cantilever staircases.pdf

26 The Structural Engineer 87 (8) 21 April 2009

Synopsis

Stone cantilever stairs are constructed in two basic forms, eitherwith plain or rebated treads. The body of work available isgenerally confined to the examination of plain treads, yet themajority of stairs in existence are of the rebated type. This paperoutlines guidelines for inspection of cantilever stairs and a methodof analysis that predicts the behaviour of a flight of rebated treadsunder a variety of loading conditions. The theoretical analysis andmodel testing provide formulae for calculation of torsion, verticaland horizontal reactions, tread rotation, displacements and shearstresses.

Historical background

The first description of cantilever stairs1 appears in AndreaPalladio’s I Quattro Libri dell’Architettura and refers to his staircasein the Academia in Venice, which was built circa 1560. InigoJones, who designed the Tulip Stair in the Queen’s House inGreenwich in 1630, is thought to have introduced them toEngland. Wren and Hooke followed in 1671 with their staircase inthe Monument to the Great Fire of London and from the early1700s onwards this type of construction was used as the standardform of stair construction in most good town houses and largecountry houses for almost 200 years. Examples of recentlyrestored cantilever staircases can be seen at City Hall2 and theArchitectural Archive in Dublin (Fig 1), City Hall (Fig 2) being aparticularly fine example. From the early part of the 20th century,

cantilever stairs fell out of favour, to the point where by the 1960sand ’70s there was no longer a clear understanding of theirstructural system. Unfortunately, when refurbishments were takingplace, these stairs were sometimes removed or provided withunnecessary additional support instead of adequate repairs,detracting from their natural elegance.

Cantilever stair structural form

The term ‘cantilever stairs’ refers to a form of stair constructionwhereby each tread is vertically supported along its front edge bythe tread below and also by the wall into which it is built on oneend. The wall also provides torsional restraint. The treads do notcantilever, but receive support along their length from the treadbelow.

The majority of cantilever stairs found in existence today aremore than 100 years old and of solid stone. Various types of stonehave been used, but Portland stone and granite are by far themost common. Reinforced concrete treads may also beencountered, although rarely. Surprisingly, one of our recentexperiences was eight sets of two-storey RC cantilever stairs in acivic building which had been installed during a refurbishment inthe 1920s after the building had been extensively damaged duringthe Civil War. Irrespective of the material, the theoretical behaviourof the stairs is the same, in terms of load transfer down tofoundation level. A portion of the load on a tread is transferred byits front edge to the tread below, with the remainder transferred tothe wall socket. Due to the deflected shape, the load transferred tothe tread below is concentrated near the free end of that tread.The reactions from the treads above and below tend to causerotation, which is resisted by the torsional restraint provided by thewall.

A refinement to this basic system, which is found on mostcantilever stairs is the provision of a rebate in the bottom edge ofthe riser, or less often, a bead protruding from the bottom of the

Paper

Stone cantilever stairs – inspection and analysis ofcantilever stairs

Philip Little, BEng Barrett Mahony Consulting Engineers

Michael Hough, BSc (Eng), Dip Struct Eng, MSc, CEng, MIEI, MIStructEBarrett Mahony Consulting Engineers

Edward Mullarkey, BE, MSc, MIEILecturer in Structural Engineering at Dublin Institute of Technology

Keywords: Stairs, Stone, Cantilever construction, Analysis, Appraising

Received: 08/08: Modified: 11/08; Accepted: 01/09

© Philip Little, Michael Hough, & Edward Mullarkey

1 Cantilever staircase in the Architectural Archive, Dublin. The first four treads and the middle tread in the upper flight were replaced and the stairs and landings subsequently load tested (Courtesy Architectural Archive)

2 Cantilever staircase in City Hall, Dublin. The flights were extensively refurbished and load tested during 1999-2001. The lighter colour of replacement treads can be seen above and below the half-landing (Courtesy Dublin City Council)

1 2

SE8 paper cantilever stone stairs inspect:Layout 3 9/4/09 12:25 Page 26

The Structural Engineer 87 (8) 21 April 2009 27

tread into a groove in the tread below. Basement flights,frequented by household staff in past times, are sometimesconstructed with plain rectangular treads, and may even use roughdressed stone. From ground floor upwards, the stairs are usually ofrebated construction, with dressed stone, often with finely shapedogee or spandrel (roughly triangular) profiles. Above first floor, thespan of the treads is sometimes reduced. This may have been tosave on construction costs, as the impression of grandeur wouldhave already been created by a wide flight at ground floor andcontinuing this on the upper, less visible, flights would beunnecessary.

With rebated cantilever stairs, as the tread rotates under load,the horizontal reaction from the rebate applies an additional torqueto the step, opposing the rotation. This reduces the overall torsionand shear stresses in the tread; the maximum shear stress in auniformly loaded flight of rebated treads is slightly less than halfthat found in a flight of plain treads. Often, some of this extra shearcapacity is sacrificed in favour of a tread with a more slenderprofile. On flights made from Portland stone, an ogee cross sectionis most common, whereas granite steps tend to be of a spandrelshape. Indeed, it may be the case that a slender profile came firstfor aesthetic reasons and the masons found that a rebate wasnecessary to cope with the increase in stress.

Theoretical model

When formulating the theory for cantilever stairs, certainassumptions and simplifications are made. These assumptionsinclude:– The bottom tread is supported over its entire base on a rigid

foundation.– Landings at the bottom of a flight are rigid and do not deflect

vertically.– The supporting wall provides only in-plane vertical, horizontal

and torsional restraint. – The supporting wall does not provide any restraint against

bending about either the vertical or horizontal in-plane axes.– The contact surface between treads, and between each tread

and the supporting wall can transfer forces in compression only.– The vertical reaction provided by the supporting wall acts

through the centreline of the tread.– The resultant of the dead and imposed loads acts through the

centreline of the tread.– The lines of action of the vertical reactions from treads above

and below are equidistant from the supporting wall throughoutthe flight.

In respect of rebated treads it is further assumed that:– The lines of action of the horizontal reactions from treads above

and below are equidistant from the supporting wall throughoutthe flight and no horizontal forces are transferred to the wall.

– Landings at either end of a flight are fixed and do not deflecthorizontally. Any resulting horizontal loads are resisted by thelandings.

– The vertical contact surface at the rebate can transfer forces incompression only.

It must be stressed that these assumptions are not alwaysborne out in practice and a thorough understanding of the effect ofany deviations is necessary in order to make a judgment of thestructural integrity of a particular flight. A good inspection willprovide the necessary information on which to make a soundappraisal and guidelines are provided later.

Plain treads

Plain treads are statically determinate and it is therefore possible tocalculate the vertical and torsional reactions from statics alone. Arecent article by Sam Price4 proposed an analysis for this type ofstair, and a similar approach is adopted here, but in addition loadsapplied to individual treads are also considered. The analysis issimplified by the assumption that vertical reactions from treadsabove and below act at a constant distance cL (taken as 0.85L)from the face of the support wall. The general equations for plainand rebated treads are the same with the exception of the torsioncomponent, which in the case of rebated treads includes anadditional term representing the horizontal reaction. As the proofsfor each set of equations are identical, equations 1-9 for plaintreads are merely stated together with force diagrams, whereasequations 11-20 for rebated treads are explained usingsummarised proofs.

The top tread of a flight of plain treads is shown in Figs 3 and 4.The centrally applied load P1 and self weight W give rise to verticalreactions RW1 and R1, and torsional reaction T1.

By standard statics

(1)

(2)

(3)

The next tread in the flight, Fig 5 is in contact with treads aboveand below. As the reaction from the tread below is a function ofthe cumulative self weight and applied loads at that level, it followsthat the loads experienced by each step, and consequently thetorsional reaction, increase down the flight. The maximum torquewill be found in the step immediately above the bottom step.

( )(1- / )R P W c1 2W1 1= +

[( ) ]/4T P W b c1 1= +

( )/2R P W c1 1= +

3 Applied forces and reactions in the first tread of a plain flight (M. Hough)

4 Applied forces, reactions and dimensions in the first tread of a plain flight (M. Hough)

5 Applied forces, reactions and dimensions in the second tread of a plain flight (M. Hough)

3 4

5

SE8 paper cantilever stone stairs inspect:Layout 3 9/4/09 12:25 7


Again from statics:

(4)

(5)

(6)

General equations

Rn: Resultant vertical reaction from below tread ‘n’.

(7)

RWn: Resultant vertical reaction at wall below tread ‘n’.

(8)

Tn: Torsional reaction at wall, tread ‘n’.

(9)

Examination of these equations reveals a range of possibilitiesfor load transfer, depending on the assumed or chosen value ofcL, the distance from the face of the supporting wall to Ri, theresultant of the reaction from the step below step ‘i’.

Possible values of c lie in the range 0.5 to 1.0 (a value of <0.5implies uplift at the supporting wall socket), with 0.5 correspondingto all load transferred to the tread below and 1.0 corresponding tothe load being transferred equally between the tread below and thewall socket. Results from physical testing of the model describedlater in this paper together with subsequent full scale load testssuggest that the resultant reaction acts approximately at a point0.85L from the supporting wall. This location may vary depending onthe stiffness of the treads and, in particular, on the condition of thegrouting. If the grouting near the free end of the tread is poor ormissing, this will decrease the value of the constant c and increasethe proportion of load transferred down the flight.

Rebated treads

Direct comparison between plain and rebated treads is not usuallypossible due to the differing geometry of the two forms ofconstruction. The rise on a flight of rebated treads is less than thatof a flight of plain treads of the same cross sectional dimensions. If

(1 / )R P W c1 2Wn n= + -] g

/2R P nW cn ii

n

1= +

=

= G/

( )(1- / )R P W c1 2W2 2= +

[(2 3 ) ]/4T P P W b c2 1 2= + +

( 2 )/2R P P W c2 1 2= + +

/ / /T P n W c P W c b1 2 2n ii

n

n1

1

#+ - + +=

-

]e ] ^g o g h> H/

the rise is preserved, a deeper tread is required to take account ofany rebate. Furthermore, as the equations used to calculate angleof twist and shear stress in a rectangular solid section aredependent on the ratio of breadth/depth, any variation in treaddepth will preclude exact comparison. These differences are smallhowever, generally less than 5%.

Figures 6 and 7 show the applied forces and reactions for thetop tread of a rebated flight. The distance of the lines of action ofHA1 and HB1 from the supporting wall are not required as theseforces only cause rotation about the centreline of the step. Itshould be noted here that the horizontal forces tend to inducetorsion of the opposite sense to that produced by the verticalreactions.

By similar approaches to those used for plain tread the followingmay be obtained

(10)

(11)

(12)

(13)

Now considering the next tread in the flight, as shown in Fig 8:

(14)

(15)

(16)

(17)

General equations

Rn: Resultant vertical reaction from below tread ‘n’.

(18)

RWn: Resultant vertical reaction at wall below tread ‘n’.

(19)

Tn: Torsional reaction at wall, tread ‘n’.

(20)

/ / /T P n W c P W c b Hd1 2 2n ii

n

n1

1

#= + - + + -=

-

]e ] ^g o g h> H/

/R P W c1 1 2Wn n= + -] ^g h

/R P nW c2n ii

n

1= +

=

= G/

/4T P P W b c Hd2 32 1 2= + + -] g6 @

/R P W c1 1 2W2 2= + -] ^g h

/R P W c1 1 2W1 1= + -] ^g h

H H H2 1= =

/R P P W c2 22 1 2= + +] g

/T P W b c H d41 1 1= + -] ]g g6 @

H H HA B1 1 1= =

//R P W c21 1= +] g

6 Applied forces and reactions in the first tread of a rebated flight (M. Hough)

7 Applied forces, reactions and dimensions in the first tread of a rebated flight (M. Hough)

8 Applied forces and reactions in the secondtread of a rebated flight (M. Hough)

6

8

7



Examination of the general equations for rebated treads revealsthat the core of each expression is identical to that for plain treads.In fact, only the formula for torsion differs, with the inclusion of theadditional term Hd, which has the opposite sense to the otherterms within equation 20. Figure 9 illustrates how the maximumtorsion in the flight is reduced, as the Torsional Moment Diagramhas been shifted down by an amount equal to Hd.

Compatibility of displacements

Cantilever stairs with rebated treads are statically indeterminate. Toenable a value to be placed on the horizontal forces, the keyconcept is compatibility of deflections. A short flight of stairs withfour treads is used to illustrate this. The analysis assumes that onlysmall angles of rotation occur.

Plain treads, vertical displacement

δ(v)i,j Absolute vertical displacement of corner j of tread i,displacement ↓ ‘+ve’.

Δ(v)i,j Vertical displacement of corner j of tread i, relative tocorner 1 of tread i, displacement ↓ ‘+ve’.

ϕi Rotation of tread i, clockwise rotation is ‘+ ve’.b Breadth of treadd Depth of tread

Considering the rotation ϕi of any tread i, as shown in Fig 10:and Δ(v)i,3 , the vertical displacement of corner 3 of tread i, relativeto corner 1 of tread i, as shown in Fig 11

(21)

The maximum vertical displacement occurs at corner 3 of step 1,as shown in Fig 12 and is denoted here by δ(v)1,3, the absolutevertical displacement of corner 3 of step 1. This is the cumulativedeflection within the flight, ignoring the bottom tread which is fullysupported by the floor and does not rotate.

(22)

Or, for the general case of a flight of non-rebated stairs with ‘n’treads

(23)

This equation is composed of the terms d, b,ϕi . Torsional theorygives us that:

, ,v v ii

n

1 3 31

1

d D==

-

] ]g g/

, , , , ,v v v v v1 3 1 3 2 3 3 3 4 3d D D D D= + + +] ] ] ] ]g g g g g

cosd d i b i,v i 3` { {D = - +] g

(24)

whereTi: Torsion in member under considerationL : Length of member. As the torsion is not constant along the

length of the tread, it is necessary here to use Leff = cL, thedistance from the supporting wall to the point of action of theresultant reaction from the tread below.

G : Shear modulus of elasticityJ : Torsional constant

Of these terms, Ti can be calculated from the general equationspreviously outlined, L may be measured directly and G – the shearmodulus of elasticity – is a material property, which is defined by:

(25)

In this paper, the values of Poisson’s ratio ν and Young’smodulus E are taken as 0.2 and 53 000N/mm2 respectively, whichare typical values for Portland stone. The remaining component, J– the Torsional constant – is a geometric property which for solidrectangular sections is given by:

(26)

The torsional coefficient kt may be found in a number oftexts5,6,7,8 on the subjects of Advanced Mechanics of Materials andis dependant on the ratio of breadth to height of section.

Using these co-efficients, ϕi may be calculated and hence thevertical displacements of each tread. The calculated deflectionscan be compared to actual results during a load test, with anupper limit set on deflection to avoid overstressing the flight.

Rebated treads, vertical displacement

To satisfy compatibility of displacements, as there is no horizontaldisplacement at either end of the flight, the sum of theanticlockwise rotations must be of the same magnitude as thesum of the clockwise rotations. Consequently, the algebraic sum ofthe tread displacements in either the horizontal or vertical directionis zero. As rotation is directly proportionate to torsion, it followsthat for a symmetrically loaded flight of treads as shown in Fig 13:

By examining the general equations for torsion in plain and

and, , , ,v v v v1 3 4 3 3 2 33D D D D=- =-] ] ] ]g g g g

J k bdt3=

/G E v2 1 += ] g

/T L GJi i{ =

9 Comparison of torsional moment diagrams for plain and rebated treads

10 Numbering of corners and rotation ϕi of tread ‘i’11 Vertical displacement of the corners of tread ‘i’12 Maximum vertical displacement, Δ(v)1,3 within a

flight of plain treads

9 10

11 12



rebated flights (eqns 9 and 20), it can be seen that to satisfycompatibility of displacements, the term Hd must be equal to theaverage torsion (rather than half the maximum torque) within aflight of plain treads of the same dimensions and mechanicalproperties subjected to the same loading.

(27)

This can best be visualised by considering a flight of treads withnon-uniform loading. If we were to simply apply an opposingtorque equal to half the value of the maximum torque then thiswould have a disproportionate effect as it would not take intoaccount treads with low torque and consequently small rotations.

Shear stresses

Equations for calculation of torsional shear stresses can be foundin texts previously referred to and the maximum shear stress isgiven by:

(28)

The maximum shear stresses in a rebated flight will beapproximately half that of a flight of plain treads. For initial analysis,typical values may be used to establish a factor of safety on thestrength of stone, based on 7N/mm2 and 10N/mm2 for Portlandstone and granite, respectively. This analysis can then be refinedon the basis of material tests conducted on stone samples takenfrom any broken treads or new stone used for repairs.

Model testing

To assess the accuracy of the mathematical model proposedabove, physical testing was carried out on a model constructedusing glulam timber treads. While the testing undertaken wasinsufficient to provide a statistically significant result, the

/T k bdmax s2x =

Hd Tave=

experimental results compared closely in both trend and value tothose obtained from the theoretical model, with results typicallymatching within 5-10%.

Individual treads were also modelled in 3D using Finite ElementAnalysis. Rectangular, ogee and spandrel treads were modelledand the results are given in Tables 1, 2 and 3. The values given arethe shear stresses due to an applied torque of 1kNm and allow acomparison to be made between rectangular treads and ogee orspandrel treads.

Sample calculation

This illustrative example is based on rectangular stair treads ofbreadth 290mm and height 180mm, extending 1200mm from theface of the wall, constructed from Portland stone, with 12 treads inthe flight. The bottom tread is fully supported by the ground floor.The assumed imposed loading is 4kN/m2, which is converted toan equivalent load of 1.46kN per tread, and the self weight of eachtread is calculated as1.38kN. The analysis follows on from thegeneral formulae already outlined.

Dimensions and propertiesb = 290mm b/d = 1.61d = 180mm ks = 0.234 (Table 4)L = 1200mm kt = 0.203 (Table 5)E = 53000 N/mm2 ν = 0.2G = 22083 N/mm2 (eqn 25) J = 3.43 ×108 mm4 (eqn 26)

Plain treads

The maximum vertical reaction within the flight occurs below tread11 (using eqn. 7):

. .

.

/1.7R R

R kN

16 06 11 1 38

18 38

max

max

11 #= +

=

]] g g

13 Deflected shape of a flight of rebated treads 13

Rectangular profile tread, torsion 1kNm

b

290 300 310 320 330

d

140 0.71 0.69 0.66 0.63 0.61150 0.63 0.61 0.58 0.56 0.54160 0.56 0.54 0.52 0.50 0.48170 0.51 0.49 0.47 0.45 0.43

180 0.46 0.44 0.42 0.41 0.39190 0.42 0.40 0.38 0.37 0.35

200 0.38 0.36 0.35 0.34 0.32

Table 1 Shear stress co-efficients for rectangular treads

Ogee profile tread, torsion 1 kNm

b

290 300 310 320 330

d

140 1.35 1.30 1.26 1.21 1.17

150 1.23 1.18 1.14 1.10 1.06160 1.14 1.09 1.05 1.01 0.97

170 1.05 1.01 0.97 0.93 0.90180 0.98 0.94 0.90 0.87 0.83

190 0.91 0.87 0.84 0.80 0.77200 0.85 0.82 0.78 0.75 0.72

Table 2 Shear stress co-efficients for ogee treads

Deep spandrel profile tread, torsion 1 kNm

b

290 300 310 320 330

d

140 1.12 1.09 1.07 1.04 1.01150 0.95 0.93 0.91 0.88 0.85160 0.84 0.81 0.78 0.76 0.73170 0.74 0.72 0.69 0.67 0.65180 0.66 0.64 0.62 0.60 0.58190 0.60 0.58 0.56 0.54 0.52200 0.55 0.53 0.52 0.50 0.48

Table 3 Shear stress co-efficients for deep spandrel treads



The vertical reaction at the wall at each tread (using eqn. 8):

The maximum torsion within the flight occurs at tread 11 (usingeqn. 9):

The maximum rotation occurs at tread 11 (using eqn. 24):

The maximum vertical deflection occurs at the top of the flight andis calculated by summing the deflection of all treads (using eqn. 23):

(at top of the flight)

The maximum shear stress is found at tread 11 (using eqn. 28):

Rebated treads

The results for plain treads can now be used to calculate thevalues for rebated treads. The maximum vertical reaction within theflight will occur below treads 1 and 11 at the top and bottom of theflight while the vertical reaction at the wall at each tread remainsconstant.

, as for plain treads

, as for plain treads

In order to find the value of the horizontal reaction throughout theflight it is first necessary to calculate all torsions within the plaintreads and average the result:

The value of the horizontal reaction throughout the flight can nowbe calculated:

. / . .H kN2 66 0 18 14 78= =

.T kNm Hd2 66ave = =

.R kN1 17W =

.R kN18 38max =

. . . / .

. radians

5 09 0 85 1 2 22083 10 3 43 10

0 0007

max

max

11

3 4# # # # #

{ {

{

= =

=

-] g

5.09 10 / 0.234 290 180

. /N mm2 32

max

max

116 2

2

# # #x x

x

= =

=

^ h

. mm1 14,max 1 3d d= =

.T kNm5 09max =

. . / . . . / . .

T T

14 6 10 1 38 0 85 1 46 1 38 1 7 0 145

max 11

# #

= =

+ + +]] ]g g g6 @

. .

.

/ .R

R kN

1 46 1 38

1 17

1 1 1 7W

W

= +

=

-] ^g h

The maximum torsion within the flight occurs at treads 1 and 11and is slightly less than half that found in a flight of plain treads(using eqn. 20):

At tread 11 (or tread 1), the rotation and deflection are:

The maximumvertical deflection of any individual tread in the flight occurs at tread11 (or tread 1)

The maximum vertical deflection of the flight occurs at the midpoint of the flight and is the sum of all individual deflections up tothat point:

The maximum shear stress is found at tread 11 (or tread 1):

In this example, it is clearly shown that the shear stresses due totorsion in a flight of rebated treads are slightly less than 50% ofthose encountered in a flight of plain treads, while the maximumdeflection is reduced by 75%. The results show a factor of safetyon the strength of stone of 3.0 and 6.4 for plain and rebated treads,respectively. This assumes a shear strength for Portland stone of7N/mm2. The most highly stressed tread in a plain flight is the lastbut one, and maximum deflection occurs at the top of the flight. Ina flight of rebated treads, maximum shear stresses are found in theuppermost tread as well as in the last tread but one, with maximumdeflection at the mid-point of the flight. This example is quiterepresentative in terms of dimensions for cantilever stairs but inpractice it is rare to find rectangular section treads. The followingsection outlines common profiles and how the analysis can berefined to take account of any given cross-section.

Tread cross section

An example of a rectangular tread is shown in Fig 14. Morecommonly, Portland stone treads are of ogee shape while granitetreads are often roughly triangular (spandrel treads) with either a

.

. . . / .

or

radi nsa0 0003

2 43 0 85 1 2 22083 10 3 43 10

max

max

11 1

3 4# # # # #

{ { {

{

=

=

=-

^

] ^

h

g h

( ) 5.09 5.09 2.66

.

T T orT Hd

T kNm2 43

max

max

11 1= = - = -

=

.

cos sinor

mm

180 180 290

0 09

, ,

,

11 3 1 3 11 11

11 3

{ {D D

D

= -

=

+^ ^ ^h h h

( ) . / .

. /

or

N mm

2 43 10 0 234 290 180

1 11

max

max

11 16 2

2

# # #x x x

x

= =

=

^ h

. mm0 28,max 6 3d d= =

Torsional co-efficients

b/d 1.0 1.2 1.5 2.0 2.5 3.0 4.0 6.0 10.0 ∞

kt 0.141 0.166 0.196 0.229 0.249 0.263 0.281 0.299 0.312 0.333

Shear co-efficients

b/d 1.0 1.2 1.5 2.0 2.5 3.0 4.0 6.0 10.0 ∞

ks 0.208 0.219 0.231 0.246 0.258 0.267 0.282 0.291 0.312 0.333

Table 5 Torsional co-efficients for solid rectangular sections

Table 4 Shear co-efficients for solid rectangular sections

14 15

14 Portland stone tread with rectangular profile (Courtesy BMCE)

15 Portland stone ogee tread (Courtesy BMCE)16 Granite spandrel tread with swept soffit

(Courtesy BMCE)

16



swept or slightly stepped soffit. Examples of these profiles areshown in Figs 15, 16 and 17. Similarly, various styles of rebate canbe found ranging from a straightforward square cut throughsplayed rebates – necessary with a swept soffit to avoid damageduring construction – and on to the less common hidden rebate,as shown in Fig 18. With the latter form, a stone bead projectsdown from the underside of the front edge of the tread andengages a groove cut in the top surface of the tread below.

Using Tables 1 to 3, it is possible to calculate the increase instress for ogee and spandrel profiles. In the sample calculation themaximum stress for the rebated rectangular tread was1.11N/mm2. Now assuming the flight has ogee treads, Tables 1and 2 allow a comparison of the stresses. The reduced self weightof the tread due to the altered cross section should also be takeninto account, and this will reduce the maximum shear stress from1.11 to 0.94N/mm2. From Tables 1 and 2, 0.98/046 = 2.13 (stressmultiplier for ogee tread b = 290, h =180). The maximum shearstress now is 2.13 × 0.94 = 2.00N/mm2

For Portland stone, this will give a factor of safety on thestrength of stone of 3.5. Typically, when analysis shows a factor ofsafety of 3 or above, a load test is not necessary, providedinspection confirms that the stairs are sound.

Inspection

When inspecting cantilever stairs, it is necessary to assess theirphysical condition plus any deviations from the theoretical model.The following is a list of points that should be considered duringinspection.– The safety of stairs in service is a priority; the stairs may be

temporarily propped or closed until such time as repairs areeffected.

– The stairs should be cleaned prior to inspection. The advice ofspecialist stone cleaners should be sought at the time ofcleaning since stone defects are easier to spot when the stoneis wet and drying.

– Survey noting if treads are rebated, span dimensions, going,rise, step profile, the arrangement and number of slabs atlandings and the number of treads in each flight.

– Determine material type. Portland stone, marble, granite,reinforced concrete etc.

– Inspect each tread to assess the stone condition. Portland stonesteps may show signs of torsional cracking close to the supportwall, particularly at the top or bottom of a flight. Often, crackswill occur close to the plaster line so ideally, skirtings should beremoved prior to inspection. There may also be evidence ofbending cracks parallel and close to the wall. This may beindicative of a high degree of embedment causing the treads totruly cantilever and in this case, it is likely that a number oftreads may have failed,

– Examine grout lines between treads. Note any missing grout orobvious tread deflection. Check the horizontal level of treadswithin a flight, if all treads are off level by a given amount, thismay be a feature of the construction. However, if the levelchanges from top to bottom, it may indicate movement in asupporting landing.

– Check for the presence of stone grafts. This relates almostexclusively to the softer wearing Portland stone steps. Wheregrafts have been inserted, the step should be replaced or astringer beam provided. Grafts cannot be taken as contributingstructurally to the step strength of the step. Often, the factor of

safety will be reduced down to or below 1.0.– Inspect the bottom tread of any flight. The importance of

landings (or the bottom tread of a ground floor flight) in terms ofresisting horizontal loads and deflections cannot be overemphasised. Assess their ability to resist vertical and horizontalloads. Usually, landings are made up of a number of individualslabs built into the wall along their outside edge, joined togetherwith halving (joggle) joints and filled with lead. Check fordamage, as often there will be excessive deflection or separationof the individual slabs. Typically, the condition of landingsdetermines the level of intervention required on a flight, as alanding which provides poor support will overstress the flightabove and the landing deflection can be directly correlated tothe level of damage suffered.

– Examine grouting at the wall support socket. It is necessary toremove the skirtings as many cracks are found close to theskirting line (due to high torsion combined with a small degree ofbending or more seriously, cantilever action linked to a highdegree of wall embedment combined with sagging of a supportlanding).

– Make a photo record of the inspection. – It is likely that some refurbishment work will have been

completed in the past. Attempts should be made to ascertainfrom records any construction details, such as the supportingwall thickness, depth of tread embedment, form of landingconstruction and any previous repairs. Previous repairs shouldbe inspected.

Landings

Landings are critical in the analysis of cantilever stairs. Usually,landings are substantial and form part of the floor slab. Anexception may be the top of flights, where judgments may benecessary as to whether adequate horizontal support is evident.Landings should be kept clear or the actual loading conditionsconsidered since landings often support photocopiers or boxes ofpaper. Provided deflection under maximum loading is minimal,landings will provide adequate support to a flight above. Halflandings, however, are normally constructed from a number ofdifferent pieces of stone and often deflect to the point where theycause failure in the lower treads. This is particularly pronouncedwhen the lower treads of a flight are embedded sufficiently deep inthe supporting wall as to render them fixed against rotation. In thiscase, if the landing deflects away from under the lowest tread, thattread will try to act as a surrogate landing and inevitably fail. As anexample, the maximum bending stresses induced in a typical flightof 12 rebated, ogee treads would be 20N/mm2; three times thestrength of Portland stone. Failure of the lowest tread wouldtransfer bending stresses to the treads above, causing progressivefailure. Flight collapse will result if the failed treads cannot beretained within the flight via a combination of tightly groutedcontact surfaces and friction along the crack line.

It is common where a landing is made from more than one pieceof stone to provide additional support, the rationale being to limitdeflection. For example, a typical Portland stone ogee tread with aspan of 1.4m fully fixed by the supporting wall can deflect by only1.7mm at its free end before failure in bending. The support maybe a steel section fitted tight to the landing soffit and forarchitectural reasons boxed out and fire protected.

Cantilever stairs and their supporting landings are particularlysensitive to any settlement and this type of stair is one of the few

17 Granite deep spandrel tread with stepped soffit (Courtesy BMCE)

18 Portland stone ogee profile with hidden rebate (Courtesy BMCE)

17 18


construction forms that does not benefit from the flexibility that theuse of lime mortar brings.

Load testing

There are a number of unknowns in a flight of stone cantileverstairs, and it is not possible to allow for all of these factors inanalysis. If analysis shows that the factor of safety falls below 3, itis common to carry out a load test.

The recommended test load 9, 10, 11, 12 is 1.0 Gk + 1.25 Qk.Loads should be applied in increments of 20-25% up to the fullload and unloading in the same decrements, with deflectionsnoted each time. The criterion for a successful test is that afterunloading, the residual deflections are less than 10% of themaximum deflection recorded during the test.

Test loads may be applied by using water or portable weights.Irrespective of how the treads are loaded, a crash deck must beprovided. This will normally be some form of scaffold system orbraced props which is located 10–15mm below the soffit of thetreads, which allows the flight to deflect during the load test butwill catch it in the event of collapse. There is a also a likelihood ofdamage to the treads if weights are dropped during loading and alayer of rigid insulation can be used to lessen the risk, this also hasthe added advantage of protecting the surface of the tread fromscratching, particularly when using concrete blocks

Conclusion

The paper’s purpose is to provide guidelines for inspection andanalysis of cantilever stairs. It must be recalled that the theoreticalmodel has been stripped down to its basics. In practice, stairsmay have been standing for decades, when a theoretical modelsuggests failure. Treads in a row are often cracked, yet the stairshave been in this condition for some time but provided loadingsare low, the grout between treads is good and there is adequateshear transfer because of interlock, this is perfectlyunderstandable, although the treads should be replaced. The skillin inspection is to understand how a stair’s physical conditiondiffers from the assumptions made in the model and to understandthe effect such differences may have.

An important point to consider is whether a stair has trulyeffective rebates. If the tread profile is ogee, with stressessignificantly greater than those in rectangular treads, then stone

strength can easily be exceeded without a functioning rebate. Clear advice must be given to building operators; it is not

enough to sign off stairs as capable of taking 4kN/m2 withoutmaking it clear what that means. Guidelines should be givendetailing the number of people that can be carried on each treadand what loadings should be avoided, such as heavy officeequipment storage or carrying heavy items that might be dropped.Stone stairs are susceptible to shock damage.

Research into cantilever stairs is ongoing in conjunction with theDublin Institute of Technology and Trinity College coveringbehaviour, analysis, material strength and non destructive testing.

Acknowledgments

We would like to thank Dr Barry Duignan of Dublin Institute ofTechnology for his considerable assistance in the preparation ofthe finite element models used in this article.

References

1 Taylor, R.: ‘Revealing mason’s mystery’, The Architects Journal, 26Sept.1990, pp 34-43

2 Hough, M., Mahony B.: ‘Restoration of City Hall- Engineering aspects’,Barrett Mahony Consulting Engineers, 2002.

3 Price, S.: ‘Stepping backwards in time’, The Architects Journal, 11Feb.1999, pp 45-46

4 Price, S., Rogers H.: ‘Stone cantilevered staircases’, The StructuralEngineer, 83/2, 2005, pp 29-36

5 Boresi, A. P., Schmidt R. J., Sidebottom O. M.: Advanced mechanics ofmaterials 5th Edition, John Wiley and Sons, Inc., 1993, pp 237-283

6 Boresi, A. P.; Elasticity in engineering mechanics, Prentice-Hall Inc., 1965,pp 164-194

7 Cook, R. D., Young W. C.: Advanced mechanics of materials, MacmillanPublishing Company, 1985, pp 283-343

8 Timoshenko S. P., Goodier J.N.: Theory of elasticity, 3rd Edition, McGraw-Hill Inc., 1984, pp 307-313

9 BS 5950-1:2000 Structural use of steelwork in building; British StandardsInstitution, 2001

10 BS 8110-1:1997 Structural use of concrete – Part 1: Code of practice fordesign and construction; British Standards Institution, 2002

11 Information paper 2/95 Guidance for engineers conducting static load testson building structures, Building Research Establishment, 1995

12 Load testing of structures and structural components; The Institution of Structural Engineers, London, 1989

13 Hume I.: ‘Cantilever or hanging stone stairs’, Context, 55, September 1997, published by the Institute of Historic Building Conservation

British Group

Date | Thursday 21 May 2009

Venue | The Institution of Structural Engineers, 11 Upper Belgave Street, London SW1X 8BH

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Engineering the 2012 Olympics by John ArmittThe delivery of the 2012 Olympics and Paralympic Games in London is dependant on the skills of engineers of all disciplines. John Armitt, Chairman of the Olympic Delivery Authority, will discuss the contribution of engineers from demolition and remediation, to Zaha Hadids iconic aquatics centre, to the complex security and communications systems necessary to deliver a 21st century Olympic Games in a sustainable manner.

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