Vertical geometric irregularity in stepped building frames
Introduction
In multi-storeyed framed buildings, damage from earthquake ground motion generally initiates at locations of structural weaknesses present in the lateral load resisting frames. In some cases, these weaknesses may be created by discontinuities in stiffness, strength or mass between adjacent storeys. Such discontinuities between storeys are often associated with sudden variations in the frame geometry along the height. There are many examples [1], [2] of failure of buildings in past earthquakes due to such vertical discontinuities.
A common form of vertical discontinuity arises from reduction of the lateral dimension of the building along its height. This building category is labelled as ‘stepped’ building in this paper. This building form is becoming increasingly popular in modern multi-storey building construction mainly because of its functional and aesthetic architecture. In particular, such a stepped form provides for adequate daylight and ventilation for the lower storeys in an urban locality with closely spaced tall buildings. This type of building form also provides for compliance with building bye-law restrictions related to ‘floor area ratio’ (practice in India). Fig. 1 shows a typical example of a stepped building located in urban India (New Delhi).
Stepped buildings are characterised by staggered abrupt reductions in floor area along the height of the building, with consequent drops in mass, strength and stiffness (not necessarily at the same rate). Height-wise changes in stiffness and mass render the dynamic characteristics of these buildings different from the ‘regular’ building. Design codes have not given particular attention to the stepped building form. This is perhaps due to the paucity of research on stepped buildings reported in the literature.
This paper discusses some of the key issues regarding analysis and design of stepped buildings. In the present study, a new approach for quantifying the irregularity in stepped building is proposed. It accounts for properties associated with mass and stiffness distribution in the frame. This approach is found to perform better than the existing measures to quantify the irregularity. The empirical equations of fundamental period given in the design codes are function of building height, which is ambiguous for a stepped building. Based on free vibration analysis of 78 stepped frames with varying irregularity and height, this study proposes a correction factor to the empirical code formula for fundamental period, to render it applicable for stepped buildings.
Section snippets
Design code perspective on stepped building
The stepped building form is recognised by several design codes, such as IS 1893:2002 [3] and ASCE 7:2005 [4], as a typical form of vertical geometric irregularity that merits special design consideration. As per IS 1893:2002, such building forms are to be treated as vertically irregular when the lateral dimension of the maximum offset () at the roof level exceeds 25% of the lateral dimension of the building at the base (), as shown in Fig. 2(a). As per ASCE 7:2005, when the horizontal
Literature review
There is very limited published literature available on the lateral load resisting behaviour of stepped moment resisting frames. An extensive literature survey has revealed only three journal papers in this area, of which two were published very recently (2008). Researchers are now beginning to give attention to this category of building frames. Two of these papers [6], [7] deal with concrete frames, while the third paper [8] deals with steel frames. All three research papers conclude that the
Proposed method of quantifying irregularity
The purpose of the present study is to address the specific stepped irregularity of the framed building. 78 building frames representing varying degree of stepped irregularity are considered for the study. All the building models considered here have four bays (in the direction of earthquake) with a uniform bay width of 6 m. It should be noted that bay width of 4 m–6 m is the usual case, especially in Indian and European practice. However, the results were checked for frames with different
Estimation of fundamental time period for stepped building
As explained earlier, design codes [3], [4], [5] recommend dynamic analysis for irregular buildings including stepped buildings, with the base shear scaled up to the value corresponding to the fundamental period as per the code specified empirical formulas. These formulas, however, have been developed for regular buildings and are a function of building height only.
For a stepped building, the height is not constant. It varies from one side to the other side of the frame. The building height at
Example of an existing building
As the above formulations are based on two-dimensional building models (plane frames), it is instructive to check the applicability of the proposed equations for a three-dimensional model of an existing building. For this purpose, Delhi Secretariat building, one of the five life-line buildings identified as part of the Delhi Government Earthquake Safety Initiative [13], is considered. It is a ten-storeyed office building located in New Delhi (Seismic Zone IV with designed PGA=0.24 g as per IS
Conclusions
Stepped building frames constitute a category of vertical irregularity, whose seismic behaviour has not received adequate attention in existing research and code formulation. In this paper, a detailed study has been carried out to address this shortcoming. The salient conclusions are as follows:
- (1)
A measure of vertical irregularity, suitable for stepped buildings, called ‘regularity index’, is proposed, accounting for the changes in mass and stiffness along the height of the building. This is
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