Arching or compressive membrane action in reinforced concrete slabs

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The strength enhancing effects of arching action in reinforced concrete floors were first recognised near the beginning of last century.Westergaard, H.M. and Slater, W.A., ' Moments and stresses in slabs', Proceedings of the American Concrete Institute, 1921, Vol. 17, pp 415–538. However, it was not until the full scale destructive load tests by OcklestonOckleston, A.J., 'Load tests on a three-storey building in Johannesburg', The Structural Engineer, 1955, Vol. 33, October, pp 304–322.Ockleston, A.J., 'Arching action in reinforced concrete slabs', The Structural Engineer, 1958, Vol. 36, No.6, pp 197–201. on the Old Dental Hospital in Johannesburg that the extent of strength enhancement caused by arching action was really appreciated. In these tests, collapse loads of between 3 and 4 times those predicted by yield-line theoryJohansen, K.W., 'Brudlinieteorier', Jul. Gjellerups Forlag, Copenhagen, 1943, 191pp (Yieldline theory', translated by Cement & Concrete Association, London, 1962). were obtained.

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Since the 1950s there have been several attempts to develop theories for arching action in both one and two-way slabs.Wood, R.H., 'Plastic and elastic design of slabs and plates', Thames and Hudson, London, 1961.Christiansen, K.P., 'The effect of membrane stresses on the ultimate strength of an interior panel in a reinforced concrete slab', The Structural Engineer, 1963, Vol. 41, No. 8, pp 261–265.Leibenberg, A.C., 'Arch action in concrete slabs', National Building Research Institute Bulletin, 1966, No. 40. CSIR Research Report No. 234, Pretoria, S. Africa. One of the principal approaches to membrane action was that due to ParkPark, R., 'Ultimate strength of rectangular concrete slabs under short-term uniform loading with edges restrained against lateral movement', Proceedings Instn. Civ. Engrs, Vol.28, June 1964, pp. 125–150. which has been used as a basis for many studies into arching action in slabs. Park's approach was based on rigid plastic slab strip theory, and required the assumption of a critical deflection of one half of the slab depth at failure. Park's approach was later extended by Park and GamblePark, R. and Gamble, W.L., 'Reinforced concrete slabs', Wiley Interscience, New York, 1980, pp 562–612. in their method for predicting the plastic load-deformation response of laterally restrained slabs.

In 1971, the American Concrete InstituteAmerican Concrete Institute, 'Cracking, deflection and ultimate load of concrete slab systems', SP-30, Detroit, 1971, 382 pp.

produced a special publication which presented the most recent research, to that time, on arching and compressive membrane action in reinforced concrete slabs.

A comprehensive review of the literature and studies of both rigid-plastic and elastic-plastic approaches to arching have been compiled by BraestrupBraestrup, M.W., 'Dome effect in reinforced concrete slabs: rigid-plastic analysis’, J. Struct. Div., Proc Am. Soc. Civ. Engrs, 1980, Vol 106, No. ST6, pp 1237–1253. and Braestrup and Morley.Braestrup, M.W. and Morley, C.T., ‘Dome effect in reinforced concrete slabs: elastic-plastic analysis’, J. Struct. Div., Proc Am. Soc. Civ. Engrs, 1980, Vol 106, No. ST6, pp 1255–1262. Lahlouh and WaldronLahlouh, E.H. and Waldron, P., ‘Membrane action in one-way slab strips’, Proc. Instn. Civ. Engrs, Structs & Bldgs, 1992, Vol 94, Nov., pp 419–428. were some of the earliest researchers to achieve a degree of success in finite element modelling of the phenomenon. In 1993, Kuang and MorleyKuang, J. S. and Morley, C. T., ‘A plasticity model for the punching shear of laterally restrained concrete slabs with compressive membrane action’, International Journal of Science, Vol. 35, No. 5, 1993, pp 371–385. presented a plasticity approach which included the effect of compressive membrane action on the punching shear strength of laterally restrained concrete slabs.

In the United Kingdom, the method developed by Kirkpatrick, Rankin & LongKirkpatrick, J., Rankin, G.I.B. and Long, A.E., 'Strength evaluation of M-beam bridge deck slabs', The Structural Engineer, Vol. 62B, No 3, Sept 1984, pp 60–68. in 1984 and substantiated by testing a full-scale bridge in 1986Kirkpatrick, J., Rankin, G.I.B. and Long, A.E., 'The influence of compressive membrane action on the serviceability of beam and slab bridge decks', The Structural Engineer, Vol. 64B, No 1, March 1986, pp 6–9 and 12. first led to the introduction of new rules for the economic design of reinforced concrete beam and slab bridge decks in Northern Ireland.Department of the Environment, Northern Ireland, 'Design of M-beam bridge decks', Amendment No. 3 to the Bridge Design Code, Northern Ireland Roads Service Headquarters, March 1986, 11.1–11.5. The concept and method were later incorporated, by the United Kingdom Highways Agency, into the UK design manual for roads and bridges, BD 81/02, 'Use of Compressive Membrane Action in Bridge Decks'.United Kingdom Highways Agency, 'Use of compressive membrane action in bridge decks', Design Manual for Roads and Bridges, Vol. 3, Section 4, Part 20, BD 81/02, 2002. Use of this CMA methodology normally results in substantial savings in reinforcement in the slab of a beam and slab bridge deck, provided certain limitations and boundary conditions are satisfied.

File:Punching failure in beam and slab bridge deck.png

File:Bridge deck - punching failure top cracking.png

File:Bridge deck - punching failure bottom cracking.png

Kirkpatrick, Rankin & Long's approach to the prediction of the enhanced punching strength of bridge deck slabs was based on the punching shear prediction equation derived by LongLong, A.E., 'A two-phase approach to the prediction of the punching strength of slabs', Journal of the American Concrete Institute, Proceedings, Vol.72, No.2, February 1975, pp 37–45. for the shear mode of punching failure, combined with an effective reinforcement ratio, which represented the arching action strength enhancement. The effective reinforcement ratio was determined from the maximum arching moment of resistance in a rigidly restrained concrete slab, which RankinRankin, G.I.B., 'Punching failure and compressive membrane action in reinforced concrete slabs', PhD Thesis, Dept of Civil Engineering, Queen's University of Belfast, 1982, 334 pp. had derived for laterally restrained concrete slabs from McDowell, McKee and Sevin'sMcDowell. E.L., McKee, K.E. and Sevin. E. 'Arching action theory of masonry walls', Journal of the Structural Division, Proceedings, American Society of Civil Engineers, 1956, 82, No. ST2, 915-1–915-18. arching action deformation theory for masonry walls. The derivation of the maximum arching moment of resistance of laterally restrained concrete bridge deck slabs utilised Rankin's idealised elastic-plastic stress-strain criterion for concrete, valid for concrete cylinder strengths up to at least 70N/mm², which he had derived on the basis of Hognestad, Hanson and McHenry'sHognestad, E, Hanson, N.W. and McHenry, D., 'Concrete stress distribution in ultimate strength design', Journal of the American Concrete Institute, Proceedings, Vol.52, No.6, December 1955, pp 455–479. ultimate parabolic stress block coefficients for concrete.

The adaptation of Kirkpatrick, Rankin & Long's punching strength prediction method for laterally restrained bridge deck slabs, given in BD 81/02, is summarised as follows:

The concrete equivalent cylinder strength, $f\_\{\backslash text\{c\}\}$, is given by:

{{NumBlk|:|$f\_\{\backslash text\{c\}\}\; =\; \backslash frac\{\; 0.8f\_\{\backslash text\{cu\}\}\; \}\; \{\backslash gamma\_\{\backslash text\{m\}\}\; \}$|{{EquationRef|Equation 1}}}}

The plastic strain value, $\backslash varepsilon\_\{\backslash text\{c\}\}$, of an idealised elastic-plastic concrete is given by:

{{NumBlk|:|$\backslash varepsilon\_\{\backslash text\{c\}\}\; =\; \{\; \backslash left(\; -400+60f\_\{\backslash text\{c\}\}-0.33f\_\{\backslash text\{c\}\}\; ^2\backslash right)\; x\; 10^\{-6\}\}$|{{EquationRef|Equation 2}}}}

The non-dimensional parameter, $R$, for the arching moment of resistance is given by:

{{NumBlk|:|$R\; =\; \backslash frac\{\; \backslash varepsilon\_\{\backslash text\{c\}\}\; L\_\{\backslash text\{r\}\}\; ^2\; \}\; \{h^2\}$|{{EquationRef|Equation 3}}}}

In order to treat the slab as restrained, $R$ must be less than 0.26. If $R$ is greater than 0.26, the deck slab shall be treated as if it were unrestrained.

The non-dimensional arching moment coefficient, $k$, is given by:

{{NumBlk|:|$k\; =\; \{\; 0.0525\backslash left(\; 4.3-16.1\backslash sqrt\{3.3\; x\; 10^\{-4\}\; +0.1243R\}\backslash right)\; \}$|{{EquationRef|Equation 4}}}}

The effective reinforcement ratio, $\backslash rho\_\{\backslash text\{e\}\}$, is given by:

{{NumBlk|:|$\backslash rho\_\{\backslash text\{e\}\}\; =\; k\backslash left\; [\; \backslash frac\{f\_\{\backslash text\{c\}\}\; \}\{240\}\; \backslash right\; ]\; \backslash left\; [\; \backslash frac\{h\}\{d\}\; \backslash right\; ]^2$|{{EquationRef|Equation 5}}}}

The predicted ultimate punching load for a single wheel, $P\_\{\backslash text\{ps\}\}$ (N), is given by:

{{NumBlk|:|$P\_\{\backslash text\{ps\}\}\; =\; \{\; 1.52\backslash left(\; \backslash phi+d\backslash right)d\backslash sqrt\{f\_\{\backslash text\{c\}\}\; \}\backslash left(100\backslash rho\_\{\backslash text\{e\}\}\; \backslash right)^\{0.25\}\; \}$|{{EquationRef|Equation 6}}}}

where:

• $d$ = average effective depth to tensile reinforcement (mm)
• $f\_\{\backslash text\{cu\}\}$ = characteristic concrete cube strength (N/mm²)
• $h$ = overall slab depth (mm)
• $L\_\{\backslash text\{r\}\}$ = half span of slab strip with boundary restraint (mm)
• $\backslash phi$ = diameter of loaded area (mm)
• $\backslash gamma\_\{\backslash text\{m\}\}$ = partial safety factor for strength

Further details on the derivation of the method and how to deal with situations of less than rigid lateral restraint are given by Rankin and Rankin & Long.Rankin, G.I.B. and Long, A.E. (1997), ‘Arching action strength enhancement in laterally restrained slab strips’, Proc. Instn. Civ. Engrs Structs & Bldgs, 122, Nov., pp 461–467. Long and RankinLong, A.E. and Rankin, G.I.B., ‘Real strength and robustness of reinforced concrete structures’, Proceedings of conference on Conservation of Engineering Structures, Institution of Civil Engineers/Royal Institute of British Architects, 1989, pp 47–58. claim that the concepts of arching or compressive membrane action in beam and slab bridge decks are also applicable to flat slab and cellular reinforced concrete structures where considerable strength enhancements over design code predictions can also be achieved.

Research into arching or compressive membrane action has continued over the years at Queen's University Belfast, with the work of Niblock,Niblock, R., 'Compressive membrane action and the ultimate capacity of uniformly loaded reinforced concrete slabs, PhD thesis, The Queen's University of Belfast, 1986.Rankin, G.I.B., Niblock, R.A., Skates, A.S. and Long, A.E., 'Compressive membrane action strength enhancement in uniformly loaded, laterally restrained slabs', The Structural Engineer, Vol 69, No. 16, 20 August 1991, pp 287–295. who investigated the effects of CMA in uniformly loaded laterally restrained slabs; Skates,Skates, A.S., Development of a design method for restrained concrete slab systems subject to concentrated and uniform loading, PhD thesis, The Queen's University of Belfast, 1987. who researched CMA in cellular concrete structures; Ruddle,Ruddle, M.E., 'Arching action and the ultimate capacity of reinforced concrete beams', PhD thesis, The Queen's University of Belfast, February 1989.Ruddle M.E., Rankin G.I.B. and Long A.E., 'Arching action–flexural and shear strength enhancements in rectangular and Tee beams', Proceedings of the Institution of Civil Engineers, Structures and Buildings Journal, 156, Issue 1, February 2003, pp 63–74. who researched arching action in laterally restrained rectangular and Tee-beams; Peel-Cross,Peel-Cross, R.J., Rankin, G.I.B., Gilbert, S.G. and Long, A.E., ' Compressive membrane action in composite floor slabs in the Cardington LBTF', Proceedings of the Institution of Civil Engineers, Structures and Buildings Journal, 146, Issue 2, May 2001, pp 217–226. who researched CMA in composite floor slab construction; TaylorTaylor, S.E., 'Compressive membrane action in high strength concrete bridge deck slabs', PhD thesis, The Queen's University of Belfast, January 2000.Taylor, S.E., Rankin, G.I.B. and Cleland, D.J., (2001) 'Arching action in high strength concrete slabs', Proceedings of the Institution of Civil Engineers, Structures and Buildings, Vol.146, Issue 4, Nov.2001 pp 353–362Taylor, S.E., Rankin, B., Cleland, D.J, and Kirkpatrick, J., 'Serviceability of bridge deck slabs with arching action', American Concrete Institute Structural Journal, Vol. 104, No.1 January–February 2007, pp 39–48. who researched CMA in high strength concrete bridge deck slabs, and ShaatShaat, A.J.S., 'The real strength of laterally restrained reinforced concrete slabs, PhD thesis, Queen's University of Belfast, 2005. who researched CMA using Finite Element Analysis (FEA) techniques. A comprehensive guide to compressive membrane action in concrete bridge decks, was compiled by Taylor, Rankin and Cleland in 2002.Taylor, S.E., Rankin, G.I.B. and Cleland, D.J., 'Guide to compressive membrane action in concrete bridge decks', Technical Paper 3, Concrete Bridge Development Group, Camberley, Surrey, 2002, 46 pp.

In North America, a more pragmatic approach has been adopted and research into compressive membrane action has primarily stemmed from the work of Hewitt and BatchelorHewitt , B.E. and Batchelor, B. de V., 'Punching shear strength of restrained slabs', J. Struct. Div., Proc. ASCE, Vol. 101, No. ST9, September 1975, pp 1837–1853. and Batchelor and TissingtonBatchelor, B. de V. and Tissington, I.R., 'Shear strength of two-way bridge slabs', J. Struct. Div., Proc. ASCE, Vol. 102, No. ST12, December 1976, pp 2315–2331. in the 1970s. They carried out an extensive series of field tests, which led to the introduction of an empirical method of design into the Ontario Highway Bridge Design Code in 1979.Ontario Ministry of Transportation and Communication, 'The Ontario highway bridge design code', 1979, Toronto, Ontario, Canada. This required minimum isotropic reinforcement (0.3%) in bridge deck slabs, provided certain boundary conditions were satisfied. In the 1990s Mufti et al.Mufti, A. A., Jaeger, L. G., Bakht, B. and Wegner, L.D., 'Experimental investigation of fibre reinforced concrete deck slabs without internal steel reinforcement,' Canadian Journal of Civil Engineering, 1993, Vol. 20, No.3, pp 398–406. extended this research and showed that significant enhancements in the durability of laterally restrained slabs can be achieved by utilising fibre reinforced deck slabs without steel reinforcement. Later, Mufti and NewhookMufti, A. A. and Newhook, J.P., 'Punching shear strength of restrained bridge deck slabs', ACI Structures Journal, 1998, 8(3), pp 375–381. adapted Hewitt and Batchelor's model to develop a method for evaluating the ultimate capacity of fibre reinforced deck slabs using external steel straps for the provision of lateral restraint.

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Category:Structural engineering