November 1990 DC 693.814.074.5 DEUTSCHE NORM Structural steelwork Analysis of safety against buckling o f linear members and frames DIN 18800 Part 2 Contents Page 1 General ....................................... 2 1. 1 Scope and field of application ................... 2 1.2 Concepts ..................................... 2 1.3 Common notation ............................. 2 1.4 Ultimat e limit state analysis ..................... 3 1.4.1 General ..................................... 3 1. 4.2 Ultimate imit state analysis by elastic theory .... 4 1. 4. 3 Ultimat e imit state analysis by plastic hinge heory 5 .2 imperfections.. ................................ 5 2.1 General ...................................... 5 2 .2 Bow imperfec tions. ............................ 5 2. 3 Sway imperfections ............................ 6 2 .4 Assumption of initial bow and coexistent initial sway imperfections . ........................ 7 3 Solid members ..... ........................ 7 3.1 General ...................................... 7 3.2 Design axial compr ession ...................... 8 3. 2.1 Lateral buckling ............................. 8 3. 2. 2 Lateral orsional buckling *) ................... 8 3.3 Bending about oneaxiswithoutcoexistentaxial orce 8 3.3.1 General ..................................... 8 3.3.2 Lateral and torsion al restrain t ................. 1 O 3. 3.3 Analysis of compression flange ................ 12 3. 3. 4 Lateral orsional buckling ..................... 12 3.4 Bending about one axis with coexist ent axial force 13 3.4. 1 Members subject ed to minor axial forces ....... 13 3. 4. 2 Lateral bucklin g ............................. 13 3. 4. 3 Lateral orsional buckling ..................... 14 3.5 Biaxial bendingwith or coexistent axial orce 15 3 .5 .2 Lateral orsional buckling ..................... 16 4 Single-span built-up members .................. 16 4.1 General ...................................... 16 4. 2 Common notation ............................. 17 4 .3 Buckli ng perpendicular o void axis .............. 17 4. 3. 1 Analysis of member .......................... 17 4. 3.2 Analysis of member componen ts .............. 17 4. 3.3 Analysis of panels of battened member s ........ 18 4. 4 Closely spaced built-up battened members ....... 19 4. 5 Struct ural detailin g ............................ 20 5 Frames.. ...................................... 20 5.1 Triangulated frames ........................... 20 3. 5.1 Lateral buck ling .... ................... 15 Page 5.1.1 General.. ................................... 20 5.1.2 Effective engths of frame members 5. 2 Frame sand laterally restrain ed continuo us beams . 22 5.2 .1 Negligible deformations due to axial force ...... 22 5.2 .2 Non-sway frames ............................ 23 5. 2.3 Design of bracing systems .................... 23 5. 2. 4 Analysi s of frames and continuo us beams. ...... 23 5. 3 Sway frames and continuou s beams subject to lateral displacement ........................... 23 5. 3. 1 Negligible deformat ions due to axial force ...... 23 5.3.2 Plane sway frames ........................... 23 5.3 .3 Non-rigid ly connected continuous beams ....... 27 6 Arches ........................................ 27 6. 1 Axial compression ............................. 27 6. 1. 1 In-p lane buckling ............................ 27 6. 1.2 Buckling n perpendic ular plane . ............... 30 6.2 In-plane bending about one axis with coexistent axial force ............ 6. 2. 1 In-pl ane buck ling .............. 6. 2.2 Out-of -plane buckling ........................ 33 6. 3 Design oading of arches ....... 34 7 Straight linear members with plan thin-wailed parts of cross section .............. 34 7.1 General ...................................... 34 7. 2 General rules relating to calculations . . 7.3 Effective width i n elastic-elastic method 7.4 Effective width in elastic-plastic metho d 7. 5 Lateral buckling ............................... 38 7.5.1 Elastic-elastic analysis ........................ 38 7.5.2 Analyses by appr oximate methods ............. 38 7.6 7.6.1 Analysis .................................... 39 7. 6. 3 Bending about one axis without coexist ent axial force .................................. 39 7. 6. 4 Bending about one axis with coexistent axial force .......................... ... 39 7. 6. 5 Biaxial bending with or without coexist ent axial force .................................. 39 Standards and other documents referred t o ........ 40 Literature.. ....................................... 40 ........ Lateral torsional buckling ....................... 39 7.6 .2 Axial compression ........................... 39 *) Te rm as used in Eurocode 3. In design analysis literature also referred t o as flexural-torsional buckling. Continued on pages 2 to 41 DIN 18800 Part 2 Engl. Price group 7 fh Verlag Gm bH. Berlin, has the exclusive righ t of sale for German Standards @IN-Normen). Sales No. 0117 4.93 Copyright Deutsches Institut Fur Normung E.V. Provided by IHS under license with DIN Not for Resale No reproduction or networking permitted without license from IHS - - `, , , `- ``, , `, , `, `, , `- -
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4.3.1 Analysis of member .......................... 174.3.2 Analysis of member components .............. 174.3.3 Analysis of panels of battened members ........ 184.4 Closely spaced built-up battened members . . . . . . . 194.5 Structural detailing ............................ 20
5.1.1 General.. ................................... 205.1.2 Effective engths of frame members
designed to resist compression. . . . . . . . . . . . . . . . 05.2 Frames and laterally restrainedcontinuous beams . 225.2.1 Negligible deformations due to axial force ...... 225.2.2 Non-sway frames ............................ 235.2.3 Design of bracing systems .................... 235.2.4 Analysis of frames and continuous beams. ...... 235.3 Sway frames and continuous beams subject to
M K i , y design buckling resistance momentaccording to elastic theory from Mywithout coexistent axial force
non-dimensional slenderness in bend-ing
internal forces resistances
and moments
according toM reduction factor for lateral torsional
buckling
Note 1. Where cross sections are non-uniform or axial
forces variable, (E. ) , NKiand SK shall be deter-mined for the point in the member for which the ulti-
mate limit analysis is to be carried out. In case of
doubt, an analysis shall be performed for more than
one point (cf. item 316).
Note 2. The reference slenderness ratio, ilaor steel ofthickness 40mm and less shall be as follows:
92,9 for ~t37where fy,k = 240 N/mm2, and75,9or St 52 where fy,k =360N/mm2.
Note 3. Calculations of in-plane slenderness ratios shall be
made using as the values O f y , ( E .1).NKi and MKiasspec i f ied in i tems116and117ei ther the i rcharac-teristic values or their design values throughout.
Note4. V K ~ hall beof thesame magnitude or all membersmaking up a non-sway frame.
Note 5. Where cross sections are non-uniform or internalforces and moments variable, M K ~hall be calculat-
ed for the point for which the ultimate limit stateanalysis is carried out. In cases of doubt, an analysis
shall be performed for more than one point.
111) Partial safety factors
YF partial safety factor for actions
YM partial safety factor for resistance parameters
Note. The values of YF and YM shall be taken from clause 7of DIN 18800 Fart 1. Thus, the ultimate limit state
analysis shall be carried out taking YM to be equalto 1,l both for the yield strength and for stiffnesses
(e.g. E .T ,E -A ,G -ASand S).
1.4 Ultimate l imit st ate analysis
1.4.1 General
112) Methods of analysis
The analysis shall be take the form of one of the methods
given in table 1, taking into account the following factors:
- plastic capacity of materials (cf. item 113);
- imperfections (cf. item 114and clause 2);
- internal forces and moments (cf. items 115 and 116);
- the effects of deformations (cf. item 1 16);
- slip (cf. item 118);
- the structural contribution of cross sections (cf. item
1 19);
- deductions in cross-sectional area for holes (ci. item
120).
As a simplification, lateral buckling and lateral torsional
buckling may be checked separately, irst carrying out theanalysis for lateral buckling and then that for lateral tor-sional buckling whereby, i n the latter case, members shall
be notionally singled out of the structural system and sub-jected to the internal forces and moments acting at the
member ends (when considering the system as a whole)
and to those acting on the member considered in isolation.
Details on whether first or second order theory is to be
applied are given together with the relevant method of
analysis.
The analyses described in clauses 3 o 7may be used as
an alternative to those listed in table 1.
Table 1. Methods of analysis
I Calculation of
Elastic-
plastic
plastic
Elastic-
plastic
plastic
Elastic
Iheory
Elastic
theory
Elastic Plastic
theory theory
Note 1. Details relating to elasto-plastic analysis are notprovided in his standard (cf. [i] ,hough this is per-
mitted in principle.
Note 2. In table 1 1 of DIN 18800Part 1, the generic term
‘stresses’ is used instead of ‘internal forces and
moments due to actions’.
Note 3.The conditions of restraint assumed when indi-
vidual members are notionally singled out of thestructural system shall be taken into account when
verifying lateral torsional buckling.
Note 4. Simplified methods substituting those set out inclauses 3 and 4 are listed in table 2.
113) Material requirements
The materials used shall be of sufficient plastic capacity.
Calculations may be based on assumptions of linear elas-tic-perfectly plastic stress-strain behaviour instead of
actual behaviour.
Note. The steel grades stated in sections 1 and 2 of item
401 of DIN 18800 Part 1 are of sufficient plasticcapacity.
114) Imperfections
Reasonable assumptions (e.g. as outlined in clause2) hall
be made in order to take into account the effects ofgeometrical and structural imperfections.
Note. Typical geometrical imperfections are accidental
load eccentricity and deviations from design
geometry. Typical structural imperfections would
be residual stresses.
115) Internal forces and moments
The internal forces and moments occurring at significant
points in the members shall be calculated on the basis of
the design actions.
As a simplification, the index d has been omitted in the
notation of internal forces and moments.
Note. Subclauses7.2.1 nd7.2.2 f DIN18800Part 1 spec-
ify rules for calculating design values of actions.
116) Effec ts of structural deformations
Calculations of internal forces and moments usually make
allowance for deformation effects on equilibrium (accord-
ing to second order theory), using as the design stiffness
values the characteristic stiffnesses obtained by dividing
the nominal characteristics of cross section and the char-acteristic elastic and shear moduli by a partial safety factor
YM equal to 1,l.
The effect of deformations resulting from stresses due to
shear forces may normally be ignored.
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The failure criterion is not higher than the design yield
strength, f y , d (elastic-elastic method), the specifica-
tions of item 117 being applied by analogy.
At isolated points, the failure criterion may be 10%
higher than design yield strength (cf. item 749 ofDIN 18800 Part 1).
The internal forces and moments (taking due consider-
ation of interaction) are within the limits specified for
the perfectly plastic state (elastic-plastic method).
Note 1. See item 746 of DIN 18800 Part 1 for f y , d .
Note 2. The elastic-plastic method allows for plastificationin cross sections with the possibility of plastichinges with full torsional restraint at one or more
pointS.This permits the plastic capaci tyof the crosssections to be fully utilized, but not that of the struc-
ture.
Note 3. The analysis shall be made using nteraction equa-tions (cf. tables 16 and 17 of DIN 18 800 Part l).
122) Internal forces and moments in bi-axial bending
Where bi-axial bending occurs with or without co-existent
axial force but without torsion, the internal transverse
forces and moments occurring may be determined bysuperimposing those internal forces due to actions which
result in momentsM y nd transverse forces V, and those
resulting in moments M, and transverse forces V,. How-ever, calculation of E for the total axial force due to all
actions is necessary in both cases.
(123) Limiting the plastic shape coefficient
In cases where the plast ic shape coeff ic ient ,apl ,associatedwith an axis of bending is greater than 1,25 and the prin-
ciples of first ordertheorycannot be applied,the resistance
moment occurring as a result of Co-existent normal and
transverse forces in a perfectly plastic member cross sec-
tion shall be reduced bya actor equal to 1,25/aPl.Thesame
principle shall be applied to each of the two moments in bi-
axial bending if apl,ys greater than 1,25 r apl,zs greater
than 1.25.
Note. Instead of reducing the resistance moment, theactual moment may be increased by a factor equal
to api/1,25.
1.4.3 Ultimate imit state analysis by plastic hinge theory
124) The loadbearing capacitymay be deemed adequateif an analysis according to plastic hinge theory shows inter-
nal forces and moments (taking into account interaction)to be within the limits specified for the perfectly plastic
state (plastic-plastic method). This only applies if thestructure is in equilibrium.
Item 123 gives information on limiting the plastic shape
coefficient.
Note. Interaction equations are given in tables 16 and 17of
DIN 18 800 Part 1.
2 Imperfections
2.1 General
201) Allowance for imperfections
Allowance shall be made for the effects of geometrical and
structural member frame imperfections if these result in
higher stresses.
For this purpose, equivalent geometrical imperfectionsshall be assumed, a distinction being made between initial
bow (see subclause 2.2) and sway imperfections (see sub-
clause 2.3).
Note 1. Equivalent geometrical imperfections may, in turn,be accounted for by assuming the corresponding
equivalent loads.
Note 2. As well as geometrical imperfections, equivalentgeometrical imperfections also cover the effect on
the mean ultimate load of residual stresses as aresult of rolling, welding and straightening proce-
dures, material inhomogeneities and the spread of
plastic zones. Other possible factors which mayaffect the ultimate load, such as ductility of fasten-
ers, frame corners and foundations, or shear defor-
mations are not covered.
In the elastic-elastic method, only two-thirds the values
specified for the equivalent imperfections in subclauses2.2
and 2.3 need be assumed. Ultimate limit state analyses ofbuilt-up members as specified in subclause 4.3 shall,
however, always be made using the full bow imperfect ion
stated in line 5 of table 3.
Note 1. A reduction by one-third takes account of the fact
that the plastic capacity of the cross section is not
fully utilized. The aim is to achieve on average the
same mean ultimate loads when applying both the
elastic-elastic and the elastic-plastic methods.
Note 2. The analyses set out in subclause 4.3 are based oncomparisons of ultimate loads obtained empirically
or by calculation, which also ustify the value of bowimperfection stated in line 5 of table 3 (cf. Note
under item 402).
The equivalent imperfections are already included in thesimplified analyses described in clauses 3 and 7.
202) Equivalent imperfections
The equivalent geometrical imperfections, assumed to
occur in the least favourable direction, shall be such that
they are optimally suited to the deformation mode asso-
ciated with the lowest eigenvalue.
The equivalent imperfections need not be compatible with
the conditions of restraint of the structure.
Where lateral buckling occurs as a result of bending about
only one axis with coexistent axial force, bow imperfectionsneed only be assumed with DO or W O in each direction in
which buckling will occur.
Where lateral buckling occurs as a result of biaxial bendingwith coexistent axial force, equivalent imperfections need
only be assumed for the direction in which buckling willoccur with the member in axial compression.
In the case of lateral torsional buckling, a bow imperfection
equal to 0,5 D O (cf. table 3) may be assumed.
203) Imperfections in special applications
Where provisions for special applications are made in otherrelevant standards,with specifications deviating from those
given in this standard, such specifications shall form thebasis of the global analysis.
Note. Imperfections relating to special applications are
not covered in clauses 3 to 7.
2.2 Bow imperfections
204) Individual members, members making up non-swayframes and members as specified in item 207, shall gen-erally be assumed to have the initial bow imperfections
given in figure 2 and table 3.
tLYJ2o I 0
Figure 2. Initial bow imperfections of member in the form
of a quadratic parabola or sine half wave
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Bow mperfections need not be assumed f members satisfythe criteria specified in item 739 of DIN 18800 Part 1.
Table 3 . Bow imperfections
5
If the criteria for first order theory set out in item 739 of
DIN18 800Part 1 are met, reductions in the sway imperfec-
tions may be assumed.
Built-up members,
with analysis as insubclause 4.3
Type of member
1
2
Solid member, of crosssection with following
buckling curve
a
b
imperfection,
WO? 0
11300t1250
3 1 I 11200
4 1 I 11150
11500
Note. See table 2 3 for bow imperfections for arch beams.
Figure 3. Equivalent stabilizing force for bow imperfec-
tions as shown in figure 2 assuming equilibrium)
Figure 4. Assumptions for bow imperfect ions
(examples)
2.3 Sway imperfections
205) Assumptions
Sway imperfections as in figure 5 hall be assumed o occur
in members or frames which may be liable to torsion after
deformation and which are in compression.
In the above figure,L or L, is the length of the member or
frame, and ppo or ~ 0 , ~ .he sway imperfection of the memberor frame.
Figure5. Ideal member or frame (chain thin line) and
member or frame with initial sway imperfection(continuous thick line)
Initial sway imperfections shall generally be calculated as
follows (cf. item 730 of DIN 18800 Part 1):a) solid members:
1po = 1 r2
200
b) built-up members as in figures 20 and 21 and sub-clause 4.3:
(2)1
po = l . 2400
where
r1 = is a reduction factor applied to mem-
bers or frames, where 1, the length of
the member,L,or frame,L,, having he
most adverse effect on the stress
under consideration, is greater than5 m;
r 2 = 1 ( í + t ) is a reduction factor allowing for IZ
independent causes of sway imper-
fection of members or frames.
2
Calculations of 12 for frames may generally assumen to bethe number of columns per storey in the plane under con-
sideration. Not included are columns subjected to minoraxial forces, ¡.e. with less than 25Oío of the axial force acting
in the column submitted to maximum load in the same
storey and plane.
Note 1. Since, in calculations of shear in multictorey
frames, initial sway imperfections are assumed to
have the most adverse effect in the storey under
consideration, the storey height, ¡.e. the total lengthof columns,L, shall be substituted for the length ofthe column in that storey for calculation of Il. In the
other storeys, he height of the structure,L,, may be
substituted for I (cf. figure 6).
Note 2 . Allowance for sway imperfections may also be
made by assuming equivalent horizontal forces.
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Note 1. In the literature, the combination of equations (3),
(241, 28) nd (29)s referred to as first order elasticanalysis with sway-mode effective length (equiva-
lent member method, for short).Note 2.Subclauses3.4.2.2,3.5.1nd 5.3.2.3 hall be aken
into consideration when applying the equivalent
member method to members notionally singled out
of the frame.
(303) Lateral torsional buckling
Members notionally singled out of the system and consid-
ered in isolation shall be analysed for lateral torsional
buckling.Their end moments may require to be determinedby second order theory.The moments in the span may then
be calculated by first order theory using these end
moments.
An analysis of lateral torsional buckling is not required forthe following:
- hollow sections:
- members with sufficient lateral or torsional restraint;- members designed to be in bending, providedthat their
non-dimensional slenderness in bending, AM, s not
more than 0,4.
Note. See subclause 3.3.2or verification of sufficient re-
straint.
a b C d
0.21 0,34 0,49 0,76
3.2 Design axial c ompression
3.2.1 Lateral buckling
(304) Analysis
The ultimate limit state analysis shall be made for the direc-
tion in which buckling will take place, using equation (3).
5 1 (3)
The reduction factor x (¡.e. xy or x,) shall be obtained bymeans of equations (4a) to (4 c) as a function of the non-
dimensional slenderness in compression,AK,and the buck-
ling curve for the particular cross section, aken from table 5.
N
x ~ N p1 ,d
AK 5 0,2 x = 1
1
k + i qAK >0,2 x =
k = 0,5I + a (XK -0,2) nK]
as a simplification, in cases where AK > 3,O:
1x = -
AK í& + a)
a being taken from table 4.
Table 4. Parameters a for calculation of
reduction factor x
Note 1. The effective length required for calculating 3~ is
given in the literature. Four simple cases are given n
figure 9, and figures 27and 29 may provide assist-
ance i n other cases. If, in certa in cases, the load on
the member changes direction when this moves
laterally,this factor shall be aken into considerationwhen determining the effective length (e.g.with the
aid of figures 36 o 38).
i I i IN
SKß= 1,0 2,O D,il 0,5
Figure9. Effective lengths of single members ofuniform cross section (examples)
Note 2.Reference shall be made to the literature (e.g. [2])
for the use of equations (4 ) to (4 ).
(305) Further provisions for non-uniform cross sectionsand variable axial forces
Where equation (3)s applied to members of non-uniform
cross section andlor variable axial forces, the analysis shall
be made using equation (3) for all relevant cross sections
with the appropriate internal forces and moments, cross
section properties and axial forces,NKi.and in addition the
following conditions shall be met:
min M , 12 0,05 an M,l (6)
3.2.2 Lateral torsional buckling
(306) Members of uniform cross section with anytype ofend support not permitting horizontal displacement, sub-
ject to constant -¡al force shall be analysed as specified insubclause 3.2 .1 .1~hall be calculated substituting for N K i
the axial force occurring under the smallest bifurcation
load for lateral torsional buckling, with the reduction factor x
being determined for buckling about the z-axis.
I sections (including rolled sections) do not require ulti-
mate limit state analysis with respect to lateral torsionalbuckling.
Note. Torsional buckling is treated here as a special type
of lateral torsional buckling.
3.3 Bending about one axis without
coexistent axial force
3.3.1 General
(307) Ultimate limit state analysis shall be carried out asspecified in subclause 3.3.4 or bending about one axis,except in cases where bending is about the z-axis or the
conditions outlined in subclause 3.3.2 r 3.3.3are met.
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k, When flanges are connected to webs by welding, nshall be further multiplied by a factor of 0,8.
Note 1. Calculation of äMis only possible where the idealdesign buckling resistance moment, M K ~ , ~ ,s
known (cf. [5] and [6]). Equation(19) r (20) may beapplied for beams of doubly symmetrical uniform
cross section.
M K ~ , , , C * NK~,,, (11, + 0,25 Z; + 0.5 zP) (19)
where
<
NK~,,, is equal to n2. . zll';
is the moment factor applicable to fork
restraint at the ends, from table 10
Io + 0,039 1 * IT
I ,
c2 =
zp is the distance of the point of transmission of
the in-plane lateral load from the centroid
(positive in tension).
t I I I
1.77- 0,77I ImaxM-1cp1
maxM
Calculations of beams not more than 60cm in
height may be simplified by substituting equation
(20)or equation (19).
1,32b * t ( E *I,)
1 * h 2Ki,y =
aI16Figure 15. Beam dimensions qualifying for simpli-
fied analysis using equation (20) or (21)
Note 2. XM may also be taken from figure 10 if the beam
coefficient, n, s equal to 2 5
Note 3. XM may be assumed to be equal to uni tyfor beamsnot more than 60cm in depth (see figure
15)nd
ofuniform cross section provided that they satisfy
equation (21):
be t 2401 5- 200-
h fy,k
f y , k being expressed in N/mm2.
Note 4. Coefficient n allows for the effect of residualstresses and initial deformations on the service oadbut not the effect of the support conditions (these
being allowed for by MKi,y).
3.4 Bending about one axis with
coexistent axial force
3.4.1 Members subjected to minor axia l forces
312) Members subjected to only minor axial forces andmeeting the condition expressed by equation (22) may be
analysed for bending without coexistent axial force, asspecified in subclause 3.3.
N< 0,l (22)
X * Npl ,d
3.4.2 Lateral buckling
3.4.2.1 Simplified method of analysis
(313) The analysis for lateral buckling of members pin-jointed on bo th sidesand subject to in-plane ateral loading
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in the form of a concentrated or line load and with a maxi-
mum moment,M, ccording to f irst order theory, may be
analysed by means of equation (3), while substituting inequation (4 b) k from equation (23).
+ a & - 0,2) + 3; +
tem 305 shall be taken into consideration.
3.4.2.2 Equivalent member method
314) Analysis
The ultimate limit state analysis shall be made applying
equation (24) and using the buckling curves specified in
subclause 3.2.1.
+-' e + A n < (24)N
.N p 1 . d M p l , d
where
x Ea reduction factor from equation (4), a function of
AK and the appropriate buckling curve (see table 5),
for displacement in the moment plane;
is the uniform equivalent moment factor for lateral
buckling taken from column 2 of table 11.
Moment factors less than 1 are only to be used formembers of uniform cross section whose end sup-
port conditions do not permit lateral displacementand which are subjected to constant compression
without in-plane lateral loading;
is the maximum moment according to first order
elastic theory, imperfections being neglected;
ßm
M
N NA n isequa l to- -- x 2 * 36,
x ' N p i , d (1 x - N p l , d )
but not more than 0,l.
Item 123 shall be aken into account when calculating M p l , d .
For doublysymmetrical cross sections with a web compris-
ing at least 18Yo of the'total area of cross section,M p l , d in
equation(24) may be multiplied by a factor of 1,l if the
following applies:
Note 1.Where the maximum moment is zero,equation (3)shall be applied instead of equation (24) for the
ultimate limit state.
Note 2. Calculations mayde simplified by substituting for
A n either 0,25 x 2 .A or 0.1.
315) Effect of transverse forces
Due account shall be taken of the effect of transverse
forces on the design capacity of a cross section.
Note. This may be achieved by reducing he internal forcesand moments in the perfectly plastic state (e.g. as
set out in tables 16 and 17 of DIN 18800 Part 1).
variable axial forces(316) Non-uniform cross section and
Where cross sections are non-uniform or axial forces vari-
able, the analysis shall be made applying equation (24) to
all key cross sections, with all relevant internal forces and
moments and cross section properties and the axial force,
NK~,ssumed as acting at these points. In addition, equa-
tions (5) and (6) in item 305 shall be met.
(317) Rigid connections
In the absence of a more rigorous treatment, rigid connec-
tions shall be calculated substituting forthe actual moment,
M , he moment in the perfectly plastic state,Mp1,d .
Note. If a more detailed analysis is required, he design of
connections shall be based on the basis of the
bending moment according to second order theory,
taking into account equivalent imperfections.
318) Portions of members not subjectedt o compression
The analysis of portions of members which are not them-
selves subject to compression but which are required to
resist moments due to being connected to members in
compression shall be by means of equation (26). The yieldstrength of cross sections not in compression shall not be
less than that of those in compression.
M
5 11,15
1--
VKi
with V K ~> 1,15
Note. A portion of a member not in compression could bea
beam connected t o columns in compression.
319) Movement of supports and temperature effects
Any effects of deformations as a result of movement of the
supports or variations in temperature shall be taken into
consideration when calculating moment M .Note. Further nformation shall be taken from the literature
k g . VI).
3.4.3 Lateral torsional buckling
320) Channels and C sections, and I sections of mono-symmetric or doubly symmetrical cross section, exhibitinguniform axial force and not designed for torsion, with relative
dimensions as for those of rolled sections,shall be analysed
for ultimate limit state by means of equation (27):
My k y < 1N
+x z Npl, d xM M p l , y , d
The following notation applies in addition to that given in
subclause 3.3.4.
x z is a reduction factor from equation 4),substituting
AK,z for buckling perpendicular to the z-axis,where
& z is equal toE he non-dimensional slenderness
associated with axial force;
N K ~ is the axial force underthe smallest bifurcation loadassociated with buckling perpendicularto he z-axisor with the torsional buckling load;
is a coefficient taking into account moment diagram
My and a K , z . It shall be calculated as follows:
k y = l -
where
ay= 0,15K,z. B M , ~O, , with a maximum of 0,9
where
M , ~s the moment factor associated with lat-eral torsional buckling, from column 3 oftable 11, aking intoaccount moment dia-
gram My.Note 1. Due regard shall be taken, particularly in the case
of channels and C sections, of the fact that this ana-lysis does not take account o f design torsion.
Note 2. Tsections are not covered by the specifications of
this subclause.
Note 3. A k, value of unity gives a conservative approx-imation.
Note 4. The torsional bending load plays a major role, forexample, in members subject t o torsional restraint.
k ,
N
xz N p l , d
ay. but not more than unity,
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h, and h , distance between centroidal axes of chords;
a length of chord between two nodes;
AG gross area of cross section of chord;
A = AG gross area of cross section of built -up member;
AD gross area of cross section of a strut;
4 smallest radius of gyration of one chord;
1 , second order moment of area of a chord crosssection about the centroidal axis parallel to thez-axis;
Y s distance of the centroid of each componentcross section from the z-axis;
I , = AG y ; + I z , ~ ) econd order moment of area ofthe gross cross section about the
z-axis (assuming rigid connection
of components, providing shear re-
sistance);
effective length of equivalent member, disre-
garding any deformation due to transverseforces;
SK,ZAK,z = lenderness ratio of the equivalent member
for battened members (disregarding defor-E mations due to transverse forces);
correction for battened members (cf. table 12);
system length (of built-up member);
s K ,z
17
Table 12. Correction, v or battened members
1 I
> 150 O
Figure18. Laced and battened members (examples)
1; A G y ; + 17. I z , ~ ) esign second order moment ofarea of the gross cross section of
battened members;
1;=2 (AG- y ; ) design second order moment of area ofthe gross cross section of laced members;
section modulus of the gross cross sec-
tion, relative to the centroidal axis of the
outermost chord;
Sz*,d design shear stiffness of the equivalentmember.
Note 1. The shear stiffness corresponds to the transverseforce resulting in an angle of shear,y, equal to unity.
Note 2. Examples of shear stiffness of laced and battened
members are given in table 13.
Note 3. The shear stiffness of battened members has been
multiplied by the factor n2/12n order to excludefailure of single panels solely due to shear.
w;=- L
YS
4.3 Buckling perpendicular o void axis
4.3.1 Analysis of member
405) Analysis of a member shall be made taking into con-sideration the conditions of restraint. The internal forces
and moments in a member designed to be in axial compres-sion, with its ends nominally pinned to prevent lateral dis-
placement will be as follows:
at member mid-point: Mz (31)N 00
N1 --
NKi z,
where1
(32)1
+ -T ~ ( E I;)d s;,d
12NKi,z, d =
n - M zat member end: max V , =-
133)
Note. The literature (e.g. [IO]) shall be consulted for inter-nal compression and design bending.
4.3.2 Analysis of member components
4.3.2.1 Chords of laced and battened members
(406) The global analysis of internal forces and momentsacting throughout the member not resistant to shear givesan axial force,NG, in the chord undermaximum stressequal
to the following:
NG shall be used for analysis of the part of a chord as spec-
ified in subclause3.2, ssuming pin-jointing on both sides.
The slenderness ratio, aK,1. shall be obtained as follows:
where
SK,1 is the effective length of the part of a chord under
maximum stress, usually aken to be the same as the
length of the chord, a, between nodeS.The effective
length of parts of laced members consisting of four
angles shall be taken from table 13.
Note. The analysis may be made as specified i n subclause
3.4 or laced members as shown in columns 4 and 5of table 13where a is subject to transverse loading.
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(407) The axial forces of web members making up lacingsystems shall be obtained from the total transverse forces,
Vy,acting in the laced member.The effective length shall be
taken from subclause 5.1.2.
Note. The total transverse force required when consider-
ing a member in axial compression, shall be ob-
tained from equation (33).
3 4 5
4.3.3 Analysis of panels of battened members
408) Panels between two battens
The panel between two battens resisting the maximum
transverse force, rnax Vv, obtained from the global calcula-
tion shall be analysed by verifying the ultimate limit state of
a chord subject to the following internal forces and
moments:
1,52 a 1.28 a
ma r Vy aMG =
r 2end moment,
a
rnax Vytransverse force, VG =
r(37)
(38)
where XB is the position of the batten
In the case of monosymmetric chord cross sections, the re-
sistance moment,M , at the ends of the part of the chordshall be obtained from the mean of the moments f Mpl,NGderived from interaction equation (38).
Note 1. The plastic design capacity of the chord cross sec-
tion as obtained from the interaction equations may
be utilized (cf. [9]and [lo]), he transverse force, VG,
normally being neglected.
on the chord.
Note 2. The moments of resistance, M, ,N~, ccurring in
the chords at their connections with battens are of
different magnitude owing to their different direc-
tions. Failure of a panel does not occur until all
M p ~ , ~ Galues have been fully util ized (cf. [9]).
Note 3. The moment axes shall also be taken to be parallel
to the void axis in the case of angle chords.
Table 13. Effectwe lengths sK,1 and equivalent shear stiffnesses, s , * , d , of laced and battened members
SK; 1
Sz,d = m . E A .cos a .sin2a
(m= number of braces normal to void axis)
a
z
y + y:rz
a
6
Battened members
a
The effective lengths,sK,l,in columns 1and 2 onlyapply to angle-sectioned chords, the slenderness ratio,ili, being calculat-I d on the basis of the smallest radius of gyration, i l .
If, in special cases, fasteners are used which are likely to slip, this may be accounted for by increasing the equivalent geo-
metrical imperfections accordingly.
The information relating to Sg,d does not apply to scaffolding,which generally makes use of highly ductile fasteners which
must be taken into account.
Note. Further information on ductili tyand slip of fasteners and on eccentricityat the connections between web members in
laced members is given in the literature (e.9. [9]).
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Battens and their connections shall be designed for shear
and the design moments (cf. table 14).
Table 14. Distribution of forces and moments in the
battens of battened members
1
Cross section of
built-up battenedmembers
Structural model
Moment diagram in
the connection
due to shear, T
Shear, T,n theconnection
2
This also applies for closely spaced built-up battened
members as shown in figures 19,20and 21.The moments inthe centroids of batten connections shall be taken into
account.
If packing plates are used to connect the main componentsin built-up battened members as shown in igures19and21,it is sufficient to design he connection for resistance o theactual shear.
4.4 Closely spaced built-up battened members
(410) Cross sections with one void axis
Built-up members with cross sections as shown n figure 19
may also be reated as solid members as set out i n clause3when calculating lateral displacement normal to the void
axis, provided that eitherof the following conditions is satis-fied:
a) battens or packing plates positioned as specified insubclause 4.5 are not more than 15 i apart;
b) continuous packing plates are used,which are connect-ed at intervals equal to 15 il or less apart.
Figure 19. Built-up memebers with a void axis and a clearspacing of main components not oronlys lightly
greater than the thickness of the gusset
Continuity of packing may be taken into considerationwhen calculating the second order moment of area. When
determining the area of cross section,A, this only applieswhen the packing is adequately connected to the gusset.
The shear in the battens, connections or packing may be
calculated fora transverse force equalling 2.5% of the com-
pressive force in the battened member.
(411) Star-battened angle members
Built-up members. consisting of two star-battened anglemembers need only be checked for lateral displacement
perpendicular to the.material axis (figure 20) by the follow-ing equation:
(39)
If the effective lengths of the two members are not the
same, the mean of the two effective lengths shall be used.
Angles with a cross section as shown in figure 20 b) may be
verified by the following equation, he radius of gyration,io,of the gross cross section relating to the centroidal axisparallel to the longer leg:
. iolY = .15
a) r = 2 b) r = 2
Figure 20. Star-battened angle members
Consecutive battens may be in corresponding or mutuallyopposed order. Shear may be determined as specified in
item 410.
Note. According to item 503, the effective lengths of diag-onals or verticals in triangulated frames differ, de-
pending on whether lateral displacement in or per-
pendicular to the plane of the frame is being consid-
ered.
(412) Cross sections with two void axes
Where built-up members as shown in figure 21 consist ofmain components with a clear spacing not or only slightly
greater than the thickness of the gusset,the specifications
applying to the built-up members in figure 19 shall be
applied by analogy to the two void axes.
r = 4
Figure 21. Closely-spaced built-up member with two
void axes
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(509) The out-of-plane effective length of solid trussmembers with elastic support at mid-length for the swaymode may be obtained by means of equation (44):
1 2
O<A , < 1 1 2
fi<3,<3,0
;2 = 0,35 + 0,753 AK
n# =0,50 0,646 AK
- Ia K = - non-dimensional slenderness of solid
il * Aa member
z system length
il minimum radius of gyration of anglecross section
(44)
where
1 is the system length;
N
c d
is the maximum compressive force acting in the
member ( N Ior N2);
is the frame stiffness with respect to lateral dis-
placement of the points of connection of solid
members and of columns forming part of the sub-
frame in the perpendicularplane,this being equal onot less than 4 N IL
Figure 23. Solid member and frame stiffness
5.1.2.4 Angles used as solid members
in triangulated frames
(510) Where angle ends are nominally pinned (e.g. bymeans of a single bolt), the effects of eccentricity shall be
taken into consideration.
Figure 24. Rigidly connected angles (examples)
If one of the two angle legs is rigidly connected at the node,
the effects of eccentricity may be disregarded and the
analysis of lateral buckling as specified in subclause 3.2.1
carried o3t using the non-dimensional slenderness inbending, Ak, from table 16.
Table 16. Non-dimensional slenderness in bending, ni<
5.2 Frames and laterally restrained
5.2.1 Negligible deformations due to axial force
(511) The specifications of subclause 5.2 may be deemedapplicable if the deformations due to axial force of the
columns of frames and bracing systems are negligible, his
being the case when equation (45) is met:
(45)
whereE . is the bending stiffness,
S is the storey stiffness,
L is the overall height (see figure 25),
of the bracing system or multistorey frame.
IfE -1or S varies over a number of storeys, heir mean maybe used.
I may be approximated using equation (46):
continuous beams
E * I > 2,5 S. 2
B2
Ali Are
I = (46)1 1-+-
the width, B, and cross-sectional areas Ali and Are of the
columns being as shown in figure 25.
Bracing system Multistorey frame
Al i
L B
Figure 25. Criteria for calculation of I by means of
It shall be presumed throughout that for the column offrames the member characteristic is not greater than unity.
Note I . Equation (45) ensures that in a cantilever member
whose low bending stiffness and storey stiffness
remain constant under an evenly distributed load,
the lateral displacement at the free end asa result of
transverse force is at least ten times that resultingfrom the bending moment.
Note 2. Equations or calculation of the stiffness of bracing
systems and of multistorey frames are given intable 17 and subclause 5.3.2.1 respectively.
equation (46)
5.2.2 Non-sway frames
512) Non-sway braced frames
In cases where the frame and the bracing components co-
operate to resist in-plane horizontal loads, the frame shallbe regarded as non-sway provided tha t the stiffness of the
bracing system,SAusst,isat least ive times that of the frame,
Sb,n the storey under considerat ion, ¡.e.
By a simplified method, equation (47) need only be applied
to the lowest storey if the stiffness conditions there are not
considerably different from those of the other storeys.
Note. Examples of stiffening elements are wall panels and
bracing.Their stiffness may be taken from table 17.
SAusst 2 5 SRa (47)
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The stiffness of beam-and-column type frames,S, is defined
by:
S = V J p (48)
Figure 26. Stiffness of beam-and-column type frames,S
As a simplified method,
in item 519, ith SAusst equal to zero.
Table 17. Stiffness of bracing systems,
may be calculated as specified
1
Bracing system
Wall panel
(e.g. masonry)
Diagonals
(one diagonaleffective)
nL
SAusst
G - t - 1
E . A sin a * cos'a
Value doubled
where bracing
sufficiently
preloaded
5.2.3 Design of bracing systems
514) Principle
Bracing systems shall be designed by second order theory
assuming all horizontal loads and uplift due to imperfec-tions for both stiffening system and frame.
515) Imperfections
Initial sway imperfect ions,q o , as specified in subclause2.3shall be assumed forall columns of frames and the bracing
system.
516) Calculation by first order theory
In the global analysis by elastic theory,first ordertheory may
be applied provided that each storey meets equation (49):
SAusst , d
N(49)
where
SAusst ,d is the total stiffness of all frame bracing systems nthe storey under consideration;
N is the total vertical load transmitted in the storey
under consideration.
If equation (49) is not met, the bracing system design shall
be based on the transverse force calculated by second
order theory.
A simpler method may also be used,in which the transverse
force according to first order theory (including any uplift,
N -PO)s multiplied by the factor a obtained by means of
equation (50).
Note. The following general case applies to bracing systems:
NKi,d = SAusst , d
5.2.4 Analysis of frames and continuous beams
517) The ult imate limit state analysis of frames and con-tinuous beams may be effected by analysing their main
components as specified in clause 3.
In the analysis of lateral buckling of non-sway frames as
specified in subclause3.4.2.2,he moment factor,&,for lat-eral buckling, aken from column 2 of table 1 1 may be usedto calculate the moment components from transverse
loads on beams.
When analysing beams by means of equation (26),he
maximum bending moment may be reduced by multiplying
by the factor (1 0,8/q~i) rovided there are no (or virtually
no) compressive forces acting in them.Note. The effective lengths required for the above check
are given in figure 27. Practical examples are givenin [ll].
5.3 Sway frames and continuous beamssubject to lateral displacement
5.3.1 Negligible deformations due to axial force
516) Item511 shall apply in the cases where the deforma-tions due to axial force are negligible.
5.3.2 Plane sway frames
Note. The use of bolts or welding for unst iffened beam-to-
column connections requires due consideration of
their structural behaviour and susceptibility to
deformations, ¡.e. their plastic design capacity com-bined with their rotation capacityand theirdeforma-
tions under service loads.
5.3.2.1 Calculation by first order elastic theory
519) Global analysis of beam-and-column type frames(regardless of the number of storeys or panels) which are
pinned or rigidly connected at their base, with columns ofequal length within a storey and nodes permitting only
lateral displacement, may be designed by first order theory,provided that each storey meets equation (51).
where
N, being the sum of all vertical loads transmitted in thert h storey.
In the above, the stiffness S, shall be obtained by means of
equations (52)o (54), sing he notation and values given
in figure 28.
In the first storey (where r =l), shall be as follows, de-
pending on the condit ions of restraint at the column bases:
rigidly connected:
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Sr ,Ausst ,d is the stif fness of any stiffening elements in ther th storey.
If an analysis of external horizontal forces by first order
theory is already provided,q ~ i , ,may also be obtained bymeans of equation 55).
55)VF
qKi,r =-pr * Nr
where
VF is the transverse force from external horizontalloads in the r th storey;
p, is the associated angle of rotation in the r th storey,obtained by first order theory.
Note 1 . In first order theory, the reduced initial sway im-
perfections p~ specified in items 729 and 730 ofDIN 18800 Part 1 shall be taken into account.
Note 2. Alternatively, K i , r may be determined with the aidof figure 29.
N K i , r , d assumed as being equal to S,d/1,2gives aconservative estimation of the design bifurcationload; examples are given in [ll].
5.3.2.2 Simplified method applying second order theory
520) Method
Calculations shall be as in first order theory but assuming
an increased transverse force in the storeys as set out initem 521 or 522.
521) Transverse force n beam-and-column ype frames
Where the member characteristic,E, of beam-and-column
type frames is less than 1,6,higher transverse forces in
the storey, V,, hall be used, to be obtained by means ofequation 56).
56)
where
VF is the transverse force in the storey due to externalhorizontal loads only;
N , is the total vertical load transmitted within the rt hstorey;
00 is the initial sway imperfection as specified in sub-clause 2.3;
pr is theangleofrotation ofthecolumnsintherth storey(calculated by the simplified second order theory
method).
Note. When applying nitial sway imperfections at the base
or top of columns, the angles of rotation, Q, (see
figure 30), being unknown, the simplified secondorder method gives an only slightly different result
than the first order method, the additional term1,2pr N , giving a decrease n he principal diagonalterms, and po N , an increase in the load terms, of
the equilibrium equations. Thus calculations are
onlyslightly more complex than by irst ordertheory.
522) Approximate calcu lation of transverse force
in beam-and-column type frames
If equation 57) s met by all storeys, equation 58) may besubstituted for 56) to obtain V,by approximation.
V,= V,H+ 90 .N r + 1,29,.NI
I
v, = VT + Co * N I
11 - -
v K i , r
5.3.2.3 Analysis by equivalent member method
523) Global method
The ultimate limit state analysis for sway frames may be car-ried out byanalysing each member separately,as specifiedin clause3, ut using the effective length of the system as a
whole.
Where, in certain cases, the compressive forces acting on
the frame are liable to change direction during buckling,this shall be taken into account when calculating the effec-tive lengths of members.
Note. Effective engths may be determined using igure 29,
or using figures 36 to 38 in cases where compres-
sive forces are liable to change direction.
524) Cross sections not in compression
Analysis by means of equation 26) or cross sections not in
compression need only be made for beams in sway frames
where Mpl of ihe beam is less than the total Mpl of the
columns meeting the beams.
525) Systems with nominally pinned columns
In global analysis by first order theory, sway systems nclud-
ing nominally pinned columns shall be calculated with an
additional equivalent load,VO obtained by means of equa-
tion 59) and illustrated in figure 30), n order to take intoaccount initial sway imperfections.
59)
where
p0,i is as specified in item 205.
VO 1 Pi.p0.i
VO = XPi V0.i90 rom figure 5.
Figure 30. Systems including nominally pinned columns:additional transverse force in a storey, VO
Note. The initial sway imperfections as specified in items
729 and 730 of DIN 18800 Part 1 need not be
assumed in addition to VO.
5.3.2.4 Analysis applying first order plastic hinge theory
(526) Beam-and-column type frames
Beam-and-column ype frames as specified in subclauses5.3.2.1,with columns having no or virtually no plastic hinge
action at their ends, may be analysed according to first
order plastic hinge theory provided that initial sway imper-fections from subclause2.3are assumed and the columns
in each storey satisfy equation 60).
60)
611
Vr
p r s l o N ,where
v,= v,H + 80.N ,
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Figure 34. Buckling coefficient, ß, for in-plane buckling of parabolic arches with m hangers (relative to the axial force at
the springing ( K ) )
602) Tied arches
In the case of tied arches where the ties are connected to
the arch by means of hangers, he ultimate limit state analy-
sis shall be carried out using the full effect ive length of thearch,since it is not usually sufficient to check the section of
arch between two hangers.
Note. Further details are given in the literature (e.g. [13]
603) Snap-through buckling of arches
Snap-through buckling will not occur in lat arches provided
that equation (65)s satisfied.
and [141).
where
E . A is the longitudinal stiffness;
E . , is the in-plane bending stiffness;
k is an auxiliary value taken from table 20.
Note. Snap-through buckling loads cannot be determined
for arches using his standard,and shall be calculat-ed applying the non-linear theory using large de-
formations.
6.1.1.2 Non-uniform cross sections
604) The ultimate limit state analysis of arches of non-uniform cross section shall be by second order theory
assuming equivalent geometrical imperfections as spec-
ified in subclause 6.2.1.
6.1.2 Buckling in perpendicular plane
6.1.2.1 Arch beams without lateral restraint
between springings
605) The ultimate limit state analysis of arch beamswithout lateral restraint between springings may be carried
out applying equation (3), sing the in-plane slenderness
ratio,AK, obtained as follows.
For parabolic arches,
where
i
p l
is the radius of gyration of the z-axis at the crown;
is the buckling coeffcient taken from table 21 (assum-ing loading to correspond to the arch form), under a
uniform vertical load distribution, with both ends of
the arch laterally restrained in the perpendicular
plane;
is the buckling coefficient taken from table 22,cover-ing the change in direction of the load in lateral buck-
ling.
For one-centred arches,
with
where
N K ~ , K ~s the axial force under the smallest bifurcationload of a one-centred arch of constant doubly
symmetrical cross section with fork restraint, sub-ject to constant radial loading corresponding to
the arch form;
is the radius of the one-centred arch;
is the angle of the one-centred arch,greaterthan O
but less than n;
r
a
6.1.2.2 Arches with wind bracing and end portal frames
606) The sway mode normal to the arch plane may becalculated by approximation, it only being necessary totake into account buckling of the portal frames.
The ultimate limit state analysis for the columns of portal
frames may be by means of equation 3), taking AK fromequation (69).
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propertiesA’,I’,etc. are assigned to b’,andA”,I“ tob”. Figure 40 b) shows a reduced cross section inelastic-elastic analysis, this applying by analogy for
elastic-plastic analysis.
Note 3. The methods of analysis set out in subclauses 3.2
to 3.5 also apply inprinciple to members with effec-
tive cross sections, subject to the modificationsspecified in subclauses7.5 and 7.6.
(706) Approximate methods
The effective cross section is obtained by reducing the zoneof tensile bending. If the cross section is not symmetrical
about the bending axis and both positive and negative
bending moments occur, the governing bending momentshall be that resulting in the smaller effective second order
moment of area.This moment shall be assumed to be con-stant over the length of the member.
Note 1. If the reduced zone of tensile bending is used, the
compressive stress, UD, may be conservatively
approximated to fy,k/YM. Iteration may be avoided’ by also making a conservative approximation of the
edge stress ratio, y.
Note 2. The zone of tensile bending is not reduced usingthis approximate method, even though compressive
stresses may occur. This approximate method is
elaborated in he literature [cf.l6],with the nclusion
of practical examples.
707) Analysis of cross section
The analyses shall be of the effective cross section. Thereduction in cross section shall be in correlation with the
direction of the actual bending moment in the bendingcompression zone of the member after deformation.
Note. In the absence of a design bending moment, thebending moment as a result of bow imperfections
shall be used. It may prove necessary to examine
both directions in he case of monosymmetric cross
sections.
708) Centroidal shif t as a result of reductionin cross section
The effect of a shift, e, of the centroid in the transition fromthe gross (¡.e. actual) to the effective cross section shall be
taken into account.
For convenience, his may be done as specified in tems 709
and 710.
(709) Increase in bow imperfection
Where members are to be assumed with an initial bow
imperfection, wo, this shall be increased by A W O from
table 25.
For a cross section symmetrical about the axis of bending,
and assuming that a compressive stress, OD,due to thepositive moment and the negative moment are of equalmagnitude,ep, , and e may also be taken to be equal.
Note. The diagrams shown in table 25 are onlyexamples ofmoments.Of significance is the occurrence of posi-
tive and negative moments.
d u e t o + M
Figure 41. Centroidal shift (examples)
710) Increase in initial sway imperfections
Where members are assumed with an initial swayimperfec-
tion PO,his shall be increased by Apo= (e, +e,)íZ if both
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715) The ultimate limit state analysis shall be madetaking UD as equal to or less than fy,d 88),here UD is themaximum compressive stress at the long edge of the thin-
walled part of the cross sect ion, calculated on the basis of
the effective cross section.The ong edge is taken to be one
of the edges of the gross part of the cross section.
The provisions of item 706 may be applied.
7.5.2 Analyses by approximate methods
7.5.2.1 Axial compression
716) The effective cross section obtained by assumingeffective widths in bending for the compression lange and,in some cases, for the web shall be taken as a basis, the
stress distribution n he web being estimated.No reduct ion
in cross section of the tension flange is to be made. Theultimate limit state analysis shall be made applying equa-
tion (89).
Table 28. Magnitude and resolution of effective
width b
N5 1
X *A . f y ,d
where.I
(89)
x = but not exceeding unity (90)
k+i
(92)
(93)
(94)
I' and A' are the second moment of area and the area ofthe effective cross section respectively;
Amo is the eccentricity as a result of a reduction incross-sectional area, to be calculated asset out
in item 709;
r D and fD are the distance of the compression edge in
bending from the centroidal axis of the gross or
effective cross section (cf. figure 40);
a is a parameter taken from table 4;
i is the radius of gyration of the gross cross sec-
tion;
SK is the effective length, calculated taking intoaccount the effective second moment of area, I
Note 1. The method of analysis specified here corre-
sponds in principle to that set out in item 304. In a
manner similar to item 313, allowance for the effectof Awo is made by substituting a supplementary
term in equation (91).
Note 2. Subclause 7.5.2.2 pecifies an alternative method
of analysis, allowance for the effect of AWO beingmade by inclusion of a bending momentMy qual o
N e Awo. In cases where this alternative method isused, the term featuring AWOshall be deleted.
717) In addition to the analysis specified in item 716, ananalysis shall be made using equation (95) on the basis ofanother effective area,A', determined assuming constant
compressive stress over the whole of the effective cross
section.
(95)
(CornDression) ,
(Tension)
k , = 18,5
k2 = 18.5
::3
L(Tension) (Compression)
kl = O
k2 = 11
i
nf Y
(Compression) (Tension)
Ei a = y c . E j
O 1 2 ? ) & 2 0
k l = {
4,56 I? j ~ ~O 2 ?) 2 -1
k , = 11
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7.5.2.2 Bending about one axis with coexistent axial force
(7l8) Analysis
The ultimate limit state analysis shall be made applying
equation (24). When determining the in-plane slendernessratio, AK, the effective second moment of area, I (cf. item
719) or I (cf. item 720) shall be taken into account.
Note. Reference may be made to the l iterature (cf. 1191)
for an alternative method of analysis.
719) Elastic-elastic methodThe analysis of bending about one axis with coexistent axialforce shall be made applying equation (24) but making the
following substitutions:
wp,d for Npl,d;
M%l,d or Mpl,d;
x for x ;
x and & being taken from item 716;
Nb1,d = A'*fy,d (96)
(971
TK foräK;
where
I'
rDMpi ,d = fy,d
720) Elastic-plastic method
The analysis of bending about one axis with coexistent axial
force shall be made applying equation (24) but making thefollowing substitutions:
Npi,d for Npl,d;
Mpi,d for Mpl,d;
x" for x ;
Tí for&.
These values shall be obtained by analogy with equations(96) and (97) and item 716,on the basis of the cross section
with an effective width b".
Note. Examples of b" are given in table 28.
7.5.2.3 Biaxial bending with or without coexistent
axial force721) The ultimate limit state analysis for biaxial bendingwith or without coexistent axial force may be made as spec-
ified in subclause 3.5.1, with subclause 7.5.2.2 applying by
analogy.
7.6 Lateral torsional buckling
7.6.1 Analysis
722) The ultimate limit state analysis for lateral torsionalbuckling may be made as specified in clause 3, but with themodifications set out in items 723 to 727.
7.6.2 Axial compression
723) The calculation of lateral torsional buckling shall bein analogywith subclause 3.2.2 and as for lateral buckling as
specified in subclause 7.5. When calc lating the non-dimensional slenderness n compression,lK, the properties
of the reduced cross section shall be taken into account for
calculation of the axial force,NKi,under the smallest bifur-
cation load in the analysis of lateral torsional buckling ac-
cording to elastic theory.
7.6.3 Bending about one axis without
coexistent axial force
7.6.3.1 Analysis of compression chord
724) Analysis of the compression chord shall be as setout i n subclause 3.3.3, but assuming k , equal to unity in
equation (13).obtaining i by means of equation (98) andsubstituting MPIJ for Mpl,y,d in equation (14).
I _ .
where
IZ,
AbA,
Note. If the elastic-plastic method is applied,
is the reduced second moment of area of the com-
pression chord about the z-axis;
is the reduced area of the compression chord;
is the gross web area.
A nd
M$,d shall be substituted for IL A; and Mgl,d,
respectively.
7.6.3.2 Global analysis
725) Design buckling resistance moment
according t o elastic theory
When calculating the design buckling resistance moment,
the moment red M K ~btained by approximation by meansof equation (99) shall be substituted for M K ~ , ~
l i
where
M ~ i , p k * Ue w (100)
this being the ideal moment relative to plate buck-
ling of the cross section or the relevant part of thecross section;
k is the buckling factor (e.g. taken from table 26);
se shall be obtained from item 712;
W is the relevant section modulus of the full cross
section.
Note 1. If a more rigorous treatment is preferred, red M K ~
shall be calculated on the basis of plate buckling of
the individual parts making up the cross section.
Note 2. A number of buckling factors of whole sections are- given in the l iterature (e.g. [17] and [18]).
726) Elastic-elastic method
When ca'culating the non-dimensional slenderness in
bending,AM, as set ou t in item 110,Mblshall be substituted
for Mpl,y, nd in the analysis using equation (16). MP1,d
obtained from equation (97) shall be substituted for Mpl,y,d.
727) Elastic-plastic method
When calculating as set out in item 110, MP1shall be
substituted for Mpl,y In the analysis using equation (16).
Mpl ,d shall be substituted for Mpl, ,d. M$ shall be obtainedby analogy from equation (97) for the effective cross sec-
tion having the width b .
7.6.4 Bending about one axis wit h coexistent axial force
728) The ultimate limit state analysis shall be madeapplying equation (27), calculating the resistance axial
force as specified in subclause 7.5.2.1 and the resistance
bending moment as specified in item 726 (when using theelastic-elastic method) or item 727 (when using he elastic-
plastic method).
7.6.5 Biaxial bending with or withoutcoexistent axial force
729) The ultimate limit state analysis may be made usingequation (30), applying by analogy provisions of sub-clause 7.6.4.
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Steel sections; hot rolled narrow flange I beams I series); dimensions, mass, limit deviations and static
values
Steel sections; hot rolled wide flange1 beams I PB andI series); dimensions, mass, limit deviations and
static values
Steel sections; hot rolled wide flange beams (IPBI series); dimensions, mass, imit deviations and static
values
Steel sections; hot rolled wide flange Ibeams (IPBv series); dimensions, mass, limit deviations and
static valuesSteel sections; hot rolled medium flange I beams (IPE series); dimensions, mass, limit deviations and
static values
Quantities, symbols and units used in civil engineering; principles
Structural steelwork; safety against buckling, overturning and bulging; design principles
Structural steelwork; safety against buckling, overturning and bulging; construction
Structural steelwork; design and construction
Structural steelwork; analysis of safety against buckling of plates
Structural steelwork; analysis of safety buckling of shells
Trapezoidal sheeting in building; trapezoidal steel sheeting; general requirements and determination of
loadbearing capacity by calculation
Trapezoidal sheeting in building; trapezoidal steel sheeting; determination of loadbearing capacity by
testing
Trapezoidal sheeting in building; trapezoidal steel sheeting; structural analysis and design
Bemessung und konstruktive Gestaltung von Tragwerken aus dünnwandigen kaltgeformten BauteilenI
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