The last earthquake events in various world areas and the resulting harms, especially human fatalities, have shown that the structures cannot withstand the earthquake loads. The large damages caused by the earthquake happened in Cairo in 1992 showed that at the construction time, the structures were designed to sustain only vertical loads and had ineffective horizontal load resistance. That expresses that, there are low ductility elements, shear resistance, and steel confinement in the plastic hinge zone that was founded in columns and beam column connections. So it is urgent to assess the seismic performance of existing structures and to constantly refresh the seismic codes for the design of the new structures.

The design of structures for seismic load resistance forced in the Egyptian design codes that motivated the Ministry of Housing and Buildings to regularly update the Egyptian codes provisions to consider the earthquake loads effect. After October 1992, a set of Egyptian codes has been released to avoid building failure and to control significant damages in structural elements. Earthquake analysis has many considerations that have been formed using the performance assessment of existing structures that have been subjected to a severe earthquake. To get a well-engineered structure, it must satisfy the seismic performance requirements that include the careful attention in analysis, design, reinforcement detailing, and good construction. The successful integration of analysis, design, and construction achieves the safety of the structure.

Krawinkler et al. [1], used the pushover analysis method to assess the building performance to get the inter-story drifts that take into account the changes in stiffness and strength, that can be used for the evaluation of P-∆ effect, determinate the effect of strength deterioration of elements on the behavior of the whole structure, get the sequence of failure of structure members and identify the weakness points in the structural members.

Maske [2], uses the nonlinear static pushover analysis, which is considered a common method for assessment of seismic performance for the new and existing structures. To discriminate the weakness zones in the building and then choose if it can be retrofitted or rehabilitated according to its level of damage. He performed the pushover analysis on multistoried frame structures by using SAP2000 software. He analyzed two framed structures with 5 and 12 floors, respectively. The results concluded from his study display that the behavior of properly reinforcement detailed reinforced concrete frame building is adequate as concluded by the capacity curve with demand curve intersection and the plastic hinges distribution in the structural members.

To perform the performance-based design, one must develop the evaluation method of the seismic resistant performance for the reinforced concrete structural members. The performance limit states are classified into three limit states, serviceability limit state, safety limit states, and damage control limit states. Each state is defined by the damages of the structural members. The yielding of reinforcing steel bars and the width of crack are used as the index of the damages. As the result of the plastic nonlinear frame analysis based on the performance-based design process method, the crack width of each member is calculated at each step [3].

Igarashi [4], developed an approach for assessment of seismic damage in reinforced concrete members which is important for exact selection of the most suitable repairing technique for structures damaged and affected by earthquakes risk. He presents the concepts and outlines of damage assessment steps of ductile reinforced concrete structural members. The suggested analytical models assess the width of crack, the length of crack, and the area of concrete that spalled in ductile column and beam. These models are planned to be applied to pushover analysis of framed structure in practical seismic design.

In the preliminary design process, equivalent linear static seismic analysis is used to get the design straining actions in structural members, and then get the strength demands for the designed structural member. One can get the equivalent static seismic forces by calculating the elastic design spectrum acceleration divided by a reduction factor that depends on the structural system that named as the ductility amount response factor (R).

In accordance with (ECP-201-2012) [5] code, the base shear force (F_b) resulted from the analysis of each horizontal direction of the structure to seismic loads is computed with the shown formula:

F b = γ × S d ( T 1 ) × λ × W / g (1)

where, S_d(T₁) is the design response spectrum ordinate at time period T₁. T₁ is the vibration time period of the structure in the direction of the horizontal load analyzed. W is the weight magnitude of the structure considering its total elements. g is the ground acceleration. γ is an important factor for the building and its value depends on the building function. λ is the modal mass correction factor. n: is the number of floors that formed the structure.

The value of the vibration time period in seconds (T₁) is computed using the shown formula:

T 1 = C t × H 3 / 4 (2)

where, C_t is a parameter depends on the structural system of the building and the material of the structure and C_t = 0.075 for a concrete framed structure and H is the total height of the building in m, from the level of footing or from the top of a rigid story.

The design response spectrum ordinate S_d(T₁), can be computed by the shown formula:

S d ( T 1 ) = [ 2.5 R ] × a g × γ × S × η [ T c T 1 ] ≥ 0.2 × a g × γ (3)

where, a_g is the equivalent design ground acceleration for the ground motion of the earthquake for a specific return period. T_c is the peak value of the constant spectral time period acceleration. Η is a damping parameter of the horizontal elastic response spectrum, where η = 1 corresponds to a normal ratio of 5% viscous damping (in the case of reinforced concrete structures). S is the parameter depends on the soil type. γ is the important factor for the building depends on the building function. R is a reduction factor depends on the structural system of the structure used to resist seismic loads, it represents the ductility amount of the structure.

The Arab Republic of Egypt is divided into five seismic zones according to the ECP code (ECP-201-2012) [5] based on the design ground acceleration as shown in Table 1, and Figure 1.

The lateral forces F_i on each story with mass m_i shall be computed as follows:

F i = [ h i × W i ∑ j = 1 n h j × W j ] × F b (4)

where, F_i is the earthquake force acting horizontally on story i. F_b is the total base shear force due to earthquake (Equation (1)). h_i and h_j are the heights of each story with masses m_i and m_j above the foundation level, respectively. W_i and W_j are the weights of masses m_i and m_j, respectively. n is the number of floors above the foundation level.

Equation (4) computes the seismic force on each floor depending only on the story height.

The Egyptian Society for Earthquake Engineering (ESEE 1988) [6] has developed regulations for the design of structures to withstand and release earthquake loads. The design criteria in this case are based on working limit state design approach. The lateral earthquake forces calculated shall be applied together at each floor and roof level.

The total base horizontal earthquake load:

Each structure shall be designed and constructed to confront and release a total horizontal earthquake load (V) in each building direction under consideration which computed by the shown formula:

V = C s × W t (5)

where, C_s is the seismic design coefficient and W_t is the total weight of the building considering dead loads and live loads.

The horizontal acceleration coefficient can be calculated using the equation mentioned below:

C s = Z × I × S × M × R × Q (6)

where, C_s is a coefficient of Seismic design and Z is the factor of seismic zoning determined from the shown formula:

Z = A × C × F (7)

where, A is the horizontal earthquake ground acceleration detected according to the building location on the seismic zoning map. C is the standardized response spectrum coefficient for average damping of 5%. F is the foundation soil factor. I is the important factor of the building. S is the factor depends on the type of the structural system, the value of (S) shall be determined separately for each direction of the building. M is the factor that depends on the construction material. R is the risk factor expresses the amount of risk exposure. Q is a factor that shows the quality of materials used and the quality of construction.

Pushover analysis is a method in which a series of incremental static horizontal load applied on the structure to get the load-displacement capacity curve of the building. This load still increasing until the structure reaches its maximum displacement (Hakim 2014) [7]. A pushover analysis results and strategy are presented in Figure 2, Figure 3.

It is a design method used for assessing the response of the building to the future seismic events and deciding whether such response meets the specific performance demands. The performance levels of the buildings due to seismic loads are as (FEMA-356-2000) [8]. According to FEMA-356, pushover analysis displays the load-displacement curve for beams and columns critical sections that form a nonlinear plastic hinge. This curve is shown in Figure 4. While Table 2 shows the Performance Level of Building and the resulted damage for each level.

According to design aids and examples in accordance with the (ECP-203-2007) [9] code when designing reinforced concrete structures, one should fulfill the following relations:

W k = β ⋅ S r m ⋅ ε s m   ( mm ) (8)

S r m = [ 50 + 0.25 K 1 K 2 ϕ ρ r ]   ( mm ) (9)

ε s m = F s E s [ 1 − β 1 β 2 ( F s r F s ) 2 ] (10)

where, W_k is the crack width value in (mm). S_rm is the Spacing between cracks in the horizontal direction measured in (mm). ε_sm is the mean steel strain under a relevant combination of loads and allowing for the effect such as tension stiffening or shrinkage. β is the Coefficient that connects the average crack width to the design crack width. Φ is the Bar diameter in (mm). β₁ is a coefficient that reflects the bond properties of the reinforcing steel bars. β₂ is a coefficient that reflects the loading duration. K₁ is a coefficient that reflects the type of steel bars. K₂ is a coefficient that shows the distribution of the strain over the subjected cross section.

K 2 = ε 1 + ε 2 / 2 ε 2 (11)

where, ε₂ and ε₁ are the minimum and maximum strain values on the subjected section, and shall be calculated according to the analysis of a cracked section.

ρ r = A s A c e f (12)

where, A_s is the area of longitudinal tension steel within the effective tension area. A_cef is the area of effective concrete section in tension = width of the section × t_cef. t_cef = 2.5 × concrete cover. F_s is the stress in longitudinal steel bars at the tension zone calculated based on the analysis of cracked section under permanent loads. F_sr is the stress in longitudinal steel bars located in the tension zone calculated according to the analysis of cracked section due to the loads that causing first crack.

Figure 4 The plastic moment in beams due to earthquake and the crack mechanism according to this moment.

The moment M_t and M_b values describe the damage level and its suitable repairing method, the normal force values in beams is small and can be neglected. By taking the average moment at the top and bottom (reversible moment). According to ECP code [9], we can get the crack width (W_k), the horizontal distance between the cracks (S_rm), and the vertical distance between the neutral axis and the maximum tensile stress. From the value of (S_rm) and the vertical distance between the neutral axis and the maximum tensile stress, the whole length of cracks that will be injected with epoxy can be calculated.

Description of the Case Study Building

The prototype building consists of 7-story framed reinforced concrete structure, with a story height of 3.0 m, the overall plan is 12 m × 12 m (144 m²). Figure 5(a) and Figure 5(b) show the typical slab layout and slab reinforcement distribution, respectively.

The structural system was designed using a design practice that considers gravity loads and linear static seismic loads according to ECP code (ECP-201-2012) [5] by using SAP2000 software(Computer and Structures 2014) [10]. A summary of the model’s assumptions is presented in Table 3. Figure 6 shows the three-dimensional model of the structure.

The cost assessment results are summarized in Table 21, and Figure 36.

The nonlinear analysis of structures designed to resist earthquakes is very important to assess the response of the structure under earthquake effect and to know the state of the building after seismic load. The ductility of building is very important but one should be careful since the large displacement will be accompanied with damage that can make the structural members irreparable and the building may lose its function. One should perform a cost assessment for each seismic mitigation design to visualize the life cycle cost of the structure to get a cost effective design, which consists of the construction cost and the repairing cost after earthquake damage. The paper presented a proposed method for seismic performance evaluation for existing and new structures depending on the width of cracks resulted from the seismic exposure. Also it helps engineers to perform a cost assessment for the reinforced concrete buildings designed to resist earthquakes to get its life cycle cost.

The steps and methodology required for structural evaluation and cost assessment mentioned in this study are summarized as follows:

1) Construct a 3-D model for structure to be analyzed by using sap 2000 software.

2) Calculate the construction cost of the designed structure by (L∙E/m²) as shown in Table 22.

3) Perform a nonlinear static pushover analysis by using SAP2000 software to locate the weakness points in the structure that appears as cracks in the structural members, if this structure is constructed and exposed to the designed seismic ground acceleration and story shear as shown in Table 23; anticipate the structure capability to undergo the deformations beyond the elastic zone determines the structural behavior of the building during seismic hazard as shown in Figure 37, and then determine the location of the plastic hinges in columns and beams as shown in Figure 38.

4) Get the plastic moment on each member as shown in Figure 39 that undergoes beyond the elastic behavior to the yielding behavior, and calculate the crack width in the structure according to its plastic moment as shown in Figure 40.

Cost of P.C = volume × (contractor fees + material cost) × % of losses and Supervision cost; Cost of R.C = volume × (contractor fees + material cost) × % of losses and Supervision cost; Cost of Reinforcement = steel weight × unit cost × % of losses and Supervision cost.

5) Select the suitable repairing method for each member according to the crack width value as shown in Table 23, stress in reinforcement steel bars, and the drift in the structure; then get the repairing cost for each member in the structure.

6) Calculate the repairing cost of the structure for each design approach by (L∙E/m²) as shown in Table 24 [13].

7) Perform a cost analysis for assessment of the cost-efficiency of seismic mitigation design based on the long term performance of the structure subjected to seismic hazard, a life-cycle cost due to the initial construction cost should be included to assess the impact of potential earthquakes that occurred during the expected life-cycle of the structure. Generally, a more resistant design with higher initial construction cost will have a lower life-cycle cost.

The authors declare no conflict of interest, financial or otherwise.

Fayed, Y., Sobaih, M.E. and El Hakem, Y. (2019) Proposed Method for Cost Assessment of Seismic Mitigation Designs for Reinforced Concrete Buildings According to ECP Code. Open Journal of Civil Engineering, 9, 319-355. https://doi.org/10.4236/ojce.2019.94023