The New Method in Calculating Columns and Beams Dimensions That Meets Requirements of The Strong Column-Weak Beam and Non-Soft Story

Situated at an earthquake prone area, buildings planning in Indonesia must implement earthquake resistant building principles. One of these principles is determining dimensions of columns and beams in the process of architectural designing.This act eventually affects the behaviour of the strong column weak beam and the probability of bending failure due to soft story. At present time, there are no simple rules architects can use in calculating the dimensions of beams and columns that meet the criteria for strong column weak beam and non soft story. This paper is an effort to provide an input to the architects in designing the dimensions of the columns and beams. This research is a review result of three theories namely: 1). The theory of columns and beams preliminary design, 2). The theory of the strong column weak beam concept, and 3). The theory of soft story and column slenderness. Those theories were then synthesized into a spreadsheet. To meet the criteria for strong column weak beam and non soft story, the following procedures must be done : 1). Determine the columns’ dimensions according to 0.15% of the columns’ cumulative tributary area, 2). Determine the beams’ dimensions according to 1/12 of the beams’ span and the beams’ plastic modulus, 3). Determine the columns’ dimensions and the columns’plastic modulus, 4). Determine the columns’ height based on the column slenderness criteria, and 5). Compare the columns plastic modulus and the beams plastic modulus and check whether they meet the criteria “the columns’plastic modulus ≥ 1.2 * the beams’ plastic modulus”.


Introduction
Indonesia is an earthquake prone area, so buildings planning in this country must be in accordance with earthquake-resistant building principles.The planning process for earthquake resistant buildings should start from its architectural design process, and the determination the columns and beams dimensions should be the first step because this important step influences the behaviour of the strong column -weak beam and the probability of bending failure as a result of the soft story.Strong column -weak beam is a concept that describes a condition where plastic hinge occurs only in beams and not in columns when a strong earthquake happens, hence avoiding sudden collapse (BSN, 2013).On the other hand, soft story is a form of geometric irregularity of a building where one story has lower lateral stiffness than the other stories (Arnold, 2001).At present there are no simple rules architects can apply in calculating the dimensions of the beams and columns to meet the criteria for strong column -weak beam and to avoid the soft story.This paper strives to provide input to architects in creating preliminary design of beams and columns dimensions that satisfies the principles of making earthquake resistant building.
The concept of a strong column -weak beam is that the calculation result of nominal moments of the upper and lower columns must be 1.2 times bigger than the sum of nominal moments of the left and right supporting beams (see figure 1.1).The purpose of this concept is to prevent the occurrence of the plastic hinge in the columns that serve as the main component of lateral load supports.Therefore, melting should only occur in the beams so that when there is very large inelastic deformation, no collapse happens.To simplify its application in the structure preliminary design, the following formula is used (Bisch et al., 2012) : Wp of columns ≥ 1.2*Wp of beams (1) In which, Wp = plastic modulus (cm3 or m3) Wp = 1/4xbxh2 b = the beams'/columns' width h = the beams/columns' height Soft story occurs if there is a story having lateral stiffness 70% (soft story) or 60% (excessive soft story) less than the lateral stiffness of the story above it (FEMA, 2007).Soft story is usually related to the height of the floor and the slenderness of the column.Soft story may occur if the height of a floor is higher than the height of other floors; in other words this floor is less stiff than the other floors (Arnold, 2001).To overcome this problem,the slenderness of the column must be limited (see table 1.1).
Tabel 1.1.The criteria for slenderness of columns (Sources : compiled from Seki 2015; Okada et al. 2005).In which, h0 = the columns'net height h = h0-h b D = the reviewed columns' width For columns, the minimum dimensions are based on the consideration of their shear strength (Ersoy, 2013) :

Columns
In which, A c = the columns' cross-sectional area A o = the columns' tributary area (see figure 1.2) ∑A o = the columns' cumulative tributary area A c ≥ 0.09 m2 (min 30x30 cm) b c = the columns' width h c = the columns' height In general, the preliminary dimensions of one-way and two-way beams can be calculated with the following formula: In Ground Floor (0.00)

Methods
This research is the result of a literature review.The theories reviewed were : 1).The theory of columns and beams preliminary design, 2).The theory of the strong column -weak beam concept, and 3).The theory of soft story and column slenderness.These theories were then synthesized into a spreadsheet to be a new proposal for calculating dimensionsof columns and beams able to fulfil the requirements for the strong column -weak beam and non -soft story.It is assumed that a building consists of 10 floors with a 5x7 meters structure module, and we need to calculate the dimensions of the beams and columns that meet the criteria for the strong column -weak beam and non -soft story.To solve this problem, we need to do following steps:  From stage 1 to stage 5, the overall results can be seen in table 3.2.From the 1 st floor to the 10 th floor, only the 1 st floor until the 5 th floor that meet the criteria for the strong column -weak beam.With the same span (Lx) = 7 m, the dimensions of the beams from the 1 st to the 10 th floors are same, namely b b = 0.3 m and h b = 0.6 m.The columns' dimensions (b c ) from1 st floor until the 5 th floor may vary according to the weight they support.The columns' dimensions can be determined based on the criteria for the strong column -weak beam and the columns' height.If it is based on the criteria of the strong column -weak beam and it is assumed that the height of the column for each floor (h) = 4 m, then only the 1 st floor to the 4 th floor that meet the criteria.Therefore, the dimensions of the columns can be simplified, for example, for the 1 st floor to the 4 th floor = 0.75x0.75m and for the 5 th floor until the 10 th floor = 0.60x0.60 m.

The Soft Story is One of The Most Dangerous Types of Building Irregularity
The aforementioned procedure of calculating columns and beams dimensions that fulfils the strong column -weak beam and non-soft requirements are expected to reduce the potential soft story because the soft story is one of the most dangerous types of building irregularity (Mezzi, 2006).Boen researches in Padang, Bengkulu and Yogyakarta (Boen, 2006(Boen, , 2007a(Boen, , 2007b) ) show that the most frequent and deadly structural failure due to large earthquakes is the soft story (see figure 3.1).

Conclusion
To make columns and beams dimensions meet the criteria for the strong column -weak beam and non -soft story, the following steps must be conducted: • Determine the columns' dimensions based on 0.15% of the columns' cumulative tributary area.• Determine beams' dimensions based on 1/12 of their span and beams' plastic modulus.
• Determine columns' dimensions and columns' plastic modulus.
• Determine columns' height based on the columns' slenderness criteria.
• Compare the plastic modulus of both columns and beams and find out whether the result satisfies the criteria "columns'plastic modulus ≥ 1.2 * the beams' plastic modulus".With this new proposed way in calculating dimensions of the columns and beams, we can expect the number of sudden collapse due to the occurrence of plastic hinge and soft stories in the columns can be reduced.

Glossary :  Column / beam plastic modulus
: used for materials where elastic yielding is acceptable and plastic behavior is assumed to be an acceptable limit.
 Plastic hinge : a yielding zone in an structural elements which generally develops at the point of maximum bending moment, support, etc.
 Slender column : if its cross-sectional dimensions are small compared with its length.
 Soft story : if there is a story having lateral stiffness 70% (soft story) or 60% (excessive soft story) less than the lateral stiffness of the story above it.

 Strong column -weak beam
: a concept that describes a condition where plastic hinge occurs only in beams and not in columns when a strong earthquake happens, hence avoiding sudden collapse.
0.25 0.0127 0.473 0.70 0.0858 4.65 3.00 Fulfil Journal of Architectural Design and Urbanism Vol 1, No 2, March 2019 The New Method in Calculating Columns and Beams Dimensions That Meets Requirements of The Strong Column-Weak Beam and Non-Soft Story On the -y axis (table 3.3), only the 1 st floor until the 7 th floor meet the criteria for strong column -weak beam.With the same span (Ly) = 5 m, the dimensions of the beams of all floors are the same, namely b b = 0.25 m and h b = 0.45 m.The column dimensions (b c ) on thex-axis and y-axis that meet the criteria for the strong column -weak beam and column height vary, so the most critical part is chosen, namely the x-axis with the columns'dimensions of the 1 st floor to the 4 th floor = 0.75x0.75m and of the 5 th floor to the 10 th floor = 0.60x0.60 m.

Figure 3
Figure 3.1.The Padang earthquake on September 30 & October 1, 2007: a).The Office of DPU Padang experienced soft story on the 1 st floor but it was still survived, b).The 3 -story-shophouse experienced severe soft story at the 1 st floor.This floor collapsed, and the 2 nd & 3 th floors fell on it, c).& d).Two commercial buildings experiencing severe soft story both at the 1st floor.The 1 st floors collapsed, and the 2 nd and 3 rd floors fell on them (source: http://www.perencanaanstruktur.com/2010/10/bentuk-keruntuhan-bangunan-saat-gempa.html).

Table 3 .
1. Determining of the columns cross-sectional area

Table 3 .
2. Determining of the beams and columns dimensions on the x-axis

Table 3 .
3. Determining of the beams and columns dimensions on the y-axis