Concept of a Pseudo-Lower Bound Solution for Reinforced Concrete Slabs

In construction industry, reinforced concrete (RC) slabs
represent fundamental elements of buildings and bridges. Different
methods are available for analysing the structural behaviour of
slabs. In the early ages of last century, the yield-line method has
been proposed to attempt to solve such problem. Simple geometry
problems could easily be solved by using traditional hand analyses
which include plasticity theories. Nowadays, advanced finite element
(FE) analyses have mainly found their way into applications of
many engineering fields due to the wide range of geometries to
which they can be applied. In such cases, the application of an
elastic or a plastic constitutive model would completely change the
approach of the analysis itself. Elastic methods are popular due to
their easy applicability to automated computations. However, elastic
analyses are limited since they do not consider any aspect of the
material behaviour beyond its yield limit, which turns to be an
essential aspect of RC structural performance. Furthermore, their
applicability to non-linear analysis for modeling plastic behaviour
gives very reliable results. Per contra, this type of analysis is
computationally quite expensive, i.e. not well suited for solving
daily engineering problems. In the past years, many researchers have
worked on filling this gap between easy-to-implement elastic methods
and computationally complex plastic analyses. This paper aims at
proposing a numerical procedure, through which a pseudo-lower
bound solution, not violating the yield criterion, is achieved. The
advantages of moment distribution are taken into account, hence the
increase in strength provided by plastic behaviour is considered. The
lower bound solution is improved by detecting over-yielded moments,
which are used to artificially rule the moment distribution among
the rest of the non-yielded elements. The proposed technique obeys
Nielsen’s yield criterion. The outcome of this analysis provides a
simple, yet accurate, and non-time-consuming tool of predicting the
lower-bound solution of the collapse load of RC slabs. By using
this method, structural engineers can find the fracture patterns and
ultimate load bearing capacity. The collapse triggering mechanism is
found by detecting yield-lines. An application to the simple case of
a square clamped slab is shown, and a good match was found with
the exact values of collapse load.

Authors:

• M. De Filippo
• J. S. Kuang

Keywords:

• Computational mechanics
• lower bound method
• reinforced concrete slabs
• yield-line.

References:

[1] Braestrup M.W., Yield-line theory and limit analysis of plates and slabs,
Magazine of Concrete Research, 22, 99-106, 1970.
[2] Code of Practice for Structural Use of Concrete, The Government of the
Hong Kong Special Administrative Region, 2013.
[3] Calladine C.R. (2008), ”Plasticity for engineers”, Hellis Horwood,
Chichister, UK.
[4] Cook R.D., Malkus D.S., Plesha M.E., Witt R.J., Concepts and
applications of Finite Element analysis, Wiley, fourth edition, 2002.
[5] Gilbert M., He L., Pritchard T., ”The yield-line method for concrete slabs:
automated at last”, The Structural Engineer, 93, 44-48, 2015.
[6] Ingerslev A. , ”The strength of rectangular slabs”, The Structural
Engineer, 1, 3-14, 1923.
[7] Jackson A.M., Middleton C.R., ”Closely correlating lower and upper
bound plastic analysis of real slabs” , The Structural Engineer, 91, 34-40,
2013.
[8] Johansen K.W., ”Yield-line theory”, London, UK: Cement and Concrete
Association, 1962.
[9] Mac Lane S., Birkhoff G. (1991) Algebra, American Mathematical
Society, p. 145, ISBN 0-8218-1646-2.
[10] Mathworks, MATLAB R2012a, The MathWorks Inc., 2012.
[11] Morley C.T., Yield criteria for elements of reinforced concrete slabs,
IABSE, 1979.