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Structural system yielding minimum differences between ordinary and staged analyses
Journal of Engineering and Applied Science volumeÂ 70, ArticleÂ number:Â 70 (2023)
Abstract
Structural engineers should appropriately design concrete structures to resist lateral loads. Determining the adequate system for resisting the expected lateral loads is important to control the building drift. Choosing the appropriate system is usually conducted assuming the predicted forces are applied to completed concrete buildings at one step which is commonly known as ordinary analysis (OA). Nevertheless, these structures are constructed sequentially which requires using staged analysis (SA) instead of OA. In this paper, a comprehensive numerical model for SA of concrete buildings, which accounts for time dependent effects, is utilized using a wellvalidated commercial software. Six reinforced concrete buildings with 10 and 20 storeys are analyzed using the developed model. Three various structural systems are considered (Rigid Frame (RF), Shear Wall (SW), and Wall Frame (WF). A comparison is conducted between the displacements and internal forces in beams and slabs obtained from the SA and OA. For a 10storeys RF building, maximum bending moment from SA is 29.9% higher than that from OA. The same conclusion was observed for the maximum shearing force with a percentage of 19.6%. Moreover, maximum bending moments and shearing forces from SA for the 20storeys RF building are, respectively, 35.0% and 23.5% larger than those from OA. The RF and WF systems provided the minimum difference in differential displacement between the OA and SA analyses. The RF system produced the least differences in internal forces from OA and SA for all studied buildings.
Introduction
Selecting appropriate structural system to resist lateral loads (SSRL) for reinforced concrete structures is critical due to having various options in practice and because this process significantly affects the construction cost. Among different types of SSRL, there is usually one system which can resist wind and earthquake loadings with minimum cost. Structural engineers usually determine the SSRL for a building based on the number of storeys, gravity and lateral loads, as well as architectural requirements. However, inadequate selection of SSRL for reinforced concrete (RC) buildings might cause severe damages or uneconomic solutions. Ordinary analysis (OA) is commonly adopted for concrete structures to select the appropriate SSRL where all design loads are applied simultaneously on the entire building. Different research works indicated that the OA is not adequate to analyze the concrete structures [1,2,3,4,5,6,7,8,9,10,11] because the dead load is usually applied on these structures in stages. Staged analysis (SA) should be utilized instead of OA to appropriately find the deformations and straining actions in these structures.
Many authors explored the selection process of an adequate SSRL [12,13,14,15] by analyzing various concrete structures having various SSRL. Taranath [12] analyzed a set of rigid frames (RF) concrete structures where the lateral loads are resisted by the moment resisting beamcolumn connections. Using this system was found suitable for concrete buildings with a small number of storeys. Based on the adequate SSRL, concrete structures were classified by Gunel and Ilgin [13] who reported that RF and shear walls (SW) systems can produce ideal solution for 20 and 30storeys concrete buildings, respectively. Elleithy [14] also selected the adequate SSRL by limiting the wind drift for various building heights using ETABS [16]. Likewise, Gunel and Ilgin [13] and Elleithy [14] recommended that RF system is adequate for 20storeys or less concrete buildings. The same limit was recommended for the SW system and extended to 40 storeys for RC buildings with WF system where the lateral loads are resisted through the interaction between the walls and frames.
However, Katkhoda and Knaa [15] recommended adopting the WF system for RC residential buildings consisting of 10 to 20 storeys. The authors reached this conclusion after performing the structural design of a number of concrete buildings using an advanced optimization technique. Multiple cycles of analysis and design were performed to determine the minimum crosssectional dimensions for all reinforced concrete elements of the studied buildings. The obtained solutions achieved significant savings in concrete and steel amounts and consequently the minimum materials cost was obtained. The recommendations by [12,13,14,15] were developed based on OA but were not validated if SA was adopted.
SA should be adopted instead of OA to accurately capture the behaviour of RC tall buildings [1,2,3,4,5,6,7,8,9,10,11]. Liu et al. [1] assessed structural performance of tall buildings by developing a performance based structural design methodology to control different structural states in various construction stages. The buildings were reported to potentially suffer from safety hazards if the construction stages are ignored in the analysis of tall buildings. The proposed methodology was illustrated by modelling of various tall buildings. The study revealed that SA yielded feasible and effective design for tall buildings.
The effect of SA was experimentally measured by Su et al. [2] who monitored floor settlements during construction of real buildings. The authors monitored settlement of tall RC buildings using SA. It was expected that using the developed approach would accurately predict the elevation of the studied building.
Fan et al. [3] used finite element (FE) models to estimate the downward displacements during the construction of buildings with a large number of storeys. The investigation was conducted on three super highrise buildings in China. The study revealed that OA yielded relative and total vertical displacements as well as straining actions significantly different from SA. Samarakkody et al. [4] studied the effect of differential axial shortening on the behaviour of tall buildings with concrete filled tube (CFT) columns. A comprehensive technique was developed and validated for differential axial shortening (DAS) estimation in tall buildings with composite CFTs. The study concluded that maximum DAS, usually occurs at mid height of the building, can be shifted to the upper floor levels due to considering creep and shrinkage effects. Significant reduction in DAS of the vertical load bearing structural components occurred due to introducing an outrigger system. It was shown that using CFT in buildings reduced the adverse effects of DAS. Correia and Lobo [5] proposed a simplified method to analyze a 45storeys building with RF and central core. The same building was analyzed using the nonlinear stagedconstruction analysis package offered by the SAP2000 software [17]. The authors found that the internal forces in interior beams of top floors obtained from the proposed simplified method were smaller than their counterparts obtained from the numerical model by 37% and 60% at levels 30 and 40, respectively.
Different researchers showed that time dependent effects should be included in the analysis of RC building to adequately evaluate the column shortenings [6,7,8,9,10,11, 18]. Yang et al. [6] developed and validated a neural network technique to estimate vertical displacements in highrise concrete buildings. The validation was conducted by comparing the numerical results with those measured experimentally for existing buildings. The developed technique provided more accurate results than conventional numerical models. Moragaspitiya et al. [7] developed a general numerical technique based on finite element modelling, to estimate differential axial shortening in RC buildings taking into account the construction sequence. The authors reported that differential axial shortenings might significantly increase due to time dependent effects of concrete. The developed technique was validated by modelling a 64 storey RC building. The study concluded that differential axial shortening between perimeter columns were influenced by the axial stiffness of the columns based on load tributary on each column.
Shrinkage, creep and temperature were found critical in evaluating the axial shortening of RC shear walls in tall buildings in the construction stage [8]. It was reported that the rate of shrinkage development between columns and shear walls can be significantly different even for elements with same volumeâ€“surface ratio. The authors studied the influence of seasonal variation of ambient relative humidity on the shrinkage and creep development of concrete. It was recommended that more experimental studies should be conducted to study the shape effect between prismatic or cylindrical specimens and shear walls. Kwak and Kim [9] analyzed a 10storey RC building to investigate the differences in structural responses between OA and SA using a numerical model considering time dependent effects of concrete. A verification was conducted by comparing the numerical results with those obtained from previous experiments. The geometric nonlinearity and the nonlinear behaviour of concrete were considered. The study concluded that SA might produce differential column shortenings and bending moments greater than those obtained from OA. These differences were reported to cause serviceability concerns in the nonstructural members located between interior and exterior columns. Elansary et al. [10] developed and validated a FE model for sequential analysis of concrete buildings and accounted for time dependent effects. Displacements, bending moments, and shear forces from SA and OA were compared. Displacements from SA were larger than those from OA by 116%â€‰~â€‰154% for buildings with various SSRL. Performance of posttensioned (PT) slabs in buildings was investigated by Elansary et al. [11] considering SA. Significant differences were detected between the service and ultimate moments, service tensile stresses, slab precompression, as well as punching stress obtained from SA and OA.
Methods
Motivated by the lack of research on selecting the SSRL for RC building that yields the minimum differences between OA and SA, this paper has the following objectives. The first is to utilize a comprehensive numerical model to estimate the differences in deformations and straining actions between OA and SA for two RC buildings with three different SSRL. The comparison includes differential displacement in vertical members, moments and shear forces in beams, as well as moments in slabs. The second objective is determining the SSRL that provides minimum differences between deformations and straining actions obtained from OA and SA for the investigated buildings. Some essential analysis assumptions, including dimensions, reinforcement, and material properties, are first reported. After that, details of the utilized finite element model (FEM) including description of the meshing and time dependent parameters are presented. Then, the settlements, shearing forces and bending moments from SA and OA are compared. Finally, the authors reported the SSRL which provides minimum differences between OA and SA for each building height.
Basic assumptions
FigureÂ 1 shows 3D views of six different reinforced concrete buildings (B_{d1}, B_{d2}, â€¦ B_{d6}) which have a floor height of 3.5Â m and various SSRL, as shown in Fig.Â 2. Details of the six buildings are presented in Table 1 while concrete dimensions of all structural elements are provided in Table 2. FigureÂ 3 shows that the footprint of the six buildings has dimensions of 30â€‰Ã—â€‰30Â m and a central 6â€‰Ã—â€‰6Â m opening. A thickness of 300Â mm is assumed for all slabs to fulfill the deflection regulations. Elleithy [14] estimated the concrete dimensions and reinforcement of the investigated buildings, according to the ACI 318R05 [19] and the ASCE/SEI 7â€“05 [20] codes. No change was applied on the cross sections of the column and wall within each five successive floors. Gravity loads (dead and live) as well as wind lateral load (wind speed of 100 mph) were considered in the design [14] where OA was adopted. The structural design under seismic loads was also checked according to the ACI 318R05 [19] and the ASCE/SEI 7â€“05 [20]. More details of the loading criteria, including partitions, finishing, cladding, live, and wind load, can be found in [14].
The concrete characteristic strength (\({\text{f}}_{\text{c}}^{^{\prime}})\), Poissonâ€™s ratio \({(\upupsilon }_{\mathrm{c}}\)), and Youngâ€™s modulus (\({\text{E}}_{\text{c}})\) are \(\text{40 MPa}\), \(0.2\), and \(\text{29,725 MPa}\), respectively. While the steel yielding stress (\({\text{f}}_{\text{y}})\), ultimate stress (\({\text{f}}_{\text{u}})\), Poissonâ€™s ratio \({(\upupsilon }_{\mathrm{s}})\), and Youngâ€™s modulus \(({\text{E}}_{\text{s}}\text{)}\) are \(\text{400 MPa}\), \(\text{520 MPa}\), \(0.3\) and \(\text{200,000 MPa}\), respectively. Parameters for the shrinkage and creep of concrete are evaluated according to the CEBFIP [26]. To consider cracking, stiffness for slabs, beams, columns, and walls were reduced by 25, 35, 70, and 70%, respectively. Fixed supports were assigned at the columnfoundation connections.
Numerical modelling
A 3D model is utilized to analyze the investigated concrete buildings using a robust finite element software (midas Gen [21]) which is selected due to the proven efficient performance in conducting SA as reported by [10, 11, 22, 23]. Two nodes 3D beam elements (Fig.Â 4a) are used to model the columns and beams whereas four node 3D plate elements (Fig.Â 4b) are utilized to simulate the slabs and walls. Each node includes six degrees of freedom (3 displacements and 3 rotations). Rotations are prevented at the between the beamcolumn and the beam wall connections.
Stiffness is updated at each load step to account for the geometric nonlinearity. A bilinear stressâ€“strain relationship is utilized for steel to consider its nonlinear behaviour [21]. The elastoplastic model of [24] is implemented to consider for the nonlinear performance of concrete under compression. The tensile strength for concrete is ignored, according to different comprehensive research work [6,7,8,9]. Timedependent behaviour of concrete is included using the parameters calculated according to [25] code and CEBFIP [26] standards. More details about the selected parameters can be found in [10].
The FEM is validated by conducting OA for the six studied RC buildings under wind loading using [21] and comparing the results with those obtained from [14] who utilized CSI ETABS software [16] in modelling the same buildings. The FEM mesh is adopted in the current study after conducting a sensitivity analysis by modelling buildings B_{d1} and B_{d5} using two different mesh sizes for each building. The number of elements in each mesh is shown in Table 3. Drift and structural period for the two buildings, B_{d1} and B_{d5} from the two mesh sizes are plotted in Fig.Â 5. For both buildings, the difference between the two meshes does not exceed 3% and 5% for the drift and structural period, respectively.
The SA is conducted for the investigated buildings using midas Gen software [21] which is selected based on the recommendation of different researchers [10, 11, 22, 23]. The software conduct accurate analysis based on current theories and numerical methods published in reputable journals. The midas Gen software [21] have been validated by numerous examples and comparisons with other engineeringÂ programs. Three consecutive floors are assumed to be supported by formwork in each stage. Table 4 shows the details of adopted SA parameters in the developed FEM.
Stagedconstruction analysis methodology
The utilized software, midas Gen [21], accounts for the stages of construction by activating/ deactivating structural elements, boundary conditions, and applied loads. Construction loads are applied to the structural models representing the various construction stages. The flooring and live loads can be applied on a completed building. Only the own weight of the structural elements is considered because live and lateral loads have insignificant effect on SA of the building [10]. FigureÂ 6 shows a flowchart summarizing the hybrid analysis steps in midas Gen [21] which includes both SA and OA. The updated concrete compressive strength is calculated at each time increment based on the timedependent, according to [27]. FigureÂ 7 shows the construction stages for the SA performed for the investigated buildings, as reported by [10]. The time for formwork installation and removal as well as floors casting is provided.
Change in straining actions due to SA
Axial loads on columns increases during the construction of a building. Axial deformations of columns lead to redistribution of forces between the columns. Consequently, cracking or severe damage might occur in the structural elements (floors, beams â€¦ etc.) and nonstructural element (brick walls, curtain walls â€¦ etc.). The damages occur due to increase of internal forces at one end of the beam/slab (Fig.Â 8). If slabs or beams are analyzed using OA, they may experience larger bending moments and shearing forces if reanalyzed using SA. Also, the other locations will be over designed where smaller internal forces will be estimated using SA compared to those from OA.
Results and discussion
In this section, the SSRL yielding the least differences between deformations and internal forces from OA and SA for each building height is provided. The differences are obtained from the elements with maximum straining actions: (B1, B2) beams and (S1, S2) strips, as shown in Fig.Â 9. B1 rests on C2 and C4 columns in buildings B_{d1} and B_{d4} (Fig.Â 9a) and also in buildings B_{d3} and B_{d6} (Fig.Â 9c). While B2 rests on C5 and C6 columns in B_{d1} and B_{d4} buildings (Fig.Â 9a) and supported on column C5 and shear wall SW in buildings B_{d3} and B_{d6} (Fig.Â 9c). The S1 strip rests on C2 and C4 columns in buildings B_{d2} and B_{d5} (Fig.Â 9b), whereas S2 strip rests on C5 column and SW shear wall in the same buildings. The following equation is utilized to estimate the difference in straining actions from OA and SA:
where X_{OA} and X_{SA} are obtained from OA and SA, respectively.
Differential displacements (DD)
FigureÂ 10 illustrates the DD between ends of the considered beams and strips of the slabs where noticeable differences are noted between OA and SA. OA results in a nonlinear distribution for displacements with a maximum value at top of the building. The SA also results in a nonlinear DD distribution, but the maximum displacement near the midheight. OA yields in a maximum DD of 1.92, 4.94, 5.11, 1.83, 4.75, and 8.62Â mm for B_{d1} through B_{d6} buildings, respectively (Fig.Â 10). SA yields maximum DDs of 1.82, 6.37, 5.40, 2.28, 3.83, and 7.25 for B_{d1} through B_{d6} buildings, respectively. One can observe that buildings B_{d1} and B_{d4} with RF system experienced the minimum DDs between beamends due to OA and SA.
Bending moments
Differences between bending moments from OA and SA are plotted in Fig.Â 11 for B1 and S1 and in Fig.Â 12 for B2 and S2. The difference in bending moments noticeably vary because shortenings of vertical elements are not equal. FigureÂ 11 shows that the bending moments developed in B1 and S1 from OA are larger than moments from SA by 41.0%, 42.4%, 58.0%, 37.7%, 38.9%, and 55.1% for buildings B_{d1} through B_{d6}, respectively. Same observation can be noted for bending moments in B2 and S2 with percentages of 16.2%, 52.3%, 52.6%, 18.6%, 57.7%, and 62.4% for buildings B_{d1} through B_{d6}, respectively (Fig.Â 12). Therefore, one can note that OA produces an uneconomic design due to utilizing overestimated bending moments. On the other hand, the bending moments from SA are higher than those from OA by 29.9%, 25.9%, 24.6%, 35.0%, 36.5%, and 22.8% for buildings B_{d1} through B_{d6}, respectively (Fig.Â 11). Similar trend is observed for the bending moments in B2 and S2 with percentages of 5.7%, 98.4%, 120.9%, 4.4%, 96.3%, and 159.2% for buildings B_{d1} through B_{d6}, respectively (Fig.Â 12). For buildings B_{d1}, B_{d2}, B_{d3}, and B_{d4}, the OA provides underestimated bending moments at the midspans. The same trend is noted for lower floors of B_{d5} and B_{d6} and overestimated values are noted at top. However, Fig.Â 12 depicts that OA overestimates the midspan bending moment of B2 and S2 for buildings B_{d2}, B_{d3}, B_{d4}, B_{d5}, and B_{d6.} The same observation is noted for upper floors of building B_{d1} and underestimated solution is obtained from the OA in the lower floors.
Shearing forces
FiguresÂ 13 and 14 show the difference in shearing force of beams B1 and B2 from OA and SA for (B_{d1}, B_{d3}, B_{d4}, and B_{d6}) buildings. The differences in shearing forces dramatically vary along the building height. Maximum difference in shearing forces in B1 from OA is higher than those from SA by 26.2%, 37.8%, 24.2%, and 35.1% for B_{d1}, B_{d3}, B_{d4}, and B_{d6} buildings, respectively. These percentages for B2 become 11.1%, 49.6%, 14.8%, and 58.9% for buildings B_{d1}, B_{d3}, B_{d4}, and B_{d6}, respectively (Fig.Â 14). It can be noticed that using OA produces uneconomical solution due to utilizing overestimated values. On the other hand, the maximum difference percentage in shearing force for B1 from SA is higher than those from OA by 19.6%, 13.9%, 23.5%, and 9.3% for B_{d1}, B_{d3}, B_{d4}, and B_{d6} buildings, respectively (Fig.Â 13). The same note is observed for the maximum shearing force of B2 with percentages of 0.75%, 60.69%, 1.36%, and 89.36% for B_{d1}, B_{d3}, B_{d4}, and B_{d6} buildings, respectively (Fig.Â 14). Therefore, OA yields unsafe design due to utilizing underestimated shearing forces.
Determining SSRL with minimum difference
The SSRL for each building is selected if it produces the smallest difference between maximum differential displacement (DD), bending moments (BM), and shearing forces (SF) from OA and SA. Table 5 shows the maximum DD between ends of the beams and strips from OA and SA. While Tables 5 and 6 show the maximum difference percentage between BM and SF obtained from OA and SA for the studied RC buildings. Comparing the DD of the 10storey buildings, one can observe that building B_{d1} has the smallest difference in maximum DD obtained from OA and SA (Table 5). The OA is noted to provide a maximum DD of 1.92Â mm at Storey 10, while the SA yield a maximum DD of 1.82Â mm at Storey 8. Comparing the DD of the 20storey buildings, one can observe that building B_{d6} has the smallest difference in maximum DD obtained from OA and SA (Table 5). The OA and SA provide a maximum DD of 8.62Â mm at Storey 20 and 7.25Â mm at Storey 13, respectively. Comparing the BM and SF of the 10storey buildings, it can be observed that building B_{d1} has the smallest difference between the values obtained from OA and SA (Tables 6 and 7). The SA yields BM and SF larger than OA by 29.9% and 19.6%, respectively both at Storey 9. Similar observations can be noted for building B_{d4} when the BM and SF for the 20storey buildings are compared. SA resulted in BM and SF larger than those from OA by 35.0% and 23.5%, respectively, both at Storey 14. The observations in this section indicate that RF system provides the minimum difference in BM and SF between OA and SA for 10 and 20storeys buildings.
Conclusions
Analysis of six concrete buildings using Ordinary Analysis (OA) and Staged Analysis (SA) is conducted in this paper using a robust numerical model which accounts for time dependent effects (SA). Moreover, the model accounts for the material and geometric nonlinearities. Two different number of storeys (10 and 20) and three various lateral load resisting systems (RF, SW, and WF) are adopted. Differential displacement (DD) and straining actions in horizontal elements from the OA and SA are compared. The following observations can be concluded:

âž¢ For 10storeys buildings, maximum DDs in beams/slabs using OA are 1.92Â mm, 4.94Â mm, and 5.11Â mm for RF, SW, and WF systems, respectively. However, SA yields maximum DDs of 1.82Â mm, 6.37Â mm, and 5.40Â mm for RF, SW, and WF systems, respectively.

âž¢ For 20storeys buildings, the maximum DDs in beams/slabs using OA are 1.83Â mm, 4.75Â mm, and 8.62Â mm for RF, SW, and WF systems, respectively. However, these differences for SA are 2.28Â mm, 3.83Â mm, and 7.25Â mm for RF, SW, and WF systems, respectively.

âž¢ For 10storeys buildings, the maximum difference in bending moment in beams obtained from SA is larger than those obtained from OA by 29.9%, 98.4%, and 120.9% for RF, SW, and WF systems, respectively.

âž¢ For 20storeys buildings, the maximum difference in bending moment in beams obtained from SA is larger than those obtained from OA by 35.0%, 96.3%, and 159.2% for RF, SW, and WF systems, respectively.

âž¢ For 10storeys buildings, the maximum difference in shearing force in beams from SA is larger than those from OA by 19.6% and 60.7% for RF and WF systems, respectively.

âž¢ For 20storeys buildings, the maximum difference in shearing force in beams from SA is larger than those from OA by 23.5% and 89.4% for RF and WF systems, respectively.

âž¢ Using RF and WF systems in 10 and 20storeys buildings, respectively, provides the minimum difference in DD between the OA and SA analyses. However, the RF system yields the minimum difference in straining actions between the OA and SA analyses for the studied buildings.
It is worth mentioning that the current paper aimed at determining the lateral load resisting system that produce the minimum differences between OA and SA. This research is currently being extended by examining the seismic behaviour of the investigated buildings.
Availability of data and materials
All data utilized to produce this research can be made available upon request.
Abbreviations
 OA:

Ordinary analysis
 SA, SCA:

Staged analysis
 SCAT:

Staged construction analysis considering timedependent effects
 RF:

Rigid frame
 SW:

Shear wall
 WF:

Wall frame
 SSRL:

Structural system to resist lateral loads
 RC:

Reinforced concrete
 FE:

Finite element
 FEM:

Finite element model
 CFT:

Concrete filled tube
 DAS:

Differential axial shortening
 \({\text{f}}_{\text{c}}^{^{\prime}}\) :

Concrete characteristic strength
 \({\upupsilon }_{\mathrm{c}}\) :

Poissonâ€™s ratio
 \({\text{E}}_{\text{c}}\) :

Youngâ€™s modulus
 \({\text{f}}_{\text{y}}\) :

Yielding stress
 \({\text{f}}_{\text{u}}\) :

Ultimate stress
 \({\upupsilon }_{\mathrm{s}}\) :

Poissonâ€™s ratio
 \({\text{E}}_{\text{s}}\) :

Youngâ€™s modulus
 CDS:

Column differential settlement
 Diff.%:

Difference in straining actions
 DD:

Differential displacements
 BM:

Bending moments
 SF:

Shearing forces
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The authors would like to thank the Structural Department at the Faculty of Engineering, Cairo University for giving the authors the opportunity to conduct this research work.
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All authors have read and approved the manuscript. The following shows some details about the contribution of each author. AE: Determined the research point, proposed modelling technique, and criticized the results. AE: Criticized the results, revised the modelling technique, revised the technical writing of the manuscript. MI: Generated the models, wrote the draft of the paper, and implemented the required technical writing revisions.
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Elansary, A.A., Metwally, M.I. & ElAttar, A.G. Structural system yielding minimum differences between ordinary and staged analyses. J. Eng. Appl. Sci. 70, 70 (2023). https://doi.org/10.1186/s44147023002433
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DOI: https://doi.org/10.1186/s44147023002433