141439606 din 18800 02 structural steelwork design construction din 1990

November 1990 UDC 693.814.074.5 DEUTSCHE NORM

Structural steelwork Analysis of safety against buckling of

linear members and frames

DIN 18800

Part 2

Contents Page

1 General ....................................... 2 1.1 Scope and field of application . . . . . . . . . . . . . . . . . . . 2 1.2 Concepts ..................................... 2 1.3 Common notation ............................. 2 1.4 Ultimate limit state analysis ..................... 3 1.4.1 General ..................................... 3 1.4.2 Ultimate limit state analysis by elastic theory .... 4 1.4.3 Ultimate limit state analysis by plastic hinge theory 5

.2 imperfections.. ................................ 5 2.1 General ...................................... 5 2.2 Bow imperfections. ............................ 5 2.3 Sway imperfections ............................ 6 2.4 Assumption of initial bow and coexistent initial

sway imperfections . ........................ 7 3 Solid members . . . . . ........................ 7 3.1 General ...................................... 7 3.2 Design axial compression ...................... 8 3.2.1 Lateral buckling ............................. 8 3.2.2 Lateral torsional buckling *) ................... 8 3.3 Bending about oneaxiswithoutcoexistentaxial force 8 3.3.1 General ..................................... 8 3.3.2 Lateral and torsional restraint ................. 1 O 3.3.3 Analysis of compression flange ................ 12 3.3.4 Lateral torsional buckling ..................... 12 3.4 Bending about one axis with coexistent axial force 13 3.4.1 Members subjected to minor axial forces ....... 13 3.4.2 Lateral buckling ............................. 13 3.4.3 Lateral torsional buckling ..................... 14 3.5 Biaxial bending with or coexistent axial force 15

3.5.2 Lateral torsional buckling ..................... 16 4 Single-span built-up members .................. 16 4.1 General ...................................... 16 4.2 Common notation ............................. 17 4.3 Buckling perpendicular to void axis .............. 17 4.3.1 Analysis of member .......................... 17 4.3.2 Analysis of member components .............. 17 4.3.3 Analysis of panels of battened members ........ 18 4.4 Closely spaced built-up battened members ....... 19 4.5 Structural detailing ............................ 20 5 Frames.. ...................................... 20 5.1 Triangulated frames ........................... 20

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

Page

5.1.1 General.. ................................... 20 5.1.2 Effective lengths of frame members

designed to resist compression. . . . . . . . . . . . . . . . 20 5.2 Frames and laterally restrained continuous beams . 22 5.2.1 Negligible deformations due to axial force ...... 22 5.2.2 Non-sway frames ............................ 23 5.2.3 Design of bracing systems .................... 23 5.2.4 Analysis of frames and continuous beams. ...... 23 5.3 Sway frames and continuous beams subject to

lateral displacement ........................... 23 5.3.1 Negligible deformations due to axial force . . . . . . 23 5.3.2 Plane sway frames ........................... 23 5.3.3 Non-rigidly connected continuous beams ....... 27 6 Arches ........................................ 27 6.1 Axial compression ............................. 27 6.1.1 In-plane buckling ............................ 27 6.1.2 Buckling in perpendicular plane. . . . . . . . . . . . . . . . 30 6.2 In-plane bending about one axis with

coexistent axial force ............ 6.2.1 In-plane buckling .............. 6.2.2 Out-of-plane buckling ........................ 33 6.3 Design loading of arches ....... 34 7 Straight linear members with plan

thin-wailed parts of cross section . . . . . . . . . . . . . . 34 7.1 General ...................................... 34 7.2 General rules relating to calculations . . 7.3 Effective width in elastic-elastic method 7.4 Effective width in elastic-plastic method 7.5 Lateral buckling ............................... 38 7.5.1 Elastic-elastic analysis ........................ 38 7.5.2 Analyses by approximate methods . . . . . . . . . . . . . 38 7.6 7.6.1 Analysis .................................... 39

7.6.3 Bending about one axis without coexistent axial force .................................. 39

7.6.4 Bending about one axis with coexistent axial force .......................... . . . 39

7.6.5 Biaxial bending with or without coexistent axial 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 7 ufh Verlag GmbH. Berlin, has the exclusive right of sale for German Standards @IN-Normen). Sales No. 0117 04.93 Copyright Deutsches Institut Fur Normung E.V.

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DIN 18800 Pari 2

1 General 1.1 Scope and field of application (101) Ultimate limit state analysis This standard specifies rules relating to ultimate limit state analysis of the buckling resistance of steel linear members and frames susceptible to loss of stability. It is to be used in conjunction with DIN 18800 Part 1.

(102) Serviceability limit state analysis Aserviceability limit state analysis need only be carried out if specifically required in the relevant standards. Note. Cf. subclause 7.2.3 of DIN 18 800 Part 1.

1.2 Concepts

(103) Buckling Buckling is a phenomenon in which displacement,v orw,of a member occurs, or rotation, 9, occurs about its major axis, or both occur in combination. A distinction is conventionally made between lateral buckling and lateral torsional buckling.

(104) Lateral buckling Lateral buckling is a phenomenon in which displacement,v or 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, of 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, internal

forces 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 fy 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 identifying it as being a limiting (¡.e. maximum permissible) value

defined in subclause 3.1 of DIN 18800 Part I.

and f y , k.

Figure 1. Coordinates, displacement parameters and

(109) Section parameters A cross-sectional area I

i = radius of gyration

IT torsion constant I , warping constant W elastic section modulus

NP1 Mp1 Mel

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 = - MP1 plastic shape coefficient Mel

Poisson’s ratio M 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 variable, Npl, Mpl and Mel at the critical point shall be calculated.

(110) Structural parameters 1 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 )

SK AK =

1

&=n/-& reference slenderness ratio

non-dimensional slenderness in com-

reduction factor according to the standard buckling curves as used in Europe

aK - = - AK = (3 NKi pression

x

member characteristic

distribution factor of system N K i , d VKi = 7 *) Translator’s note. Common term as used in design

analysis. In Eurocode 3 termed ‘buckling length’.

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DIN 18800 Part 2

Method

MKi,y design buckling resistance moment according to elastic theory from My without coexistent axial force

non-dimensional slenderness in bending

internal forces resistances and moments

according to XM reduction factor for lateral torsional buckling

Note 1. Where cross sections are non-uniform or axial forces variable, (E. I ) , NKi and SK shall be determined for the point in the member for which the ultimate 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, ila, for steel of thickness 40mm and less shall be as follows: 92,9 for ~t 37 where fy,k = 240 N/mm2, and 75,9 for St 52 where fy,k = 360 N/mm2.

Note 3. Calculations of in-plane slenderness ratios shall be made using as the values Of f y , ( E . 1). NKi and MKi asspecifiedinitems116and117eithertheircharac- teristic values or their design values throughout.

Note4. V K ~ shall beof thesame magnitude for all members making up a non-sway frame.

Note 5. Where cross sections are non-uniform or internal forces and moments variable, M K ~ shall be calculated for the point for which the ultimate limit state analysis 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 7

of DIN 18800 Fart 1. Thus, the ultimate limit state analysis shall be carried out taking YM to be equal to 1,l both for the yield strength and for stiffnesses (e.g. E . T , E - A , G -AS and S).

1.4 Ultimate limit state 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 114 and clause 2); - internal forces and moments (cf. items 115 and 116); - the effects of deformations (cf. item 1 1 6); - slip (cf. item 118); - the structural contribution of cross sections (cf. item

1 1 9); - deductions in cross-sectional area for holes (ci. item

120). As a simplification, lateral buckling and lateral torsional buckling may be checked separately, first carrying out the analysis for lateral buckling and then that for lateral torsional buckling whereby, in the latter case, members shall be notionally singled out of the structural system and subjected 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 to 7 may 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 I theory Elastic theory

Elastic Plastic

theory theory

Note 1. Details relating to elasto-plastic analysis are not provided in this standard (cf. [i]), though this is permitted in principle.

Note 2. In table 11 of DIN 18800 Part 1, the generic term ‘stresses’ is used instead of ‘internal forces and moments due to actions’.

Note 3. The conditions of restraint assumed when individual members are notionally singled out of the structural system shall be taken into account when verifying lateral torsional buckling.

Note 4. Simplified methods substituting those set out in clauses 3 and 4 are listed in table 2.

(11 3) Material requirements The materials used shall be of sufficient plastic capacity. Calculations may be based on assumptions of linear elastic-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 plastic capacity.

(1 14) Imperfections Reasonable assumptions (e.g. as outlined in clause 2) shall be made in order to take into account the effects of geometrical 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. Subclauses 7.2.1 and 7.2.2 of DIN 18800 Part 1 spec-

ify rules for calculating design values of actions.

(116) Effects of structural deformations Calculations of internal forces and moments usually make allowance for deformation effects on equilibrium (according to second order theory), using as the design stiffness values the characteristic stiffnesses obtained by dividing the nominal characteristics of cross section and the characteristic 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. Simplified ultimate limit state analyses

Lateral buckling

Lateral buckling

Internal forces and moments

4.3 31 to

4.3 38

Solid members I I I

I I Bu i l t -uprmbers

I 10 I N + M ,

Simplified analyses as in Failure 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 according to second order theory, for example, the member characteristic,s,and the distribution factor, ~ j - ~ i . shall 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 1 when deciding whether to base calculations on second order theory.

Note 3. Deformations also occur as a result of joint ductility.

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 actions multiplied by YM

As a departure from the specifications of items 115 and 11 6, internal forces and moments and deformations may also be calculated using the designvalues of actions multiplied bya partial safetyfactoryM of l,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) for the design resistances (denoted by subscript d) in the equations in clauses 3 to 7.

Note 1. Calculations of e and v ~ i shall be made, for example, using the characteristic stiffness, (E. I)k.

Note2. The alternative procedure set out in this item is especiallysuitable forthe global analyses described in clauses 5,6 and 7 but may also be used by analogy in clauses 3 and 4, giving the same results as would be obtained if yM were assigned to the resistance.To preclude the risk of confusion, it shall be stated explicitly in the analysis that this alternative procedure has been used.

Note 3. See subclause 7.3.1 of DIN 18800 Part 1 for resistance parameters.

(E * I)d.

(118) Slip Account shall be taken of slip in shear bolt or preloaded shear bolt connections in members and frames susceptible to 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 taken into consideration, their geometry shall be such that the grenz (blt) and grenz (dit) values specified in DIN 18 800 Part 1 are complied with. If,for thin-walled members,these values are not complied with, the analyses shall be of lateral buckling with 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(blt) values differ according to the

method of analysis selected (see table 1).The grenz (blt) values for individual parts of plane cross sec- tionsare given in tables12,13,15and 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 determining internal 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 to elastic theory shows the structure to 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 specifications of item 117 being applied by analogy. At isolated points, the failure criterion may be 10% higher than design yield strength (cf. item 749 of DIN 18800 Part 1). The internal forces and moments (taking due consideration 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 plastification in cross sections with the possibility of plastic hinges with full torsional restraint at one or more pointS.This permits the plastic capacityof the cross sections to be fully utilized, but not that of the structure.

Note 3. The analysis shall be made using interaction equations (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 by superimposing those internal forces due to actions which result in moments My and 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 plastic shape coefficient,apl,associated with an axis of bending is greater than 1,25 and the principles 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 section shall be reduced bya factor equal to 1,25/aPl.The same principle shall be applied to each of the two moments in biaxial bending if apl,y is greater than 1,25 or apl,z is greater than 1.25. Note. Instead of reducing the resistance moment, the

actual moment may be increased by a factor equal to api/1,25.

1.4.3 Ultimate limit state analysis by plastic hinge theory (124) The loadbearing capacitymay be deemed adequate if an analysis according to plastic hinge theory shows internal 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 the structure is in equilibrium. Item 123 gives information on limiting the plastic shape coefficient. Note. Interaction equations are given in tables 16 and 17 of

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 imperfections shall be assumed, a distinction being made between initial bow (see subclause 2.2) and sway imperfections (see subclause 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, equivalent geometrical imperfections also cover the effect on the mean ultimate load of residual stresses as a result of rolling, welding and straightening proce- dures, material inhomogeneities and the spread of plastic zones. Other possible factors which may affect the ultimate load, such as ductility of fasteners, frame corners and foundations, or shear deformations are not covered.

In the elastic-elastic method, only two-thirds the values specified forthe equivalent imperfections in subclauses2.2 and 2.3 need be assumed. Ultimate limit state analyses of built-up members as specified in subclause 4.3 shall, however, always be made using the full bow imperfection 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 on comparisons of ultimate loads obtained empirically or by calculation, which also justify the value of bow imperfection stated in line 5 of table 3 (cf. Note under item 402).

The equivalent imperfections are already included in the simplified 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 associated 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 imperfections need only be assumed with DO or WO in each direction in which buckling will occur. Where lateral buckling occurs as a result of biaxial bending with coexistent axial force, equivalent imperfections need only be assumed for the direction in which buckling will occur with the member in axial compression. In the case of lateral torsional buckling, a bow imperfection equal to 0,5 DO (cf. table 3) may be assumed.

(203) Imperfections in special applications Where provisions for special applications are made in other relevant standards,with specifications deviating from those given in this standard, such specifications shall form the basis 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-sway frames and members as specified in item 207, shall generally be assumed to have the initial bow imperfections given in figure 2 and table 3.

-t LYJ2 "o I "0

Figure 2. Initial bow imperfections of member in the form of a quadratic parabola or sine half wave



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DIN 18800 Part 2

Bow imperfections need not be assumed if members satisfy the 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 DIN 18 800 Part 1 are met, reductions in the sway imperfections may be assumed.

Built-up members, with analysis as in subclause 4.3

Type of member

1

2

-

Solid member, of cross section with following buckling curve

a

b

imperfection, WO? u0

11300 t 11250

3 1 I 11200

4 1 I 11150

11500

Note. See table 23 for bow imperfections for arch beams.

Figure 3. Equivalent stabilizing force for bow imperfections as shown in figure 2 (assuming equilibrium)

Figure 4. Assumptions for bow imperfections (examples)

2.3 Sway imperfections (205) Assumptions Sway imperfections as in figure 5 shall be assumed to 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 , ~ . the sway imperfection of the member or frame.

Figure 5. 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:

1 po = - r1 r2

200

b) built-up members as in figures 20 and 21 and subclause 4.3:

(2) 1

po = - r l . r2 400 where

r1 = is a reduction factor applied to members or frames, where 1, the length of the member, L, or frame, L,, having the most adverse effect on the stress under consideration, is greater than 5 m;

r 2 = 1 ( í + t ) is a reduction factor allowing for IZ independent causes of sway imperfection of members or frames.

2

Calculations of 12 for frames may generally assume n to be the number of columns per storey in the plane under consideration. Not included are columns subjected to minor axial 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 length of columns,L, shall be substituted for the length of the column in that storey for calculation of Il. In the other storeys, the height of the structure,L,, may be substituted for I (cf. figure 6).

Note 2. Allowance for sway imperfections may also be made by assuming equivalent horizontal forces.



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DIN 18800 Part 2

1

200

1

200

100.2 = rp -with n = 2

po,~ = r 2 -with n = 4

E rn Po.1 970,l

VI < -

21 Vo.1

4 - 970 = r1Zö 1

970=r1 Töö I!! Single

member

fTfl

%.2

970?2 ' 970.2

%*2 (P0.2

-_ u \ V I

\ " Variant

I 1 2oo P0,2 = r 2 - rl n = 2 POSI = r2 - 200

Figure 6. Initial sway imperfections in frames (examples)

Figure 7. Equivalent horizontal forces substituting initial sway imperfection 100 (assuming equilibrium)

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

Note 4. The reduction factorr2 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 of bracing systems shall be as those for the columns of sway beam-and-column type frames. The same applies 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 imperfections after deformation and have a member characteristic, &, of more than 1,6, shall be assumed with both initial sway and bow imperfections in the most unfavourable direction.

Figure 8. Assumption of initial bow and coexistent initial sway imperfections (examples)

3 Solid members 3.1 General (301) Scope The analyses specified in subclauses 3.2 to 3.5 apply for individual members and frame members which are notionally singled out of the system and considered in isolation forthe purposes of the analysis. Lateral buckling and lateral torsional 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 subclauses 3.2 to 3.5 already includes both types of imperfection and second order effects, the initial forces and momentsfromfirst ordertheoryshall betaken asa basisfor calculations. Note 1. In the literature, the combination of equations (3),

(241, (28) and (29) is referred to as first order elastic analysis with sway-mode effective length (equivalent member method, for short).

Note 2. Subclauses 3.4.2.2,3.5.1 and 5.3.2.3 shall be taken 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 considered in isolation shall be analysed for lateral torsional buckling.Their end moments may require to be determined by 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 for the following: - hollow sections: - members with sufficient lateral or torsional restraint; - members designed to be in bending, providedthat their

non-dimensional slenderness in bending, AM, is not more than 0,4.

Note. See subclause 3.3.2 for verification of sufficient restraint.

a b C d

0.21 0,34 0,49 0,76

3.2 Design axial compression 3.2.1 Lateral buckling (304) Analysis The ultimate limit state analysis shall be made forthe direction in which buckling will take place, using equation (3).

5 1 (3)

The reduction factor x (¡.e. xy or x,) shall be obtained by means of equations (4a) to (4 c) as a function of the non- dimensional slenderness in compression,AK,and the buckling curve for the particular cross section, taken from table 5.

N

x ~ Np1,d

AK 5 0,2 : x = 1

1

k + i q AK >0,2 : x =

k = 0,5 [I + a (XK - 0,2) + nK] as a simplification, in cases where AK > 3,O:

1 x = -

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 in figure 9, and figures 27 and 29 may provide assist- ance in other cases. If, in certain cases, the load on the member changes direction when this moves laterally,this factor shall be taken into consideration when determining the effective length (e.g.with the aid of figures 36 to 38).

i" I" i" IN

SK ß= 1,0 2,O D,il 0,5

Figure 9. Effective lengths of single members of uniform cross section (examples)

Note 2. Reference shall be made to the literature (e.g. [2]) for the use of equations (4 a) to (4 c).

(305) Further provisions for non-uniform cross sections and variable axial forces

Where equation (3) is 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 man M,l (6)

3.2.2 Lateral torsional buckling (306) Members of uniform cross section with anytype of end support not permitting horizontal displacement, subject to constant -¡al force shall be analysed as specified in subclause 3.2.1.1~ shall 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 ultimate limit state analysis with respect to lateral torsional buckling. 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 as specified in subclause 3.3.4 for bending about one axis, except in cases where bending is about the z-axis or the conditions outlined in subclause 3.3.2 or 3.3.3 are met.



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DIN 18800 Part 2

Table 5. Buckling curves

1 2 3

Buckling about axis

Buckling curve Type of cross section

Hollow sections

a Y - Y 2 - 2

z Hot rolled

Y - Y 2 - 2

b Cold formed Z

Welded box sections

eN@i Y - Y 2 - 2

b

Thick welds and

h,lty < 30 Y - Y 2 - 2

C

Rolled I sections

hlb > 1.2; t s 40 mrn Y - Y 2 - 2

a

b

hlb > 1.2; 40 e t 5 80 rnm

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

b

C

Y - Y 2 - 2

Y - Y

2 - 2 t>80mrn d

Welded I sections b

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.



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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 sufficient lateral restraint if the thickness of the masonry is not less than 0.3 times the height of cross section of the member.

Masonry, 2

Compression flange

Figure 11. Lateral restraint (masonry bracing)

If trapezoidal sheeting to DIN 18 807is connected to beams and the condition expressed by equation (7) is met, the beam at the point of connection may be regarded as being laterally restrained in the plane of the sheeting.

Tt2

12 + GIT + EI, - 0,25

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. S shall be substituted for S. Note. Equation (7) may also be used to determine the lateral

stability of beam flanges used in combination with types 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 dimensions as for rolled sections complying with the DIN 1025 standards series shall be considered as being torsionally restrained (¡.e. due to their axes of rotation being restrained) 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 if the beam is free to move laterally,orfrom column 3of table 6 if 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.



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DIN 18800 Part 2

Bolting to Position of profile

top bottom TOP Bottom flange flange

Line

Note 2. When determining the actual effective torsional restraint,cb,k, any deformations at the point of connection between the supported beam and the supporting 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 obtained

by means of equation (10) from the bending stiffness of the supporting member (a), assuming a rigid connection:

Bolt spacing, Washer diameter, inmm in C'A,k7 kNmim

b,') 1 2 b,')

(1 O)

where k is equal to 2 in the case of single-

span or two-span beams or 4 in the case of continuous beams with three or more spans:

( E . r a ) k is the bending stiffness of the supporting member;

a is the span of the supporting member;

CfiA,k is the torsional restraint due to deformation of the connection, that of trapezoidal sheeting being obtained by means of equation (11 a) or (11 b), substituting ?@&k from table 7;

vorh b 1 O0

with - I 1,251

vorh b 1 O0

with 1,25 - I 2,o

where vorh b is the actual flange width of the

beam, in mm.

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

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

Note 3. Instead of applying equation (81, the actual effective torsional restraint, C@,k, may also be considered when determining the ideal design buckling resistance moment, M K ~ , ~ , the check then being carried out as specified in subclause 3.3.4.

Table Z Characteristic torsional 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

I

max bt3), in mm

40

40

40

40

120

120

40

40

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

3) bt - flange 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,O mm thick, with a vulcanized neoprene backing.



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DIN 18800 Part 2

Axial force diagram

i Ip"

kC

Figure 12. Torsional restraint (example)

I I I I I u I

Figure 13. 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 a compression flange which is laterally restrained at a number 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:

0,843 M~ 5 1

' Mpl,y,d

where

My is the maximum moment;

x isareductionfactorasafunctionofbuckling c_urvec or d, obtained by means of equation (4), for A. from equation (13), buckling curve d being selected for beams otherthan the rolled beams in line 1 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 fori,,g the radius of gyration of the whole section, i,.

3.3.4 Lateral torsional buckling

(311) The ultimate limit state analysis of I beams, channels and C sections not designed for torsion shall be by means 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 with 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 ,



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DIN 18800 Part 2

Line Moment diagram

Table 9. Beam coefficient, n

r I Type of section

Rolled

Welded

Castellated

Notched

Haunched *)

-r

min h max h

2 0,25

n

2.5

2.0

min h max h

0,7 + 1.8

k, When flanges are connected to webs by welding, n shall be further multiplied by a factor of 0,8.

Note 1. Calculation of äM is only possible where the ideal design buckling resistance moment, M K ~ , ~ , is known (cf. [5] and [6]). Equation (19) or (20) may be applied for beams of doubly symmetrical uniform cross section.

MK~, , , = C * NK~,,, (11,' + 0,25 Z; + 0.5 zP) (19)

where

< NK~,,, is equal to n2. E . Izll';

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 - 0,77 I I - pmaxM -1cp1 I 1 maxM

Calculations of beams not more than 60cm in height may be simplified by substituting equation (20) for equation (19).

1,32 b * t ( E * I,) 1 * h 2 MKi,y = -a V I

16) Figure 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, is equal to 2 5

Note 3. XM may be assumed to be equal to unityfor beams not more than 60cm in depth (see figure 15) and of uniform cross section provided that they satisfy equation (21):

be t 240 1 5- 200 -

h fy,k

f y , k being expressed in N/mm2.

Note 4. Coefficient n allows for the effect of residual stresses and initial deformations on the service load but not the effect of the support conditions (these being allowed for by MKi,y).

3.4 Bending about one axis with coexistent axial force

3.4.1 Members subjected t o minor axial forces (312) Members subjected to only minor axial forces and meeting the condition expressed by equation (22) may be analysed for bending without coexistent axial force, as specified in subclause 3.3.

N < 0,l (22)

X * Npl,d

3.4.2 Lateral buckling 3.4.2.1 Simplified method of analysis (313) The analysis for lateral buckling of members pin- jointed on both sidesand subject to in-plane lateral loading



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DIN 18800 Part 2

in the form of a concentrated or line load and with a maximum moment, M, according to first order theory, may be analysed by means of equation (3), while substituting in equation (4 b) k from equation (23).

+ a (& - 0,2) + 3; + -

Item 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.

+- 'DI e + An < 1 (24) N

. Np1.d Mpl ,d where x E a 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 for members of uniform cross section whose end support conditions do not permit lateral displacement and 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 N An isequal to- -- x2 * 36,

x ' N p i , d (1 x - N p l , d ) but not more than 0,l.

Item 123 shall be taken into account when calculating Mpl,d.

For doublysymmetrical cross sections with a web compris- ing at least 18Yo of the'total area of cross section, Mpl,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 An either 0,25 x2 .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 the internal forces

and 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 variable, 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~, assumed as acting at these points. In addition, equations (5) and (6) in item 305 shall be met.

(317) Rigid connections In the absence of a more rigorous treatment, rigid connections shall be calculated substituting forthe actual moment, M , the moment in the perfectly plastic state, Mp1,d.

Note. If a more detailed analysis is required, the 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 subjected t 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 yield strength of cross sections not in compression shall not be less than that of those in compression.

M

5 1 d

1,15 1--

VKi with V K ~ > 1,15

Note. A portion of a member not in compression could bea beam connected to 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 information shall be taken from the literature

k g . VI). 3.4.3 Lateral torsional buckling (320) Channels and C sections, and I sections of monosymmetric or doubly symmetrical cross section, exhibiting uniform 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 < 1 N +

xz ' Npl, d xM ' Mpl,y, d The following notation applies in addition to that given in subclause 3.3.4. xz is a reduction factor from equation (4), substituting

AK,z for buckling perpendicular to the z-axis, where

& z is equal to E - the non-dimensional slenderness

associated with axial force; N K ~ is the axial force underthe smallest bifurcation load

associated with buckling perpendicularto the z-axis or 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 = 0,15 jK,z. B M , ~ -O,%, with a maximum of 0,9

where & M , ~ is the moment factor associated with lat-

eral torsional buckling, from column 3 of table 11, taking intoaccount moment diagram My.

Note 1. Due regard shall be taken, particularly in the case of channels and C sections, of the fact that this analysis does not take account of design torsion.

Note 2. Tsections are not covered by the specifications of this subclause.

Note 3. A k, value of unity gives a conservative approximation.

Note 4. The torsional bending load plays a major role, for example, in members subject to torsional restraint.

k,

N

xz * Npl , d ay. but not more than unity,



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DIN 18800 Part 2

3.5 Biaxial bending with or without coexistent axial force

3.5.1 Lateral buckling

(321) Method of analysis 1 The ultimate limit state analysis shall be made applying equation (28):

k, I 1 (28) N MY M,

x *Npl,d Mpl,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 and M , are the maximum moments in first order theory (disregarding imperfections); is a coefficient taking -into account moment diagram My and AK,y It shall be calculated as follows:

kY

Table 11. Moment factors

1

Moment diagram

3 d moments

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

Moments from in-plane ateral loading

flQ

Moments from in-plane lateral loading with end moments

2

Moment factors, ßm.

for lateral buckling

&,,, ,,, = 0,66 + 0,44 y

1 but not below 1 - -

VKi'

with a minimum of 0,44.

N k , = 1 - ay, with a maximum

"Y NpLd of 1,5 where

ay = & y ( 2 ß ~ , ~ - 4) + - 1). With a maximum of 0,8 where ßM,,and ßM,z are the moment factors

ßM associated with lateral torsional buckling, from column 3 of table 11; taking into account moment diagrams My and M,;

apl,y and ctPl,, are plastic shape coefficients associated with moment M y or M,. (Item 123 is not applicable here.)

3

Moment factors,

for lateral torsional buckling ßMs

= 1,8 - 0,7 y

MQ = 1 max M 1 from in-plane lateral loading only

Imax MI where no alternating 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 diagram M, and a K , p It shall be calculated as follows:

k,=1 - a,, with a maximum xz ' NpLd of 1,5

where

a, = &,, (2ßM,z - 4) + (spi,, - 1). with a

is design moment M, in the perfectly plastic 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 one

axis and coexistent axial force, x shall be the reduction factor for the plane of bending under consideration.

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,,overtheeffective 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) N ßm,, . M y ßm,z * M, + x . Npi, d Mpl,y,d ky + Mpl,z,d where x=rnin (xy, xJ 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 xy;

c, = 1

CY -

My and M,

fim,, and fim,,

are the maximum moments in first order theory (disregarding imperfections); are the moment factors for lateral buckling, from line 2 of table 11, taking into account moment diagram My or M,.

Item 314shall be referred to fOrAn,SUbStitUting~KaSSOCiat- ed with x , the other items of subclause 3.4.2.2 applying 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 is substituted for x .

3.5.2 Lateral torsional buckling (323) Monosymmetric or doubly symmetrical I sections with relative dimensions as for those of rolled sections,subject 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. A k, 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 axis*) Built-up members having cross sections with one material axis shall be dealt with as solid members as specified in clause 3 when calculating lateral displacement perpendicular to the material axis. For compression and design bending moment, My, this only applies when there is no design bending moment M,.

(402) Buckling perpendicular t o the void axis **) Calculation of lateral displacement perpendicular to the void axis may be bythe equivalent method,in which built-up members of uniform cross section are dealt with as solid members,with both deformations due to moments and those occurring as a result of transverse forces being taken into consideration. In this method, the design of each component shall be based on the global analysisofthe total internal forces and moments present (see subclauses 4.3.2 and 4.3.3). Note. Frames may also be analysed on the basis of all of

their components. Analysis by the equivalent member method assuming solid members is specified for battened members with two chords. The literature shall 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 two 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 18800 Part 2

4.2 Common notation (404) 1 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 , G second order moment of area of a chord cross

section about the centroidal axis parallel to the z-axis;

Y s distance of the centroid of each component cross section from the z-axis;

I , = AG , y ; + I z , ~ ) second order moment of area of the gross cross section about the z-axis (assuming rigid connection of components, providing shear resistance);

effective length of equivalent member, disregarding any deformation due to transverse forces;

SK,Z AK,z = - slenderness 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);

sK,z

17

Table 12. Correction, v, for battened members

77

1 I

> 150 O

Figure 18. Laced and battened members (examples)

1; AG .y; + 17. I z , ~ ) design second order moment of area of the gross cross section of battened members;

1; = 2 (AG - y ; ) design second order moment of area of the gross cross section of laced members;

section modulus of the gross cross section, relative to the centroidal axis of the outermost chord;

Sz*,d design shear stiffness of the equivalent member.

Note 1. The shear stiffness corresponds to the transverse force 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 n2/12 in order to exclude failure of single panels solely due to shear.

w;=- I L YS

4.3 Buckling perpendicular to void axis 4.3.1 Analysis of member (405) Analysis of a member shall be made taking into consideration the conditions of restraint. The internal forces and moments in a member designed to be in axial compression, with its ends nominally pinned to prevent lateral displacement will be as follows:

at member mid-point: Mz = (31) N 00

N 1 --

NKi, z, d where

1 (32)

1 +- Tt' ~ ( E I;)d s;,d

12 NKi, z, d =

n - M z at member end: max V, = -

1 (33)

Note. The literature (e.g. [IO]) shall be consulted for internal 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 moments acting throughout the member not resistant to shear gives an axial force,NG, in the chord undermaximum stressequal to the following:

NG shall be used for analysis of the part of a chord as specified in subclause 3.2, assuming 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 taken 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 in subclause 3.4 for laced members as shown in columns 4 and 5 of table 13 where a is subject to transverse loading.



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DIN 18800 Part 2

1 2

4.3.2.2 Lacing systems

(407) The axial forces of web members making up lacing systems 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 considering a member in axial compression, shall be obtained 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 calculation 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

mar Vy a MG = - -

r 2 end moment,

a

rnax Vy transverse force, VG = ~

r (37)

(38)

where XB is the position of the batten

In the case of monosymmetric chord cross sections, the resistance moment, M , at the ends of the part of the chord shall be obtained from the mean of the moments f Mpl,NG derived from interaction equation (38).

Note 1. The plastic design capacity of the chord cross section as obtained from the interaction equations may be utilized (cf. [9] and [lo]), the transverse force, VG, normally being neglected.

on the chord.

Note 2. The moments of resistance, M,!,N~, occurring in the chords at their connections with battens are of different magnitude owing to their different directions. Failure of a panel does not occur until all M p ~ , ~ G values have been fully utilized (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 . sin2 a (m = number of braces normal to void axis)

a

z

y + y r:r z

a

6

Battened members

a

The effective lengths,sK,l,in columns 1 and 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 geometrical 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 ductilityand slip of fasteners and on eccentricityat the connections between web members in laced members is given in the literature (e.9. [9]).



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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 battened members

Structural model

Moment diagram in the connection due to shear, T

Shear, T, in the connection

2

This also applies for closely spaced built-up battened members as shown in figures 19,20and 21.The moments in the centroids of batten connections shall be taken into account. If packing plates are used to connect the main components in built-up battened members as shown in figures 19 and 21, it is sufficient to design the connection for resistance to the actual shear.

4.4 Closely spaced built-up battened members (410) Cross sections with one void axis Built-up members with cross sections as shown in figure 19 may also be treated as solid members as set out in clause 3 when calculating lateral displacement normal to the void axis, provided that either of the following conditions is satisfied: a) battens or packing plates positioned as specified in

subclause 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 clear spacing of main components not oronlyslightly greater than the thickness of the gusset

Continuity of packing may be taken into consideration when calculating the second order moment of area. When determining the area of cross section,A, this only applies when 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 compressive force in the battened member.

(41 1) Star-battened angle members Built-up members. consisting of two star-battened angle members need only be checked for lateral displacement perpendicular to the.material axis (figure 20) by the following 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, the radius of gyration, io, of the gross cross section relating to the centroidal axis parallel to the longer leg:

. io lY = -

1.15

a) r = 2 b) r = 2

Figure 20. Star-battened angle members

Consecutive battens may be in corresponding or mutually opposed order. Shear may be determined as specified in item 410.

Note. According to item 503, the effective lengths of diagonals or verticals in triangulated frames differ, de- pending on whether lateral displacement in or perpendicular to the plane of the frame is being considered.

(412) Cross sections with two void axes Where built-up members as shown in figure 21 consist of main components with a clear spacing not or only slightly greater than the thickness of the gusset,the specifications applying to the built-up members in figure 19 shall be applied by analogy to the two void axes.

r = 4

Figure 21. Closely-spaced built-up member with two void axes



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DIN 18800 Part 2

4.5 Structural detailing (413) Retention of cross-sectional shape Where member cross sections have two void axes, the rectangular cross-sectional shape shall be retained by means of cross-stiffening. Note. Cross-stiffening may take the form of bracing,plates

or frames.

(414) Arrangement of battens and packing plates Battened members shall be connected at the ends by battens.This also applies to laced members unless cross bracing 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 possible, the 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) shall be satisfied:

a

i1 - 5 70 (41)

5 Frames 5.1 Triangulated frames 5.1.1 General (501) Calculation of forces in triangulated

frame members The forces acting in the members making up a triangulated frame may be calculated assuming nominally pinned member ends.Secondary stresses as a result of nodes may be disregarded. Where the cross sections of compression chords are non- uniform over their length,any load eccentricity in individual members may be disregarded if 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 in clause 3,4 or 7.

5.1.2 Effective lengths of frame members designed to resist cornpression

5.1.2.1 General (503) Rigidly connected members In the absence of a more rigorous treatment, the effective length, 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 providing non-rigid connection, is a function of the structural detailing involved. Noie. The effective length, S K , ~ , of triangulated frame

members as shown in figure 22 for the sway mode in the perpendicular plane may be determined by means of the diagrams in figure 27.

(505) Members with one end allowing lateral dlsplacement 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 effective length in that plane may be determined as 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'

/ /

Ib( Vertical 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 supported by another triangulated frame member

(506) Connection at intersection At intersections, members shall be connected directly or via a gusset. if both members are continuous, the connection between them shall be designed to withstand a force acting in the perpondicu!ar plane equal to 10% of the greater compressive force.

(507) In-plane effective length The effective length for the sway mode in the plane of the triangulated 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 forthe sway mode in the perpendicular plane appropriate to the structural detailing involved may be taken from table 15.



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DIN 18800 Part 2

Table 15. Out-of plane effective lengths of triangulated frame members of uniform cross section in the perpendicular plane

1 2 I 3

3 z - 1 1 - _ ~

4 N - 1 ,

I , 13

I . 1: 1 + -

SK = 1

but not less than 0,5 Z

N - 1,

1 +- I , 13 Y - I . 1:

SK = 1

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

Il vhere - N . 4 z - 1

i r where the following applies:

Dut not less than 0.5 1

N



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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 truss members with elastic support at mid-length for the sway mode may be obtained by means of equation (44):

1 2

O<A,<112

fi<3,<3,0

;2# = 0,35 + 0,753 AK

n# = 0,50 + 0,646 AK

- I a K = - non-dimensional slenderness of solid

il * Aa member z system length il minimum radius of gyration of angle

cross section

(44)

where

1 is the system length;

N

c d

is the maximum compressive force acting in the member ( N I or N2);

is the frame stiffness with respect to lateral displacement of the points of connection of solid members and of columns forming part of the subframe in the perpendicularplane,this being equal to not less than 4 NIL

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. by means 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 in bending, 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 deemed applicable if the deformations due to axial force of the columns of frames and bracing systems are negligible, this being the case when equation (45) is met:

(45) where E . I is the bending stiffness, S is the storey stiffness, L is the overall height (see figure 25), of the bracing system or multistorey frame.

If E -1 or S varies over a number of storeys, their mean may be used. I may be approximated using equation (46):

continuous beams

E * I > 2,5 S. L2

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

Ali

L B

Figure 25. Criteria for calculation of I by means of

It shall be presumed throughout that for the column of frames 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 resulting from the bending moment.

Note 2. Equations for calculation of the stiffness of bracing systems and of multistorey frames are given in table 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 shall be regarded as non-sway provided that the stiffness of the bracing system,SAusst,is at least five times that of the frame, Sb, in the storey under consideration, ¡.e.

By a simplified method, equation (47) need only be applied to the lowest storey if the stiffness conditions there are not considerably different from those of the other storeys. Note. Examples of stiffening elements are wall panels and

bracing. Their stiffness may be taken from table 17.

SAusst 2 5 SRa (47)



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DIN 18800 Pari 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, with 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 diagonal

effective)

n L

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 imperfections for both stiffening system and frame.

(515) Imperfections Initial sway imperfections, qo, as specified in subclause 2.3 shall 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 in

the 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) is 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 ultimate limit state analysis of frames and continuous 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 subclause 3.4.2.2, the moment factor,&,for lateral buckling, taken from column 2 of table 11 may be used to calculate the moment components from transverse loads on beams. When analysing beams by means of equation (26), the maximum bending moment may be reduced by multiplying by the factor (1 -0,8/q~i) provided 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 given in [ll].

5.3 Sway frames and continuous beams subject to lateral displacement

5.3.1 Negligible deformations due to axial force

(516) Item 51 1 shall apply in the cases where the deformations due to axial force are negligible.

5.3.2 Plane sway frames Note. The use of bolts or welding for unstiffened 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 of equal 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 the rth storey.

In the above, the stiffness S, shall be obtained by means of equations (52) to (54), using the notation and values given in figure 28. In the first storey (where r =l), S, shall be as follows, de- pending on the conditions of restraint at the column bases: rigidly connected:



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DIN 18800 Part 2

Special design situations

- $--- Cu= o

1 c, =

2 11 1s 1 + - - 3 Is 4

In all three cases:

e 1 c,, =

2 12 15 I+- -

Figure

Nominally pinned

t k - - E u) L

5 a c O c .- 3 Y

3 o O

O

L

O

Rigid

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

SK = BIs

Division of non-sway frame into subframes with only one column, for application of diagram below q K i =

K i + K ë = K6

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

(Resolution of K3 and K6 may be freely selected.)

27. Diagram to determine the distribution factor, q ~ i , and effective length, SK, for columns of non-sway frames where seam is not greater than 0,3



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Storey r + 1

l I R

& 1s

Beam r

Storey r

Figure 28. Notation and values for calculation of & d

Special design situations

DIN 18800 Part 2

Cr+l = ...

1 er = - C l5 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)- Nominally connected pinned SK = ß J ! S

~ ~ \ z E I s N,i ißk,

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

-0 Ca KO l + -

Ks + 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



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DIN 18800 Part 2

nominally pinned:

(53)

In the other storeys:

where Sr,Ausst,d is the stiffness of any stiffening elements in the

r t h storey. If an analysis of external horizontal forces by first order theory is already provided, q ~ i , , may also be obtained by means of equation (55).

(55) VF

qKi,r = - pr * Nr

where

VF is the transverse force from external horizontal loads 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 imperfections p~ specified in items 729 and 730 of DIN 18800 Part 1 shall be taken into account.

Note 2. Alternatively, q K i , r may be determined with the aid of figure 29. N K i , r , d assumed as being equal to S,d/1,2 gives a conservative estimation of the design bifurcation load; 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 in item 521 or 522.

(521) Transverse force in beam-and-column type frames Where the member characteristic, E, of beam-and-column type frames is less than 1,6, higher transverse forces in the storey, V,, shall be used, to be obtained by means of equation (56).

(56) where VF is the transverse force in the storey due to external

horizontal loads only; N , is the total vertical load transmitted within the r th

storey; 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 initial sway imperfections at the base or top of columns, the angles of rotation, Q, (see figure 30), being unknown, the simplified second order method gives an only slightly different result than the first order method, the additional term 1,2 pr + N , giving a decrease in the principal diagonal terms, and po N , an increase in the load terms, of the equilibrium equations. Thus calculations are onlyslightly more complex than by first ordertheory.

(522) Approximate calculation of transverse force in beam-and-column type frames

If equation (57) is met by all storeys, equation (58) may be substituted for (56) to obtain V, by approximation.

V, = V,H + 90 . N r + 1,2 9,. . N I

I v, = - (VT + Co * NI) 1

1 - - vKi , r

5.3.2.3 Analysis by equivalent member method (523) Global method The ultimate limit state analysis for sway frames may be carried out byanalysing each member separately,as specified in clause 3, but 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 effective lengths of members. Note. Effective lengths may be determined using figure 29,

or using figures 36 to 38 in cases where compressive forces are liable to change direction.

(524) Cross sections not in compression Analysis by means of equation (26) for 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 including nominally pinned columns shall be calculated with an additional equivalent load, VO (obtained by means of equation (59) and illustrated in figure 30), in order to take into account initial sway imperfections.

(59) where p0,i is as specified in item 205.

VO = 1 (Pi . p0.i)

VO = XPi V0.i 90 from 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 type frames as specified in subclauses 5.3.2.1, 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 imperfections from subclause 2.3 are assumed and the columns in each storey satisfy equation (60).

(60)

(61 1

Vr prs loN,

where v, = v,H + 80. N ,



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DIN 18800 Part 2

where V r is the transverse force in the storey due to external

horizontal loads only; N , is the total vertical load transmitted within the r th storey; 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

E or 1

subject to axial compression when considering out-of- plane buckling.

Note. In the case of bridges, elastic support is usually provided 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 positioned between two subframes may be averaged to give a constant value, the chord cross section being taken to include the chords plus one fifth of the web.

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

Trusses and solid web beams with subframes in perpendicular plane

&+i- 1

141 I

Figure 31. Notation used in equation (62)

If the height of nominally pinned columns, Z,, is not the same as the height of the frame columns, h, the vertical loads on the nominally pinned columns shall be multiplied by the factor hll, for calculation of N . Note. This specification may give very conservative results

since 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 order elastic theory as specified in subclause 5.3.2.2 assuming transverse forces in the storey as obtained by means of equation (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 method shall 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 continuous beams 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 Axial compression 6.1.1 In-plane buckling 6.1.1.1 Arches of uniform cross section (601) Analysis The ultimate limit state analysis shall be made by applying equation (3), N being the value at the springing.

Plan view

7 Y

Figure 32. Arch axes

Note. Figure 33 shows buckling coefficients obtained by means of equation (63) for various types of symmetrical arch systems, all of which assume that deformations due to axial forces can be disregarded.

(63)

where sK is the effective length and s half of the beam length, /? is used to calculate the axial force at the springing, N K ~ , under the smallest bifurcation load (see equation 64):

SK ß = - S

I * \2



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DIN 18800 Part 2

Table 19. Spring stiffness of triangulated structures without verticals

1

Typical Warren truss 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 only resistant to bending, adjacent bottom chords only resistant to torsion.

A + B - 2 D Spring stiffness: C - 2 ( E * 1u)d

d - ~ . ~ - ~

h2 - 1, d 3 . I , b' - u B = ~ +'+- Ur Idr 3

1

6 D = - 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 ITr are the torsion constants of the adjacent bottom chord members.

If the half-wave coefficient, rn, of 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.



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DIN 18800 Part 2

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

t ß

t ß

t ß

Antimetric buckling

Antimetric buckling

t P

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 to 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)



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DIN 18800 Part 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, the ultimate limit state analysis shall be carried out using the full effective length of the arch,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 flat arches provided that equation (65) is satisfied.

and [141).

where E . A is the longitudinal stiffness; E . I , 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 this standard, and shall be calculated applying the non-linear theory using large deformations.

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 specified 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 beams without lateral restraint between springings may be carried out applying equation (3), using the in-plane slenderness ratio, AK, obtained as follows. For parabolic arches,

where i, pl

is the radius of gyration of the z-axis at the crown; is the buckling coeffcient taken from table 21 (assuming 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, covering the change in direction of the load in lateral buckling.

For one-centred arches,

with

where N K ~ , K ~ is the axial force under the smallest bifurcation

load of a one-centred arch of constant doubly symmetrical cross section with fork restraint, subject 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 be calculated by approximation, it only being necessary to take 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 from equation (69).



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DIN 18800 Part 2

5

Any transverse loads (such as wind load) shall be checked separatelytaking into account bending moments as set out in item 314.

6 7

Note 1. Buckling coefficients may be taken from the literature (cf. [15]) and figures 36 to 38 which cover loading corresponding to the arch form, not just in portai frames of arches.

Note 2. h, as featured in figures 36 to 38 shall be obtained by multiplying the averaged hanger length, h ~ , by the factor l ls in 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 10 2 Two-hinged arch

3 Rigidly connected arch 97 42 I 13

Table 21. Buckling coefficient, ß, 7 0,50 0,s

- 4

0.2

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

I, constant

with 1- t - I _ f 0,59

Table 22. Buckling coefficient, ß2

I Loading I ß 2 Notation

1 I Corresponding to arch form I 1

q =total load

q H = load component, transmitted by hangers

qst = load component, transmitted by columns

Via hangers 2 1 9H 1 - 0,351 -

4

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

9 I I

l) The deck is fixed to the arch crown.

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



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DIN 18800 Part 2

h lh , -

hlh, -

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 portal frames with columns connected by two beams of equal stiffness



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DIN 18800 Part 2

2

Form of equivalent geometrical imperfection (sinusoidal or parabolic)

In figure 39, a is the angle of the arch equal to 2 slr but not less than O or more than TE.

3

WO 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 analysed for ultimate limit state using one of the methods listed in table 1, assuming equivalent geometrical imperfections from table 23 occurring 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 equivalent imperfections.

sc ß = -

liK1 The following applies for arches in compression:

a Kt = 2,47 - (3 + 0,21 k ) -

100

+ (700 - 6 k + 0.08 k2) - (1:o)' (73)

The following applies for arches in tension: 9 58 7/58

Ki = - 0,036 + - + - 10+ k (10 + k)2

Note 1. SK may be derived from equation (63) in 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-plane buckling of arches may be carried out as specified in subclause 6.1.2.

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

tion with figure 33.

method.

(609) Laterally restrained arches with the static system as shown in figure 39 may be given a simplified treatment using equation (3) and employing the in-plane slenderness ratio, AK, obtained by means of equation (71), to determine K.

- ß . s

i, . la & = - (71)

- (0,226 - - 13,4 1,94 k +F) (i) (74)

where E . I , k = L G ' 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 I cross section may be carried out usingequation (27) and employing the in-plane slenderness ratio,iK,obtained from equation (75) to determine K .

- ß . s AK = - (75) i, . A,

where a is the angle of the arch,equal to 2 sir but not less than O

or more than T E ; 2n

(76) ß is the buckling coefficient, equal to - I'K1 where (TE2 - a 2 ) 2

Ki = (77) n2 + a2 . k where

E . I , k = - G * 1, Figure 39. Static system for laterally restrained arches

Table 23. In-plane equivalent geometrical imperfections in arches

1

Three-hinged arch in symmetrical buckling

S S S S

300 250 200 1 50

I I I I

2 Two-hinged arch, three-hinged arch, fixed-ended arch in antimetric buckling



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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 'curve geometrical imperfections (cf. table 5)

in horizontal direction (sinusoidal or parabolic) a

I

I ' ' - r 6 '

k

M K ~ , ~ , required for calcuJation of the reduction factor X M from equation (18) usingAM from item ll0,shall be obtained by means of equation (78).

where - E * I , . T C ~ C =

r 2 . a2

In equation (78), there shall be a plus sign before the root if My 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 loading of arches (611) The ultimate limit state analysis shall normally be made 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 imperfections acting in a single (i.e.the most unfavourable) direction, 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 in the state of deformation. Note. Design loading plays a significant role in arches

exposed to outdoor conditions due to the possible effect of wind acting transverse to the arch plane. In this case, the loading conditions set out in subclauses 6.1 and 6.2 are not met.

7 Straight linear members with plane thin-walled parts of cross section

7.1 General (701) Field of application This clause shall apply in cases where the grenz (bit) values for 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. The grenz (bl 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 behaviour of the member

b C d

as a whole by causing a reduction in its stiffness and a 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 its viability is given sufficient practical backing.

Note 2. In subclauses7.2 to7.6,the effect of buckling ofthe individual parts of cross section on member buckling as a whole is taken into account.

(703) Effect of shear stresses In cases where subclauses 7.2 to 7.6 are applied, shear stresses when analysing plate buckling of 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, to be determined as specified in DIN 18800 Part 3.

If equations (79) and (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 of also 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 following forms: 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) orb', issubstitutedfortheactualwidth,b,of the thin-walled part of the cross section. The resulting effective cross section is taken as the basisforcalculations.



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DIN 18 800 Part 2

Y 5 N

Table 25. Increase in bow imperfection, A W O

1 2 3 I Moment diagram

e, = centroidal shift due to positive moment en = centroidal shifi due to negative moment

4 M ; = M , + N e

a) Gross cross section b) Reduced effective cross section as a result of buckling of upper flange

Figure 40. Effective cross section (example)

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

Note 2. Provisions for the calculation of b’ orb” are made in subclauses 7.3 (elastic-elastic method) and 7.4 (elastic-plastic method). Accordingly, cross section properties A’, I’, etc. are assigned to b’, and A”, I“ to b”. Figure 40 b) shows a reduced cross section in elastic-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 in principle to members with effective cross sections, subject to the modifications specified in subclauses 7.5 and 7.6.

(706) Approximate methods The effective cross section is obtained by reducing the zone of tensile bending. If the cross section is not symmetrical about the bending axis and both positive and negative bending moments occur, the governing bending moment shall be that resulting in the smaller effective second order moment of area.This moment shall be assumed to be constant 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 using this approximate method, even though compressive stresses may occur. This approximate method is elaborated in the literature [cf.l6],with the inclusion of practical examples.

(707) Analysis of cross section The analyses shall be of the effective cross section. The reduction in cross section shall be in correlation with the direction of the actual bending moment in the bending compression zone of the member after deformation.

Note. In the absence of a design bending moment, the bending moment as a result of bow imperfections shall be used. It may prove necessary to examine both directions in the case of monosymmetric cross sections.

(708) Centroidal shift as a result of reduction in cross section

The effect of a shift, e, of the centroid in the transition from the gross (¡.e. actual) to the effective cross section shall be taken into account. For convenience, this may be done as specified in items 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 AWO from table 25. For a cross section symmetrical about the axis of bending, and assuming that a compressive stress, OD, due to the positive moment and the negative moment are of equal magnitude, ep, e, and e may also be taken to be equal.

Note. The diagrams shown in table 25 are onlyexamples of moments. Of significance is the occurrence of positive 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, this shall be increased by Apo = (e, + e,)íZ if both



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DIN 18800 Part 2

Table 26. Buckling factors, k

7,81

7,81 - 6,29 + 9,78 W 2

23,s

Type of support

Stress distribution

1,70 0,57

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

23,8 0,85

q = 1

1 > q > o

?#=O

O > W > - l

q = -1

1 2

At one end

~~~

4 I 0,43

82 W + 1,05

0,578 + 0,34

0,57 - 0,21 q + 0,07 W2

ends are restrained and moments with different signs are liable to occur here. If one of the ends is nominally pinned, ep or 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 in 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 the basis 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 width The effective width shall be determined by means of equation (81) for cases in which plates (web or flange) are supported on both sides with constant compression and equation (82) for support on only one side. The assumption of support on both sides presupposes that the supporting construction is of adequate stiffness.

b’=b for Apo g 0,673

(1 - 0,22/äp0)

APO ob

for npo > 0,673

(82) 0,7

APO b = : 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 relating to plate buckling, obtained by means of equation (83):

U

= /G ue = 189800 - , in N/mm2; (ir t is the thickness of the thin-walled part of the cross

section;



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DIN 18800 Part 2

k is the buckling factor from table 26, the edge stress ratio, y. being a function of the stress distribution in the effective cross section. Where plates are supported 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 to second 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 expressed in N/mm2.The long 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 shall be substituted for fy,d in the analyses specified in subclauses 7.5.2.1 to 7.5.2.3. Note 1. 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 stiffness of the supporting constructions for plate edges.

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

Note 3. u, shall be obtained thus:

u

5 ~ ' * E . t 2

12 b2 (1 - ,U') u, =

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

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

H

(Y2.u Effective web width with u and y = y2 2 y1

. .

JL Effective cross section

Figure 42. Determination of effective cross section of an I 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 at

national and international level, the procedure described here has been modified somewhat in

Table 27. Resolution of effective width -

m U C (u

O a m

O Q

3 v)

f c

c L

n

F a> (u C O

m O CL 0. J

W I

c

c L

-

-1 5 * 5 1 b; = Q - b - k , b > = Q + b . k ,

where Q =

1 =- [(0.97 + 0,03 W ) - (OJ6 + 0,06 tp)/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-1 sion)

P - 7 I G b -i -1 < * < l

comparison with line 3 of table 1 of DIN 18 800 Part 3 and table 12 of DIN 18 800 Part 1 in that the factorc is not applied for y equal to O but not greater than 1.

Note 2. Calculation of the e, kl and k2 values is such that the buckling factor, k , can be determined as specified in item 712.

7.4 Effective width in elastic-plastic method (T14) The effective width shall be calculated using one of equations (85) to (87).Coefficients k , and k2 and resolution of the effective width shall be as in table 28, ensuring that

(84) ZN¡= 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.



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DIN 18800 Pari 2

7.5 Lateral buckling 7.5.1 Elastic-elastic analysis (715) The ultimate limit state analysis shall be made taking UD as equal to or less than fy,d (88), where UD is the maximum compressive stress at the long edge of the thin- walled part of the cross section, calculated on the basis of the effective cross section.The long 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 assuming effective widths in bending for the compression flange and, in some cases, for the web shall be taken as a basis, the stress distribution in the web being estimated.No reduction in cross section of the tension flange is to be made. The ultimate limit state analysis shall be made applying equation (89).

Table 28. Magnitude and resolution of effective width b"

N 5 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 of the effective cross section respectively;

Amo is the eccentricity as a result of a reduction in cross-sectional area, to be calculated as set 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 into

account 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 effect of Awo is made by substituting a supplementary term in equation (91).

Note 2. Subclause 7.5.2.2 specifies an alternative method of analysis, allowance for the effect of AWO being made by inclusion of a bending moment My equal to N e Awo. In cases where this alternative method is used, the term featuring AWO shall be deleted.

(717) In addition to the analysis specified in item 716, an analysis shall be made using equation (95) on the basis of another 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 :1 (Tension) (Compression)

kl = O k2 = 11

::i

::n

f Y #

(Compression) (Tension)

Ei H Ea=yc.E j

O 1 2 ? ) & 2 0

k l = {

4,56 - I ? j ~ ~ I O 2 ?)& 2 -1

k, = 11



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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 slenderness ratio, 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 literature (cf. 1191)

for an alternative method of analysis.

(719) Elastic-elastic method The analysis of bending about one axis with coexistent axial force shall be made applying equation (24) but making the following substitutions:

wpi,d for Npl,d;

M%l,d for Mpl,d; x' for x ;

x' and & being taken from item 716;

Nb1,d = A'*fy,d (96)

(97 1

TK foräK;

where

I'

rD Mpi,d = 7 ' 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 the following 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 force

(721) The ultimate limit state analysis for biaxial bending with or without coexistent axial force may be made as specified 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 torsional buckling may be made as specified in clause 3, but with the modifications set out in items 723 to 727.

7.6.2 Axial compression (723) The calculation of lateral torsional buckling shall be in analogywith subclause 3.2.2 and as for lateral buckling as specified in subclause 7.5. When calc!lating the non- dimensional slenderness in compression,lK, the properties of the reduced cross section shall be taken into account for calculation of the axial force, NKi, under the smallest bifurcation load in the analysis of lateral torsional buckling according to elastic theory.

7.6.3 Bending about one axis without coexistent axial force

7.6.3.1 Analysis of compression chord (724) Analysis of the compression chord shall be as set out in subclause 3.3.3, but assuming k , equal to unity in

equation (13).obtaining i,, by means of equation (98) and substituting MPIJ for Mpl,y,d in equation (14).

I _.

where

IZ, g

Ab A, Note. If the elastic-plastic method is applied,

is the reduced second moment of area of the compression chord about the z-axis; is the reduced area of the compression chord; is the gross web area.

A% and M$,d shall be substituted for IL,,, A;! and Mgl,d, respectively.

7.6.3.2 Global analysis (725) Design buckling resistance moment

according to elastic theory When calculating the design buckling resistance moment, the moment red MK~ obtained by approximation by means of 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 buckling of the cross section or the relevant part of the cross 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 MK~ 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 literature (e.g. [17] and [18]).

(726) Elastic-elastic method When ca'culating the non-dimensional slenderness in bending, AM, as set out in item 110, Mbl shall be substituted for Mpl,y, and 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, MP1 shall be substituted for Mpl,y In the analysis using equation (16). Mpl,d shall be substituted for Mpl , ,d . M$ shall be obtained by analogy from equation (97) for the effective cross section having the width b .

7.6.4 Bending about one axis with coexistent axial force (728) The ultimate limit state analysis shall be made applying 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 the elastic-elastic method) or item 727 (when using the elastic- plastic method).

7.6.5 Biaxial bending with or without coexistent axial force

(729) The ultimate limit state analysis may be made using equation (30), applying by analogy provisions of subclause 7.6.4.



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DIN 18800 Part 2

Standards and other documents referred to DIN 1025 Part 1

DIN 1025 Part 2

DIN 1025 Part 3

DIN 1025 Part 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-Richtlinie O16

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 and I B series); dimensions, mass, limit deviations and static values Steel sections; hot rolled wide flange I beams (IPBI series); dimensions, mass, limit deviations and static values Steel sections; hot rolled wide flange I beams (IPBv series); dimensions, mass, limit deviations and static values Steel 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 Bauteilen I)

Literature ECCS-CECM-EKS, Publication No. 33. Ultimate limit state calculation of sway frames with rigid joints, Brussels, 1984. Stahl im Hochbau (Steel construction), 14th ed., vol. I, Part 2, Düsseldorf: Verlag Stahleisen mbH, 1986. Lindner, J.; Gregull, T. Drehbettungswerte für Dachdeckungen mit untergelegter Wärmedämmung (Values of torsional restraint for roof coverings with thermal insulation), Stahlbau, 1989: 58,173-179. Lindner, J. Stabilisierung von Biegeträgem durch Drehbettung - eine Klarstellung (Stabilization of beams by torsional restraint), Stahlbau, 1987: 56, 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 of structures), 2nd ed., Braunschweig, Wiesbaden: Friedr. Vieweg und Sohn, 1982. Roik, K.; Kindmann, R. Das Ersatzstabverfahren - Tragsicherheitsnachweise für Stabwerke beieinachsiger Biegung und Normalkraft (The equivalent member method: ultimate safety analyses of frames subjected to bending about one axis and coexistent axial force), Stahlbau, 1982: 51, 137-145. Lindner, J.; Gietzelt, G. Zweiachsige Biegung und Längskraft - ein ergänzender Bemessungsvorschlag (Biaxial bending and coexistent axial force. A supplementary design proposition), Stahlbau, 1985: 54, 265-271. Ramm, W.; Uhlrnann, W. Zur Anpassung des Stabilitätsnachweises für mehrteilige Druckstäbe an das europäische Nachweiskonzept (Bringing into line stability analyses of built-up compression members with the European concept), Stahlbau, 1981: 50,161-172. 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 18800 Part Z) , Stahlbau, 1989: 58,103-109. Rubin, H. Das Drehverschiebungsverfahren zur vereinfachten Berechnung unverschieblicher Stockwerkrahmen nach Theorie I . 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: 59, 467-475. Palkowski, S. Stabilität von Zweigelenkbögen mit Hängern und Zugband (Stability of two-hinged arches with hangers and ties), Stahlbau, 1987: 56, 169-172. Palkowski, S. Statik und Stabilität von Zweigelenkbögen mit schrägen Hängern und Zugband (Statics and stability of two-hinged tied arches with diagonal hangers), Stahlbau, 1987: 56, 246-250. Dabrowski, R. Knicksicherheit des Portalrahmens (Safety against buckling of portal frames), Bauingenieur, 1960: 35, 178-182.

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



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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. Bemessung von Dachplatten und Wandriegeln aus Kaltprofilen (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 bei 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 editions DIN 4114 Part 1: 0 7 . 5 2 ~ ~ : DIN 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, the resistance to buckling of linear members and frames, of plates and of shells now

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

on structural steelwork.

being dealt with in different Parts of DIN 18800.

Explanatory notes 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 for 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.

international Patent Classification E 04 B 1/19 E 04 B 1124 G O1 B 21/00 G O1 N 3/00



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141439606 din 18800 02 structural steelwork design construction din 1990

Documents

lateral buckling

analysis of frames

lateral torsional buckling

buckling of linear members

analysis of panels

buckling perpendicular

lateral displacement

coexistent axial force