Department of Civil Engineering and Construction, Bradley University, USA
This research is done to understand the slenderness effects of Reinforced Concrete (RC) columns in a frame structure through a parametric study considering column design inputs and using a Finite Element Method (FEM) modelling procedure. Three ten-
Keywords: Column, Finite Element Modelling, Non-
Slender columns are the structural members whose ultimate load carrying capacities are affected by the slenderness effect and produce additional bending stresses and instability of columns due to excessive deflection. A slender column has less strength than a short column of the same sectional area, and hence it carries a lesser load as compared to a short column (Kumar, 2005). The slenderness of a column dramatically increases due to increase in column height and buckling under gravity or horizontal loads (Halder, 2007). Therefore, evaluation of a slender column involves consideration of the column height, its cross-
Slender columns exhibit deflections when being subjected to eccentric loads. These deflections produce additional flexural stresses due to the increase in eccentricity by the amount of transverse deflection (∆). This phenomenon is known as the P-
If the total moment including the secondary moment reaches the ultimate capacity of a section, the column fails to owe to material failure. Parameters such as column-
If a column is very slender, it becomes unstable before reaching material failure, and such instability failure is observed in those columns (Bazant & Know, 1994). Slenderness effects are more pronounced in a column of a unbraced frame than a braced frame. Frames that do not have adequate bracing against lateral loads show excessive sway, which jeopardizes the stability of columns. Adequate bracing in a frame helps to stabilize secondary deformations at column ends and produces more stable columns. Columns are treated differently depending on the bracing conditions in their frames because of different behavior between a braced and an unbraced frame.
The vertical extension of the buildings is now essential for Bangladesh due to deficiency of land, cost of property, and to accommodate a vast number of growing human population in small areas. Most of the buildings are concrete beam-
However, with the increasing use of high-
2. Objective and Methodology
Considering the effect of slenderness in column design is becoming very important especially with the rapid increase in materials properties in high-
In this research, one typical model of a high-
It should be noted that ACI code (318-
3. Slender Column Design Procedures
As suggested in the ACI Commentary 10.11.4, a compression member can be assumed braced if it is located in a story in which the bracing elements (i.e., shear walls) have a stiffness that is substantial enough to limit lateral deflection to the extent that the column strength is not substantially affected. Visual inspection can often make such a determination. If not, the ACI Code 10.11.4 provides two quantitative criteria for determining if a story is treated as non-
where Q is the stability index, is the total factored vertical load in the story, Vu is the total factored shear load in the story, lc is the height of the column measured center-
A frame could have both non-
If a column is slender, it will fail by buckling into the shape of a sine wave when the load reaches a particular value Pc, which is called the Euler buckling load or critical load that is given by the equation (2),
where E is the elastic modulus of the column and Imin is the minimum moment of inertia of the column. It is seen that the buckling load decreases rapidly with the increase in the Slenderness Ratio (kl/r).
If a column falls under either a non-
where Ec is the elastic modulus of the column and Ig is the effective moment of inertia of the girder, which is equal to 0.35Igross, where Igross is the gross moment of inertia of the girder, Pu is the ultimate load of the column, and Cm is a factor which is a function of the column moments. The column’s original moments are then multiplied by the δns to get the moment required for the calculation of steel ratio. The δns value could be less than 1.0, and for that case, it is assumed as 1.0. If δns is greater than 1.0, then the column is considered to have a slenderness effect. However, a column could have a δns value of less than 1.0, but the kl/r crosses the limiting value or vice-
The step required to calculate a Sway Moment Magnification Factor (δs) is shown in the equation (7),
Pc is calculated for each column in the story of the column being designed and then ∑Pc is calculated for the given story. Similarly, Pu is calculated for each column in the story of the column being designed and then ∑Pu is calculated for the given story.
In a RC frame, columns are rigidly attached to girders and adjacent columns. The effective height of a particular column between stories will depend on how the frame is braced and on the bending stiffness of the girders. For frames braced against side sway, k varies from 1/2 to 1, whereas for laterally unbraced frames, it varies from 1 to ∞, depending on the degree of rotational restraint at both ends.
It should be noted that ACI code describes two methods for determining EI for a column. However, it has been observed that ETABS uses equation (3) to calculate EI for the slender column design (Hossain, 2008; Computers & Structures, 2003).
ETABS Version 8.4.6 is chosen for the study. Figures 1(a), (b), and (c) present the plan view of the ground floor of the models consisting of 15'x15' (4572 mm x 4572 mm), 20'x20' (6096 mm x 6096 mm), and 25'x25' (7620 mm x 7620 mm) slab panels respectively. For simplicity, all the beam-
Gravity loads include the live loads on the building and the self-
The wind and earthquake loads are calculated following the BNBC standard for a commercial building in Dhaka. From the BNBC table titled “Basic wind speeds for selected locations in Bangladesh,” the Basic Wind Speed (Vb) is chosen as 130.5 mph (210 km/hr). The exposure type is selected as “Exposure A” because the building is sited in the urban area. The windward coefficient is calculated as 1.4 from the BNBC article titled “Overall pressure coefficient (Cp) for rectangular building with flat roofs.” As the structure is ranked as the standard occupancy structure, the corresponding value of the Structural Importance Coefficient (CI) equals to 1.00. The wind load is assumed as acting on the four faces of the building.
From the Seismic Zone Map of Bangladesh, the Seismic Zone Coefficient (Z) is taken as 0.15 corresponding to the seismic zone 2. In the basic design information, it has been acknowledged that the building system is a dual system defined as the concrete with concrete IMRF (Intermediate Moment Resisting Frame). So, from the BNBC, the Response Modification Coefficient for the Structural Systems (R) is taken as 9 for both horizontal directions of the building. The Structural Importance Coefficient is taken as 1.00 for the earthquake analysis, and it is same as the wind load analysis. The earthquake force is acting from both horizontal directions of the building. The Structural Period is intended from “Method A” that is described in the BNBC.
According to the BNBC, the site soil characteristic is considered as S3. As it is a commercial building, the BNBC classified it as a standard occupancy structure and ranked the Structural Importance Category as IV. In the model, the following loads are used: dead load (DL), live load (LL), wind load from X direction (WLx), wind load from Y direction (WLy), earthquake load from X direction (EQx), and earthquake load from Y direction (EQy). The Wind and the earthquake loads are applied perpendicular to the building axis. No eccentric load is applied to the model for simplicity. The ETABS version 8.4.6 uses the default cases of load combination of ACI code (318-
4.1 Parametric Study
Table.1 shows the floor panel sizes, column positions, column heights, column sizes, and periphery beam sizes, which are used for the parametric study. In this parametric study, total 15 models (3 models for each floor panel size with 5 varying column heights) are generated for the beam-
For reinforced concrete column structure, a slab panel size larger than 25' (7620 mm) is not a common scenario. The slab panel greater than 25′ (7620 mm) needs special design requirements. A slab panel smaller than 15' (4572 mm) does not frequently use in commercial buildings because the column spacing is small for ground floor car parking. The beam dimensions were kept the same for all the models because of the consistency in the analysis. After each model is analyzed using ETABS, the moment values of the selected column end and design load for each targeted column are extracted from the software. All the parameters are calculated using a spreadsheet and cross-
5. Results and Discussions
5.1 Critical Buckling Load
The limiting value of Slenderness Ratio (kl/r) is checked with the recommended ACI code guideline. According to the ACI guideline, a column should be treated as a slender column if the value of the kl/r is greater than 34-
Figure 2 represents Slenderness Ratio (kl/r) for three column locations with corresponding Critical Buckling Load (Pc). According to Figure 2(a), the value of kl/r is greater or equals to 40 for six corner columns. The Pc value decreased by around 25% when the column height increased by 2.5' (762 mm).
Pc verses kl/r for the edge column is illustrated in Figure 2(b). It can be seen that the kl/r is greater than 40.0 for three edge columns. The 20' (6096 mm) edge column for the 25'x25' (6096 mm x 6096 mm) panel size showed slenderness effect. The size of the edge column is 17"x17" (432 mm x 432 mm) for the 25'x25' (7620 mm x 7620 mm) size panel. The steel ratio increases by more than 8% in some cases if the column size is reduced from the 17"x17" (432 mm x 432 mm) dimensions. The kl/r is greater than 40.0 for one column in the 15'x15' (4572 mm x 4572 mm) size panel.
Figure 2(c) shows the kl/r for the inner columns. It is seen that three inner columns showed slenderness effect. No column showed slenderness behavior for the 25'x25' (7620 mm x 7620 mm) slab panel. In the FEM model, the steel ratio increased such that the column fails due to over reinforcement if the column dimension is reduced for the 25'x25' (7620 mm x 7620 mm) slab panel. The column section for the 25'x25' (7620 mm x 7620 mm) slab panel is increased compared to the 20'x20' (6096 mm x 6096 mm) slab panel to keep the steel ratio in the tolerable limit. For this reason, the column of this slab panel does not show slenderness compared to the lower sized slab panels. For all three panels, the Pc value decreases by around 25.0% while the column height is increased from 17.5' (5334 mm) to 20.0' (6096 mm). Therefore, it will not be wise to judge slenderness by the only variation in the PC and the kl/r. From the above discussions and observations, it can be concluded that corner columns are most sensitive to become slender.
5.2 Ration of Ultimate Load to Critical Buckling Load
Figure 3 shows three column locations to understand the influence of loads in column buckling. Figure 3(a) represents the ratio of Pu/Pc with kl/r. According to Figure 3(a), the slenderness needs to be considered when the design load (Pu) reaches 30% of the critical buckling load (Pc). According to Figures 3(b) and 3(c), the edge and the corner columns showed a slenderness behavior when Pu is increased by more than 40% of Pc. The above observation also proves that the corner columns showed a higher slenderness behavior when compared to the edge and inner columns.
Figure 4(a) shows the δns variations for the corner columns. The δns value is increased by about 43.0% when column height is increased from 17.5' (5334 mm) to 20.0' (6096 mm) for the 15'x15' (4572 mm x 4572 mm) slab panel. The δns value is increased by about 25.0% when column height is increased from 15.0' (4572 mm) to 17.5' (5334 mm) for the 15'x15' (4572 mm x 4572 mm) slab panel. Also the δns is increased by about 59.0% for the same increment in column height (17.5' to 20.0') for the 20'x20' (6096 mm x 6096 mm) slab panel and δns is increased by about 72.0% for the 25'x25' (7620 mm x 7620 mm) slab panel. So, there is a significant increase in the δns value which is observed when the column height increases from 17.5' (5334 mm) to 20.0' (6096 mm).
According to Figure 2(b), the value for kl/r is higher than 40.0 for one column in the 15'x15' (4572 mm x 4572 mm) panel but according to Figure 4(b) and the ETABS calculation, the δns value is lower than 1.0. If the design engineers only rely on the ETABS analysis and design output, then it is highly likely that it may result in an under reinforced column design. According to the ACI code approach, the 17.5' (5334 mm) column should be considered as a slender column, but ETABS does not consider it as a slender column as δns value is lower than 1.0. It is seen that δns for a corner column exceed 1.0 only for three columns even though from slenderness point of view six columns should be designed as slender columns. The variation between ETABS and ACI analysis shows that the ACI code design approach is conservative.
Figure 4(b) is illustrated for the δns against the kl/r for the edge columns of the three panels. The δns value is increased by about 74.0% while the column height is increased from 17.5' (5334 mm) to 20.0' (6096 mm) for the 15'x15' (4572 mm x 4572 mm) panel. The δns value is increased by about 84.0% while column height is increased from 17.5' (5334 mm) to 20.0' (6096 mm) for the 20'x20' (6096 mm x 6096 mm) slab panel and it is increased by about 36.0% for the 25'x25' (7620 mm x 7620 mm) panel. The δns value is not significantly high for the edge column for the 25'x25' (7620 mm x 7620 mm) panel compared to the other two panels. Similar to the corner column, a drastic change is observed when column height is increased from 17.5' (5334 mm) to 20.0' (6096 mm) for the edge column.
Figure 4(c) is illustrated for the δns against kl/r for the inner columns of the three slab panels. The δns value is increased by about 45.0% while the column height is increased from 17.5' (5334 mm) to 20.0' (6096 mm) for the 15'x15' (4572 mm x 4572 mm) slab panel size. The δns is increased by about 43.0% for the column while the height is increased from 17.5' (5334 mm) to 20.0' (6096 mm) for the 20'x20' (6096 mm x 6096 mm) slab panel size. The δns is increased by about 20.0% for the 25'x25' (7620 mm x 7620 mm) slab panel for the same height increment of the column. Compared to the other two column locations, the value of δns is not much higher for the inner column for all the panel size. For this reason, it is concluded that the inner column is not slender susceptible compared to the corner and the edge column.
5.4 Sway Moment Magnification Factor
There is no straightforward procedure for designing a sway column considering Sway Moment Magnification Factor (δs) in ETABS. For sway frame, the magnified moment in ETABS is determined after performing the P-
The limiting value of the kl/r is checked with the recommended ACI code guidelines. In the case of a sway frame, according to the ACI code, the column should be treated as a slender column if the kl/r value is greater than 22.0. It is observed that the value for kl/r was higher than 22.0 for all the 15 corner columns. Therefore, the columns which are neglected due to a lower value (i.e., less than 1.0) of δns must be considered carefully for the sway moment effect even if the value is low. It has been checked that these 15 columns Stability Index (Q) are lower than 5.0% for the five columns, but according to Figure 5(a), it is seen that even they are in the non-
Figure 5(b) presents δs versus kl/r for the 15 edge columns for the three panels, and all the columns have a sway effect similar to the corner columns. The kl/r is more than 100.0 for the 20' (6096 mm) column for the 25'x25' (7620 mm x 7620 mm) slab panel. So, the column needs second order computer analysis. From comparing Figures 5(a) and (b) it is seen that value of the δs varies from 1.0 to 1.18. The 15′x15′ (4572 mm x 4572 mm) size panel showed higher δs values compared to the 25′x25′ (7620 mm x 7620 mm) size panel but another way around while compared the kl/r. For the 25'x25' (7620 mm x 7620 mm) panel, the kl/r values are higher in comparison to the other two panels. Therefore, considering only δs or kl/r would not be sufficient to compute column slenderness.
In Figure 5(c), the δs value is plotted against the kl/r for the inner column. Except for the 20′ (6096 mm) column, no kl/r value exceeds 100.0, but every column has a slenderness effect. The increment of δs in a column for the 15'x15' (4572 mm x 4572 mm) slab panel is steeper than the other two panels. From the preceding description of δns, it was concluded that slenderness effect in the inner column is as important as it is for other two locations.
5.5 ACI Code Limitations
According to the ACI code, for compression members in the non-
According to the ACI code, for the compression member in a sway frames, the effect of slenderness may be neglected when the klu/r value is less than 22.0. So, if the value of (klu/r)/22 is lower than 1.0, then the column will not show any slenderness effect. Figure 6(b) is a plot of δs versus (klu/r)/22 for the beam-
6. Conclusions and Recommendations
The objective of this study is to understand the effects of column design inputs on the slenderness of RC columns for beam-
1. Every RC frame in a structure needs to be treated as a sway frame since ground floor columns in the structure have a Sway Moment Magnification Factor (δs) of varying magnitudes. So, if a designer uses ETABS, then it is necessary to run the P-
2. Corner columns are more sensitive to becoming slender in comparison to the edge and the inner columns. For a corner column, the bi-
3. The effect of Non-
4. The structural models are developed considering a commercial building located in a moderate earthquake influence zone and occupancy. In Bangladesh as other two earthquakes influence zones are present, so it is a need to study the effect of slenderness in columns in different earthquake zones and another occupancy, i.e., residential.
5. The study is done considering increment of all columns of a particular floor level. In Bangladesh, it is observed that only columns in one or two frame make double height while other columns keep in nominal height. So, the influence of slenderness in this type of building requires a thorough study.
6. The 15 models developed by ETABS are square shape, and all the columns considered in these models have a square cross-
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