INTRODUCTION OF STRIP DESIGN METHOD:
Scope
The strip method is in principle a method for designing slabs so that the safety against bending failure is
sufficient. As opposed to the yield line theory it gives a safe design against bending failure. The strip method
does not in itself lead to a design which is close to that according to the theory of elasticity, nor does it take shear
or punching failure into account.
The development of the rules for practical application involved in many cases rather complicated discussions and
theoretical derivations, which were necessary in order to prove that the resulting practical rules rested on a solid
theoretical basis
In order to make the application of the method to practical design as simple as possible some approximations
have been used which have been estimated to be acceptable even though the acceptability has not been strictly
proved. Even with these approximations the resulting designs are probably safer than many accepted designs
based on yield line theory, theory of elasticity or code rules.
Strip method versus yield line theory
The yield line theory is based on the upper bound theorem of the theory of plasticity. This means in principle that
a load is found which is high enough to make the slab fail, i.e. the safety in the ultimate limit state is equal to or
lower than the intended value. If the theory is correctly applied the difference between the intended and the real
safety is negligible, but there exists a great risk that unsuitable solutions may be used, leading to reduced safety
factors, particularly in complicated cases like irregular slabs and slabs with free edges.
With the strip method the solution is in principle safe, i.e. the real safety factor is equal to or higher than the
intended. If unsuitable solutions are used, the safety may be much higher than the intended, leading to a poor
economy. From the point of view of safety the strip method has to be preferred to the yield line theory.
As the yield line theory gives safety factors equal to or below the intended value, whereas the strip method gives
values equal to or above the intended value, exactly the intended value will be found in the case where the two
solutions coincide. This gives the exact solution according to the theory of plasticity. Exact solutions should in
principle be sought, as exactly the intended safety gives the best economy. How close a strip method solution is
to the exact solution can be checked by applying the yield line theory to the found solution.
When comparing the strip method and the yield line theory it should be noted that the strip method is a design
method, as a moment distribution is determined, which is used for the reinforcement design. The yield line theory
is a method for check of strength. When the yield line theory is used for design, assumptions have to be made for
the moment distribution, e.g. relations between different moments. In practice the reinforcement is often
assumed to be evenly distributed, which as a matter of fact may not be very efficient. The strip method in most
applications leads to a moment distribution where the reinforcement is heavier at places where it is most
efficient, e.g. along a free edge or above a column support. As the strip method thus tends to use the
reinforcement in a more efficient way, strip method solutions often give better reinforcement economy than the
yield line solution, in spite of the fact that the strip method solution is safer. The reinforcement distribution
according to the strip method solution is often also better from the point of view of the behaviour under service
condition.
A reinforcement design does not only mean the design of the sections of maximum moments, but also the
determination of the lengths of reinforcing bars, and the curtailment of the reinforcement. As the strip method
design is in principle based on complete moment fields, it also gives the necessary information regarding the
curtailment of reinforcement. With the yield line theory it is very complicated to determine the curtailment of
reinforcement in all but the simplest cases. The result from the application of the yield line theory may be either
reinforcing bars which are too short or unnecessarily long bars, leading to poor reinforcement economy, as the
length in practice is based on estimations, due to the complexity of making the relevant analyses.
From the above it seems evident that the strip method has many advantages over the yield line theory as a
method for design of reinforced concrete slabs. In a situation where the strength of a given slab has to be
checked, the yield line theory is usually to be preferred.
Strip method versus theory of elasticity
It is sometimes stated that the strip method is not a very useful practical design method today, as we are able to
design slabs by means of efficient finite element programs, based on the theory of elasticity. This point of view is
worth some discussion.
A finite element analysis gives a moment field, including torsional moments which also have to be taken into
account for the determination of the design moments for the reinforcement. The design moment field is usually
unsuitable for direct use for the design of the reinforcement. The moments have a continuous lateral variation
which would require a corresponding continuous variation of the distances between reinforcing bars. This is of
course not possible from a practical point of view.
One solution to this problem is to design the reinforcement for the highest design moment within a certain width.
This approach is on the safe side, but may lead to poor reinforcement economy, for example, compared to a strip
method solution.
A correct solution according to the theory of elasticity sometimes shows very pronounced moment
concentrations. For instance, this is the case at column supports and supports at reentrant corners. It is in practice
not possible to reinforce for these high local moments.
In order to avoid poor reinforcement economy and high reinforcement concentrations the reinforcement may be
designed for an average design moment over a certain width. As a matter of fact this approach is based on the
theory of plasticity, although applied in an arbitrary way. It may lead to results which are out of control regarding
safety. In this averaging process some of the advantages of the theory of elasticity are lost. In an efficient use of
finite element-based design some postprocessing procedure has to be used for the averaging. The result, e.g.
regarding safety, economy, and properties in the service state, will depend on this postprocessor.
Efficient use of the finite element method, with due regard to economy and safety, may thus necessitate the use
of rather sophisticated programs including postprocessors. The cost of using such programs has to be compared
to the cost of making a design by means of the strip method. In most cases the time for making a strip method
design by hand calculation is so short that it does not pay to use a sophisticated finite element program.
A hand calculation by means of the strip method can probably in many cases compete favourably with a design
based on finite element analysis.
It should also be possible to write computer programs based on the strip method, although such programs do not
so far seem to have been developed.
Fundamentals of the strip method
General
The strip method is based on the lower bound theorem of the theory of plasticity. This means that the
solutions obtained are on the safe side, provided that the theory of plasticity is applicable, which is the case for
bending failures in slabs with normal types of reinforcement and concrete and normal proportions of
reinforcement. As the theorem is usually formulated its purpose is to check the loadbearing capacity of a given
structure. In the strip method an approach has been chosen which instead aims to design the reinforcement so
as to fulfil the requirements of the theorem. The strip method is thus based on the following formulation of the
lower bound theorem:
Seek a solution to the equilibrium equation. Reinforce the slab for these moments.
It should be noted that the solution has only to fulfil the equilibrium equation, but not to satisfy any
compatibility criterion, e.g. according to the theory of elasticity. As a slab is highly statically indeterminate this
means that an infinite number of solutions exist
The complete equilibrium equation contains bending moments in two directions, and torsional moments with
regard to these directions. Any solution which fulfils the equation can, in principle, be used for the design, and
thus an infinite number of possible designs exist. For
practical design it is important to find a solution which is favourable in terms of economy and of behaviour under
service conditions.
From the point of view of economy, not only is the resulting amount of reinforcement important, but also the
simplicity of design and construction. For satisfactory behaviour under service loading the design moments used
to determine the reinforcement should not deviate too much from those given by the theory of elasticity.
Torsional moments complicate the design procedure and also often require more reinforcement. Solutions
without torsional moments are therefore to be preferred where this is possible. Such solutions correspond to
the simple strip method, which is based on the following principle:
In the simple strip method the load is assumed to be carried by strips that run in the reinforcement
directions. No torsional moments act in these strips.
The simple strip method can only be applied where the strips are supported so that they can be treated like
beams. This is not generally possible with slabs which are supported by columns, and special solution
techniques have been developed for such cases. One such technique is called the advanced strip method. This
method is very powerful and simple for many cases encountered in practical design, but as hitherto presented
it has had the limitation that it requires a certain regularity in slab shape and loading conditions. It has here
been extended to more irregular slabs and loading conditions.
An alternative technique of treating slabs with column supports or other concentrated supports is by means of
the simple strip method combined with support bands, which act as supports for the strips. This is the most
general method which can always be applied and which must be used where the conditions that control the use
of other methods are not met. It requires a more time-consuming analysis than the other methods.
The rational application of the simple strip method
In the simple strip method the slab is divided into strips in the directions of the reinforcement, which carry
different parts of the total load. Usually only two directions are used, corresponding to the x- and
y-directions. Each strip is then considered statically as a one-way strip, which can be analysed with ordinary
statics for beams.
The load on a certain area of the slab is divided between the strips. For example, one half of the load can
be taken in one direction and the other half in another direction. Generally, the simplest and most economical
solution is, however, found if the whole load on each area is carried by only one of the strip directions. This
principle is normally assumed in this book. We can thus formulate the following principle to be applied in
most cases:
The whole load within each part of the slab is assumed to be carried by strips in one reinforcement direction.
In the figures the slab is divided into parts with different loadbearing directions. The relevant direction within
each area is shown by a double-headed arrow (see Fig. 2.2.1).
The load is preferably carried with a minimum of cost, which normally means with a minimum amount of
reinforcement. As a first approximation this usually means that the load should be carried in the direction that
runs towards the nearest support, as this results in the minimum moment and the minimum reinforcement
area. From the point of view of economy, the lengths of the reinforcing bars are also important. Where the
moments are positive, the length of the bars is approximately equal to the span in the relevant direction. In
such cases, therefore, more of the load should be carried in the short direction in a rectangular slab.
A consequence of these considerations is that a suitable dividing line between areas with different loadbearing
directions is a straight line which starts at a corner of a slab and forms an angle with the edges. Fig. 2.2.1 shows
a typical simple example, a rectangular slab with a distributed load and more or less fixed edges. The dividing
lines are shown as dash-dot lines.
Fig. 2.2.1
The dividing lines are normally assumed to be lines of zero shear force. Along these lines the shear force is thus
assumed to be zero (in all directions). The use of lines of zero shear force makes it possible to simplify and
rationalize the design at the same time, as it usually leads to good reinforcement economy. For the choice of
positions of the lines of zero shear force the following recommendations may be given.
A line of zero shear force which starts at a corner where two fixed edges meet may be drawn approximately to
bisect the angle formed by these edges, but maybe a little closer to a short than to a long edge.
line of zero shear force which starts at a corner where two freely supported edges meet should be drawn
markedly closer to the shorter edge. The distances to the edges may be chosen to be approximately proportional
to the lengths of the edges, for example.
Where a fixed edge and a freely supported edge meet, the line of zero shear force should be drawn much
closer to the free edge than to the fixed edge.
The economy in reinforcement is not much influenced by variations in the positions of the lines of zero shear
force in the vicinity of the optimum position. In cases of doubt it is fairly easy to make several analyses with
different assumptions and then to compare the results.
The use of lines of zero shear force is best illustrated on a simple strip (or a beam). The slab strip in Fig. 2.2.2 is
acted upon by a uniform load q and has support moments ms1 and ms2 (shown with a positive direction,
though they are normally negative). The corresponding lines for shear forces and moments are also shown. The
maximum moment mf occurs at the point of zero shear force.
The parts to the left and right of the point of zero shear force can be treated as separate elements if we know
or assume the position of this point. These separated elements are shown in the lower figure. The following
equilibrium equation is valid for each of the elements:
(2.1)
with the indices 1 and 2 deleted.
The beam in Fig. 2.2.2 can thus be looked on as being formed by two elements, which meet at the point of zero
shear force. This corresponds to strips in the y- direction in the central part of the slab in Fig. 2.2.1.
In many cases the loaded elements of the strips do not meet, but instead there is an unloaded part in between.
This is the case for thin strips in the x-direction in Fig.
2.2.1. Such a strip is illustrated in Fig. 2.2.3. It can be separated into three elements, the loaded elements near
the ends and the unloaded element in between.The unloaded element is subjected to zero shear force, and
thus carries a constant bending moment mf. The span moments mf must be the same at the inner ends of the
two load-bearing elements in order to maintain equilibrium.
In a rigorously correct application of the simple strip method we would study many thin strips in the
x-direction in Fig. 2.2.1 with different loaded lengths and different resulting design moments. This leads to a
very uneven lateral distribution of design moments and a reinforcement distribution which is unsuitable from a
practical point of view. For practical design the average moment over a reasonable width must be considered.
In the first place, therefore, the analysis should give the average moments.
To calculate these average moments we introduce slab elements, which are the parts of the slab bordered
by lines of zero shear force and one supported edge. Each slab element is
actively carrying load in one reinforcement direction, the direction shown by the double-headed arrow. Thus
the slab in Fig. 2.2.1 is looked upon as consisting of four slab elements, two active in the x-direction and two in
the y-direction. The load within each element is assumed to be carried only by bending moments
corresponding to the reinforcement direction. Such elements are called one-way elements. Each one-way
element has to be supported over its whole width. The average moments and moment distributions are
discussed in Sections 2.3–5. Cases which can be treated by means only of one-way elements can be found by
the simple strip method.
Even though the dividing lines between elements with different load-bearing directions, shown as
dash-dot lines, are normally lines of zero shear force, there are occasions when the analysis is simplified by
using such lines where the shear force is not zero. It is then on the
Fig. 2.2.3
safe side to undertake the analysis as if the shear force were zero along such lines, provided that the strip of
which the element forms part has a support at both ends.
This is explained in Fig. 2.2.4, which shows a strip loaded only in the vicinity of the left end. The loaded part
corresponds to an element. The right-hand end of the loaded part corresponds to a dividing line between
elements. The upper curve shows the correct moment curve, whereas the lower curve is determined on the
assumption that the shear force is zero at the dividing line. The moments according to the lower curve are
always on the safe side compared to those according to the correct curve. Where this approximation is used
the economical consequences for the reinforcement are usually insignificant.
Fig. 2.2.4
Average moments in one-way elements
General
The load on a one-way element is carried to the support by bending moments in one direction, corresponding
to the direction of the main reinforcement. This direction is shown in figures as a short line with arrows at both
ends, see e.g. Fig 2.2.1. This type of arrow thus indicates that the element is a one-way element and also shows
the direction of the loadbearing reinforcement.
A one-way element is supported only along one edge. Normally the shear force is zero along all the other
edges. The formulas given below refer to this case. The edges with zero shear force are shown as dash-dot
lines.
In most cases the reinforcement direction is at right angles to the
supported edge. This case will be treated first The load per unit area
is denoted q.
Theoretically, a one-way element consists of many parallel thin strips in the reinforcement direction. The
moment in each thin strip can be calculated and a lateral moment distribution can thus be determined. The
principle is illustrated in Fig. 2.3.1, which shows a triangular element with load-bearing reinforcement in the
x-direction and with one side parallel to that direction.
The element carries a uniform load q and is divided into thin strips in the x-direction. Each strip is assumed to
have zero shear force at the non-supported end. The length of a strip is yc/l, and the sum of end moments in
each thin strip can be written
Fig. 2.3.1
(2.2)
The corresponding lateral moment distribution is shown in the figure. The average moment is qc2/6.
Formulas are given below for average moments in typical elements with distributed loads. These formulas are
utilised in the numerical examples. The theoretical moment distributions are illustrated. In practical design
applications the moment distribution is simplified as discussed in Section 2.4.
Combining elements to form strips
The elements into which the slab is divided have to be combined in such a way that the equilibrium conditions are
fulfilled. These conditions are related to the bending moments and the shear forces at the edges of the elements.
Torsional moments are never present at the edges of the elements.
Shear forces are in most cases assumed to be zero at those edges of the elements, which are not
supported, but in some applications non-zero shear forces may appear at such an edge.
One-way elements with the same load-bearing direction are often not directly connected to each other, but
by means of a constant moment transferred through elements with another load-bearing direction, typical
example is shown in Fig. 2.7.1, of a rectangular slab with a uniform load, two simply supported edges and two
fixed edges. The positive moments in elements 1 and 3 have to be the same, and this moment is transferred
through 2 and 4, which in practice means that the reinforcement is going between 1 and 3 through 2 and 4.
Fig. 2.7.1
The choice of c-values is based on the rules in Section 2.2. It may in practice also be influenced by rules for
minimum reinforcement if the optimum reinforcement economy is sought.
In flat slabs corner-supported elements are often directly connected with each other and with rectangular
one-way elements into continuous strips as in Fig. 2.7.2. Each such strip acts as a continuous beam, and can
be treated as such. The slab in Fig. 2.7.2 thus has a continuous strip in three spans with width wx in the
x-direction and a continuous strip in two spans with width wy in the y-direction. The design of the slab is based
on the analysis of these two strips, taking into account the rules for reinforcement distribution for
corner-supported elements. The average design moments in the triangular elements near the corners of the
slab are equal to one-third of the corresponding moments in the adjoining rectangular elements.
Fig. 2.7.2
The strip method
The strip method is based on the lower bound theorem of the theory of plasticity, which means that it in
principle leads to adequate safety at the ultimate limit state, provided that the reinforced concrete slab has a
sufficiently plastic behaviour. This is the case for ordinary under-reinforced slabs under predominantly static loads.
The plastic properties of a slab decrease with increasing reinforcement ratio and to some extent also with
increasing depth. With a design based on the recommendations in this book,, the demand on the plastic
properties of the slab is not very high. The solutions should give adequate safety in most cases, possibly with the
exception of slabs of very high strength concrete with high reinforcement ratios.
As the theory of plasticity only takes into account the ultimate limit state, supplementary rules have to be given to
deal with the properties under service conditions,
i.e. deflections and cracks
The lower bound theorem of the theory of plasticity states that if a moment distribution can be found which fulfils
the equilibrium equations, and the slab is able to carry these moments, the slab has sufficient safety in the ultimate
limit state. In the strip method this theorem has been reformulated in the following way:
Find a moment distribution which fulfils the equilibrium equations. Design the reinforcement for these
moments. The moment distribution has only to fulfil the equilibrium equations, but no other conditions, such as the
relation between moments and curvatures. This means that many different moment distributions are possible, in
principle an infinite number of distributions. Of course, some distributions are more suitable than others from
different points of view.
2
 2 mxy
 2 mx  m y
+
−2
= −w
2
2
x
y
xy
• Where w = the external load per unit area
• mx, my = Bending Moments per unit width in the x and y directions and
• mxy = the twisting moment
So according to the lower bound theorem, any combination of mx, my, and mxy that satisfies the
equilibrium at all points in the slab and that meets boundary conditions is a valid solution, provided that
the reinforcement placed to carry these moments
The basis for the simple strip method is that the torsional moment is chosen equal to zero; no load is
assumed to be resisted by the twisting strength of the slab
→ mxy = 0
The equilibrium equation then reduces to:
2
2
y
x
2
2
 m
 m
+
= −w
x
y
This equation can be split conveniently into 2 parts, representing twist less beam strip action.
𝜕 2 𝑚𝑥
= −𝑘𝑤
𝜕𝑥 2
𝜕 2 𝑚𝑦
−= −(1 − 𝑘)𝑤
𝜕𝑦 2
Where the proportion of load taken by the strips is k in the x-direction and (1-k) in the y-direction
(concept of load dispersion)
In many regions in slabs, the value k will be either 0 or 1, i.e., load is dispersed by strips in x or in y
direction
In other regions, it may be reasonable to assume that the load is divided equally in the two directions,
i.e. k=0.5
Choice of load distribution
Theoretically, the load w can be divided arbitrarily b/n x and y directions.
Different divisions will, of course, lead to different patterns of reinforcement, and all will not be equally
appropriate
The desired goal is to arrive at an arrangement of steel that is safe and economical and will avoid problems
at service load level associated with excessive cracking or deflections.
In general, the designer may be guided by his knowledge of the general distribution of elastic moments.
To see an example of the strip method and to illustrate the choices open to the designer, consider the
square, simply supported slab shown below, with side length a and a uniformly distributed factored load
w per unit area.
The simplest load distribution is obtained by setting k=0.5 over the entire slab, as shown in Figure 2.
The load on all strips in each direction is thus w/2 ( with k=0.5), as illustrated by the load dispersion
arrows
This gives maximum design moments mx = my = wa2/16, implying a constant curvature for all strips in the
x-direction at mid-span corresponding to a constant moment wa2/16 across the width of the slab (see
fig. 2
So what we need is a type of load distribution (dispersion) which can give a moment distribution that
gives rise to greater curvatures in strips near the middle of the slab and less near the ends
Try the alternative,
Here the regions of different load dispersion separated by the dash-doted discontinuity lines follow the
diagonals, and all of the load on any region is carried in the direction giving the shortest distance to the
nearest support.
k=0 or 1 in the different regions
The lateral distribution of moments shown in Fig (3) would theoretically require a continuously variable
bar spacing → impractical
A practical solution would be to reinforce for the average moment over a certain width, approximating the
actual lateral variation in Fig. (4) in a stepwise manner
• Hillerborg notes that this is not strictly in accordance with the equilibrium theory and that the design
is no longer certainly on the safe side, but other conservative assumptions, e.g., neglect of
membrane strength in the slab or strain hardening of the reinforcement, would compensate for the
slight reduction in safety margin
• A third alternative is with discontinuity lines parallel to the edges.
• Here again the division is made so that the load is carried to the nearest support, as before, but load
near the diagonals is divided with one-half taken in each direction.
• Thus k is given the values 0 or 1 along the middle edges and 0.5 in the corners and center of the slab
•
•
•
•
•
Two different strip loadings are now identified, strip along A-A and along B-B.
This design leads to practical arrangement, one with constant spacing through the center strip of
width a/2 and a wider spacing through the outer strips, where the elastic curvatures and moments
are known to be less.
The averaging of moments necessitated in the second solution is avoided here, and the 3rd (Fig. 4)
solution is fully consistent with the equilibrium theory.
The three examples also illustrate the simple way in which moments in the slab can be found by strip
method, based on familiar beam analysis.
It is important to note too that the load on the supporting beams is easily found because it can be
computed from the end reactions of the slab-beam strips in all cases.
Rectangular slabs with simple support
•
•
•
•
•
•
•
Discontinuity lines parallel to the edges as shown in the figure
In the x-direction:
Side strips: mx = w/2×b/4×b/8 = wb2/64
Middle strips: mx = w×b/4×b/8 = wb2/32
In the y-direction
Side strips: my = wb2/64
Middle strips: my = wb2/8
Figure 5 Rectangular slab with discontinuity
parallel to the edges
Figure 6 Rectangular slab with discontinuity lines
originating at the corners.
INTRODUCTION YIELD LINE METHOD:
The yield line theory of analysis is a factored or ultimate load method of analysis. The yield line theory is
conducted based on the bending moment of the structural element at its collapse state. Rectangular one
way or two way slabs under normal uniform loading can be analyzed and then designed using coefficients
obtained from Tables published for this purpose.
In a situation where irregular shapes, varied support conditions, presence of openings, varied loading and
more complex conditions are encountered, the established theory of elasticity or plasticity cannot be
employed straight. For these circumstances, the yield line theory is found useful.
The yield line theory is an ultimate load method of analysis of slab, i.e. the BM at the verge of collapse is
used as the basis for design. At collapse loads, an under reinforced slab begins to crack with the
reinforcement yielding at points of high moment. The crack lines or the yield lines propagate with the
increase in deflection until the slab is broken in to a number of segments.
A yield line is a line in the plane of the slab across which reinforcing bars have yielded and about which
excessive deformation (plastic rotation) under constant limit moment (ultimate moment) continues to occur
leading to failure.
Upper and lower bound theorem:
Plastic analysis methods such as the yield line theory derived from the general theory of structural plasticity,
which states that the ultimate collapse load of a structure lies between two limits, an upper bound and a
lower bound of the true collapse load.
The lower bound and upper bound theorem, when applied to slabs, can be stated as follows:
Lower bound theorem: If, for a given external load, it is possible to find a distribution of moments that
satisfies equilibrium requirements, with the moment not exceeding the yield moment at any location, and if
the boundary conditions are satisfied, then the given load is a lower bound of the true carrying capacity.
Upper bound theorem: If, for a small increment of displacement, the internal work done by the slab,
assuming that the moment at every plastic hinge is equal to the yield moment and that boundary conditions
are satisfied, is equal to the external work done by the given load for that same small increment of
displacement, then that load is an upper bound of true carrying capacity.
If the lower bound conditions are satisfied, the slab can certainly carry the given load, although a higher load
may be carried if internal distributions of moment occur. If the upper bound conditions are satisfied, a load
greater than the given load will certainly cause failure, although a lower load may produce collapse if the
selected failure mechanism is incorrect in any sense. Accordingly the yield line method of analysis for slabs is
an upper bound method, and consequently the failure load calculated for a slab with known flexural
resistance may be higher than the true value.
The yield line phenomenon involves:
• a slab under increasing loads where cracking and reinforcement yielding occur in the most highly
stressed zone (i.e. around maximum moment)
• the highly stressed zone normally acts as a plastic hinge where the subsequent loads are distributed
to other region of the slab
• cracks develop forming patterns of yield lines until a mechanism is formed,
• collapse is then indicated by increasing deflection under constant load
Fig. 2.1 Deformation of slab with yield lines
Sign convention
positive yield linebuilt in
negative yield linesimply
rotationpoint load
unsupported edgecolumn
edge
supported edge axis of
support
Plastic Analysis:
Plastic behavior of the material describes the deformation of the body undergoing nonreversible
changes of shape in response to applied forces. The main task of the structural engineering is to
design the structural members so they can carry the loads under all possible conditions including
ultimate limit states. The elastic distribution of stresses can be obtained by solution of the elasticity
problem. However the structural elements do not behave elastically near ultimate load and bending
capacity of section is based on a plastic analysis.
Characteristic features of yield lines:
• yield lines are generally straight and end at a slab boundary or the intersection of other yield
lines
• axes of rotation generally lie along lines of support
• Axes of rotation pass over column supports
• For a mechanism to develop, the yield line may pass through the intersection of the axes of
rotation of adjacent segments
The
Page 14
Yield Lines in Concrete Slabs
As mentioned before, the analysis is conducted based on the collapse load of the slab under
consideration. Under this load, cracks are formed. This crack formation is seen in under reinforced
concrete slabs and the yield line theory considers only under reinforced concrete slabs. During the
crack formation in the slabs, the reinforcement starts to yield. This yielding happens at the point of
maximum bending moments. As the cracks propagates, the yield lines started to develop. Finally,
under uncontrollable load, the slab will collapse leaving maximum yield line representing the
achievement of maximum bending moments. The main principle of yield line theory is to determine
the location of appropriate yield lines. In the case of two-way slab systems, which are statically
indeterminate, detailed inelastic analysis (yield line analysis) is conducted. The moment coefficients
given in the Code IS 456:200. for two-way rectangular slabs with various possible edge conditions
are based on Yield Line Analysis. ‘Yield line analysis’ is the equivalent for a two-dimensional flexural
member (plate or slab) of the limit analysis of a one-dimensional member (continuous beam).
Characteristics of Yield Lines in R.C Slabs
The following are the characteristics of yield lines formed in reinforced concrete slabs under
ultimate loads: i) Yield lines are straight ii) Yield lines end at supporting edges of slab iii) Yield lines
passes through intersection of axis of rotation of adjacent slab elements iv) Axis of rotation lies along
lines of supports and passes over columns
Fig.1: Typical interior panel in a two-way slab system
fig.2: Yield line pattern under uniformly distributed collapse load
Assumptions of Yield Line Theory
Page 15
The following are the assumptions of the yield line analysis of reinforced concrete slabs:
• At the collapse stage, the steel reinforcement will be fully yielded along the yield lines.
• During the collapse, the slab is deformed plastically and they gets separated into segments.
Elastic behaviour is followed by the individual segments.
• In yield theory, only plastic deformations are taken into consideration. The elastic
deformations are neglected. The so called deformations are happening along the yield lines.
Even in the collapse state, the elements will remain plane.
• Along the yield lines, there is uniform distribution of bending and twisting moments. The
amount of reinforcement in the section will reflect the capacity that will give the maximum
moment capacity of the section.
• The yield lines are lines of intersection of two planes, hence they will remain straight.
• Rules of yield lines
• The positive and negative yield lines, are used to designate the yield lines for positive
bending moments having tension at the bottom and negative bending moments having
tension at the top of the slab, respectively. The guidelines for predicting the yield lines and
axes of rotation are mentioned below:
• Straight yield line occure between two intersecting planes.
• In one way slabs, the positive yield line occur at the mide span.
• At the supports, the yield lines are negative in addition to the mid positive yield lines for one
way continuous slabs.
• For slabs with a point load, the yield lines will be projecting out of the point of application.
• For two planes intersecting, the yield line will pass through the intersection of axes of
rotation of two planes.
• The end of yield lines are either at the boundary of the slab or at another yield line.
• The axes of rotation is represented by yield lines
• The edges of support too act as axes of rotation. If the support is fixed, the support will have
negative yield lines. This negative yield line will have resistance to rotation. In the case of
simply supported edges, the axes of rotation at the support will not have resistance to
rotation.
• For a column support, the axis of rotation pass over it. The orientation will depend on other
considerations
• Methods of Analysis in Yield Line Theory
Based on the rules and the assumptions, a general load pattern and the axis of rotation is located.
The below mentioned methods can be used to determine the final pattern, axes of rotation and the
collapse load. The methods used are:
Method of Segmental Equilibrium
Method of Virtual Work
1.Methods of Segmental Equilibrium
As the name says, the collapse of the slab under certain mechanism will give several segments. The
equilibrium of the segments is taken into consideration and hence to arrive at set of simultaneous
equations. The solution of the simultaneous equations will give certain parameter values that will
help us to finalise the yield pattern. This will also give us a relation between the load capacity and
the moment value. As shown in figure-4, each segment is studied as a free body. Each segment is
under equilibrium under the action of moment, loads along the yield lines. The reactions and shear
will be along the supports. Here, the twisting moments along the yield lines will be zero. From figure4,Considering segments AOB, about axis of rotation 1-1
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Fig.4:Yield Line Pattern of a Two Way Simply Supported Slab
The vector sum of moments along AO & OB = Moments due to the loads on the segment AOB For a
small element of length dx and dy, the governing equation of equilibrium is given:
Here, External load
acting per unit area = w Bending Moments per unit width x and y = Mx & My Twisting Moment =
Mxy
Equilibrium method of Analysis:
it makes use of the equilibrium equations for individual segment to obtain the collapse load. The
FBD represented by each collapsing segment is in equilibrium under - applied loads, - yield moments
and - Reactions or shears along support lines. The method of segment equilibrium should not be
confused with a true equilibrium method such as then strip method. A true equilibrium method is a
lower bound method of analysis. Essentially, the yield lines form at lines of maximum moment
where neither shear nor torsion is typically present at positive yield lines. For demonstration
purpose consider the one way slab uniformly loaded and is continuous as shown in Fig. Below
Let the slab with span L is reinforced to provide resistance of m2 KN.m per m through the span and
m1 and m3 KN.m per m at the two supports. Suppose it is desired to determine the collapse load
wu.
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2.Method of Virtual Work
The principle of virtual work is followed in this method. Virtual Work principle states that, The
external work done by the loads to cause small virtual deflection is equal to the small internal work
done by the yield moments to make a rotation to accommodate the virtual deflection. Here, the slab
is assumed with a yield pattern and axes of rotation. The system is in equilibrium with the moments
and loads acting on it. This system is subjected to an infineitesimal increase in load so that the
structure will undergo further delflection. Based on the principle, the internal and the external works
is equated hence giving a relationship between the applied loads and the ultimate resisting
moments. The work equation is given by:
Where, Collapse Load = w Vertical Deflection along which load “w” moves = D Moment capacity per
unit length of the section = M Rotation of the slab segment = Length of Yield Line = l
Virtual work method of Analysis:
Based on principle that work done by external forces in undergoing a small virtual displacements is
equal to the internal virtual work done in rotations along yield lines, the ultimate load which the slab
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can sustain is determined. In other words, the work during small motion of collapse mechanism is
equal to the work absorbed by the plastic hinges formed along the yield lines. In here, the segment
of the slab with in the yield lines is assumed to go through rigid body displacement with the collapse
load acting on the structure. WE = WI i.e. Work done by external forces = Energy absorbed by the
hinges (internal work)
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References#
1]https://books.google.com.pk/books?id=vp2obAZjHLAC&printsec=frontcover&dq=strip+design+me
thod+example+problems
2] https://slideplayer.com/slide/10003763/
3] https://fdocuments.in/download/chapter-2-strip-method-for-slabs
4] https://www.icevirtuallibrary.com/doi/pdf/10.1680/iicep.1969.7388
5] https://theconstructor.org/structural-engg/yield-line-theory-for-slab-design-assumptionsmethods-of-analysis/6839/
6] https://uomustansiriyah.edu.iq/media/lectures/5/5_2019_09_07!01_47_26_PM.pdf
7] http://www.geocities.ws/drsuhaimi03/YLineEx.pdf