**3. Optimal Criteria and Design Procedure**

#### *3.1. Optimal Design Criteria*

The successful design of a composite structure demands efficiency and safety during operation. Thus, optimizing material usability and preventing sudden failure for the retrofitted slabs are considered essential criteria in the optimal design procedure. The enhanced efficiency of the retrofit system would stem from the high compressive strength of the HPC overlay and the high tensile strength of FRP. The HPC overlay significantly enhances flexural strength at the mid-span section and shear strength at the support. On the other hand, the retrofit system does not focus on exploiting the high tensile strength of FRP at the mid-span section due to its location near the neutral axis. Consequently, FRP contributes a relatively small amount to flexural strength at the mid-span section, whereas it is the main factor in improving flexural strength at support.

For retrofitted slabs, overly thick FRP will result in an excessive enhancement of flexural strength over shear strength at support, resulting in shear failure. A too-thick overlay can excessively improve the mid-span flexural strength over the support and increase the slab's self-weight, which does not take advantage of the structure carrying capacity. Ideally, the moment-carrying-capacity ratio of the mid-span section to the support section should be equivalent to the corresponding proportion of factored moments. For symmetric continuous slabs, the positive to negative moment ratios at the end and interior spans subjected to a uniform distributed load can be computed using ACI 318M as follows:

$$\frac{\text{M}\_{\text{n,Pe}}}{\text{M}\_{\text{n,N1e}}} = \frac{\text{C}\_{\text{m,Pe}}}{\text{C}\_{\text{m,N1}}} = \frac{1/14}{1/16} = 1.14\tag{14}$$

$$\frac{\text{M}\_{\text{n,Pe}}}{\text{M}\_{\text{n,N2e}}} = \frac{\text{C}\_{\text{m,Pe}}}{\text{C}\_{\text{m,N2}}} = \frac{1/14}{1/10} = 0.71\tag{15}$$

$$\frac{\mathbf{M}\_{\rm n,Pi}}{\mathbf{M}\_{\rm n,N1i}} = \frac{\mathbf{M}\_{\rm n,Pi}}{\mathbf{M}\_{\rm n,N2i}} = \frac{\mathbf{C}\_{\rm m,Pi}}{\mathbf{C}\_{\rm m,N}} = \frac{1/16}{1/11} = 0.69\tag{16}$$

The ratios of positive and negative factored moments range from 0.69 to 1.14. Nevertheless, the moment ratio of 1.14 of Equation (14) is not a typical value for a continuous multi-spans RC slab because it is only related to the N1e section of the end span. The average moment ratio of 0.7, derived from Equations (15) and (16), should be used to optimize RC slab performance. The design approach based on failure limit methodology can achieve ductile failure and the desired moment ratio for a strengthened slab with a retrofit system by adjusting the increase of positive and negative moment carrying capacity separately. In addition to meeting the guidelines of ACI committee 440, an optimal retrofit system can be founded once the conditions for ductile failure and optimal moment ratio are satisfied.
