Structural Analysis of Historical Constructions - Modena, Lourenço & Roca (eds) © 2005 Taylor & Francis Group, London, ISBN 04 1536 379 9

Staircases as cantilevers or arches? - a question for limit analysis

E.A.W. Maunder Department of Engineering, University of Exeler, UK

ABSTRACT: The questions of determining the load capacity and possible modes ofload transfer are considered for "cantilever" staircases, i.e. where the treads appear to cantilever from a side wall, but where such a description is inadequate or inappropriate on closer inspection. Altemative modes of action are proposed in terms of arch actions in a flight oftreads with discontinuous thrust lines. The discontinuities are balanced by torsional moments from the side wall. The questions are then addressed via limit analysis as an optimization problem with an implicit appeal to the lower bound theorem ofplasticity. The problem is posed as a non-linear one and solutions, as exemplified for granite staircases in Castle Drogo, are presented based on a generalized reduced gradient algorithm using a spreadsheet formal.

INTRODUCTION

Staircases are a vital component of multi-storey struc­tures. They may be purely functional or form spec­tacular aesthetic features, and they can involve many different structural systems from simply supported inclined spans to dog-Ieg frames and curved spirals. This paper is concemed with a particular structural form which relies on interactions between treads as voussoirs combined with torsional reactions to individual treads from a stairwell.

It may be speculated that such a form originated in the 16th century in the works of the Italian renais­sance architect Andrea Palladio with his development of systematic geometric design principies in three dimensions. In the modem translation by Tavemor & Schofield (1997) it is clear that Palladio recognized and advocated staircases supported by an outer wall without support in the middle. Similar structural con­cepts appearto have been extended to spectacular stair­cases with large oval stairwells in the 18th century, e.g. in the Greenwich Naval Hospital as recently discussed by Eatherley & Brady (2004), and Sharpham House in Devon as discussed by Cherry & Pevsner (1989).

Price (1996) has provided good explanations of the structural principies and the important roles of torsional moments provided to the treads by the sup­porting wall of the stairwell, and the interactions between treads even ifthese are only provided by small horizontal overlaps as observed in Palladio's design for the monastery of the Carita in Venice. Price (1996) also explains the structural significance of rebates between treads, in that they allow inclined thrusts to be transmitted in a similar way as occurs between

voussoirs of an arch. Indeed arching action alone in the manner of a flying buttress has been noted by Sutherland (1994).

However there is a danger that the underlying struc­tural concepts could be misunderstood by present day architects and others involved with maintenance and refurbishrnent. The term "canti lever" is a commonly used adjective, e.g. Cherry & Pevsner (1989), and whilst this term may be used as a generic one for the type of staircase under consideration there is an obvi­ous risk that the term may be taken toa literally as indicated by the recent collapses reported by Parker (2004).

The question in the title ofthis paper is a somewhat oversimplified one, since the torsional moments allow cantilever and arch modes of action to share the load with considerably more capacity than can be justified by either of the separate modes. Such questions are reminiscent ofthose raised in the debate on dam design in the early 20th century, Smith (1972), when arch and/or gravity (cantilever) modes ofaction were being considered. It appears that, without recourse to elastic theories, European designers at least decided them­selves how a dam should work by placing reliance, perhaps unwittingly, on the lower bound theorem of plasticity.

These questions can be answered using non-linear numerical optimization techniques in the context of a limit analysis. This paper considers the optimization problem as one of maximizing a load facto r applied to live loads with ali other variables being static ones, i.e. stress resultants which are sufficient to determine ali stresses of interest, subject to constraints which are generally non-linear. To the author's knowledge the

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staircase problem has not been considered before in this way although the underlying philosophy is similar to that discussed by Maunder & Harvey (2001).

The opportunity has also been taken to demonstrate optimization using a spreadsheet and a generalized reduced gradient algorithm as available in Excel. For relatively small scale problems such tools appear to be very suitable and they have the advantage ofproviding a transparent approach to computational aspects with both the intermediate workings and the final results displayed along with appropriate graphical output -ali at the control of the engineer, Gottfried (1998).

The paper continues in Section 2 to describe an equilibrium model for limit analysis. An example of a typical granite staircase in Castle Drogo, the most recent castle to have been built in the UK, is consid­ered in Section 3 where optimized solutions for load transfer are illustrated with thrust line diagrams as for an arch. This example is also considered in the context of a parametric study, as advocated by Eatherley & Brady (2004), when there are uncertainties in the data, e.g. the depths of embedment and the safe bearing stresses. Conclusions and proposals for further work are contained in Section 4.

2 AN EQUILIBRlUM MODEL

2.1 A tread as a cantilever

Each tread is embedded into a supporting wall which provides as reactions a bending moment M, a vertical shear force V and a torsional momentT. The moment M and shear Vare assumed to be transmitted via contact with the horizontal surfaces of a rectangular section. The torsion T is assumed to be transmitted via both horizontal and vertical surfaces. Ali transmission is assumed via uniform normal bearing stresses (1v or (1h

a

Figure I. Vertical contact forces on a tread.

on allocated contact areas. These areas are somewhat arbitrary, but are selected and optimized to maximize the stress resultants.

The horizontal areas for (1v are indicated in Figure I where d denotes the embedment length. The stress resul tants must satisfy the equilibrium Equations (I).

Fa (d - 0.5a)- Fb O.5b = M (I)

where z is the lever arm of the torsional couple Tv formed by the pair of forces Fc. These equations can be put into non-dimensional forms as in Equation (2).

(2)

where

and

V M 4~ ri = --; r2 = ri - = ri r; r3 = --2 -

avwd Vd avw d

where w denotes the width of a tread. The torsion com­ponent Tv is maximized in the above equation with the relevant contact areas extending over halfthe width of the tread and z = 0.5w.

For a fixed ratio r combined shear and torsion becomes limited by Equation (3).

(3)

Equation (3) represents an arc of a circle in r i , r3 space illustrated in Figure 2, with the maximum magnitude

-I

Figure 2. Shear/torsion interaction diagramo

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of r3 being 1 when r, = O, and the maximum value of r, being given by Equation (4) when c = r3 = O.

( 2 )0.5 r, = 2+4r+4r -2r-1 (4)

A torsion component Th is provided by bearing stresses a" on the vertical faces as indicated in Figure 3 and defined in Equation (5).

(5)

T" is maximized with the vertical contact areas extend­ing over height e = 0.5h, and then T" is given by Equation (6).

(6)

2.2 A flight oftreads as an arch

Treads butt together at waist sections like voussoirs of an arch or flying buttress with adjacent landings acting

When the maximum stress ali applies, the stress resultants are related by Equation (7).

r =~=~~(1-~)= rJ (I-r) 4 tNu 2 Nu Nu 2 3

(7)

where r3 and r4 are the non-dimensional terms rep­resenting N and M respectively, and NII = ali ti where I denotes the length of the outstand of a tread from the wall. Equation (7) defines limiting curves in r3, r4 space as illustrated in Figure 6.

as abutments. This mode of action is shown in Figure 4 Figure 4. Continuous thrust line for arch action. with a typical thrust line to support the weight ofthe treads.

However for the thrust line to be admissible it needs to lie within the waist sections without overstress­ing the contact areas between adjacent treads. With reference to Figure 5, and again assuming uniform compressive stress on the contact area with a maxi­muro value of ali, the axial stress resultant is equivalent to an axial force N and a bending moment M about the central axis.

e

Figure 3. Torsional contact forces on a tread.

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Figure 5. Detail oftread interaction.

Figure 6. Thrust/moment interaction diagramo

Figure 7. A discontinuous thrust line solution.

In general thrust line solutions can only be made admissible by allowing for discontinuities wilhin the treads which are balanced by lorsional moments intro­duced from the wall as indicated in Figure 7.

2.3 Non-/inear optimization

The optimization problem is now posed which seeks to combine the modes of action described above in Sections 2.1 and 2.2 so as to maximize the live load capacity within the stress limits O'v, O'h, and O'u.

The total vertical loads on the stairs are repre­sented by the combination of dead and live loads in Equation (8), where À denotes the live load factor.

(8)

The objective is to maximize À. The variables other than À are static quantities comprising: the shear forces V supported by cantilever action; the torsional moments Tv and Th provided by the wall; plus three biactions X selected to render lhe arch statically determinate when X = O.

The linear constraints correspond to the limits imposed on Th by Equation (6) and to limits imposed by friction between the treads and/or the treads and the supporting landings.

Non-linear constraints in quadratic form corre­spond to the limits imposed on combinations of V and Tv by Equation (3), and to the limits imposed on combinations of N and M by Equation (7).

3 NUMERICAL EXAMPLES BASED ON STAIRCASES IN CASTLE DROGO

Castle Drogo, near Drewsteignton in Devon, is gener­ally considered to be the last castle built in England in the early part of the 20th century according to Cherry & Pevsner (1989). Its structure is largely one of granite although it also contains early examples of reinforced concrete. The staircase that has been studied

w 150 rnrn

200 rnrn

Figure 8. Cross-sections of a tread.

is contained within the North Tower which was origi­nally used as a service wing with a separate staircase. Hence it was ensured that the palhs ofthe servants and family remained separate! The staircase is an impres­sive granite structure rising up through five storeys, constructed with large granite landing slabs and flights of four treads formed from large blocks of solid gran­ite which are referred to by Cherry & Pevsner (1989) as "cantilevered stone steps". The treads and land­ings receive support from an enclosing masonry wall. Through the centre ofthe staircase is an oak balustrade lhat stands independently from the stairs themselves.

Each tread has a cross-section which approximates to a right-angled triangle, however where a tread is embedded in a wall the section changes to one which approximates to a rectangle. The key dimensions are given in Figure 8 afier minor simplification.

Following the work of Parker (2002) the length of embedment into a wall is estimated at d = 230 mm, and the selfweight of a tread is estimated at 1.042 kN with its line of action inset 100 mm from the leading edge. For simplicity live loads are assumed to have the same line of action.

A typical flight of 4 treads is shown in Figure 4. It should be observed that the treads are not rebateel, but the interfaces at the waist are formed on planes normal to the plane ofthe soffit and so, with the aid offriction, allow for inclined thrusts. For subsequent reference the treads are numbered 1 to 4 from the top of a flight downwards, and the waist sections are numbered 1 to 5 from the top downwards.

The arch, as a 2D structure, is three times statically indeterminate. The release system adopted for the def­inition of biactions consists of: the axial force N3 at section 3, and the moments M1 and Ms at sections 1 and 5. The basic number of static variables is thus 3 + 4 x 3 = 15, however lhe optimization algorithm requires the use of non-negative variables. Conse­quently the number ofmoment variables is doubled to include moments ofboth senses leading to a total num­ber of non-negative static variables = 5 + 4 x 5 = 25.

The total number of constraints applied is 22 = 5 + 3 x 4 + 5 made up from 5 for the arch sections, 12

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30

-- N(l )=0.0 kN

-+- N( I)=IO.O kN

...... N(l )=20.0 kN O~~~~~~~~~~~~~

O 5 10 15 20 25 30 35 40 45 50

Number of iterations

Figure 9. Convergence of live load facto r À.

50

~ 40

~ 30 ~

:§ 20 '" ' ;(

< 10

-- N(I )=O.O kN

-,0- N(l )=IO.O kN

...... N(l )=20.0 kN

O

O 10 15 20 25 30 35 40 45 50

Number of iterations

Figure 10. Convergence ofaxial force NJ .

for the treads, and 5 to account for friction at sections I to 5. The constraints are functions of three normal stress components which are initially considered at the following values:

au = 1.0 Nlmm 2• a v = ali = 0.5 Nlmm 2 (9)

These values have been set at much lower values than would correspond to the strength ofthe granite in order to account for the lower strengths normally expected at mortared joints, e.g. see Genna & Ronca (200 I).

Due to the non-linear nature of some of the con­straints there is a choice for solution method between using linearised constraints in a linear programme, as in Maunder (1993), or to use a non-linear programme. The Excel spreadsheet has the facility to invoke a non­linear generalized reduced gradient (GRG) algorithm which appears to be well suited to the staircase prob­lem - the non-linear constraints being well defined in convex quadratie forms.

3.1 Convergence studies

Figures 9 and 10 illustrate typical eonvergenee behaviour when the stress limits are as given in

Equation (9), the friction eoefficient J..L = 0.5, and the live load is restricted to tread 2.

All variables are initially set to zero except for the axial force N3 in section 3. This force serves as a me a­sure of the arching action. The initial value has the physical significance of an axial prestressing force which is then allowed to vary. The initial values are shown as N (I) in Figures 9 and 10.

The operational parameters of GRG were set to an absolute value for the constraint tolerance of 10- 6 ,

and a relative value of I in 106 for the convergence eriterion, i.e. the iterations stop when the relative changes in the objective function À are less than 10- 6 over 5 iterations. These parameters lead to the same solutions within 45 iterations with monotonic convergence, although as can be seen in Figure 9 the solution initiated with N(l) = 20 kN converged to a lower value when the convergence criterion was raised to 10- 5 . Clearly the value of initial prestress has a sig­nificant effect on the number of iterations required. However as soon as À > O, the self weight beeomes supportable and ali subsequent solutions are feasible even ifnot optima!. This reflects one ofthe benefits of an optimization method based on equilibrium .

Figure 10 shows non-monotonic convergence ofthe axial force from various initial values. When N (I) = O, almost half the number of iterations is needed to mobilize significant arching action. A check on con­vergence is provided by the evidence in the spreadsheet that arching and cantilever actions are fully mobilized and only the frictional constraints may not be attained. The final solution shows À = 28.0 and only 23.3% of the totalload is supported by the treads as cantilevers, the remaining 76.7% being supported by the areh.

3.2 Sensitivity studies

A number of parameters, e.g. the depth d of embed­ment of the treads, the limiting bearing and contaet stresses, and the friction eoefficient between the treads are likely to have uncertain values. In which case it is prudent to eonsider ranges ofvalues in sensitivity anal­yses, and such analyses are now demonstrated for the example of Castle Drogo.

With live load on tread 2 on ly, Figures II to 14 illustrate the effeets on live load factors À and the per­centage ofthe totallimit load supported by cantilever action as parameters are varied from a standard set of values with:

d = 225 mm; av = ali = 0.5 N/mm2, and

The following observations can be made:

- 80th the load factor and the proportion of cantilever aetion are approximately linearly dependent on d .

573

30

25 i5 " o .9 ." 20 u

'" u ~ ..'!

~li 15 0'=

~ ~ 10 :.:: ~

O O 50 100 150 200 250

depth of embedment mm

Figure li. Sensitivity with respect to depth of tread embedment.

50

.... "40 o .s .... -~ ~ 30 ~ ~ ~ ~ 20 .9§ ., u

& ~ lO -Ioad factor

--canti lever action Ot---~~~~~~~~

O 0.2 0.4 0.6 0.8

Bearing stress in wall N/mm2

Figure 12. Sensitivity with respect to the bearing stress in thewall.

90

80

70 i5 c: .9 t 60 § '" ~ 50 ..'! " >

load factor

- cantilever action

-o " 40 ~:: - c: " '" 30 > u

:'::: çf2 20

10

O O 0.2 0.4 0.6 0.8 1.2

bearing stress between treads N/mm2

Figure 13. Sensitivity with respect to the contact stress between treads .

50

40

i5 c: o

§ .~

30 ~ ..'! >

"l6 ~ .!2 E 20

" fl ~ >fi

10 - load factor - cantilever action

O O 0.2 0.4 0.6 0.8

friction coeffi cient between treads

Figure 14 . Sensitivity with respect to friction between the treads .

Figure 15. Complete optimized solution fo r equal loads on ali treads. The landing slabs are indicated by the gray areas as abutments .

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Figure 16. Complete optimized solution for live load on tread 2 only.

- Up to a bearing stress of about 0.8 N/mm2 the load factor increases approximately linearly with stress and the proportion of cantilever/arch action tends towards a constant value. Higher values of bearing stress permit greater torsional restraints and poten­tially greater arching action, but this type of action then becomes constrained by friction in both sec­tions for the top tread. Consequently further load requires a greater use of the cantilever actions.

- Up to a contact stress of about 0.6 N/mm2 the load facto r increases and the cantilever action decreases. These changes are mainly influenced by frictional constraints for the top tread. Friction tends not to be fully mobilized as the contact stress increases further, but the arch action then becomes limited by the torsional restraints from the wall, and little further load can be taken for contact stress higher than about 1.0 N/mm2 .

- For low values of the fTiction coefficient j..t up to about 0.15 the limit loads are constrained by fTiction on both sections ofthe top tread. Higher values of j..t allow a rei ative increase in arch action and a greater load with friction still mobilized on section I. For j..t> 0.5 no frictional constraints are mobilized, and the load and mode of action do not change.

3.3 Optimized solutions

Figures 15 and 16 and Table 1 detail optimized solu­tions for equalloads on all treads, and with live load on

Table I. Stress-resultants for complete optimized solutions.

Stress-resultant

WI kN Wz kN W3 kN W4 kN VI kN Vz kN V3 kN V4 kN TI kNm Tz kNm T3 kNm T4 kNm P I kN Pz kN P3 kN P4 kN Ps kN

Equalloads on each tread

11.l4 11.l4 11.14 11.14 0.44 2.60 2.61 0.00

- 3.433 -1.320 - 0.004

3.656 34.87 36.48 39.83 44.58 52.25

Live load only on tread 2

1.04 28.35

1.04 1.04 0.00 2.43 2.61 2.14

- 3.656 -1.787

1.127 2.194

32 .21 32.26 42.66 41.64 40.95

tread 2 only, based on the standard set of parameters. For the equal load case it is observed that self-weight could be supported by (a) cantilever action alone with a load factor of about 2.50, or (b) arch action alone with a load factor of about 1.0, or (c) the optimized solution as illustrated in Figure 15 with a load factor of about 10.0.

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In the unequalload case it is observed that (a) about 1.57 kN of live load could be supported by cantilever action alone, or (b) zero live load could be supported by arch action alone, or (c) about 27.30kN can be supported by the optimized solution illustrated in Fig­ure 16, i.e. the strength ofthe whole is much more than the strength of the parts!

4 CONCLUSIONS & FUTURE DIRECTIONS

Exploration of structural actions by limit analy­ses confirms that torsional restraint provided to the treads by a stairwell is of great significance to assessing load capacity. Although these moments may be relatively small, they are essential to allow­ing discontinuous arch thrust lines to be accommo­dated within the waist. It has been demonstrated that the statical, or lower bound, limit analysis is a useful tool for carrying out parametric studies in order to judge the sensitivity of load capacity to uncertain data, e.g. the depth of embedment oftreads, the safe bearing stresses, and the amount of friction that can be mobilized. In general the use ofthe GRG algorithrn within the Excel spreadsheet appears to be well suited to the limit analysis problem for masonry stairs assum­ing relatively simple non-linear constraints. For the problem considered convergence was rapid with results displayed "instantaneously" using a PC with a Pentium 4 processor. However, it should be noted that the algorithrn as used in Excel provides a "black box" type of computational tool without access to its inner work­ings. Experience and understanding ofthe physical problem are essential to judge the type of conver­gence that is c1aimed by the software. This type of algorithrn suffers from questions regarding the nature of the optimized solution, i.e. is a solution a local or global optimum? For most ofthe solutions obtained in this paper the bearing stress constraints are fully utilized, and the solutions appear to be independent of the initial value of the axial force variable. Such results give the user confidence that the global optimum has been found. However when the stress constraints are not fully utilized the

question becomes more problematic, particularly when friction is mobilized for both interfaces of a tread. In such cases global optima have been sought by trying a range ofinitial values ofthe variables. It is intended to extend the limit analyses as pre­sented in this paper to inc1ude landings, curved or oval staircases, and balustrades.

- An alternative genetic algorithrn is available for use with the Excel spreadsheet, and this should be inves­tigated for the staircase optimization problem -both to verify the results obtained by the GRG algorithrn, and to check its suitability for this type ofproblem.

REFERENCES

Cherry, B. & Pevsner, N. 1989. Devon: The Buildings of England. 2nd edition, London: Penguin Books.

Eatherley, M. & Brady, M. 2004. Royal Naval College. The Struetural Engineer 82(1): 17- 19.

Genna, F. & Ronca, P 2001. Numerical Analysis of Old Masonry Buildings. In J.W Buli (ed), Computational Modelling of Masonry, Briekwork and Bloekwork Strue­tures: 22 1- 272. Stirling, Saxe-Coburg Publications.

Gottfried, B.S. 1998. Sp readsheet Toolsfor Engineers: Exeel 97 Version. Boston: WCB/MeGraw-Hill.

Maunder, E.A.W 1993. Limit analysis of masonry struc­tures based on diserete elements. In C.A. Brebbia & RJ.B. Frewer (eds), Struetural Repair and Maintenanee of Historie Buildings III: 367- 374. Southampton, Com­putational Mechanies Publieations.

Maunder, E.A.W & Harvey, WJ. 2001. Historie Masonry Struetures. In J.W Buli (ed) , Computational Modelling of Masonry, Briekwork and Bloekwork Struetures: 273- 3 12. Stirling, Saxe-Coburg Publications.

Parker, A.P. 2002. Granite struetures within Castle Drogo. Project Report, Department ofEngineering, University of Exeter.

Parker, D. 2004. Fears for eantilever stairs afler eollapse. New Civil Engineer (issue 1524): lO.

Priee, S. 1996. Cantilevered staireases. Arehiteetural Researeh Quarterly l(eonstruetion): 76--87.

Smith, N. 1972. A History of Dams. New Jersey: Citadel Press.

Sutherland, R.J.M. 1994. Active engineering history. The Struetural Engineer 72(13): 205- 212.

Tavernor, R. & Sehofield, R. 1997.ANDREA PALLADIO: lhe four books on arehiteeture. Cambridge: The MIT Press.

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