## 4 Analytical Approach of Punching Shear Strength  
In 1961, [PERSON] [16] proposed that punching shear strength of a slab could be determined from its flexural strength. [PERSON] suggested the punching shear strength, \(V_{u}\), as  
\[V_{u}=\frac{1.25\big{(}1-0.075\frac{5}{d}\big{)}\sqrt{f_{c}^{\prime}}}{1+\frac {0.44u_{0}d\sqrt{f_{c}^{\prime}}}{V_{flex}}}u_{0}d \tag{1}\]  
where \(f_{c}^{\prime}\) is the compressive strength of concrete, in MPa; \(d\) is the effective depth of the slab section; \(c\) is the side length of a square column; \(u_{0}\) is the perimeter of a square column; and \(V_{flex}\) is the shear strength corresponding to the flexural punching failure of the slab. The \(V_{flex}\) was calculated as  
\[V_{flex}=8\bigg{(}\frac{b}{s-c}-0.172\bigg{)}M_{n} \tag{2}\]  
where \(b\) is the side dimension of a square slab; \(s\) is the side dimension between supports of a square slab; and \(M_{n}\) is the flexural strength of the slab. The \(V_{flex}\) was derived based on the formation of a yield line mechanism of a slab at ultimate strength.  
In order to take into account the contribution of the CFRP laminates, among the variables in Equations (1) and (2), the effective depth of the slab section, \(d\), and the flexural strength, \(M_{n}\), needed to be amended to reflect the strengthening effect of the CFRP laminates [7]. To calculate the flexural strength of the slab strengthened with CFRP laminates, the fundamental theory to calculate the flexural strength of a reinforced concrete member was followed. Flexural strength was determined based on the deformation compatibility and force equilibrium. Figure 8 shows the strain, stress and force distributions in the slab section.  
The calculation of the flexural strength of the slab strengthened with laminates are as follows:  
1. Linear strain distribution across the depth of the section was assumed, and the strains and stresses on the concrete, steel reinforcement, and CFRP laminates were determined accordingly.
2. The flexural strength was achieved when extreme concrete compressive strain attained 0.003 or the CFRP laminates developed its ultimate strain.
3. Consequently, the compressive force in the concrete, \(C_{c}\), the tension force in the steel reinforcement \(T_{s}\), and the tension force in the CFRP laminates, \(T_{\text{CFRP}}\), were able to be determined.
4. Through the force equilibrium, the neutral axial depth, \(c\), could be confirmed.
5. Further, the flexural strength of the specimens strengthened with CFRP laminates was calculated as follows. \[M_{n}=T_{s}\Big{(}d-\frac{a}{2}\Big{)}+T_{\text{CFRP}}\Big{(}h-\frac{a}{2} \Big{)}\] (3) where \(h\) is the thickness of the slab; and \(a\) is the depth of the rectangular concrete stress block.  
Moreover, the flexural strength could be also determined as follows if an equivalent effective depth, \(d_{eqv}\), was assumed.  
\[M_{n}=(T_{s}+T_{\text{CFRP}})\Big{(}d_{eqv}-\frac{a}{2}\Big{)} \tag{4}\]  
Therefore, the equivalent effective depth was determined by the following equation.  
\[d_{eqv}=\frac{M_{n}}{T_{s}+T_{\text{CFRP}}}+\frac{a}{2} \tag{5}\]  
Instead of \(d\), substituting \(d_{eqv}\) to Equation (1), the punching shear strengths were calculated based on [PERSON]'s equations. Table 4 summarizes the equivalent effective depth, and the ultimate strengths of the tests and analytical calculations. The increase of the effective depth owing to the contribution of the CFRP laminates on the flexural strength was observed. Moreover, increasing the amount of the CFRP laminates resulted in an increase in the equivalent effective depth. The calculated punching shear were in good agreement with the test results. The average ratios of the tested punching shear strength to analytical calculations was 1.01 and the coefficient of variation was 0.05.  
Figure 8: Strain, stress and force distributions in the section at ultimate strength.