design of portal frames - notes
TRANSCRIPT
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MSc Design of Steel Buildings Design of Portal Frames 1
MSc Design of Steel Buildings
Design of Portal Frames
1 INTRODUCTION
In most European countries, steel construction is used for the majority of non-domestic
single-storey buildings. This is due to the ability to design relatively light, long span, durable
structures in steel, which are easy to erect safely and quickly. The capacity to provide spans
up to 60 m, but more commonly around 30 m, using steel has proved very popular for
commercial and leisure buildings. The lightness and flexibility of this kind of steel structure
reduces the sizes and the costs of foundations and makes them less sensitive to the
geotechnical characteristics of the soil.
The brief for the design of the majority of single storey industrial buildings is essentially to
design a structure with a limited number of internal columns. In principle, the requirement is
for the construction of four walls and a roof for a single or multi-bay structure. The walls canbe formed of steel columns with cladding of profiled or plain sheet. The designer considers a
system of beams or frameworks (latticed or traditional) in structural steel to support the
cladding for the roof. Use is made of hot rolled hollow sections (circular, rectangular) and
traditional sections (I sections, angles, etc.) and also cold formed sections which, in many
cases, provide the most efficient and economic solution.
In the following, after mentioning the main components of common single storey steel
buildings, principles and detailed rules for designing traditional steel portal frames, which
represent the main structural systems in about 50% of single storey structures in the UK, will
be presented. Reference will be made to modern codes of practice, in particular to Eurocode 3
(EN 1993-1-1) and BS5950.
2 COMPONENTS IN SINGLE STOREY BUILDINGS
The skeleton of a typical single-storey building is shown in the Figure 1. It consists of
three major elements: cladding for both roof and walls, secondary steel members to support
the cladding and form framing for doors and windows and the main frame of the structure,
including all necessary bracing.
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MSc Desi
Figure 1:
2.1
The
appearan
structure
has emerusual su
sheets a
There ar
2a), ii) d
composi
Single-s
insulatio
Figure 2
to 35 m
insulatio
span of
typically
secret fi
incorpor
fasteners
seam sy
with exp
in the w
gn of Steel B
Component
ladding
primary fu
ce. The cla
and it is lik
ged as thestrate with
e formed f
e four main
ouble skin c
e panels.
in sheetin
is require
. The cladd
trapezoida
material,
uch system
in the orde
x systems
tes a clipp
and impro
tems may b
osed fastene
ather sheet.
uildings D
of a portal s
ction of th
dding there
ely to be so
ost populaaluminium
om a subst
categories
ladding (Fi
is widely
. The sheet
ing is gener
l profile de
spacer sys
s is limited
of 2 m to
use a speci
d joint bet
es the weat
e used on v
rs). Insulate
Standing se
sign of Port
tructure.
e cladding
fore repres
e 50% of
choice sinas a more
rate with la
f cladding
. 2b), iii) st
used in a
ng is fixed
ally made fr
th. Double
em and an
by the spa
2,5 m depe
lly designe
een adjac
er tightnes
ery low roo
d panel syst
am sheeting
l Frames
is to provi
nts one of
he total cos
e its introdexpensive
yers of gal
systems: i)
anding sea
ricultural
directly to t
om 0,7 mm
skin claddi
uter metal
ning capab
ding on th
d profile fo
nt sheets.
s of the clad
f slopes (do
ems are als
can be man
de shade, s
the most i
. Cladding
ction in thsecond cho
anising, pr
ingle skin t
/secret fix
nd industr
e purlins a
gauge pre-
g consists
heet, as ill
lity of the
applied lo
the weath
his elimina
ding syste
wn to 1 co
available
ufactured fr
helter, and
mportant el
ormed fro
1970's. Stece. Steel-b
imer and c
rapezoidal
ladding (Fi
al structur
d side rail
oated steel
f a metal li
strated in F
cladding sh
ading. Stan
r (external)
tes the nee
. Conseque
mpared to
ith a standi
om steel or
an attracti
ement of t
metal shee
el is the mosed claddi
lour coatin
ladding (Fi
g. 2c) and i
s, where
, as shown
ith a 32 m
er, a layer
igure 2b. T
ets, which
ing seam
sheet, whi
for expos
ntly, standi
for syste
ng seam joi
aluminium.
2
e
e
ts
stg
g.
g.
)
o
n
m
of
e
is
r
h
d
g
s
t
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MSc Desi
Figure 2
seam/sec
2.2 S
For s
or sheet
between
member
gn of Steel B
: Cladding s
et fix.
econdary
teel portal f
are norma
the rafters
is to transf
uildings D
stems: (a)
elements
amed indu
ly supporte
and colum
er load fro
sign of Port
ingle skin t
trial type b
d by a syst
ns respecti
the claddi
l Frames
apezoidal, (
ildings wit
m of light
ely. The p
g to the pri
) double ski
low pitch
steel purlin
rimary fun
mary steel
(a)
(b)
(c)n trapezoida
oofs, the cl
s and side r
tion of the
rame, inclu
l, (c) standi
dding pane
ails spanni
se seconda
ing claddi
3
g
ls
g
y
g
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MSc Desi
self-wei
The purl
to transf
Cold
sizes (Fi
profile t
high len1-3, hen
Figure 3:
The
from thprovide
costs, it
the prim
forces a
provided
sheeting,
the roof
predomi
Figure 4:
In or
purlins
that the r
Figure 5.
least 25
the purli
gn of Steel B
ht, wind lo
ns and side
r horizontal
-formed pur
. 3). The d
ickness var
th/thicknese section pr
Common ty
urlins and
cladding,estrain to ra
s common
ry steelwor
ising from
that the fo
ii) there is
sheeting is
antly roof l
Details of c
der to provi
r cladding r
estraining
As a rule o
of the dep
s and side
uildings D
ds and, for
rails may al
loads into t
ins and sid
pth of the
ing betwee
values, areoperties wil
es of cold-f
ladding rail
and to tranfter and col
ractice to
k. It is gen
the lateral
llowing co
bracing of
capable o
oads.
lumn and ra
de the requ
ails must p
ember will
f thumb, it i
th of the m
ails will be
sign of Port
roofs, imp
so be used t
he bracing s
rails are a
ection typi
n 1,2 mm a
typically clbe need to
rmed purlin
s need to be
sfer theseumns. In fa
se the seco
rally accept
restraint o
ditions are
dequate sti
acting as
ter stay and
red level o
ssess suffi
bend and al
s normally
mber being
sufficiently
l Frames
sed loads d
o provide re
ystem.
ailable in a
ally lies bet
d 3,2 mm.
assed as Clbe based on
.
designed to
oads into tt, to achiev
dary steel
ed that purl
rafters in
met: i) the
ffness in th
stressed-s
onnection.
torsional r
ient flexura
ow the restr
dequate to
restrained.
stiff for por
ue to snow
straint to th
variety of s
ween 120
urlins and
ss 4 sectioeffective v
carry all of
he structure savings in
ork (the pu
ins and rail
either roof
purlins are
plane of t
in diaphra
straint to t
l stiffness.
ained mem
provide a p
In practice,
tal frames
and mainte
e rafters and
apes and a
m and 340
ide rails, be
s as definelues.
the loads a
l frame.material an
lins and rai
need not b
trusses or
adequately
e rafters o
m, iii) the
e rafters or
therwise,
ers to rotat
rlin or clad
this general
ith spans u
nance acces
columns a
ide range
mm, with t
cause of th
in EN 199
plied to the
oreover thd therefore
ls) to restra
e checked f
ortal fram
restrained
alternative
rafters car
columns, t
here is a ri
, as shown
ding rail of
ly means th
to 40 m a
4
s.
d
f
e
ir
3-
m
yn
n
r
s
y
y
y
e
k
in
at
at
d
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MSc Desi
frame sp
the rafte
insuffici
Figure 5:
Cold
bending
they are
supporti
up the
load/spa
producindegree o
conditio
effect o
greater l
and the s
of a wall
sheet or
generall
general,
should, t
Figure 6:
For
restraint
restraine
shown i
gn of Steel B
acings of 6
r size incre
nt to provid
Purling stiff
formed ste
action, but
adequately
g steelwor
hole syste
tables, w
g their desif restraint t
s. These a
the design
vel of restr
pacing of th
), restraint i
insulated p
capable o
perforated
erefore, be
Purlin restra
ind uplift
directly to
by a com
Figure 6.
uildings D
o 8 m. Ho
ases relativ
e adequate
ess effects.
l purlins a
they are su
restrained.
relies on t
. Purlins a
ich are d
n data, allat is availa
sumptions
resistance
aint is achi
e fasteners.
provided d
nel, as sho
providing
iners are n
designed as
int.
(or negativ
the compr
ination of l
s an alterna
sign of Port
ever, as the
e to that o
orsional res
d cladding
ceptible to
gain the e
e interacti
d cladding
rived from
urlin manuble from th
re central
f the purli
ved in pra
In the gravi
irectly to th
wn in Figu
sufficient l
ot consider
unrestraine
e pressure
ssion flang
teral restrai
tive to cold
l Frames
span increa
the purlin
traint and s
rails are ext
failure thro
conomic an
n between
rails are no
analytical
acturers hae cladding s
o the desig
or rail. It i
tice. This
y load case
top flange
e 6. Built-
ateral restra
d to be re
d members.
1 Later
compres
2 Claddi
tension
restraint
on a wall),
e. In this c
nt to the te
formed stee
ses relative
), the purl
ould, there
remely effi
ugh lateral-
safe desig
he individu
rmally sele
models su
e to make aystem unde
n model a
s therefore
ill depend
(or positive
of the purli
p cladding
int for the
straining an
al restraine
ion flange b
g provides l
flange and
the claddi
ase, the pu
sion flange
l, purlins a
to the fram
n stiffness
ore, be chec
ient at carr
torsional b
n of the cla
al compone
ted from
ported by
judgementr gravity a
d can have
essential th
n the choi
wind press
(or side rai
and insulat
gravity loa
d the supp
d provided
cladding
ateral restrai
partial torsi
g cannot p
rlin (or cla
and torsion
d cladding
spacing (a
may beco
ked.
ing loads
ckling unle
dding and i
nts that ma
anufacturer
test data.
regarding td wind upl
a significa
t an equal
e of sheeti
re in the ca
l) by the lin
ed panels a
ding case.
rting purli
to
t to
onal
rovide later
ding rail)
l restraint,
ails may al
5
d
e
y
ss
ts
e
s
n
eft
t
r
g
e
er
e
n
s
al
is
s
o
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MSc Desi
be made
industria
a higher
generall
more. T
common
suitablepurlins (
2.3
Steel
building
configur
Figure 7beam), (c
external
The
when th
necessar
of steel
foundati
gn of Steel B
from hot-r
l buildings,
load-carryi
used at mu
is wide spa
ly require r
for elasticbove 8 m).
ain fram
portal fra
, because
tions can b
: Various tyPortal with
ffices.
ost popul
re is no c
to support
results, in
ns of the s
uildings D
olled steel
often used i
g capacity
ch greater s
ing makes
straint to th
rames and
e
es are wi
they com
designed u
es of portalinternal offic
r solutions
ane to be
crane loads
any case,
pports can
(a)
(c)
(e)
sign of Port
sections. I
n conjuncti
than larges
pacings tha
hem unsuit
e rafters at
also for sp
ely used
ine structu
sing the sa
frame: (a)es, (d) Portal
are pitch p
supported,
and to obta
from fixed
e higher th
l Frames
the past,
n with steel
t cold form
their cold
ble for plas
pproximate
ans beyond
s the main
ral efficien
e structural
itch portal fwith crane, (
ortal frame
r the fully
in smaller h
based fra
n the savin
(
his type o
roof trusse
d purlins.
ormed cou
tically desi
ly 1,8 m int
the range
structural
cy with f
concept as
ame, (b) Ce) Two-span
either wit
rigid versi
orizontal di
es, but th
of steel. T
(
(
f)
purlin wa
s. Hot-rolle
his means
terparts, ty
ned portal
ervals. How
of standard
system in
nctional f
shown in Fi
rved portalportal frame,
pinned ba
on (Fig. 8b
splacements
additional
e frames a
)
)
common
purlins ha
that they a
ically 3 m
rames, whi
ever, they a
cold form
single stor
rm. Vario
ure 7.
rame (cellul(f) Portal wi
ses (Fig. 8a
), when it
. Less weig
cost of t
e construct
6
n
e
e
or
h
e
d
y
s
arth
),
is
t
e
d
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MSc Design of Steel Buildings Design of Portal Frames 7
from I-section (UKB, universal beam) rafters and columns with haunches at the connections
at the eaves as illustrated in Figures 9-10 . The haunch length is approximately 10% of the
span and can be formed from welded plate or more commonly a cutting from a rolled section.
The depth at the column face is typically slightly deeper than the rafter section. Portal frames
can be also built with tapered rather than haunched sections. Frames of this type are common
in the USA and are being used more frequently in Europe. The sections are fabricated from
plate on automated welding machines. The ability to vary web thickness, flange dimensionsand section depth results in high material efficiency. Deep slender sections are used to
maximise economy.
Figure 8: Arrangements for portal frames.
A single-span symmetrical portal frame (as shown in Figure 9) is typically of the
following proportions:
a span between 15 m and 50 m (25 m to 35 m is the most efficient). An eaves height (base to rafter centreline) of between 5 and 10 m (7,5 m is commonly
adopted). The eaves height is determined by the specified clear height between the top
of the floor and the underside of the haunch.
A roof pitch between 5and 10 (6 is commonly adopted). A frame spacing between 5 m and 8 m (the greater frame spacing being used in longer
span portal frames).
Members are hot rolled I sections rather than H sections or UKB (universal beam),because they must carry significant bending moments and provide in-plane stiffness.
Sections are generally S235 or S275. Because deflections may be critical, the use ofhigher strength steel is rarely justified.
Haunches are provided in the rafters at the eaves to enhance the bending resistance ofthe rafter and to facilitate a bolted connection to the column. Small haunches are
provided at the apex, to facilitate the bolted connection.
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MSc Desi
Figure 9:
Figure 1
The
directionbe at bot
vertical
Figure 1
gn of Steel B
Single-span
: Details of
ortal fram
, stability ish ends of t
racing by a
: (a) horizon
uildings D
symmetrical
eam-to-colu
in-plane s
provided be building,
hot-rolled
tal bracings,
sign of Port
portal frame.
n connecti
ability is p
vertical bror in one b
ember at e
(b) longitudi
l Frames
n with haunc
ovided by
acing in they only (Fig
ves level.
al vertical b
hes.
rame conti
elevations.. 11). Each
acings.
uity. In th
The verticarame is co
(a)
(
longitudin
bracing mnected to t
)
8
al
ye
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MSc Design of Steel Buildings Design of Portal Frames 9
3 DESIGN OF PORTAL FRAMES
When designing portal frames, different steps are usually considered: i) determine
possible loading conditions, ii) calculate factored design load combination(s), iii) estimate
the element cross sections, iv) analysed the frame for each loading condition and select the
sections and determine connections and v) check secondary modes of failure. In the
following, after mentioning the main loads to be considered when designing portal frames(Fig. 12), different analysis methods, characterized by different levels of accuracy and
complexity are reported. Then an effective design approach, which is based on the rigid-
plastic analysis, is described. This represents the current practice in the UK, which leads to
the lightest and hence the most economical form for a portal frame.
Simplified formulations for accounting for frame stability are considered in the last part of
the note. As plastic design methods result in relatively slender frames, checking frame
stability is a basic requirement; thus in-plane and out-of-plane stability of both the frame as a
whole and the individual members must be considered.
3.1 Loading
External Gravity Loads
The dominant gravity load is from snow. The general case is the application of a basic
uniform load, but with sloping roofs having multiple spans and parapets, the action of drifting
snow has to be considered. The design for main portal frames can be carried out using the
uniform load case, but the variable loads caused by drifting are to be applied to cladding and
purlins. For portal frames, the structure capacity is usually determined by the snow load case,
unless the eaves height is large in relation to span.
Figure 12: In-plane loads to be considered in portal frames design.
Wind Loads
With lightweight cladding and purlins and rails, wind loads are important. Cladding and
its fasteners are designed for the local pressure coefficient. Care must be taken to include the
total effect of both internal and external pressure coefficients.
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MSc Design of Steel Buildings Design of Portal Frames 10
Internal Gravity Loads
Service loads for lighting, etc., are reasonably assumed to be globally 0,6kN/m2. As
service requirements have increased, it has become necessary to consider carefully the
provision to be made. Most purlin manufacturers can provide proprietary clips for hanging
limited point loads to give flexibility of layout. Where services and sprinklers are required, it
is normal to design the purlins for a global service load of 0,1 - 0,2kN/m2
with a reducedvalue for the main frames to take account of likely spread. Particular items of plant must be
treated individually.
Cranes
Where moving loads such as cranes are present, in addition to the gravity loads, the
effects of acceleration and deceleration have to be taken into account in the design. A quasi-
static approach is generally used in which the moving loads are enhanced and treated as static
loads in the design. The enhancement factors to be used, depend on the particular plant and
its acceleration and braking capacity. Manufacturers must be consulted where heavy, high
speed or multiple cranes are being used. To take into account dynamic effects due to cranes,
the maximum vertical loads and the horizontal forces are increased by specific factors. Therepeated movement of a crane gives rise to fatigue conditions. Fatigue effects are restricted to
the local areas of support, the crane beam itself, support bracket and the connection to main
columns. It is not normal to design the whole frame for fatigue as the stress levels due to
crane travel are relatively low.
Other Actions
In certain areas, the effects of earthquakes should be considered. In those countries
affected, there are maps which identify the seismic level of each zone and standards to
evaluate structural behaviour (Eurocode 8).
In common practice, it is not necessary to take into account differential settlement of less than
2,5cm. If differential settlement exceeds 2,5cm, its effects must be examined, both from thestructural and functional points of view. In less ductile structures, such as those constructed
with sections not in Class 1 or 2, it is always important to evaluate the sensitivity of the
structure in relation to possible differential settlement.
It is also general practice not to take into account the effects of temperature when the
maximum dimension of the building is less than 40 to 50 m, or when expansion joints have
been used which separate the structure into zones which do not exceed this dimension.
Elsewhere, it is important to evaluate the effects of variations in temperature. It is also
necessary to ensure that the characteristics of the finished structure, both the systems of
fastening and the seals in the envelope, are compatible with the inevitable deformations due
to change in climate.
3.2 Methods of analysis
According to current codes of practice, structures must generally be checked at two
different Limit States: at Ultimate Limit State (ULS) and at Serviceability Limit State (SLS).
In the case of portal frames, ULS is the most critical condition.
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MSc Design of Steel Buildings Design of Portal Frames 11
Figure 13: Curves representing the frame response determined using different approaches.
At ULS different methods can be employed for determining internal forces for each
individual structural member or critical loads for the structural system as a whole. These
techniques include first and second order approaches based on both elastic and plastic theory.
In particular eight different procedures can be employed: i) first order elastic analysis, ii)first-order rigid-plastic analysis, iii) elastic critical load analysis, iv) second-order elastic
analysis, v) second-order rigid-plastic analysis, vi) first-order, elastic-plastic theory and vii)
second-order, elastic-plastic analysis and viii) second-order, plastic zone analysis.
Conversely, when calculating deflections at working load levels for the purpose of checking
serviceability (SLS), it is usual to employ only linear elastic analysis.
First-Order Elastic Analysis
In first-order elastic analysis a linear relationship between the applied loading F and the
deformations () is considered. The internal force distribution in the frame is assumed to beunaffected by the displacements in the frame. Frame analysis can therefore be conducted
according to linear elastic principles. The frame responds according to line 1 in Figure 11.
First-Order Rigid-Plastic Analysis
Rigid-plastic analysis neglects the effects of elastic deflections and assumes that all
structural deformation takes place in discrete regions, called plastic hinges, where plasticity
has developed. When using first-order, rigid-plastic theory only the collapse condition is
addressed. This condition occurs when sufficient plastic hinges are assumed to have formed
to convert the structure into a mechanism. Thus the path by which this stage is reached, i.e.
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MSc Design of Steel Buildings Design of Portal Frames 12
the sequence of formation of the hinges and any intermediate distributions of internal forces,
are not considered. Curve 2 in Figure 4 gives the frame response according to this approach.
Due to the form of the analysis, no information is provided on the magnitude of the
deflections. The analysis gives only that all stiffness is lost at the collapse load and
deflections therefore (in theory) become uncontrolled.
Elastic Critical Load Analysis
Using elastic critical load analysis, it is possible to calculate the buckling loads for frames
under specific loading conditions. Depending upon the content and complexity of the frame,
several different buckling modes, each with its associated elastic vertical load, may be
possible. Once again the analysis provides no information on the magnitude of the
deflections; it simply identifies a particular load level. The curve 3 in Figure 4 gives the
representation of the critical load obtained by an elastic buckling analysis.
Second order elastic analysis
In second-order elastic analysis the effect of elastic deformations on the internal force
distribution is taken into account. The result is a transition from the linear analysis line 1 atlow loads to the elastic critical line 3 at large deflections. For frames the second-order effects
may be separated into 2 parts: i) reduction in the effective bending stiffnesses of individual
members due to compressive loading, ii) a destabilising effect due to the overturning moment
produced by the vertical loads acting through the horizontal deflections caused by the lateral
loads.
Second-order rigid-plastic analysis
If the deformations, that may develop as a result of the formation of the plastic collapse
mechanism are allowed for when formulating the equilibrium of the frame, then the result is
the developing mechanism curve of line 5 in Figure 4. This curve shows that equilibrium can
only be maintained with a reduction in the level of the applied loads.
First-order, elastic-plastic theory
If a linear elastic analysis is modified to allow for reductions in frame stiffness with the
progressive formation of plastic hinges at increasing levels of the applied load, then the
response curve of line 6 is obtained. This line exhibits progressive loss of stiffness as each
plastic hinge is formed and eventually merges with the rigid-plastic line 2.
Second-Order, Elastic-Plastic Analysis
When the analysis that traces the formation of plastic hinges also allows for the effects of
deformations in setting up the governing equations, then line 6 is modified somewhat intoline 7. Line 7 initially follows the first-order elastic line 1, but diverges from this line to
follow the second-order elastic line 4 as destabilising effects become more significant.
Formation of the first plastic hinge - which occurs at a slightly lower applied load than is the
case with the first-order, elasto-plastic analysis due to the larger deformation associated with
second-order analysis - further reduces the stiffness, causing line 7 to diverge from line 4.
This divergence becomes more pronounced as more plastic hinges form. The peak of this
curve corresponds to the failure load predicted by this type of analysis. At large deformations
line 7 tends to merge with the curve for the mechanism, line 5.
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MSc Design of Steel Buildings Design of Portal Frames 13
Second-Order, Plastic Zone Analysis
If the spread of plasticity both through the cross-section and along the member length is
taken into account, instead of assuming that it is concentrated into the desirable regions of the
plastic hinges, then the resulting type of analysis is usually termed plastic zone theory. It
provides an even closer representation of actual behaviour and leads to a curve similar to line
7. In principle, any of the above approaches to frame analysis may be adopted. In practice,some of the effects may be found to be of little real significance for certain classes of
structure, e.g. for many low-rise frames second-order effects are very small and may
reasonably be neglected. Certain cases may also arise where particular forms of response
should be avoided, e.g. for buildings containing heavy cranes which will cause repeated
loading, elastic design is normally employed. The more complex approaches will almost
certainly require the use of suitable computer software to implement the volume of
calculation. It is therefore important to select an approach which is compatible with both the
accuracy required and the level of importance of the project under consideration. In the case
single-storey pitch portal frames, it has been shown that the use of rigid-plastic analysis leads
to significant saving in material. Therefore rigid plastic analysis can be easily and effectively
used to determined the required capacity for rafter and stanchions. However, further checks
are required to consider the second-order effects.
3.3 Rigid-plastic analysis
The rigid-plastic analysis allows us to calculate the plastic collapse load (p in Fig. 13) ofstructures when it is related to indefinite development of deflection under constant load.
When the load at ULS is given, the rigid-plastic procedure can be used as a design method to
determine the required plastic bending capacity for the beams and columns.
The plastic collapse method does not assess deflections and does not deal with the stability of
individual members or the structure as a whole. Therefore, it should be used as a design
approach only when the strength is the overriding design criterion. Compared to the
traditional elastic approach, plastic collapse analysis can lead to a more effective use ofstructural materials and cost savings, is more easily applicable to a wide range of common
structures, and, unlike the elastic method, the results achieved are independent of initial
imperfections (lack of fit, settlements etc.). The fundamental concepts of the plastic collapse
approach were initially derived from observations of experimental tests (Fig. 14) and not
from the application of a sound mathematical theory as in the case of elastic methods.
Figure 14: Fix-ended beam test: stress concentration and plastic hinge concept (Kazinczy, 1914).
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MSc Design of Steel Buildings Design of Portal Frames 14
Basic Hypotheses
When using plastic methods for designing and analysing framed structures, we rely on the
ability of the joints to transmit bending moments and on the resistance to bending of the
members. Consequently, assuming that the flexural capacity of joints is always greater than
that of the joined elements, the basic hypotheses for the plastic collapse theory refer to the
relation between bending moment and curvature for beams and columns, which is closely
associated with the behaviour of the component material. In this respect, the basic physical
property exploited in the plastic method is ductility, in the sense that the material is assumed
to be capable of deforming well into the plastic range without fracture and significant
strength degradation under constant or slowly increasing loads (Fig. 15). High ductility must
also characterize the behaviour of steel sections, which should experience large rotations
without degradation in strength.
Eurocode 3 introduces some restrictions on steel and cross sections to be used in
structures designed by means of plastic methods of analysis. These restrictions are needed in
order to guarantee that sections at least at the locations at which the plastic hinges may form,
have sufficient rotation capacity to permit all the plastic hinges to develop throughout the
structure. In particular, at plastic hinges, the code allows only for the use of structural steel
with an elongation at failure 15% and Class 1 member sections. Conversely, in the otherparts of the structure, section Class 2 may also be employed.
Plastic hinge concept
The plastic hinge concept can be explained by analysing the behaviour of a simply
supported beam loaded by a vertical force (Fig. 15). When the load is increased until reaching
the plastic moment Mp at midspan, plastic deformations extend over a region where the
bending moment exceeds the elastic moment My.
Figure 15: Simply supported beam at collapse: distribution of plastic deformations, (M-) relationand load-displacement curve (W-).
Because of the shape of the M- diagram, the curvature k remains very small outside theelasto-plastic region. Conversely, close to the point where the load is applied, the curvature is
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MSc Design of Steel Buildings Design of Portal Frames 15
very high. The beam therefore deforms very nearly as if it consisted of two rigid portions
connected by a hinge, a plastic hinge. The plastic hinge behaves like a real but rusty hinge,
needing (non-zero) momentp
M M= to rotate it.
Figure 16: Plastic collapse mechanism for a simply supported beam loaded at midspan.
The plastic limit analysis is aimed at determining the plastic collapse loading condition,
focusing on the evaluation of plastic mechanisms. Therefore, when using such approach in
designing and analysing structures, a rigid-plastic approximation for the (M-) relationshipcan be considered (Fig. 16). The elasto-plastic law (Fig. 15) is fundamental only when
analysing the nonlinear elasto-plastic structural response, for instance trough an incremental
elasto-plastic procedure.
Theorems of Plastic Collapse Analysis
When calculating the collapse load of structures characterized by more than one potential
mechanism, it becomes necessary to identify the actual collapse mechanism. The basic
theorems for plastic collapse1
(Greeberg, Prager, Horne) give us the principles governing the
plastic collapse in formal statements, which can be used to define the actual collapse
mechanism and then to calculate the true collapse load, or, alternatively, determine the
required plastic bending capacity.
Static or lower bound theorem
If, at any load factor , it is possible to find a statically admissible and safe BMD(satisfying equilibrium and yield conditions), then is either equal to or less than the loadfactorc at collapse.
Corollary 1: the collapse load of a structure cannot be decreased by increasing the strength of
any part.
1Horne,M.(1971)PlasticTheoryofStructures,MITPress,Cambridge,Massachusetts.
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MSc Design of Steel Buildings Design of Portal Frames 16
Kinematic or upper bound theorem
If, for any assumed plastic mechanism (mechanism condition), the external work done by
the loads (at a positive load factor) is equal to the internal work at the plastic hinges, then is either equal or greater than the load factorc at collapse.
Corollary 2: the collapse load of a structure cannot be increased by decreasing the strength of
any part.
The importance of the second theorem is obvious, for it follows that if the values of external
loads (load factors) corresponding to all the possible collapse mechanisms are found, the
actual collapse load (load factor) will be the smallest of these values.
Uniqueness theorem
If, at any load factor , a BMD can be found which satisfies the three conditions ofequilibrium, mechanism, and yield, then that load factor is the collapse load factorc.
Corollary 3: the initial state of stress has no effect on the collapse load.
Corollary 4: if a structure is subjected to any programme of proportional or non-proportional
loading, collapse will occur at the first combination of loads for which a BMD satisfying the
conditions of equilibrium, mechanism, and yield can be found.
Summary of basic theorems for plastic collapse analysis:
Uniqueness
theorem c =
MECHANISM CONDITION
EQUILIBRIUM CONDITION
YIELD CONDITION
c - kinematic theor.
c
- static theorem
Static approach
Collapse in statically determinate structures occurs when a plastic hinge forms in the most
stressed section. Therefore the ultimate load assessment and plastic design can be carried out
using simple equilibrium considerations and the static theorem.
In the case of statically indeterminate structures, as the load increases beyond the elastic
limit, which is attained when the first hinge forms, plastic hinges appear in succession at
sections where the absolute value of bending moments has a local maximum equal to the
plastic bending capacity (plastic moment), until the structure turns into a mechanism at
collapse. Also in these situations, the ultimate load can be calculated using static methods,
which investigate the collapse state considering equilibrium and yield conditions. In this
respect, the equilibrium diagram method, which uses the graphical superposition offree andreactant bending moment diagrams, represents an effective strategy.
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MSc Design of Steel Buildings Design of Portal Frames 17
Analysis (c unknown)by statics at collapse:
c pM WL =
Design (p unknown)by statics at collapse:
p cM WL=
Analysis (c unknown)by statics at collapse:
c p4M WL =
Design (p unknown)by statics at collapse:
p cM WL 4=
Figure 17: Collapse load for statically determinate structures calculated using the static approach.
Yield condition: MMp
Equilibrium condition: BMD is in equilibrium with external loads.
Mechanism condition: plastic hinges at critical sections.
By inspection:
A p
B p
A BC p
p
p c
M M
M M
M 2M2WLM M
3 33M2WL
2M W3 L
=
=
+ = + =
= =
Other examples:
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MSc Design of Steel Buildings Design of Portal Frames 18
A p
B p
2
A BC p
2p
p c 2
M M
M M
M MwLM M
8 2
16MwL2M w
8 L
=
=
+= + =
= =
A p
B
A BC p
p
p c
M M
M 0
M MWLM M
4 2
6MWL 3M W
4 2 L
=
=
+= + =
= =
Figure 18: Collapse load for simple statically indeterminate structures calculated using the static
approach.
The same procedure can be used for investigating portal frames. In this case different
collapse mechanisms, which define plastic hinge position, should be considered (Fig. 19).
After fixing the position of plastic hinges (collapse mechanism), the moment diagram at
collapse can be drawn and the yield condition can be checked.
Consider the frame in Figure 20, it is made statically determined so that free bending diagram
can be drawn. The reactant diagram may be constructed by considering the effect of
redundant actions destroyed by cutting the frame. Assuming that mechanism 1 is the true
collapse mechanisms, we can write the following simultaneous equations, which define
plastic hinges at B,C,D, and E:
At B: 2 pVl Sl
M Rh M4 2
+ =
At C:p0 M M =
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MSc Design of Steel Buildings Design of Portal Frames 19
At D: 2 pVl Sl
M Rh M4 2
+ + =
At E: ( )1 1 2 pVl Sl
Hh M R h h M4 2
+ + + =
It gives: 1 2p p 1
1 2 1 2
h HhVl 1 VlM ; M=M ; S=0; R= Hh
h h 8 2 h h 8
= + + +
Figure 19: Basic collapse mechanisms for a pitch frame.
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MSc Design of Steel Buildings Design of Portal Frames 20
Figure 20: Free and reactant diagrams.
The calculated Mp is a safe value only is the bending moment diagram at collapse
satisfies yield condition. It can be checked considering the value for bending moment at A:
( )A 1 2 pVl Sl
M M R h h M4 2
= + +
Substituting the values for S, R and M, we found the relation:1
0 Hh Vl 4 , which is
usually satisfied for frames of a wide range of span to height ration.
3.4 Graphical method for plastic design of portal frames
A simplified graphic procedure, based on the reactant and free moment distribution, can
be effectively used in the case of pitch frame, when the snow load determines the collapse
condition.
Figure 21: Pitch frames with pinned base columns.
For the governing load case we require a bending moment diagram that:
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MSc Design of Steel Buildings Design of Portal Frames 21
satisfies mechanism, equilibrium and yield condition (Uniqueness Theorem). In practicethis can be achieved by utilising hinges at: i) maximum moment near apex of rafter, ii)
maximum moment in stanchion at the bottom of the haunch (Fig. 21);
gives a haunch region that remains elastic. The haunch is usually a tapered section cutfrom the same UKB as the rafter, giving a maximum depth at the end plate connection of
just less than twice the rafter depth. The rafter end of the haunch must remain elastic andtherefore the moment at this point should remain less than Mpr/1.15 where Mpr is the
moment resistance of the rafter. 1/1.15 comes from the shape factor for a UKB;
gives a practical eaves bolted connection. This normally requires a limit on the moment atthe rafter-stanchion connection. Typically the notional moment at this point should be
around 1.5Mpr. In general the moment should be within the range: Mps/2.5 < Mpr< Mps,
where Mps is the moment resistance of the stanchion. Well proportioned designs will be
somewhere close to the middle of this range.
We can adopt a graphical technique to determine the moments for design of members
relying in the first instance on the application of a release to reduce the problem to a statically
determinate one. We release the horizontal reaction at one of the column bases, this meansthere are no moments in the stanchions and the free bending moment in the rafter is the same
as for a beam, thus giving the maximum free bending moment diagram: 2M wL 8= .
Then applying a horizontal reactant force to the roller gives a bending moment diagram in the
frame proportional to the height above the reactant force. This gives the reactant bending
moment diagram. Figure 22 illustrates the concept.
Figure 22: Free and reactant bending moment.
Thus the practical application of the graphical method goes as follows:
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MSc Design of Steel Buildings Design of Portal Frames 22
Draw the portal frame to scale and either back project the roof line to the horizontal toobtain the point A referred to in Figure 22. This point can also be determined using the
relationship:2 1 1 2L L h h= , where L1, h1 and h2 are defined in Figure 22.
On a separate diagram mark out the horizontal distances that you obtained from the firststep. Now draw a parabola as accurately as possible to represent the free bending moment
to a convenient vertical scale. The concept is illustrated in Figure 23.
Project a line from point A with a clear ruler such that the difference between the reactantmoment line and the free moment parabola satisfies the conditions of a haunched region
that remains elastic and a practical eaves bolted connection as discussed previously. it is
important to note that it is you who decides on the gradient of this line.
Figure 23: Graphical procedure to determine rafter and stanchions plastic bending capacity.
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MSc Design of Steel Buildings Design of Portal Frames 23
Once the reactant moment line is established, pick off values for rafter and stanchion
moment capacities and choose appropriate sections for both. From the figure you should also
be able to check the minimum haunch length to ensure that the haunch tip remains elastic
(Mpr/1.15). If using a haunch length of span/10 the check can still be applied to the tip of the
haunch. It can also be checked that the moment in the stanchion is within the range discussed.
Note that the position of the maximum moment near the apex varies as the slope of the
reactant line varies. In practice it can only occur at one of the purlins - where load istransferred to the rafter - generally the first or second from the apex. However the error is
insignificant and it is assumed to be capable of continuously variable location.
Approximate tabular method
Based on the assumptions introduced for the graphical design method, a number of design
charts for estimating member sizes have been produced. They can be used to determine
quickly the sizes of simple pinned based frame elements assuming that:
i) the depth of the rafter is approximately span/55 and the depth of the haunch below
the eaves intersection is 1.5 times the rafter depth.ii) The moment in the rafter at the tip of the haunch is 0.87Mp,r, so that it is assumed
that the haunch remains elastic.
Figure 24: Rise/span versus horizontal base force for various values of span/eaves height.
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MSc Design of Steel Buildings Design of Portal Frames 24
Figure 25: Rise/span versus required Mp of rafter for various values of span/eaves height.
The graphs in Figures 24,25,26, which can be used for the range of span/eaves height
between 2 and 5 and rise/span of 0 to 0.2 (where 0 is a flat roof), give: i) the horizontal force
at the base of the frame as a proportion of the total factored load wL, where w is the load per
unit length of rafter and L (=2L1) is the span of the frame (Fig. 24), ii) the required moment
capacity of the rafters as a proportion of wL2
and iii) the required moment capacity of the
stanchions as a proportion of wL2. In practical calculations, after determining the ratios
span/height (2L1/h1), rise/span (h2/2L1) and the total load (wL and wL2), tables in Figs 24, 25
and 26 can be used to calculate horizontal thrust at base, Mpr required for rafter and Mps
required for stanchion respectively.
Figure 26: Rise/span versus required Mp of stanchion for various values of span/eaves height.
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MSc Design of Steel Buildings Design of Portal Frames 25
Further considerations on rigid- plastic analysis
As it has been mentioned before, when designing pitched roof portal frames, attention is
mainly paid on the snow load, because it determines the frame sizes. Therefore the frame is
analysed under a uniformly distributed load, also neglecting the notional horizontal loads
which account for practical imperfections such as lack of verticality (Fig. 27). Such
simplification is admissible, as geometrical imperfections have a significant influence only onthe design of structures which are sensitive to second-order sway effects. Typical portal
frames are not particularly sensitive to such sway modes, and notional horizontal loads
usually have a less than 1% effect on the required plastic moment of resistance and can be
ignored in member sizing. The bending moment distribution considered in the analyses is
therefore symmetrical, thus leading to a symmetrical collapse mechanism (Fig. 28). This
mechanism is overcomplete and should not be analysed. However, if the constraint of
symmetry is applied, so that the apex moves down vertically without rotating, this
mechanism becomes 'complete' and can be considered in the calculations. Moreover even
though the position of the rafter hinges is unknown, in practical design it can be assumed at
the apex (see the graphical method in 3.4). In reality, as distributed load from the roof is
usually applied to the frame as a series of point loads through purlins, the rafter hinges
always form under the first or second purlin down from the apex (Fig. 28).
Sway imperfection Equivalent forces (notional horizontal loads)
Figure 27: Frame imperfection and notional horizontal loads (EC3).
Figure 28: Symmetrical collapse mechanism.
F2
F1 F1
F2
F1
F2
(F1+F2)/2 (F1+F2)/2
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MSc Design of Steel Buildings Design of Portal Frames 26
After determining required Mpr and Mps by using the graphical approach or the
approximate method with tables, actual sections for stanchions and rafters can be determined,
choosing among Class I sections. A potential reduction in plastic bending capacity caused by
axial forces should be considered, as well as the frame under wind loads should be analysed.
However, both checks are generally not relevant for common pitch frames. Finally the actual
plastic load factor p*, associated with the most critical load combination, which usually
includes factored dead and snow loads (w) can be determined (Fig. 29). It is p* >1, as itreflects the excess capacity due to the choice of discrete member sizes in some arbitrary way.
In the calculation, the actual plastic capacity of stanchions Mp,s* and rafters Mp,r* (in case
reduced because of axial force) are considered, as well as it is assumed that plastic hinges at
collapse form in the columns below the connections to the rafter first and then at the apex.
( )
( )
( )
p,s*
1
p,r* 1 2
p* 2
1
MR
h b
M R h h
w 2L 8
=
+ + =
Figure 29: Calculation of the load factorp associated with rigid-plastic analysis.
3.5 Second-order effects
First-order rigid-plastic analysis provides only an unsafe assessment of the actual capacity
of framed structures, as is does not account for second-order effects (Fig. 30). This is shown
in Figure 13, where the maximum capacity f of a structure, calculated using an accuratesecond order elasto-plastic procedure, is compared against the load factor at collapse p,which is determined by means of a rigid-plastic approach. It appears that the load factorffalls always well below p, thus preventing the use of simple plastic theory in practicalapplications. However, many common structures, as single-storey portal frames, are not
usually excessively slender and the strain hardening contribution, which ensures that plastic
hinges do not rotate at a constant moment but rather have a rising moment-rotation
relationship, is often sufficient to overcome the destabilising effect of axial compressiveloads or at least to ensure that the shortfall offbelow p is not excessive. However, in a safeassessment and design, there is a need for methods allowing for simple estimates of f, thusleaving the use of sophisticated nonlinear analysis procedures only for investigating unusual
structures or for slender structures, where the difference between fand p is significant.
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MSc Design of Steel Buildings Design of Portal Frames 27
where: h is the height from the column base to the inflexion point
is the sway relative to the column base of the inflexion point.
Figure 30: First and second order moments in a beam-column.
Merchant-Rankine approach
The Merchant-Rankine formula provides the most effective approximate method for
estimating the load factorfat failure. It is employed by current codes of plastic (Eurocode 3,BS5950) and represents an extension of the Rankine equation for the failure load of a column
under compression to frames:
f p cr
1 1 1= +
(1)
which leads to:
p
f
p cr1
=
+ (2)
where f is the actual load factor at collapse, p is the rigid-plastic load factor and cr is theelastic critical load factor.
Equation (1) was initially suggested on a purely empirical basis and only later Horne showed
that it has a theoretical justification provided that the critical buckling mode is similar to the
plastic collapse mode. While, in other situations, it is likely to be conservative. In the case of
single-story portal frames, which are relatively stocky structures, strain hardening in plastic
hinges and a small amount of restraint from the cladding can be sufficient to overcome the
destabilising effects of axial compressive loads. Recognising these factors, Wood suggested a
modification of eq. (2) to give:
p crf
p cr p
crf p
p
when 4 100.9
when 10
=
+
= >
(3)
P
H
x
M(h) =Hh +P
M(x) = Hx +P +P x / h
P
H
h
x
M(h) =Hh
M(x) =Hx
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MSc Design of Steel Buildings Design of Portal Frames 28
In the case of slender frames, whencr p 4.6 < according to BS5950, more accurate
nonlinear second order elasto-plastic analyses should be carried out.
Figure 31: Critical load and buckling mode for a pitch frame.
It is evident that any approach based on the Merchant-Rankine formula requires a good
estimate of the elastic critical load factorcr(Fig. 31). For multi-storey frames such value can
be calculated quite easily, using different alternative methods. Conversely the evaluation ofthe critical load for pitched roof portal frames is not simple and requires a specific treatment.In this case, the destabilising effect of the axial thrust in the rafter must be considered.
Davies developed an effective method, which is remarkably simple and give results thatare sufficiently accurate for all practical purposes. According to this approach, the criticalmode is assumed to be anti-symmetrical sway, with a corresponding deflection of the rafter(Fig. 31). Davies determined the critical load, solving the buckling problem in Figure 32,
where a small disturbing moment M at eave which gives rise to a rotation initiates framebuckling.
1
Figure 32: Elastic critical load calculation for a pitched roof frame.
An explicit expression forcr can be derived as a function of the axial load in the column P cand along the rafter Pf:
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MSc Design of Steel Buildings Design of Portal Frames 29
( )
( )
( )
c rcr 22
cr r r
r c
cr
r c r r
5E 10 R I Lwith R for fixed base frames
2RP h5P L I h
I I
5E 10 R for pinned base frames
L P h 0.3P L
+ = =
+
+ =
+
(4)
where Ic and Irare the second moment of era of columns and rafter respectively.
An estimate of the axial loads Pc and Pr, which is sufficiently accurate for practical design can
be obtained considering the notations in Figure 33 using the expressions:
c r
wL M wLP , H= and P H cos sin
2 h 2= = + (5)
Figure 33: Evaluation of the axial loads Pc and Prat collapse.
Typical portal frames, when well-designed, are characterised by values ofcrwhich are about5, so the consideration of second-order effects is extremely important. Being in the rage of
intermediate slenderness, the modified Merchant-Rankines formula (Eq. 3) can be
employed. This required that the load factor for plastic design should be increased
accordingly to the above equation forp and that all of the internal forces obtained by first-order analysis should be amplified in proportion.
3.6 Member stability
At ULS, it is essential that all members of the frame remain stable. In general, the
member stability checks should be carried out using final bending moment diagram for the
loading combination used to the to determine the member sizes. In structures designed using
elastic methods, this means ensuring that the frame elements are stable against both in-plane
and out-of-plane (lateral torsional buckling) failure. While, when using plastic design, also
different requirements need to be satisfied, to guarantee large rotational capacity at plastic
hinge position. The first requirement implies to use only Class I (plastic) sections at plastic
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MSc Design of Steel Buildings Design of Portal Frames 30
hinges location. This guarantees that the elements are sufficiently stocky for plastic hinges to
develop, without premature loss of strength due to local buckling. Moreover codes of practice
(e.g. Eurocode 3, BD5950) require to provided torsional restraints at all plastic hinge
positions, with adjacent lateral restraints within a specific distance Lm from the hinge
restraint. The Lm value can be calculated using the expression:
zm 1/ 222
pl, y yEd
2
1 t
38iL
W f1
57.4 756 C AI 235
= +
(6)
where Ed is the average compressive stress [N/mm2] due to axial load, fy is the design
strength [N/mm2], iz is the radius of gyration [mm] about the minor axis, I t is torsional
constant and C1 is a factor depending on the loading and end condition.
In a portal frame, purlins and sheeting rails are assumed to provide lateral restraint to the
flange to which they are connected. On their own, they do not provide either torsional
restraint to the section or lateral restraint to the remote flange if this flange is in compression.
Torsional restraints hold both flanges of a beam in position relative to each other (in the
lateral direction). The most usual way of achieving either of these forms of bracing is to use aknee brace connected to a purlin or sheeting rail as shown in Figure 34. Past research
indicates that each knee brace should be designed for a compression load equal to 2% of the
compression flange yield load and should have a stiffness that is given by a slenderness ratio
of at least 100.
Figure 34: Typical knee brace.
There are three different regions of the frame which require checking: the apex region, the
haunch zone and the columns. The apex region of the rafter contains a length of almost
uniform bending moment and this is the most difficult distribution to stabilise. However, the
most critical load condition places the outer flange in compression and this is laterally
restrained by the purlins. The requirements therefore depend critically on whether it has beenpossible to prove that this part of the frame is entirely elastic at the ultimate limit state. If this
part of the rafter contains a plastic hinge, then the purlin spacing is limited to Lm. If this part
of the rafter is elastic, and in any event for parts of the rafter remote from the plastic hinge,
then the standard requirements for elastic design apply. The purlin spacing is then restricted
to a value of LE (effective length depending upon the amount of end restraint and loading
condition). In the haunch region of the rafter, the purlins restrain the tension flange. If this
region contains a plastic hinge, then the purlin spacing in the vicinity of the hinge is again
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MSc Design of Steel Buildings Design of Portal Frames 31
limited by Lm. Provided that every length of purlin has at least two bolts in each purlin-rafter
connection; and the depth of the purlin section is not less than 0.25 times the depth of the
rafter, the bottom flange of the rafter can also be assumed to be restrained at the point of
contraflexure. If the design procedure suggested in previous sections has been followed, then
the haunch region of the rafter will generally be elastic. In this case the maximum purlin
spacing can be increased to that which results in a value of LE which satisfies standard lateral
torsional buckling checks.
Figure 35: Spacing for lateral and torsional restraints in columns and haunched rafter.
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Under the dominant load condition, the column will also usually be restrained at intervals
along the tension flange by the sheeting rails. In a plastically designed frame, there will
almost always be a plastic hinge below the haunch. This will, of course, require a torsional
restraint. According to current codes, the distance to the next restraint should be L m.
However, this value should be modified to account for the actual bending moment profile,
which is not constant.