Paper: ODwyer/OBrien
Paper
Design and analysis of concrete slabs using a modified strip
methodD. W. ODwyer, BE, MEngSc, CEng, MIEI Trinity College, Dublin
Professor E. J. OBrien, BE, MEngSc, PhD, CEng, MIStructENational
University of Ireland
Synopsis This paperdescribes a technique for the ultimate limit
state analysis or design of reinforced concrete slabs. The modified
strip method presented is a development of the strip technique
using linear and quadratic programing optimisation. The first stage
of the procedure uses linear programing to find the distribution of
loads between the strips which, in the analysis case, leads to the
maximum collapse load factor or, in the design case, the minimum
reinforcement requirement. The load distributions leading to the
largest collapse load factor or minimum reinforcement requirement
are not unique. The second stage of the procedure uses quadratic
programing to search among the optimal distributions to find the
load distribution which minimises the sum of squares of the
differences between the elastic moments and the strip moments. This
results in a solution which does not deviate excessively from the
results of a linear elastic analysis. The method is of particular
use in the design of slabs which have irregular patterns of
loading, irregular shape, irregular support conditions or which
include openings. IntroductionThe design of reinforced concrete
slabs an everydaystructural engineeris ing task. Where the slab
tobe designed is rectangular and where the support conditions
areregular, a popular approach is to use graphs ortables to
determine the distribution moments. Such tables are of generally
theresult of linear elasticanalyses for uniformly distributed loads
acting on slabs of constant thickness whose supports, whether
pinned or encastr6, uniform are and infinitely stiff. Although
these conditions met in practice, designare not ers often assume
that irregularities in the support conditions, loading and slab
stiffness can be accommodated by small plastic deformations in the
slab. Such techniques are useful for the design of standard slabs
but are clearly unsuited for more complex situations.
at reinforcement top and bottom, and encastrk its edges, is
illustrated in Fig 1. For the mechanism assumed in 1 (a), the
relationshipbetween thecolFig lapse load,qc,and the plastic moment
capacity, % is2: ,
Fig I (a).Hinge patternfor simple yieldline analysis
Plastic assumption and yieldline methodThe assumption that
reinforced concrete slabs behave plastically justified is because
they are generally under-reinforced2. Load tests on
under-reinforced concrete slabs3 indicated that failure occurs
after formation have the of yieldlines, alongwhich the tensile side
of the slab cracks and the reinforcement yields. If the shape,
loading and support conditions of a slabare regular, it is possible
topredict the mechanism (the pattern of yieldlines), by which it
will faiP4. By examining the virtual work equations associatedwith
any given mechanism, it is possible calculate imposed load required
cause to the to the mechanism to form. The mechanism will form when
therate of change in potential energyof the loaddue to formation of
the mechanismexceeds the or equals the rate at which energy is
expended in forming it. The yieldline method has significant
disadvantages. Firstly, it generates of an upper bound on the
collapse load. For any assumed pattern yieldlines, the predicted
collapse load will exceed or equal the true value. The difference
between the two is a function of how closely the assumed pattern of
yieldlines agreeswith the pattern at failure. If the critical
yieldline pattern has been identified, the calculated collapseload
will be correct. A second disadvantage is that the yieldline
technique does not give the support reactions along the slabs edge.
This a particular disadvantage is where the slab is supported by
edge beams whose designis dependent onhow the slab transfers load
onto them. pattern of true collapse mechanism The yieldline
analysis a square, isotropically reinforced slab, with equal Fig
I(6).Form of hinge of
The Structural EngineerVolume
76/No 17
1 September 1998
329
Paper: ODwyer/OBrien
&=4xmP
....(1)
The relationship between the collapse load and the plastic
moment capacity calculated using the correct hinge pattern,
illustrated in Fig 1(b), gives5:4 L2 C= 42.85 1 mP
....(2)
For this example, the yieldline analysis based on the simple
hinge pattern overestimates the collapse load by 12%. The critical
yieldline patterns can be readily predicted if the slabs shape,
loading and support conditions are straightforward. However, when
the slab shape or the support or loading conditions are complex,
identifying the critical pattern becomes more difficult.
The strip methodThe strip method, introduced by Hillerborg6 and
developed by Wood & Armer, is also based on the assumption that
reinforced concrete slabs behave plastically. However, the strip
method, unlike yieldline analysis, provides a lower bound on the
collapse load. The strip technique is based on the lower bound or
safe theorem of plasticity. It involves findinga set of moments
which are in equilibrium with the loads on the structure and which
do not exceed the plastic moment capacity of the slab at any point.
The safe theorem ensures that the collapse load factor associated
with any statically admissible set of moments will be less than, or
equal to, the true collapse load factor. Consider the segment of
slab and the associated moments and shear forces illustrated in Fig
The condition which must be satisfied for the ele2. ment to be in
vertical equilibrium is:...(3) &y a where m, and my are
moments/unit length the X andY faces, respectively, on represents
the torsional momenthnit length. Hillerborg suggested and mXy m,
and my accordingly. setting the torsional term to zero and choosing
Setting the torsional term in eqn (3) to zero gives:ax2
Fig 3 . Hillerborgs strip method
d2mx +-+2--=-4 d2my
d2mxy
ayLv2
other direction. Further, the value is not restricted to being
between zero of a and unity. However, when Hillerborgs method is
appliedhand, the value by of a for each sectionof slab is often set
to either zero or one. Once the load has been allocated between the
notional slabs spanning in the two orthogonal directions, the
one-way-spanning slabs are subdivided into strips and each strip is
analysed as an independent beam. As they are assumed to behave
independently, the with the lowest collapse load facstrip tor
dictates the overall collapse load - e.g. symmetry would suggest
sharing the load on the slab of Fig I equally between the two
orthogonal directions, i.e. a = X. This assumption leads to the
following relationship between the collapse load and the plastic
moment: &=32mP
....(6)
...which can be rewritten as:
which is 75% of the true collapse load. In contrast,linear
elastic analysis* a gives the relationship between the maximum
moment and the applied uniform load as:-- - 19.34 qL2mmax
....(7 )
...(5 )d=-(l-a)qay2
where a is a factor reflecting the degree of loadsharing between
the two orthogonal directions. Hillerborgs technique involves
sharing the load at each point on the slab between two notional
one-way spanning slabs which span in the reinforcement directions
which are usually orthogonal, illusas trated in Fig 3. The value of
a determines how the loadis divided between the X andY directions.
Hillerborgs approach does not require that the load on a given
element be carried by spanning exclusively in either one or the
1
which is 45% of true collapse load and only of the
Hillerborgstrip the 59% result. If the objective is design rather
than analysis, eachstrip is designed as an independent beam
and,this manner, the reinforcement requirements in in the X and Y
directions in the real two-way spanning slab are calculated. Both
the yieldline method and the Hillerborg strip technique are listed
as acceptable design methods inBS 8 1 10 and the draft Eurocode
EC2.
Analysis using the modified strip methodIf a slab is subdivided
into a grid of rectangular elements as illustrated in then,
depending on how the load is shared between the andY strips X Fig
4,intersecting at each element (i.e. the value of a at each
element), a differ-
myx +
-6Y 6V
6myx
mxy + -6x6X
6mxy
Fig 2. Slab element
Fig 4. Subdivision of slab into a series of orthogonal
strips
330
TheStructuralEngineerVolume
76/No 17
1
September 1998
Paper: ODwyedOBrien
ent allowable load will be obtained. Assuming infinite
ductility, any manTABLE l ner of sharing the load will generate an
acceptable lower-bound solution to the collapse load. The objective
in analysis find the pattern load disis to of tribution which
givesthe maximum collapse load factor,h. Eqn (4), the basis of
Hillerborgs analysis, is the vertical equilibrium equation for a
grillage model in which the torsional capacity is ignored. Applying
numerical approximations for the derivatives in eqn 4 at(ij) node
and including the load factor gives:
- Form
of additional constraints for each type of edge condition
- MP= < Edge
moments = < MP
Simply supported edge Edge moments = 0Free edge Moments
perpendicular to the edge at 1st internal node = 0
where mXij myij are the moments/m in theX and Y directions,
respecand tively, evaluated at node (ij) and ijy and 6y are the
corresponding intervals (i,j). between nodes. Wij is the portion of
vertical load applied to node This equation must be satisfiedfor
node i,j to be in vertical equilibrium. A similar equilibrium
equation can be formulated for each node in the grillage, i.e. for
each rectangular element in the In matrix form these equations
slab. can be expressed as: ....(9) where[mX]and [my]
[l wmatrix theis [Cl
are the matrices of moments/m in the X and Y directions,
respectively, evaluated at the nodes the matrix is of appliedand
loads of coefficients:00
-2 1 0 . . 1-2 1 0 0 1 - 2 . 0 [c]=0 0 1 0 . . 1 0 0 -2 0 0 0
1-
.
.
1
2
Clearly, at the boundaries, the relevant equations will be
replaced by the boundary conditions. Any statically admissible
setof moments must satisfy all of these equilibrium equations. In
addition, for a set of moments to be acceptable, the magnitude of
moment at each node, in each direction, must not exceed the plastic
moment capacity of the slab at that point. Therefore, in addition
to the equilibrium equations, there are two additionalof
constraints at each sets node:,.(lo)
ject to the assumptions of plasticity, and ensures the safety of
the slab. However, the moments which the linear programing
algorithm finds may differ greatly from the moments that actually
occur. This leads to two poten of tial difficulties - first, the
designer will be sceptical the results and, secondly, the real
reinforced concrete slab may not be able to accommodate th plastic
rotations requiredto achieve the optimised moments. It is clearly
of interest to find a solution that similar to what will actually
occur and that is requires a minimum amount plastic hinge rotation.
of This information cannot be obtained directly from the linear
programing algorithm because the number optimal solutions may be
very large or infiof nite. However, once the optimum collapse load
factoris known, an approby priate distribution of moments can be
found reformatting the problem. One approach to minimising the
needplastic hinge rotation and which for should find a distribution
moments similar to the moments obtained from of a linear elastic
analysis, and FzYi,j,is to minimise the magnitude of the greatest
difference between the linear programing moments and the elastic
moments. This could be formulated as a linear programing problem.
An alternative and more traditional approach whichakin to curve is
fitting is to minimise the sumof the squares of the differences
between the linear programming and the elastic moments. Thus the
objective function in secthis ond stage of the optimisation is to
minimise:
...( 1) 1
subject to eqn(9) and inequalities (10) and ( l 1) and subject
to the equality that h equals its collapse load value. This is a
quadratic programing problem and can be solved formulatby ing the
Kuhnnucker conditions and using a modified linear programing
technique to find a feasible solution... The value of the collapse
load factor h may have to be reduced if the amount of
redistribution is excessive; BS 8 110 requires that the maximum
elastic moments are not reduced more than 30% through
redistribution. by
The modifiedstrip method was used to calculate the collapse load
factor for a uniformly distributed load applied to the slab shown
in Fig The slab is 5. 5m square and simply supported on three sides
the fourth side with free. The uniform load was modelled as a
series point loads appliedat the centres of of one 100 0.5m x 0.5m
square elements.The point loads were250N, repof resenting a
uniformly distributed load 1kN/m2. An equilibrium equation with the
form (8) was formulatedfor each of eqn loaded node. The moments all
nodes, both loaded and support, were conat Linear and quadratic
optimisation strained to be less than the known local moment
capacity the slab; conof Only the moments and the collapse load
factor in eqn (9) are unknown. straints with the form of eqns (10)
and (1 1). The slab was assumed to be Therefore, the problem that
of maximising the collapse load factor, is h, subisotropically
reinforced with a plastic moment capacity of 20kNm/m in ject to the
linear eqns (9) and inequalities (10) and (11). This can
bereadiboth hogging and sagging. ly formulated as a standard linear
programing problem and, once the The free edge incorporatedby
setting the mXij was moments to zero at the problem is expressed in
this form, standard packagescan be used to find the optimum
collapse load. Additional constraints must be added to the formu-
slabs free right-hand edge andat the loaded node nearest this edge,
since no vertical reactionis applied at the edge nodes. lation to
take account of boundary conditions at the slabs edges and supThe
initial phase of the optimisation found the maximum collapse load
ports. The form of the additional constraints for each typeof edge
condition factor to beh = 14.4, i.e. a maximum UDLof 14.4kN/m2. is
summarised in Table 1. The second stage in the procedure found,
from among the optimal soluLinear programing will yield the maximum
collapse load factor h subject to the given constraints. However,
the set of moments which the linear tions, the setof moments for
which the sumof squares of differences from the elastic
distribution of moment was a minimum. elastic distribution The
programing algorithm finds correspondingto the optimum collapse
loadh used was for a UDL 14.4kN/m2 and included an allowance the
twistof for is often not unique, i.e. there may be a large number
of possible sets of ing moments, mXYij accordance with the in Wood
& Armer equations7.The moments which satisfy the equilibrium
equations and moment constraints elastic moment distribution is
illustrated in Fig 6 and the optimal solution and which result in
the maximumh. Each of these solutions is valid, sub-
where [mxIhog, [myIhog and [mXlsag, [mYISag moment capacitiesat
the are the nodes for hogging and sagging moment, respectively. set
of moments Any which satisfies eqn (9) and does not violate
constraints(10) and (1 1) at all the nodes in the slab is a
statically admissible solution and guarantees a h, lower bound. The
object to find the maximum value the load factor, is of for which
an acceptable solution exists, i.e. as high a lower bound on the
collapse load factoras possible.
Analysis example using the modified strip method
TheStructuralEngineerVolume
76INo 17
1
September 1998
331
Paper: ODwyerlOBrien
closest tothis elastic distribution is shown graphically in
Fig7. The elastic moment distribution had a peak moment of
31.2kNm/m which is 56% would limit the maxgreater than the
allowable moment. An elastic analysis imum allowable load to
9.2kN/m2. The limits onmoment redistribution in BS 8110 would limit
the maximum allowable peak elastic moment to 20kNm/m + 30% or
26kNm/m which corresponds to h = 12 and a maximum UDL of 12
kN/mz.
R
l
-4 A d
l
!
Design usingthe modified strip methodThis modified strip
technique can be used to greatestadvantage in design. The design
formulation is similar to the analysis formulation except that the
design objective is to minimise the amount of reinforcement
required. When a slab thicknessis adopted, the moment capacity at
any point is a function of the area reinforcement at that location.
Hence minimising the amount of of reinforcement required is
approximately equivalent to minimising the sum of absolute values
of the moments. The sum of themoments throughout a slab is
sometimes referred to as the moment volume7. The equilibrium
condition must be satisfied at eachnode but the design load is
known.Therefore theload factor isremoved from eqn (9) to give:
...Fig 6(a).Elastic distributiono moments inX direction f
Constraint eqns (10)and ( l 1) remain unchanged, with the
important difference that the plastic hogging and sagging moment
capacities [mxIhog, [myIhog [mx]%,[my]% are now the unknowns. The
objective function, and for the most general case, is thesum over
allnodes of the absolutevalues of the four moment capacities at
each node.
Allowing the plastic moment capacities to fluctuate from onenode
to the next may not be feasible in practice. It is up to the
designerto dictate the level of variation. sensible approach to
adopt is to A allow the reinforcement in each strip tovary
independently of adjacent strips and to allow the reinforcement
within the strip to vary according to the reinforcement details.
The pattern of reinforcement will often be obvious before bar sizes
and spacings are calculated. Controlling the level of variation is
achieved by limiting the number of unknown moment capacity
variables in eqns (10)and (1 l), i.e. a single variable may be used
to describe themoment capacity at allnodes in a strip. By providing
reinforcement such that the moment capacity of the moments, the
safetheorem guaranslab equalsor exceeds the equilibrium tees the
safety of the slab. Wood & h e r 7 examined the strip method
and found that, if the reinforcement in the slab, which dictates
the ultimate moment field, is provided to correspond precisely with
the equilibrium
Fig 6(b).Elastic distributionof moments inY direction
Fig 5. Square slab with uniform reinforcement top and bottom
encastre on three sides
moment field, the collapse load will equal thedesign load. Hence
this optimisation technique can lead tovery efficient designs. The
requirement that the amount of redistribution is limited can be
achieved by adding constraints which ensure that the plastic moment
of resistance provided in any strip is not more than 30% less than
the maximum elastic moment in the strip. The second, quadratic,
stage the optimisation is of particular imporof tance where the
method is used in design. In the design formulation, the initial
linear programing analysis calculates the minimum amount of
reinforcement required. However, as was the case in the
analysisformulation, there is morethan one way to reinforce the
slab the minimum amount with of reinforcement while achieving the
design load. Of the many possible ways of reinforcing the slab
using the minimum amount of reinforcement, the designer must
findthat pattern which will give satisfactory performance at
serviceloads. As with the analysis case,this can be achieved by
finding the set moments for which the sum of the squares of
differences from the of elastic distribution of moment is a
minimum, i.e. eqn (12). . Consider applying the modified strip
method todesign the perforated slab shown in Fig 8. The positions
of the holes in the slab dictate the way in which this slab can be
reinforced. This is clear without evenconsidering how the load
islikely to be distributed. Fig 8(a)shows the reinforcement bands
in the X direction, while Fig 8(b) shows those for theY direction.
Within each band the curtailment details will dictatewhich sections
have similar moment capacities. Using this procedure, the practical
reinforcement details
332
TheStructuralEngineerVolume
76/No 17
1 September 1998
Paper: ODwyedOBrien
X direction
_ _ 1 (
Fig 8(a).Design strips spanning in theX direction
Fig 7(a). Optimum distributionof moments in X direction
Fig 8(b).Design strips spanning in theY direction
Fig 7(b).Optimum distributionof moments inY direction
of the holes, and using the techniqueto size the reinforcement.
The modified strip method can also generate the reactions applied
to the supports, which is important for designing edge beams,
internal columns, and support structures.
dictate the reinforcement pattern, and the lineadquadratic
programing modified strip method provides the optimum area of
reinforcement, A,, for each section of each band. It is not
essential the mesh be very dense forthis procedure to give that
good results. The improvement gained by increasing the number of
nodes/ strip is similar to the improvement in the accuracy of a
bending moment diagram for a uniformly distributed loading as the
number of point loads used to model the loading is increased.
References1. Reynolds, C. E., Steadman,J. C.: Reinforced
concrete designershandbook, loth ed., Spon, 1988 2. Moy, Stuart S .
J.: Plastic methods for steel and concrete structures, Macmillan
Publishers Ltd, 1981 3. Armer, G . S . T.: Ultimate load tests of
slabs designed by the strip method, Proc. ICE, 41, October 1968, pp
3 13-331 4. Jones, L. L., Wood, H.: Yieldlineanalysis of slabs,
London, Thames R. & Hudson, 1967 . 5 . Fox, E.N : Limit
analysis for plates: The exact solution for a clamped square plate
of isotropic homogeneous material obeying the square yield
criterion and loaded by uniform pressure, Proc. Royal Society A,
277, August 1974, pp 121-155 6. Hillerborg, A.: Strip method of
design, Viewpoint Publications, 1975 7. Wood, R. H., Armer, G. S.
T.: The theory of the strip method for design of slabs, Proc. ICE,
41, October 1968,pp 285-31 l 8. Timoshenko, S. P.,
Woinowsky-Krieger, S.: Theory of plates and shells, 2nd ed., New
York,McGraw-Hill Kogakusha Ltd, 1959 9. Taha, H. A.:
OperationsResearch, 4th ed., London, Collier MacMillan Publishers,
1987 10. Saaty, L ,and Bram, Non-linear mathematics, . J.: New
York, McGraw Hill Book Company Ltd, 1964 11. Schrage, Li.: UNDO -
An optimisation modelling system, 4th ed., Scientific Press.
1991
ConclusionsThe modified strip method presented here isa
development of the standard strip technique. The method uses
standard mathematical programing techniques to calculate the
optimum collapse load factor when used to analyse a slab and the
minimum reinforcement requirement when used to design a slab. The
modified strip method is based on the same assumptions of
plasticity as the stripmethod and thus generates a lower-bound
estimate on the collapse load factor. Since it makes no new
assumptions and is simply an optimisation of the standard strip
method, it is allowed by many design Codes including,the draft
Eurocode, EC2, subject limits on the to amount of redistribution.
The technique is ideally suited to the design of slabs with holes
or with complex support conditions or shape. It does not require
the designer to assume either thepattern of load distribution or
the moment capacities of the slab. It can be used most efficiently
by having the designer input the pattern of reinforcement, which in
a perforated slab is dictated by the positions
The Structural EngineerVolume
76/NO 17
1 September 1998
333
