Simplified equilibrium approach to the design of free standing stairs based on finite element analysis

Abstract

Because of the absence of any specific code, the process of design of a free standing stair is dependent on some approximate analytical methods. These methods fail to recognise the variation of stress resultants across any cross section of the stair slab and require a lot of computational works. Amanat[4] made an extensive study on free standing stairs using the idealisation of Ahmad's[2] thick shell element and proposed a simplified guideline for easy analysis of free standing stairs. His approach also recognises the stress variation across different sections of the free standing stair, based on finite element analysis, and developed seven empirical design equations. If the loadings and the geometric conditions of the free standing stairs are symmetric, then, only two redundants, namely the lateral shear and the bending moment at the mid landing section are present. Hence, the number of empirical equations can be reduced to two for finding the values of these two redundants. The moments and forces at the other critical locations can be calculated by using equilibrium equations. With a view to developing a more rational and realistic approach by reducing the number of equilibrium equations, and also for increasing the range of applicability of Amanat's equations, this research was undertaken. Sensitivity analysis of the different geometric parameters is carried out over an extended range for two cases of loading. It is found that, in the higher range, the linear equations proposed by Amanat fail to represent the moments and forces as seen from the results of finite element analysis. Based on this study, two sets of empirical equations are developed for two cases of loadings. Other necessary design parameters are obtained from' equilibrium equations which reduce the total number of empirical relations, developed iby Amanat, from seven to four. In this research, detailed study is carried out to select the equations which nearly represent the actual situation. Ten prototype stairs are solved by using the proposed equations, Amanat's equations, and finite element analysis. These results establish the acceptability of the proposed equations. The proposed method has made the calculations for forces and moments, required for design of free standing stairs, very easy, and its estimation is found to be in the acceptable limits for commonly used dimensional ranges of stairs. Recommendations regarding the reinforcement layout have also been made.