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2. Results

The confinement demand of the boundary element of a wall, and the confinement capacity of the optimal solution (solution with a minimum number of stirrups meeting the confinement demand) are presented in Table 1. Similarly, the reinforcement detailings are presented in Figure 2.

I. Confinement Demand (C_d)
The value of confinement demand is determined using Eq (1). $$ {C_{d} = \frac{30 \,(v_{d} + w_{d})\, μ_{ϕ} \times 0.002 \,L_{be}}{b_{o}}-0.035}\tag{1}$$ 'L_be' is the length of the boundary element of the wall calculated from Eq (3), $$ {L_{be} = Max (1.5\,b_{c},0.15\,L_{w})}\tag{3}$$ 'b_o' is the distance confined by the stirrups as shown in Figure 1 and is calculated from Eq (3), $$ {b_{o} = L_{be} - c - φ_{s}}\tag{3}$$ 'w_d' is the mechanical volumetric ratio of vertical reinforcement as calculated from Eq (4), $$ {w_{d} = \frac{ρ\,f_{yd}}{100\,f_{cd}}}\tag{4}$$ ρ is the percentage of vertical reinforcements at the boundary element determined using Eq (5), $$ {ρ = \frac{[2\,n_{L} +2(n_{b}-2)] \times \frac{\pi \,φ_{v}^2}{4} \times 100}{L_{be}\,b_{w}}}\tag{5}$$ n_L is the number of vertical reinforcement along the length of the boundary element determined from Eq (6). $$ {n_{L} \ge \frac{b_{o} - φ_{s} - φ_{v}}{200}}\tag{6}$$ The values of normalised design axial force 'v_d ', length of wall 'L_w', thickness of wall 'b_w', number of vertical reinforcement along the thicknness 'n_b', concrete cover 'c', and diameter of stirrups 'φ_s' are defined in Section 1 (Input Parameters).

Curvature Ductility Factor (μ_ϕ) is determined from the value of behaviour factor (q_o),
         For T₁ ≥ T_C , $$ {μ_{φ} =2qₒ - 1}\tag{7a}$$          For T₁ < T_C , $$ {μ_{φ} = 1 + 2(qₒ - 1) \times \frac{T_{C}}{T_{1}}}\tag{7b}$$ where,
The result of confinement demands (C_d) are presented in Table 1. Figure 1. Details of the area of confined concrete and various confinement parameters (Daniel et al., 2020).

II. Confinement Capacity (C_c)
To determine the confinement capacity, the confinement parameters for the case of stirrups spacing not exceeding the minimum of (b_o/2 , 175 mm, and 8φ_v) are determined using Eq (9-12). Finally, the confinement capacity is calculated by substituting the values of the confinement parameters in Eq (8). The detail of the optimal solutions along with the confinement capacity is shown in Table 1. $$ {C_{c} = α_{n} \times α_{s} \times ω_{wd}}\tag{8}$$ $$ {α_{n} = 1- \frac{(n_{L}-1)b_{i}^2 +(n_{b}-1)h_{i}^2}{6\,b_{o}\,h_{o}}}\tag{9}$$ $$ {α_{s} = (1- \frac{s}{2b_{o}})(1- \frac{s}{2h_{o}})}\tag{10}$$ $$ {ω_{wd} = \frac{(n_{L}\,b_{o} + n_{b}\,h_{o})a_{s}\,f_{yd}}{b_{o}\,h_{o}\,s\,f_{cd}}}\tag{11}$$ where, $$ {h_{o} = b_{w} - 2c - φ_{s}}\tag{12}$$ $$ {b_{i} = \frac{b_{o} - n_{L}φ_{v}}{n_{L}-1}}\tag{13}$$ $$ {h_{i} = \frac{h_{o} - n_{b}φ_{v}}{n_{b}-1}}\tag{14}$$ $$ {a_{s} = \frac{\pi \,φ_{s}^2}{4}}\tag{15}$$ $$ {f_{cd} = \frac{α_{cc} \times{f_{ck}}} {γ_{c}}}\tag{16}$$ Where,
'α_n' is the cross-sectional area reduction factor,
'α_s' is the elevational reduction factor,
'ω_wd' is the mechanical volumetric ratio,
'b_i' and 'h_i' are clear distance between consecutive longitudinal rebars constrained by the stirrups, 'a_s' is the area of individual stirrup,
'f_cd' is design concrete cylinder strength,
'γ_c' is the partial safety factor for concrete and a value of 1.5 is adopted in this program, and
'α_cc' is the coefficient taking account of long term effects on the compressive strength and of unfavourable effects resulting from the way the load is applied. α_cc of 0.85 is adopted in this program.

Table 1: Confinement parameters, and comparison of the confinement capacity of the optimal solutions with the confinement demand.

Figure 2. Cross-section of the boundary element of the wall showing details of the vertical reinforcements and stirrups.

Height of confinement/critical region, $$ {h_{cr} = max (l_{w}, \,h_{w}/6)}\tag{17}$$ but $$ {h_{cr} \le 2\,l_{w}}$$ $$ {h_{cr} \le 2\,h_{s}}$$ 'l_w', 'h_w', and 'h_s' are defined in the input section.

3. References:

1. Department of Standards Malaysia (2017). MS EN 1998-1:2015 Malaysia National Annex to Eurocode 8: Design of structures for earthquake resitance – Part 1: General rules, seismic actions and rules for buildings.
2. European Commitee for Standardization (2004). EN 1998-1:2004 Eurocode 8: Design of structures for earthquake resistance - Part 1 : General rules, seismic actions and rules for buildings.
3. Department of Standards Malaysia (2017). MS EN 1998-1:2010 Malaysia National Annex to Eurocode 2: Design Of Concrete Structures. Part 1-1 : General Rules And Rules For Buildings.
4. European Commitee for Standardization (2004). EN 1992-1-1:2004 Eurocode 2: Design of concrete structures - Part 1-1: General rules and rules for buildings.
5. Looi, D.T.W.; Tsang, H.H. and Lam, N.T.K. (2020). SIMPLIFYING EUROCODE 8 DUCTILE DETAILING RULES FOR REINFORCED CONCRETE STRUCTURES.