Title No. 63-29

Experimental Study of aFree-Standing Staircase

By A. R. CUSENS and JING-CWO KUANG

Describes loading tests to failure on a half-scale model of a symmetricalreinforced concrete slab-type, free-standing staircase. Methods of analysisare compared in the light of experimental results and general design recom-mendations are made.

Key words: analysis; cantilever staircase; design; reinforced concrete:research; staircase; torsion.

n THEREINFORCED CONCRETE FREE-STANDING STAIRCASE (Fig.l) has becomepopular with architects in recent years. The cantilevered flights andlanding have obvious structural and aesthetic advantages particularlywhen used for multiflight stairs. In consequence the design of thistype of structure has considerable interest for structural engineers.

Liebenbergl first introduced the concept of the space interaction ofplates for the design of this type of staircase. His analysis was madefor a statically indeterminate structure on the assumption that torsionaleffects were negligible. SieP extended the theory to include the deter-mination of the secondary stresses resulting from the compatibilitycondition at the intersection between the flights and the landing. Heconcluded that the torsional moments were usually small and may beconsidered as secondary stresses; for most practical purposes it wassufficient to compute primary stresses.

Fuchssteinefl simplified the basic staircase by considering it to be arigid space frame. The flights were considered as sloping cantileverbeams and the landing as a semicircular bow girder (Fig. 2). Sauter*recently published an analysis employing the principle of least work,using a frame identical to Fuchssteiner’s assumption; it is the opinionof the authors that the bow girder does not provide an adequate repre-sentation of the structural behavior of the landing. Gould5 and Talebemade simplified analyses neglecting the bending moments along theline of intersection of the flights and landing.

The authors7 have recently made an analysis of free-standing stairsassuming that the structural behavior could be simulated by the skeletal

508 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE May 1966

ACI member A. R. Curenr is professor and head, Department of Civil Engineering, Uni-versity of St. Andrews, Queen’s College, Dundee, Scotland. He has recently returned toBritain after 5 years on the faculty at the SEATO Graduate School of Engineering, Bangkok,

Thailand. Professor Cusens is the author of numerous technical contributions.

ACI member Jinggwo Kuang is a graduate student, Technological Institute, NorthwesternUniversity, Evanston, Ill. He received his undergraduate training in Taiwan and later studiedat the SEATO Graduate School of Engineering, Bangkok, Thailand, where he received hismaster’s degree in structural engineering in 1964.

rigid frame shown in Fig. 3. The frame is cut at 0 as shown, andhorizontal restraining forces H and moments M, are applied to the twohalves of the staircase. Each half of the structure is now statically deter-minate and equations may readily be written for bending and torsionalmoments, and axial and shearing forces in the various structural mem-bers.

Neglecting the effects of axial and shearing forces on deformation,the partial differential coefficients of strain energy with respect to Hand M, are equated to zero. From the two simultaneous equations thus

Fig. I - isometric sketch of free-standing stairs

FREESTANDING STAIRCASE 589

obtained, the values of M, and H may be found and individual momentsand forces evaluated. The principal equations are given in the Appendix.

PREVIOUS EXPERIMENTAL STUDIES

The only tests reported on a reinforced concrete free-standing stair-case have been made by Hajnal-K6nyP on a full-sized structure. Deflec-tions were measured for various arrangements of load below the designvalue.

Mitchell and Shaw9 used a loaded plexiglass model and measuredstrains with electrical resistance gages. From these values, momentswere calculated at various points.

LiebenberglO describes an experiment on a model staircase of epoxyresin, employing photoelastic methods to determine the stresses in theflights under a symmetrical loading on the landing.

The experimental values from these studies are compared with theauthors’ results in a later section.

Fig 2 - Fuchssteiner’s assumed form for the free&ending staircase

DESIGN AND CONSTRUCTION OF A MODEL STAIRCASE

A prototype staircase, having the dimensions shown in Fig. 4, hasbeen analyzed by the methods of Siev2 (including secondary stresses),Sauter* and the authors7 The resulting values of bending and torsionalmoment are shown in Table 1. It may be seen that there is little dif-ference between the values obtained by the methods of Siev and theauthors; the greatest discrepancy occurs at the cantilever supports.At the time of the test, the authors’ method of analysis had not beencompleted and Siev’s values were used for design purposes. Reinforce-ment at the various sections of the prototype was designed using theconventional modular ratio method of design. Plain mild steel rein-forcing bars with hooks were used throughout.

The staircase built for testing was a half-scale model of the prototype.The same percentage of reinforcement was maintained as in the originaldesign for the prototype; thus, in the various sections of the model,the area of steel was 25 percent of the value calculated for the prototype.The arrangement of the reinforcement in the model is shown in Fig. 5.

Fig. 3 -Skeletal rigid frame representing the fro+standing staircase

FREESTANDING STAIRCASE

TABLE I -COMPARISON OF COMPUTEDVALUES OF MOMENT (FT-LB\ IN STAIRS

(a) With both flights and landing loaded

Sauter

- 1 6 6 0- 6 3 6 0- 6 4 0 0- 450f 4.530MO.330

Authors

- 2 9 2 0- 6 3 6 0-11,580- 3 0-+- 3,350229,810

(b) With live load on both flights only

AB

MI, 0D

Mr B AM, B A

Sauter Authors

- 2 3 5 0- 4 1 5 0- 4 7 0 0+ 200* 3,330c14,930

- 3660- 4 1 5 0- 8 8 8 0+ 710-c 2,760k22.600

Mn = bending moment about the horizontal axisM. = bending moment about the vertical axisMt = torsional moment

Fig. 4- Dimensions of prototype staircase

2-1/2"$+1-1/4y 2-l/4.@ 2-1/2"~tI-1/4"~

Section A-A

Fig. 5 -Arrangement of reinforcement in model s 8taircase

-

The base of the model, representing the lower floor level, was a 6 in.reinforced concrete slab, with plan dimensions 3 ft 3 in. x 2 ft 9 in. Itwas heavily reinforced with mats of %-in. bars at 4-in. centers ineach direction, positioned both at the top and the bottom of the slab.The edges of the slab were cast into mild steel channel sections, boltedrigidly to the laboratory floor. The upper floor level was representedby an identical slab. The channels at the edges of this slab were sup-ported by four vertical steel columns. The bases of these columns werebolted rigidly to the laboratory floor. High tensile steel wires were usedto brace the columns supporting the upper floor slab. The general ar-rangement may be seen in Fig. 6 and 7.

The concrete used in constructing the model had an aggregate-cementratio of 5 by weight and a water-cement ratio of 0.45. Natural riversand and crushed limestone of 3/s in. maximum size were used with arapid hardening (ASTM Type III) portland cement. To check thequality of the concrete, 6 in. test cubes were made from each batch.The average cube strength at the time of test was 5420 psi.

TESTING

To measure vertical and horizontal displacements under load, 22 dialgages with magnetic bases were mounted on an independent steel frame.Electrical resistance strain gages were mounted on the model by epoxyglue at various positions on the underside of the staircase. Gage pointswere also fixed at either side of two flights.

Load was applied in increments by placing pig-iron bars on the stepsand landing according to the following program:

1. Half design live load on upper flight only.2. Half design live load on right-hand portion of landing only.3. Full design live load on upper flight and right-hand portion of landing.4. Full design live load on whole landing only.

Fig . 6 -Model free-standing staircaseat an early stage of the loading test

Fig. ‘br Model staircase after failure,with the load removed

594 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE

Fig. 8 -View of the torsional cracks atthe top of the lower flight of stairs

/ \ after failure of the staircase

f xv ”

5. Full design live load on the staircase (1210 lb).6-24. Load increased in increments of half the design load until collapse.

Fig. 6 shows the staircase under load during this stage of the test pro-gram.

The first crack was observed at the outside edge of the junction ofthe lower flight with the ground floor. The crack extended diagonallytowards the inner edge of the lower flight. The corresponding liveload was 3825 lb. With increase of load, a similar diagonal crack occurredtowards the top of the lower flight (see Fig. 8) and at the base of theupper flight. At a live load of 5800 lb, cracks were visible at the junctionof the upper flight and upper floor, and also at the intersection betweenthe landing and flights.

Final failure occurred along the line of the intersection of the landingand the flights when the live load was 13,800 lb. If the dead load of 1100lb is included, this represents a load factor of 6.48. Fig. 7 shows thestaircase after failure with the load removed.

At the later stages of loading, cracks developed in the upper faceof the landing near the junctions with the flights. The cracks wereconsiderably wider near the intersection with the flights, narrowingto insignificance at the free edge of the landing. A sketch of the crackpattern is shown in Fig. 9.

. -

:,

7

Fig. 9-Crack pattern in the landingat ultimate load

\

FREESTANDING STAIRCASE

During the test it was found that although the upper floor supportswere braced, the steel columns had horizontal displacements which musthave had some effect on the distribution of moments in the structure andwere certainly responsible for the unsymmetrical cracking of the struc-ture.

ANALYSIS OF RESULTS

Displacement measurementsThe displacement profiles of the stairway give a very clear concept

of the structural behavior under load. Fig. 10 shows the profiles of theedges of the upper and lower flights under load. The spacing of thedial gages does not permit the plotting of precise profiles in Fig. 10but the positive moments, predicted by Siev2 for the midspan of theflights, are not apparent from the behavior of the model. Fig. 11 showsan isometric view of the deflected form of the staircase under the de-sign loading.I The displacements and stresses in the line of the intersection of theflights and landing of free-standing stairs have provoked some discus-sion in previous papers. In their analysis, the authors have consideredthe landing to act as a beam concentrated at the line of intersection.Since the landing slab is commonly tapered in section as in the modeland the reinforcement tends to be concentrated towards the line of in-tersection, this is not an unreasonable assumption. The effective canti-lever span of the landing was assumed to be one half of the actual value,as in Siev’s analysis. Treating Beam CBOB’C’ as a member of a skeletal

I n n e r e d g e

(in.1

i

00.10.2030405

(in.)

1

0010.2030405

(ill.)I

00.10.20.3a40.5

Fig. IO - Profiles of edges of flights under load

JOURNAL OF THE AMERICAN CONCRETE INSTITUTE

Fig. I I - Deflected form of model staircase under design loaa

rigid frame, the bending moment diagram is as shown in Fig. 12a. If thechange of bending moment at B and B’ are taken to occur linearly acrossthe widths of the respective flights this diagram is modified to the formof Fig. 12b. This may be compared with the bending moment diagramfrom Siev’s analysis, and with the interpretation of resistance straingage readings from the plexiglass model tested by Mitchell and Shaw:in Fig. 12~ and 12d.

Double integration of moment diagrams for the model staircase yieldsthe general forms of the deflection diagrams which are compared withthe experimental results in Fig. 13. Here it may be seen that the authors’simplified skeletal structure gives analytical results which have reason-able parity with the test results from the model, provided that thechange in bending moment at the joint of flight and landing is dis-tributed across the flight. Vertical displacements may be computed di-rectly from the theoretical skeletal rigid frame more simply, but resultin much higher values; only the calculated displacement of the centralpoint 0 corresponds with the experimental value.

FREE-STANDING STAIRCASE

It may be observed from Fig. 13 that the experimental deflectioncurve for the line of intersection of the flights and landing is not sym-metrical about Point 0. The vertical deflection at C’ is some 20 per-cent higher than at C. Part of this may be attributed to the displacementat Support A. However it is felt that the tendency of the landing torotate in an anticlockwise direction about 0 is a contributory factorin the asymmetry of the curve.

Horizontal displacements are of considerable significance in free-standing stairs. The significance is not primarily a matter of structuralsafety but more of psychological concern. A stairway may be of ade-quate structural strength but if the horizontal deflection is large thenthe stairs feel unsafe to the user. The authors know of at least one casewhere free-standing stairs had to be strengthened due to this feelingof insecurity, although there was no doubt as to the structural safetyof the stairs. Subjective tests on the model stairway revealed that hori-zontal deflections of the order of 0.02 in. were sufficient to cause someconcern. However, on the longer spans of a full-sized stairway thismight be too stringent as a limiting value. It is therefore felt that if arestriction of horizontal deflection at design load is to be put forward,it should be expressed in terms of the span (a + c) of the staircase. Ona purely empirical basis, a limiting value of (a + c) /5000 is suggested

Fig. 12a -Moment diagram for beam Fig. I2b- Moment diagram for beamof skeletal rigid frame with change of moment taken linearly

across joint with flight

C’ 0’

P

0 c

0’ 0 0 C

IFig. I SC - Siev’s analysis

Fig. I2d- Comparable results fromMitchell and Shawls experiments

(inch)0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.06

0.09

0.10

C' B' 0 B ’ C

I

I/ \

/c-Theoretical values \/ \

// \

/\

\

/I\

/\

/\

Fig. 13- Deflections at intersection of landing and flights from theory andexperiment

and should be checked as a normal part of the design procedure. In thecase of the model staircase, this function gives a value of approximately0.04 in., and for the prototype, 0.02 in. It is sufficiently accurate for de-sign purposes to compute horizontal displacements for the skeletal rigidframe without correction of the bending moment diagram.

Strain measurements

The measured strains were generally smaller than predicted strains.There was considerable scatter in the results of measurements made withboth Demec and electrical resistance gages but in general they confirmthe values of moment computed analytically by the authors’ method.For example on the underside of the midspan of the flights at the center,the electrical gages gave negligible strains, showing that the momentwas small, but giving little indication as to the direction of the moment.

Liebenberg,lO as a result of photoelastic tests, predicted considerablestress concentrations in the region of the intersection of the inner edgeof the flights and the inner edge of the landing. Sievll expresses theview that these concentrations are of secondary importance.

FREESTANDING STAIRWE 599

There was no experimental evidence of stress concentrations havingany serious effect at the intersection of the flights and landing. Anelectrical gage mounted at the center of the inner edge of the landingshowed a steadily increasing tensile stress of the order expected fromthe calculated negative moment in the beam BCOC’B’.

Failure of staircaseInitial cracks in the staircase occurred through combined bending

and torsion and this focuses attention on the difficulty in providingreinforcement. to resist torsion in shallow-wide sections. This factoralone may be responsible for a thicker stair section than would other-wise be necessary. Kemp, Sozen and SiesP express the view that crack-ing in reinforced concrete members under torsion depends almost en-tirely on the geometry of the cross section and the concrete strength.In this test the first crack (at 1.6 times the design load) occurred at aslightly lower load than expected, but this may be, at least partially,attributed to the displacement of the supports of the upper flight.

Although the first cracks appeared in the flights the collapse of thestaircase occurred in the landing, which failed as a simple cantilever.The load factor was satisfactory in this case but it is obviously desirableto restrict the width c of the landing as much as possible in practice.The shape of the transverse cracks in the landing provide an approximateverification of the assumption that one half of the width of the landingshould be considered as effective in the consideration of the staircaseas a rigid frame.

Final failure occurred at a load factor of 6.48 times the design liveplus dead load which may be regarded as satisfactory.

CONCLUSIONS

The following conclusions may be drawn from this study:

1. For general design purposes the analytical methods of Siev2 and theauthors’ may be recommended. For the staircase tested, the authors’method gives a more accurate prediction of the interaction of the flightsand landing.

2. The transverse reinforcing steel in the landing should have a majorconcentration in the vicinity of the line of intersection of the flights andthe landing.

3. Large torsional moments are present in the flights of free-standingstairs and a proper thickness of concrete must be chosen to resistthese moments, due to the difficulty of reinforcing shallow-wide sec-tions against torsion.

4. It is desirable to restrict the landing length c as much as prac-ticable.

600 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE May 1966

5. Horizontal displacements should be checked in design to insure thatuncomfortable lateral sway does not take place A suggested empiricalupper limit of horizontal movement is (a + c)/5000.

ACKNOWLEDGMENT

The work described in this paper was performed in the Structural ResearchLaboratory of the SEATO Graduate School of Engineering, Bangkok.

APPENDIX

METHOD OF ANALYSIS

The staircase of Fig. 1 is simplified for the purposes of analysis to the rigidframework ABOB’A’ shown in Fig. 3. Projected views are given in Fig. Al.

The positive vectors for moments are given in Fig. A2, and the usual right-hand rule is applied. The frame is cut at 0 and the horizontal restraining forcesH and the restraining moments MO are applied to the two halves of the stair-case as shown in Fig. 3 and Al.

The bending and torsional moments along the members of the upper half ofthe frame are:

Member OB

Mr = - MO - T . . . . . . . . . . . .._........................... (Al)

M, = - Hy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A2)

Aft=-- . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A3)

Member BC

M,.P--!!!.2 -$ + bl - y)2 . .._...._................ (A4)

MS=0 .,_., (A5)

Mt=+ ++ bl--1J>

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A6)

Member AB

Mr=Hssina-W (-!& +bl)scosa-T ($ +bl) - ws2~2a,,. (A7)

MS=- T cosa-MMosina+ W ($+ br )-&{( bl+ %) - b / sina

Mt=+ sina+Mocosa-W (-$+bl)+{(bl+$) - b /Lo?’(A9)

where w and W are the loads per unit length of the flights and landing,respectively.

ELEVATION

Awu

END ELEVATIONA’

ta cow I: C

1

PLAN

Fig. Al - Projected views of the staircase

602 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE

auXT= J

b'2 MS aMs dy+J

a Mr aMr +.+- -J

a MS aMs ds- -o EII’ aH ,, El2 aH ,, Ele’ aH

+ .I a Mt aMt ds- -,, GJ2 aH

whereII, II’ = second moments of area of the landing section about horizontal and

vertical axes, respectivelyIz, I2’ = second moments of area of the flight section about horizontal and

vertical axes, respectivelyJi, J2 = polar second moments of the landing and flight sections, respectivelyIf the staircase consists entirely of shallow-wide sections, then Ii’ is much

greater than II and Is’ is much greater than 12, so that expressions containingl/Ii’ and l/12’ may be neglected.

Then if aU/aH = 0:

au- - = &-[l sina[an

HS2sinu-W(-~-+bl)s2cosn-WWs(g+bl)~

7us3ya 1 b- - - sin a2

q sin a + M. cos a

+T (++bi)($-b+osa]ds=Q

whence:

1_-232

-%+bl) (-~P-bi)+2M.-HHhtana] =O (A101

Also:

au- =J

b'2 M, aMr &,+o EII aMo J

a !kaM,&+ - -aMo ,, Elz’ aMo

The second term is neglected and then:

++bl)(+l)

+2Mo-Hbtana 1 =o (All)

Appropriate values of a, b, bt, c, Ii, 12, J2, W, w, and a may be substitutedinto Eq. (AlO) and (All) which are then solved for H and MO. The moments(and shears and thrusts) may then be evaluated for the various members ofthe structure and appropriate reinforcement designed.

The prototype staircase was analyzed using the above equations to deter-mine the redundants. The stairs were designed for a live load of 40 lb per sq ft.The design load was low as tests were to be carried out on a half-scale modelof the stairs and it was necessary to keep the ultimate load within reasonablebounds. With the dead loads included, the total design loading along the flightswas 597 lb per ft (w), and 528 lb per ft (W) along the landing.

FREE-STANDING STAIRCASE 603

Y

M t

Fig. A2 - Direction of positive momentvectors Mr 4=

2’X

sMS

To determine the effective second moment of area of the member OBC withrespect to the x-axis, one half the actual width of the landing was considered.Sievs made a similar assumption in his calculations.

The following values were used for the prototype:

a = 9.17 ft G 5s = 1305 in.4b = 5.90 ft

- = 0.435E zz tan-l 0.60

;= 4.59 ft I1 = 786 in.4 Tf = 12,100 in.4

1 = 2.30 ft Is = 274 in.4 Iz’ = 54,900 in.4

Replacing the appropriate values of the terms in Eq. (AlO) and (All), twosimultaneous equations were obtained as follows:

0.0614H - 0.00452Mo - 514.99 = 0 (AX!)

0.0182H + 0.01356Mo + 11.80 = 0 (A13)

whence

Mo = 11,770 ft-lb

H = 9235 lb

These values are then substituted in Eq. (Al) to (A9) to determine thebending and torsional moments given in Table 1. Equations for shear and nor-mal forces may also be written for each member and evaluated similarly. Adiagrammatic sketch of bending moments Mr is given in Fig. 12. New equationswould be required to be formulated from Eq. (AlO) and (All) using W = 344lb per ft for the case of loading on the flights only. The resulting values ofmoment are shown in Table 2.

REFERENCES1. Liebenberg, A. C., “The Design of Slab Type Reinforced Concrete Stair-

ways, ” The Structural Engineer (London), V. 38, No. 5, May 1960, pp. 156-164.2. Siev, A., “Analysis of Free Straight Multiflight Staircases,” Proceedings,

ASCE, V. 88, ST3, June 1962, pp. 207-232.3. Fuchssteiner, W., “Die Freitragende Wendeltreppe,” Beton-und Stahlbeton-

bau (Berlin), V. 49, No. 11, Nov. 1954, pp. 256-258.4. Sauter, F., “Free-Standing Stairs,” AC1 JOURNAL, Proceedings V. 61, No. 7,

July 1964, pp. 847-870.5. Gould, P. L., “Analysis and Design of a Cantilever Staircase,” AC1

.JOURNAL, Proceedings V. 60, No. 7, July 1963, pp. 881-899.6. Taleb, N. J., “The Analysis of Stairs with Unsupported Intermediate Land-

ings,” Concrete and Constructional Engineering (London), V. 59, NO. 9, Sept.1964, pp. 315-320.

604 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE

7. Cusens, A. R., and Kuang, Jing-Gwo, “Analysis of Free-Standing Stairsunder Symmetrical Loading,” Concrete and Constructional Engineering (London),V. 60, No. 5, May 1965, pp. 167-172.

8. Hajnal-K6nyi, K., “Test of a Staircase,” Concrete and Constructional En-gineering (London), V. 54, No. 1, Jan. 1959, pp. 25-27.

9. Mitchell, L. H., and Shaw, F. S., “Columnless Stairs,” ArchitecturalScience Review (Sydney), V. 5, No. 2, July 1962, p. 80.

10. Liebenberg, A. C., Discussion of “Analysis of Free Straight MultiflightStaircases,” Proceedings, ASCE, V. 89, ST5, Oct. 1963, pp. 251-254.

11. Siev, A., Closure to discussion of “Analysis of Free Straight MultiflightStaircases,” Proceedings, ASCE, V. 89, ST5, Oct. 1963, pp. 251-254.

12. Kemp, E. L.; Sozen, M. A.; and Siess, C. P., “Torsion in Reinforced Con-crete,” Structural Research Series No. 266, Civil Engineering Studies, Univer-sity of Illinois, Sept. 1961, 126 pp.

Received by the Institute Nov. 23, 1965. Title No. 63-29 is a part of copyrighted JOURNALof the American Concrete Inst i tute, Proceedings V. 63, No. 5, May 1966. Separate pr ints 81e

available at 60 cents each, cash with order.

American Concrete Inst i tute, P.O. Box 4754, Redford Station, Detroit , Michigan 48219

Discussion of this paper should reach ACI headquarters in triplicateby Aug. 1, 1966, for publication in the December 1966 JOURNAL.

(See p. iii for details.)

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Les auteurs decrivent les essais de chargement B rupture d’un mod&le B echelle$ d’un escalier autoportant en bbton arm4 B deux volees adjacentes B paillasseporteuse. Les methodes d’analyse sont comparCes g la lumi&re des rbultats ex-perimentaux et des recommandations generales pour le dimensionnement sontdon&es.

Experimentalstudie an einer freitragenden Treppe

Dieser Bericht beschreibt Belastungsproben bis zum Zerbrechen am Model1(Masstab 1: 2) einer symmetrischen, freitragenden Treppe aus Eisenbeton coder:aus Eisenbetonteilen). Die Untersuchungsmethoden werden im Licht der ex-perimentellen Ergebnisse verglichen; ferner werden allgemeine Konstruktions-vorschlHge gegeben.