Din 18800-Part2 English Language

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November 1990DC 693.814.074.5 DEUTSCHE NORM

Structural steelworkAnalysis of safety against buckling of

linear members and frames

DIN18800

Part 2

ContentsPage

1 General ....................................... 21.1 Scope and field of application . . . . . . . . . . . . . . . . . . . 21.2 Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Common notation ............................. 21.4 Ultimate limit state analysis ..................... 31.4.1 General ..................................... 3

1.4.2 Ultimate imit state analysis by elastic theory . . . . 41.4.3 Ultimate imit state analysis by plastic hinge heory 5

.2 imperfections.. ................................ 52.1 General ...................................... 52.2 Bow imperfections. ............................ 52.3 Sway imperfections ............................ 62.4 Assumption of initial bow and coexistent initial

sway imperfections . ........................ 7

3 Solid members . . . . . ........................ 73.1 General ...................................... 7

3.2 Design axial compression ...................... 83.2.1 Lateral buckling ............................. 83.2.2 Lateral orsional buckling *) ................... 83.3 Bendingaboutoneaxiswithoutcoexistentaxial orce 83.3.1 General ..................................... 83.3.2 Lateral and torsional restraint . . . . . . . . . . . . . . . . . 1O3.3.3 Analysis of compression flange . . . . . . . . . . . . . . . . 123.3.4 Lateral orsional buckling ..................... 123.4 Bending about one axis with coexistent axial force 133.4.1 Members subjected to minor axial forces . . . . . . . 133.4.2 Lateral buckling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.4.3 Lateral orsional buckling ..................... 143.5 Biaxialbending with or coexistent axial orce 15

3.5.2 Lateral orsional buckling ..................... 16

4 Single-span built-up members .................. 164.1 General ...................................... 164.2 Common notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.3 Buckling perpendicular o void axis .............. 17

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 Frames.. ...................................... 205.1 Triangulated frames ........................... 20

3.5.1 Lateral buckling . . . . ................... 15

Page

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

lateral displacement ........................... 235.3.1 Negligible deformations due to axial force . . . . . . 235.3.2 Plane sway frames ........................... 235.3.3 Non-rigidly connected continuous beams . . . . . . . 27

6 Arches . . . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . 276.1 Axial compression ............................. 276.1.1 In-plane buckling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276.1.2 Buckling n perpendicular plane.

. . . . . . . . . . . . . . .30

6.2 In-plane bending about one axis withcoexistent axial force . . . . . . . . . . . .

6.2.1 In-plane buckling . . . . . . . . . . . . . .6.2.2 Out-of-plane buckling ........................ 336.3 Design oading of arches . . . . . . . 34

7 Straight linear members wit h planthin-wailed parts of cross section . . . . . . . . . . . . . . 34

7.1 General ...................................... 347.2 General rules relating to calculations . .

7.3 Effective width in elastic-elastic method7.4 Effective width in elastic-plastic method7.5 Lateral buckling ............................... 387.5.1 Elastic-elasticanalysis ........................ 387.5.2 Analyses by approximate methods . . . . . . . . . . . . . 387.67.6.1 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

7.6.3 Bending about one axis without coexistent

axial force .................................. 397.6.4 Bending about one axis with coexistentaxial force .......................... . . . 39

7.6.5 Biaxial bending with or without coexistentaxial force .................................. 39

Standards and other documents referred t o . . . . . . . . 40

Literature.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

. . . . . . . .

Lateral torsional buckling ....................... 39

7.6.2 Axial compression ........................... 39

*) Term as used in Eurocode 3. In design analysis literature also referred to as flexural-torsional buckling.

Continued on pages 2 to 41

DIN 18800 Part 2 Engl. Price group 7fh Verlag Gm bH. Berl in, has the exclusive righ t of sale for German Standards @IN-Normen).

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DIN 18800 Pari 2

1 General

1.1 Scope and field of application

101) Ultimate limit sta te analysis

This standard specifies rules relating to ultimate limit state

analysis of the buckling resistance of steel linear members

and frames susceptible to lossof stability.It s to be used n

conjunction with DIN 18800Part 1.

(102) Serviceability limit sta te analysis

Aserviceability limit state analysis need only be carried out

if specifically required in the relevant standards.

Note. Cf. subclause 7.2.3of DIN 18 800 Part 1.

1.2 Concepts

103) Buckling

Buckling s a phenomenon n which displacement,v orw,ofa member occurs, or rotat ion,9,occurs about its major axis,

or both occur in combination.

A distinction is conventionally made between lateral buck-ling and lateral torsional buckling.

(104) Lateral buckling

Lateral buckling is a phenomenon n which displacement,vor w, of a member occurs,or both occur in combination,any

rotation,9, about its major axis being neglected.

(105) Lateral torsional buckling

Lateral torsional buckling is a phenomenon in which dis-placements,u and w , f a member occur in combination

with rotation, 4, about its major axis, consideration of the

latter being obligatory.

Note. Torsional buckling, in which virtually no displace-

ments occur, is a special form of lateral torsional

buckling.

1.3 Common notation

106) Coordinates, displacement parameters, internalforces and moments, stresses and imperfections

axis along the member (major axis)

axis of cross section

(In solid members, I, shall be not less than Iz.)

displacement along axes x, y and z

rotation about the x-axis

initial bow imperfections in unloaded state

initial sway imperfection of member or frame in

unloaded state

axial force (positive when compression)

bending moments

shear forces

(107) Subscripts and prefixes

k

d

grenz

vorh actual

red reduced

Note. The terms ‘characteristicvalue’and design value’are

(108) Physical parameters

E elastic modulus

G shear modulus

f y yield strength

Note. See table 1 of DIN 18800 Pari 1 for values of E , G

characteristic value of a parameter

design value of a parameter

prefix to a parameter dentifying it as being a limit-

ing (¡.e. maximum permissible) value

defined in subclause 3.1of DIN 18800Part I.

and f y ,k.

Figure 1. Coordinates, displacement parameters and

(109) Sect ion parameters

A cross-sectional area

I

i = radius of gyration

IT torsion constant

I , warping constant

W elastic section modulus

NP1

Mp1Mel

internal forces and moments

second order moment of area

axial force in perfectly plastic state

bending moment in perfectly plastic state

bending moment at which stress u reaches

yield strength in the most critical part of cross

section

apl= P1 plastic shape coeff icient

MelPoisson’s ratio

v moment ratio

Note. The term ‘perfectly plastic state’ applies when the

plastic capacity is fully utilized, although in certain

cases (e.g. angles and channels), pockets of elastic-ity may still be present. Where cross sections are

non-uniform or internal forces and moments vari-

able, Npl,Mpl and Mel at the critical point shall becalculated.

(110) Structural parameters

system length (of member)

NKi

s K = i T ; y , associated with N K ~

axial force at the smallest bifurcation

load, according to elastic theory

effective length *) of a linear member

slenderness ratio

7 ~ * E * I )

SKAK =

1

& = n / - & reference slenderness ratio

non-dimensional slenderness in com-

reduction actor according to the stand-

ard buckling curves as used in Europe

aK = K = 3NKi pression

x

member characteristic

distribution factor of systemK i , d

VKi =

*) Translator’s note. Common term as used in designanalysis. In Eurocode 3 termed ‘buckling length’.

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DIN 18800Part 2

Method

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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DIN 18800 Part 2

Table 2. Simplif ied ultimate limit state analyses

Lateral buckling

Lateral buckling

Internal forces

and moments

4.3 31to

4.3 38

Solid membersI I I

I I Bui l t -uprmbers

10 I N + M ,

Simplified analysesas inFailure mode

Lateral buckling 3.2.1

Lateral torsional buckling 3.2.2 3

3.3.2, 7, 8,

3.3.4 16, 21

Lateral torsional buckling 3.3.3, 12, 14,

Lateral buckling I 3.4.2 I 24

Lateral buckling 3.4.2 24

Lateral torsional buckling 3.4.3 27

Lateral buckling 3.5.1 28.29

Lateral torsional buckling I 3.5.2 I 30

Note 1. In calculations of internal forces and moments ac-

cording to second order theory, for example, themember characteristic,s,and the distribution factor,

~ j - ~ i .hall be determined using the design stiffness,

Note 2. Reference shall be made to the criteria set out in

item 739 of DIN 18 800 Part 1when deciding whether

to base calculations on second order theory.

Note 3. Deformations also occur as a result of joint ductil-

ity.

Note 4. Deformations resulting from stresses due to shear

forces shall be taken into account as specified in

clause 4 for built-up compression members.

117) Analysis on the basis of design actionsmultiplied by YM

As a departure from the specifications of items 115 and 116,

internal forces and moments and deformations may also becalculated using he designvalues of actions multiplied bya

partial safetyfactoryM of ,l,in which case the ultimate limit

state analysis shall be carried out using the characteristic

strengths and stiffnesses, substituting these (denoted by

subscript k) or the design resistances (denoted by sub-script d) in the equations in clauses 3 to 7

Note 1. Calculations of e and v ~ ihall be made, forexample, using the characteristic stiffness, (E.)k.

Note2. The alternative procedure set out in this item isespeciallysuitable forthe global analyses described

in clauses5,6 and 7 but may also be used by analogyin clauses 3 and 4, giving the same results as would

be obtained if yM were assigned o the resistance.To

preclude the risk of confusion, it shall be statedexplicitly in the analysis that this alternative proce-

dure has been used.

Note 3. See subclause 7.3.1 of DIN 18800 Part 1 or resist-

ance parameters.

(E I)d.

118) Slip

Account shall be taken of slip in shear bolt or preloaded

shear bolt connections in members and frames susceptibleto loss of stability, using the values specified in item 813 of

DIN 18800 Pari 1.

Note. Due account shall be taken of slip if this greatly

increases the risk of loss of stability.

119) Effective cross section

If the full cross section of parts in compression is aken into

consideration, their geometry shall be such that the grenz(blt) ndgrenz (dit)values specified in DIN 18 800 Part 1are

complied with. If,for thin-walled members,these values are

not compl ied with, the analyses shall be of lateral bucklingwith coexistent plate buckling of individual members, or of

lateral torsional buckling with coexistent plate buckling, as

specified in clause 7 of DIN 18800 Part 3 or Part 4.

Note 1. The grenz(bl t ) values differ according to the

method of analysis selected (see table 1).The grenzblt)values for individual parts of plane cross sec-

tionsare given n ables12,13,15and 18of DIN 18800

Part 1.

Note 2. The grenz (dlt) values for circular hollow sections

are given in tables 14,15and 18 of DIN 18800 Pari 1.

Methods of analyses of circular hollow sections the

geometry of cross section of which does not comply

with these limits are not covered in this standard.

120) Deductions for holes

Deductions for holes need not be made when determininginternal forces and moments and deformations if it can be

ruled out that premature local failure occurs as a result.

1.4.2 Ultimate limit state analysis by elastic theory

121) Analysis

The loadbearing capacity may be deemed adequate if an

analysis of the internal forces and moments according toelastic theory shows the structure t o be in equilibrium and

either one of the following applies.

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DIN 18800 Part 2

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





DIN 18800 Part 2

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.




DIN 18800 Part 2

1

200

1

200

100.2 = p -with n = 2

po,~ r 2 -with n =4

Ern Po.1 970,l

VI<-

21Vo.1

470= r Zö

1970=r1Töö

Iingle

member

f fl.2

970?2 970.2

%*2 (P0.2

- _\ V I

\

Variant

I 1

2oo P0,2 = r 2 l n = 2OSI= r2200

Figure 6. Initial sway imperfections in frames (examples)

Figure 7. Equivalent horizontal forces substitut ing initialsway imperfection 100 (assuming equilibrium)

Note 3. Sway imperfections due to slip of screws may also

Note 4. The reduction factor r2 may be used byanalogyfor

require consideration (cf. item 118).

roof bracing providing extra stability to beams.

206) Sway imperfections for analysis

The initial sway imperfections assumed for the columns ofbracing systems shall be as those for the columns of sway

beam-and-column type frames. The same appl ies for any

suspended columns connected to, and thus given extra

stability by, the bracing system.

of bracing systems

2.4 Assumption of initial bow and coexistent

initial sway imperfections

(207) Members in frames, which may exhibit sway imper-fections after deformation and have a member character-

istic, &, of more than 1,6,shall be assumed with both initialsway and bow imperfections in the most unfavourable

direction.

Figure 8. Assumption of initial bow and coexistent initial

sway imperfections (examples)

3 Solid members3.1 General

(301) Scope

The analyses specified in subclauses 3.2 to 3.5 apply for in-

dividual members and frame memberswhich are notionally

singled out of the system and considered in solation forthe

purposes of the analysis. Lateral buckling and lateral tor-

sional buckling are dealt with separately.

Note. If members are notionally singled out, allowance

shall be made of the actual conditions of restraint

relating to the particular member.



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DIN 18800 Part 2

Buckling curve

a

302) Lateral buckling

Since the analysis of lateral buckling specified in sub-

clauses3.2 o 3.5 lready includes both types of imperfec-

tion and second order effects, the initial forces and

momentsf romfirst ordertheoryshall betaken asa basisforcalculations.

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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DIN 18800Part 2

Table 5. Buckling curves

1 2 3

Buckling

about

axis

Buckling

curveype of cross section

Hollowsections

aY - Y2 - 2

z Hot rolled

Y - Y

2 - 2

bold formed

Z

Welded box sections

eN@i- Y

2 - 2

b

Thick welds and

h,lty < 30 Y - Y

2 - 2

C

Rolled I sections

hlb > 1.2; t s 40 mrnY - Y

2 - 2

a

b

hlb > 1.2; 40 e 5 80 rnm

hlb 5 1,2; t 5 8 0 m m

b

C

Y - Y

2 - 2

Y - Y

2 - 2t>80mrn d

Welded I sectionsb

C

Y - Y

2 - 2

Y - Y

2 - 2

C

d

Channels, L,T and solid sections

C

z z Y - Y

2 - 2

plus built-up members to subclause 4.4

Sections not included here shall be classified by analogy, taking into consideration the likely residual stresses

and plate thicknesses.

Note. Thick welds are deemed to have an actual throat thickness, a, which is not less than min t.





Pagel0 DIN 18800 Part 2

Lateral torsional buckling

0.8

\ - a

I -

Figure 10. Reduction factors x for lateral buckling (buckling curves a, b, C and d) and XM for lateral torsional buckling,

obtained by equation (18) with n equal to 2,5

3.3.2 Lateral and torsional restraint

(308) Lateral restraint

Members with masonry bracing permanently connected to

the compression flange may be considered to have suffi-cient lateral restraint if the thickness of the masonry s not

less than0.3 times the heightof cross section of the member.

Masonry, 2

Compression flange

Figure 11. Lateral restraint (masonry bracing)

If trapezoidal sheeting to DIN18807is connected to beams

and the condition expressed by equation (7) is met, thebeam at the point of connection may be regarded as being

laterally restrained in the plane of the sheeting.

Tt2

12+ GIT+ E I , - ,25

S being the shear stiffness provided by the sheeting for

beams connected to the sheeting at each rib.

If sheeting is connected at every second rib only, 0,2.

shall be substituted for S.

Note. Equation (7) may also be used o determine he lateral

stability of beam flanges used in combination withtypes of cladding other than trapezoidal sheeting,

provided that the connections are of suitable design.

(309) Torsional restraint

I beams of doubly symmetrical cross section with dimen-

sions as for rolled sections complying with the DIN 1025standards series shall be considered as being torsionally

restrained (¡.e. due to their axes of rotation being restrai-

ned) if the condition expressed by equation (8) is met.

where

k , is equal to unity for the elastic-plastic and plastic-plastic methods or 0,35 for the elastic-elastic

method;

is to be taken from column 2 of table 6 f the beam is

free to move laterally,orfrom column 3of able 6 f the

beam is laterally restrained at its top flange.

ka

Table 6. Coefficients ko

Note 1. Equation (8) is a simpler check which makes use of

the characteristic values.




DIN 18800 Part 2

Bolting toPosition of profile

top bottomTOP Bottom flange flange

Line

Note 2. When determining the actual effective torsional

restraint,cb,k, any deformations at the pointof con-nection between the supported beam and the sup-

porting member shall be taken into consideration,e.g. by means of equation 9).

1 1 1 1

C@,k C8M,k COA,k C@P,k+-+- 9)- --

where

cg,k is the actual effective torsional restraint;

CbM,k is the theoretical torsional restraint obtainedby means of equation 10) from the bending

stiffness of the supporting member (a),

assuming a rigid connection:

Bolt spacing, Washer

diameter,nmm inC'A,k7Nmim

b, ) 2 b,')

1O)

where

k is equal to 2 in the case of single-

span or two-span beams or 4 in the

case of continuous beams withthree or more spans:

( E . a ) k is the bending stiffness of the sup-

porting member;

a is the span of the supportingmember;

CfiA,k is the torsional restraint due to deformationofthe connection, that of trapezoidal sheeting

being obtained by means of equation 11a)

or 11b), substituting ?@&k from table 7;

vorh b

1O0with- ,251

vorh b

1O0with 1,25 - 2,o

where

vorh b is the actual flange width of thebeam, in mm.

Cf. [3] or further details on the use of C@A,k.

Cbp,k is the torsional restraint due to deformationof the supported beam section (cf. [4]).

Note 3. nstead of applying equation (81,he actual effec-tive torsional restraint, C@,k,may also be considered

when determining the ideal design buckling resist-ance moment,M K ~ , ~ ,he check then being carriedout as specified in subclause 3.3.4.

Table Z Characteristic tors ional restraint values for trapezoidal steel sheetins connections, assuming a flange width,

I I Sheeting subjected to suction

7 X X X 16

8 X X X 16

max bt3),

in mm

40

40

40

40

120

120

40

40

l) , - rib spacing.

2, Ka - washer diameter irrelevant; bolt head to be concealed using a steel cap, not less than 0,75mm in wall thickness.

3) bt - lange width of sheeting.

The values stated apply to bolts not less than 6,3mm in diameter, arranged as shown in figure 13, used with steel

washers not less than 1,Omm thick, with a vulcanized neoprene backing.



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DIN 18800 Part 2

Axial force diagram

iI p

kC

Figure 12. Torsional restraint (example)

IIIII

Figure13. Arrangement of screws in connections between

beams and trapezoidal sheeting (example)

3.3.3 Analysis of compression flange

(310) I beams symmetrical about the web axis, with acompression flange which is laterally restrained at a num-

ber of points spaced a distance c apart, do not require a

detailed analysis for lateral torsional buckling if

1 2)

Asimplified method using equation (14) may be used where

equation (12) is not met:

0,843M~5 1

' Mpl ,y ,d

where

My is the maximum moment;

x isareductionfactorasafunctionofbuckling _urvec

or d, obtained by means of equation (4), for A. fromequation 13),buckling curve d being selected for

beams otherthan the rolled beams n line1 oftableg,

which are subject to in-plane lateral bending on

their top flange. Equation 15)shall also be met by

beams coming under this category:

5 4 4 -t

h being the maximum beam depth;

t being the thickness of the compression flange.

Buckling curve c may be used in all other cases.

Note. Calculations may be simplified bysubstituting ori,,gthe radius of gyration of the whole section, i


(311) The ultimate limit state analysis of I beams, chan-nels and C sections not designed for torsion shall be bymeans of equation 16):

where

My

XM

is the maximum moment as specified in item 303;

is a reduction factor applied to moments as a

function of AM;

where

II is the beam coefficient from table 9.

Where there are moments My ith a moment ratio, W ,

greaterthan 0,5,the beam coefficient,n,shall be multiplied

by a factor k , from figure 14.

*-Figure 14. Beam coefficient and associated factor k ,




DIN 18800Part 2

Line Moment diagram

Table 9. Beam coefficient, n

r Iype of sect ion

Rolled

Welded

Castellated

Notched

Haunched*)

-r

min h

max h2 0,25

n

2.5

2.0

min h

max h0,7 + 1.8

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


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


3.4.2.1 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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DIN 18800Part 2

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).


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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DIN 18800 Part 2

3.5 Biaxial bending with or wi thoutcoexistent axial force


(321) Method of analysis1

The ultimate limit state analysis shall be made applyingequation (28):

k, 1 28)N MY M,

x *Npl,dMpl,y,d MpL z, d

+ -. ky +

where

x =min (xy, x is a reduction factor for the relevant buck-ling curve, from equation (4);

My nd M , are the maximum moments in first order

theory (disregarding imperfections);

is a coefficient taking -into accountmoment diagram My and A K , y It shall be

calculated as follows:

kY

Table 11. Moment factors

1

Moment diagram

3 d moments

y,, .:;. .. . ,, . .... , . .::s ..... *- l 1. . .

Moments from in-plane

ateral loading

f lQ

Moments from in-plane lateral

loading with end moments

2

Moment factors,

ßm.

for lateral buckling

& , = 0,66 0,44 y

1but not below 1-

Ki'

with a minimum of 0,44.

Nk , = 1 - ay, with a maximum

Y NpLd of 1,5

where

ay = & y ( 2 ß ~ , ~4) + - ). With a

maximum of 0,8

where

ßM,,and ßM,z are the moment factorsßM associated with

lateral torsional buck-ling, from column 3 f

table 11 taking intoaccount moment dia-

grams My and M,;

apl,y nd ctPl,, are plast ic shape co-efficients associatedwith moment M y r

M,. (Item 123 is not

applicable here.)

3

Moment factors,

for lateral torsional bucklingßMs

= 1,8- 0,7

MQ= max M from in-plane lateral

loading only

Imax MI where noalternating

moments OCCUI

A M =Imax MI + Imin Ml where

alternating

moments OCCUI



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DIN 18800 Part 2

k , is a factor taking into account moment

diagramM, and a K , p It shall be calculated

as follows:

k , = 1 - a with a maximumxz NpLd of 1,5

where

a = &,, 2ßM,z- ) + (spi,,- ). with a

is design moment M, n the perfectlyplastic state, disregarding item 123.

N

maximum of 0,8

Mpl,z,d

Item 305 shall be taken into consideration.

Note 1. If equation 28) is applied for bending about oneaxis and coexistent axial force, x shall be the reduc-

tion factor for the plane of bending under consid-eration.

Note 2. The actual increase in the internal forces and

moments in second order theory is accounted for

'by calcuLating the non-dimensional slendernesses

AK,yandaK,,overtheeffect ive lengthsforthe whole

structure (cf. [8] .

322) Method of analysis 2

The ultimate limit state analysis by method 2 shall be made

using the following equation:

k,+ A n j l 29)ßm,, .M y ß m , z * M,+

x .Npi,d Mpl,y,d ky Mpl,z,d

where

x = r n i n (xy, J is the reduction factor for the relevant

buckling curve, obtained using equation

(4);

k, shall be equal to unity and k , = c with

xy < x,;

k, and k , shall be equal to unity, with

xy= x

k, shall be equal to cy and k , equal to

unity, with x , c y;

c =1

CY

-

My and M,

fim and f im

are the maximum moments in first order

theory (disregarding imperfections);

are the moment factors for lateral buck-

ling, from line 2 of table 11 , taking into

account moment diagramM y r M,

Item 314shall be referred o O r A n , S U b S t i t U t i n g ~ K a S S O C i a t -

ed with x , he other items of subclause3.4.2.2applying by

analogy.

Note. If there is only one moment, equation (24) shall be

substituted for equation 29) where the reduction

factor in the plane of bending under consideration s

substituted for x .


323) Monosymmetric or doubly symmetrical I sectionswith relative dimensionsas for those of rolled sections,sub-

ject to axial force shall be analysed for the ultimate limit

state by means of equation (30):

Other notation is explained in subclauses 3.3.4,3.4.3 and

3.5.1.

Note 1. This analysis does not take account of design

Note 2. Tsections are not covered bythe specifications of

Note 3. Ak , value taken to be equal to unity and a k , value

torsion.

this subclause.

of 1,5 give a conservative approximation.

4 Single-span built-up members

4.1 General

401) Buckling perpendicular t o the material ax is*)

Built-up members having cross sections with one material

axis shall be dealt with as solid members as specified in

clause3 when calculating lateral displacement perpendic-

ular to the material axis. For compression and design bend-ing moment, My,his only applies when there is no design

bending moment M,

402) Buckling perpendicular t o the void axis**)

Calculation of lateral displacement perpendicular to thevoid axis may be bythe equivalent method,in which built-up

members of uniform cross section are dealt with as solidmembers,with both deformations due to moments and those

occurring as a result of transverse orces being taken intoconsideration. n this method, he design of each component

shall be based on he global analysisofthe otal nternal orces

and moments present (see subclauses 4.3.2 and 4.3.3).

Note. Frames may also be analysed on the basis of all oftheir components. Analysis by the equivalent mem-

ber method assuming solid members s specified for

battened members with two chords. The literatureshall be referred to for information on members with

more than two chords [91.

r = 2 r = 2

Figure 16. Built-up members with cross sections having

one material axis (y-axis) (examples)

403) Cross sections with tw o void axes

The following information applies by analogy to both axes

for cross sections with two void axes.

r = 4

Figure 17. Built-up member with a cross section having

two void axes (y-and z-axes) (example)

ky and k , being taken from item 320 and item 321

respectively.

*) Axis intersecting with components.

**) Axis between components.


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DIN 18800Part 2

4.2 Common notation

404)

r number of chords;

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.



--` , , ,`

-` -` , ,` , ,` ,` , ,` ---



DIN 18800 Part 2

1 2

4.3.2.2 Lacing systems

(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]).




DIN 18800 Part 2

(409) Battens

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



--` , , ,` -` -` , ,` , ,` ,` , ,` ---



DIN 18800 Part 2

4.5 Structural detailing

413) Retention of cross-sectional shape

Where member cross sections have two void axes, the rec-

tangular cross-sectional shape shall be retained by means

of cross-stiffening.

Note. Cross-stiffening may take the form of bracing,platesor frames.

414) Arrangement of battens and packing plates

Battened members shall be connected at the ends by bat-tens.This also applies to laced members unless cross brac-

ing is used instead.

If built-up members are connected at the same gusset,due

account shall be taken of the fact that the gusset will also

function as an end batten or end packing plate.

The other battens shall be spaced as equally apart as pos-sible, he use of packing plates being permitted instead for

the members shown in figures 19 and 21. The number of

panels shall be not less than three, and equation (41) shallbe satisfied:

a

i1

- 70 (41)

5 Frames

5.1 Triangulated frames5.1.1 General

501) Calculation of forces in triangulatedframe members

The forces act ing in the members making up a triangulated

frame may be calculated assuming nominally pinned

member ends.Secondary stresses as a result of nodes maybe disregarded.

Where the cross sections of compression chords are non-

uniform over their length,any oad eccentricity in individual

members may be disregarded f the mean centroidal axis of

each cross section coincides with the centroidal axis of the

compression chord.

502) Analysis of compression members

Analysis of compression members shall be as specified inclause 3,4 or 7

5.1.2 Effective lengths of frame membersdesigned to resist cornpression

5.1.2.1 General

(503) Rigidly connected members

In the absence of a more rigorous treatment, the effectivelength,SK,of frame members which are rigidly connected

using at least two bolts or by welding shall be 0.9 I for in-

plane buckling (42) and equal to unity for out-of-plane

buckling (43).

504) Non-rigidly connected members

In the absence of a more rigorous treatment, the analysis

for the sway mode of vertical and diagonal members held

horizontally by cross beams or transverse members provid-ing non-rigid connection, is a function of the structuraldetailing involved.

Noie. The effective length, S K , ~ ,of triangulated frame

members as shown n figure 22 for the sway mode inthe perpendicular plane may be determined by

means of the diagrams in figure 27.

(505) Members with one end allowing late raldlsplacement and one or two non-rigidly

connected ends

Where verticals and diagonals in main triangulated frames

also act as the columns of sway portal frames,and thsirbot -

tom chords are in the perpendicular plane, the effectivelength in that plane may be determinedas for compressive

forces which do not always act in the same direction.

Note 1. Chords may be held in the perpendicular plane by

Note 2. The effective length can be determined with the

a road deck, for example.

aid of figures 36 to 38.

/

N A

/

/

IbVertical member held horizontally,

non-rigidly connected at one side

Vertical member held horizontally,non-rigidly connected at both sides

Figure 22. Non-rigidly connected triangulated frame

members for out-of-plane buckling

5.1.2.2 Triangulated frame members supportedby another triangulated frame member

(506) Connection at intersec tion

A t intersections, members shall be connected directly or

via a gusset.

if both members are continuous, he connection between

them shall be designed to withstand a force acting in the

perpondicu ar plane equal to 10% of the greater compres-sive force.

(507) In-plane effective length

The effective length for the sway mode in the plane of thetriangulated member shall be assumed to be the system

length to the node of the intersecting members.

(508) Out-of-plane effective length

The effective length fort he sway mode in the perpendicular

plane appropriate to the structural detailing involved maybe taken from table 15.




DIN 18800Part 2

Table 15. Out-of plane effect ive lengths of triangulated frame members of uniform cross section i n the perpendicular

plane

1 2 I 3

3 z - 11 - _

4 N - 1 ,

I , 13

I . :

1 +-SK = 1

but not less than 0,5 Z

N 1,

1 +-I , 13Y . :

SK =

but not less than 0,5 Z

Continuous compression member

but not less than 0,5 I

1 +-

but not less than 0,5 1

Nominally pinned compression member

where

Ilhere. 4

z -i r where the following applies:

Dut not less than 0.5 1

N


--`,,,`-`-`,,`,,`,`,,`---



DIN 18800 Part 2

1

2

3

5.1.2.3 Solid truss members with elastic support

at mid-length

(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)




DIN 18800Pari 2

513) Stiffness of beam-and-column type frames

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:



--` , , ,` -` -` , ,` , ,` ,` , ,` ---



DIN 18800 Part 2

Special design situations

-$---Cu= o

1c =

2 11 1s1 + -

Is 4In all three cases:

ec,, =

2 12 15I+-

Figure

Nominally

pinned

tk-Eu

L

5acO

c.-

33o

O

O

L

O

Rigid

3 1s 12 Rigid c, or c, (whichever greater)- Nominallypinned

SK = Is

Division of non-sway frame into subframes with only

one column, for application of diagram belowK i =

K i + K ë = K6

Kb i K i i s i K:" = K3

(Resolution of K3 and K6

may be freely selected.)

27. Diagram to determine the distribution factor, q ~ i ,nd effective length, SK, for columns of non-sway frames

where seam is not greater than 0,3



--` , , ,` -` -` , ,` , ,` ,` , ,` ---



Storey r + 1

lI R

&s

Beam r

Storey r

Figure 28. Notation and values for calculation of & d

Special design situations

DIN 18800 Part 2

Cr+l ...

1er 5

hr

kI-l= ...

1

1 + 2 -

In all six cases

(disregarding a ~ ) :

C =

Cu =

Il 's

I s 12

1

1 + 2 -12 s Tr5

Is 12 O 0,i 0,2 0.3 0,L 0,s 0,6 0,7 0,ô 0,9 1

Rigidly c, or c, (whicheber greater)- Nominallyconnected pinnedK = ß J S

~ ~ \ z E I sN,i ißk,

q K i = N = For multistorey frames, calculate c and c as follows:

-0

CaKOl + -

K s + KS.0

1

Storey under

consideration

Figure 29. Diagram to determine distribution factor, I;IK~, and effective length, sK,for columns of sway frames where

is not greater than 0.3




DIN 18800 Part 2

nominally pinned:

53)

In the other storeys:

where

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 ,



--`,,,`-`-`,,`,,`,`,,`---



DIN 18800 Part 2

where

V r is the transverse force in the storey due to externalhorizontal loads only;

N , is the total vertical oad ransmitted within he r thstorey;

pr is the angle of rotation of the columns in a storey(ca1-culated according to first order plastic hinge theory).

Note, Formula for calculating pr for single-storey frames

(527) Single-storey frames

First order plastic hinge theory may be applied for the

frames shown in figure 31 provided that there are no (or

virtually no) plastic hinges at their ends and equation (62)is satisfied:

are given in the literature (cf. 1121).

1+-I R *h

where

a is equal to 3 or 6 for nominally pinned or rigidly con-

nected bases respectively;

N is the total vertical load.

I

Eor 1

subject to axial compression when considering out-of-

plane buckling.

Note. In the case of bridges, elastic support is usually pro-vided by subframes (cf. table for spring stiffness of

such frames).

531) Averaging of compressive force

For solid web beams, the axial force of the compression

chord posi tioned between two subframes may be averaged

to give a constant value, the chord cross section beingtaken to include the chords plus one fifth of the web.

Table 18. Examples of spring stiffness, Cd, of a subframein trough bridges

Trusses and solid web beams with subframesin perpendicular plane

+i-

141 I

Figure 31. Notation used in equation (62)

If the height of nominally pinned columns, Z is not thesame as the height of the frame columns, h, he vertical

loads on the nominally pinned columns shall be multipliedby the factor hll, for calculation of N .

Note. This specification may give very conservative resultssince it covers the whole range of possible plastic

hinge configurations.

5.3.2.5 Simplified calculation according to second

order plastic hinge theory

528) The simplified method according to second orderelastic theory as specified in subclause 5.3.2.2 assuming

transverse forces in the storey as obtained by means ofequation (56), may be adopted as it stands in plastic hinge

theory provided that there are no (or virtually no) hinges at

columns.The angle of rotation of the column according to

the present simplified second order plastic hinge methodshall be substituted for qr in equation (56).

5.3.3 Non-rigidly connected continuous beams

5.3.3.1 General

529) Analysis of non-rigidly connected continuousbeams may be on the lines of subclause 3.4.2.

5.3.3.2 Compression chords with elastic lateral support

530) Trusses and solid web beams

The compression chords of trusses or solid web beams may

be dealt with as non-rigidly connected continuous beams

N

6 Arches

6.1 Axia l compression

6.1.1 In-plane buckling

6.1.1.1 Arches of uniform cross section

601) Analysis

The ultimate l imit state analysis shall be made by applyingequation (3), being the value at the springing.

Plan view

Figure 32. Arch axes

Note. Figure 33 shows buckling coefficients obtained by

means of equation (63) for various types of sym-metrical arch systems, all of which assume that de-formations due to axial forces can be disregarded.

(63)

where sK is the effective length and s half of thebeam length, ? s used to calculate the axial force at

the springing, N K ~ ,nder the smallest bifurcation

load (see equation 64):

SKß = -

S

I * \2



--`,,,`-`-`,,`,,`,`,,`---



DIN 18800 Part 2

Table 19. Spring sti ffness of triangulated structures without verticals

1

Typical Warrentruss bridges

Through bridge design on which

analysis based (cf. figure 18)

2

C

Subframes in Warren truss bridges

A*) Hinge allowing for torsion

System on which analysis based. Bottom chord of centre panel onlyresistant to bending, adjacent bottom chords only resistant to torsion.

A + B - 2 DSpring stiffness: C - 2 ( E 1 u ) d

d - ~ . ~ - ~

h2 1, d 3 . , b uB = +

U r Idr 3

1

6D = a . b - u

Any areas resistant to bending at member ends shall be deducted from dl, d,, a, b, u and b , and those resistant

to torsion, from u1 and u

Idl , Idr and I, are second order moments of area of the diagonals and bottom chord with respect to bending

perpendicular to the main beam.

Z,l and I,, are second order moments of area of the cross beams at the left and right of the panel with respect to

bending of the deck.

Z T ~and I T r are the torsion constants of the adjacent bottom chord members.

If the half-wave coefficient, rn, f the bending curve due to buckling of the top chord is less than a half the number

of panels, reduced spring stiffness shall be assumed by calculating the second order moments of area, I,, of all

inner cross members with only half their values.



--`,,,`-`-`,,`,,`,`,,`---



DIN 18800 Part 2

Buckling coefficients, /3, for in-plane buckling of arch

tß

tß

tß

Antimetric buckling

Antimetric buckling

tP

Symmetric buckling

f / l-Pa: parabola; Ke: catenary; Kr: circle

Loads (e.g. hydrostatic pressure) shall be assumed to correspond to the arch form in the case of arches of the parabolic or

catenary type but t o act linearly in the case of one-centred arches.

Figure 33. Buckling coefficients, ß, for in-plane buckling of arches loaded in their thrust line (deformations due to axial

forces being neglected)




DIN 18800Part 2

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).



--`,,,`-`-`,,`,,`,`,,`---



DIN 18800 Part 2

5

Any transverse loads (such as wind load) shall be checked

separatelytaking into account bending moments as set outin item 314.

6 7

Note 1. Buckling coefficients may be taken from the litera-

ture (cf. [15]) and figures 36 to 38 which cover load-

ing corresponding to the arch orm, not ust in portai

frames of arches.

Note 2. h, as featured in figures 36 o 38 shall be obtained

by multiplying the averaged hanger length,h ~ ,y

the factor llsin Czk, a k being the angle between the

sloping columns of the frame and the beam.h, shall

be assumed to be negative where the deck is on

J z *a

where

ß is the buckling coefficient;

h

i

is the in-plane height of the column of the portal frame;

is the radius of gyration of the z-axis of the portal

frame column. supports.

Table 20. Auxiliary value, k

I I 1 1 2 1 3 4

0,075

23 17 I 10Two-hinged arch

3 Rigidly connected arch 97 42 I 13

Table 21. Buckling coefficient, ß,

0,50 0,s

0.2

0,65/z,,, (at crown)

I, constant

with 1 t I _ f,59

Table 22. Buckl ing coeff icient, ß2

I Loading I ß 2 Notation

1 I Corresponding to arch form I 1

q =total load

q H = oad component, transmitted by hangers

qst = oad component, transmitted by columns

Via hangers2 9H1- ,351

4

Via columns')2 l 9%1 + 0.45-

9I I

l The deck is fixed to the arch crown.

Deck

Figure 35. Braced arches with end portal frames and suspended deck





DIN 18800 Part 2

h l h ,-

h lh ,-

hlh,-

Figure 36.

Buckling coefficients for portal

frames with nominally pinned

column bases

Figure 37.

Buckling coefficients for portal

frames with rigidly connected

column bases

Figure 38.

Buckling coefficients for por tal

frames with columns connected

by two beams of equal stiffness


--`,,,`-`-`,,`,,`,`,,`---



DIN 18800 Part 2 Page33

2

Form of equivalentgeometrical imperfection

(sinusoidal or parabolic)

In figure 39, s the angle of the arch equal to 2 lr but not

less than O or more than TE.

3

W O for cross sections

with buckling curve(cf. table 5)

a b C d

6.2 In-plane bending about one axis

with coexistent axial force

6.2.1 In-plane buckling

607) The in-plane buckling of the arch shall be analysedfor ultimate limit state using one of the methods listed in

table 1, assuming equivalent geometrical imperfections

from table 23occurring in the most unfavourable direction.

The effective length of arches of uniform cross section with

in-plane buckling, satisfying equation (io), may be calculated

by first order theory without taking into account equivalentimperfections.

scß = -

liK1The following applies for arches in compression:

aKt = 2,47 - (3+ 0,21 )

100

+ (700 - 6 k + 0.08 k2 ) (73)

The following applies for arches in tension:

9 8 7/58K i =- 0,036-

0+ (10 + k)2Note 1. SK may be derived from equation (63)n conjunc-

Note 2. Cf. item 201 when applying the elastic-elastic

6.2.2 Out-of-plane buckling

6.2.2.1 General

608) The ultimate limit state analysis for out-of-planebuckling of arches may be carried out as specified in sub-clause 6.1.2.

6.2.2.2 One-centred arches of uniform rectangularor I cross section, with their chord in tension

or compression

tion with figure 33.

method.

609) Laterally restrained arches with the stat ic system asshown in figure 39may be given a simplified treatment using

equation (3) nd employing the in-plane slenderness ratio,AK, obtained by means of equation (71), to determine K.

- ß . s

i .la= (71)

- (0,226 3,4 1,94

k +F i ) (74)

whereE . I ,

k = LG IT

6.2.2.3 One-centred arch sections of uniform

i cross section, with fork restraint

(6103 An approximate ultimate limit state analysis of one-centred arch sections of uniform cross section may be car-

ried out usingequation (27) and employing the in-plane slen-

derness ratio,iK,obtained from equation (75)to determineK .

- ß . sAK = (75)

i .A,

where

a is the angle of the arch,equal to 2 sir but not less than Oor more than T E

2n(76)is the buckling coefficient, equal to

I'K1where

(TE2- 2)2

Ki = (77)n2+ a2 .k

where

E . ,k = -

G 1,igure39. Static system for laterally restrained arches

Table 23. In-plane equivalent geometrical imperfections in arches

1

Three-hinged arch insymmetrical buckling

S S S S

300 250 200 150

I I I I

2 Two-hinged arch, three-hingedarch, fixed-ended arch in

antimetric buckling



--` , , ,` -` -` , ,` , ,` ,` , ,` ---



DIN 18800 Part 2

Table 24. Out-of-plane equivalent geometrical imperfections of the arch

Two-hinged arch,three-hinged arch,

fixed-ended arch

Form of equivalent u0 for cross sections with buckling 'curvegeometrical imperfections (cf. table 5 )

in horizontal direction

(sinusoidal or parabolic) a

I

I ' ' - r 6 '

M K ~ , ~ ,equired for calcuJation of the reduction factor XM

from equation (18)usingAM rom item ll0,shall be obtained

by means of equation (78).

where

- E I , . T C ~C =

r 2 a

In equation (78), there shall be a plus sign before the root i fMy results in tension on the inside of the arch.

Note. Equation (78) assumes fork restraint perpendicular

to the plane of the arch.

6.3 Design loadingof arches

611) The ultimate limit state analysis shall normally bemade by the elastic-elastic method, assuming feasible

equivalent geometrical imperfections in addition to the

design loads. In the absence of lateral restraint of arches

between springings, the equivalent imperfections may be

taken from table 23 or 24.k is sufficient to assume imper-fections acting i n a single (i.e.the most unfavourable) direc-

tion, either in or perpendicular to the plane of the arch.

Where there is transfer of loads via hangers or columns, it

shall be assumed that these retain their design direction inthe state of deformation.

Note. Design loading plays a significant role in arches

exposed to outdoor conditions due to the possibleeffect of wind acting transverse to the arch plane.

In this case, the loading conditions set out in sub-

clauses 6.1 and 6.2 are not met.

7 Straight l inear members with plane

thin-walled parts of cross section

7.1 General701) Field of application

This clause shall apply in cases where the grenz (b i t )valuesfor individual parts of a cross section are exceeded, which

then requires the effect of plate buckling of such parts on

the buckling behaviour of the member as a whole to be

taken into account when calculating both internal forces

and moments and resistances.

Note 1. Thegrenz ( b l t )values shall be taken from tables 12,13 and 15 of DIN 18800 Part 1.

Note 2. Plate buckling of individual parts of a cross section

usually affects the buckling behaviourof the member

b C d

as a whole by causing a reduction in ts stiffness anda redistribution of stresses within a cross section to

parts exhibiting greater stiffness or less subject to

stress.

(702) Analysis

The ultimate limit state analysis shall be by the elastic-

elastic or elastic-plastic method.

The analysis may take the form of the approximate methods

set out in subclauses 7.2 to 7.6.Note 1. The application of plastic hinge theorywill not be

possible until ts viability is given sufficient practical

backing.

Note 2. In subclauses7.2 to7.6,the effect of buckling of the

individual parts of cross section on member buck-

ling as a whole is taken into account.

(703) Effect of shear stresses

In cases where subclauses 7.2 to 7.6are applied, shear

stresses when analysing plate buckling o f thin-walled parts

of cross section are so minor that they can be disregarded,

¡.e. if they meet the following conditions:

pi,d is the ideal buckling stress in plates due solely to

edge stresses t, o be determined as specified inDIN 18800 Part3.

If equations 79) nd (80) are not met, allowance for the

additional effect of shear stresses may be made as set out

in DIN 18 800 Part 3.This does not affect the necessity ofalso taking into account the overall reduction in stiffness of

the member.

(704) Permitted sections

The provisions of subclauses 7.2 to 7.6 shall only apply to

members of uniform cross section taking the followingforms: hollow rectangular sections, doubly symmetric or

monosymmetric I sections, channels, C sections,Zsections

and trapezoidal hollow ribs.

Note. Hollow sections are considered rectangular where

blr is not less than 5 (cf. figure 40).Circular cross

sections and T sections are not dealt with.

7.2 General rules relating to calculations

(705) Effective cross section (model)

In a model of the effective cross section,an effective width,

b'(cf.figure 40) rb', i ssubs t i tu ted fo r theac tua lwid th ,b ,o fthe thin-walled part of the cross section. The resulting

effective cross section is taken as the basisforcalculations.




DIN 18 800 Part 2

Y5N

Table 25. Increase in bow imperfection, A W O

1 2 3IMoment

diagram

e , = centroidal shift due to positive moment

en= centroidal shifi due to negative moment

M ; = M , + N e

a) Gross cross section b) Reduced effective crosssection as a result of

buckling of upper flange

Figure 40. Effective cross section (example)

Note 1. Thus, all cross section properties of the effectivecross section require to be determined.

Note 2. Provisions or the calculation ofb’orb” are made n

subclauses 7.3 (elastic-elastic method) and 7.4

(elastic-plastic method). Accordingly, cross section

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





DIN 18800 Part 2

Table 26. Buckling factors, k

7,81

7,81 6,29 + 9,78 2

23,s

Type ofsupport

Stress

distribution

1,70 0,57

1,70 5 + + 17) W 2 0,57 0,21? f 0,07@

23,8 0,85

q =

1 > q > o

?#=O

O > W > - l

q = 1

1 2

A t one end

~~~

4 I 0,43

82

W + 1,05

0,578

+ 0,340,57 0,21q +0,07 2

ends are restrained and moments with different signs are

liable to occur here. If one of the ends is nominally pinned,

epor e (see item 709) is equal to zero at this end.

Note. An additional imperfection is to be assumed as a

result of this increase n sway imperfection when the

equivalent member method is applied.

7.3 Effective width in elastic-elastic method

711) Stress distribution

In the elastic-elastic method, calculations shall be on thebasis of a linear stress distribution in the effective cross

section.

Note. This is an assumption only,and is not based on actual

fact since the actual stress distribution is non-linear.

(712) Determining the effective wid th

The effective width shall be determined by means of equa-

tion (81)or cases in which plates (web or flange) are sup-

ported on both sides with constant compression and equa-

tion (82)or support on only one side. The assumption of

support on both sides presupposes that the supportingconstruction is of adequate stiffness.

b ’ = b

for Apo g 0,673

1 - ,22/äp0)

APO

ob

for npo > 0,673

82)0,7

APO

b = , but not exceeding b

where

b

ripa

is the width of the thin-walled part of the cross

section from table 26;

is the non-dimensional slenderness elating to plate

buckling, obtained by means of equation (83):

U

= G

ue = 189800 - in N/mm2;i rt is the thickness of the thin-walled part of the cross

section;




DIN 18800 Part 2

k is the buckling factor from table 26, the edge stress

ratio,y. eing a function of the stress distribution inthe effective cross section. Where plates are sup-

ported on both sides,y may be calculated on the

basis of the gross cross section of the part under

consideration.

The stress distribution shall be calculated on the

basis of all internal forces and moments;

is the maximum compressive stress according t osecond order theory acting at the long edge of the

thin-walled part of the cross section, calculated on

the basis of the effective cross section, and ex-

pressed in N/mm2.The ong edge is taken to be an

edge of the gross part of the cross section.

if. in equation (83),u is assumed to be less than fy,d. u shallbe substituted for fy,d in the analyses specified in sub-

clauses 7.5.2.1 to 7.5.2.3.

Note1. Reference may be made to,forexample,subclause

3.10.2 of the DASt-Richtlinie DASt Code of practice)

016 Bemessung und konstruktive Gestaltung von

Tragwerken aus dünnwandigen kaltgeformten Bau-teilen (Design and construction of structures with

cold formed, thin-walled sections) for suitable stiff-

ness of the supporting constructions for plate

edges.

Note 2. Where u is equal to fy,d, npo is equal to Xp fromtable 1 of DIN 18 800 Part 3.

Note 3. u, shall be obtained thus:

u

5 ~ * E . 2

12b2 (1- U)u, =

inserting a Poisson's ration, ,u, equal to 0,3.

Effective flange width with u and VI= - 1,0

(Y2.u

Effective web width with u and y = 2 2 y1

. .

JLEffective cross section

Figure 42. Determination of effective cross section of anI section with bending about one axis

(713) Resolution of effective width

Resolution of the effective width, b',shall be as in table 27.

Note 1. As a simplification, and in line with provisions atnational and international level, the procedure de-

scribed here has been modified somewhat in

Table 27. Resolution of effective width-

mUC(u

Oa

m

OQ

v )

f

n

Fa>

(u

C

O

m

OCL0J

W I

c

-

-1 5 5 1b; = Q b k ,

b > = Q + . k ,

where

Q =1=- [(0.97 0,03 ) - (OJ6 + 0,06 p /IpJ

k , = -0.04 q2 OJ2 I #+0,42

k2 = +0,04@ - 0,12 I#+ 0,58

&o

0 w(Compression)

u+W(Tension)Compression)

-1 < ? p i o

@a(Compres-

x sion)

P - 7G b -i

- 1 < * < l

comparison with line3 of table 1of DIN 18 800 Part3and table 12 of DIN 18 800 Part 1 n that the factorc is

not applied for y equal to O but not greater than 1.

Note 2. Calculation of the e ,k l and k2 values is such that

the buckling factor, k , can be determined as spec-

ified in item 712.

7.4 Effective width in elastic-plastic method

T14) The effective width shall be calculated using one ofequations (85) to (87).Coefficients k , andk2 and resolution

of the effective width shall be as in table 28, ensuring that

(84)N¡= and b = Z b i , but with b 2 b

i being between unity and 3.

bi = kl * t

(87)

Note. Iteration is usually required for calculation of the

effective width.




DIN 18800 Par i 2

7.5 Lateral buckling

7.5.1 Elastic-elastic analysis

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



--`,,,`-`-`,,`,,`,`,,`---



DIN 18800 Part 2

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


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.





DIN 18800 Part 2

Standards and o ther documents r eferred to

DIN 1025 Part 1

DIN 1025 Part 2

DIN 1025 Part 3

DIN 1025Part 4

DIN 1025Part5

DIN 1080Part 1

DIN 4114Part1

DIN 4114 Part 2

DIN 18800 Part 1

DIN 18800 Part 3

DIN 18800 Part 4

DIN 18807 Part 1

DIN 18807 Part 2

DIN 18807 Part 3

DASt-RichtlinieO16

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

Literature

ECCS-CECM-EKS, Publication No.33.Ultimate limit state calculation of sway frames with rigid oints, Brussels,1984.

Stahl m Hochbau (Steel construction), 14th ed., vol. I Part 2,Düsseldorf: Verlag Stahleisen mbH, 1986.

Lindner,J.; Gregull,T. Drehbettungswerte ür Dachdeckungen mit untergelegter Wärmedämmung (Values of torsional

restraint for roof coverings with thermal insulation), Stahlbau,1989: 8,173-179.

Lindner, J. Stabilisierung von Biegeträgem durch Drehbettung- ine Klarstellung (Stabilization of beams by torsional

restraint), Stahlbau,1987: 6, 365-373.

Roik, K.; Carl, J.; Lindner, J. Biegetorsionsprobleme gerader dünnwandiger Stäbe (Problems with flexural torsion of

straight thin-walled linear members), Berlin, München, Düsseldorf: Ernst& Sohn, 1972.

Petersen, Chr. Statik und Stabilität der Baukonstrukrionen (Static and stability o f structures), 2nd ed., Braunschweig,

Wiesbaden: Friedr. Vieweg und Sohn, 1982.Roik, K ;Kindmann, R. Das Ersatzstabverfahren- Tragsicherheitsnachweise ür Stabwerke beieinachsiger Biegung und

Normalkraft (The equivalent member method: ultimate safety analyses of frames subjected t o bending about one axis

and coexistent axial force), Stahlbau, 1982: 1, 137-145.

Lindner,J.; Gietzelt,G. weiachsige Biegung und Längskraft- in ergänzenderBemessungsvorschlag Biaxial bending

and coexistent axial force. A supplementary design proposition), Stahlbau, 1985: 4, 265-271.

Ramm, W.; Uhlrnann, W. Zur Anpassung des Stabilitätsnachweises für mehrteil ige Druckstäbe an das europäische

Nachweiskonzept (Bringing into line stabi lity analyses of built-up compression members with the European concept),

Stahlbau, 1981: 0,161-172.

Vogel, U.; Rubin, H.Baustatik ebener Stabwerke (Statics of plane frames), Stahlbau-Handbuch, vol. 1, Köln: Stahlbau-

Verlag, 1982.

Rubin, H. Näherungsweise Bestimmung der Knicklängen und Knicklasten von Rahmen nach ?-DIN 18800 Teil 2

(Approximate determination of effective lengths and buckling loads of frames to draft Standard DIN 18800Part Z) ,

Stahlbau, 1989: 58,103-109.

Rubin, H. Das Drehverschiebungsverfahrenzur vereinfachten Berechnung unverschieblicher Stockwerkrahmen nachTheorie . undII. Ordnung (The method using initial sway imperfections for simplified calculation of non-sway beam-

and-column type frames by first and second order theory), Bauingenieur, 1984: 9, 467-475.

Palkowski,S. Stabilität von Zweigelenkbögen mit HängernundZugband (Stability of two-hinged arches with hangers

and ties), Stahlbau,1987: 6, 69-172.

Palkowski,S.Statik und Stabilität von Zweigelenkbögen mit schrägen Hängern und Zugband (Statics and stabi lity of

two-hinged tied arches with diagonal hangers), Stahlbau, 1987: 6, 246-250.

Dabrowski, R. Knicksicherheit des Portalrahmens (Safety against buckling of portal frames), Bauingenieur,1960: 5,

178-182.

Obtainable from Deutscher AusschuB für Stahlbau, Ebertplatz i, -5000 Köln 1.





DIN 18800 Part 2

[16]

[17]

Rubin,H. Bed-Knick-Problem eines Stabes unterDruck und Biegung (The problem of plate-bucklinglbuckling of linear

members subject to bending and compression), Stahlbau, 1986:55, 79-86.

Schardt, R.; Schrade, W. Bemessungvon Dachplatten undWandriegeln aus Kaltprofi len (Design of roof plates and wall

girders with cold-formed sections), Forschungsbericht des Ministers für Landes- und Stadtentwicklung des Landes

Nordrhein-Westfa\en (Research report issued by the Nordrhein-Westfalen Ministry for Urban and Rural Planning),

Technische Hochschule Darmstadt (Darmstadt Polytechnic), 1981.

Bulson, P.S., The stability of flat plates, London: Chatto and Windus Ltd., 1970.

Grube, R.; Priebe,J.Zur Methode der wirksamen Querschnitte be i einachsiger Biegung mit Normalkraft (Effective cross

section-method for bending about one axis and coexistent axial force), Stahlbau, 1990: 59, 141-148.

[18]

1191

Previous edition s

DIN 4114 Part 1: 0 7 . 5 2 ~ ~ : IN 4114 Part 2: 02.52~.

Amendments

The following amendments have been made to the July1952 edition of DIN 4114 Part 1 and February1953 edition of DIN 4114

Part 2.

a) The number and title of the standard have been changed to bring them into line with the reorganized system of standards

b) The material has been rearranged, he resistance to buckling of linear members and frames, of plates and of shells now

c) The standard has been revised, bringing it in to line with the current state of the art.

on structural steelwork.

being dealt with in different Parts of DIN 18800.

Explanatory not es

The revision of the content of the DIN 18800 standards series has been accompanied by a redesign of their layout in an

attempt to improve their clarity and make them more convenient to use.

The new layout is based on the type employed by the Deutsche Bundesbahn or its regulations covering construction work

while keeping to the rulesformulated in DIN 820.As well as the conventional division into clauses and subclauses,the text is

subdivided into smaller ‘items’ each of which contains a piece of self-contained information which can be incorporated into

other standards.

internati onal Patent Classification

E 04 B 1/19

E 04 B 1124

G O1 B 21/00

G O1 N 3/00

Din 18800-Part2 English Language

Documents

Transcript of Din 18800-Part2 English Language