Structural analysis and design of the 32 models shown in Tables 1 and 2 are required to obtain interstory drifts, rebar requirements, and structure costs, according to the ACI 318R-19 (2019) Special Moment Frame section and the NEC-SE-HM (2014). Seismic design is completed as per NEC specifications. Here, the seismic demand is obtained by the elastic response spectra of Guayaquil city, assuming a soil type: D, (NEC, 2014). The base shear force is determined with an importance factor of 1, a mass source equal to 100% of the total dead load, and a spectral acceleration according to the fundamental period, obtained from NEC (2014), which is 0.397 s. In order to evaluate the behavior of both systems, all models are analyzed with the same spectral acceleration. The structural analysis and design are conducted considering seismic response modification factors of eight and five for drop and hidden beams, respectively.

The structural systems above are modeled using ETABS (Version 19.0.0). Material parameters f c ′ = 21  MPa and f _y = 420 MPa are selected. For this research study, a two-way ribbed slab is modeled as a “thin shell” type element of 0.25 m total depth, as presented in Figure 2. The reason for this design decision is to transmit torsion to the beams and, thus, determine if the beams comply with shear stress due to shear force and torsion requirements of ACI 318R-19 (2019). The moment of inertia is modified according to the structural element, 0.3 for hidden beams and two-way ribbed slabs (ACI 318R-19, 2019), 0.5 for drop beams and 0.8 for columns, to include cracked sections according to NEC guidelines. Gravitational loads of 6 kN/m² and 2 kN/m² are defined for superimposed dead and live loads, respectively.

Although all 32 models are loaded considering the same gravitational loads, the structural elements’ self-weight varies since they have different cross-sections in the columns and beams. Therefore, the seismic weight depends on each model, and the lateral seismic force is assigned to rigid diaphragms. Here, a perfect fixed restraint at the base is assumed. The connections between beams and columns are considered with a rigid-zone factor of 50% at the end-length offsets.

All the modeled structures are analyzed to obtain the total cost. The cost index is used for comparison purposes to simplify the data obtained from the estimated budget. It represents values between 1 and 2. Quantity 1 represents the lowest budget of all the analyzed structures, whereas quantity two represents the highest budget. The slab structure is composed of 21 MPa concrete, hollow blocks for weight-lightening purposes, and top and bottom layers of one 12 mm fy = 420 MPa steel rebar per rib. Additionally, the cost reduction on slab construction produced by the shoring procedure on hidden beam structures is considered in this analysis. A summary of concrete slab material costs (labor included) for hidden and drop beams is shown in Table 3. The structural costs for beams and columns are estimated according to Ecuadorian unit costs from Arciniega Larrea and Suárez Coba (2016) (Table 4).

Once the longitudinal and transverse reinforcement of all the structural elements of the models have been obtained, plastic hinges are generated when a nonlinear static pushover analysis is conducted. The ETABS software package is used for the assignment of automatic plastic hinges, based on the ASCE 41–17 standard, (ASCE, 2017), at the beginning and end of the clear length of each structural element of the building models. A nonlinear static dead load case is generated to perform the nonlinear static pushover. This procedure pushes the structure laterally with a load pattern related to the elastic lateral seismic force corresponding to the model under analysis.

A series of pushover curves are obtained With all the settled information on the models. Moreover, the corresponding response elastic spectrum is also computed using the FEMA 440 method. As a result, the performance points of analysis models with H35 × 25, B25 × 35, H45 × 25, and B25 × 45 are calculated according to FEMA-440 (2005); Velásquez Londoño (2017). In addition, the base shear, roof displacement, vibration period and effective damping are obtained. This information is then used to compare the global behavior of buildings with hidden and drop beam systems.

The linear static procedure is to analyze each model’s beam with the highest rebar area. With this information, the moment-curvature diagrams for beams from models 9, 12, 25, and 28 are obtained. Then, the local behavior of the hidden and drop beams is compared. To analyze the local curvature ductility of the plastic hinge zone on hidden and drop beam elements, stress-strain diagrams of the confined and unconfined concrete are determined according to the model of Kent and Park (1971) of Eq. (1). Moreover, the stress-strain diagram of reinforcement steel was defined by the model developed by Park and Paulay (1991). Consequently, the static equilibrium equations are obtained by considering the deformation compatibility conditions of the unconfined, and confined concrete and rebar steel. Then for each concrete unit strain, the nominal moment and curvature of the section are determined, thus obtaining the moment-curvature diagram (Paulay and Priestley, 1992; Córdova, 2015). f c ′ = f c ′ 2 ε ε co − ε ε co 2 0 ≤ ε ≤ ε co f c ′ 1 − Z ε − ε co ε co ≤ ε ≤ ε 20c 0.2 f c ′ ε ≥ ε 20c . (1) All the variables and the adopted values appearing in Eq. 1 are defined in Table 5. Regarding reinforcement steel stress-strain curve, it is defined by Eqs. 2–4. f s = m ε s − ε s h + 2 60 ε s − ε s h + 2 + 60 − m ε s − ε s h 2 30 r + 1 2 f y (2) where m and r are: m = f s u / f y 30 r + 1 2 + 60 r − 1 15 r 2 (3) r = ε s u − ε s h . (4) Similarly, the variables in Eq. 2 are defined in Table 5.

The ductility due to the curvature of the different sections is computed through: U = φ u φ y , (5) where φ _u is the ultimate curvature and φ _y is the yield curvature of the section. According to Paulay and Priestley (1992), the yield curvature considers the first yield of the tensile rebar, and for the ultimate curvature, this corresponds to the failure point, that is the last point in the moment-curvature diagram.

Since seismic-resistant design codes emphasize that beams should not fail due to shear actions, there are some types of failure that control the flexural behavior of beam sections, these are: failure due to crushing of the concrete by rupture of the stirrups, which means that concrete loses the confinement and fails due to crushing, and tensile rupture of the longitudinal rebar. To consider the latter, the moment-curvature diagram must be interrupted when the unit strain of the tensile rebar exceeds ɛ _su . While considering the former, the moment-curvature diagram must be interrupted when the unit strain of the compression concrete exceeds ɛ _cu , Paulay and Priestley (1992) which is established by: ε c u = 0.004 + 1.4 p s f y h ε s m f c c ′ , (6) where p _s is the volumetric ratio of confining steel for rectangular sections p _S = p _x + p _y . In this case, since we are using the Kent and Park model for confined concrete, thus f c c ′ = f ′ c . Kent and Park model is commonly used on elements with high levels of confinement, such as columns. However, Delalibera (2002); Nogueira and Rodrigues (2017) used this model to consider the confinement on beams bending caused by the transversal reinforcement.

The flexural capacity and curvature ductility along plastic hinge zones, are compared between hidden and drop beams according to the structural design from Subsection 2.1. Longitudinal and transverse reinforcement is computed and thus used to obtain the flexural capacity and curvature ductility. It is expected to have higher capacity and ductility on drop beams than on hidden beams due to their depth-width relation, (Aguiar, 2003).

Interstory drift analysis was conducted in all the considered structural models, using the linear static analysis approach (Table 1, 2), and their results are summarized in Table 6.

As shown in Figure 3 (drop beams), the maximum interstory drift value occurs in Model #1 with 1.66%, and the minimum value of 0.6% appears in Model #16. The influence of increasing the beam section depth is equivalent to increasing the column width and depth, which is manifested in isoline slopes, which suggests the effectiveness of drop beams when compared with hidden beams. For the case of hidden beams, the analysis results are presented in Table 6. Here, the maximum interstory drift value appears in Model #17, with 3.25%, and the minimum of 1.81% occurs in Model #32. The data suggest that increasing the column section is more effective than increasing the beam width.

A closer look at the data presented in Figures 3, 4 indicates that a 0.05 m depth increase in drop beams is equivalent to a 0.15 m width increase in hidden beams. These results show that the influence of drop beams’ depth is threefold over hidden beams’ width (Samir, 2021). It should be noted that the present study considers interstory drift limits as per the NEC: all drop beam models have a maximum interstory drift less than 2%, which is the maximum allowed by the referred code. Regarding the hidden beam system, only three models comply with the Ecuadorian Code: Model #24, Model #28, and Model #32. Based on the obtained results from the elastic analysis, it was found that the buildings containing drop beams have greater lateral stiffness, which decreases the maximum interstory drift, with respect to the results obtained in buildings containing hidden beams. The maximum drifts obtained in drop beam models were two to three times lower than those obtained in hidden beam models.

The maximum reinforcement ratio of each model is shown in Figure 5. From a seismic design standpoint, this parameter has much relevance, as ductility mainly depends on it. For hidden beam models, the reinforcement ratio is between 0.72% and 1.1%, whereas for drop beam models, the ratio lies between 0.35% and 0.66%. It proves that hidden beams require 90% more longitudinal steel than drop beams, which in turn means that the ductility will be lower and the cost higher in terms of rebar steel (Paulay and Priestley, 1992; Samir, 2021).

During the design process, some hidden beam models presented problems with torsion and shear design. The obtained results reveal that Model #17 has a 47.22% of beams that do not comply with the torsion/shear interaction. Similarly, Model #18 has a 7.78%, and Model #21 presents an 11.11% non-compliance. Nevertheless, this could be solved by reducing the torsional constant, which was not considered in this study. Hidden beam buildings do not meet allowable criteria, requiring an increased lateral stiffening of the whole structure, either by increasing the section of beams or columns, to achieve compliance. The shear design of beams suggests that drop beams have a better performance against shear and torsion forces than hidden beams.

According to the structural design, columns C40 × 40 and C45 × 45 have a longitudinal reinforcement of 20.32 cm², equivalent to eight 18 mm diameter bars, and columns C50 × 50 and C55 × 55 have a rebar area of 30.48 cm², equivalent to twelve 18 mm diameter bars. Table 7 summarizes the longitudinal reinforcements of columns.

Figure 6 shows the cost index for different structural sections. The results presented here indicate that cost increases as structural sections increase. The maximum cost index is 1.89 for a combination of a C55 × 55 column paired with a B25 × 50 beam, and the minimum cost index of 1 occurs for a combination of a C40 × 40 column with a B25 × 35 beam. An important finding was identified in the change of column sections, where it can be observed that the cost index increased around 30% between C45 × 45 and C50 × 50. The main reason for this is the change of the minimum longitudinal reinforcement on the columns (Table 7).

A similar situation is observed in the hidden beam case, whose cost index chart is shown in Figure 7. The most significant difference is the maximum cost index, which reaches a value of two for a combination of C55 × 55 column with an H50 × 25 beam. According to the interstory drift analysis, the models that comply with the Ecuadorian code are Model #24, Model #28, and Model #32. Cost differences with drop beam equivalent models are between 1% and 2% (Table 8), representing a negligible cost factor. A summary of superstructure costs is shown in Table 9.

A comparison of interstory drifts and costs shows that Model #1 and Model #32 evidence similar interstory drifts. From a seismic design standpoint, these two models also have similar behavior, with a difference of 9% between interstory drifts but with a 31% difference in costs. Since the results show that constructing buildings with drop beams instead of hidden beams is more economical and the interstory drift is small, the data implies that it is more cost-effective and technically feasible. At the same time, a lower interstory drift implies that the repair costs of the building after a seismic event are also lower.

Nonlinear static analysis was performed using the ASCE 41–17 modeling parameters for beams and columns (ASCE, 2017). A FEMA 440 equivalent linearization was used to determine the performance points and global parameters from the nonlinear static analysis (FEMA-440, 2005). The results and global parameters are shown in Table 10. Drop beam models have higher values for damping and ductility ratio than hidden beam models. Also, the effective period is higher on hidden beam models due to their low stiffness.

Next, the global ductility ratios are compared between hidden beam (red) and drop beam (blue) models. The results in Figure 8 suggest that drop beam models have a ductility ratio between 2.75 and 3.31. On the other hand, for hidden beams, the ductility ratio is approximately two on all the analyzed models. The ductility ratio is 50% higher in models of buildings containing drop beams than in those with hidden beams. This result is probably due to the greater height and the lower amount of rebar required by the drop beams for the same bending demand. Thus, drop beam structures have better seismic behavior when compared to hidden beam models, (Sanchez Aguilar, 2010). Changes in the width of hidden beams do not affect the system ductility. On the contrary, increasing the drop beam depth will significantly increase the system ductility ratio.

The effective period is a parameter also considered in this study. Figure 9 compares the effective period for different beam/column combinations, where B refers to a drop beam and H to a hidden beam. Hidden beam models show a higher effective period than drop beam models. This result suggests that buildings with hidden beams are more flexible since they have a greater effective period than buildings with drop beams. This demonstrated that buildings with hidden beams deform more laterally in earthquakes, given their flexibility. The highest period for hidden beam models is 1.25 s, and for drop beam structures is 0.97 s, which represents a 29% difference.

Figure 10 shows the differences in damping ratios for different beam/column combinations, using the same letter scheme as Figure 9. It can be observed that for drop beam models, the damping ratios are above 14%, and for hidden beam models, the damping ratios drop below 10%. According to these results, the percentage of damping is higher in models containing drop beams (133%) than in buildings containing hidden beams (63%). This allows hidden beam model buildings to better dissipate the effects of earthquakes in the nonlinear range.

Performance points obtained from the nonlinear static analysis following FEMA-440, exhibit differences in base shear and displacement values, according to the pushover curves of Figure 11. Values of base shear for H35 × 25 hidden beam section are between 907.70 kN and 1,272.70 kN, which can be compared with the B25 × 35 drop beam pushover curve that shows base shear values between 951.93 kN and 1,667.62 kN. According to these results, drop beam systems present higher base shear values, with a mean difference of 5.5%. In terms of displacement, drop beam systems evidence lower displacements with a mean difference of 15.6%.

In contrast with the H45 × 25 hidden beam and B25 × 45 drop beam models, higher differences are observed in the pushover curves presented in Figure 12. Here, the base shear of the H45 × 25 models is between around 999.68 kN and 1,394.60 kN, which is 23.8% lower than that of B25 × 45 drop beam model. Therefore, drop beam models show higher base shear values, and the differences increase as the beam depth increases. Regarding displacements, it can be observed that drop beam models have 20% lower displacements than hidden beam models. These results demonstrate a better seismic performance of drop beam systems instead of hidden beam systems.

A summary of performance points is presented in Figure 13, where F _x is the base shear obtained by the model and F _min is the minimum base shear obtained by the 16 analysis models, of which 8 are hidden beam models and the other eight are drop beam models, see models in Table 10. On the other axis, U _x and U _min are the roof displacement and minimum roof displacement of the 16 models, respectively. The analysis shows that the effect of stiffness on beam sections and column sections represents a decrease in displacements and an increase in base shear values. Based on the nonlinear static pushover analysis, these results show an improvement in the seismic behavior of buildings with drop beams instead of those with hidden beams. The displacement ratio defined by U _x /U _min shows that buildings with hidden beams deform more significantly than buildings with drop beams, that is to say, about 55% more with respect to the model with beam section B25 × 45 and columns of C55 × 55. At the same time, the stiffer buildings support higher basal shears with lower deformations, i.e., they have better seismic behavior, similar to Samir (2021) results.

To compare ductility and flexural capacity for beams with the highest longitudinal reinforcement ratios of the models shown in Table 11, several moment-curvature diagrams were generated and overlapped in Figure 14. The main factors that affect moment-curvature diagrams are the beam cross-section, longitudinal reinforcement ratio, and stirrup spacing. Results suggest that drop beams have 20%–30% higher maximum moment values than hidden beam sections. As expected, section ductility in drop beams is higher than in hidden beams because of having a lower longitudinal reinforcement ratio. Drop beam sections’ ductility is approximately 30% higher than hidden beam sections, which demonstrates its advantages in terms of stiffness, ductility, steel reinforcement, and flexural capacity, (Paulay and Priestley, 1992; Aguiar, 2003; Córdova, 2015).