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 V_{n} = 0.17√f_{c}′bd + f_{yt}A_{v}d/s, may not be directly used for the design of the belt walls (b and d = width and effective depth, respectively, and f_{yt} and s = yield strength and spacing of shear reinforcement, respectively). Furthermore, the shear stress level of the belt wall, defined as τ_{u} = V_{u}/[l_{w}t_{w}], is mostly higher than the allowable maximum stress, 0.83√f_{c}′, specified in concrete design codes such as ACI 31814 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 highstrength 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 highstrength 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 E_{c} is adopted for design though the actual behavior is nonlinear. The tensile strength of the concrete, f_{ct}, is taken as 0.33√f_{c}′ (Eom et al. 2018). For the highstrength prestressing (PS) strands, as shown in Fig. 9b, a bilinear behavior following the elastic modulus E_{ps} = 195 GPa and postyield modulus E_{pp} = 0.05E_{ps} is assumed. The yield and ultimate strengths of the PS strands are f_{pu} = 1860 MPa and f_{py} = 1674 MPa, respectively. For simple calculation, the behavior of the PS strands is idealized as a bilinear relationship and the yield strength f_{py} is taken as 90% of the ultimate strength (i.e. f_{py} = 0.9f_{pu}, 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. f_{cx} and f_{cy} are the normal stresses acting along the x and y axes, respectively, and v_{cxy} is the shear stress. ε_{cx}, ε_{cy}, and γ_{cxy} are the concrete strains corresponding to f_{cx}, f_{cy}, and v_{cxy}, respectively. f_{c1}, ε_{c1}, f_{c2} and ε_{c2} 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 f_{c1} − ε_{c1} and f_{c2} − ε_{c2} 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.

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.

The PS strands are placed along the x and y axes as reinforcements. The spacing and crosssectional area of the PS strands in both axes are the same. The prestressing force applied to each strand by posttensioning is also the same as f_{pe}, where f_{pe} 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 posttensioned to develop the effective prestress f_{pe}, as shown in Fig. 11, the concrete is compressed by an initial stress f_{cx} = f_{cy} = f_{ci}. By letting the reinforcement ratio of PS strands ρ_{p}, the initial stress and strain of the concrete, f_{ci} and ε_{ci}, 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 f_{ci} 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 posttensioning of the PS strands, the concrete panel of the belt wall is subjected to a pure shear stress, v = V/[l_{w}t_{w}]. Under the pure shear, the stress and strain circles of the uncracked concrete enlarge around the initial point (i.e. f_{ci} and ε_{ci}) 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 f_{cx} = f_{cy} = f_{ci} and ε_{cx} = ε_{cy} = ε_{ci}. Then, when the principal stress in tension, f_{c1} (= f_{ci} + v) reaches the tensile strength f_{ct}, shear cracking occurs in the belt wall. Thus, the shear stress and strain of the concrete at shear cracking, v_{cr} and γ_{cr}, 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 ε_{ct} is the cracking strain of the concrete, taken as f_{ct}/E_{c}.
Since the normal strains of the concrete remained constant as ε_{cx} = ε_{cy} = ε_{ci}, 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 f_{c1} decreases to 0 and the compressive principal stress f_{c2} 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, ε_{cx} and ε_{cy}, increase from ε_{ci} 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, ε_{py} (= f_{py}/E_{ps}).
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. − ρ_{p}f_{py}) should be in equilibrium with the internal resultant force of the concrete (f_{cx} or f_{cy}). Thus, by taking the normal stresses of the concrete as f_{cx} = f_{cy} = − ρ_{p}f_{py} and by letting the shear stress v_{cxy} be equal to f_{cx} or f_{cy} (θ = 45°, refer to Fig. 12c1), the belt wall shear stress at strand yielding, v_{y}, 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, γ_{y}, can be computed as
$$\gamma_{y} = 2\gamma_{c} = 2\left( {\Delta \varepsilon  \varepsilon_{c2} + \varepsilon_{ci} } \right)$$
(6)
In Eq. (2), ε_{ci} is equal to − ρ_{p}f_{pe}/E_{c}. Δε is taken as [ε_{py }− ε_{pe}] or [f_{py }− f_{pe}]/E_{ps}, as shown in Fig. 11. In addition, the compressive principal stress f_{c2} is equal to 2f_{cx} = − 2 ρ_{p}f_{py} (refer to Fig. 12c1), and thus ε_{c2} (= f_{c2}/E_{c}) can be approximated as − 2ρ_{p}f_{py}/E_{c}. 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 n_{p} is the elastic modulus ratio of the PS strands to concrete (= E_{ps}/E_{c}). γ_{y} 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 f_{c2} (i.e. the compressive stress of diagonal concrete struts) should not exceed the effective compressive strength, f_{ce} = 0.85β_{s}f_{c}′, specified in concrete design codes such as ACI 31814 and KCI 2012.
$$\left {f_{c2} } \right = 2\rho_{p} f_{py} \le 0.85\beta_{s} f^{\prime}_{c}$$
(8)
where β_{s} 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 v_{cr} and v_{y} by the crosssection area of the belt wall, l_{w}t_{w}, the belt wall shear strength at shear cracking and yielding, V_{cr} and V_{y}, 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, V_{cr} and V_{y} 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, V_{u}, transferred via the floor slab should not exceed ϕV_{cr} or ϕV_{y} (ϕ = 0.75).
As shown in Eq. (9), the effective prestress and reinforcement ratio of the PS strands, f_{pe} and ρ_{p}, need to be increased to secure a greater resistance against shear cracking under service loads. However, excessively large f_{pe} and ρ_{p} are not desirable for the design of belt walls because brittle failure such as crushing of concrete strut can occur. Thus, when posttensioning the PS strands used for the belt walls, the effective prestress and reinforcement ratio of the PS strands, f_{pe} and ρ_{p}, should be limited as follows.

In the viewpoint of practical application, the shear cracking strength V_{cr} might not be greater than the shear yield strength V_{y}. By taking V_{y} ≥ V_{cr} in Eqs. (9) and (10), the effective prestress f_{pe} is limited to
$$f_{pe} \le f_{py}  \frac{{f_{ct} }}{{\rho_{p} }}$$
(11)

To prevent early concrete crushing, the compressive stress f_{c2} of diagonal concrete struts should not exceed the effective compressive strength f_{ce}, as discussed in Eq. (8). Thus, if the factor β_{s} is taken as 0.6 in Eq. (8) for conservative design, the reinforcement ratio ρ_{p} is limited to (ACI 31814 and KCI 2012)
$$\rho_{p} \le 0.51\frac{{f^{\prime}_{c} }}{{f_{py} }}$$
(12)