Behavior of Reinforced Concrete Members Prone to Shear Deformations: Part I-Effect of Confinement

Behavior of Reinforced Concrete Members Prone to Shear Deformations: Part I-Effect of Confinement

Powanusorn, Suraphong

Confinement due to transverse reinforcement is acknowledged to have a positive influence on enhancing the overall performance of reinforced concrete (RC) members in terms of strength and deformability. Significant research in the literature has concentrated on the effect of confinement in RC members subjected to flexural or combined flexural/axial forces. Relatively little has been done to evaluate the effects of confinement on shear-dominated RC members. A new analytical model under the context of the Modified Compression Field Theory (MCFT) is developed to incorporate the beneficial effect of confinement due to transverse reinforcement. Results from the analytical investigation show good correlation with available experimental results on RC bent caps, where shear deformations were considered the dominant action. In addition, the confinement provided by the out-of-plane horizontal legs of the transverse reinforcement has a significant effect on the strength and deformability on shear-dominated RC members.

Keywords: bent caps; confined concrete; shear deformations.

(ProQuest-CSA LLC: … denotes formulae omitted.)

INTRODUCTION

Concrete strength, either in uniaxial or multiaxial loading, is strongly dependent on the hydrostatic pressure. Generally, the compressive strength and deformability of concrete increases as the applied hydrostatic pressure increases. In seismically active zones, use of seismic hoop reinforcement in structural joint regions, columns, and beams is required in modern building codes (ACI Committee 318 2005). The primary purpose of the seismic hoop reinforcement is to confine the core concrete and thereby create increased hydrostatic pressures that allow for significant energy dissipation by longitudinal reinforcement yielding. Numerous research studies have focused on the use of passive and active measures to provide confinement to reinforced concrete (RC) members for enhancing their axial and flexural performance, either for rehabilitation or retrofit of existing structures (Mander et al. 1988). Relatively little has been done to evaluate the effect of confinement on RC members prone to shear deformations.

An experimental program was conducted at Texas A&M University to evaluate the effect of reinforcement details on the structural performance of RC bent caps where shear was considered to be the dominant action in the load transfer mechanism (Bracci et al. 2000; Young et al. 2002). Figure 1 shows the general test setup and existing reinforcement details for the specimen RC bent caps. The failure mechanism observed with typical bent cap reinforcement details was predominantly due to shear failure, although the main longitudinal reinforcement (top steel) was loaded slightly beyond the yield limit at failure. When the amount of shear stirrup reinforcement was increased (using overlapping stirrups, as shown in Fig. 2), however, the failure load and deformability of the RC bent cap increased, given the same longitudinal reinforcement. Because overlapping stirrups have twice the reinforcement area as a single stirrup, it can be argued that the increased specimen strength and ductility is attributed to the stirrups themselves. The data from strain gauges installed on the stirrups, however, indicate that the stirrups themselves did not significantly contribute to the strength of RC bent caps. This result is in agreement with the conclusions made by Ferguson (1964) on an experimental program of similar bent caps. Therefore, it should be concluded that the overlapping stirrups helped enhance the performance of the RC bent caps through mechanisms other than being directly engaged with the applied external force demand. It is then hypothesized that the additional amount of stirrups had no direct influence on the failure mechanism of RC bent caps and the observable increase in member strength and deformation was attributed to the effect of confinement due to the transverse reinforcement as shown in Fig. 2 (Young et al. 2002). The effect of confinement helps increase the strength and deformability of concrete in compression, which helps delay the onset of compressive failure of concrete on the compression face. Given the same amount of longitudinal reinforcement, confined specimens are able to sustain larger deformations which, in turn, ensure a level of overstrength of members due to an increase in concrete strength from confinement and in longitudinal reinforcing steel strength due to strain hardening.

RESEARCH SIGNIFICANCE

The results described in this paper show that the member strength predicted by the Modified Compression Field Theory (MCFT) is slightly conservative compared to the experimental results of 16 RC bent caps with varying levels of confinement. A significant improvement in the forcedeformation response can be obtained, however, by incorporating the effect of confinement due to transverse reinforcement into the MCFT. The newly proposed model follows the same set of assumptions as the MCFT. The peak stress and strain of concrete in compression, however, are modified to account for the multiaxial state of stress in concrete. For the RC bent caps considered in this paper, results show that the analytical model provides a better estimate of the member strength and deformability. The model, however, underestimates the post-cracking stiffness of the RC bent cap members, which is addressed in a companion paper (Powanusorn and Bracci 2006).

PROPOSED ANALYTICAL MODEL

To verify the underlying hypothesis, a constitutive model for RC members under the context of the MCFT (Vecchio and Collins 1986) is proposed to incorporate the effect of confinement due to transverse reinforcement. The newly proposed constitutive model adopts the same assumptions as the MCFT that can be viewed as a smeared rotating crack model. The following assumptions are used for establishing the proposed constitutive model of RC members:

1. Directions of principal stresses and strains coincide;

2. Principal stresses can be expressed as a function of principal strains; and

3. Reinforcing steel is perfectly bonded to the adjacent concrete. Any inherent bond-slip between the concrete and reinforcement is implicitly taken into account by tension-stiffening.

Equilibrium and compatibility conditions

Equilibrium and compatibility between the principal directions and the global x- and y-directions can be expressed by using the stress-strain transformation relationships or two-dimensional Mohr’s Circles of stress and strain, as shown in Fig. 3 and 4. The following relationship can be derived.

Equilibrium-

… (1)

where σ^sub x^ is the total applied stress in the global x-direction; σ^sub y^ is the total applied stress in the global y-direction; τ^sub xy^ is the total applied shear stress in the global x- and y-directions; σ^sub c1^ is the principal concrete stress in the major direction; σ^sub c2^ is the principal concrete stress in the minor direction; σ^sub sx^ and σ^sub sy^ are the steel stresses in the global x- and y-directions, respectively; ρ^sub x^ and ρ^sub y^ are the reinforcement ratios in the global x- and y-directions, respectively; and θ is the angle between the global x-direction and direction of minor principal concrete stress or strain.

Compatibility-

… (2)

where ε^sub x^ is the strain in the global x-direction; ε^sub y^ is the strain in the global y-direction; γ^sub xy^ is the shear strain in the global x- and y-direction; ε^sub 1^ is the concrete strain in the major principal direction; and ε^sub 2^ is the concrete strain in the minor principal direction.

Constitutive model for concrete

Figure 5 shows an RC panel subjected to a biaxial state of principal stresses. In essence, the total average principal strains ε^sub 1^ and ε^sub 2^ are the superposition of the stress-induced strains ε’^sub 1^ and ε’^sub 2^ and expansion strains v^sub 12^ε’^sub 2^ and v^sub 21^ε’^sub 1^ as shown in Eq. (3)

… (3)

The stress-induced strains ε’^sub 1^ and ε’^sub 2^ are used to define the state of principal stresses. Therefore, the principal stresses in concrete are defined based on the algebraic sign of the stressinduced strains ε’^sub 1^ and ε’^sub 2^.

For uncracked concrete in tension, a linear elastic model is assumed

σ^sub t^ = E^sub c^ε^sub t^, ε^sub t^

where σ^sub t^ and ε^sub t^ are the tensile stress and strain, respectively; E^sub c^ is the initial modulus of elasticity of concrete which equals 4730 [the square root of]f’^sub c^ (in MPa); and ε^sub cr^ is the cracking strain associated with the cracking stress of concrete f^sub cr^ that is assumed to equal 0.33 [the square root of]f’^sub c^ (in MPa).

The post-cracking tensile stress-strain relationship of concrete, however, is significantly different for reinforced and unreinforced regions. Hordjik (1991) conducted an experimental program on unreinforced concrete subjected to uniaxial tension and proposed a post-cracking concrete tensile stress-crack width relationship as

… (5)

where σ^sub t^ is the calculated concrete stress in tension; w^sub c^ is the crack opening at the complete release of stress (in mm) w^sub c^ = 5.14(G^sub F^/f^sub cr^); G^sub F^ is the fracture energy of concrete (in MN/m) required to create a unit area of stress free crack, which is equal to the area under the curve of tensile stress and crack width (G^sub F^ = 0.000025f^sub cr^); w is the crack opening associated with the concrete is tension (in mm); and c^sub 1^ and c^sub 1^ are the material constants 3.0 and 6.93, respectively.

The crack opening w is a product between the cracking strain and the length of the localized zone, which is equal to the characteristic length of the element in FEM applications. Cracking strain is obtained from the concept of decomposition of the total strain into the concrete elastic strain and cracking strain as shown in Fig. 6.

On the other hand, the tension-stiffening field is expected for concrete in a relatively well-reinforced region. An expression proposed by Collins and Mitchell (1987) is used

… (6)

Because the unrealistic perfect bond assumption is employed, the total stress may exceed the yield stress of reinforcing steel at a crack location. Certain measures must be implemented to correct this deficiency such that the errors due to the model assumption can be controlled. Vecchio and Collins (1986) proposed the crack check process by reducing the concrete stresses such that, for a given average strain beyond the yielding strain of reinforcing steel, the total stress cannot exceed the yield stress of the reinforcing steel. This process is explained in a one-dimensional setting as shown in the following

… (7)

where ε^sub t^ is the average tensile strain; f^sub y^ is the yield stress of reinforcing steel; and σ^sub s^ is the steel stress corresponding to the average strain ε^sub t^.

For a biaxial state of stress involving reinforcing steel in both the x- and y-directions, the following equation was proposed (Vecchio and Collins 1982)

… (8)

where ρ^sub x^ are ρ^sub y^ are the reinforcement ratios in the x- and y-directions; f^sub yx^ and f^sub yy^ are the yield stresses of reinforcing steel in x- and y-directions; σ^sub sx^ and σ^sub sy^ are the calculated reinforcing steel stresses based on the given strain ε^sub x^ and ε^sub y^, respectively; and θ is the angle between the global x-direction and direction of minor principal concrete stress or strain.

On the contrary, Belarbi and Hsu (1994), following the method presented by Tamai et al. (1987), proposed to decrease the steel stresses for the crack check, while maintaining the tension-stiffening for the concrete constitutive relationship. The procedure involves modifying the yield stress and postyield stiffness for the embedded reinforcement. The mathematical form of the apparent yield stress of the embedded reinforcement is as follows

f’^sub y^ = (0.93 – 2B)f^sub y^

… (9)

where f’y apparent yield stress of embedded reinforcement and ρ is the steel reinforcement ratio.

Belarbi and Hsu (1994) used the value of 2% of the initial elastic modulus of reinforcing steel for the post-yield stiffness. This post-yield stiffness of an embedded bar is also affected by the tension stiffening of concrete. The complete constitutive relationship of embedded reinforcement, including the preand post-yield range is given by (Belarbi and Hsu 1994)

… (10)

where σ^sub s^ and ε^sub s^ are the average stress and strain in the embedded reinforcement, respectively.

Note that Eq. (10) and (9) are presented in generic forms of ε^sub s^ and ρ, respectively. The average stresses of the embedded bars in the x- and y-directions can be obtained by substituting ε^sub x^, ρ^sub x^, or ε^sub y^, ρ^sub y^ for ε^sub s^ and ρ in Eq. (9) and (10). Eq. (10), however, is derived using tension stiffening different from Eq. (6) and the post-peak stiffness of 2% of initial stiffness for the reinforcement.

Based entirely on numerical facilitation, the method proposed by Belarbi and Hsu (1994) is implemented in this work in favor of the crack-check process proposed by Vecchio and Collins (1982). From FEM analyses performed during the initial studies of the work presented herein, however, the two methods yield comparable results.

For concrete stress-strain behavior in compression (σ^sub c^ – ε^sub c^), an equation proposed by Mander et al. (1988) is adopted and given by

… (11)

where f’^sub cc^ is the peak concrete stress considering the effect of confinement; x = ε^sub c^/ε^sub cc^; ε^sub cc^ is the concrete strain at peak concrete stress; r = E^sub c^/(E^sub c^ – E^sub sec^); and E^sub sec^ is the secant modulus of elasticity determined at the peak stress = f’^sub cc^/ε^sub cc^.

For a state of biaxial stress as shown in Fig. 5, the effect of confinement is mobilized only when both stress-induced strains are in compression. In cases where the other component of strain is in tension, there is no effect of confinement and f’^sub cc^ = f’^sub c^. As reported by Vecchio and Collins (1986), however, concrete stress must decrease as a result of the effect of the transverse tensile strain. Therefore, Eq. (11) is modified as

… (12)

where β = peak stress softening factor =

… (13)

Determination of peak stress and strain of concrete in compression

According to Eq. (11), the most essential element to incorporate the effect of confinement for concrete subjected to multiaxial state of compressive stress is to identify the peak stress and strain for the base curve. Vecchio (1992) defined the peak stress and strain for concrete in compression using the failure envelope for concrete subjected to biaxial compressive stress. A provision was also proposed to extend the biaxial failure envelope to triaxial state of stress. The strength enhancement factor K is defined as the ratio between the adjusted peak compressive stress and unconfined compressive stress of concrete. Vecchio (1992) also adopted the same factor to modify the base curve peak strain. A more general approach to define the peak stress under a triaxial state of stress is possible, however, using a three-dimensional failure surface (Mander et al. 1988).

In this research, the five-parameter failure surface proposed by Willam and Warnke (1974) is adopted to define the peak stress of concrete under multiaxial compressive stress. The triaxial test results from Schickert and Winkler (1977) are used to define the failure envelope. Detailed derivations can be obtained in Powanusorn and Bracci (2003).

Currently, however, there is insufficient test data to uniquely quantify this peak strain enhancement factor. Vecchio (1992) applied the same stress enhancement factor to modify the peak strain for incorporating the confinement effect into the MCFT. Nonetheless, many experiments show that, in general, the peak strain enhancement factor is larger than the stress enhancement factor (Kupfer et al. 1969). Based on this evidence, the strain enhancement factor adopted in this research is based on the work of Mander et al. (1988)

… (14)

where ε^sub o^ is the strain associated with the peak stress from a uniaxial compression test.

Constitutive relationship for reinforcing steel

An elastic plastic strain-hardening assumption is used for modeling the constitutive relationship for the reinforcing steel. The post-yield tangent modulus of the reinforcing steel is assumed to be 3.0% of the initial elastic modulus.

NUMERICAL IMPLEMENTATION

There are two significant steps in the numerical implementation in FEM: 1) evaluation of tangent or secant constitutive matrix[D^sub sec^]; and 2) determination of return stress vector {σ}.

Secant Stiffness Matrix Formulation

Concrete-For softening materials like concrete, the tangent constitutive matrix may become negative after the peak stress is reached. This negative value of tangent constitutive matrix can introduce numerical instability into the model, which leads to bizarre results such as force convergence oscillation or, in some cases, non-convergence. Vecchio (1989) proposed the use of secant constitutive matrix for concrete in favor of the tangent stiffness.

Consider a given state of principal strain (ε^sub 1^ and ε^sub 2^), the total strain in each direction (shown in Eq. (3)) can be decomposed into two components: 1) the stress-induced part; and 2) the part due to the Poisson’s effect.

By defining E^sub c1^ = σ^sub c1^/ε’^sub 1^ and E^sub c2^ = σ^sub c2^/ε’^sub 2^, Eq. (3) can be rewritten as

… (15)

By inverting Eq. (15), the secant stiffness matrix of concrete in the principal direction is

… (16)

As treated by Crisfield and Wills (1989) and Willam et al. (1987), an additional relationship for shear stiffness is required to account for possible crack rotation. Vecchio (1989) proposed the use of … in the early model, which neglects the Poisson’s effect. Zhu and Hsu (2002) showed that a secant shear-stiffness for fixed crack model had the same form as the tangent shear stiffness of the rotating crack angle model defined as G^sub c^ = (σ^sub c1^ – σ^sub c2^)/(2(ε^sub 1^ – ε^sub 2^)). In this research, the tangent shear stiffness is also used. Preliminary numerical studies show that the use of this tangent shear stiffness results in a more numerically stable solution. The algorithmic secant constitutive matrix becomes

… (17)

where …

Note that the secant stiffness of concrete shown in Eq. (17) is, in general, nonsymmetric and represents the secant constitutive matrix in terms of the principal coordinate. The secant constitutive matrix in the global x-y coordinate is obtained through coordinate transformation

[D^sub c^]^sup XY^^sub sec^ = [T]^sup T^[d^sub c^]^sup 12^^sub sec^ [T] (18)

where

… (19)

Reinforcing steel-The secant stiffness matrix for smeared reinforcing steel can be obtained in a straightforward manner. By definition, the secant stiffness E^sup sec^^sub s^ is defined as the ratio between the current stress and strain. Taking into account the reinforcement ratio in each direction, the secant constitutive matrix for smeared reinforcing steel is

… (20)

Total-The total secant constitutive of an RC membrane is obtained by adding Eq. (18) and (20)

[D]^sup XY^^sub sec^ = [D^sub c^]^sup XY^^sub sec^ + [D^sub s^]^sup XY^^sub sec^ (21)

Return stress vector

For the case of the stress-strain relationship of concrete, the magnitude and direction of the principal strain corresponding to the i-th iteration can be derived from the Mohr’s Circle of strain and is given by

… (22)

where ε^sub 1^ is the major principal strain; and ε^sub 2^ is the minor principal strain,

The original MCFT defined the principal stresses σ^sub c1^ and σ^sub c2^ in terms of the magnitude of the two principal strains obtained. Vecchio (1992) proposed a model that, in effect, decreases the magnitude of parameter ε^sub 1^ by taking into account the expansion due to the presence of compression in the perpendicular direction (Poisson’s effect). Figure 5 shows the effect of compression on the magnitude of tensile strain in the perpendicular direction. In essence, the total average strain, ε^sub 1^ and ε^sub 2^, are the superposition of the true applied strain, ε’^sub 1^ and ε’^sub 2^, and the expansion strain, v^sub 21^ε’^sub 1^ and v^sub 1^ε’^sub 2^, as shown in Eq. (3).

Concrete expansion model-Vecchio (1992) proposed the following model for Poisson’s ratio ? used for calculating the transverse expansion strain due to compression

… (23)

… (24)

where v^sub 0^ is the initial Poisson’s ratio of uncracked concrete, taken as 0.2.

Similarly, the effect of tensile strain also results in contraction in the transverse direction. For uncracked concrete, the magnitude of contraction can be determined by the value v0 as defined previously. For cracked concrete, experimental results (Van Mier 1986) show that, on average, the lateral contraction decreases as soon as cracking occurs, and the value asymptotically reaches zero at sufficiently high tension. Van Mier (1997) also showed that for members subjected to uniaxial tension, strains are localized at the vicinity of the crack while the remainder of the specimen unloads. Unloading of the majority of the specimen resulted in a decrease in lateral contraction with the same poisson’s ratio as uncracked concrete. For simplicity, the lateral contraction due to tensile strain of cracked concrete is assumed to be zero in this study. Vecchio (1992) and Zhu and Hsu (2002) adopted the same assumption in their analytical studies for RC membranes. Therefore, for a panel with biaxial tension-compression, the parameter ε^sub c^ or ε^sub 2^ in the case of biaxial tension-compression) used in the constitutive relationships in compression (Eq. (11) and (12)), in general, does not require any modification.

For the state of biaxial compression stress, however, Poisson’s effect contributes to the overall average strain in the transverse direction. Therefore, the variable εc in Eq. (11) should also be modified to account for the transverse expansion due to compression. Equation 3 is used to decrease the magnitude of ε^sub 1^ and ε^sub 2^ (both in compression for this case). The effect of lateral expansion and contraction is also used in the secant stiffness matrix as defined by Eq. (17).

Out-of-plane confinement due to transverse reinforcement-Consider an RC column subjected to uniaxial compression, as shown in Fig. 7. Poisson’s effect causes transverse strain perpendicular to the direction of the axial load. For members without hoop or cross-tie reinforcement, this transverse expansion occurs in a stress-free condition in the direction of the transverse strain. The presence of transverse reinforcement, however, prevents this free expansion because this transverse expansion causes tension in the transverse reinforcement. This results in a self-equilibrated stress condition in the transverse direction because the sum of concrete and reinforcing steel stress must be zero, as shown in Fig. 7. Therefore, compressive stress occurs in concrete to balance out the tensile stress in the transverse reinforcement. In other words, the presence of transverse reinforcement causes confining stress to the inner core concrete when the member is subjected to an axial load. Mander et al. (1988) quantified the effect of this confining stress on the performance of columns with varying reinforcement ratios and configurations, and concluded that the overall stress-strain relationships of the inner core concrete are enhanced. Analytical expressions of the effect of confinement proposed by Mander et al. (1988) are shown in the previous section. The magnitude of the confining stress due to transverse reinforcement, however, is based on the yield stress of the transverse reinforcement. For a low level of applied uniaxial stress, this assumption may be violated. Nonetheless, the presence of the confining stress does not have a significant influence on the change in pre-peak stress-strain curve at low levels of applied stress. Therefore, Mander’s method generally yields good results for practical applications when the main object of the study is near the post-peak performance of an RC column.

In this study, the procedure proposed by Mander et al. (1988) is extended to two-dimensional stress analysis and the confining stress is determined by the amount of Poisson’s expansion in the transverse direction. Because the model formulation is based on two-dimensional stress analysis, the restraining effect due to the vertical legs of the stirrups (refer to Fig. 2) is automatically satisfied. The effect of the horizontal legs that provide the out-of-plane confinement, however, requires further clarification. A special effort is required to incorporate the out-of-plane horizontal stirrup legs into the constitutive model. By extending Eq. (4) in three-dimensions, the following result is obtained

… (25)

Using equilibrium and compatibility conditions in the out-of-plane direction, the strain of smeared reinforcement steel must be equal to ε3, while the summation of stress in this out-of-plane direction must vanish. In other words

σ^sub c3^ + σ^sub s3^ = 0 (26)

Theoretically, the concrete stress-induced strain ε’^sub 3^ can be solved iteratively. To facilitate the numerical implementation, however, it is assumed that ε’^sub 3^ equals 0. This assumption is justified only in cases where the reinforcement ratio in the out-of-plane direction is low, which implies that confinement stress in the out-of-plane direction is also low. From this assumption, the confinement stress σ^sub c3^ can be determined directly from the total out-of-plane strain ε^sub 3^.

Note that Vecchio (1992) also proposed a slightly different expression for the out-of-plane expansion involving the out-of-plane reinforcement ratio. This simplified assumption should also yield reasonable results because the out-of-plane reinforcement is relatively low.

After taking into account the effect of concrete expansion and confinement, the magnitude of concrete stress in the principal direction can be obtained by substituting the appropriate parameters into Eq. (5) and (6), and (11) to (13). The concrete return stress in the global x-y coordinate is determined by the stress transformation relationship, as shown in Eq. (1).

In summary, the determination of the concrete return stress from a given state of strain ε^sub x^, ε^sub y^, γ^sub xy^ becomes more involved that the originally proposed MCFT. In general, the process is similar to that proposed by Vecchio (1992), taking into consideration the effect of concrete expansion and confinement. Figure 8 shows a flowchart summarizing the steps for calculating the concrete return stress.

Note that in the step for determining σ^sub c1^ and σ^sub c2^ from the given ε’^sub 1^, ε’^sub 2^, and σ^sub c3^, three possible scenarios arise:

1. Both ε’^sub 1^ and ε’^sub 2^are tensile. In this case, concrete tensile stress in the 1- and 2-directions can be obtained directly by Eq. (5) or (6);

2. ε’^sub 1^ is tensile and ε’^sub 2^ is compressive. This corresponds to the biaxial compressive stress condition and one tension in the triaxial stress state. The concrete tensile stress in the 1-direction is obtained as in the first case. The compressive stress in the 2-direction, however, requires the use of the out-of-plane confining stress σ^sub c3^ to determine the peak stress and strain taking into account the effect of confinement (Powanusorn and Bracci 2003). The unsoftened concrete stress is calculated from Eq. (11). The final stress σ^sub c2^ is obtained by applying the softening factor (Eq. (13)) to the unsoftened concrete stress; and

3. Both ε’^sub 1^ and ε’^sub 2^ are compressive. The determination of σ^sub c2^ from Eq. (11) requires a prior knowledge of two confining stresses: (a) the known out-of-plane confining stress σ^sub c3^; and (b) the unknown stress σ^sub c1^, or vice versa. Therefore, the process essentially requires iteration. A simplified procedure is used in this research to avoid possible numerical difficulties:

a. Use out-of-plane confining stress σ^sub c3^ to determine the stress and strain enhancement factors in the smaller compressive strain direction ε’^sub 1^ and calculate temporary σ’^sub c1^ from Eq. (11) using ε’^sub 1^; and

b. Use σ’^sub c1^ from (a) and σ^sub c3^ to determine the stress and strain enhancement factors. Use these stress and strain enhancement factors to calculate σ^sub c1^ and σ^sub c2^ from Eq. (11) using ε’^sub 1^ and ε’^sub 2^, respectively.

Steel return stress-For smeared reinforcements in the definition of MCFT at a given state of strain{ε^sub x^, ε^sub y^, γ^sub xy^}^sup i^ of the i-th iteration, the return stress can be obtained directly using modified forms of Eq. (9) and (10) to take into account the effect of tension stiffening.

Total return stress vector-The total return stress vector {σ} is essentially the algebraic sum of the concrete return stress and steel return stress, as shown in Eq. (1).

RESULTS

The constitutive model derived in the previous section was implemented in FEM analyses for the RC bent caps tested by Bracci et al. (2000). A commercial FEM analysis program, with the ability to incorporate a user-defined material subroutine, was used to perform this analytical study. Only the return stress vector and the secant stiffness matrix are defined within this user-defined subroutine. The nonlinear incremental analysis with convergence checks for both residual force vectors and incremental displacement are handled internally by the FEM analysis program.

FEM mesh

Sixteen RC bent caps were modeled using two-dimensional finite element analysis. Three-noded and four-noded plane stress elements were used for the concrete, while the longitudinal reinforcement was modeled by two-noded bar elements. Vertical stirrups and longitudinal skin reinforcement was smeared into the concrete model and their contribution to stiffness and strength were modeled internally by the userdefined material subroutine within the FEM analysis program environment. The out-of-plane horizontal legs of transverse reinforcements, as shown in Fig. 2, are also modeled by the smearing technique. Note that the out-of-plane stirrups have no direct contribution to the stiffness matrix of the members. Their presence, however, contributes to the confinement of concrete and; therefore, they indirectly provide strength and deformability to RC members.

Figure 1 shows the geometry of the RC bent caps and Fig. 9 shows the idealized finite element mesh used for these analytical studies. Specimens 1A and 1B employed slightly different element material properties from the remaining 14 specimens because of the different skin reinforcement detailing. Specimens 1A and 1B adopted four No. 5 bars distributed evenly throughout the member depth as shown in Fig. 9(a). Therefore, an average value of 0.001 is used for the skin reinforcement ratio embedded in all smeared reinforced concrete elements in Specimens 1A and 1B. On the other hand, the remaining 14 specimens used six No. 4 bars distributed only in the tension zone. In these cases, an average skin reinforcement ratio of 0.0015 is used for elements in the upper 533.4 mm, as shown in Fig. 9(b).

The tension-stiffening effect only occurs in adequately reinforced regions and where the bond between the concrete and reinforcing steel is properly mobilized. In relatively low reinforcement regions, however, the proper development of the tension-stiffening field is rather dubitable. Therefore, different tensile characteristics for reinforced concrete should also be taken into consideration based on the aforementioned criteria. In these numerical simulations, the tensile constitutive model for concrete elements near the top longitudinal reinforcement, as shown in Fig. 10, is represented by the stress-strain relationship, as proposed in Eq. (6). In other regions, the effect of tension-stiffening is ignored and only tension-softening is considered. Therefore, Eq. (5) is adopted for tensile stress-strain relationships in these relatively low reinforcement regions, as shown in Fig. 10.

Because of the effect of tension stiffening, the yield strength and post-yield stiffness of the longitudinal reinforcing steel should be adjusted to the apparent yield strength, and post-yield stiffness for embedded reinforcement. Note that the adjustment of yield strength is only required in regions where tension stiffening is expected, that is, for the longitudinal reinforcing steel and the smeared reinforcement in the top region, shown in Fig. 10. Because the tensile strength of concrete in the strain-softening zone is practically negligible near the yield strain of the reinforcement, no adjustment for the apparent yield stress and the post-yield stiffness is required. By manual adjustment, the apparent yield strength for the embedded reinforcement is approximately 54 ksi for all specimens. This value is calculated based on the assumption that the yield strength of reinforcing steel is 60 ksi. The postyield stiffness of embedded reinforcement slightly increases from 3 to 4% of the bare bars.

Strength comparison

Table 1 and Fig. 11 and 12 show that the proposed constitutive model is capable of accurately predicting the failure load for the RC bent caps tested. The unconfined model tends to underestimate the RC bent cap strength. On average, the performance of the confined model is superior to the unconfined model regarding the strength prediction.

Comparing the experimental results of the varying transverse reinforcement details, it is clearly evident that the overlapping stirrups help improve the performance of the RC bent cap specimens. Strain gauge data revealed that only one or two stirrups along the member length actually participated in resisting the vertical load. This result confirms the conclusions made by Ferguson (1964) for a similar experimental program on RC bent caps. Ferguson (1964) showed that with the same amount of longitudinal reinforcing steel, vertically unreinforced bent caps and bent caps with single hoop stirrups had the same level of strength. The current experimental results and the proposed analytical model suggest, however, that the use of overlapping stirrups have a significant effect by enhancing the strength and ductility of the RC bent cap members. The analytical results show that one of the possible explanations for this phenomenon is through the effect of confinement caused by the out-of-plane horizontal legs of the transverse reinforcement, as shown in Fig. 2.

For specimen Series 4 and 6 that used an equivalent area of larger diameter longitudinal bars, the confined model slightly overpredicts the ultimate actuator load and the unconfined model either underpredicts (Series 6) or accurately predicts (Series 4) the ultimate actuator load. Experimentally, these specimens failed immediately after the first reinforcement yielding (Young et al. 2002). The use of larger bar sizes may cause higher interfacial stresses between the concrete and reinforcing steel, given the same reinforcement ratio. Therefore, it is possible that the rupture or slip between concrete and reinforcement interface may contribute to premature failure of these specimens. Table 2 and 3 show the analytical prediction of ultimate actuator forces using the confined and unconfined models on the RC bent caps based upon the longitudinal reinforcement bar size. The confined model performs slightly better than the unconfined model for specimens reinforced with No. 7 and 8 bars, but results in a slight overestimation for specimens reinforced with No. 10 bars. For all bar sizes tested, the unconfined model generally underpredicts specimen strength.

Load-deformation prediction

Figure 11 and 12 show the average actuator forcedisplacement history for Specimens 2A, 2B, 8G, and 8H. These four specimens have eight No. 8 bars as the top longitudinal reinforcement. Specimens 8G and 8H, however, have overlapping stirrups as the transverse reinforcement, while Specimens 2A and 2B have single stirrups. Numerical simulations satisfactorily predicted the load-displacement diagrams for these specimens. All four specimens exhibited the yield plateau where the main longitudinal reinforcement reached their yield strength prior to specimen fracture. Analytical results predict a yield strength of approximately 1500 kN for all specimens, which agrees well with the experimental results. The load-displacement relationship obtained from the experimental results, however, show a more gradual transition between the pre- and post-yielding behavior.

Figure 11 and 12 highlight the capability of the confined model to predict the specimen strength and deformability. Recall that the model assumes that the stress of the out-ofplane horizontal legs of the transverse reinforcement depends upon the amount of lateral expansion due to compression. Because the lateral expansive strain is low at the lower load regime, the effect of confinement is imperceptible at the lower load regime. Figure 11 and 12 show that the predicted load-deformation curves for both confined and unconfined models follow essentially the same path at low load levels. At higher load levels, the effect of confinement becomes significant as the confining stress due to out-of-plane transverse reinforcement increases, which helps the confined models carry additional load to larger deformation while the unconfined model fails prematurely by numerical instability.

Comparing Fig. 11 for the Specimen 2 series with Fig. 12 for the Specimen 8 series, experimental results show that they both give practically the same magnitude of actuator forces at first reinforcement yielding of approximately 1500 kN. The Specimen 8 series, however, is capable of sustaining higher levels of force and ductility. It is clearly evident that the use of overlapping stirrups significantly improved the structural performance as both strength and deformability increased.

Numerical simulations are shown to yield satisfactory results for the predicted load-displacement diagrams for Specimens 2A, 2B, and 8G, while excellent results are obtained for Specimen 8H. On average, however, the analytical results underestimate the deformation of RC bent caps. The stiff analytical prediction was also obtained for the remaining 12 specimens. The confined models consistently predict higher levels of load and deformation than the unconfined models.

SUMMARY AND CONCLUSIONS

This paper presents a new constitutive relationship for shear-dominated RC members that incorporates the effect of confinement using a similar set of assumptions as the MCFT. The proposed constitutive relationships are verified with experimental results from a series of RC bent cap tests with varying transverse reinforcement details. Results show that the confinement provided by the out-of-plane horizontal legs of the transverse reinforcement has a significant effect on the strength and deformability of RC bent caps. At low levels of stress, the effect of confinement is not evident, as the simulated results for the confined and unconfined models are virtually identical under small stress. Both models also yield an identical result on the prediction of the first cracking load. At higher levels of stress, however, the effect of confinement is shown to be a contributing factor for increasing the deformability of RC members prone to shear deformations. This, in turn, improves the ductility of the RC bent cap members by delaying the brittle fracture of the concrete in compression. In addition, the effect of confinement also leads to a noticeable increase in the strength of RC members through strain hardening of the reinforcement, which is a consequence of the increased deformability of the sections. The proposed analytical model, however, overestimates the post-cracking stiffness of the RC bent cap specimens, which is further addressed in a companion paper (Powanusorn and Bracci 2006).

ACKNOWLEDGMENTS

Funding for this research was provided by the National Science Foundation under Grant No. CMS 9733959, the Texas Department of Transportation (Project 0-1851), and the Department of Civil Engineering of Texas A&M University. This support is gratefully acknowledged. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the view of the sponsors.

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Suraphong Powanusorn is a Structural Engineer with Thai Nippon Steel Engineering and Construction Co. Ltd. in Chachuengsao, Thailand. He received his PhD in civil engineering from Texas A&M University, College Station, Tex.; his MS in civil engineering from the University of New South Wales, Australia; and his BS from Chulalongkorn University, Bangkok, Thailand.

Joseph M. Bracci is a Professor and Head of the Construction, Geotechnical, and Structural Engineering Division in the Zachry Department of Civil Engineering at Texas A&M University. He received his PhD from the State University of New York at Buffalo, Buffalo, N.Y. His research interests include experimental testing, analytical modeling, and performance-based design of structures.

Copyright American Concrete Institute Sep/Oct 2006

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