KSCE Journal of Civil Engineering (0000) 00(0):1-7 Copyright ⓒ2015 Korean Society of Civil Engineers DOI 10.1007/s12205-015-0289-0

Structural Engineering

pISSN 1226-7988, eISSN 1976-3808 www.springer.com/12205

TECHNICAL NOTE

Formulation for Free-standing Staircase Saeid Zahedi Vahid*, Mohammad Ali Sadeghian**, Siti Aminah Bt. Osman***, and Abdul Khalim Bin Abdul Rashid**** Received May 21, 2013/Revised August 5, 2014/Accepted August 20, 2014/Published Online February 6, 2015

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Abstract In the analysis of a free-standing staircase with slab elements, approximate analytical methods are sometimes used because of the absence of specific code provisions due to their inherent limitations. These approaches, however, cannot predict the actual threedimensional (3D) behavior of the stair slab system. In addition, analytical methods cannot predict the stress resultant distribution in any section. These drawbacks highlight the need for the development of more rational but simple analysis methods for design purposes. Thus, in this study, an extensive numerical study on the behavior of free-standing staircases was performed using the finite element package SAP2000. A sensitivity study on the different geometric parameters and material properties that affect the design force and moments was also conducted. As a result of this study, semi-empirical equations are proposed from which the design forces and moments can be calculated in a single step. The accuracy of the equations within an acceptable limit is established by comparison with the results of rigorous FE analyses. Finally, the proposed design equations lead to a simple, straightforward and safe design while simultaneously representing the true behavior of 3D free-standing stair slabs. Keywords: concrete, stair, free-standing, analysis, design ··································································································································································································································

1. Introduction Stairs are essential features of all residential and commercial buildings. A staircase is constructed with steps rising without a break from floor to floor or with steps rising to a landing between floors, with a series of steps rising further from the landing to the floor above. Although many types of stairs can be planned and designed in concrete, steel or timber. For free-standing stairs, none of the available analytical approaches which can categorized into two type idealizes the stair slab as a “ space frame” and “ space plate” is readily suitable for practical design because of considerable calculations; an inadequate and simple method of analysis has restricted their use and has hindered architects and engineers from more widely adopting this impressive stair design. Unfortunately, apparent complications derived from space action have compelled the use of empirical design methods and the introduction of undesirable and unnecessary simplifications, with consequent loss of economy and slenderness. Due to the lack of a simple rational design code, designers are

forced to use a conservation design, resulting in an unnecessarily heavy-looking structure. Although the code provisions for ordinary dog-legged type stairs based on rigorous analytical studies were developed by Ahmed et al. (1995; 1996), the leading codes of practice, e.g., ACI or British code, do not provide any guideline regarding the analysis and reinforcement design of this type of concrete structure. Hence, there is a need for further improvement in the analysis, and the design procedures of free-standing stairs need to be investigated using rigorous finite element analysis. Many researchers have developed different concepts for analytical, numerical, design and experimental assessments of freestanding staircases. The available simplified analytical approaches can be categorized into two types. The first approach idealized the stair slab structure as a space frame. The methods by Fuchsteiner (1954), Saouter (1964), Cusens and Kuang (1965), Taleb (1964) and Gould (1963) fall into this category. One limitation of such an idealization is that the methods fail to predict the variation of the stress resultants across any section of the flights or landing. In the second type, the space plate configuration of the stair

*Graduate Research Assistant, Dept. of Civil And Structural Engineering, Faculty of Engineering, Universiti Kebangsaan Malaysia, 43600 Ukm Bangi, Selangor, Malaysia (Corresponding Author, E-mail: [email protected]) **Ph.D. Student, Dept. of Civil and Structural Engineering, Faculty of Engineering, Universiti Kebangsaan Malaysia, 43600 Ukm Bangi, Selangor, Malaysia (E-mail: [email protected]) ***Associate Professor, Dept. of Civil and Structural Engineering, Faculty of Engineering, Universiti Kebangsaan Malaysia, 43600 Ukm Bangi, Selangor, Malaysia (E-mail: [email protected]) ****Associate Professor, Dept. of Civil and Structural Engineering, Faculty of Engineering, Universiti Kebangsaan Malaysia, 43600 Ukm Bangi, Selangor, Malaysia (E-mail: [email protected]) −1−

Saeid Zahedi Vahid, Mohammad Ali Sadeghian, Siti Aminah Bt. Osman, and Abdul Khalim Bin Abdul Rashid

slab is retained; however, the configuration is made determinate based on some assumptions. The overall structure rigidity resulting from the indeterminacy is lost when such assumptions are made. The method of Siev (1962) and Liebenberg (1960) fall into this category. Smith (1980) discussed the behavior of a 90° free-standing staircase using finite element analysis. Ng and Chetty (1975) analyzed a three-flight free-standing staircase. Amanat and Ahmad (2001) studied the behavior of a free-standing staircase using finite element analysis. In this study, a full three-dimensional static linear finite element analysis was carried out to investigate the behavior of a freestanding staircase. The effect of various parameters such as material and geometry on the forces and moments of a reinforced concrete free-standing staircase under symmetrical and nonsymmetrical loading was also investigated. The objective of this research was to investigate the free-standing staircase and to propose a formula for the design of forces and moments.

Fig. 1. Finite Element Model of Standing Staircase

2. Numerical Modeling In this study, 23 staircases were modeled using the finite element software package SAP2000 under different combinations of loading, with the loads depending on the position of the live load shown in Table 1. The staircase consists of two flights and a landing and was built monolithically and symmetrically loaded. The flights have the same dimensions and are held fixed at floor levels. The front, back, left and right sides of the staircase and the inside and outside of it are arbitrarily defined as illustrated in Fig. 1. According to a study by Amanat and Ahmad (2001), the various dimensions of the prototype staircase are A = 305 mm, B = 1220 mm, C = 1220 mm, L = 2550 mm and H = 3050 mm, as indicated in Fig. 2. The thicknesses of the flight slabs and the landing slab are the same, i.e., T1 = T2 = 125 mm. Although the stairway is a concrete structure, the material is assumed to be linearly elastic, homogeneous and isotropic. The Poisson's ratio is assumed to be 0.15, and the modulus of elasticity of concrete is calculated from the ACI formula, E = 4700 fc ′ MPa, ( fc ′ is the ultimate strength of concrete in MPa). Both live load and dead loads are applied as gravity loads. A live load of 0.48 × 10− 2 MPa is used. The unit weight of the material is 2.356 × 10−3 N/ mm3. Additional weight due to the steps is also considered. This is where the line numbering ends within the text. Make sure there is a section break before the references section so that the references are not line-numbered.

Fig. 2. Stair Slab Geometry: (a) Elevation (b) Plan

To ensure that the simulation process is correct, this verification process is needed in this study. A reinforced concrete freestanding staircase that was analyzed by Amanat and Ahmad (2001), as illustrated in Fig. 1, was selected and analyzed. According to their study, the bending moment at the point of support, mid-span and kink obtained from the paper was arbitrary, -10.56, 1.34, and -7.23, respectively, while the bending moment obtained from the SAP2000 was -10.56, 1.20, and -6.33, respectively. Therefore, the difference is less than 10% and acceptable. Fig. 3 plots the bending moment as a function of distance at the centerline of the upper flight, proving that SAP 2000 is reliable in predicting the behavior of free-standing reinforced concrete. As the mesh used for all the models was

Table 1. Combinations of Loading on the Position of the Live Load Load case i ii iii iv v

Position Live load over the entire stairway Live load on flights only Live load on landing slab only Live load over upper flight Live load over lower flight

Fig. 3. Bending Moments at the Centerline of the Upper Flight vs. Distance in Load Case i −2−

KSCE Journal of Civil Engineering

Formulation for Free-standing Staircase

Table 2. Dimensions and Properties used for the Stairs Model 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

A (mm) 200 305 400 500 305 305 305 305 305 305 305 305 305 305 305 305 305 305 305 305 305 305 305

B (mm) 1220 1220 1220 1220 900 1050 1400 1220 1220 1220 1220 1220 1220 1220 1220 1220 1220 1220 1220 1220 1220 1220 1220

C (mm) 1220 1220 1220 1220 1220 1220 1220 900 1050 1400 1220 1220 1220 1220 1220 1220 1220 1220 1220 1220 1220 1220 1220

H (mm) 3050 3050 3050 3050 3050 3050 3050 3050 3050 3050 3500 4000 4500 3050 3050 3050 3050 3050 3050 3050 3050 3050 3050

T (mm) 125 125 125 125 125 125 125 125 125 125 125 125 125 100 150 175 125 125 125 125 125 125 125

L (mm) 2550 2550 2550 2550 2550 2550 2550 2550 2550 2550 2550 2550 2550 2550 2550 2550 3000 3500 4000 2550 2550 2550 2550

DL (N/mm2) 0.00295 0.00295 0.00295 0.00295 0.00295 0.00295 0.00295 0.00295 0.00295 0.00295 0.00295 0.00295 0.00295 0.00295 0.00295 0.00295 0.00295 0.00295 0.00295 0.002 0.004 0.00295 0.00295

LL (N/mm2) 0.0048 0.0048 0.0048 0.0048 0.0048 0.0048 0.0048 0.0048 0.0048 0.0048 0.0048 0.0048 0.0048 0.0048 0.0048 0.0048 0.0048 0.0048 0.0048 0.0048 0.0048 0.004 0.006

auto meshed using the SAP2000 auto mesh, no convergence study was required.

3. Results and Discussions 3.1 Finite Element Analysis The deflected shape of the stairway obtained from FEM analysis is presented in Fig. 4. Figs. 5 and 6 reveal that the total bending moment at the support and mid-span of the flight is maximum for load case ii, while the kink moment is maximum for load cases iii. Fis. 7 and 8 demonstrate that the torsion moment of flight is maximum at load case i. Fig. 9 presents a plot of the plate stress resultant (moment/unit width) across the section at the support of the upper flight. Fig. 10 plots the distribution of the same stress resultant for the landing across the mid-landing section. These figures demonstrate that the stress resultants are non-uniformly distributed across the section, clearly indicating that the free-standing stair slab, which is primarily a three-dimensional plate structure, cannot be simplified to a skeletal frame structure or a determinate slab system. The bending moment at other critical sections is not distributed uniformly across the section except at the mid span of the flight section, which is depicted with a white line in Fig. 4(a). Examining the stresses at these locations, 70% of the total bending moment at the support section of the flight occurs within the outer half of the width of that section. The inner half of the section, on average, carries the remaining 30% of the total bending. In contrast, the inner points at the flight landing junction are stressed higher than Vol. 00, No. 0 / 000 0000

Fig. 4. Deflected Shape and Vertical Deflection of the Stairway: (a) Deflected Shape at the Mid Span of the Flight, (b) Vertical Deflection of the Stairway

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Saeid Zahedi Vahid, Mohammad Ali Sadeghian, Siti Aminah Bt. Osman, and Abdul Khalim Bin Abdul Rashid

Fig. 11. Bending Moments at Intersection of Landing and Flights

Fig. 5. Bending Moments in Upper Flight

Fig. 6. Bending Moments in Lower Flight

Fig. 7. Torsion Moments in Upper Flight

the outer points on average, with the inner half of the section carrying 60% of the total kink moment and the outer half supporting the rest, as illustrated in Fig. 11. At the mid-landing section, the bending moment varies for the inner one-third of the sections shown in Fig. 10, and the bending stresses do not vary significantly across the remainder of the section. On average, 50% of the total bending at the mid-landing section is resisted by the inner one-third of the section, and the outer two-thirds carry the remainder. Lateral distribution is important in designing the reinforcement layout. Fig. 11 plots the bending moment for the intersection of the landing and the flights. The stairway behaves as a three-dimensional plate structure, which is clearly indicated by its deflected shape. Except at the midspan of the flights, the bending moment at other critical sections is not distributed uniformly across the width of the section. The moment is concentrated near the outer edge of the support and near the inner edge of the kink and mid-landing section. The deflected profile of the staircase on the horizontal plane clearly indicates that the effects of axial forces on the flights (elongation of the upper flight and shortening of the lower flight) are more than offset by the effect or in-plane moments that causes lateral sway of the entire staircase toward the upper flight, as illustrated in Fig. 4.

Fig. 9. Bending Moments in Support of Upper Flight

3.2 Sensitivity Analysis To develop a straightforward method for determining design forces and moments, a sensitivity analysis was performed. Note that although an analytical method or finite element analysis enables us to determine forces and moment at any section of the stairway, only a few of these quantities at some critical locations are necessary to design the stairway. The sensitivity of these quantities to the various geometric parameters was investigated. For the purpose of analysis, a default stairway with initial values of A = 305 mm, B = 1220 mm, C = 1220 mm, H = 3050 mm, T = 125 mm, L = 2550 mm, DL = 0.00295 N/mm2 and LL = 0.0048 N/mm2 was established. Then, each of these geometric parameters was varied in turn; e.g., when A was varied, the other parameters retained their default value.

Fig. 10. Bending Moments in Mid-landing Section

3.3 New Design Rationale We can now formulate explicit expressions using multiple linear regressions for a rational and safe estimation of the moments and forces at various critical locations within the acceptable limits of accuracy, which will greatly expedite the calculations and design process. Multiple linear regression attempts to model

Fig. 8. Torsion Moments in Lower Flight

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KSCE Journal of Civil Engineering

Formulation for Free-standing Staircase

the relationship between two or more explanatory variables and a response variable by fitting a linear equation to the collected data. Every value of the independent variable x is associated with a value of the dependent variable y. The population regression line for p explanatory variables x1, x2,..., xp is defined to be µ3 = β0 + β1x1 + β2x2 +L+ βpxp. This line describes how the mean response µ3 changes with the explanatory variables. The observed values for y vary about their means µ3 and are assumed to have the same standard deviation σ. The fitted values b0, b1, ..., bp estimate the parameters β0, β1, ..., βp of the population regression line. Because the observed values for y vary about their means µ3, the multiple regression model includes a term for this variation. In other words, the model is expressed as DATA = FIT + RESIDUAL, where the “FIT” term represents the expression β0 + β1x1 + β2x2 + L βpxp. The “RESIDUAL” term represents the deviations of the observed values y from their means µ3, which is normally distributed with mean 0 and variance σ. The notation for the model deviations is ε. Formally, the model for multiple linear regression, given n observations, is yi = β0 + β1xi1 + β2xi2 + ... βpxip + βi for i = 1, 2, ... n. In the least-squares model, the best-fitting line for the observed data is calculated by minimizing the sum of the squares of the vertical deviations from each data point to the line (if a point lies precisely on the fitted line, then its vertical deviation is 0).

Because the deviations are first squared and then summed, there are no cancellations between positive and negative values. The least-squares estimates b0, b1, ... bp are usually computed using statistical software. The values fit by the equation b0 + b1xi1 +L+ bpxip are denoted y'i, and the residuals ei are equal to yi – y'i, the difference between the observed and fitted values. The sum of the residuals is equal to zero. The variance σ2 may be estimated Σe 2i by s² = -------- , also known as the Mean-Squared Error (MSE). The np1 estimate of the standard error s is the square root of the MSE. The models were analyzed based on the data in Table 4 with SAP2000, and a value for Mmax1 was obtained. This parameter was then determined using the regression values provided in Table 4, and the coefficients were derived from multiple linear regressions and are presented in Table 3. Using this method, given some parameters and equations, we can plot all the figures Table 3. Coefficients of Regression Coefficients Intercept A B C H T L DL LL

30.9156 -0.00343 0.00269 -0.00296 -0.00077 -0.01073 -0.01023 -1198.04 -1819.47

Table 4. Bending Moments Model 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

Fig. 12. Residual Plot

Fig. 13. Line Plot

Fig. 14. Normal Probability Plot Vol. 00, No. 0 / 000 0000

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Regression Mmax1 -12.13 -12.49 -12.81 -13.16 -13.35 -12.94 -12.00 -11.54 -11.98 -13.02 -12.83 -13.22 -13.60 -12.22 -12.76 -13.02 -17.09 -22.21 -27.32 -11.35 -13.74 -11.03 -14.67

SAP 2000 Mmax 1 -12.35 -12.38 -13.15 -13.11 -13.40 -12.76 -11.92 -11.67 -11.96 -13.23 -12.70 -13.15 -13.68 -12.12 -12.68 -13.01 -17.78 -21.50 -27.57 -11.23 -13.64 -10.90 -14.59

Difference % 1.8 0.8 2.6 0.3 0.7 1.3 0.6 1.1 0.1 1.6 1.0 0.5 0.5 0.8 0.6 0.0 4.0 3.1 0.9 1.0 0.7 1.1 0.5

Saeid Zahedi Vahid, Mohammad Ali Sadeghian, Siti Aminah Bt. Osman, and Abdul Khalim Bin Abdul Rashid

related to the variables. As an example, for the maximum bending moment in the support flight (Mmax1), residual and line fit plots for all the different variables as well as a normal probability plot can be produced, as demonstrated in Figs. 12-14. From the regression coefficient data, the maximum bending moment in the support flight can be derived: Mmax1 = −.00343A + .002692B − .00296C − .00077H − .01073T − .01023L − 1198.04DL − 1819.47LL + 30.91564 KN.mm/mm

(1)

Based on the detailed sensitivity analysis, approximate expressions for the forces and moment at critical locations are proposed in terms of the various dimensions of the stairway, which can completely define the geometry of a free-standing staircase. These expressions are valid within the usual ranges of the geometric parameters.

Fig. 15. Mmax1, Mmax2, Vmax, Tmax, for Flight and Landing Predicated using the Proposed Equations

3.4 Proposed Equation Based on the present study, a simple and straightforward method of determining the required design forces and moments has been developed. Mmax1, Mmax2, Vmax and Tmax are the negative moment, positive moment, shear moment and torsion moment, respectively. All of the equations for the flights are as follows: In flights, i. Mmax1 = −.00343A + .002692B − .00296C − .00077H − .01073T − .01023L − 1198.04DL − 1819.47LL + 30.91564 KN.mm/mm

(1)

ii. Mmax2 = − .00224A − .00208B − .00092C + .00074H − .00464T + .005625L + 100.6893DL + 808.1385LL − 11.681 KN.mm/mm

(2)

Fig. 16. Positions of the Forces and Moments in the Landing

iii.Vmax = − 5.8E-07A − 2.4E-06B + 1.59E-06C − 6.5E-08H + 3.02E-06T + 8.06E-06L + 1.697078DL + 2.678156LL −.01935 KN/mm (3) iv.Tmax = .005544A + .001997B + .003612C + 5.11E-05H + .005184T − .00032L + 917.2982DL + 586.8903LL − 9.98993 KN.mm/mm (4) In landing, i. Mmax1 = − .00306A − .01656B − .02909C − .00112H + .051901T − .00504L − 7320.07DL − 4574.51LL + 76.70828 KN.mm/mm

(5)

ii. Mmax2 = − .0088A + .005846B − .01279C − .00018H + .001066T − .00267L − 2148.38DL − 1362.52LL + 21.66841 KN.mm/mm

(6)

iii.Vmax = − .00014A − 2.8E-05B − 3.8E-05C − 9.5E-06H + .000183T − 6.4E-06L − 13.7538DL − 8.80052LL + .16955 KN/mm (7) iv.Tmax = .00331A + .005331B + .006495C + .000533H − .01922T + .001563L + 1855.02DL + 1177.53LL − 21.4839 KN.mm/mm

(8)

A comparison of the proposed equations with the finite element results was performed to verify their accuracy. of the proposed equations, values given by these equations are compared with the corresponding values obtained from finite element analysis. In all cases, the proposed equations provided reasonably accurate results on the safe side, thus establishing their acceptability. From these equations, we can determine Mmax1, Mmax2, Vmax, Tmax, for the flight and landing, as demonstrated in Figs. 15 and 16. The effect of various geometric parameters on the design forces and moments were investigated. Based on the findings of this study, guidelines for direct analysis of the staircase were developed. The design forces and moments can be rapidly and easily determined from the suggested equations without requiring any formal analysis. These equations yield results that are on the conservative side but are within acceptable limits of accuracy. The transverse reinforcing steel in the landing should be significantly concentrated in the vicinity of the line of intersection of the flights and the landing. Large torsion moments are present in the flights of free-standing stairs, and an appropriate thickness of concrete must be selected to resist these moments due to the difficulty of reinforcing shallow-wide sections against torsion. Analytical approaches are not practically suitable for the analysis of free-standing stairs as far as economy and efficiency in design are concerned. These methods fail to simulate the actual interaction of plates in three dimensions. In addition, these approaches cannot demonstrate the variation of stress resultants across any cross section.

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KSCE Journal of Civil Engineering

Formulation for Free-standing Staircase

4. Conclusions Based on the analysis performed in the present study, the following conclusions can be drawn with respect to the result obtained for the response of various staircase models under symmetrical static loading. The test results of 23 free-standing staircases with various geometries were presented. Important parameters, such as the bending moment at flights, bending moment at landings, torsion moment at flights, torsion moment at landings, shear force at flights, shear force at landings and the deflection shape of stair are described in detail. The effects of length, depth, and location on the live load behavior of the staircase are discussed. Within the scope of this investigation, the following conclusions may be drawn: 1. The required forces and moments can be obtained rapidly and easily from the suggested equations without formal analysis. This approach will relieve the designer from the rigorous calculations required even for the approximate analytical methods. The entire process of analysis can be further simplified by creating separate small computer programs or spreadsheets. 2. The required forces and moments produced by the given equations are always on the conservative side but are within acceptable limits of accuracy. 3. The resulting proportions of the stair slab structure are optimum. Thus, economy is achieved by avoiding an overly conservative design.

Acknowledgements The authors are grateful for the sponsorship and financial support given by The Ministry of Higher Education Malaysia through grant ERGS/1/2012/TK03/UKM/02/4.

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