FINITE ELEMENT ANALYSIS OF

SLABLESS TREAD RISER STAIRS

A DISSERTATION

submitted in partial fulfilment of the requirements for the award of the degree

of MASTER OF ENGINEERING

in CIVIL ENGINEERING

(With Specialization in Computer Aided Design)

by

SAN DEEP SHARMA

...................... • • .. .,.. , , .,,

....° 1...:e • • . ..." o J ,), , • .

,.- 4 cf *-. 2 473 A e N 5 8 i :,

`, ■ r 7 , 'e '̂Set r,'No.t.:

•• ';'t ‘.1'-',....,74'

."....„....

DEPARTMENT OF CIVIL ENGINEERING UNIVERSITY OF ROORKEE ROORKEE-247667 (INDIA)

FEBRUARY, 1993

DEDICATED

To

My Parents

CANDIDATE'S DECLARATION

I hereby certify that the work which is being presented in

• this dissertation entitled FINITE ELEMENT ANALYSIS OF sLikpLEss,

TREAD RISER STAIRS in partial fulfilment of the requirements for

the award of the degree of MASTER OF ENGINEERING with

specialization in COMPUTER AIDED DESIGN IN CIVIL ENGINEERING

submitted in the Department of Civil Engineering, University of

Roorkee, Roorkee, is an authentic record of my own work carried

out for a period of about eight months from June, 1992 to

January, 1993 under the supervision of DR. G.C. NAYAK, Professor,

Department of Civil Engineering, University of Roorkee, Roorkee,

India.

The matter embodied in this dissertation has not been

submitted by me for the award of any other degree or diploma.

Dated: 42.2.93

SAIK404.0 ( SANDEEP SHARMA )

This is to certify that the above statement made by the

candidate is correct to the best of my knowledge.

( G.C. NAYAK )

Professor Deptt. of Civil Engg.

University of Roorkee

Roorkee, India.

ACKNOWLEDGEMENT

The author would like o express his profound gratitude to

Dr.G.C. Nayak, Professor, Depdrtment of Civil Engineering,

University of Roorkee, for his expert guidance, helpful suggestion

and indefatigable interest shown in the preparation of this

dissertation.

The author. is also thankful to Mt. Nirendra Dev, Research .

Scholar,. whose moral support in the times of stress has always

provided. the uplift needed.

The HP FORTRAN compiler has been an Invaluable ally.

Debugging is. fun. AlSo the .superb diagnostics have often,

suggested More intelligent -coding. For this and other gifts,

thanks go to CAD Centre, Department of Civil ' Engineering,

University of . Roorkee, Roorke,

The author likes to thank to staff of CAD Centre, especially

Mr. M.S. Rawat whose help was an indirect spur 'to the efforts.

The author would like to express his personal appreciation to

his class mates, especially Mr. D.K. Pachauri, Mr. G.B. Modha, Mr.

Ajay Kumar Mittal, Mr. I.H. Shamsi etc., who helped directly or

indirectly in the preparation of this dissertation.

Thanks are also due to 1.1r: Vinod Kumar for his devotion to

the excellent typing work of this disSertation.

..;14)NAft&i.0 &Witrun4 ( SANDEEP SHARMA )

ABSTRACT

The Slabless tread riser stairs,'due to their application in

the construction of modern commercial and residential building,

has had a wide acceptance. The construction of some stairwayt

this type with approximately plane orthopolygonal axes, has

generated the interest of many architects and engineert.

In this dissertatton, the Finite ElementAethod has been used

to analyse the Slabless tread riser stairs with extraordinary

architectural possibilities. The Slabless tread riser stairs has

been analysed by the finite eleMent method to study the state of

strest.in the structure for different parametric cases. 7 The

problem has been analysed as a two dimensional, plane strain case

using the parabolic quadrilateral isoparametric .element f

serendipity family. This study showt the presence of axial

stresses, horizontal displacement§ which are completely being

ignored in coventional study. This study also thoWt that 'the

fixed end moments are less than those obtained froM conventional:

methods and vertical reactions are unequal at two ends contrary to

conventional method of analysit. The variation of axial force,

thear force. and bending moments are plotted. The resultt for BM

are compared with the conventional analysis.

TheJT software named PSPSAS has been used for finite element

analysis that runs_ on HP . Syttem 9000 series in UNIX.

Pre-proce§sor has: the capability of automatic generation of. FE.

meshet and.pott processor offers graphic modules to interpret the

analysis results. Post processor can produce methet, deformed

shape, stress vector; stress contours and stress resultant

variations. At the junction of tread and riser, a detailed stress

distribution is obtained by finite element analysis.

In addition to these, conventional analysis of these stairs

is performed for three different support condition cases and the

present analysis technique has been critically examined. A

modified analysis method is also suggested to incorporate the

effect

axial deformation whiCh introduces the horizontal

thrust. The detailed analysis thus reveals'that the stairs are

folded plate like structure and membrane action causes tens.lon.

The horizontal displacement of structure plays a vital role in its .

structural behaviour.

A special study of •nonorthopolygonal stair has been carried

out by using F.E. program.

CONTENTS

Page No.

LIST OF FIGURES ix

LIST OF TABLES xi

CHAPTER 1 INTRODUCTION 1

1.1 GENERAL 1

1.2 BASIC STRUCTURE 2

1.3 LOADINGS AND PARAMETERS 3

1.4 OBJECTIVES 4

1.5 SCOPE OF THE THESIS

CHAPTER 2 FINITE ELEMENT TECHNIQUE 9

2.1 GENERAL 9

2.2 BASIC STEPS IN FINITE ELEMENT METHOD 9

2.3 FINITE ELEMENT FORMULATION FOR PLANE PROBLEM 11

2.3.1 Description of Geometry 11

2.3.2 Variation of Unknown Function 11

2.3.3 Gradient Relationship 13

2.3.4 Macroscopic Constitutive Law 14

2.3.5 Element Characteristic Equations 15

2.4 ISOPARAMETRIC ELEMENT 16

CHAPTER 3 COMPUTER PROGRAMS 23

3.1 INTRODUCTION 23

3.2 PSPSAS PROGRAM 23

3.3 COMNIX PROGRAM 30

CHAPTER 4 CONVENTIONAL ANALYSIS STUDIES 37

4.1 GENERAL 37

vi

4.2 REVIEW OF LITERATURE .37

4.3 BASIC ASSUMPTIONS 43

4.4 ANALYSIS TECHNIQUE BY MOMENT DISTRIBUTION 44

4.4.1 Stairs with Intermediate Supports 45

4.4.2 'Stairs without Intermediate Supports 47

4.5 ILLUSTRATIVE EXAMPLES 49

4.5.1 Stair with Intermediate Supports 49

4.5.2 Stair without Intermediate Supports 56

4.5.3 Modified Conventional Technique 58

CHAPTER 5 PARAMETRIC STUDIES OF STAIR CASE 67

5.1 DISCRETIZATION AND IDEALIZATION OF DOMAIN 67

5.2 CASES STUDIED 68

5.2.1 Stairs without Intermediate Supports 68

5.2.2 Stairs with Intermediate Supports' 85

5.2.3 Non-orthopolygonal Stairs 108

5.3 REVIEW OF•RESULTS 108

5.3.1 Displacements 123

5.3.2 Axial Force 123

5.3.3 Shear Force 124

5.3.4 Bending Moment 124

5.3.5 Stress Concentration 125

CHAPTER 6 SUMMARY AND CONCLUSIONS 131

6.1. SUMMARY OF RESULTS 131

6.1.1 Conventional Analysis 131

6.1.2 Finite Element Analysis 132

6.1.3 Comparative Study of Results 134

vii

6.2 CONCLUSIONS 141

6.3 FUTURE SCOPE FOR STUDY 142

REFERENCES 143

APPENDIX " 145

viii

LIST OF FIGURES

Page No.

Fig. 1.1 Longitudinal section of Slabless tread

riser stair

Fig. 2.1 Isoparametric mapping of a parabolic element le

Fig. 2.2 Typical Shape functions for parabolic element

of serendipity family 19

Fig. 3.1 Flow chart for PSPSAS program 25

Fig. 3.2 Flow chart of plot program 'COMNIX' 31

Fig. 4.1 Plotted values of the variation of k with n 41

Fig. 4.2 An illustrative example 51

Fig. 4.3 Unsupported stair with both ends hinged Se

Fig. 4.4 Free body diagram of hinged case 63

Fig. 5.1 Deflected shape of stair for case (A) 73

Fig, 5,2 Stress resusltant diagram for case (A) 77

Fig. 5.3 Deflected shape of stair for case (B) 79

Fig. 5.4 Stress resultant diagram for case (B) 83

Fig. 5.5 Deflected shape of stair for case (C) 87

Fig. 5.6 Stress resultant diagram for case (C) 91

Fig. 5.7 Deflected shape of stair for case (=A') 95

Fig. 5.8 Stress resusltant diagram for case (A') 99

Fig. 5.9 Deflected shape of stair for case (8') 101

Fig. 5.10 Stress resultant diagram for case (B') 105

Fig. 5.11 Deflected shape of stair for case (C') 109

Fig. 5./2 Stress resultant diagram for case (C') 113

Fig. 5.13 Deflected shape of Non-orthopalygonal Stair 117

ix

Fig. 5.14 Stress resultant diagram for

Non-orthopolygonal stair 121

Fig. 5.15 Longitudinal section of stair at central zone 127

Fig.. 5.16 Normal stress distribution diagram at section

B-B 127

Fig. 5.17 Normal stress distribution diagram at section

A-A 129

Fig. A.1 Reinforcement with bars and stirrups 147

Fig. A.2 Reinforcement with bars and inclined bars 147

Fig. A.3 Alternative reinforcement 149

Fig. A.4 Chamfered entrant corners. 149

LIST OF TABLES

Page no.

Table 4.1 Variation of k with n 39

Table 4.2

Table 5.1

Table 5.2

Table 5.3

Table 5.4

Table 5.5

Table 5.6

Table 5.7

Table 5.8

Table 5.9

Table 5.10

Table 6.1

Table 6.2

Table 6.3

Coefficients for solving stairs with

2 to 20 treads 39

Parameters and mesh details for section

5.2.1 71

Stress resultants for case A 75

Stress resultants for case B 81

Stress resultants for case C 89

Parameters and mesh details for section

5.2.2 93

Stress resultants for case A' 97

Stress resultants for case B' 103

Stress resultants for case C' 111

Parameters and mesh details for Non-

orthopolygonal stair 115

Stress resultants for Nonorthopolygonal

stair 119

Comparative study for case (A) 135

Comparative study for case (A') 137

Comparative study for case (B) 139

xi

CHAPTER 1

INTRODUCTION

1.1 GENERAL

A Slabless tread-riser stair is, as its name suggests, .a

stair in which the stresses due to the external loads are resisted

purely by the treads and risers which form in themselves .a plane

orthopolygonal structure. These type of stairs are also regarded

as ORTHOPOLYGONAL STAIRS.

The stairs are provided in buildings to afford the means of

ascent and descent between various floors of a building. With the

increasing participation of Architects in the field of building

construction, a building and it's components are being much and

much sounder by asthetic point of view. ,Slabless stair is a type

of stair with extra ordinary architectural possibilities. These

stairs; due to their light and slender form have aroused the

interest of many architects and engineers and their application in

the construction of modern commercial and residential building has

had a wide acceptance.

The design of coventional stairs are being done by assuming

that the waist slab, generally called as slab, carries all types

of loads so in these type of stairs the reinforcement is provided

in slab while in Slabless tread-riser stairs the loads are being

carried by treads and risers, therefore, the reinforcement is

provided in treads and risers in the form of loops.

1

There are various adventages of orthopolygonal stairs over

conventional stairs such as it is asthetically much superior

provides more head room, requires less concrete and reinforcement

but laying out difficulties, requirement of special shuttering and

skilled labour are the main disadventages. The present study is

aimed at using finite element method to evaluate the stress

resultants and it's behaviour at various cross sections of

Slabless tread-riser stairs. A comparison between conventional

method analysis and finite element anlysis for these stairs is

also presented . A critical examination of present conventional

analysis techniques is also performed.

1.2 BASIC STRUCTURE

These stairs form an orthopolygonal structure in plane. A

plane orthopolygonal structure is one whose axis lies within a

plane and consists of a continuous broken line whose segments form

approximately right angle between them.

These stairs may have different support condition and end

condition as well. In general the ends of a stair between a

flight is embedded in the walls so the ends may be treated as

fixed end however it is not true each time. If a rigid beam

exists at the beginning of the landing or if there is a thick slab

at the ends then also the ends may be assumed to be fixed. If

the ends of the stairs are supported on bearing wall then the ends

may be assumed as hinged.

2

Generally a concrete beam is provided beneath the landing to

support it. This type of case is called intermediately supported

stair case. If the support beam does not exist it is called as

unsupported stair. In fact in supported case the bending moments,

which is main criterion for designing, significantly reduce .

For the simplification of structural analysis by conventional

methOds,these types of stairs are treated as fixed or continuous

beam but this idealization ignores some fundamental aspect of

axial deformation which are discussed later An the present study

and various different combinations of end and support conditions

are also discussed-in this study.

1,3 LOADINGS AND PARAMETERS

Basically two types of loads are considered for designing a

stair i.e. Dead load, . live load. As outlined in IS:

875-1964(3) ,for stairs in residential building, office building,

etc'. where there is no possibility of over crowding, the live load

may be taken as 3 kN/m2. For other public building, the live load

may be taken to be 5 kN/m2 .

The dead load comparises of self weight of treads, risers

and landings. It is in the form of uniformly distributed load but

it may be assumed, without appreciable loss in accuracy,

consisting of a number of equivalent point load concentrated at

riser.

The various design parameters of a Slabless tread riser

stairs are as shown in Fig. 1.1 and their general variations are

as follows.

Parameters Variations in mm

LANDING (L) 900 to 1500

TREAD (T) 230 to 300

RISE (R) 150 to 200

THICKNESS OF TREAD (th) 100 to 150

THICKNESS OF RISER(tv) 100 to 150

NUMBER OF.STEPS (N) 7 to 12 (in number)

The thickness of tread and thickness of riser usually kept

same for asthetic reason and also due to the fact that they resist

almost equal amount of moments.

1.4 OBJECTIVES

The study is aimed at realising following objectives

(i) To find out stress resultants at various cross-sections for

different end conditions and support conditions and their

combinations.

(ii) To study the stress distributions at different cross sections

of stairs through the concept of stresses at Gaussian points

in finite element analysis.

(iii)To compare the results of finite element analysis and

conventional methods for these stairs and critically examine

the results.

(iv) To prepare an auto-mesh generation programme to generate the

nodal coordinate and element connectivity itself.

1-7

•-1

5

(v) To prepare the programme for calculating the nodal loads due

to dead and live load itself by input parameters.

(vi) To study non orthopolygonal stairs and stress concentration

at tread riser junctions.

1.5 SCOPE OF THE THESIS

The scope of the thesis itself may be' seen from the table of

contents. The thesis consists of six chapters. The first chapter

introduces the stairs, their structure, design parameters and

objectives. In chapter 2, the finite element method is discussed

and the formulation of finite element analysis is presented.

Isoparameteric elements are also discussed in same chapter. In

chapter 3, two very general program PSPSAS and COMMNIX are

discussed with their entire structure and flow charts are also

presented. In chapter 4, various parameteric case studies have

been performed and ,their analysis have been presented with

numerical values as well as graphically. In chapter 5,

conventional methods with their formulations are discussed and

three general cases have been solved. In chapter 6, the results

of conventional analysis are compared with finite element

analysis. The modification in conventional methods are suggested.

The scope for future work is alSo discussed in this chapter.

CHAPTER 2

FINITE ELEMENT TECHNIQUE

2.1 GENERAL

The finite element method as known today has been presented

in 1956 by Turner, Clough, Martin and ToPP(17) • In their paper

they presented the application of constant strain triangle for the

analysis of aircraft structure. Since then much progress has been

made and this method is now firmly established as a powerful

numerical analysis technique for obtaining appoximate solution to

complex engineering problems due to it's utmost importance, both

conceptually and from the computational point of view. The

fundamental concept of the finite element method is that any'

continuous quantity, such as temperature, pressure, displacement

etc. can be approximated by a discrete model composed of a set of

piece wise continuous function defined over a finite number of

subdomains. The general applicability of the finite element

method can be seen by observing the strong similarities that

exist between various type of engineering problem.

In the following section a brief description of the finite

elemnet method and it's formulation for the structural analysis of

plane stress/strain case is presented.

2.2 BASIC STEPS IN FINITE ELEMENT METHOD

The basic steps for obtaining the finite element solution to

9

a structural analysis problem are(18)

(i) Discretization of the continuum

The continuum is seperated by imaginary lines or surfaces

into a number of finite elements. The elements are assumed to be

interconnected at a discrete number of nodal points situated on

their boundary.

(ii) Selection of displacement model

A Set of function is chosen to define uniquely the state of

displacement within each finite element in terms of it's nodal

displacements. The displacement function now define uniquely the

state of strain within an element in terms of the basic unknown

parameters (i.e. nodal displacement) of the problem. Usually

polynomials are selected as interpolation function.

(iii) Derivation of element stiffness matrices and load vectors

From the assumed displacement model, the stiffness matrix and

the load vector of each element are derived by using either

equilibrium condition or a suitable variational principle.

(iv) Assemblage of element equations

The individual element stiffness matrices and load vectors

are assembled _in a suitable manner to obtain global stiffness

matrix and load vector. The overall equilibrium equations have

been modified to account for the boundary conditions of the

problem.

(v) Solution of equations

The linear simultaneous equations resulting from the assmebly

process are solved for the unknown nodal displacements. Any

10

efficient solution technique can be used for this.

(vi) Computation of element strains and stresses

From the known nodal displacements, strains can be calculated

using the strain displacement relationship. Stresses then can be

calculated from strains using stress-strain relationship.

2.3 FINITE ELEMENT FORMULATION FOR PLANE PROBLEM

Let us consider an elastic continuum in the X-Y reference

plane for which a plane stress/strain analysis is to be performed.

The finite element formulation can be done under following steps.

2.3.1 Description of Geometry

The first step in finite element formulation is to divide the

continuum into.a number of finite elements. These element may be

of different shape (1. . triangular, rectangular etc.) and of

different family (i.e. serendipity, Lagrangian etc.). Let each

element has n nodal points.

2.3.2 Variation of Unknown Function

The unknown function (displacements) can be uniquely defined

by assuming same interpolation formulae, within each element

terms of parameters generally associated with the values of the

function at the nodes of the element.

For a two dimensional problem the displacement variations are

written as

= [NI{a)e = E v (2.1)

1=1 evt}

11

where Ni

is the shape function corresponding to node i and it is

dependent on the spatial coordinates of the elements. Eq. (2.1)

can also be written as

u = [I211 I2

N2"..I2NnI {8}e

(2.2)

where

vector {S}e = {u1,v

1,u2'

v2' , u n,v n}

T

[1 r 1 0 21 L 0 1

u and v are the horizontal and vertical displacements respectively

of the point within the element.

The shape 'functions Ni should be such that they give

appropriate nodal displacement when coordinates of the appropriate

nodes are inserted in Eq.(2.1). In general,

Ni (xl,yi) = 1

N (x ,y ) = 0 where J i

n

and E Ni (x,y) = 1 1=1

(2.3)

Now the chosen displacement function automatically guarantees

continuity of displacements with adjacent elements because the

displacements vary by the same order along any side of the

element, and with identical displacement imposed at the nodes, the

12

0 8u ax

ev ay

au.o.av ay ax 1

{c} =

a ax

o a ay

a a ay ax

same displacement will clearly exists all along an interface.

2.3.3 Gradient relationship

For plane stress/strain problems, with displacements known

within the element the infinitesimal strain at any point can be

written as

(2.4)

i.e.

{e} = ILI ful

where EIJ is the displacement differential operator matrix.

Substituting Eq.(2.1) in Eq.(2.4).

= [L]M (8)e = [B] {3)e

= fB1, 132. Bill (8)e

(2.5)

where [B] is strain displacement matrix, the matrix [BO is given

by

[13iI =

aNi

8x 0

aNI

0 ay

8Ni

aNI

ay 8x

(2.6)

2.3.4 Macroscopic constitutive law

For linear elastic material, the relationship between the

stress and strain will be of the form

iml = [D] ({c}-ic01)+icrol (2.7)

where [D] is an elasticity matrix consisting of the appropriate

material property, (mo} and (co) are the initial stresses and

initial strains respectively and for plane stress/strain problem

(m) = (2.8)

where ox, oy and Txy are the three components of stresses

corresponding to three component of strains defined in Eq.(2.4).

For plane stress cases (i.e. mz =0) in a isotropic material

the elasticity matrix is given by

[D] -

1

0 •••••

v

1

0

0

0 1-v

(2.9) 2 1-v 2

and for plane strain cases

the elasticity matrix is

E(1-v) [D] -

(i.e. cz

=0)

given by

1 •■••••

0

1-2v -27-1=17)

in an isotropic material

(2.10)

v

1-v

0

(1-v)(1-2v) 1 1-v

0

and

. az = v +T ) -aEAT x y

In these expressions, E is the modulus of elasticity, v is

the Poisson's ratio of the material, a is the coefficient of

thermal expansion and AT is the change in temperature.

2.3.5 Element Characteristic equations

These expressions for element characteristics can be derived

using the principle of virtual work method. Let us consider an

element in equalibrium under the action of applied concentrated

load (I)c}e, initial stresses (a ), initial strains {co}, body

forces per unit volume {b} and distributed loads per unit area

{p}.

Now applying virtual work principle (i.e. internal virtual

work is equal to external virtual work) following equations can be

obtained

[K] {8} =

- where [K] is the global stiffness matrix given by

ne ne [K] = E [K]e = E [B)T[D] [B] dv

e=1 e=1 v

(2.11)

(2.123

and ne

= E {R}e

e=1

where

{R}e= (Fr + {F}e + {F}e + {F}e + {F}e (2.13) a0

c0

and Nconc.

{F}e = E EN )T {Pci}e

(2.14)

1=1

IFIZ = fv EN]T {b} dv (2.15)

{F}e p = iv EN)T {p} dA (2.16)

{F}e =-fv '[B]T {o-o} dv (2.17)

v {F}e = ENT [D]{eo}dv (2.18) c

In the above given Eqs.(2.14) to (2.18) the right hand side is

the summation of equivalent point load, Nconc. are the number of

nodes where concentrated loads are applied and Ni is their shape

functions at corresponding nodes.

Now solving the Eq. (2.11) thc- nodal displacement can be

obtained. By using Eq. (2.5) and Eq. (2.7) the stresses at any

point can be obtained from the relation

{o} = ED] ([8){8}e - {c0}) {o-o} (2.19)

2.4 I SOPARAMETRI C ELEMENT

It is well known that for higher order curved elements the

evaluation of integrals in Eq.(2.12) and Eqs. (2.15) to (2.18) can

be easily carried out numerically by using Gauss-Legendre

16

quadrature rule . This reguires change of function of variables

in x and y to new functions of variables in local coordinate e and

n whose values lie between -1 and +1.

The basic idea underlying the isoparametric element is to use

the same shape functions to define the element shape or geometry

as well as,unknown functions within the element. To derive the,

isoparametric element equations, a local curvilinear coordinate

system is first introduced, then the shape functions has to be

expressed in terms' of local coordinates (i.e. g; n) in two

dimensional analysis.

Thus, for an isoparametric element, the displacement

function is defined as

u(g,n) = E ui 1=1

n v(,T1) = N (g n

'

1=1

and the geometry is defined as

x(c,n) = E (,n) xi i=1 .h

y( 07) = E 141(e, n) Yi

1=1

(2.20)

( 2.21)

'In Eqs. (2.20) and (2.21), u and v are the displacement 4'

components and (x,y) are the coordinates of any point -(,11).

17

Similarly (xi, yi) are the coordinates of node i while (uv) are

the displacement components.

The representation of geometry in terms of shape functions

can be considered as a mapping procedure which transforms a square

2x2 parent element in local coordinate system into a distorted

shape (i.e. a curved sided quadrilateral) in the global cartesian

coordinate system (Fig. 2.1). In present study parabolic

isoparametric element of serendipity family is used. These

parabolic elements are extremely versatile and are well tested.

For these elements, the shape functions are as follows (Fig.2.2):

For corner nodes

Ni = 1/4 (1+ 0) (1+n0) (c0+770-1) (2.22)

mid side nodes

Ni = 1/2 (1-2) (1+10) for gi= o

(2.23)

Ni = 1/2 (1-n2) (1+%) for ni. o (2.24)

where

= no = nni

For mapping the element geometry the coordinates Jacobian matrix

(J) is required. Using Eq.(2.20).

rax ax 7

aNi oN Eat xi x Eat i

[x, [..11 =

y.r]

ON v E— v E i a 71

(2.25)

ay ay _ag an

18

L

PARENT ELEMENT

2-DIMENSIONAL MAP

FIG.2.1- ISOPARAMETRIC MAPPING OF A PARABOLIC ELEMENT

FIG.2.2-TYPICAL SHAPE FUNCTIONS FOR PARABOLIC ELEMENT OF SERENDIPITY FAMILY

19

By the chain rule of partial differentiation

{71 aNil aNi

ae ' an 7 ax ' ay e (2.26)

Eq. (2.23) gives the cartesian derivatives of shape functions

required for matrix [B] in Eq.(2.19).

aN

8x

ONi

ay .4

(2.27)

The explicit form of inverse coordinate Jacobian matrix is

[ ag ag 8x ay

an an ex ay

[J]-1 =

:

8x 1

5771

-an ay 8x ag ag

(2.28)

and it's determinant

= IJT

I = ax ay ay ax ag 8n ag n

(2.29)

For two dimensional problem the element area is given by

dA = dx dy = IJI dg do

dv = tdxdy = tIJI dg do (2.30)

Where t is the thickness of element.

In above given expression x and y are substituted from Eq.(2.21)

21

and it is used for evaluating the stiffness, equivalent nodal

loads etc. from Eqs. (2.12) and (2.13), and for expressions

involving integral over the element. These can be expressed in

terms of the local coordinates with an appropriate change of the

limits of integration (-1 to +1). Thus the element stiffness now

can be written substituting the volume from Eq.(2.30).

+1 {1C}e = f+1 1 [13]

T [D] [B] t pi soldn

-1 -1 (2.31)

Using the numerical integration technique (Gaussian-Legendre

quadrature rule) Eq.(2.31) can be rewritten as

m n

= E E [8]T [D] [B] t IJI cicj (2.32)

1=1 1=1

where m and n are the number of Gaussian points in e and 71

directions respectively, and Ci and C are the Gaussian weights.

In Eq.(2.29) [B] and [J] have to be evaluated at each Gaussian

pointsi' ). By substituting the appropriate values the nodal

displacement can be calculated.

22

CHAPTER 3

COMPUTER PROGRAMS

3.1 INTRODUCTION

For conducting the studies and achieving the objects listed

in section 1.4, a general purpose program is required. Since

finite element analysis requires a lot of computations and

manipulation so high speed digital computers and efficient

programs are the basic necessity. The expanding use of finite

element method to various branches of engineering, science and

field problem creates the necessity of pre-processors and post

processors to avoid the inaccurate input and output.

Fortunately PSPSAS, a very general purpose program, is available'

. to do the finite element analysis for plane stress/plane strain

and axisymmetric problems. COMNIX is another program which serves

as postprocessor of PSPSAS: For the analysis of Slabless

tread-riser stairs these prOgrams are used.

3.2 PSPSAS PROGRAM

This program is a general purpose .finite element program

solving problems of plane stress, plane strain and axisymmetric

cases subjected to concentrated loads, gravity and centrifugal

loads; pressure loads and thermal loads.

It is written by Dr. G.C. Nayak for linear, quadratic and

cubic quadrilateral isoparametric element of serendipity family

23

and is being used extensively all over the world. The program

details are given in book(7)

. This program uses the Gaussian

quadrature technique for numerical integration.

The program consists of a main program and 16 subroutine.

Two subroutines namely meshg and moshth have been added to analyse

the Slabless tread riser stairs, specifically. The order and

description of main program and subroutines are outlined below.

(1) MAIN PROGRAM

This program consists the main body of the program. It reads

the number of problem to be solved along with number of load

•

cases, degree of freedom at each node, counter for type of

isoparametric element, order of Gaussian integration, problem type

indicator, material properties, some counters controlling output

etc. It calls various subroutine for the solution of problem.

The flow chart of MAIN program is given in Fig. 3.1.

(a). Subroutine MESHG

This subroutine reads the various parameters of stairs (i.e.

landing, rise, tread etc.), live load, number of steps and

parametric case counter. This subroutine is called in MAIN and

discretizes the Slabless stairs into two dimensional quadrilateral

elements of serendipity family with quadratic variation. This

discretises the stairs automatically depending upon rise and

thickness of tread ratio. It also provides the boundary condition

and supporting condition by restricting the corresponding nodes in

appropriate direction. The loading data are supplied to LDATA

24

A

FEH

STIFFNESS FORMULATION

A HOD

•04sFRI

ITE X1 1FEM

H

I ROSE.]

DISK DATA SET 1

SOLUTION SUBROUTINES

DI SK DATA

SOLVE I-•- SET I I -a• BSUB RESOLV

A

DI SK DATA SET II

(LOAD CASE NUMBER

wari

gou RHO mvu allow

2 17ri

V

PROBLEM DATA DEFINED

MESHG

NODEXY

G DATA GAUSSP I

SET I

LOAD VECTOR FORMATION

FOIT1

D SFR

A

STRESS CALCULATION

745T51 SFR

A

STRP

STRESS RESULTANT CALCULATIONS N

SFR

AUX

FIG. 3.1 FLOW CHART FOR PSPSAS PROGRAM

25

T

E S

0 S H T

simultaneously.

(b) Subroutine GDATA

This subroutine gets the nodal coordinates, boundary

conditions and element connectivity from subroutine MESHG. If

MESHG is not called it reads the above data. Midside nodes

coordinates are calculated by calling subroutine NODEXY and the

Gaussian integration constants are obtained by calling subroutine

GAUSSP.

(c) Subroutine NODEXY

It calculates the unspecified coordinates of midside nodes

lying on the straight edge of parabolic and cubic elements.

(d) Subroutine GAUSSP

It provides the Gaussian sampling points and corresponding

weight functions for numerical integration.

(e) Subroutine LDATA

In this subroutine all the loading data are converted into

equivalent nodal loads. It also reads thermal, centrifugal loads

etc. It can deal with concentrated load acting within the

element. For. influence surface problem, the pinch load can be

generated automatically.

(f) Subroutine MOD

It forms the elasticity matrix [D] for plane stress, plane

strain and axisymmetric cases. which ever is applicable

corresponding to a suitable counter.

(g) Subroutine SFR

This contains explicit expressions of shape functions and

their first local derivatives for all three types of elements.

(h) Subroutine AUX

This subroutine computes the coordinate Jacobian matrix [J],

it's determinant and inverse as well. It also calculates

cartesian derivatives of shape function using the Jacdbian

transformation.

(i) Subroutine FEM

This subroutine computes the gradient matrix [B] and then

computes {D} [B] dv.

(J)- Subroutine SET

This subroutine forms an array IONARY of element numbers

which are arranged to get minimum band width. This then computes

the band width and the total number of elements in the banded

matrix.

(k) Subroutine STIFM

This subroutine calculates element stiffness matrices by

using numerical integration. It calls subroutine ROSB to carry

out transformation for sloping boundary, if required.

(1) Subroutine ROSB

This subroutine is called in subroutine STIFM to make

operations over the elements of stiffness matrix for sloping

boundary case.

(m) Subroutine SOLVE

This subroutine acts as a solver of linear simultaneous

equations using Gaussian elimination method. Assembly and forward

elimination is carried out and the coefficient for back

substitution are stored. The most economic use of core storage is

made by storing banded form of matrix in one dimensional array.

(n) Subroutine BSUB

This subroutine does back substitution operations to give

displacements and reactions.

(o) Subroutine RESOLV

This subroutine is called if the problem is analysed for more

than one load case. In such cases there is no need to recompute

element stiffness. Only force vector needs to be changed and this

subroutine does the same.

(p) Subroutine STRP

This subroutine computes the strains and stresses at any

point. It is called in subroutine STRESS.

(q) Subroutine STRESS

In this subroutine strains and stresses are seperately

calculated at' required points (i.eGaussian points, Nodal

points). It calls the subroutine STRP. Stresses and strains

calculations along any desired coordinate axis can be obtained if

the rotation of the areas and element number are specified.

2

(r) Subroutine MOSHTH

This subroutine calculates the bending moment, shear force

and axial force. It is called in MAIN program and this subroutine

uses the results at Gaussian integration points to evaluate the

stress resultants from the stress which it gets from subroutine

STRESS. It calls subroutine SFR and uses Jacobian transformation.

3.3 COMMIX PROGRAM

This program is originally contains the subroutines of

'PLOT88'. To make it compatible with HP system it is converted to

make use of 'STARBASE'. This program serves as a post processor

of PSPSAS program by interpreting output in different ways and

drawing different plots of the structure. It consists of one main

program and a number of subroutines.

The order and description of them are outlined below.

(1) MAIN PROGRAM

The main program reads number of problems to be plotted and

number of type of plots for one structure only. It also reads

control data and calls for the required subroutine to plot the

desired figure. The flow chart of main program is given in

Fig.3.2.

(a) Subroutine GDATA

This subroutine reads the coordinates of corner nodes. of

straight line element and also mid side nodes for curved arms as

it's geometric input data. The element connectivity and local

30

IMAXHIN I

IKESH I

II HAG

trraoD1

111(1.14st-1.1 'MESH I

IBONDRY J

TEXT STRING PLOT

MORE THAN ONE TYPE OF PLOTS OF ONE STRUCTURE (NPR > 1

1

I 0

2 CD 2 t1

1-< h::I

0 zj

2

tTj 2 t:f txi 2

1-1:1 0 tcl tzj

pc) 0 tri V

•

FIG. 3.2 FLOW CHART OF PLOT PROGRAM 'COMMIX '

31

NTEXT 0

I PR 1 I PROBLEM DATA DEFINED

INTERIOR BOUNDARY

I NTBOD I Ifoxm

-1INI

1.P2117

0

INTERIOR MESH PLOT

KAXMI N ,

I NTMSK IKESIii STVEC

STRUCTURE MESH PLOT

FMES I NODELI

STRUCTURE BOUNDARY

I MESH 0'

NBC X

NGASP 0 NSTS x 0

DEFORMED PROFILE PLOT NSHAPE 0 0

IMAGE PLOT

LI MAC

CONTOUR POINTS PLOT

STCONT .

I CAUSSP I SFR

rEffn

11.0qd SOVNI SNO

V

—11E5.11? X 0

M

A

IMAGE > 1

NSTCR 0

BONDRY 1

D

E

F

S

GDAT A I IMAXMI N

TEXT

coordinates (e,n) of the nodes of element are also read In this

subroutine.

(b) Subroutine IMAG

This subroutine consists the facility of plotting the four

images of any structure or •of a part of It in all the four

quadrent in one single plot.

(c) Subroutine INTMSH

This subroutine facilitates, as per requirement of user, to

get a plot of the mesh of a certain portion. This subroutine is

called in MAIN program, subroutine DEFSHP and subroutine STVEC if

corresponding counter IMESH is given a non zero value.

(d) Subroutine INTBOD

ThIS surboutine provides the facility of plotting only, The

boundary of the part of structure: This is called in. MAIN program

and subroutine DEFSHP by giving a non-zero value to the. counter

INBC.

,(e) Subroutine DEFSHP

This subroutine reads the displaceMent of all nodes, the

magnification factor of displacement. The coordinates of all the

nodes are modified by just adding the magnified deformation values

to the initial coordinate values and. than subroutine MESH is

called to plot the deformed shape of the structure. Deformed

shape of various section may also be 'plotted as it calls

subroutines INTMSH, INTBOD, BONDRY etc.

33

(f) Subroutine BONDRY

This subroutine provides the facility to plot the outer

boundary of the entire. structure. This reads the total number of

nodes on outer boundary, their individual number in a specific

direction (i.e. clockwise or anticlockwise).

(g) Subroutine MESH

This subroutine is called in main program to plot the

original mesh of entire structure. It is also called in

subroutines DEFSHP, INTMSH INTBOD and BONDRY to plot the

respective shape and portion of the structure. This subroutine

also has the facility of calculating the total number of points

that must be calculated to get a smooth curve by joining all these

points by straight lines.

(h) Subroutine MAXMIN

This subroutine is useful if the program is being executed on

PLOTIO graphic terminal. This subroutine brings out the maximum

and minimum values of the coordinates of portion of the mesh which

is of the interest of user.

(i) Subroutine STVEC

This subroutine plots stress/strain vector on a suitable

scale of the desired Gaussian points. Arrow heads are placed at

the end of one vector out of two indicating tensile or compressive

stress/strain.

34

(i) Subroutine SFR

This subroutine contains the shape function of serendipity

family for ltnear parabolic and cubic quadrilateral element and

their first local derivatives.

(k) Subroutine GAUSSP

This subroutine gives the Gaussian sampling point and their

weighting functions upto.six points using Gaussian-Legendre rule

for numerical integration.

• (1) Subroutine AUX

The subroutine computes the coordinate of the Gaussian points

and provides them when called in subroutines STVEC and STCONT.'

(m) Subroutine .STCONT

This subroutine plots the contours of stress/strain in the

entire structure or a part of it. It also prints the values of

the contours.

(n) Subroutine NODEL

This subroutine plots the node number and element number on

the mesh of structure or a part of it when corresponding counters

NNOD and NELE are assigned a non zero value.

(o) Subroutine TEXT

This subroutine provides the facility of writing any text in

the figure for Its recognition, anywhere on the figure. -Counter'

NTEXT in main programtakes care of calling.,this subroutine.'

35

CHAPTER 4

CONVENTIONAL. ANALYSIS STUDIES

4.1 GENERAL

The analysis of Slabless tread-riser stairs, due to its

broken line form, is a cumbersome and time consuming process in

the majOrity of cases. These stairs are normally statically

indeterminate and their analysis may be approached , by using

various methods which are available for the analysis of

indeterminate structure.

In the following section present conventional analysis

technniques are dlscusSed and the stairs for three- cases are

analysed by conventional methods with the same parameters and

loads as analysed in following chapter.

4.2 REVIEW OF LITERATURE.

In their paper, Saenz and MartinJ14)

, .presented the elastic

analysis of'orthopolygonal.stairs in a plane, fixed at both' ends

by two different methods namely ColuMn analogy and second

difference method. They simplified the problem .by taking some

assumption as presented at the end of this section: Professor

Saenz had als6 presented the analySis in a book(13) W.

Benjamin(2) , gave .the formulae to determine the fixed end moment

37

of these stairs without landing as given below.

R/T)

M - P.n.T (n

2 + I

-1) r /I

t 12 n

(4.1) n-1 R/T

1 + n Ir/I

t]

where P is the concentrated load acting at the edge of riser, n is

the number of steps. T is the tread, R is the rise. Ir

is the

moment of inertia of riser and It

is the moment of inertia of

tread.

The Eq.(4.1) may also be written as

M = M'k

(4.2)

where k is the factor by which fixed end bending moment for a case

of fixed beam,.with identical condition, has been increased. The

value of this factor was also given by him in tabular form as

shown in Table 4.1, and in graphical form as shown in Fig. 4.1.

Benjamin(1)

also discussed the stairways with equal landing

at both ends and gave following expression assuming length of

landing is a multiple integer of the tread width

- P.n.T(n2-1)

12 n [

1+p -pA 1+n- 1

p-pB (4.3)

where

R/T P I

r/I

t

A(2m)(m+1)(3n-2m-1); B = 2m

- n(n

2-1)

38

TABLE 4.1: VARIATION OF k WITH n

R/T n

0.4 0.5 0.6 0.7 0.8 0.9 1.0

2 1,1667 1,2000 1,2308 1,2593 1,2857 1,3103 1,3333

3 1,1052 10 250 1,1429 1.1591 1,1739 1,1875 1.2000

4 1,0769 1,0909 1,1034 1,1148 1,1250 1.1343 1.1429

5 1,0606 1,0714 1,0811 1,0897 1,0978. 1,1046 1,1111

6 1,0503 1,0588 1,0667 10737 1,0804 1,0857 1,0909

7 1,0425 1,0500 1,0566 1,0625 1,0678 1,0726 1,0769

8 1,0370 1,0415 1,0492 1,0543 1,0588 1,0629 1,0667

.9 1,0328 1,0385 1,0435 1,0479 1.0519 1,0556 1,0588

10 1,0294 1,0345 1,0390 1,0429 1,0465 1,0497 1.0526

11 1,0267 1,0312 1,0353 1,0389 1,0421 1,0450 1,0476

12 1,0244 1,0285 1,0323 1,0355 1,0384 1,0411 1,0435

13 1,0224 1,0263 1,0297 1,0327 1,0354 1,0378 1,0400

14 1,0209 1,0244 1,0275 1,0303 , 1,0328 1,0350 1,0370

15 1,0194 1,0227 1,0257 1,0282 1,0306 1,0326 1,0344

16 1,0182 1,0212 1,0240 1,0264 1,0286 1,0305 1,0323

17 „ 1,0171 1,0200 1,0226 1,0242 1,0268 1,0286 1,0303

18 1,0161 1,0189 1,0213 1,0234 1,0253 1.0274 1,0286

19 1.0153 1,0178 1,0201 1,0221 1,0239 1.0255 1,0270

20 1,0145 1,0169 1,0191 1,0210 1,0227 1,0243 1,0256

TABLE 4.2: COEFFICIENTS FOR SOLVING STAIRS WITH 2T0 20 TREADS

Nktolbcr of rcpt "H" Coeffl. cienIS

3 4 5 6 7 R 9 10 11 12 13 14 15 16 17 IR 19 20

A 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4,50 5.00 5.50 6.00 6.50 7.nn 7.50 R.00 8.50 9.00 9,50 10.00

II 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00 5.50 6.00 6.50 7.00 7.50 8.00 8.50 9.00 9.50

C 0.25 0.50 1.00 1.50 2.25 3.00 3.00 5.00 6.25 7.50 9.00 10.50 12.25 1.1.00 16.00 18.00 20.25 22.50 25.00

0.00 0.00 0.50 1.00 2.50 4.00 7.00 10.00 15.00 20.00 27.5n 75.00 .15.50 56.00 70.00 84.00 101.00 120.00 142.50

39

n" = No. of Step& or treads.

k = Factor to be:muitiplied to the

Equivoient Stab Ftxed End Bending

Moment. Nei° to get the Actual Fixed.

End Bending Moment '/Y1.

0 For all Curves tvr.th so

that Iv/Ih= 1•00.

4 6 g to 10. i6 la 90 u- i -u

No. of Steps ‘ri. 11■••

I-

0

0

FIG. 4.1: PLOTTED VALUES OF THE VARIATION OF k WITH n

41

[

pi, , n(n2-1)(1+k1) FEM -

12 n+(n=l)kl

(4.4)

M is the number .of landing 'treads' and other variables are .-same

as mentioned earlier.

DeschapelleS also studied the. case and preSented the formulae

to calculate the fixed end moment for stairs without landing as

giVen below

where k1 = R/T.

His analytical study published in Havana(4),

The paper by Dianu, et al.(5) treated the matter objectively

and gave the analytical solution procedure by two methods. i.e.

column analogy and moment distribution method. They presented the

Solution steps for odd number and even number of ste0s, with and

without landings; with and without intermediate support. . They

also discussed the design stages and reinforcement detailS.

4.3 BASIC ASSUMPTIONS

Present conventional analysis techniques are based on certain:

assumptions. Following assumptions have been adopted in these

analysis methods.

1. The structure is plane (two dimensiOnal), neglecting possible

three dimensional stress. interrelations

2. Dead and live loads are concentrated on the edge of riser

L+3

3. Risers deflect vertically (relative rotaton of the two ends

of the riser being equal to zero)

4. External load is carried only by the treads and risers.

4.4° ANALYSIS TECHNIQUE BY MOMENT DISTRIBUTION

The analytic solution technique is presented by Dianu et

( al.

5) in their paper and is being presented in this section.

By using the stress-strain method the expression for the

fixed end moment for stairs without landing and with an even or

odd number of steps can be derived. The expression may be stated

as

FEM = 2 CPT-M (4.5)

where FEM is the fixed end moment of the stairs and M is the

moment at the centre of span of a statically indeterminate

structure (i.e. stairs) subject to actual loads and given as

C+D(1+k) M = PT A+KB (4. 6)

The coefficients used in Eqs.(4.5) and (4.6) for an even

number of steps, are

n-1,....= n2

D _ n(n -1)(n -2) A= 2; B= 2 ; 16,

2' 48 (4.7)

If the stair consists of an odd number of steps, Eqs.(4.5) and

(4.6) are valid but the coefficients will be modified as

n-1 (n+1)(n-1) B = C - 16

D - (n-1)(n+1)(n-3) 48

(4.8)

A =

44

EI t K1-1'

= K - '-1 T(A+KB) (4.11)

In both the cases

R K- It

r (4.9)

n —N+2

and N is number of steps-or treads.

The coefficients A,B,C and D for stairs with steps varying.

from 2 to 20 may be found from Table 4.2.

The cases with equal. landings at both ends may have different

support condition as outlined in following section. The sign

convention is kept such that the clock-wise moments at Joints are

assumed as positive and vice versa. The mid span moment are

positive with tension at the bottom fibre.

4.4.1 Stairs with Intermediate Supports

This case of stair is shown in Fig. 4.2. The fixed end

moment at 1 and 1' for the section 1-1' may be determined, for

stairs with an even or odd number of steps, from Eq. (4.5).

The fixed end moment at 1 and 2 or 1' and 2' for the section 1-2

or 1'-2' may be found as

= _ q(L-T)2

MI-2 142-1 12 (4.10)

Now moment distribution can be applied as for a continuous beam.

The rigidity of section' 1-1' is found from .

".5

where A,B and K are coefficients with the value as given in Eqs.

(4.7) or (4.8) and (4.9). The rigidity of landing is found from

4E11

K - 1-2

K2-1 - L-T

(4.12)

The distribution coefficients are

4 r1-2 - KL 4+

r1-1'

1 - r1-2 -

where

1 (4.13) 4(A+KB)

KL

A+KB

(L-T) It KL

T I

R It

and K = - . T

The transmission coefficients are

- r1-2 r1-1'

1-2 2 ' Y1-1' 2

r

(4.14)

The structure being symmetric, the moment distribution may be

done for only half of the structure.

After p operation of distribution and transmission, the final

bending moments at different points are found as

M1-1'

= M1-1' -(M

1-1'+M

1-2)

1=1

46

M1-2 = M1-2 -(M1-1,+M1-2 r1-

1=0

M2-1 = M -(M +M r1-2 r1-1'

2-1 1-1' 1-2 2 21

r1-2.r

ti7 1_1, Mc m m

1-1' ,1-2 2 21 1=1

1=0

(4.15)

Here M is computed from Eq. (4.6) while p is taken depending upon

desired accuracy.:

The reaction at Support.. 1- and •2 is given by -

R1 = AP q(L-T)

M +M

2 - (L-T)

R = AP -+ q(L-T) 2 2

(4.16)

(4.17)

where

A = 2-

4.4.2 Stairs Without intermediate supportS

For the analysis of this case of stairs same procedure as

given in section 4.4.1 may be followed but the effect Of.supports

would be eliminated. By virtue of load symmetry the:reactiOn R1

and R are zero and the vertical displacements of points l and 1'

are equal. However; the elimination of supports causes variation

in the moments'of the structure.

The fixed end moment may be determined by allowing for

deformation symmetry. For this purpose a yield y is imparted to

the supports so that a unit moment will appear in each landing.

The unit moments introduced by yield are then balanced for each

joint and then additional reactions are determined using

expression given below

i=p r1-1' MY1-2 = 1 -

i!0

21 '

MY =

r1-2 1p r

1-1' 2-1 2

1=0 21

i=p r MY ,= - E

1-1' 1-1 1=1 21

MY +MY y 1-2 2-1 R1 -

L-T (4.18)

Since the reaction R1 does not exists so the increments of moments

are

AM2-1

- 2

flr

Ry -1 1

R1 my

AM

1-1'- y -1' R

1

(4.19)

4-8

and the final moments at the ends and mid span .of stairs are

M = M + AM 2-1 2-1 2-1

M1- '= M1-1'

+ AM1-'

=11' "M - .c 1-1'

( 4 . 20)

4.5 ILLUSTRATIVE. EXAMPLES

In this cases two cases of'Slabless tread-riser stairs are

solved using conventional analysls methods. The, two cases have,

same end condition i.e. fixed. at both end but one is

intermediately supported and in other case there is no

intermediate support. These cases have been analysed with same

parameters and loads as in case A 'and case A' to make .

Comparative study between conventional and finite.eleMeht analysis .

Methods_ One case of stairs with both ends hinged_ and without

intermediate Support has been also analysed by modified

conventional technique.

4.5.1 Stair with intermediate supports.

The stair (Fig. 4.2) is analysed under following step by step

procedure

(a) Data

Length of landing L = 1.2 m; Tread T = 0.255 m, Rise R =

0.180 m;

Thickness of tread th

= 0.15 m; thickness of riser tv = 0.15

49

m; number of steps n = 11

Live load = 4kN/m2

(b) Calculation for load P and.q

One meter width is considered and load per riser is

calculated as

Dead load of tread = 0.255x0.15x25 = 0.956 kN/m

Dead load of riser = 0.180x0.15x25 = 0.675 kN/m

Live load per step = 4x0.255 = 1.020 kN/m

So load on edge of each riser P = 2.651 kN

Uniformly distributed load on landing

q = 44.(1.0x0.15x25) = 7.75 kN/m

(c) Calculation of Constants

11 11-1 (11+1)(11-1) A = = 5.5; B - 5; c - 7.5; 2 2 16

(11-1)(11+1)(11-3) D -

- 20; 48

h =

Ih

= Iv

0.180 - 0.7059 K - 0.255

50 -24 S 73

51

ILLU

STRA

TIVE

(d) Cal-Culation of::MOMents

M C+D(1+K)

A+KB

Indeterminate moment at the centre of:..§pan of stair

= 2.651x0.255x7.5+ 20(1+0.7059) 5.5+5x0.7059

M = 3:416 kN-m

P1xed end moment- at support 1 and 2

M.1-1' = 2CPT-M

2x7.5x2.65x0.255 - 2.1 16

= 7....0241(Wm

= _ _7.75x(1.2-0.2255)2

1

M1-2 = 0.576 kN-m.

(1.20.225512 m -775 x. 2-1 =

1

M2-1 = 0.576 kN-m

(e) Calculation of distribution coefficients

r 1-2 +KB

53

where

KL = (L-T) _ 1.20-0.255 - 3.706 0.255

r1-2 - 4

4+ 3.706 5.5+5x0.7059

r1-2

= 0.9069

r1-1' = 1 - r1-2 = 1-0.9069

= 0.093 ri_l,

Transmission coefficients are

r1-2 0.9069 Y =

1-2 2 2

Y1-2 = 0.45395

r1-1' 0.093 Y1-1' 2 2

Y1-1' = 0.0465

(f) Calculation of final moments

Using Eq. (4.15)

M1-1'= 7.024-(7.024-0.576) [0.093'

X0.093)2+(0.093)3}. ' 2 22 "

23

M1-1' = 6:709 kN-M

54

_._ 0.093+ ( 0.093)2(O. 09j) 2 2 3 +' • 2 2

M- =-0.576- ( 7.024-0.576 ) (0.9069)

1-2 =-6.709 kN-M

(0.9069 ) , +0.093+ ( 0.093 ) M2_1 = O. 576-(7.024-0.576) 2 2 22 0.093)34..

23

=-2,490 kI471,1

3: 116+ ( 7.-024-0.575 )

3.259 kN-M

) Calculation of StippOrt Reactions

= 5 5 x 2.651 7.75 (1.2-0.255) (-6.709 - 2.490) .

= 27.977 04

= AP + q(L-T) - R

= 5.5 x 2.651 + 7.75 (4.2-0.255) - 27.977

55

(1.2 - 0.255) 2

4.5.2 Stair Without Intermediate Supports

For the analysis of this case the parameters and loads are

assumed to be same as given in preceding section. The procedure

steps will be same from step (a) to step (g). The further analysis

steps are outlined below

(h) Determination of additional reaction

using Eq. (4.18) following may be find out

0.093 (0.09342 (0.093)3 M = 1-0.9069 [1+

1-2 2 22 23

y M

= 0.0489 1-2

y 0.9069 0.093 (0.093)2 (0.093)3

M2-1 = 1 - [1 + 22 2

+

2 2

y M = 0.5244 2-1

y 0.093 (0.093) 2(0.093) 3

1-1' 2 22 23 •

MY = - 0.04828 1-1'

56

Additional Reaction

• y = 0.0489 + 0.5244 R 1 (1.20 - 0.255)

y R1 = 0.6067 kN

(i) Calculation of final moments.

Since Reaction R1 does not exits, the increments of moments

are.

AM2-1 - 27.977

0.6067

x 0.5244

AM2-1 = 24.1819 kN-M

27.977 AM1-1' 0.6067

x (-0.04828)

AM1-1'

= -2.2264 kN-M

Final moments are as given

M2-1 = -2.490 + 24.1819 = 21.6919 kN m

M1-1'

= 6.709 - 2.2264 = 4.4826 kN m

M

= 3.259 + 2.2264 = 5.4854 kN m

57

(j) Calculation of Reactions

R2 = AP + q(L-T)

R2 = 5.5 x 2.651 + 7.75 (1.2-0.255)

R2 = 21.904 kN

4.5.3 Modified Conventional Technique

It may be observed that stairs, due to membrane action,

subjected to thrust. The fixed end moments will also be less

than those calculated from present analysis techniques. The

present conventional analysis techniques assume the stairs as a

straight line structure, neglecting the bent partion of stairs

thus giving larger fixed end moments and ignoring horizontal

thrust.

Here a case of intermediately unsupported stairs with both

end hinged is solved by using strain energy method (Fig. 4.3).

The method gives more reliable stress resultants.

The problem is solved under following step by step procedure

(a) Data

Length of landing L = 0.9 m; Tread T = 0.28 m; Rise R = 0.15m;

Thickness of tread th=0.1Sm ; Thickness of riser tv = 0.15 m ;

Number of steps = 9 ; Live load = 3.5 kN/m2

(b) Calculation for loads

One meter width is consideredOniformly distributed load on straight

portion

q1=3.6+(1.0x0.15x25)=7.25 kN/m

58

59

H C/D

C

LZ

E-1

C

U)

Uniformly distributed load on bent portion

(0.28x0.15x25+0.15x0.15x25+3.5x0.28) - 9.2589 kN/m

(c) Calculation of Reactions

Considering the free body diagram (Fig. 4.4)

and applying strain energy method to bent portion.

Taking moment about c

HB x 1.2 + 9.2589 x 2.52 x (2.52/2) - VB x 2.52 = 0

HB

= 2.1 VB - 24.4990

(4.21)

Moment Ms at a distance s from point B along the member is given

by

Ms = 10.5321s - 3.7738 s2

Since moment is independant of reaction the bending strain energy

due to reaction will be absent

Axial force Ps

at a distance s from point B along the member is

given by

Ps

= 0.9029HB + 0.4299 V

B - 9.2589 x 0.9029 x 0.9029 S

Replacing the HB by V2 from Eq. (4.21)

PS = 2.326 VB - 29.6683 s

a PS = 2.326

a vB

q2 0.28

61

auBC 1

8VC AE

2.7911

so 5.4103 VB - 69.0084 s

[15.1007 VB - 268.7960]

From minimum energy principle

avBC - 0 0 Vc

15.1007 VB - 268.7960 = 0

VB = 17.800 kN

Substituting this value in Eq.(4.21)

HB=12.881 kN

From static equalibrium

VC = 9.2589 x 2.52 - 17.801

VC = 5.532 kN

and HB = HC = 12.881 kN

From static equalibrium of portion AB

VA = 7.25 x 0.62 + 17.800

VA = 22.295 kN

HA = H = 12.881 kN

1

AE

62

411. 141 1-4 410 ,4111 '407 114 ,44111,

4110 4111 114 4P. 4110 114 141 4111 1.11 411 1141,s, 114 0-4411. 141481

0-44111 411 1.4 1-44 441

4110 41, 114 4111 411. 4111. 411 4610 I-4 1141

111 0-44111 0-41410

40 114 1110 10-4 410 14011 0.1_14 -41P 111 114

1141 10-4 461, 141 411 114 gte, 74-W' 114

410 114 4110 1.4 400 141

0 0 01 W Cf)

C.)

C

O

C <14

0

Fx4

C

(24

63

From static equalibrium of portion CD

Vd = 7.25 x 0.62 + 5.532

Vd = 10.027 kN

Hd = HC = 12.881 kN

Moment at the centre of span

Mc=VAx1.88-q1 x0.62x1.57-q2x1.26x0.63-HAx0.6

M =22.295x1.88-7.25x0.62x1.57-9.2589x1.26x0.63-12.881x0.6

Mc =19.779 kN-m.

65

CHAPTER 5

PARAMETERIC STUDIES OF STAIRCASE

6.1 DISCRETIZATION AND IDEALIZATION OF DOMAIN

For the finite element analysis of Slabless tread-rise stairs

the solution region is discretized in a number of elements. The

element used is two dimensional eight noded quadrilateral

isoparametric element of serendipity family. In the

discretization process the thickness of landing and thickness of

tread is divided into two parts to get a more accurate solution.

Because of the identical cross-section throughout landing,

dividing the landing into more than two elements along the depth

will not produce more accurate results but will increase

computational efforts.

Although the stair is a three dimensional structure yet it is

considered as a two dimensional plane strain case since geometry

and loading do not vary significantly in third dimension. For

analysis purpose the length in third direction is assumed to be

uniform (1 m).

The structure is discretized in, such a way that a node must

necessarily appears under a concentrated load, at a re-entrant

corner or at the point where geometry changes abruptly. The

boundary conditions have been imposed by restricting the nodal

displacements in corresponding directions.

67

The finite element solution based on the shear deformation

theories suffers from locking for thin laminates(6). To circumvent

this difficulty reduced integration is used. The order of

integration used is reduced from 3x3 to 2x2, the reduction in the

order of integration in computing stiffness matrix of an

isoparametric element generally has the effect to reduce the

values of the stiffness of an element below the value that the

exact integration produces.

5.2 CASES STUDIED

A number of cases have been solved by finite element analysis

for different support and end conditions. For all cases following

material properties have been taken

1. Grade of concrete = M20

2. The Young's modulus of elasticity (E) = 2.55x107 kN/m2

3. The Poisson's ratio (v) = 0.15

4. The density (p)

= 25.0 kN/m3

The cases may be broadly categorised into two as given below.

5.2.1. Stairs without Intermediate Supports

These type of stairs are not supported elsewhere except at

the ends. The ends of the stairs may be supported on a concrete

beam'or on a bearing wall. The landing slab may also be continuous

with floor slab so these different end conditions effect the

stress resultants in the stairs. An extensive study has been done

to observe the behaviour of stairs under different end conditions.

68

These cases with different end conditions are discussed as given

below.

Case A: Stairs with both ends fixed

This is the case when ends of the stairs are supported by

concrete beam or the landing slabs are continuous with the floor.

In this case the nodes situated at the boundary are assumed to be

restricted in both direction (i.e. x,y) so the ends of the stairs

have neither displacement nor. rotation. The various parameters

and mesh details for this case are given in Table 5.1. The mesh

for.this case has 88 elements and 387 nodes.

The deflected shape for this case is shown in Fig. 5.1. The

displacements (deformations) are 10 times magnified to plot this

deflected shape. It is to be noted that the stair deform in such

a way that it takes a shape like a catenary therefore, catenary

action is present there producing thrust in the stair.

All the three stresses and principal stresses are calculated

and from these stresses stress resultants have been computed in

the program. The stress resultants, namely bending moment, shear

force and axial tension have been computed at various cross

sections on which Gaussian points lie. In this case there are 34

such cross sections and at these cross sections the stress

resultants are given in Table 5.2. The Table 5.2 also contains

the reactions, fixed end moments and horizontal reactions at the

ends.

69

The variations of stress resultants along the longitudinal

section of stair have been plotted and shown in Fig. 5.2.

Case B: Stairs with both ends hinged

If the landings rest on exterior bearing wallsor on small

beams, the end conditions of the stairs are changed and the stairs

behave like a simply supported structure having both ends hinged.

To provide this boundary condition, the centre node of the ends

are restricted in both directions and other nodes are restricted

in y direction only. The ends of the stairs, now, will have

rotation at the support. 6

The various parameters and mesh details for this case are

given in Table 5.1. The mesh for this case has 58 elements and

253 nodes.

The deflected shape for this case is shown in Fig. 5.3. The

deformations are magnified for 10 times to plot this deflected

shape. The stresses and strains have been studied at each

Gaussian points and from these stresses, stress resultants at

various cross sections computed and presented along with support

reactions and horizontal reactions at the ends of stair in

Table 5.3.

The variations of stress resultants along the longitudinal

section of stair have been plotted and shown in Fig. 5.4.

70

TABLE 5.1 : PARAMETERS AND MESH DETAILS FOR SECTION 5.2.1

S. No. Particulars Case A Case B Case C

1 Parameters :

(a) Landing (m.) 1.200 0.900 1.500

(b) Tread (m. ) 0.255 0.280 0.300

(c) Rise (m. ) 0.180 0.150 0.200

(d) Thickness of tread

(e) 'Thickness of riser

(f) Number of steps

(g) Overall span of stair

(h) Flight span

(m. )

(m. )

(N)

(m. )

(m. )

0.150

0.150

9

4.695

1.80

0.150

0.150

7

3.760

1.20

0.150

0.150

10

6.000

2.20

(i) Live load (kN/m2) 4.00 3.50 4.50

2 Mesh details :

(a) Elements (NE) 88 58 95

(b) Nodes (NP) 387 253 418

(c) Boundary nodes

(NB) 10 10 10

73

all

co co O I

rr rr 11 I 0- W 03 Z Z Z

IFT I:

I Ir

U FOR

CAS

E D

E FLE

CTE

D S

H AP

E

TABLE 5.2 : STRESS RESULTANTS FOR CASE A

S. No x coordinate

of cross section

(m.)

Shear force

(kN/m)

Thrust

(kN/m).

Bending moment

(kN-m/m)

1 0.000 (25.683) (-5.923) (-22.016) 2. 0.054 26.165 -5.923 -20.595 3. 0.201 25.024 -5.923 -16.827 4. 0.309 24.211 -5.923 -14.174 5. 0.456 23.069 -5.923 -10.693 6. 0.564 22.298 -5.923 -8.298 7. 0.711 21.153 -5.923 -5.050 8. 0.803 20.461 -5.921 -3.137 9. 0.907 19.661 -5.925 -1.052 10. 0.967 18.057 -5.925 0.091 11. 1.028 18.047 -5.925 1.186 12. 1.222 15.281 -5.907 3.369 13. 1.283 15.274 -5.904 4.295 14. 1.477 12.205 -5.946 5.911 15. 1.538 12.220 -5.927 6.651 16. 1.732 9.405 -5.894 7.681 17. 1.793 9.399 -5.888 8.250 18. 1.987 6.293 -5.861 8.728 19. 2.048 6.284 -5.868 9.109 20. 2.242 3.461 -5.860 9.004 21. 2.303 3.442 -5.866 9.213 22. 2.497 0.555 -5.875 8.544 23. 2.558 0.573 -5.889 8.579 24. 2.752 -2.146 -5.957 7.344 25. 2.813 -2.141 -5.955 7.213 26. 3.007 -4.798 -5.988 5.466 27. 3.668 -4.773 -5.996 5.175 28. 3.262 -7.475 -6.013 2.919 29. 3.323 -7.467 -6.015 2.467 30. 3.549 -10.242 -6.019 -0.670 31. 3.696 -10.230 -6.017 -2.177 32. 3.788 -11.807 -6.013 -3.172 33. 3.892 -12.603 -6.014 -4.440 34. 3.984 -13.281 -6.014 -5.629 35. 4.131 -14.418 -6.018 -7.668 36. 4.239 -15.231 -6.017 -9.266 37. 4.386 -16.374 -6.017 -11.592 38. 4.494 -17.203 -6.017 -13.402 39. 4.641 -18.345 -6.018 -16.019 40. 4.695 (-18.128) (-6.065) (-17.020)

Notes:(i) (-) indicates 0 shear, tensile thrust and hogging moment

(ii) Stress resultants in parentheses have been calculated from reactions at nodes situated at the ends of stair.

75

STRESS RESU

LTAN

T

BEN

DIN

G M

OME

NT

( k N

- m

im

THRU

ST( k N

/m )

7

DIS

TANCE

• "

• 1 .••

I. 1-1-.11...1-I..1 .1-1...1.4.1 I L.i LJ. I -I 1.1.111S)-.1.

SINUITS38 SS3ES

77

CU in

1-4 LL

Ca

1-4

U)

b

DEFL

ECTE

D SHA

PE

SI

to In

79

TABLE 5.3 : STRESS RESULTANTS FOR CASE B

S. No x coordinate

of cross section

(m.)

Shear force

(kN/m)

Thrust Bending moment

(kN/m) (kN-m/m)

1. 0.000 (20.904) (-13.669) (0.000) 2.. 0.059 20.770 -13.664 1.242 3. 0.221 19.596 -13.664 4.505 4. 0.352 18.676 -13.673 7.012 5. 0.548 17.238 -13.671 10.537 6. 0.647 15.267 -13.680 12.176 7. 0.723 15.286 -13.670 13.323 8. 0.927 12.538 -13.668 14.139 9. 1.003 12.566 -13.664 15.083 10. 1.207 9.907 -13.693 15.349 . 11. 1.283 9.922 -13.695 16.092 12. 1.487 7.458 -13.745 15.823 13. 1.563 7.462 -13.739 16.382 14. 1.767 4.953 -13.855 15.585 15. 1.843 4.996 -13.862 15.958 16. 2.047 2.281 -13.858 14.632 17. 2.123 2.302 -13.851 14.804 18. 2.327 -0.401 -13.866 12.947 19. 2.403 -0.422 -13.850 12.915 20. 2.607 -3.172 -13.816 10.459 21. 2.683 -3.215 -13.818 10.219 22. 2.919 -5.690 -13.819 7.050 23. 3.081 -5.635 -13.816 6.134 24. 3.212 -7.631 -13.816 5.271 25. 3.408 -9.086 -13.816 3.631 26. 3.539 -9.977 -13.825 2.381 27. 3.701 -11.158 -13.825 0.673 28. 3.760 (-11.418) (-13.826) (0.000)

Notes:(i) (-) indicates it shear, tensile thrust and hogging moment

(ii) Stress resultants in parentheses have been calculated from reactions at nodes situated at the ends of stair.

81.

•

STRESS RESULTANT DIAGRAM FOR CASE

in

• •

THRU

ST (k

N/m

)

t

7'% Eli

1 . . • ES)

511E1198 MEC 83

Case C: Stairs with one end fixed and other hinged

This is a case when two ends of stair have different end

conditions. A case has been taken when upper end of stair is

assumed to be fixed due to reasons discussed for case A and lower

end is assumed to be hinged due to reasons discussed for case B.

The appropriate boundary conditions are applied. The various

parameters and mesh details for this case are given in Table 5.1.

The mesh for this case has 95 elements and 418 nodes.

The deflected shape for this case is shown in Fig. 5.5. The

deformations .are magnified 10 times to plot the deflected shape.

The stress resultants at various cross sections passing through

Gaussian points along with support reactions, fixed end moments

and horizontal reactions at the ends of stair has been presented

in Table 5.4.

The variations of stress resultants along the longitudinal

section of stair have been plotted and shown in Fig. 5.6.

5.2.2 Stairs with Intermediate Supports

In these type of stairs, which are most common, the landings

are supported on beams. These beams provide support to the

landings and thus reduce the bending moment in the structure

significantly. This case is analogous to the continuous beam

supported at two points. These beams are provided at a distance

equal to tread from the beginning of landing. A number of cases

have been analysed by finite element analysis to study the

85

behaviour for different end conditions of stairs with intermediate

supports. Three cases have been discussed in this section, with

different end conditions, as given below.

Case A': Stairs with both ends fixed

The end conditions are same as in case A. The stair is

supported at two intermediate points at landings. The various

parameters and mesh details are given in Table 5.5. The mesh for

this case has 88 elements and 387 nodes.

The deflected shape for this case is shown in Fig. 5.7 by

magnifying the deformations 250 times. It is to be noted that in

this case the deformations are much less in comparison of case A.

The stresses and strains are find out at each Gaussian

point. These quantities and stress resultants as well, are

greatly reduced for this case.

The stress resultants along with support reactions, fixed end

moments and horizontal reactions at the ends of stair are shown

in Table 5.6. Variations of st ress resultants along the

longitudinal section of stairs are shown in Fig. 5.8.

Case B': Stairs with both ends hinged

In this case the end conditions are similar to as that of

case B, but in addition to the end supports it is having

intermediate supports also. These intermediate supports are

86

87

F OR C ASE

DEFLECTED S

HAPE OF STAIR

1•I ft

it

• a

1 t i

t-i

I T

TABLE 5.4 : STRESS RESULTANTS FOR CASE C

x coordinate S. No of

cross section (m.)

1. 0.000 2. 0.063 3. 0.237 4. 0.363 5. 0.537 6. 0.663 7. 0.837 8. 0.963 9. 1.137 10. 1.232 11. 1.318 12. 1.532 13. 1.618 14. 1.832 15. 1.918 16. 2.132 17. 2.218 18. 2.432 19. 2.518 20. 2.732 21. 2.818 22. 3.032 23. 3.118 24. 3.332 25. 3.418 26. 3.632 27. 3.718 28. 3.932 29. 4.018 30. 4.232 31. 4.318 32. 4.563 33. 4.737 34. 4.863 35. 5.037 36. 5.163 37. 5.337 38. 5.463 39. 5.637 40. 5.763 41. 5.937 42. 6.000

Shear force

(kN/m)

Thrust

(kN/m)

Bending moment

(kN-m/m)

(54.936) (-43.058) (-55.387) 54.248 -43.057 -51.931 52.821 -43.057 -42.659 51.764 -43.055 -36.029 50.350 -43.056 -27.186 49.257 -43.056 -20.871 47.831 -43.055 -12.463 46.745 -43.048 -6.465 45.248 -43.040 1.501 43.076 -43.031 5.718 43.097 -43.039 9.450 39.708 -42.966 9.686 39.689 -42.982 13.127 36.789 -43.109 12.655 36.702 -43.136 15.840 33.596 -43.161 14.729 33.614 -43.169 17.638

- 29.877 -43.123 15.775 29.822 -43.133 18.360 26.805 -43.206 15.803 26.801 -43.201 18.123 23.954 -43.060 14.956 23.999 -43.028 17.032 20.652 -42.881 13.290 20.748 -42.888 15.082 17.297 -42.962 10.523 17.229 -42.989 12.018 14.394 -42.907 6.836 14.413 -42.921 8.082 10.942 -42.919 2.189 11.022 -42.893 3.141 7.403 -42.927 -3.232 7.356 -42.936 -1.953 5.411 -42.910 -1.127 3.995 -42.905 -0.314 3.285 -42.881 0.149 1.827 -42.878 0.592 0.760 -42.860 0.755 -0.668 -42.857 0.763 -1.634 -42.856 0.617 -3.061 -42.856 0.211 -3.565 -42.858 0.000

Notes:(i) (-) indicates IT shear, tensile thrust and hogging moment

(ii) Stress resultants in parentheses have been calculated from reactions at nodes situated at the ends of stair.

89

-

- T

HRU

ST (

k W

m)

BEN

DIN

G M

OMEN

T ( kN

-W

m)

FIG

. 5.6 :S

TRES

S RESU

LTANT D

IAG

RAM

FO

R CAS

E

CM

Litlalitiiiu..tilthiliallitailsiaLlatuatttilentitt.artili 'el iltitst,14tuluiduilltaidttel!

q eA22`ncniRrii Banns3a SS3ES

91

TABLE 5.5 : PARAMETERS AND MESH DETAILS FOR SECTION, 5.2.2

S.No. Particulars Case A' Case B' Case C'

1 Parameters :

(a) Landing (m.) 1.200 0.900 1.500

(b) Tread (m.) 0.255 0.280 0.300 .

(c) Rise (m.) 0.180 0.150 0.200

(d) Thickness of tread

(e) Thickness of riser

(f) Number of steps

(g) Overall span of stair

(h) Flight span

(m. )

(m.)

(N)

(m.)

(m.)

0.150

0.150

9

4.695

1.80

0.150

0.150

7

3.760

1.20

0.150

0.150

10

6.000

2.20

(i) Live load (kN/m2) 4.00 3.50 4.50

2 Mesh details :

(a) Elements (NE) 88 58 95

(b) Nodes (NP) 387 253 418

(c) Boundary nodes

(NB) 12 12 12

93

II

CL w Z Z Z

FOR CASE ( A' )

DEFLECTED SHAPE OF STRIR

FIG. 5.7:

95

TABLE 5.6 : STRESS RESULTANTS FOR CASE A'

S. No x coordinate

of cross section

( m. )

Shear force

(kN/m)

Thrust

(kN/m)

Bending moment

(kN-m/m)

1. 0.000 (-4.928) (-10.095) (2.024) 2. 0.054 -5.349 -10.095 1.747 3. 0.201 -6.490 -10.095 0.876 4. 0.309 -7.327 -10.095 0.131 5. 0.456 -8.468 -10.095 -1.032 6. 0.564 -9.307 -10.095 -1.989 7. 0.711 -10.448 -10.095 -3.444 8. 0.803 -11.162 -10.095 -4.437 9. 0.907 -11.967 -10.095 -5.639 10. 0.967 20.821 -10.095 -5.638 11. 1.028 20.820 -10.095 -4.375 12. 1.222 18.164 -10.092 -2.403 13. 1.283 18.164 -10.091 -1.302 14. 1.477 15.499 -10.098 0.153 15. 1.538 15.502 -10.095 1.092 16. 1.732 12.848 -10.087 2.029 17. 1.793 12.846 -10.091 2.808 18. 1.987 10.184 -10.082 3.228 19. 2.048 . 10.184 -10.078 3.846 20. 2.242 7.481 -10.087 3.750 21. 2.303 7.480 -10.079 4.204 22. 2.497 4.824 -10.067 3.584 23. 2.558 4.820 -10.067 3.876 24. 2.752 2.202 -10.061 2.739 25. 2.813 2.202 -10.059 2.873 26. 3.007 -0.423 -10.062 1.233 27. 3.068 -0.419 -10.061 1.207 28. 3.262 -3.067 -10.059 -0.941 29. 3.323 -3.065 -10.060 -1.127 30. 3.549 -5.716 -10.062 -3.973 31. 3.696 -5.716 -10.062 -4.815 32. 3.788 10.458 -10.062 -4.719 33. 3.892 9.653 -10.062 -3.674 34. 3.984 8.939 -10.062 -2.820 35. 4.131 7.798 -10.062 -1.588 36. 4.239 6.962 -10.062 -0.792 37. 4.386 5.821 -10.062 0.149 38. 4.494 4.985 -10.062 0.731 39. 4.641 3.844 -10.062 1.381 40. 4.695 (3.424) (-10.062) (1.577)

Notes:El) (-) indicates It shear, tensile thrust and hogging moment

(ii) Stress resultants in parentheses have been calculated from reactions at nodes situated at the ends of stair.

97

—to

DIST

ANCE

C M.

)

E

6

ffi TH

R UST

(k W

m)

BEND

ING

MOME

N T (k

N- m

/m)

■._1..4 A ttLi... t- s-J- L. J -L. I.-t .4--t. L. _I ..1_ L.1_1

SINti1TS38 SaLLS

99

LU tn a: U re

z CY

CC

(1) LU

LU

CY

CY

• •

I. II I. w m

z z z

DEFLECTED SHAPE

FIG. 5.9:

TABLE 5.7 : STRESS RESULTANTS FOR CASE B'

x coordinate S. No of

cross section

(m.)

Shear force

(kN/m)

Thrust Bending moment

(kN/m) (kN-m/m)

1. 0.000 (-4.747) (-7.884) (0.000) 2. 0.059 ' -5.175 -7.884 -0.294 3. 0.221 -6.347 -7.884 -1.225 4. 0.352 -7.298 -7.884 -2.119 5. 0.548 -8.721 -7.884 -3.691 6. 0.647 15.022 -7.884 -3.924 7. 0.723 15.022 -7.884 -2.796 8. 0.927 12.423 -7.883 -1.166 9. 1.003 12.424 -7.883 -0.233 10. 1.207 9.832 -7.884 0.865 11. 1.283 9.832 -7.884 1.602 12. 1.487 7.240 -7.888 2.171 13. 1.563 7.237 -7.888 2.714 14. 1.767 4.629 -7.897 2.747 15. 1.843 4.630 -7.897 3.095 16. 2.047 2.044 -7.902 2.595 17. 2.123 2.045 -7.901 2.748 18. 2.327 -0.540 -7.903 1.717 19. 2.403 -0.540 -7.903 1.677 20. 2.607 -3.135 -7.900 0.116 21. 2.683 -3.136 -7.899 -0.120 22. 2.919 -5.716 -7.900 -2.393 23. 3.081 -5.715 -7.900 -3.317 24. 3.212 - 7.622 -7.900 -3.089 25. 3.408 6.199 -7.900 -1.732 26. 3.539 5.249 -7.900 -0.982 27. 3.701 4.077 -7.900 -0.229 28. 3.760 (3.650) (-7.900) (-0.000)

Notes:(i) (-) indicates It shear, moment

tensile thrust and hogging

(ii) Stress resultants in parentheses have been calculated from reactions at nodes situated at the ends of stair.

103

- - -

T HRU

ST ( k

N/m

BEN

DIN

G M

OMENT ( k N- m/m)

Ls...LA..11.1...1_1.1-1 I\

4..1

urq ;4 5S3NIS

STRESS RESULTANT DIAGRAM FOR CASE

• II

105

provided by the beams. The parameters and mesh details of this

case are given in Table 5.5. The mesh for this case has 58

elements and 253 nodes.

The deflected shape for this case is shown in Fig. 5.9. The

deformations have a magnification factor 250. In this case also

the stress resultants are significantly reduced due•to support

conditions. These resultants, along with fixed end moments,

support reactions and horizontal reactions at the ends of stair

are given in Table 5.7. The variations of stress resultants along

the longitudinal section for this case are plotted and shown in

Fig. 5.10.

Case C': Stairs with one end fixed and other hinged

The support conditions'of this case is identical to case C

except two additional supports, provided intermediately in the

form of beams. The parameters and mesh details for this case are

shown in Table 5.5. The mesh of this case has 95 elements and 418

nodes.

The deflected shape of this case, with deformations magnified

125 times, is shown in Fig. 5.11.

The stress resultants along with support reactions, fixed end

• moments and horizontal reactions at the ends of stair are

presented in Table 5.8 and their variations along the longitudinal

section of stair are shown in Fig. 5.12.

107

5.2.3 Non-orthopolygonal Stairs

In these type of stairs, riser are made inclined and the

stairs are not truely orthopolygonal. In these type of stairs, the

tread increases for the same flight span and same number of steps

in comparison with that of an orthopolygonal stairs. In fact in

these stairs the tread overlaps by some amount providing more

space to rest the foot. These type of stairs are mainly being

constructed in Rajasthan state in India.

These type of stairs are also analysed using finite element

method and parameters and mesh details for a specific problem are

given in Table 5.9. The mesh for this case consists of 74

elements and 325 nodes.

The deflected shape with a magnification factor 10 is shown

in Fig.5.13. .The stresses and strains for this case is also find

out at each Gaussian point and the stress resultants are computed

by using these values. The stress resultants along with reactions

at support, fixed end moments and horizontal reactions at the ends

of stair are shown in Table 5.10. The variationsof stress

resultants along the longitudinal section of stair are shown in

Fig. 5.14.

5.3 REVIEW OF RESULTS

The finite element analysis results present some interesting

phenomenon. These are being presented in this section.

108

W 4f1

I II I

Z Z Z

109

FOR C

ASE

DEF

LECTED

SH

APE OF STAIR

:

TABLE 5.8 : STRESS RESULTANTS FOR CASE C'

S. No x coordinate

of cross section

(m.)

Shear force

(kN/m)

Thrust

(kN/m)

Bending moment

(kN-m/m)

1. 0.000 (-7.059) (-16.494) (3.697) 2. 0.063 -7.560 -16.494 3.234 3. 0.237 -8.989 -16.494 1.801 4. 0.363 -10.034 -16.494 0.595 5. 0.537 -11.464 -16.494 -1.266 6. 0.663 -12.507 -16.494 -2.786 7. 0.837 -13.936 -16.494 -5.076 8. 0.963 -14.982 -16.494 -6.910 9. 1.137 -16.411 -16.494 -9.628 10. 1.232 29.077 -16.494 -9.764 11. 1.318 29.078 -16.494 -7.245 12. 1.532 25.865 -16.490 -4.683 13. 1.618 25.868 -16.491 -2.443 14. 1.832 22.584 -16.498 -0.566 15. 1.918 22.580 -16.482 1.390 16. 2.132 19.366 -16.480 2.576 17. 2.218 19.360 -16.480 4.252 18. 2.432 16.082 -16.485 4.731 19. 2.518 16.077 -16.484 6.123 20. 2.732 13.046 -16.509 5.937. 21. 2.818 13.055 -16.508 7.068 22. 3.032 9.960 -16.486 6.233 23. 3.118 9.950 -16.484 7.095 24. 3.332 6.733 -16.429 5.596 25. 3.418 6.756 -16.430 6.180 26. 3.632 3.484 -16.452 3.981 27. 3.718 3.462 -16.453 4.282 28. 3.932 0.256 -16.436 1.392 29.. 4.018 0.259 -16.437 1.415 30. 4.232 -2.990 -16.430 -2.160 31. 4.318 -2.991 -16.434 -2.419 32. 4.563 -6.207 -16.434 -6.885 33. 4.737 -6.204 -16.435 -7.960 34. 4.863 11.392 -16.435 -7.614 35. 5.037 9.963 -16.435 -5.765 36. 5.163 8.908 -16.435 -4.569 37. 5.337 7.480 -16.435 -3.149 38. 5.463 6.438 -16.435 -2.267 39. 5.637 5.009 -16.435 -1.276 40. 5.763 3.963 -16.435 -0.707 41. 5.937 2.534 -16.435 -0.144 42. 6.000 (2.016) (-16.435) (0.000)

Notes:(1) (-) indicates 0. shear, tensile thrust and hogging moment

(ii) Stress resultants in parentheses have been calculated from reactions at nodes situated at the ends of stair.

111

E z Y

E z

tV Sri

kits.istullitdmiltis.doutghtntlittlitillstutlustiu.$44ittlatsittai:RtatttilittIlts■olmoluttlatialettlitnii

Si LTS"k SSM.S 113

TABLE 5.9:PARAMETERS AND MESH DETAILS FORNON-ORTHOPOLYGONAL STAIR

S.No. Particulars Magnitude

1. Parameters :

(a) Landing (m) 1.320'

(b) Tread (m) 0.300

(c) Rise (m) 0.150

(d) Thickness 0.100 of (m) tread

(e) Thickness 0.100 of

(m) riser

(f) Number 7 of

(N) steps

(g) Overall span of 4.500 stair (m)

(h) Flight span (m) 1.200

(i) Overlap (m) 0.030 length

(j) Live load (kN/m2) 4.00

2. Mesh details :

(a) Elements (NE) 74

(b) Nodes (NP) 325

(c) Boundary (NB) 10 nodes

115

ry

z 0

0

I f-CE 0 z 0

0

cn

(.)

117

TABLE 5.10 : STRESS RESULTANTS FOR NONORTHOPOLYGONAL STAIR

S. No x coordinate

of cross section

(m.)

Shear force

(kN/m)

Thrust

(kN/m)

Bending moment

(kN-m/m)

1. 0.000 (19.374) (-5.180) (-15.649) 2. 0.054 19.282 -5.249 -14.809 3. 0.201 18.311 -5.248 -12.042 4. 0.309 17.604 -5.249 -10.107 5. 0.456' 16.631 -5.250 - 7.586 6. 0.564 15.956 -5.252 - 5.830 7. 0.711 14.984 -5.253 - 3.553 8, 0.819 14.245 -5.253 - 1.978 9. 0.966 13.264 -4.258 0.047 10 1.059 11.966 -4.884 1.274 11. 1.167 9.303 -4.202 2.414 12. 1.384 8.654 -4.044 3.952 13. 1.662 6.574 -4.349 5.414 14. 1.939 4.236 -4.700 6.271 15. 2.217 2.133 -4.923 6.490 16. 2.494 - 0.082 -5.253 6.059 17.. 2.772 - 1.840 -5.486 5.045 18. 3.049 - 4.693 -5.958 3.364 19. 3.293 - 6.773 -6.092 1.095 20. 3.474 - 7.918 -5.510 -0.191 21. 3.594 - 9.287 -5.260 -1.219 22. 3.741 -10.246 -5.261 -2.657 23. .3.849 -11.039 -5.263 -3.803 24. 3.996 -12.005 -5.267 -5.500 25. 4.104 -12.684 -5.262 -6.830 26. 4.251 -13.654 -5.263 -8.769 27. 4.359. -14.353 -5.260 -10.278 28. 4.506 -15.323 -5.260 -12.463 29. 4.560 (-15.471) -5.191 (-13.123)

Notes:(i) (-) indicates 0 shear, tensile thrust and hogging moment

(ii) Stress resultants in parentheses have been calculated from reactions at nodes situated at the ends of stair.

119

STR

ESS

RE

SU

LTAN

T D

I AG

RA

M FO

R N

ON

OR

THO

POLY

GO

NAL

STA

IR

L1_.1.1-1..1_11.1..1_1.1.. .1.1.1..1-1..1..1..1

U

/-.1..1 1.1..4 1.1-1 I.I I.t..I . 1t.L14.I.I.1 LJ

ts' ENIS

121

5.3.1. Displacements

The stairs in each case deform to take a catenary shape. The

maximum displacement occures at centre of span. The displacement

can be broken into two parts, one horizontal displacement and

other vertical displacement. The vertical displacement is almost

same as of a beath of equal span and with same end conditions but

the horizontal displacement distinguishes the stairs behaviour to

the behaviour of a beam with same end conditons and of same span.

The maximum horizontal displacement occures at the centre of the

span of stairs. Displacement depends upon the rigidity of the

structure. Stair with fixed end conditions has smaller

displacement than the stairs with hinged end condition. The

intermediate supports also reduce the displacements significantly.

In finite element analysis the displacements also depend upon the

order of integration used for numerical integration technique.

Displacements will be less for 2x2 Gaussian integration than the

displacements obtained from 3x3 Gaussian integration.

5.3.2 Axial Force

Since the ends of stairs are not at the same level so this

difference generates axial force in the stairs throughout. Infact

it is the membrane action that causes tension in the stairs. The

axial force remains constant throughout the structure in each

case.

The stress in x direction is composed of direct stress and

bending stress. These direct stress provides axial force.

123

This concept of horizontal thrust is being neglected in

present analysis technique but it may play a significant role as

evident from case C where the order of axial stress Is 285 kN/m2.

5.3.3 Shear Force

It is observed that upper end reaction is always greater than

the reaction at lower end. The outward horizontal reaction at the

ends of stairs form a couple and to balance this couple a moment

in opposite direction is required and there lies the basic

difference in reactions at the end supports. In the cases of

intermediately supported stairs most of the applied load is

transferred through the intermediate supports and a negligible

amount is transferred through end supports. Even in some cases it

is in negative direction.

Hence it is observed that shear force diagram for stairs,

with both end conditions being same, is not perfectly mirror

imaged about the centre as is expected to be in conventional

analysis methods. The order of shear stress in all cases except •

one, with fixed at one and hinged at other end, solved in this

study is less than 0.2 N/mm2 which may be safely carried out by

M20 concete. For stairs with different end conditions at both

ends (i.e. case C) shear force is more at fixed support requiring

shear reinforcement at the ends.

5.3.4 Bending Moment

The fixed end moments obtained from finite element analysis

are less than those of obtained from conventional analysis

124

methods.The fixed end moments which are main design criterian,

will be less and thus a less depth is required for treads and

risers. If the inclined portion of slab is considered than the

fixed end moments will be less in conventional analysis methods

also.

It is obvious from the stress resultants variationsdiagram,

that the shape of the bending moment diagram is equivalent to that

of a beam with same supported conditions, but the sagging moment

at the centre being larger and the fixed end moments being smaller

in comparison to conventional analysis method results.

5.3.5 Stress Concentration

Observations show that the state of stress as found from

finite element analysis corresponds to the actual state of stress,

the prevailing loading being of the bending stress type.

Predictably stress concentrations tend to appear at the

re-entrant corners of the stair soffit. The concentration of the

stresses near the corners(12) is a well known fact and a slightly

increased depth of the section, by beveling the corners, is a

sound practice.

Stress distribution in the plane A-A and B-B for the central

zone of 'case A' stair (Fig. 5.15) are studied and normal stress

distribution at the section B-B (at the junction of tread and

riser) is shown -in Fig. 5.16. The normal stress distribution at

the section B-B is obtained by computing the normal stresses at

three points which lie on the plane. Computation of normal

125

stresses at more points for the plane B-B will give a smooth

stress distribution curve.

The normal stress distribution at the section A-A in shown in

Fig. 5.17.

In other zones of the stair the stress distribution type are

similar to that of central zone while the stress in the elements

near the ends doesnot generally differ from the state of stress of

a straight beam with same parameters and same supported

conditions.

126

FIG. 5.15: LONGITUDINAL SECTION OF STAIR AT CENTRAL ZONE

--GOE12

-5032

-4001W

-30E12

-2MM

."-----.--.. ja i-1.4.4-+4-4-4-1-4-1-4-4-1-4-4-1-1-44-4-4.1•44-44-1-1-44-1-4-4-1-4-1-4-1-4-1-4-1-1-4-0-4-44-4-4-4 4 ' 14 ' 141-4-4.-4-4-4-4-14-1-4 1-4-4-1-4-6-144-4-4-6-1-0-1-4-44-4-4-4-4-1-4-4-4-4-4-

A9 22 .04 26 as AO .12 14 19 .1B 20 .....

MO IE-

21 I E'

amm ... i

4M2 : DISTRNCE Cm.) ( - ) TENSILE .

mum L (+) COMPRESSIVE E

MM2 L

FIG. 5.18 : NORMAL STRESS DISTRIBUTION DIAGRAM AT SECTION B-8

127

CE

AT SE

CT

ION

CE CZ

CE

DIS

TR

IBU

TION

13

.cr

DIST

ANCE

(m.)

L1111.11 lialmitittiluilimtlimiltlilmt1111111t111111W.m.w.trl.nintudmi111.1ILI.Hilititilmuirl.r.d

Q m © a cs? o d d d d o d a a a Q7 tn

a

Nr- a NI fll q i : I I 1 7

csi cv 1 co 0

SSA 1I -um 129

CHAPTER 6

SUMMARY AND CONCLUSIONS

6.1 SUMMARY OF RESULTS

The results obtained from the Conventional Analysis and

Finite Element Analysis of Slabless tread riser stairs are

summarised in this section. A Comparative study is also presented

in the tabular form.

6.1.1 Conventional Analysis

The Conventional Analysis Method results of the stairs,having

both end fixed and with intermediate supports, may be summarised

as given below for the parameters as assumed in section 4.5.1.

Fixed end moment.at both ends 2.490 kN-m (sagging)

Bending moment at the intermediate = 6.709 kN-m (hogging)

supports

Bending moment at the centre of = 3.259 kN-m (sagging)

span

Reaction at both ends = 6.073 kN (downward)

Reaction at the intermediate supports= 27.977 kN (upward)

For the same parameters the. Conventional Analysis Method results

of the stairs, having both ends fixed and without intermediate

support, may be summarised as

Fixed end moment at both ends = 21.692 kN-m (hogging)

Bending moment at support *locations= 4.483 kN-m (hogging)

Bending moment at the centre of span= 5.485 kN-m (sagging)

Reaction at both ends = 21.904 kN (upward)

The Modified Conventional Analysis Method results of the stairs

131

having both ends hinged and without intermediate supports for the

parameters as assumed in section 4.5.3 may be summarised as

Bending moment at the centre of span= 19.779 kN-m (sagging)

Reaction at upper end = 22.295 kN (upward)

Reaction at lower end 10.027 kN (upward)

Horizontal reaction at both ends = 12.881 kN (outward)

6.1.2 Finite Element Analysis

The results of Finite Element Analysis may be summarised by

categorising it.Into four parts as:-

(a) Stairs with both ends fixed

For these stairs, without intermediate supports (case A) and

parameters as assumed in section 5.2.1 the fixed end moments are

-22.016 kN-m and -17.020 kN-m for upper end and lower end

respectively. For the same parameters but with intermediate

supports (case A ) these fixed end moments reduce to 2.024 kN-m

and 1.577M-In for upper end and lower end respectively. The

vertical reactions which are 25.68 kN and 18.13 kN at respective

end for case A become -4.93 kN and -3.42 kN. The supports mainly

carry the load in this case by providing reactions equal to 34.38

kN and 17.78 kN at the upper support and lower support

respectively. The maximum sagging moment which is 9.22 kN-m at the

centre of span for case A reduces to 4.21 kN-m for case A'. For

case A' the bending moment at the upper intermediate support is

maximum and it is equal to -5.638 kN-m. The horizontal axial

tension increases from 5.99 kN for case A to 10.08 kN for case A .

The maximum vertical and horizontal displacement, which are

132

-2.00 mm and -0.103 mm respectively for case A, become -0.390 mm

and -0.256 mm correspondingly for case A'.

(b) Stairs with both ends hinged

For these stairs, without intermediate supports (case B) and

the parameters as assumed in section 5.2.1 the vertical reactions

are -20.90 kN and -11.42 kN at upper end and lower end

respectively reduce to -4.75 kN and -3.65 kN correspondingly for

case B'. The horizontal axial tension reduces from 13.75 kN for

case B to 7.89 kN for case 8'. The maximum sagging moment also

reduces from 16.00 kN-m for case B to 3.10 kN-m for case B'.

Maximum bending moment for case B' occures at upper intermediate

support and it is equal to -3.92 kN-m.

The maximum vertical and horizontal displacement of stair,

which are -3.400 mm and -0.765 mm respectively for case B reduces

to -0.212 mm and -0..105 mm correspondingly for case B'.

(c) Stairs with one end fixed and other hinged

For these stairs without intermediate supports (case C), with

the parameters same as assumed in sections 5.2.1 the fixed end

moment is computed as -55.387 kN-m. This moment reduces to 3.697

kN-m for case C' with same parameters.

The vertical reactions, mainly provided by upper support, are

54.93 kN and 3.57 kN at upper end and lower end respectively for

case C reduce to -7.06 kN and -2.02 kN at respective end for case

C'. The vertical reactions in case C' are mainly provided by

supports and their values are 47.78 kN and 19.79 kN for upper

133

support and lower support respectively. The maximum sagging moment

at the centre of span is 18.40 kN-m for case C and 7.10 kN-m for

case C'. Maximum bending moment for case C' occures at upper

intermediate support and it is equal to -9.76 kN-m. The

horizontal reaction in outward direction considerably reduces from

42.96 kN to 16.47 kN for case C to case C'.

The maximum vertical and horizontal displacement which are

-7.14 mm and -2.26 mm respectively for case C reduces to -1.11 mm

and -0.694 mm correspondingly for case C'.

(d) Non-orthopolygonal stairs

For these stairs, having both ends fixed and without

intermediate supports, with the parameters as assumed in section

5.2.3 the fixed end moments are -15.649 kN-m and -13.123 kN-m at

upper end and lower end respectively. The maximum sagging moment

is 6.06 kN-m.

The vertical reactions at the upper end and lower end are

19.374 kN and 15.471 kN respectively. The horizontal axial

tension in the stair is 5.186 kN.

The maximum vertical and horizontal displacement for this

case are -4.61 mm and -1.10 mm.

6.1.3 Comparative Study of Results

The results obtained from two methods, namely Conventional

Analysis Method and FEM, are compared for case A (Table 6.1), case

A (Table 6.2) and case C (Table 6.3). Notations are as shown

in Fig. 4.2.

134

TABLE 6.1: COMPARATIVE STUDY FOR CASE(A)

S.No. Stress Resultants Conventional FEM Study Difference

Study Results Results

1. M2-1 (kN-m) -21.692

2. M1-2

=M1-1'(kN-m) -4.483

3. M111

=M1'-2'

(kN-m) -4.483

4. 2' -1' (kN-m) -21.692

5. Mc (kN-m) 5.485

6. R2 (kN-m) 21.904

7. R2 (kN-m) 21.904

8. H

(kN) 0.000

-22.016

-0.328

-2.761

-17.020

9.220

25.683

18.128

-5.994

-1.49

92.68

38.41

21.54

-68.09

-17.25

17.24

-100

Notes: (1) (-) indicates tension, hogging moment and downward reaction.

(ii) Positive difference indicates that the conventional study results are more in magnitude.

135

TABLE 6.2: COMPARATIVE STUDY FOR CASE (A').

S.No. Stress Resultants Conventional Study Results

FEM Study Results

Difference ( % )

1. M2-1 (kN-m) 2.490 2.024 18.71

2. M1-2 =M1-1 4(kN-m) -6.709 -5.638 15.96

3. M1'1 =M "

1-2(kN-m) -6.709 -4.755 29.13

4. pf

2- 4

1 4 (kN-m) 2.490 1.577 36.67

5. Mc (kN-m) 3.259 4.210 -29.18

6. R2 (kN) -6.073 -4.928 18.85

7. RI (kN) 27.977 34.381 -22.89

8. R1' (kN) 27.977 17.781 36.44

9. R2

(kN) - 6.073 - 3.424 43.62

10. H (kN) 0.000 kN -10.079 -100

Notes: (i) (-) indicates tension, hogging moment and downward reaction.

(ii) Positive difference indicates that the conventional study results are more in magnitude.

137

TABLE 6.3: COMPARATIVE STUDY FOR CASE (B)

S. No. Stress Conventional FEM Study Difference

Resultants Study Results Results ( % )

1. Mc 19.779

16.000

19.10

2. 2 22.295

20.904

9.87

3. R2 10.027

11.418 -13.87

4. H -12.881 -13.748 -6.73

Notes: (i) (-) indicates tension, hogging moment and downward reaction.

(ii) Positive difference indicates that the conventional study results are more in magnitude.

139

6.2 CONCLUSIONS

1. The analysis of Slabless tread riser stairs has been

developed independantly by Finite Element Analysis and the

results have been compared with that of Conventional

Analysis. It is found that the results differ appreciably.

The following differences have been noticed :-

(a) The important point which came into picture by Finite Element

Analysis is the horizontal displacement. In fact the stairs

deform like a catenary having both horizontal and vertical

displacements. The effect of horizontal displacement is being

ignored in Conventional Analysis Methods. The horizontal

displacement between ends of risers differentiate it's

behaviour from the behaviour of a straight line structure

(i.e. beam).

(b) Conventional Analysis Procedures completely ignore the axial

stress developed in the stairs. These axial thrust phenomenon

is explicity highlighted by Finite Element Studies.

(c) The reactions at the ends are unequal being higher at the

upper end. These differences are due to difference in level

of supports. Similarly, fixed end moment at upper end is

more than the fixed end moment at lower end. Both the fixed

moments are less than those obtained from Conventional

Analysis.

2.

Slabless tread riser stair is a folded plated/shell like

structure and it should be analysed by keeping this fact into

consideration.

3. Present Conventional Ana lytical Procedures ignore the

141

inclined portion of stairs. In fact the membrane action of

this portion causes tension.

4.

Finite Element Method is more suitable method since it may

analyse unsymmetric stairs, Unsymmetrically loaded stairs

while Conventional Analysis Methods are not capable of doing

so without a cumbersome and time consuming process. The axial

load effect may also be considered in FEM enhancing its

importance for the analysis of Slabless tread rise stairs

than any conventional analysis method.

6.3 Future Scope for Study

Besides the studies conducted in this dissertation few other

studies can be carried out on the Slabless tread riser stairs.

These include :-

(I) Stress concentration study at the conner of steps and

reduction of stress concentration coefficient after beveling

the corners.

(ii) A detailed study of Non-orthopolygonal stairs (i.e. stairs

with inclined risers) with different parameters and

different supported conditions.

(iii)This study may be extended to the study of spiral Slabless

tread riser stairs by using general shell elements.

142

REFERENCES

1. Benjamin, B.S., 'Analysis of Slabless Tread Riser Stairs',

Indian Concrete Journal, Dec. 1962, pp. 443-447.

2. Benjamin, B.S., 'Discussion on Slabless Tread Riser Stairs,'

Journal of American Concrete Institute, June 1962, Part 2,

Proc. V.59, No.6, pp. 837-864.

3. 'Code of practice for structural safety of building

IS:875-2964,' Indian Standard Institution, New Delhi.

4. Deschapelles, B., 'Analisis de los Momentos de Empotramiento

en una Escalera de eja ortopoligonal', Revista de la Sociedad

Cubana de Ingenieros (Havana), Mar. 1958, pp. 173-177.

5. Dianu, V., Istrate, M. and Momanu, Gh., 'Orthopolygonal

Stairs', Journal of International Civil Engineering, July

1971, Vol.II, No.1, pp. 8-16.

6. FEICOM-85, 'Proceedings of the International Conference held

at the Indian Institute of Bombay, India, 2-6 December, 1985.

Vol.1,Finite Element in Computational Mechanics Edited by

Tarun Kant,pp.93-101.

7. Hinton, E. and Owen, D.R.J., 'Finite Element Programming',

Academic Press (1977).

8. Irons, B. and Ahmad, S., 'Techniques of Finite Elements',

Ellis Horwood Limited, Chichester, England, 1986.

9. Mittal, S., 'Adaptive Finite Element Analysis,' M.E. Thesis,

University of Roorkee, Roorkee, 1983.

143

10. Nayak, .G.C., 'Concept of Generalised Modulus and Generalised

Stresses in Finite Element Analysis', International

Conference on Finite Element in Structural Mechanics,

Coimbatore, India 1974.

11. Reference Manual, 'HP FORTRAN77', Hewlett-Packard Company,

Cupertino 1987.

12. Sachnowski, K.W., 'Stahlbetonkonstruktionen', VEB Verlag

Technik (Berlin), 1956, p. 424.

13. Saenz," L.P., 'Calculo de Estructururas Planas de eje

ortopoligonal', College of Engineering,. Havana University,

1952, Published by Compania Editora de libros y Folletos.

14. Saenz, L.P. and Martin, I., 'Slabless Tread Riser Stairs',

Journal of American Concrete Institute, October, 1961, Proc.

V.58, No.4, pp. 353-366.

15. 'Starbase Graphic Techniques for HP 9000 Computers', Hewlett

Packard Company, Colorado, USA, 1986.

16. 'Starbase Reference Manual', Hewlett-Packard Company,

Colorado, USA, 1985.

17. Turner, M.J., Clough, R.W., Martin, H.C. and Topp, L.J.

'Stiffness and Deflection Analysis of Complex Structures',

Journal of Aeronautical Sciences, 23 (1956) pp. 805-824.

18. Zienkiewicz, 0.C., 'The Finite Element Method', Third

edition, Tata McGraw Hill, New Delhi, 1986.

110

APPENDIX - A

REINFORCEMENT DETAILS

The reinforcement of this type of stairs are interesting and

typical one. Although the bending moments vary in magnitude along

the length of the stairways, yet in practice it is decided that

it is more economical to reinforce all the steps in the same way.

Stairs of this type may be reinforced in two ways(5)

1. With continuous reinforcement bars and stirrups (Fig. A.1)

2. With continuous reinforcement bars and inclined additonal

bars (Fig. A.2)

Reinforcement of the first type is less economical than the

second. Inclined reinforcement bars carry the load at entrant

corners on the lower face of the stairs and aid in taking possible

loads from local three dimensional effects.

If the second method of reinforcement is to be used, the

continuous bars should be dimensioned so as to carry the loads

resulting from the calculated bending moment. While for the

second case of reinforcement it is necessary to provide for a

cross section equal to that to the continuous reinforcement.

Some authorities suggest provision, for engineering purposes

of a low reinforcement percentage on the compressed surface of the

particular structural element (Fig. A.3) through mounting

continuous reinforcement bars. Another type of reinforcement is

provided by forming rectangular frames which are joined to each

14.5

other by other bars and which also serve as temperture

reinforcement.

In order to minimize stress concentrations it is advisable

to round or to chamfer the entrant corner. The introduction of

haunches make it possible to achieve the most efficient and

convenient system for reinforcement with straight bars for taking

up tensile stresses ( Fig. A.4).

146

o

o

1

0

FIG. A.1: REINFORCEMENT WITH BARS AND STIRRUPS

0

FIG.A.2: REINFORCEMENT WITH BARS AND INCLINED BARS

147

FIG. A.3: ALTERNATIVE REINFORCEMENT

FIG. A.4: CHAMFERED ENTRANT CORNERS

149