Structural system yielding minimum differences between ordinary and staged analyses

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 well-validated 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 10-storeys 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 20-storeys 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 beam-column 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 30-storeys concrete buildings, respectively. El-leithy [14] also selected the adequate SSRL by limiting the wind drift for various building heights using ETABS [16]. Likewise, Gunel and Ilgin [13] and El-leithy [14] recommended that RF system is adequate for 20-storeys 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 cross-sectional 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 high-rise 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 45-storeys building with RF and central core. The same building was analyzed using the nonlinear staged-construction 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 high-rise 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 volume–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 10-storey 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 non-linearity and the non-linear 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 non-structural 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% ~ 154% for buildings with various SSRL. Performance of post-tensioned (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 pre-compression, 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

Figure 1 shows 3D views of six different reinforced concrete buildings (B_d1, B_d2, … B_d6) which have a floor height of 3.5 m and various SSRL, as shown in Fig. 2. Details of the six buildings are presented in Table 1 while concrete dimensions of all structural elements are provided in Table 2. Figure 3 shows that the footprint of the six buildings has dimensions of 30 × 30 m and a central 6 × 6 m opening. A thickness of 300 mm is assumed for all slabs to fulfill the deflection regulations. El-leithy [14] estimated the concrete dimensions and reinforcement of the investigated buildings, according to the ACI 318R-05 [19] and the ASCE/SEI 7–05 [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 318R-05 [19] and the ASCE/SEI 7–05 [20]. More details of the loading criteria, including partitions, finishing, cladding, live, and wind load, can be found in [14].

Fig. 1

Three-dimensional view of studied buildings

Fig. 2

SSRL in the studied buildings

Table 1 Number of storeys, height, and structural system of the investigated buildings

Table 2 Concrete dimensions of structural elements for the investigated buildings (in cm)

Fig. 3

Plan view of the studied buildings [14]

The concrete characteristic strength (${\text{f}}_{\text{c}}^{^{\prime}})$, Poisson’s ratio ${(\upupsilon }_{\mathrm{c}}$), and Young’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 (${\text{f}}_{\text{u}})$, Poisson’s ratio ${(\upupsilon }_{\mathrm{s}})$, and Young’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 CEB-FIP [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 column-foundation 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. 4a) are used to model the columns and beams whereas four node 3D plate elements (Fig. 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 beam-column and the beam wall connections.

Fig. 4

Elements utilized in the numerical model (a) Beam element (b) Plate element [21]

Stiffness is updated at each load step to account for the geometric non-linearity. A bi-linear stress–strain relationship is utilized for steel to consider its nonlinear behaviour [21]. The elasto-plastic model of [24] is implemented to consider for the non-linear performance of concrete under compression. The tensile strength for concrete is ignored, according to different comprehensive research work [6,7,8,9]. Time-dependent behaviour of concrete is included using the parameters calculated according to [25] code and CEB-FIP [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. 5. For both buildings, the difference between the two meshes does not exceed 3% and 5% for the drift and structural period, respectively.

Table 3 Number of elements and number of degrees of freedom for the studied buildings for both analyses

Fig. 5

Results of sensitivity analysis for buildings B_d1 and B_d5 (a) storey drift (b) structural period

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 engineering programs. Three consecutive floors are assumed to be supported by form-work in each stage. Table 4 shows the details of adopted SA parameters in the developed FEM.

Table 4 SA parameters implemented in the model

Staged-construction 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]. Figure 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 time-dependent, according to [27]. Figure 7 shows the construction stages for the SA performed for the investigated buildings, as reported by [10]. The time for form-work installation and removal as well as floors casting is provided.

Fig. 6

Erection sequence steps in [21]

Fig. 7

Construction schedule adopted for SA [10]

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 … etc.) and non-structural element (brick walls, curtain walls … etc.). The damages occur due to increase of internal forces at one end of the beam/slab (Fig. 8). If slabs or beams are analyzed using OA, they may experience larger bending moments and shearing forces if re-analyzed using SA. Also, the other locations will be over designed where smaller internal forces will be estimated using SA compared to those from OA.

Fig. 8

Bending moments and shearing forces from OA and SA

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. 9. B1 rests on C2 and C4 columns in buildings B_d1 and B_d4 (Fig. 9a) and also in buildings B_d3 and B_d6 (Fig. 9c). While B2 rests on C5 and C6 columns in B_d1 and B_d4 buildings (Fig. 9a) and supported on column C5 and shear wall SW in buildings B_d3 and B_d6 (Fig. 9c). The S1 strip rests on C2 and C4 columns in buildings B_d2 and B_d5 (Fig. 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:

$$Diff. \%= \frac{{X}_{SA}-{X}_{OA}}{{X}_{OA,max}}\times 100\%$$

(1)

where X_OA and X_SA are obtained from OA and SA, respectively.

Fig. 9

Plans of studied buildings; (a) B_d1 and B_d4 (b) B_d2 and B_d5 (c) B_d3 and B_d6

Differential displacements (DD)

Figure 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 mid-height. OA yields in a maximum DD of 1.92, 4.94, 5.11, 1.83, 4.75, and 8.62 mm for B_d1 through B_d6 buildings, respectively (Fig. 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 beam-ends due to OA and SA.

Fig. 10

Differential displacement in columns and walls from OA and SA: (a) B_d1 (b) B_d2 (c) B_d3 (d) B_d4 (e) B_d5 (f) B_d6

Bending moments

Differences between bending moments from OA and SA are plotted in Fig. 11 for B1 and S1 and in Fig. 12 for B2 and S2. The difference in bending moments noticeably vary because shortenings of vertical elements are not equal. Figure 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. 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. 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. 12). For buildings B_d1, B_d2, B_d3, and B_d4, the OA provides underestimated bending moments at the mid-spans. The same trend is noted for lower floors of B_d5 and B_d6 and overestimated values are noted at top. However, Fig. 12 depicts that OA overestimates the mid-span 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.

Fig. 11

Difference in bending moments in B1 and S1 between OA and SA: (a) B_d1 (b) B_d2 (c) B_d3 (d) B_d4 (e) B_d5 (f) B_d6

Fig. 12

Difference in bending moment for beam B2 and slab S2 between OA and SA: (a) B_d1 (b) B_d2 (c) B_d3 (d) B_d4 (e) B_d5 (f) B_d6

Shearing forces

Figures 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. 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. 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. 14). Therefore, OA yields unsafe design due to utilizing underestimated shearing forces.

Fig. 13

Difference percentage in shearing force in B1 between OA and SA: (a) B_d1 (b) B_d3 (c) B_d4 (d) B_d6

Fig. 14

Difference percentage in shearing force in B2 between OA and SA: (a) B_d1 (b) B_d3 (c) B_d4 (d) B_d6

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 10-storey 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 1.92 mm at Storey 10, while the SA yield a maximum DD of 1.82 mm at Storey 8. Comparing the DD of the 20-storey 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 8.62 mm at Storey 20 and 7.25 mm at Storey 13, respectively. Comparing the BM and SF of the 10-storey 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 20-storey 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 20-storeys buildings.

Table 5 Maximum DD from OA and SA

Table 6 Maximum difference percentage between BMs in beams and slab strips from OA and SA

Table 7 Maximum difference percentage between SFs in beams from OA and SA

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 non-linearities. 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:

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

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

• ➢ For 10-storeys 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.

• ➢ For 20-storeys 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.

• ➢ For 10-storeys 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.

• ➢ For 20-storeys 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.

• ➢ Using RF and WF systems in 10 and 20-storeys 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.