2024年4月1日发(作者：无鸿云)
8 Yield Line Theory
8.1 The essentials
Yield Line Design is a well-founded method of designing reinforced concrete slabs, and
similar types of elements. It uses Yield Line Theory to investigate failure mechanisms at
the ultimate limit state. The theory is based on the principle that:
work done in yield lines rotating = work done in loads moving
Yield Line Design is easy to grasp but there are several fundamental principles that need to
be understood. Yield Line Design is a plastic method: it is different from ‘normal’
elastic methods.
8.1 .1 A short history of Yield Line Theory
Yield Line Theory as we know it today was pioneered in the 1940s by the Danish engineer
and researcher K W Johansen.
As early as 1922, the Russian, A Ingerslev presented a paper to the Institution of
Structural Engineers in London on the collapse modes of rectangular slabs. Authors such
as R H Wood, L L Jones, A Sawczuk and T Jaeger, R Park, K O Kemp, C T Morley, M
Kwiecinski and many others, consolidated and extended Johansen’s original work so
that now the validity of the theory is well established making Yield Line Theory a
formidable international design tool.
In the 1960s 70s and 80s a significant amount of theoretical work on the application
of Yield Line Theory to slabs and slab-beam structures was carried out around the world
and was widely reported.
To support this work, extensive testing was undertaken to prove the validity of the
theory. Excellent agreement was obtained between the theoretical and experimental Yield
Line patterns and the ultimate loads.
The differences between the theory and tests were small and mainly on the
conservative side. In the tests where restraint was introduced to simulate continuous
construction, the ultimate loads reached at failure were significantly greater
than the loads predicted by the theory due to the beneficial effect of membrane
forces.
8.1 .2 What is a yield line?
A yield line is a crack in a reinforced concrete slab across which the reinforcing bars have
yielded and along which plastic rotation occurs.
8.1 .3 What is Yield Line Theory?
Yield Line Theory is an ultimate load analysis. It establishes either the moments in an
element (e.g. a loaded slab) at the point of failure or the load at which an element will fail.
It may be applied to many types of slab, both with and without beams. Consider the case
of a square slab simply supported on four sides as illustrated by Figure 8.1.1. This slab is
subjected to a uniformly distributed load, which gradually increases until collapse occurs.
Initially, at service load, the response of the slab is elastic with the maximum steel stress
and deflection occurring at the centre of the slab. At this stage, it is possible that some
hairline cracking will occur on the soffit where the flexural tensile capacity of the concrete
has been exceeded at midspan.
Figure 8.1.1 Onset of yielding of bottom reinforcement at point of maximum deflection in a
simply supported two-way slab
Increasing the load hastens the formation of these hairline cracks, Increasing the load
further will increase the size of the cracks further and induce yielding of the reinforcement,
initiating the formation of large cracks emanating from the point of maximum deflection.
On increasing the load yet further, these cracks migrate to the free edges of the slab at
which time all the tensile reinforcement passing through a yield line yields.
At this ultimate limit state, the slab fails. As illustrated by Figure 8.1.2, the slab is divided
into rigid plane regions A, B, C and D. Yield lines form the boundaries between the rigid
regions, and these regions, in effect, rotate about the yield lines. The regions also pivot
about their axes of rotation, which usually lie along lines of support, causing supported
loads to move. It is at this juncture that the work dissipated by the hinges in the yield lines
rotating is equated to work expended by loads on the regions moving. This is Yield Line
Theory.
Under this theory, elastic deformations are ignored; all the deformations are assumed to
be concentrated in the yield lines and, for convenience, the maximum deformation is given
the value of unity.
For a more detailed appraisal of the situation at corner see Section ?
Figure 8.1.2 The formation of a mechanism in a simply supported two-way slab with the
bottom steel having yielded along the yield lines
8.1 .4 What is a yield line pattern?
When a slab is loaded to failure, yield lines form in the most highly stressed areas and
these develop into continuous plastic hinges. As described above, these plastic hinges
develop into a mechanism forming a yield line pattern.
Yield lines divide the slab up into individual regions, which pivot about their axes of
rotation. Yield lines and axes of rotation conform to rules given in Table 8.1.1, which help
with the identification of valid patterns and the Yield Line solution.
Table 8.1.1 Rules for yield line patterns
• Axes of rotation generally lie along lines of support and pass alongside any columns.
• Yield lines are straight.
• Yield lines between adjacent rigid regions must pass through the point of intersection of
the axes of rotation of those regions.
• Yield lines must end at a slab boundary.
• Continuous supports repel and simple supports attract positive or sagging yield lines.
8.1 .5 What is a Yield Line solution?
In theory, there may be several possible valid yield line patterns that could apply to a
particular configuration of a slab and loading. However, there is one yield line pattern that
gives the highest moments or least load at failure. This is known as the yield line solution.
The designer has several ways of determining the critical pattern and ensuring safe
design:
• From first principles, e.g. by using The Work Method
• Using formulae for standard situations.
It will be noted that valid yield line patterns give results that are either correct or
theoretically unsafe. These ‘upper bound solutions’ can deter some designers but, as
discussed later, this theoretical awkwardness is easily overcome by testing different
patterns and by making suitable allowances (see 10% rule later).
8.1 .6 How do you select relevant yield line patterns?
A yield line pattern is derived mainly from the position of the axes of rotation, (i.e. the lines
of support) and by ensuring that the yield lines themselves are straight, go through the
intersection of axes of rotation and end at the slab boundary, i.e. conform to the rules in
Table 8.1.1. Some simple examples are shown in Figure 8.1.3 Considering a slab to be a
piece of pastry laid over supports may help designers to visualise appropriate yield line
patterns.
Figure 8.1.3 Simple Yield Line patterns
The aim of investigating yield line patterns is to find the one pattern that gives the critical
moment (the highest moment or the least load capacity). However, an exhaustive search is
rarely necessary and selecting a few simple and obvious patterns is generally sufficient as
their solutions are within a few percent of the perfectly correct solution.
8.1 .7 What is a fan mechanism?
Slabs subjected to heavy concentrated loads may fail by a so-called fan mechanism, with
positive Yield Lines radiating from the load and a negative circular Yield Line centred under
the point load. This mechanism is shown in Figure 8.1.4. It is rare for this form of failure to
be critical but nonetheless a check is advised where large concentrated loads are present
or for instance in flat slabs where the slab is supported on columns.
Figure 8.1.4 Fan collapse pattern for a heavy concentrated load onto a reinforced slab
The mechanism for a slab supported by a column is the same shape but with the positive
and negative yield lines reversed.
8.1 .8 What is the Work Method?
The Work Method (or virtual Work Method) of analysis is the most popular (and most easy)
way of applying Yield Line theory from first principles. Indeed, many experienced users of
Yield Line theory of design choose to use the Work method because it is so very easy. The
fundamental principle is that work done internally and externally must balance.
In other words, at failure, the expenditure of external energy induced by the load on the
slab must be equal to the internal energy dissipated within the yield lines. In other words:
External energy = Internal energy
by loads moving = dissipated by rotations about yield lines
Expended = Dissipated
E = D
Σ (Ν x δ)
for all regions
= Σ (m x l x θ)
for all regions
where
N = load(s) acting within a particular region [kN]
δ = the vertical displacement of the load(s) N on each region expressed as a fraction of
unity [m]
m = the moment in or moment of resistance of the slab per metre run [kNm/m]
l = the length of yield line or its projected length onto the axis of rotation for that region
[m]
θ = the rotation of the region about its axis of rotation [m/m] By way of illustration,
consider the slab shown in Figure 8.1.2. Figure 8.1.5 shows an axonometric view of this
two-way simply supported slab that has failed due to a uniformly distributed load. Note
that:
• The triangular regions A, B, C and D have all rotated about their lines of support.
• The loads on the regions have moved vertically and rotation has taken place about the
yield lines and supports.
• The uniformly distributed load on each of these regions will have moved on average 1/3
of the maximum deflection.
The rotation of the regions about the yield lines can be resolved into rotation about the
principal axes of rotation, and thereby measured with respect to the location and size of
the maximum deflection.
Figure 8.1.5 Deformed shape at failure
This, fundamentally, is the ‘Work Method’. Any slab can be analysed by using the principle
of E = D. Some judgement is required to visualise and check likely failure patterns but
absolute accuracy is rarely necessary and allowances are made to cover inaccuracies.
Once a yield line pattern has been selected for investigation, it is only necessary to specify
the deflection as being unity at one point (the point of maximum deflection) from which all
other deflections and rotations can be found.
8.1 .9 Formulae
Rather than go through the Work Method, some practitioners prefer the even quicker
method of using standard formulae for standard types of slab. The formulae are
predominantly based on the work method and they are presented in more detail in Chapter
3. As an example, the formula for one-way spanning slabs supporting uniformly distributed
loads is as follows:
per unit width
where
m = ultimate sagging moment along the yield line [kNm/m]
m’ = ultimate support moment along the yield line [kNm/m]
n = ultimate load [kN/m2]
L = span [m]
i1 , i2 = ratios of support moments to mid-span moments. (The values of i are chosen by
the designer: i1= m’1/m, i2= m’2/m)
Where slabs are continuous, the designer has the freedom to choose the ratio of hogging
to sagging moments to suit any particular situation. For instance, the designer may choose
to make the bottom span steel equal to the top support steel (i.e. make sagging moment
capacity equal support moment capacity.)
Failure patterns for one-way spanning slabs are easily visualised and the standard
formulae enable the designer to quickly determine the span moment based on any ratio of
hogging moments he or she chooses to stipulate (within a sensible range dictated by codes
of practice). Formulae are also available for the curtailment of top reinforcement.
Formulae for two-way spanning slabs supported on two, three or four sides are also
available for use. These are a little more complicated due to the two-way nature of the
problem and the fact that slabs do not always have the same reinforcement in both
directions. The nature of the failure patterns is relatively easy to visualise and again the
designer has the freedom to choose fixity ratios.
8.1 .10 Is Yield Line Theory allowable under design codes of practice?
Yes. Any design process is governed by the recommendations of a specific code of practice.
In the UK, BS 8110 clause 3.5.2.1 says ‘Alternatively, Johansen’s Yield Line method ….
may be used…. for solid slabs’. The proviso is that to provide against serviceability
requirements, the ratio of support and span moments should be similar to those obtained
by elastic theory. This sub-clause is referred to in clauses 3.6.2 and 3.7.1.2 making the
approach also acceptable for ribbed slabs and flat slabs.
According to Eurocode 2, Yield Line Design is a perfectly valid method of design. Section
5.6 of Eurocode 2 states that plastic methods of analysis shall only be used to check the
ultimate limit state. Ductility is critical and sufficient rotation capacity may be assumed
provided x/d ≤0.25 for C50/60. A Eurocode 2 goes on to say that the method may be
extended to flat slabs, ribbed, hollow or waffle slabs and that corner tie down forces and
torsion at free edges need to be accounted for.
Section 5.11.1.1 of EC2 includes Yield Line as a valid method of analysis for flat slabs. It is
recommended that a variety of possible mechanisms are examined and the ratios of the
moments at support to the moment in the spans should lie between 0.5 and 2.
8.1.11 Yield Line is an upper bound theory
Yield line theory gives upper bound solutions - results that are either correct or
theoretically unsafe, see Table 8.1.2. However, once the possible failure patterns that can
form have been recognised, it is difficult to get the yield line analysis critically wrong
Table 8.1.2 Upper and lower bound ultimate load theories
Ultimate load theories for slabs fall into two categories:
• upper bound (unsafe or correct) or
• lower bound (safe or correct).
Plastic analysis is either based on
• upper bound (kinematic) methods, or on
• lower bound (static) methods.
Upper bound (kinematic) methods include:
• plastic or yield hinges method for beams, frames and one-way slabs;
• Yield Line Theory for slabs.
Lower bound (static) methods include:
• the strip method for slabs,
• the strut and tie approach for deep beams, corbels, anchorages, walls and plates loaded
in their plane.
The mention of ‘unsafe’ can put designers off, and upper bound theories are often
denigrated. However, any result that is out by a small amount can be regarded as
theoretically unsafe. Yet few practising engineers regard any analysis as being absolutely
accurate and make due allowance in their design. The same is true and acknowledged in
practical Yield Line Design.
In the majority of cases encountered, the result of a Yield Line analysis from first principles
will be well within 10%, typically within 5%, of the mathematically correct solution. The
pragmatic approach, therefore, is to increase moments (or reinforcement) derived from
calculations by 10%. This ‘10% rule’ is expanded upon later.
There are other factors that make Yield Line Design safer than it may at first appear, e.g.
compressive membrane action in failing slabs (this alone can quadruple ultimate
capacities), strain hardening of reinforcement, and the practice of rounding up steel areas
when allotting bars to designed areas of steel required.
The practical designer can use Yield Line Theory with confidence, in the knowledge that he
or she is in control of a very useful, powerful and reliable design tool.
8.1.12 Corner levers
‘Corner levers’ describes the phenomenon in two-way slabs on line supports where yield
lines split at internal corners. This splitting is associated with the formation of a negative
yield line across the corner which ‘levers’ against a corner reaction (or holding down force).
Corner levers particularly affect simply supported slabs and Figure 8.1.5 shows the effect
corner levers can have on a simply supported square slab. It should also be noted that the
sagging moment m in an isotropic slab increases with decreasing corner fixity. Table 8.1.3
illustrates the effects of continuity on both the extent of the corner levers and on positive
moments [13]. At an average fixity ratio of 1.0 the effects are minimal. Nonetheless, if the
corners are left unreinforced, span moments increase.
Figure 8.1.5 The effect of corner levers on a simply supported square slab where corners
are held down and prevented from lifting.
Table 8.1.3 Effects of corner continuity on corner levers in a simply supported square slab
Corner fixity
i = m’/m
0
0.25
0.50
1.00
x h
m
[kNm/m]
Na2/22
Na2/23
Na2/23.6
Na2/24
Positive moment
increase in the slab
due to corner lever
9.0%
4.3%
1.7%
-
0.159a
0.110a
0.069a
0
0.523a
0.571a
0.619a
-
For simplicity in the analysis, yield line patterns are generally assumed to go into corners
without splitting, i.e. corner levers are ignored and an allowance is made for this. This
simplification is justified for three principle reasons:
• The error for neglecting corner levers is usually small.
• The analysis involving corner levers becomes too involved.
• Corner levers usually bring out the beneficial effects of membrane action that negate
their impact.
All methods and formulae are based on straight-line crack patterns that go into the corners.
The values of the moments obtained in this way are only really valid if the top
reinforcement provided in the corners is of the same magnitude as the bottom steel
provided in the span. If this is not the case, as generally assumed, then the straight line
pattern will not form and some type of corner lever will appear depending on the amount
of top reinforcement provided, if any. This in turn leads to additional moment to be added
to the calculated positive (sagging) moment.
The exact amount of increase depends on a number of parameters, but generally about
4% to 8% is assumed for rectangular two-way slabs. At worst, for simply supported square
slabs, the increase is approximately 9%. The effects of corner levers in slabs supported on
four sides diminishes in rectangular slabs and begin to die out at a ratio of sides greater
than 3:1. In triangular slabs and slabs with acute corners, the straight-line mechanism into
the corners can underestimate the positive moment by 30% 35%.
The effects of corner levers have to be recognised. For regular slabs their effects are
allowed for within ‘the 10% rule’.
Despite this, it is good practice, and it is recommended, to specify and detail U-bars,
equivalent to 50% of the span steel around all edges, including both ways at corners.
8.1.13 The 10% rule
A 10% margin on the design moments should be added when using the Work
Method or formulae for two-way slabs to allow for the method being upper
bound and to allow for the effects of corner levers
The addition of 10% to the design moment in two-way slabs provides some leeway where
inexact yield line solutions have been used and some reassurance against the effects of
ignoring corner levers (see above). At the relatively low stress levels in slabs, a 10%
increase in moment equates to a 10% increase in the designed reinforcement.
The designer may of course chase in search of a more exact solution but most pragmatists
are satisfied to know that by applying the 10% rule to a simple analysis their design will be
on the safe side without being unduly conservative or uneconomic. The 10% rule can and
usually is applied in other circumstances where the designer wants to apply engineering
judgement and err on the side of caution.
The only situations where allowances under this ‘10% rule’ may be inadequate relate to
slabs with acute corners and certain configuration of slabs with substantial B point loads or
line loads. In these cases guidance should be sought from specialist literature.
8.1 .14 Yield Line Design has the advantages of:
• Economy
• Simplicity and
• Versatility
Yield Line Design leads to slabs that are quick and easy to design, and are quick and easy
to construct. There is no need to resort to computer for analysis or design. The resulting
slabs are thin and have very low amounts of reinforcement in very regular arrangements.
The reinforcement is therefore easy to detail and easy to fix and the slabs are very quick to
construct. Above all, Yield Line Design generates very economic concrete slabs, because it
considers features at the ultimate limit state.
Yield Line Design is a robust and proven design technique. It is a versatile tool that
challenges designers to use judgement. Once grasped, Yield Line Design is an exceedingly
powerful design tool.
Yield Line Design demands familiarity with failure patterns, i.e. knowledge of how slabs
might fail. This calls for a certain amount of experience, engineering judgement and
confidence, none of which is easily gained.
8.2 The Work Method of analysis
8.2 .1 General
Before explaining how to apply the Work Method of analysis it may help to review the
stages involved in the failure of a slab:
• Collapse occurs when yield lines form a mechanism.
• This mechanism divides the slab into rigid regions.
• Since elastic deformations are neglected these rigid regions remain as plane areas.
• These plane areas rotate about their axes of rotation located at their supports.
• All deformation is concentrated within the yield lines.
The basis of the Work Method is simply that at failure the potential energy expended by
loads moving must equal the energy dissipated (or work done) in yield lines rotating. In
other words:
External energy = Internal energy
by loads moving = dissipated by rotations about yield lines
Expended = Dissipated
E = D
Σ (Ν x δ)
for all regions
= Σ (m x l x θ)
for all regions
where
N is the Load(s) acting within a particular region [kN]
δ is the vertical displacement of the load(s) N on each region expressed as a fraction of
unityC [m]
m is the moment or moment of resistance of the slab per metre run represented by the
reinforcement crossing the yield line [kNm/mD]
l is the length of yield line or its projected length onto the axis of rotation for that region
[m]
θ is the rotation of the region about its axis of rotation [m/m]
Once a valid failure pattern (or mechanism) has been postulated, either the moment, m,
along the yield lines or the failure load of a slab, N (or indeed n kN/m
2
), can be established
by applying the above equation.
This, fundamentally, is the Work Method of analysis: it is a kinematic (or energy) method
of analysis.
8.2 .2 Principles
To illustrate the principles, two straightforward examples are presented. Consider a
one-way slab simply supported on two opposite sides, span, L and width w, supporting a
uniformly distributed load of n kN/m
2
.
Figure 8.2.1 A simply supported one-way slab
Therefore:
Which is rather familiar!
The same principles apply to two-way spanning slabs. Consider a square slab simply
supported on four sides. Increasing load will firstly induce hairline cracking on the soffit,
then large cracks will form culminating in the yield lines shown in Figure 8.2.2.
Figure 8.2.2. Simply supported slab yield line pattern
Diagonal cracks are treated as stepped cracks, with the yield lines projected onto parallel
axes of rotations.
Assuming the slab measures L x L and carries a load of n kN/m
2
:
Therefore:
8.2 .3 Design procedure
When applying the Work Method the calculations for the expenditure of external loads and
the dissipation of energy within the yield lines are carried out independently. The results
are then made equal to each other and from the resulting equation the unknown, be it the
ultimate moment ‘m’ generated in the yield lines or the ultimate failure load ‘n’ of the slab,
evaluated.
Calculating expenditure of energy of external loads: E
Having chosen a layout of yield lines forming a valid failure pattern, the slab is divided
into rigid regions that rotate about their respective axes of rotation along the support lines.
If we give the point of maximum deflection a value of unity then the vertical displacement
of any point in the regions is thereby defined. The expenditure of external loads is
evaluated by taking all external loads on each region, finding the centre of gravity of each
resultant load and multiplying it by the distance it travels.
In mathematical terms: E = Σ (Nδ) for all regions The principles are illustrated in Figure
8.2.3. Having chosen a valid pattern and layout the points of application of all resultant
loads are identified. Points 1-8 are the points of application of the resultant of the
uniformly distributed loads in the individual regions bounded by the yield lines. Point ‘P’ is
the point of application of the point load P.
Figure 8.2.3 Principles of expenditure of external loads: E
Calculating dissipation of energy within the yield lines: D
The dissipation of energy is quantified by projecting all the yield lines around a region onto,
and at right angles to, that region’s axis of rotation. These projected lengths are multiplied
by the moment acting on each length and by the angle of rotation of the region. At the
small angles considered, the angle of rotation is equated to the tangent of the angle
produced by the deflection of the region. The sense of the rotations is immaterial.
In mathematical terms: D= Σ (m l θ) for all regions.
Figure 2.2.4 (see over) is a graphical presentation of the terms involved in the dissipation
of internal energy along the yield lines, (assuming an isotropic layout of reinforcement). In
region D, for instance, the projection of the positive (sagging) yield line of value ‘m’
surrounding that region a-b-e onto its axis of rotation, a-b, has the length a-b, shown as
length ‘Lx’. Similarly the yield lines d-f-c around region A are projected onto d-c and has
the length of ‘Lx’.
In region C, the projection of the positive (sagging) yield line of value ‘m’ surrounding that
region b-e-f-c onto its axis of rotation, b-c, has the length b-c, shown as length ‘Ly’. This
side also has continuous support and a negative (hogging) yield line, of value m’, that
forms along the support. As this yield line already lies on the axis of rotation, it has a
projected length equal to the length of the side b-c, again shown as length ‘Ly’. The angle
of rotation of region C affecting both these moments is shown in section 1-1. It will be seen
that, by definition, the angle of rotation ,φc, equals 1/hC. A similar procedure is applied to
the other regions. The yield lines a-e-f-d around region B would be projected onto a-d. In
this case as it is a simple support no negative moment would develop at the support.
Figure 8.2.4 Principles of dissipation of internal energy, D
8.2 .4 Orthotropic slabs
2024年4月1日发(作者：无鸿云)
8 Yield Line Theory
8.1 The essentials
Yield Line Design is a well-founded method of designing reinforced concrete slabs, and
similar types of elements. It uses Yield Line Theory to investigate failure mechanisms at
the ultimate limit state. The theory is based on the principle that:
work done in yield lines rotating = work done in loads moving
Yield Line Design is easy to grasp but there are several fundamental principles that need to
be understood. Yield Line Design is a plastic method: it is different from ‘normal’
elastic methods.
8.1 .1 A short history of Yield Line Theory
Yield Line Theory as we know it today was pioneered in the 1940s by the Danish engineer
and researcher K W Johansen.
As early as 1922, the Russian, A Ingerslev presented a paper to the Institution of
Structural Engineers in London on the collapse modes of rectangular slabs. Authors such
as R H Wood, L L Jones, A Sawczuk and T Jaeger, R Park, K O Kemp, C T Morley, M
Kwiecinski and many others, consolidated and extended Johansen’s original work so
that now the validity of the theory is well established making Yield Line Theory a
formidable international design tool.
In the 1960s 70s and 80s a significant amount of theoretical work on the application
of Yield Line Theory to slabs and slab-beam structures was carried out around the world
and was widely reported.
To support this work, extensive testing was undertaken to prove the validity of the
theory. Excellent agreement was obtained between the theoretical and experimental Yield
Line patterns and the ultimate loads.
The differences between the theory and tests were small and mainly on the
conservative side. In the tests where restraint was introduced to simulate continuous
construction, the ultimate loads reached at failure were significantly greater
than the loads predicted by the theory due to the beneficial effect of membrane
forces.
8.1 .2 What is a yield line?
A yield line is a crack in a reinforced concrete slab across which the reinforcing bars have
yielded and along which plastic rotation occurs.
8.1 .3 What is Yield Line Theory?
Yield Line Theory is an ultimate load analysis. It establishes either the moments in an
element (e.g. a loaded slab) at the point of failure or the load at which an element will fail.
It may be applied to many types of slab, both with and without beams. Consider the case
of a square slab simply supported on four sides as illustrated by Figure 8.1.1. This slab is
subjected to a uniformly distributed load, which gradually increases until collapse occurs.
Initially, at service load, the response of the slab is elastic with the maximum steel stress
and deflection occurring at the centre of the slab. At this stage, it is possible that some
hairline cracking will occur on the soffit where the flexural tensile capacity of the concrete
has been exceeded at midspan.
Figure 8.1.1 Onset of yielding of bottom reinforcement at point of maximum deflection in a
simply supported two-way slab
Increasing the load hastens the formation of these hairline cracks, Increasing the load
further will increase the size of the cracks further and induce yielding of the reinforcement,
initiating the formation of large cracks emanating from the point of maximum deflection.
On increasing the load yet further, these cracks migrate to the free edges of the slab at
which time all the tensile reinforcement passing through a yield line yields.
At this ultimate limit state, the slab fails. As illustrated by Figure 8.1.2, the slab is divided
into rigid plane regions A, B, C and D. Yield lines form the boundaries between the rigid
regions, and these regions, in effect, rotate about the yield lines. The regions also pivot
about their axes of rotation, which usually lie along lines of support, causing supported
loads to move. It is at this juncture that the work dissipated by the hinges in the yield lines
rotating is equated to work expended by loads on the regions moving. This is Yield Line
Theory.
Under this theory, elastic deformations are ignored; all the deformations are assumed to
be concentrated in the yield lines and, for convenience, the maximum deformation is given
the value of unity.
For a more detailed appraisal of the situation at corner see Section ?
Figure 8.1.2 The formation of a mechanism in a simply supported two-way slab with the
bottom steel having yielded along the yield lines
8.1 .4 What is a yield line pattern?
When a slab is loaded to failure, yield lines form in the most highly stressed areas and
these develop into continuous plastic hinges. As described above, these plastic hinges
develop into a mechanism forming a yield line pattern.
Yield lines divide the slab up into individual regions, which pivot about their axes of
rotation. Yield lines and axes of rotation conform to rules given in Table 8.1.1, which help
with the identification of valid patterns and the Yield Line solution.
Table 8.1.1 Rules for yield line patterns
• Axes of rotation generally lie along lines of support and pass alongside any columns.
• Yield lines are straight.
• Yield lines between adjacent rigid regions must pass through the point of intersection of
the axes of rotation of those regions.
• Yield lines must end at a slab boundary.
• Continuous supports repel and simple supports attract positive or sagging yield lines.
8.1 .5 What is a Yield Line solution?
In theory, there may be several possible valid yield line patterns that could apply to a
particular configuration of a slab and loading. However, there is one yield line pattern that
gives the highest moments or least load at failure. This is known as the yield line solution.
The designer has several ways of determining the critical pattern and ensuring safe
design:
• From first principles, e.g. by using The Work Method
• Using formulae for standard situations.
It will be noted that valid yield line patterns give results that are either correct or
theoretically unsafe. These ‘upper bound solutions’ can deter some designers but, as
discussed later, this theoretical awkwardness is easily overcome by testing different
patterns and by making suitable allowances (see 10% rule later).
8.1 .6 How do you select relevant yield line patterns?
A yield line pattern is derived mainly from the position of the axes of rotation, (i.e. the lines
of support) and by ensuring that the yield lines themselves are straight, go through the
intersection of axes of rotation and end at the slab boundary, i.e. conform to the rules in
Table 8.1.1. Some simple examples are shown in Figure 8.1.3 Considering a slab to be a
piece of pastry laid over supports may help designers to visualise appropriate yield line
patterns.
Figure 8.1.3 Simple Yield Line patterns
The aim of investigating yield line patterns is to find the one pattern that gives the critical
moment (the highest moment or the least load capacity). However, an exhaustive search is
rarely necessary and selecting a few simple and obvious patterns is generally sufficient as
their solutions are within a few percent of the perfectly correct solution.
8.1 .7 What is a fan mechanism?
Slabs subjected to heavy concentrated loads may fail by a so-called fan mechanism, with
positive Yield Lines radiating from the load and a negative circular Yield Line centred under
the point load. This mechanism is shown in Figure 8.1.4. It is rare for this form of failure to
be critical but nonetheless a check is advised where large concentrated loads are present
or for instance in flat slabs where the slab is supported on columns.
Figure 8.1.4 Fan collapse pattern for a heavy concentrated load onto a reinforced slab
The mechanism for a slab supported by a column is the same shape but with the positive
and negative yield lines reversed.
8.1 .8 What is the Work Method?
The Work Method (or virtual Work Method) of analysis is the most popular (and most easy)
way of applying Yield Line theory from first principles. Indeed, many experienced users of
Yield Line theory of design choose to use the Work method because it is so very easy. The
fundamental principle is that work done internally and externally must balance.
In other words, at failure, the expenditure of external energy induced by the load on the
slab must be equal to the internal energy dissipated within the yield lines. In other words:
External energy = Internal energy
by loads moving = dissipated by rotations about yield lines
Expended = Dissipated
E = D
Σ (Ν x δ)
for all regions
= Σ (m x l x θ)
for all regions
where
N = load(s) acting within a particular region [kN]
δ = the vertical displacement of the load(s) N on each region expressed as a fraction of
unity [m]
m = the moment in or moment of resistance of the slab per metre run [kNm/m]
l = the length of yield line or its projected length onto the axis of rotation for that region
[m]
θ = the rotation of the region about its axis of rotation [m/m] By way of illustration,
consider the slab shown in Figure 8.1.2. Figure 8.1.5 shows an axonometric view of this
two-way simply supported slab that has failed due to a uniformly distributed load. Note
that:
• The triangular regions A, B, C and D have all rotated about their lines of support.
• The loads on the regions have moved vertically and rotation has taken place about the
yield lines and supports.
• The uniformly distributed load on each of these regions will have moved on average 1/3
of the maximum deflection.
The rotation of the regions about the yield lines can be resolved into rotation about the
principal axes of rotation, and thereby measured with respect to the location and size of
the maximum deflection.
Figure 8.1.5 Deformed shape at failure
This, fundamentally, is the ‘Work Method’. Any slab can be analysed by using the principle
of E = D. Some judgement is required to visualise and check likely failure patterns but
absolute accuracy is rarely necessary and allowances are made to cover inaccuracies.
Once a yield line pattern has been selected for investigation, it is only necessary to specify
the deflection as being unity at one point (the point of maximum deflection) from which all
other deflections and rotations can be found.
8.1 .9 Formulae
Rather than go through the Work Method, some practitioners prefer the even quicker
method of using standard formulae for standard types of slab. The formulae are
predominantly based on the work method and they are presented in more detail in Chapter
3. As an example, the formula for one-way spanning slabs supporting uniformly distributed
loads is as follows:
per unit width
where
m = ultimate sagging moment along the yield line [kNm/m]
m’ = ultimate support moment along the yield line [kNm/m]
n = ultimate load [kN/m2]
L = span [m]
i1 , i2 = ratios of support moments to mid-span moments. (The values of i are chosen by
the designer: i1= m’1/m, i2= m’2/m)
Where slabs are continuous, the designer has the freedom to choose the ratio of hogging
to sagging moments to suit any particular situation. For instance, the designer may choose
to make the bottom span steel equal to the top support steel (i.e. make sagging moment
capacity equal support moment capacity.)
Failure patterns for one-way spanning slabs are easily visualised and the standard
formulae enable the designer to quickly determine the span moment based on any ratio of
hogging moments he or she chooses to stipulate (within a sensible range dictated by codes
of practice). Formulae are also available for the curtailment of top reinforcement.
Formulae for two-way spanning slabs supported on two, three or four sides are also
available for use. These are a little more complicated due to the two-way nature of the
problem and the fact that slabs do not always have the same reinforcement in both
directions. The nature of the failure patterns is relatively easy to visualise and again the
designer has the freedom to choose fixity ratios.
8.1 .10 Is Yield Line Theory allowable under design codes of practice?
Yes. Any design process is governed by the recommendations of a specific code of practice.
In the UK, BS 8110 clause 3.5.2.1 says ‘Alternatively, Johansen’s Yield Line method ….
may be used…. for solid slabs’. The proviso is that to provide against serviceability
requirements, the ratio of support and span moments should be similar to those obtained
by elastic theory. This sub-clause is referred to in clauses 3.6.2 and 3.7.1.2 making the
approach also acceptable for ribbed slabs and flat slabs.
According to Eurocode 2, Yield Line Design is a perfectly valid method of design. Section
5.6 of Eurocode 2 states that plastic methods of analysis shall only be used to check the
ultimate limit state. Ductility is critical and sufficient rotation capacity may be assumed
provided x/d ≤0.25 for C50/60. A Eurocode 2 goes on to say that the method may be
extended to flat slabs, ribbed, hollow or waffle slabs and that corner tie down forces and
torsion at free edges need to be accounted for.
Section 5.11.1.1 of EC2 includes Yield Line as a valid method of analysis for flat slabs. It is
recommended that a variety of possible mechanisms are examined and the ratios of the
moments at support to the moment in the spans should lie between 0.5 and 2.
8.1.11 Yield Line is an upper bound theory
Yield line theory gives upper bound solutions - results that are either correct or
theoretically unsafe, see Table 8.1.2. However, once the possible failure patterns that can
form have been recognised, it is difficult to get the yield line analysis critically wrong
Table 8.1.2 Upper and lower bound ultimate load theories
Ultimate load theories for slabs fall into two categories:
• upper bound (unsafe or correct) or
• lower bound (safe or correct).
Plastic analysis is either based on
• upper bound (kinematic) methods, or on
• lower bound (static) methods.
Upper bound (kinematic) methods include:
• plastic or yield hinges method for beams, frames and one-way slabs;
• Yield Line Theory for slabs.
Lower bound (static) methods include:
• the strip method for slabs,
• the strut and tie approach for deep beams, corbels, anchorages, walls and plates loaded
in their plane.
The mention of ‘unsafe’ can put designers off, and upper bound theories are often
denigrated. However, any result that is out by a small amount can be regarded as
theoretically unsafe. Yet few practising engineers regard any analysis as being absolutely
accurate and make due allowance in their design. The same is true and acknowledged in
practical Yield Line Design.
In the majority of cases encountered, the result of a Yield Line analysis from first principles
will be well within 10%, typically within 5%, of the mathematically correct solution. The
pragmatic approach, therefore, is to increase moments (or reinforcement) derived from
calculations by 10%. This ‘10% rule’ is expanded upon later.
There are other factors that make Yield Line Design safer than it may at first appear, e.g.
compressive membrane action in failing slabs (this alone can quadruple ultimate
capacities), strain hardening of reinforcement, and the practice of rounding up steel areas
when allotting bars to designed areas of steel required.
The practical designer can use Yield Line Theory with confidence, in the knowledge that he
or she is in control of a very useful, powerful and reliable design tool.
8.1.12 Corner levers
‘Corner levers’ describes the phenomenon in two-way slabs on line supports where yield
lines split at internal corners. This splitting is associated with the formation of a negative
yield line across the corner which ‘levers’ against a corner reaction (or holding down force).
Corner levers particularly affect simply supported slabs and Figure 8.1.5 shows the effect
corner levers can have on a simply supported square slab. It should also be noted that the
sagging moment m in an isotropic slab increases with decreasing corner fixity. Table 8.1.3
illustrates the effects of continuity on both the extent of the corner levers and on positive
moments [13]. At an average fixity ratio of 1.0 the effects are minimal. Nonetheless, if the
corners are left unreinforced, span moments increase.
Figure 8.1.5 The effect of corner levers on a simply supported square slab where corners
are held down and prevented from lifting.
Table 8.1.3 Effects of corner continuity on corner levers in a simply supported square slab
Corner fixity
i = m’/m
0
0.25
0.50
1.00
x h
m
[kNm/m]
Na2/22
Na2/23
Na2/23.6
Na2/24
Positive moment
increase in the slab
due to corner lever
9.0%
4.3%
1.7%
-
0.159a
0.110a
0.069a
0
0.523a
0.571a
0.619a
-
For simplicity in the analysis, yield line patterns are generally assumed to go into corners
without splitting, i.e. corner levers are ignored and an allowance is made for this. This
simplification is justified for three principle reasons:
• The error for neglecting corner levers is usually small.
• The analysis involving corner levers becomes too involved.
• Corner levers usually bring out the beneficial effects of membrane action that negate
their impact.
All methods and formulae are based on straight-line crack patterns that go into the corners.
The values of the moments obtained in this way are only really valid if the top
reinforcement provided in the corners is of the same magnitude as the bottom steel
provided in the span. If this is not the case, as generally assumed, then the straight line
pattern will not form and some type of corner lever will appear depending on the amount
of top reinforcement provided, if any. This in turn leads to additional moment to be added
to the calculated positive (sagging) moment.
The exact amount of increase depends on a number of parameters, but generally about
4% to 8% is assumed for rectangular two-way slabs. At worst, for simply supported square
slabs, the increase is approximately 9%. The effects of corner levers in slabs supported on
four sides diminishes in rectangular slabs and begin to die out at a ratio of sides greater
than 3:1. In triangular slabs and slabs with acute corners, the straight-line mechanism into
the corners can underestimate the positive moment by 30% 35%.
The effects of corner levers have to be recognised. For regular slabs their effects are
allowed for within ‘the 10% rule’.
Despite this, it is good practice, and it is recommended, to specify and detail U-bars,
equivalent to 50% of the span steel around all edges, including both ways at corners.
8.1.13 The 10% rule
A 10% margin on the design moments should be added when using the Work
Method or formulae for two-way slabs to allow for the method being upper
bound and to allow for the effects of corner levers
The addition of 10% to the design moment in two-way slabs provides some leeway where
inexact yield line solutions have been used and some reassurance against the effects of
ignoring corner levers (see above). At the relatively low stress levels in slabs, a 10%
increase in moment equates to a 10% increase in the designed reinforcement.
The designer may of course chase in search of a more exact solution but most pragmatists
are satisfied to know that by applying the 10% rule to a simple analysis their design will be
on the safe side without being unduly conservative or uneconomic. The 10% rule can and
usually is applied in other circumstances where the designer wants to apply engineering
judgement and err on the side of caution.
The only situations where allowances under this ‘10% rule’ may be inadequate relate to
slabs with acute corners and certain configuration of slabs with substantial B point loads or
line loads. In these cases guidance should be sought from specialist literature.
8.1 .14 Yield Line Design has the advantages of:
• Economy
• Simplicity and
• Versatility
Yield Line Design leads to slabs that are quick and easy to design, and are quick and easy
to construct. There is no need to resort to computer for analysis or design. The resulting
slabs are thin and have very low amounts of reinforcement in very regular arrangements.
The reinforcement is therefore easy to detail and easy to fix and the slabs are very quick to
construct. Above all, Yield Line Design generates very economic concrete slabs, because it
considers features at the ultimate limit state.
Yield Line Design is a robust and proven design technique. It is a versatile tool that
challenges designers to use judgement. Once grasped, Yield Line Design is an exceedingly
powerful design tool.
Yield Line Design demands familiarity with failure patterns, i.e. knowledge of how slabs
might fail. This calls for a certain amount of experience, engineering judgement and
confidence, none of which is easily gained.
8.2 The Work Method of analysis
8.2 .1 General
Before explaining how to apply the Work Method of analysis it may help to review the
stages involved in the failure of a slab:
• Collapse occurs when yield lines form a mechanism.
• This mechanism divides the slab into rigid regions.
• Since elastic deformations are neglected these rigid regions remain as plane areas.
• These plane areas rotate about their axes of rotation located at their supports.
• All deformation is concentrated within the yield lines.
The basis of the Work Method is simply that at failure the potential energy expended by
loads moving must equal the energy dissipated (or work done) in yield lines rotating. In
other words:
External energy = Internal energy
by loads moving = dissipated by rotations about yield lines
Expended = Dissipated
E = D
Σ (Ν x δ)
for all regions
= Σ (m x l x θ)
for all regions
where
N is the Load(s) acting within a particular region [kN]
δ is the vertical displacement of the load(s) N on each region expressed as a fraction of
unityC [m]
m is the moment or moment of resistance of the slab per metre run represented by the
reinforcement crossing the yield line [kNm/mD]
l is the length of yield line or its projected length onto the axis of rotation for that region
[m]
θ is the rotation of the region about its axis of rotation [m/m]
Once a valid failure pattern (or mechanism) has been postulated, either the moment, m,
along the yield lines or the failure load of a slab, N (or indeed n kN/m
2
), can be established
by applying the above equation.
This, fundamentally, is the Work Method of analysis: it is a kinematic (or energy) method
of analysis.
8.2 .2 Principles
To illustrate the principles, two straightforward examples are presented. Consider a
one-way slab simply supported on two opposite sides, span, L and width w, supporting a
uniformly distributed load of n kN/m
2
.
Figure 8.2.1 A simply supported one-way slab
Therefore:
Which is rather familiar!
The same principles apply to two-way spanning slabs. Consider a square slab simply
supported on four sides. Increasing load will firstly induce hairline cracking on the soffit,
then large cracks will form culminating in the yield lines shown in Figure 8.2.2.
Figure 8.2.2. Simply supported slab yield line pattern
Diagonal cracks are treated as stepped cracks, with the yield lines projected onto parallel
axes of rotations.
Assuming the slab measures L x L and carries a load of n kN/m
2
:
Therefore:
8.2 .3 Design procedure
When applying the Work Method the calculations for the expenditure of external loads and
the dissipation of energy within the yield lines are carried out independently. The results
are then made equal to each other and from the resulting equation the unknown, be it the
ultimate moment ‘m’ generated in the yield lines or the ultimate failure load ‘n’ of the slab,
evaluated.
Calculating expenditure of energy of external loads: E
Having chosen a layout of yield lines forming a valid failure pattern, the slab is divided
into rigid regions that rotate about their respective axes of rotation along the support lines.
If we give the point of maximum deflection a value of unity then the vertical displacement
of any point in the regions is thereby defined. The expenditure of external loads is
evaluated by taking all external loads on each region, finding the centre of gravity of each
resultant load and multiplying it by the distance it travels.
In mathematical terms: E = Σ (Nδ) for all regions The principles are illustrated in Figure
8.2.3. Having chosen a valid pattern and layout the points of application of all resultant
loads are identified. Points 1-8 are the points of application of the resultant of the
uniformly distributed loads in the individual regions bounded by the yield lines. Point ‘P’ is
the point of application of the point load P.
Figure 8.2.3 Principles of expenditure of external loads: E
Calculating dissipation of energy within the yield lines: D
The dissipation of energy is quantified by projecting all the yield lines around a region onto,
and at right angles to, that region’s axis of rotation. These projected lengths are multiplied
by the moment acting on each length and by the angle of rotation of the region. At the
small angles considered, the angle of rotation is equated to the tangent of the angle
produced by the deflection of the region. The sense of the rotations is immaterial.
In mathematical terms: D= Σ (m l θ) for all regions.
Figure 2.2.4 (see over) is a graphical presentation of the terms involved in the dissipation
of internal energy along the yield lines, (assuming an isotropic layout of reinforcement). In
region D, for instance, the projection of the positive (sagging) yield line of value ‘m’
surrounding that region a-b-e onto its axis of rotation, a-b, has the length a-b, shown as
length ‘Lx’. Similarly the yield lines d-f-c around region A are projected onto d-c and has
the length of ‘Lx’.
In region C, the projection of the positive (sagging) yield line of value ‘m’ surrounding that
region b-e-f-c onto its axis of rotation, b-c, has the length b-c, shown as length ‘Ly’. This
side also has continuous support and a negative (hogging) yield line, of value m’, that
forms along the support. As this yield line already lies on the axis of rotation, it has a
projected length equal to the length of the side b-c, again shown as length ‘Ly’. The angle
of rotation of region C affecting both these moments is shown in section 1-1. It will be seen
that, by definition, the angle of rotation ,φc, equals 1/hC. A similar procedure is applied to
the other regions. The yield lines a-e-f-d around region B would be projected onto a-d. In
this case as it is a simple support no negative moment would develop at the support.
Figure 8.2.4 Principles of dissipation of internal energy, D
8.2 .4 Orthotropic slabs