressive stress block to neutral axis depth; fse is the effective stress in prestressing steel wire rope (after allowance for all prestress losses); Es is the modulus of elasticity of reinforcement; Eps is the modulus of elasticity of prestressed steel wire ropes

Stiffness analysis of cross-section Characteristics of load–midspan deflection curves

The load–midspan deflection curves of each beam are presented in Figure 6. For the unstrengthened Beam L1, the first flexural crack appeared at the midspan when the load value reached 45 kN. When the load value reached 124.7 kN, the longitudinal bar began to yield. The beam yielded and deformation rapidly increased when the load value was approximately 150 kN. The beam arrived at its ultimate load 159.4 kN and the concrete compression zone crushed under continuous loading. Thus, Beam L1 experienced a typical failure for adequately reinforced concrete beams. Beam L4 was strengthened with one-layer prestressed steel wire ropes without mortar. It can be found that the stiffness was larger than Beam L1 at the initial stage. The first flexural crack appeared at midspan when the load value reached 95 kN. When the load value reached 210 kN, the longitudinal bar yielded. The beam yielded and stiffness rapidly decreased when the load value reached approximately 230 kN. Because the stress of the lateral steel wire ropes kept increasing, the beam was capable to arrive at its ultimate load of 244.1 kN. As the load continued to increase, the concrete compression zone began to be crushed and both the stress of the steel wire ropes and deformation of the beam increased. Because the steel wire ropes and concrete on the beam were unbonded, the steel wire ropes could be tensioned freely, so the strain was uniformly distributed along span, and thus the curve shows better ductility. Finally, the concrete in the compressive zone was crushed, the steel wire ropes ruptured in the anchorage zone and the beam failed. Beam L5 was strengthened by one-layer steel wire ropes with mortar and a much larger initial stiffness than Beam L4 for the existence of mortar on the beam


Structure and Infrastructure Engineering soffit. The first crack appeared in the concrete of the beam soffit when the load value reached 90 kN. When the load value reached approximately 220 kN, the structure yielded. As the load value continued to increase slowly, the displacement increased rapidly. When the load increment was carried to a maximum of 243.8 kN and the midspan deflection reached 18.6 mm, the concrete compression zone was crushed, the steel wire ropes ruptured and the load value decreased rapidly and finally remained stable at approximately 147 kN. Unlike Beam L4, which was strengthened by unbonded steel wire ropes, the steel wire ropes at the main crack at midspan of Beam L5 had the largest strain, so the steel wire ropes ruptured there. Beam L9 was strengthened by two-layer steel wire ropes with mortar on the surface. The first concrete crack appeared at the midspan when the load value reached 125 kN. When the load value reached 280 kN, flexural cracks at midspan remained the same, while those at the ends continued to extend. When the load increment was increased to 320 kN, the displacement at midspan also increased, and damages appeared in the concrete compression zone. When the load value arrived at 338 kN, the steel wire ropes on the beam soffit towards the right loading point fractured. The load value decreased rapidly to 180 kN and the beam failed. Beam L7 was strengthened by one-layer steel wire ropes after preloading, whose load–midspan deflection curves are similar to those of Beam L4. Both Beams L7 and L4 suffered from ductile failure consisting of steel yielding, crushing of concrete and fracture of steel wire ropes. To conclude, the load–displacement curve of beams strengthened with P-SWR consists of two straight lines and one non-linear section. The first straight line starts from the coordinating origin and shows the elastic working stage before cracking. The second straight line stands for the elastic working stage after cracking. The transition from the second stage to the third stage is caused by the steel rebar yielding, which is presented by an abrupt turning point, corresponding to the nonlinear stress–strain relationship.

5.2. Stiffness-influencing parameters of cross-section 5.2.1. Influence of prestress on stiffness of RC beams The effects of applying prestress to RC beams to improve stiffness can be explained with the aid of Figure 8. Members A and B represent the same parameters concerning reinforcement and cross-section areas, but different degrees of prestress, with Member B having the greater prestress. From Equation (1), the

167

Figure 8. Influence of prestressing and ratio of reinforcement on cross-sectional stiffness. (a) Influence of prestressing; (b) influence of reinforcement ratio on crosssection.

cracking moment of Member B Mcr,B is larger than that of Member A Mcr,A, that is Mcr,B>Mcr,A. The two members have the same reinforcement ratio, their moment–curvature curves are almost parallel to each other on the second stage after cracking. Provided with the same moment Mk, the cross-section stiffness of Member B is larger than that of Member A, or tgbB>tgbA. Thus, an increase in the cracking moment will result in an increase of stiffness when the reinforcement ratio is the same. 5.2.2. Influence of reinforcement ratio on stiffness of RC beams In Figure 8(b), Members A and B have the same degree of prestress but different reinforcement ratios. It is known from Equation (1) that they also have the same cracking moment Mcr. The straight-slope of the second working stage is mainly concerned with reinforcement ratio r: the larger r is, the larger the slope of the line. In other words, if rB> rA, the stiffness of Member B on the second stage will be larger than that of Member A. Therefore, provided with the same cracking

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moment Mk, the cross-section stiffness of Member B will be larger than that of Member A, or tgbB>tgbA.


5.2.3. Influence of bond behaviour on stiffness of RC beams Compared with bonded PRC beams, unbonded PRC beams have comparatively lower stiffness, given the absence of bonding between high strength steels and concrete. This means that the bond characteristic of prestressed steel must be considered. Thus, for unbonded PRC beams, the ratio of the unbonded reinforcement parameter and the composite method needs to be considered as one of the cracking stiffness parameters that refer to the formula for bonded PRC beams (Rao & Dilger, 1992a, 1992b; Scholz, 1991). 5.3. Influence of prestressed steel wire ropes on stiffness of RC beams Load–deflection curves from cracking load Pcr to 2.2Pcr of Beams L1, L4, L5 and L9 are shown in Figure 9(a). When the load was 145 kN, the displacements of Beams L1, L4, L5 and L9 were 4.81, 3.21, 3.15 and 2.05 mm, respectively, which demonstrate an obvious stiffness improvement after strengthening. The main

reasons are: (1) application of prestress increased the cracking load, which consequently improved the stiffness; and (2) strengthening with steel wire ropes was equivalent to an increase in the reinforcement ratio of the member, thus improving the stiffness through improving the stiffness on the second stage. Based on previous analysis, the moment–curvature relationship of beams strengthened with high strength steel wire ropes is presented in Figure 9(b). Each steel wire rope of Members A and B had the same effective stress, while Member B had more steel wire ropes. Hence, better stiffness of Member B was mainly evidenced by its larger cracking load and high cross-section reinforcement ratio. In addition, by comparing test curves of Beams L5 and L9 with those of Beams L4 and L5, the degree of prestress has a greater influence on the stiffness of beams strengthened with prestressed steel wire rope than the reinforcement ratio and bond behaviour. This is because the area of the steel wire ropes was smaller than that of the existing steels of ordinary strengthened beams. 5.3.1. Short-term stiffness evaluating method of crosssection of strengthened beams (Chinese code) Based on the above analysis, the short-term stiffness evaluation equation of cross-section of beams strengthened with high strength steel wire rope should take the following points into consideration: (1) the contribution of prestress to stiffness; (2) the contribution of all longitudinal tensile reinforcement to stiffness and (3) the influence of bond behaviour between steel wire ropes and concrete on stiffness. For bonded and unbonded prestressed RC beams strengthened with high strength steel wire rope, postcracking Ie can be calculated through Equation (6): Ie ¼ o¼

1:0 þ

0:85Ig kcr þ ð1 kcr Þo

ð6Þ

0:21 1 þ 0:45gf 0:7 aE re

ð7Þ

re ¼

As þ kp Aps bd

As Es þ Aps Eps aE ¼ As þ Aps Ec

Figure 9. Cross-sectional stiffness of beams strengthened by P-SWR. (a) Test results; (b) theoretical curves.

ð8Þ ð9Þ

where kcr is Mcr/Mk, which is the ratio of the cracking moment Mcr to the moment on standard load effect combination Mk (when kcr>1.0, assume 1.0.); and o is a synthesis parameter reflecting reinforcement and cross-section characteristics of the member, which can be calculated by Equation (7) (MOHURD, 2002).


Structure and Infrastructure Engineering Moreover, some revisions are provided: (1) the contribution of prestress to stiffness is demonstrated through kcr, which is Mcr/Mk in the equation, with Mcr calculated by Equation (1); (2) the contribution of longitudinal tensile reinforcement to stiffness is considered through the total reinforcement ratio re; (3) a bonding property adjustment coefficient kp is suggested as a coefficient to describe the influence of bond behaviours of steel wire ropes and concrete on stiffness, whose value is 0.5 when steel wire ropes and concrete are totally unbonded, 1.0 when they are excellently bonded, and between 0.5 and 1.0 otherwise; (4) the value of o is calculated by Equation (7), where gf is the ratio of the cross-sectional area of tensile flange to the effective crosssectional area of the web. gf is equal to 0 for a rectangular cross-section, and aE is the ratio of transformed elastic modulus of reinforcement to concrete elastic modulus, which can be calculated by Equation (9). 5.3.2. Modified ACI code The ACI Code stated that reduction in flexural stiffness of the member due to cracking can be taken into account in the deflection calculation by the wellknown ‘effective moment of inertia’ Ie method. Effective moment of inertia Ie can be calculated by Equation (10) (Rao & Dilger, 1992b): ( " # ) Mcr 3 Mcr 3 Ig þ 1 Icr Ig Ie ¼ Mk Mk

ð10Þ

The calculation of Icr is a complicated procedure because the distance y from the top of section to the neural axis should be solved through a quadratic equation. Once the neutral axis is located, the moment of inertia Icr may be found using the familiar transfer formulation from engineering mechanics.

169

Taking a rectangular section strengthened with PSWR, for example, the neutral axis is located as follows with reference to the transformed section shown in Figure 10 (Spiegel & Limbrunner, 1998): ! rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e ns As þ nps Aps 1 þ n A 2bd 1 ð s s þnps Aps Þ y¼ ð11Þ b The moment of inertia of the cracked section may now be found as follows: Icr ¼

by3 þ ns As þ nps Aps ðde yÞ2 3

ð12Þ

where ns is the modular ratio Es/Ec; nps is the modular ratio Eps/Ec and de is the distance from extreme compression fibre to the transformed centroid of the existing reinforcing steel bars and prestressed steel wire ropes. If the beam cross-section contains compressed steel, this steel may also be transformed and the neutral-axis location and cracked moment-of-inertia calculations can be carried out as before. 5.4. Comparison between calculated value and test value Once the effective moment of inertia is determined, the member deflection may be calculated using standard deflection expressions. For example, the deflection of three dividing point loading beams in this paper can be obtained using Equation (13). D¼

23Fl3 1296Ec Ie

ð13Þ

Figure 10. Method of transformed section and effective area. (a) Beam cross-section; (b) transformed cross-section. (c) Effective area.

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where F is concentrated force and l is the distance between two bearings as shown in Figure 5; Ec is the modulus of elasticity for the concrete. A comparison between the calculated values suggested by this article and the test values of each beam when the moments at the midspans were 1.4, 1.8 and 2.2Mcr is shown in Table 4, where deflection of the unstrengthened beam is calculated with the standard RC formula. It can be seen that calculated values are similar to the test values, with a discreteness ratio within 10%. Moreover, after the value of short-term stiffness of cross-section of beams strengthened with steel wire rope is calculated, the long-term stiffness of RC beams can settle according to the suggested method on current code, and consequently deflection can be computed according to the ‘minimum stiffness principle’. 6.

reinforced concrete members with deformed bar reinforcement. It is a general agreement that stress in reinforcement, cover, type of reinforcement and its distribution, and area of concrete in tension are some of the important variables to be considered in estimating crack width of reinforced members. According to classical slip theory, crack width, which is related to the relative slip of rebar between cracks and concrete, is mainly influenced by the ratios of steel diameter to reinforcement ratio db/r (db is the nominal diameter of steel bar and r is the reinforcement ratio) and fy/Es. However, according to no-slip theory, crack width, which is related by asymmetry deformation of steel around reinforcement rebar in cross-section of cracks, is influenced mainly by concrete cover depth dc and fy/Es. The Chinese Code considers both of these two theories (MOHURD, 2002). Two important parameters, average crack spacing and stress of steel bars are included. The average crack spacing considers the influence of db/r and concrete cover depth dc, and the stress of non-prestressed steel bars or stress increment of prestressed steel bars considers the influence of the parameter fy/Es. The ACI Building Code uses stress in reinforcement, cover and concrete area per bar as the main variables.

Analyses of flexural cracks

6.1. Test results Compared with those on the unstrengthened beam, cracks on beams strengthened with high strength steel wire rope extended more slowly, but occurred more often with a smaller width under the same load (Wu et al., 2010). Figure 11 shows the average maximum width of four cracks on Beams L1, L4, L5 and L9 in the pure bending section at midspan. When the load value was 120 kN, the average crack widths of Beams L1, L4 and L5 was 0.15, 0.09 and 0.06 mm, respectively. When the load value was 200 kN, Beam L1 failed and the average crack widths of Beams L4, L5 and L9 was 0.205, 0.155 and 0.08 mm, respectively. Therefore, prestressed steel wire rope strengthening has an evident effect on the extension of cracks.

6.2.1. Average crack width Figure 12 shows the crack distribution of some beams. For the unstrengthened Beam L1, crack distribution was recorded when the load value was 100 kN, and for Beams L4 and L5, crack distribution was recorded when the load value was 140 kN, when almost all the cracks had appeared. According to the statistics, average crack spacing of Beams L1, L4 and L5 was 143, 125 and 109 mm, respectively. The influence of strengthening on average crack spacing was well displayed by these changes. Beam L4 was strengthened by unbonded steel wire ropes; its value of r increased, thus resulting in smaller crack spacing than Beam L1. For Beam L5,

6.2. Influence of strengthening on crack width Experimental studies by many researchers have provided vital clues to cracking the behaviour and parameters that affect the maximum crack width of Table 4.

Comparison between test results and theoretical values of midspan deflection (Unit: mm).

Beams

Loads

Test value

Method 1

Error %

Method 2

Error %

Beam L1

1.4Mcr 1.8Mcr 2.2Mcr 1.4Mcr 1.8Mcr 2.2Mcr 1.4Mcr 1.8Mcr 2.2Mcr 1.4Mcr 1.8Mcr 2.2Mcr

1.35 2.10 2.80 2.70 4.02 5.40 2.40 3.72 5.14 3.02 4.92 6.73

1.37 2.04 2.72 2.79 4.09 5.40 2.60 4.00 5.32 3.40 5.16 6.76

71.8 2.6 3.0 73.2 71.8 0 78.3 77.9 73.5 712.6 74.8 70.4

1.26 1.92 2.53 2.58 3.85 5.04 2.44 3.65 4.77 3.30 4.87 6.32

6.7 8.8 9.8 4.5 4.2 6.7 71.7 1.9 7.1 79.3 1.1 6.1

Beam L4 Beam L5 Beam L9

Structure and Infrastructure Engineering after being strengthened with steel wire rope to increase the value of r, a high performance mortar on beam soffit added the depth of concrete cover dc, which made crack spacing even smaller than Beam L4. These results also imply that bond characteristics of non-prestressed reinforcement should be taken into account in the crack-width prediction equation. 6.2.2. Steel stress The stress of prestressed steel bars or stress increment of non-prestressed steel bars is a main factor that influences the crack width of RC beams and is directly related to the area of reinforcement rebar. Figure 13

shows the load–strain curves of steel bars of Beams L1, L4, L5 and L9 at the normal use stage. When the external load reached 130 kN, the strains of steel bars of Beams L1, L4, L5 and L9 were 1945, 710, 1060 and 523 me , respectively. The strain of the steel bars of Beam L4 was smaller than that of Beam L5 because of its larger initial prestress. This trend was also found to be true under other load values, which means that the steel bars of Beam L1 had the fastest stress increment, while Beam L9 had the slowest. In short, the prestress steel wire rope strengthening postponed the appearance and extension of cracks, and also enabled slower stress increase and smaller average crack width and spacing under the same load. 6.3.


171

6.3.1.

Calculation of crack width Modified Rao & Dilger method

Rao and Dilger (1992a) collected test data on 245 beams and obtained a simple expression for estimating the maximum crack width in cracked prestressed members using statistical analysis: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ð14Þ wmax ¼ kfs dc At As þ Aps

Figure 11.

Load–crack width curves of beams.

Figure 12. Average crack spacing of beams L1, L4 and L5.

Figure 13.

Load–steel strain curves of beams.

where k is the coefficient considering bond characteristics of different types of reinforcement. For example, it can be taken as 3 6 1076 for combination of deformed bars and strands and for strands only. fs is the steel stress after decompression; dc is concrete cover to centre of the nearest reinforcement bar; At is concrete area in tension below neutral axis; and As and Aps are the areas of non-prestressed and prestressed steel, respectively. This method takes into consideration the bond characteristics of both prestressed and non-prestressed reinforcement in estimating the maximum crack width and can be applied to all practical combinations of prestressed (pretensioned or post-tensioned, bonded or unbonded, strands or wires) and non-prestressed reinforcement (deformed bars, strands or wires). The disadvantage of this method is that fs and the neutral axis depth must be obtained by a rational crackedsection analysis. Though good accuracy can be obtained using the modified Rao and Dilger method, stress analysis of the cross-section of cracks is needed to calculate the stress of prestressed steel wire ropes or the stress increment of non-prestressed steel bars, which is quite complicated because the position of neutral axis is a function of load for some cross-sections of cracks. Despite the availability of the simplified methods, the procedure remains complicated (Rao & Dilger, 1992a). Therefore, it is necessary to put forward simplified calculating methods with comparatively good accuracy.

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6.3.2. Hypothetical tension stress method The estimation of steel stress on a cracked prestressed member is more complicated than on reinforced members. Disregarding the presence of steel in uncracked section analysis, a simple crack-width formula could be evolved from the hypothesis that the tension force of an uncracked unreinforced concrete section must be carried by both bonded non-prestressed and prestressed reinforcement based on a set of assumptions called the ‘hypothetical tension stress method’. The following equation is suggested by Scholz (1991): At ft0 wmax ¼ zs P 2 As Es

f0t ¼

where As is As þ Aps and represents the area of all bonded tension steel and f t0 is the hypothetical tension stress of concrete (Wu et al., 2010). Further simplifications are possible for a symmetrical section. For instance, it is reasonable to assume that At represents the area within the lower 40% of the section depth. In addition, from experimental data of partially presented members, a mean crack spacing z s of 110 mm may be substituted. Scholz (1991) also stated that the value z s could be updated provided with a more precise prediction method. Because the mean crack spacing z s is mainly related to the thickness of the concrete cover, steel diameter and reinforcement ratio, the Chinese code (MOHURD, 2002) is adopted in this paper since lcr (Equation (16)) considers the influence of cover, steel diameter and reinforcement ratio: deq rte

ð16Þ

where deq is the equivalent diameter of tensile longA þk A itudinal reinforcement; rte ¼ s Atp ps , which is calculated based on the effective concrete tensile section area. For a rectangular section, rte can be taken as 0.5bd. The average crack width of Beams L4 and L5 is 121 and 110 mm, respectively, according to the equation, while the test values are 125 and 109 mm, respectively, which are in good agreement with the calculated values. Equation (17) is suggested to predict the maximum crack width of RC beams strengthened with P-SWR: At ft wmax ¼ lcr P 2 As Es þ Aps Eps

ð18Þ

where Atr and Itr are the area and moment of inertia of the uncracked transformed section. The simplicity of this method is evident in the use of uncracked section properties, even though the section is cracked under the action of loads. 6.3.3.

lcr ¼ 1:9dc þ 0:08

Pe Pe e Mk þ yb yb Atr Itr Itr

ð15Þ

P


sufficient strength to stay uncracked. The fictitious concrete stress is obtained as (Rao & Dilger, 1992a):

ð17Þ

In this approach, a crack control is achieved by limiting the concrete stress that would occur at the extreme tensile fibre of the member with concrete of

Simplified evaluating method

Based on previous analysis, factors affecting the crack width of RC beams strengthened with steel wire rope are mainly: (1) the concrete on beam soffits has to counteract the pre-pressure set in the tensile zone before cracking; (2) an area increase of steel wire ropes brings a slow stress-increment for non-prestressed steel bars; (3) for the existence of mortar, bonded steel wire rope strengthening is able to restrict crack extension and decrease average crack spacing better than unbonded steel wire rope strengthening does. Based on these factors, a simplified formula, Equation (19), is proposed. As M0 a 1 wmax ;0 ð19Þ wwax ¼ kp Aps þ As Mk where wmax,0 is the maximum crack width of the corresponding unstrengthened beam calculated by the RC standard formula; M0 is the moment when the precompressive stress of concrete on the tensile edge of the cross-section is counteracted to be zero, i.e. the value of Mk when f0t ¼ 0 in Equation (18). When M0 ¼ 0, the beam is a non-prestressed RC beam, so wmax ¼ wmax,0; when M0 is the maximum moment Mk under the service load–effect combination, wmax ¼ 0. The stress of steel bars is calculated according to that of RC beams, which cuts down the workload to a great extent. One point to be noticed is that stress of steel bars used is a nominal stress value while calculating wmax,0 of the unstrengthened beam; thus, it is allowed to exceed the value of its yield strength. a is uncertain and is approximately 2.0 according to the test data of this paper. For the remainder of this paper, it is assumed to have the value 2.0, but a more accurate value needs to be statistically analysed on more test data. This method is suitable for members of whose maximum crack width is known before strengthening; we can estimate the maximum crack width after strengthening quickly, and then the scheme is analysed and determined.

Structure and Infrastructure Engineering Table 5.

Comparison between test results and theoretical values of mean crack width (Unit: mm). BeamL1

Load (kN)


60 80 100 120 140 160 180 200 220 240 260 280

173

BeamL4

BeamL5

BeamL9

wt

wc,1

wt

wc,1

wc,2

wc,3

wt

wc,1

wc,2

wc,3

wt

wc,2

wc,2

wc,3

0.08 0.10 0.11 0.15 0.20 – – – – – – –

0.08 0.11 0.13 0.17 0.20 – – – – – – –

– – – 0.09 0.11 0.14 0.18 0.21 0.26 – – –

– – – 0.09 0.11 0.13 0.16 0.18 0.22 – – –

– – – 0.08 0.10 0.12 0.15 0.18 0.20 – – –

– – – 0.09 0.11 0.14 0.17 0.20 0.23 – – –

– – – 0.06 0.08 0.11 0.13 0.16 0.23 – – –

– – – 0.08 0.10 0.12 0.14 0.17 0.21 – – –

– – – 0.07 0.09 0.11 0.13 0.15 0.18 – – –

– – – 0.08 0.10 0.13 0.16 0.18 0.22 – – –

– – – – 0.03 0.05 0.06 0.08 0.11 0.13 0.17 0.23

– – – – 0.06 0.07 0.09 0.12 0.14 0.16 0.18 0.21

– – – – 0.05 0.07 0.09 0.11 0.12 0.14 0.16 0.18

– – – – 0.06 0.07 0.09 0.12 0.14 0.17 0.19 0.22

6.4. Comparison between calculated value and test value A comparison between calculated values and test values can be seen in Table 5, where wt is the test value of the average maximum crack width of each beam; wc,1 is the calculated value according to the Modified Rao & Dilger method; wc,2 is the calculated value according to the hypothetical tension stress method and wc,3 is the calculated value according to the simplified calculating method. Test values are the average width of four cracks in pure bending section at midspan. Hence, the calculated value of the maximum crack width under corresponding load using methods above should be divided by 1.5 (MOHURD, 2002) to get the calculated value of average crack width under each degree of load, the coefficient of 1.5 was based on the statistical regression of a large number of test data of the maximum crack width and average width. As shown in the table, though with some exception, most of the calculated values of crack width correspond well with the test values. From the comparison, we can show that Method 1 has better precision, but the calculation is complicated since stress of the reinforcement and height of the cracked section are calculated by non-linear analysis of the cracked section. Method 2 is suitable for engineering application with simple calculation and good accuracy. Method 3 is more useful for comparison and determination of a strengthening scheme based on calculation data of unstrengthened beam, obtained by the maximum crack width of the unstrengthened beam timing and the corresponding reduction factor. In addition, for beams strengthened with steel wire rope, mortar cover often cracks earlier than concrete because of its weaker tensile strength, and corrosion has more influence on steel wire ropes for their small diameters and tension endured than on steel bars in

RC beams. This is important to consider during strengthening design and construction, where relevant measures may be taken to ensure safety of structure. 7.

Conclusions

Strengthening reinforced concrete structures with PSWR with large ultimate strain and low relaxation can improve their crack load and ultimate bearing capacity, while still following the classical theories for RC or PRC beams. The nominal yield strength can be assumed to be the strength of the steel wire rope while calculating ultimate bearing capacity. Furthermore, various factors contribute to crosssection stiffness of beams strengthened by prestressed steel wire rope, such as degree of prestress, crosssection reinforcement ratio, and bonding properties of steel wire rope and reinforced concrete. It is suggested that the contribution of degrees of prestress to stiffness be measured by coefficient kcr and the contribution of longitudinal tensile steel bars after strengthening and bonding properties of steel wire ropes and reinforced concrete to stiffness be represented by the equivalent reinforcement ratio re. In this way, the midspan deflection of beams strengthened by prestressed steel wire rope can share the method for calculating deflection of bonded prestressed RC beams, which is easy and whose results correspond well with test results. Steel wire rope strengthening can slow down the stress increment of steel bars, reduce the average crack spacing and decrease the crack width. Based on the parametric studies, the entire comparatively accurate evaluation system, which includes the methods of modified Rao and Dilger calculation and hypothetical tension stress, as well as a simplified method for predicting the maximum crack width, is proposed.

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Acknowledgements The authors would like to acknowledge financial support from the National Natural Science Foundation of China (No. 51078077, No. 51178099), the Project funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD) and the Fok Ying-Tong Education Foundation in the Higher Education Institutions of China (No. 122011).


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Lorenc, W., & Kubica, E. (2006). Behavior of composite beams prestressed with external tendons, experimental study. Journal of Constructional Steel Research, 62, 1353– 1366. Martin, J.A., & Lamanna, A.J. (2008). Performance of mechanically fastened FRP strengthened concrete beams in flexure. Journal of Composites for Construction, ASCE, 12, 257–265. MOHURD. (2002). Code for design of concrete structures. Beijing: Ministry of Housing and Urban-Rural Development of the People’s Republic of China. [in Chinese]. Rao, S.V.K.M., & Dilger, W.H. (1992a). Control of flexural crack width in cracked prestressed concrete members. ACI Structural Journal, 89, 127–138. Rao, S.V.K.M., & Dilger, W.H. (1992b). Evaluation of short-term deflections of partially prestressed concrete members. ACI Structural Journal, 89(1), 71–78. Scholz, H., (1991). Simple deflection and cracking rules for partially prestressed members. ACI Structural Journal, 88, 199–203. Spiegel, L., & Limbrunner, G.F. (1998). Reinforced concrete design (4th ed.). Englewood Cliffs, NJ: Prentice Hall. Thanoon, W.A., Jaafar, M.S., Kadir, M.R.A., & Noorzaei, J. (2005). Repair and structural performance of initially cracked reinforced concrete slabs. Construction and Building Materials, 19, 595–603. Teng, J.G., De Lorenzis, L., Wang, B., Li, R., Wong, T.N., & Lam, L. (2006). Debonding failures of RC beams strengthened with near surface mounted CFRP strips. Journal of Composites for Construction, ASCE, 10, 92– 105. Wu, G., Wu, Z.S., Jiang, J.B., Tian, Y., & Zhang, M. (2010). Experimental study of RC beams strengthened with distributed prestressed high strength steel wire rope. Magazine of Concrete Research, 62, 253–265. Wu, Z.S., Yuan, H., Asakura, T., Yoshizawa, H., Kobayashi, A., Kojima, Y., & Ahmed, E. (2005). Peeling behaviour and spalling resistance of bonded bidirectional fiber reinforced polymer sheets. Journal of Composites for Construction, ASCE, 9, 214–226.

Structure and Infrastructure Engineering

Structure and Infrastructure Engineering

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