Calculation of Axial Loads on Columns by Tributary Area Method and Finite Element Method

Mohd Faiz Mohammad Zaki 1, a, Mohd Zulham Affandi Mohd Zahid1, b, Afizah Ayob 1, c and Tee Chin Fang 1,d

1School of Environmental Engineering, Kompleks Pusat Pengajian Jejawi 3, Universiti Malaysia Perlis, 02600 Arau, Perlis, Malaysia

afaizzaki@unimap.edu.my, bmohdzulham@unimap.edu.my, c afizah@unimap.edu.my, dteechinfang@gmail.com

Keywords: Tributary area method, axial load, finite element method

Abstract. Basic concept of structural design is to transmit the loading from superstructure to substructure. This idea normally required sound knowledge of structural design and rational engineering judgments. Recently, there have several techniques that can be utilized to determine the superstructure loading, such as finite element method and tributary area method. However, the compatibility of both methods in order to determine the loading from superstructure is prime important and has been investigated in this research framework. Axial loading, represented as products from dead load and service load, which are imposed on the top of slab is directly transmit to the column nearby and modelled through computer simulation. Models of slab were then varies and studies through comparison with broad dimensions of slab thickness, ranging in 100 mm to 600 mm. Results has shown the increasing of slab thickness will indirectly increases the rigidity characteristic of slab and potential to distribute the axial load equally for all column members. Axial load against slab thickness on corner, edge, center, outer and inner column demonstrated the incompatibility for both methods, finite element method and tributary area method in determining the axial loading from superstructure.

Introduction

Comprehensive understanding of the structural building analysis is mandatory when attempting to evaluate and determine the structural loading before commences any works in building constructions, particularly for slab, beam and column. The first phase of any evaluation is to identify the impose load on these structure elements. This stage required great attention from structural engineer to perform the load analysis which is contributed by load from superstructure and design to transmit safely this load to foundation. The accurate computation of axial loads is crucial for determining the size and strengths of the structure support. It is a complicated task because time-dependent deformations of concrete and the way the building is constructed affect the way the gravity loads are carried by them (Kurc and Lulec, 2013).

This research focused on the calculation of axial loading in column structure. The calculation

methods that adopted in this research are tributary area method and finite element method. These methods are initiated through computing the forces on columns due to the load applied on the slab by calculating the surrounded area of columns and multiply this area with the load on the slab. Thus, resulted in area (m2) × floor load (kN/m2) = loading on column (kN). The floor load can be divided into dead load (such as finishes and self weight) and live load. Live loads are produced by the usage and occupancy of area for the structure (Ebrahimpour and Sack, 1992). Based on this study, the main purpose is to analyze the axial loadings that transmitted to column by using tributary area method and compare with the analysis using finite element method.

Material and Methodology

There are two cases studies that have been selected in this research. The first case study is a square flat slab structures modelled with various different slab thickness. Second case study is a circular flat slab structure and developed to simulate the water tank. These models then have undergone the process of determination axial loading by using two different methods, tributary area method and finite element method. To check the compatibility for both methods, a ratio study was conducted. The ratio of FEM (Finite Element Method) to TAM (Tributary Area Method) is calculated based on the first and second cases with different slab thickness ranging from 100 mm to 600 mm, associated with varies span length, 3m to 7m for columns interval. Tributary Area Method demonstrated an easy manual calculation and time consuming. However, this method has the limitations, for example, tributary area method unable to consider accurately the wind load and seismic loading. First step is to dividing the area into several parts which is assumes the load dividing equally into the column support surround (Fig 1).

Figure 1: Area divided into several parts based on columns location In order to analyze both models based on finite element method, SAFE v8 has been utilized and

performs through computer simulation.

Results and Discussions

Case Study I is a square flat slab with 9 columns (4 Corner columns, 4 edge columns and 1 center column). Figure 2 below show the result of deformed shape for case I with 100 mm thickness slab and 600 mm thickness slab. When the slab was assigned as 100 mm thickness, the maximum deflection is occurred in between the column which is 10.7 mm. When the slab was assigned as 600 mm thickness, the maximum deflection is occurred in middle of modal, which is 7 mm. In thin slab, the deformed shape is show more obviously. But the thick slab only deflected a bit. This is due to the thinner the slab, the more flexible the slab. But, a thicker slab, it is hard and rigid to be deflected.

Figure 2: Deformed shape for Case Study I

Figure below shows the result for Case I by using tributary area method and finite element method. The similar calculations are performed for different slab thickness to obtain the axial load on column. Tributary area method, totally depend on the area of slab surrounding by the column. Axial load carried by edge column presented as approximately similar value for TAM and FEM. However, for axial load on corner columns are identical for the slab thickness 300 mm. Axial load carried by centre of column demonstrated the constant regardless the thickness of slab based on FEM. However, the tributary area method shows the axial load carried by center of column was increased dramatically.

Figure 3: Result for Case I, TAM (Tributary Area Method) and FEM (Finite Element Method)

Case Study II modelled as circular flat slab with 9 columns. The result in Figure 4 below presented

the deformed shape for case II. The maximum deflection for slab 100 mm thicknesses is 33.91 mm. The maximum deflection is occurred in middle of modal for slab 600 mm, which is 7.5 mm. The deformed shape for 600 mm thickness slab is not obviously if compare with 100 mm slab. This is due to its rigid characteristic. It also carries higher dead load, contributed by self weight. Thus, the whole slab sank downward together. This shown similar result with case study I.

Figure 4: Deformed shape for Case Study II

The loading that assumed for this case is high due to the circular slab are always designed for water tank. Thus, the loading assigned in this analyse take into account the weight of water in a water tank with 5m height. The results shown the axial load carried by the centre column is highest due to its area. The same calculations are carried out for different slab thickness from 100 mm to 600 mm, and obtain the axial load on column.

Figure 5: Result for Case II, TAM (Tributary Area Method) and FEM (Finite Element Method)

Based on the Figure 5 above, the value of axial load on outer column is approximately similar for both methods, FEM and TAM against the slab thickness. Inner column, however, decreased significantly using finite element method for the slab thickness higher than 300 mm. The axial load carried by inner column decreased from 1105.07kN to 823.8kN according to FEM result. This is due to the axial load carried by inner column is potential to distributed the load to outer column when the slab become rigid. Conclusion

Increasing of slab thickness will indirectly increases the rigidity characteristic of slab and distributed the axial load equally for all columns member. This has been presented through the analyses of finite element method above. The deformed shape for the both cases shown the slab is difficult to deflect when the thickness of slab increase and proved the slab thickness is able to controls the deflection of a flat slab. The graph of axial load against slab thickness on corner, edge, center, outer and inner column showed the results are significantly different for both methods.

References

[1] Al-Qasem, I. A. A. & Abdulwahid, M. Y. (2012). Comparison of calculation of axial loads on columns by tributary area method and 3D modeling by SAP2000. International Journal of Civil and Structural Engineering, 3, pp.336-345.

[2] Bryant, Ebrahimpour, A. & Sack, R. L. (1992). Design Live Loads for Coherent Crowd Harmonic Movements. Journal of Structural Engineering, 118, pp.1121-1136.

[3] EN1991-1-1 (2002) Eurocode 1: Actions On Structures - Part1-1: General actions-Densities, Self-weight, Imposed Loads for Buildings.

[4] Kurc, O. & Lulec, A. (2013). A comparative study on different analysis approaches for estimating the axial loads on columns and structural walls at tall buildings. The Structural Design of Tall and Special Buildings, 22, 485-499.

Axial Load on Outer Column Axial Load on Inner Column