As discussed in the previous section, the belt wall is subjected to pure shear. Thus, the conventional shear strength equations for structural walls subjected to combined bending and shear, such as Vn = 0.17√fc′bd + fytAvd/s, may not be directly used for the design of the belt walls (b and d = width and effective depth, respectively, and fyt and s = yield strength and spacing of shear reinforcement, respectively). Furthermore, the shear stress level of the belt wall, defined as τu = Vu/[lwtw], is mostly higher than the allowable maximum stress, 0.83√fc′, specified in concrete design codes such as ACI 318-14 and KCI 2012.
To address such high shear demand, in this study, it is proposed into use a prestressed concrete (PSC) system for the belt walls. Figure 8 shows a belt wall reinforced horizontally and vertically with high-strength strands. By tensioning the strands placed in both directions, it is possible to prevent concrete cracking from occurring early under lateral loading. After concrete cracking, the high-strength strands themselves act as a shear reinforcement for the belt wall. In the construction viewpoint, placement and tensioning of strands are not difficult because the belt walls are distributed separately.
In this section, the shear strengths of the proposed PSC belt walls at concrete cracking and reinforcement yielding are estimated based on the compression field theory as follows (Collins and Mitchell 1980; Vecchio and Collins 1986).
3.1 Material Models and Basic Assumptions
For the concrete, as shown in Fig. 9a, a linear elastic behavior following the elastic modulus Ec is adopted for design though the actual behavior is nonlinear. The tensile strength of the concrete, fct, is taken as 0.33√fc′ (Eom et al. 2018). For the high-strength prestressing (PS) strands, as shown in Fig. 9b, a bilinear behavior following the elastic modulus Eps = 195 GPa and post-yield modulus Epp = 0.05Eps is assumed. The yield and ultimate strengths of the PS strands are fpu = 1860 MPa and fpy = 1674 MPa, respectively. For simple calculation, the behavior of the PS strands is idealized as a bilinear relationship and the yield strength fpy is taken as 90% of the ultimate strength (i.e. fpy = 0.9fpu, European Committee for Standardization 2004; Han et al. 2018; Lee et al. 2018).
The biaxial stresses and strains of the concrete are presented in Fig. 10. fcx and fcy are the normal stresses acting along the x and y axes, respectively, and vcxy is the shear stress. εcx, εcy, and γcxy are the concrete strains corresponding to fcx, fcy, and vcxy, respectively. fc1, εc1, fc2 and εc2 are the concrete stresses and strains along the two principal directions. θ is the inclination angle of the principal direction. For the stresses and strains of the concrete, positive and negative signs indicates tension and compression, respectively. It is noted that only the fc1 − εc1 and fc2 − εc2 relationships follow the uniaxial behavior presented in Fig. 9a.
For simple formulation, basic assumptions regarding application of the compression field theory are made as follows.
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Since the belt wall is confined by the left and right perimeter columns and by the top and bottom floor slabs including spandrel beams, uniform stress and strain field is assumed for the internal concrete panel of the belt wall. Thus, the behavior of the concrete panel can be represented as the stresses and strains of an element, shown in Fig. 10.
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The PS strands are placed along the x and y axes as reinforcements. The spacing and cross-sectional area of the PS strands in both axes are the same. The prestressing force applied to each strand by post-tensioning is also the same as fpe, where fpe is the effective prestress. Based on these conditions, a constant inclination angle of the principal stresses, θ = 45°, is assumed.
In fact, the stresses and strains of the belt wall are not uniform because the confinement effects are different at the corner and center. Such local behavior is investigated further by nonlinear finite element analysis in the next section.
3.2 Prestressing of Strands: Initial State
If the PS strands are post-tensioned to develop the effective prestress fpe, as shown in Fig. 11, the concrete is compressed by an initial stress fcx = fcy = fci. By letting the reinforcement ratio of PS strands ρp, the initial stress and strain of the concrete, fci and εci, respectively, can be expressed as
$$f_{ci} = - \rho_{p} f_{pe} \quad {\text{and}} \quad \varepsilon_{ci} = \frac{{f_{ci} }}{{E_{c} }} = - \frac{{\rho_{p} f_{pe} }}{{E_{c} }}$$
(2)
Since the concrete is compressed by the same fci both in the x and y axes, the biaxial stress and strain of the concrete is represented as a point, as shown in Fig. 12a1, a2.
3.3 Shear Cracking: Behavior of Uncracked Concrete
If a lateral shear force V is applied after the post-tensioning of the PS strands, the concrete panel of the belt wall is subjected to a pure shear stress, v = V/[lwtw]. Under the pure shear, the stress and strain circles of the uncracked concrete enlarge around the initial point (i.e. fci and εci) with no change of their centers, as shown in Fig. 12b1, b2. Furthermore, since the inclination angle of the principal axis is θ = 45°, the normal stresses and strains remains constant as fcx = fcy = fci and εcx = εcy = εci. Then, when the principal stress in tension, fc1 (= fci + v) reaches the tensile strength fct, shear cracking occurs in the belt wall. Thus, the shear stress and strain of the concrete at shear cracking, vcr and γcr, can be computed as follows.
$$f_{ci} + v_{cr} = f_{ct} \quad {\text{or}} \quad v_{cr} = f_{ct} - f_{ci} = f_{ct} + \rho_{p} f_{pe}$$
(3)
$$\gamma_{cr} = 2\gamma_{c} = 2\left( {\varepsilon_{ct} - \varepsilon_{ci} } \right) = 2\left( {\frac{{f_{ct} + \rho_{p} f_{pe} }}{{E_{c} }}} \right)$$
(4)
where εct is the cracking strain of the concrete, taken as fct/Ec.
Since the normal strains of the concrete remained constant as εcx = εcy = εci, as shown in Fig. 12b2, the stress and strain of the PS strands do not change until the shear cracking occurs.
3.4 Yielding of PS Strands: Behavior of Cracked Concrete
As the lateral force of the belt wall increases further after shear cracking, the tensile principal stress fc1 decreases to 0 and the compressive principal stress fc2 increases in magnitude (see Fig. 12c1). In addition, the concrete dilates as the width of shear cracks increase. Consequently, the normal strains of the concrete, εcx and εcy, increase from εci in compression to a positive value in tension by Δε, as shown in Fig. 12c2, and the strain of the PS strands also increase by the same amount, as shown in Fig. 11. In the end, it is considered that the yielding of the PSC belt wall occurs when the strain of the PS strands is equal to the yield strain, εpy (= fpy/Eps).
As shown in Fig. 10, there is no external load applied within the concrete panel of the belt wall. This means the confining force provided to the concrete by the PS strands (i.e. − ρpfpy) should be in equilibrium with the internal resultant force of the concrete (fcx or fcy). Thus, by taking the normal stresses of the concrete as fcx = fcy = − ρpfpy and by letting the shear stress vcxy be equal to fcx or fcy (θ = 45°, refer to Fig. 12c1), the belt wall shear stress at strand yielding, vy, can be computed as
$$v_{y} = v_{cxy} = - f_{cx} = \rho_{p} f_{py}$$
(5)
As shown in Fig. 12c2, the shear strain of the belt wall at strand yielding, γy, can be computed as
$$\gamma_{y} = 2\gamma_{c} = 2\left( {\Delta \varepsilon - \varepsilon_{c2} + \varepsilon_{ci} } \right)$$
(6)
In Eq. (2), εci is equal to − ρpfpe/Ec. Δε is taken as [εpy − εpe] or [fpy − fpe]/Eps, as shown in Fig. 11. In addition, the compressive principal stress fc2 is equal to 2fcx = − 2 ρpfpy (refer to Fig. 12c1), and thus εc2 (= fc2/Ec) can be approximated as − 2ρpfpy/Ec. Therefore, Eq. (6) can be rewritten as
$$\gamma_{y} = 2\left( {\frac{{f_{py} }}{{E_{ps} }}\left[ {1 + 2n_{p} \rho_{p} } \right] - \frac{{f_{pe} }}{{E_{ps} }}\left[ {1 + n_{p} \rho_{p} } \right]} \right)$$
(7)
where np is the elastic modulus ratio of the PS strands to concrete (= Eps/Ec). γy estimated by Eq. (7) is based on the assumption that yielding of the PS strands precedes crushing failure of the concrete. Thus, the compressive principal stress fc2 (i.e. the compressive stress of diagonal concrete struts) should not exceed the effective compressive strength, fce = 0.85βsfc′, specified in concrete design codes such as ACI 318-14 and KCI 2012.
$$\left| {f_{c2} } \right| = 2\rho_{p} f_{py} \le 0.85\beta_{s} f^{\prime}_{c}$$
(8)
where βs is the factor addressing the effects of cracking and reinforcements on the effective strength of the concrete strut.
3.5 Shear Strength of PSC Belt Walls
Multiplying vcr and vy by the cross-section area of the belt wall, lwtw, the belt wall shear strength at shear cracking and yielding, Vcr and Vy, respectively, are computed as.
$$V_{cr} = \left( {f_{ct} + \rho_{p} f_{pe} } \right)l_{w} t_{w}$$
(9)
$$V_{y} = \rho_{p} f_{py} l_{w} t_{w}$$
(10)
When designing the PSC belt walls, Vcr and Vy can be used for strength check such as the serviceability and ultimate limit states. In this case, the horizontal shear force of the belt wall, Vu, transferred via the floor slab should not exceed ϕVcr or ϕVy (ϕ = 0.75).
As shown in Eq. (9), the effective prestress and reinforcement ratio of the PS strands, fpe and ρp, need to be increased to secure a greater resistance against shear cracking under service loads. However, excessively large fpe and ρp are not desirable for the design of belt walls because brittle failure such as crushing of concrete strut can occur. Thus, when post-tensioning the PS strands used for the belt walls, the effective prestress and reinforcement ratio of the PS strands, fpe and ρp, should be limited as follows.
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In the viewpoint of practical application, the shear cracking strength Vcr might not be greater than the shear yield strength Vy. By taking Vy ≥ Vcr in Eqs. (9) and (10), the effective prestress fpe is limited to
$$f_{pe} \le f_{py} - \frac{{f_{ct} }}{{\rho_{p} }}$$
(11)
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To prevent early concrete crushing, the compressive stress fc2 of diagonal concrete struts should not exceed the effective compressive strength fce, as discussed in Eq. (8). Thus, if the factor βs is taken as 0.6 in Eq. (8) for conservative design, the reinforcement ratio ρp is limited to (ACI 318-14 and KCI 2012)
$$\rho_{p} \le 0.51\frac{{f^{\prime}_{c} }}{{f_{py} }}$$
(12)