## Friday, May 10, 2013

### THE EFFECT OF TENSION STIFFENING ON THE BEHAVIOUR OF REINFORCED CONCRETE MEMBERS

1.INTRODUCTION
The bending stiffness of reinforced concrete beams under service loads is considerably smaller than the stiffness calculated on the basis of uncracked cross sections. This is because the beam contains numerous tensile cracks. Yet, at the same time, the stiffness is significantly higher than that calculated when the tensile resistance of concrete is neglected. This phenomenon, often termed tension stiffening, is attributed to the fact that concrete does not crack suddenly and completely but undergoes progressive micro cracking. Immediately after first cracking, the intact concrete between adjacent primary cracks carries considerable tensile force, mainly in the direction of the reinforcement, due to the bond between the steel and the concrete. The average tensile stress in the concrete is a significant percentage of the tensile strength of concrete. The steel stress is a maximum at a crack, where the steel carries the entire tensile force, and drops to a minimum between cracks. The bending stiffness of the member is considerably greater than that based on a fully cracked section, where concrete in tension is assumed to carry zero stress. This tension stiffening effect may be significant in the service load performance of beams.
Previous numerical studies have found that the tension stiffness of high-strength concrete (HSC) is lower than that of normal-strength concrete (NSC). Other variables such as the percentage and distribution of reinforcing steel, bar size, bond properties and shrinkage of concrete are also reported to have an effect on tension stiffening.
Tension stiffening is particularly significant in relatively lightly reinforced members, where the actual stiffness may be several times larger than the stiffness calculated on the basis of fully-cracked cross-sections, where the tensile concrete is ignored and only the embedded tensile reinforcement is considered. Tension stiffening increases with an increase in tensile stress in the concrete. Conversely tension stiffening decreases when the tensile stress in the concrete drops and, under constant load, this is caused either by cracking, by tensile creep or by a time-dependent deterioration of bond. Cracking can be caused by external loads or by restraint to imposed deformations, such as drying shrinkage. The tension stiffness of concrete plays an important role in the deformation behavior of the reinforced concrete (RC) structures in the post-cracking region of concrete.
2.MODELS FOR TENSION STIFFENING
Various methods have been proposed to account for tension stiffening in the analysis of concrete structures. These range from simple empirical estimates of the flexural rigidity of a member to assumed unloading stress-strain relationship for concrete in tension. Techniques involving an adjustment to the stiffness of the tensile steel to account for tension stiffening have also been used . An alternative approach for modelling tension stiffening is to assume that an area of concrete located at the tensile steel level is effective in providing stiffening. Figure 2.1 shows an average cross-section of a singly reinforced member. The properties of this average section are between those of the fully cracked cross section and the uncracked cross section between the primary cracks. The tensile concrete area Act, which is assumed to contribute to the beam stiffness after cracking, depends on the magnitude of the maximum applied moment M, the area of the tensile reinforcement Ast, the amount of concrete below the neutral axis, the tensile strength of the concrete (the cracking moment Mc), and the duration of sustained load. Bazant and Byung proposed a simplified equivalent transformed cross section, taken from their proposed model which is derived from the intrinsic material properties of concrete, particularly the strain-softening properties. In this simplified model, which is used in this study, the tensile resistance of concrete distributed over the tension side of the neutral axis is neglected and an equivalent tensile area Aeq and an equivalent tensile stress of concrete in this area σeq, which would yield about the same beam curvature κ, is determined. The centroid of this equivalent area coincides with that of tensile reinforcement.
Fig 2.1 Average section after cracking
The equivalent tensile area can be obtained from:
Aeq = [b(dn)2/2 + nAs1 (dn - ds1 )] x (ds2 –dn )-1 - nAs2 Eq.(2.1)
In which As1 and As2 are the area of compression and tension reinforcement, respectively, n = Es / Ec. The value of dn needed in Equation 2.1 has to be calculated from the condition of the same curvature κ. Two cases is distinguished depending on whether the tensile stress in concrete at tensile face is zero or finite1. With considering these cases, Aeq and dn could be evaluated in an iterative manner for a given bending moment. Once dn and Aeq are determined, the inertia moment of the transformed cross section can be evaluated.
3.SHORT-TERM ANALYSIS OF CROSS SECTIONS
Fig 3.1 Equivalent transformed cross section
Consider the equivalent transformed cross section in Figure 3.1. The top surface of the cross section is selected as the reference surface. The position of the centroidal axis depends on the quantity of bonded reinforcement and varies with time owing to the gradual development of creep and shrinkage in the concrete. Therefore, it is convenient to select a fixed reference point that can be used in all stages of analysis.
3.1 UNCRACKED SECTION
In Figure 3.2, the strain at a depth y below the top of the cross section is defined in terms of the top fibre strain ε0i and the initial curvature κi , as follows:
εi = ε0i + y κi Eqn (3.1)
The initial concrete stress at y below the top fibre is:
σi = Ec εi = Ec (ε0i + y κi) Eqn(3.2)
Fig 3.2 Uncracked section analysis
Integrating the stress block over the depth of the section, horizontal equilibrium requires that:
Ni = ∫ σi dA
= Ec ε0i ∫ dA + Ec κi ∫ y dA
= Ec ε0i A + Ec κi B Eqn(3.3)
where A (= ∫ dA) is the area of the transformed section and B (= ∫ ydA) is the first moment of the stress block about the top surface of the section. If the first moment of the stress block about the top fibre is integrated over the depth of the section, the resultant moment about the top surface, Mi , is found. Therefore,
Mi = ∫ σi y dA
= Ec ε0i ∫ y dA + Ec κi ∫ y2 dA
= Ec ε0i B + Ec κi Ī Eqn(3.4)
where Ī ( = ∫ y2 dA ) is the second moment of the transformed area about the top surface of the transformed section. By re arranging Eqs.3.3 and 3.4, expressions are obtained for the initial top fibre strain and curvature :
ε0i = B Mi / [Ec (B2 – A Ī )] Eqn(3.5)
κi = -A Mi / [Ec (B2 – A Ī )] Eqn(3.6)
3.2 CRACKED SECTION
The instantaneous strains and stresses on a cracked section are shown in Figure2.3
Horizontal equilibrium dictates that:
Fig 3.3 Cracked section analysis
Ts – Cc – Cs + Tc = 0 Eqn(3.7)
and moment equilibrium requires that
M = Cc dz + Cs ds1 – Tc ds2 – Ts ds2 Eqn(3.8)
where Cc , Cs , Tc , and Ts can be expressed as functions of dn and ε0i :
Cc = σ0i bdn /2 = Ec ε0i bdn /2 Eqn(3.9)
Cs = Es As1 [ε0i (dn – ds1)/dn] Eqn(3.10)
Ts = Es As2 [ε0i (ds2 – dn)/dn] Eqn(3.11)
Tc = σeq Aeq = Ec Aeq [ε0i (ds2 – dn)/dn] Eqn(3.12)
By substituting Eqs. (3.8-3.12) into Eqs. (3.7) and (3.8) and solving the simultaneous equations,
ε0i and dn are found with an iterative manner. Based on the values of ε0i and dn the curvature can be calculated:
κi = - ε0i / dn Eqn(3.13)
4.TIME-DEPENDENT ANALYSIS OF CROSS-SECTIONS
During any time period, creep and shrinkage strains develop in the concrete. The time dependent change of strain at any depth y below the top of the cross section, Δε, may be expressed in terms of the change in top fibre strain, Δε0 , and the change of curvature, Δκ :
Δε = Δε0 + y Δκ Eqn(4.1)
The increments of top fibre strain, Δε0 , and curvature, Δκ , may be obtained from the following equations :
Δε0 = (Ýe δM - Īe δN) / [Ēe (Ýe2 - Āe Īe)] Eqn(4.2)
Δκ = ( Ýe δN - Āe δM) / [Ēe (Ýe2 - Āe Īe)] Eqn(4.3)
where Āe is the area of the age-adjusted equivalent transformed section and Ýe and Īe are the first and second moments of the area of the age-adjusted equivalent transformed section about the top surface. For the determination of Āe , Ýe , and Īe the age-adjusted effective modulus Ēe is used . Therefore, the total strain and curvature may be obtained from:
ε = ε0i + Δε
κ = κi + Δκ
The deflection δ at any point along a beam can be calculated by integrating the curvature κ(x) over the length of the beam:
δ = ∫∫ κ(x) dx dx
Figure 4.1 Deflection of a typical beam
Consider the beam shown in Figure 4.1. If the variation in curvature along the member, subjected to uniform load q is parabolic, then the deflection at mid-span, δC , is given by:
δc = (κA + 10 κC + κB) L2/ 96 Eqn(4.4)
where κA and κB are the curvature at each end of the span and κC is the curvature at midspan.
5.BEHAVIOUR OF HIGH STRENGTH CONCRETE TENSION MEMBERS
Reinforced high-strength concrete (RHSC) is being increasingly used in buildings and bridges because it enables the use of smaller cross-sections, longer spans, reduction in girder height and improved durability. The recent trend of employing RHSC with a concrete strength of over 100 MPa has resulted in smaller member sizes which lead to higher tension stress in the reinforcement.
Direct tension tests were performed on reinforced high-strength concrete (RHSC) members. The test results showed not only were splitting cracks along the reinforcement more extensive, but also the transverse crack spacing became smaller. Thereby, the reduction in the tension stiffening effect in high-strength concrete (HSC) is much greater than that would be expected.
5.1 TENSION STIFFENING PREDICTION
A number of empirical relationships have been proposed for tension stiffening, where the loss of rigidity in a cracked member can be taken into account for the stress-strain response of the steel or an average stress-strain response for concrete in the post-cracking range. Collins and Mitchell considered a load-sharing concept to account for tension stiffening, where axial load N is carried by both the steel and concrete.
(εcr/ εm)0.4 Eqn(5.1)
β=[1+√500εm]-1 Eqn(5.2)
where, β is normalized stress, εcr is cracking strain, εm is axial member strain.
5.2 METHODOLOGY
5.2.1 Details of Materials and Specimens
Testing was carried out on ten specimens that were axially loaded. Fig. 5.1 shows the geometry and instrumentation for a typical test specimen. All of the specimens had a length of 1200 mm. A single deformed steel bar, with a minimum concrete cover of 40 mm, was provided. Tension stiffening was evaluated for NSC (40 to 60 MPa) and HSC (100 to 150 MPa) using reinforcement ratios (ρ) of 2.252% respectively. The yield strength and Young’s modulus of steel were 722 MPa and 202.5 GPa respectively. The concrete properties are shown in Table 4.1. The diameters of reinforcement bars were selected to prevent the effect of splitting cracks since they are not significant when concrete cover to bar diameter (c/db) is larger than 2.5 . Details of specimens are given in Table 5.1.
5.2.2 Loading Method
Specimens were loaded vertically through one-axial tension rods. Two linear variable displacement transducers (LVDT) were clamped to the steel reinforcing bar just outside of the concrete to measure the total elongation of the reinforced concrete specimen (Fig. 5.1). At each load stage, the cracks were measured using pi-gauges. The complete response of each specimen was described by plotting the applied tension against the average member strain.
Fig 5.1 Test Setup
Average early-age shrinkage was determined for all concretes from strain measurements on 100x100x400 mm shrinkage specimens that had the same moisture curing conditions as the tension specimen. Shrinkage was included in analysis of the member response by using the calculated shrinkage strain value from the early-age shrinkage specimens to determine the initial strain for each tension specimen (Table 5.1). The initial strain was taken as an offset strain equal to; where εsh is concrete early-age shrinkage, n is modular ratio (Es/Ec), ρ is reinforcing steel ratio, Es is Young’s modulus of steel, and Ec is elastic modulus of concrete . Shrinkage strains were assumed to be uniform over the cross section. Subtracting shrinkage strains from the member response gives an idealized concrete member strain, equivalent to a member with no shrinkage.
5.3 CRACKING BEHAVIOUR
The average tensile strength of NSC and HSC was used to gain a better understanding of cracking behavior. According to tension members’ test results, the average tensile strengths of NSC and HSC (> 100 MPa) were 0.37f’c0.5 and 0.32f’c0.5 (MPa) respectively. Also, the corresponding elastic modulus of Ec for NSC and HSC was equal to 4030f’c0.5 and 3270f’c0.5 (MPa) respectively. This results in an estimated cracking strain of 92με for NSC and 98με for HSC. However, observed cracking strains were 90με on average for NSC and 100με for HSC.
The HSC specimens exhibited a larger cracking load than NSC specimens. As the concrete strength increased roughly four-fold, from 40 MPa to 145 MPa, the transverse cracking load increased by approximately 1.3 times. However, the splitting cracking loads decreased by approximately 40%. According to the experimental results, the relative transverse cracking load in the specimens matched the increase in the splitting-tensile strength of concrete. Therefore, transverse cracks were directly affected by concrete tensile strength. However, splitting cracking strength was not affected by concrete splitting-tensile strength.
Table 5.1: Specimen properties and test results
db : Steel bar diameter
c: Concrete cover
f’c : Compressive strength of concrete
ft : Splitting tensile strength of concrete
εsh: Early age shrinkage
ε’: Offset strain
T: Transverse crack
S: Splitting crack
L: Average transverse crack spacing
As the compressive strength of concrete increased from 40 MPa to 145 MPa, the average transverse crack spacing decreased by 10-50 % in both steel bar sizes.
5.4 INFLUENCE OF CONCRETE STRENGTH ON TENSION STIFFENING
HSC specimens with a strength of 90 MPa exhibit a larger tension stiffening after cracking than the NSC specimens. This behavior is clearly attributed to the early splitting cracks and excessive progress along the rebar with the increase in load. At the concrete splitting cracks along the rebar, the bond between the bar and the concrete was diminished. Therefore, the concrete is no longer able to share the tensile force, in turn resulting in a large deformation with a small stiffening effect.
The tension stiffening reduction in HSC members can be explained using elastic theory. According to elastic theory, the bond behavior of HSC can be quantitatively drawn. Since the elastic modulus of concrete is a function of compressive strength, while that of steel remains constant, the composite structural system consisted of reinforcement and concrete is altered with concrete strength, which results in a different stress state in the interface. Furthermore, HSC is more brittle than NSC and in turn, less stress redistribution can take place at the ultimate loading stage. These two material properties in HSC change the crack spacing and tensile stiffness of tension members.
Fig.5.2 shows a comparison of the measured normalized stress (β) response with the curves predicted by Eq. (5.1) and (5.2). The predictions by Eq. (5.1) underestimates the normalized stresses of NSC members, while overestimate that of HSC (f’c>100 MPa) members. The model proposed by Collins and Mitchell also overestimates the normalized stresses of HSC members. Also, it overestimates member stresses of NSC members during the early cracking and underestimates the member stresses once cracking has stabilized. From the above results and discussion, it can be concluded that the effect of material characteristics of HSC is not accounted for by Eq. (5.1) and (5.2).
5.5PROPOSED MODEL FOR HIGH STRENGTH CONCRETE
The present experimental results show that the tension stiffness of axially loaded members is highly dependent on concrete strength. According to Fig. 8, tension stiffness of HSC members after cracking cannot be sufficiently predicted by available models. Therefore, a new model is proposed to predict normalized stress, β, of HSC tension members. A best fit to the test results is obtained by using the following prediction equation.
β = 2.5( 1- 0.135 ln εm) Eqn(5.3)
This model is valid for HSC tension members with a concrete strength over 100 MPa. Eqn (5.3) was examined by comparing with the experimental normalized stress and load-deformation curves and it can be seen that the new model provides more accurate predictions for HSC tension members.
Fig.5.2 Normalized tensile behavior of concrete specimens
6. CONCLUSION
After cracking, the concrete between the cracks carries tension and hence stiffens the response of a reinforced concrete member subjected to tension. This stiffening effect, after cracking, is refered to as tension stiffening. A simple formulation is proposed for study of short-term and long-term behaviour of reinforced concrete beams. The results of a test series of five tension specimens were analyzed and observed that the tension stiffening effect is highly dependent on concrete strength when it is greater than 100 MPa. The crack spacing between the adjacent transverse cracks becomes narrower as higher concrete strength is used, and a reduction in crack spacing was also observed when the compressive strength of concrete increased. The present tension stiffening prediction equations by Belarbi and Hsu, Collins and Mitchell are not accurate enough and needs to be modified. A more accurate tension stiffening prediction equation is suggested for the design of RHSC members with steel bar diameter of 25mm and concrete strength of over 100 MPa.
REFERENCES
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