Eurocode 3 Stress Ratio Presentation 5 by Lee CK

RSA Symposium, 5 August 2013, NTU

Regency Steel Asia Symposium on Impact of Structural Eurocodes on Steel and

Regency Steel Asia Symposium on Impact of Structural Eurocodes on Steel and

Composite Structures

Surviving Class 4 Slender Section inSurviving Class 4 Slender Section inComposite Structures

Surviving Class 4 Slender Section inSurviving Class 4 Slender Section inSurviving Class 4 Slender Section in Surviving Class 4 Slender Section in EurocodeEurocode 33

Surviving Class 4 Slender Section in Surviving Class 4 Slender Section in EurocodeEurocode 33

Associate Professor Lee Chi KingAssociate Professor Lee Chi KingDivision of Structures and Mechanics

School of Civil and Environmental Engineering, N T h l i l U i iNanyang Technological University

5 August 2013

1


Topics of presentationIntroduction Classification of steel sections under EC3 Part 1-1 Recall what you learnt in Year 1 Mechanics of Materials Recall what you learnt in Year 1 Mechanics of MaterialsClass 4 sections under EC3 Part 1-5 Why Class 4 sections are troublesome? Plate-like y

buckling and effective width Calculation of Class 4 sections properties according to

EC3 Part 1-5EC3 Part 1 5 Comparison with BS 5950Examples of section properties calculations A plate girder with Class 4 web A Class 4 box section Implications on buckling strength calculations Implications on buckling strength calculationsSummary and Conclusions

22


Introduction

Classification of steel sections under EC3 Part 1-1

Recall what you learnt in Year 1 yMechanics of Materials

33

RSA Symposium, 5 August 2013, NTU Introduction

Classification of steel sectionsCl 1 t Cl 4 St l S tiClass 1 to Class 4 Steel Sections

Classification is important as it determines how resistances are calculated in manyhow resistances are calculated in many design procedures.

Class 1: can develop plastic hinge with the rotation capacity required for plastic analysisrotation capacity required for plastic analysis without any reduction of resistance.

Class 2: can develop plastic moment i t b t li it d t ti itresistance but limited rotation capacity.

Class 3: can only develop elastic distribution where extreme fiber stresses can reach yield but local buckling prevents development of the full plastic moment resistance.

Class 4: develops local buckling Class 4: develops local buckling before attainment of yield.

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Classification of steel sectionsClasses 1 - 4 Steel Sections properties

Classes 1 and 2: Plastic deformation, load independentEffective area under compression A Gross area AEffective area under compression Aeff Gross area, AG.Bending resistance related to the section Plastic Modulus,

Wpl (from handbook or simple calculations). Class 3: Elastic deformation load independent Class 3: Elastic deformation, load independentEffective area under compression Aeff Gross area AG.Bending resistance related to the section Elastic Modulus,

W (from handbook or simple calculations)Wel (from handbook or simple calculations).Class 4: Elastic deformation, load dependent!Effective area under compression Aeff < Gross area AG.Bending resistance related to the section Effective

Section Modulus, Weff which in general requires iterative calculations to establish.Additional moments could be generated due to shift in

section centroid.

55


References References

Eurocode 3: Design of Steel Structures Part 1-1 General rules for buildings (BS EN1993-1-1:2005) [EC3 Part 1-1]

Eurocode 3: Design of Steel Structures Part 1-5 Plated Structural Elements (BS EN1993-1-5:2005) [EC3 Part 1-5]

Darko Beg et. al. Design of plated Structures Eurocode 3: Design of steel structures: Part 1-5- Design of plated structures, ECCS d E t & S h 2010 [TA684 DA457 f] [B ]ECCS and Ernst & Sohn, 2010 [TA684.DA457sf] [Beg]

Lee C. K. and Chiew S. P., 2013, An efficient modified flanges only method for plate girder bending resistance calculationonly method for plate girder bending resistance calculation , Journal of Constructional Steel Research, Vol. 89, pp. 98-106 [Lee and Chiew, 2013][ ]

66


Recall what you learnt in Year 1 MoMCentroid, 1st and 2nd moments of area For a given section, the first and second moments of area above the

d th d fi d zy-y and the z-z axes are defined as

==A

z

A

y ydaS zdaS

zdaz

yG

==A

2z

A

2y

AA

dayI dazIy y

z Sy and Sz are both zero if the y-y and the z-z axes pass through the

centroid G. For a rectangular section I is given by

AA

bd For a rectangular section, Iy is given by

d th ll l i th h ld th t

y yGd

y yh12

bdI3

y =and the parallel axis theorem hold so that y y

bdA ,AhII 2yy =+=7

7


Recall what you learnt in Year 1 MoMElastic Modulus and bending stresses Bending stresses (tension ve) and moment:

Z

TopII

Bottom

B

ybottommax,y

T

yTopmax,y z

IM and zIM ==

f and f ybottommax,yTopmax, ZT and ZB are the distances between the top and bottom fibres and the natural axis (NA).

Re rite

Bottomy,yp,

WMandWM Rewrite .are the elastic moduli at the top and

bottom fibres of the section.

Bel,bottommax,yTel,Topmax,y WMand WM ==ByBel,TyTel, zIW andzI W ==

For unsymmetrical section that ZT > ZB then . The top fibre will reach fy first while the bottom fibre stress is still < fy The stress ratio is defined as

Bel,Tel, W W Plate-like buckling. Geometrically prefect plate => Pre and post critical behaviors are obvious but more

b

gradual for imperfect plate. For shorter plate with lower a/b value, the post-buckling resistance gradually

diminishes => 2D behavior changes to 1D (column) like behavior.

a

bg ( )

[Beg: 2.4.1, Fig. 2.12][EC3 Part 1-5: 4.4 Fig. 4.3]

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Note: In EC3 Part 1-5, b is always the dimension of theedge where the direct stress is applied

RSA Symposium, 5 August 2013, NTU Class 4 sections under EC3 Part 1-5

Plate-like buckling and effective widthEffective widthp for thin plate

After cr (elastic critical stress) is reached, resistance of the plate is not exhausted. Stress re distribution occurs and ultimate resistance reached after f is reached at Stress re-distribution occurs and ultimate resistance reached after fy is reached at

the two sides near the supports. The non-uniform stress distribution (act) is not convenient for design and the

Eff ti Width M th d d t i l i EC3 P t 1 1 d P t 1 5Effective Width Method are used extensively in EC3 Part 1-1 and Part 1-5. Reduce the gross width to an appropriate Effective Widthp beff adjacent to the

edges and assume that fy is reached there.

reduced crosssection method

b /2b /2

fyy

actlim f

reduced stress method

beff /2beff /2

ylim f

b

ylim fa

[Beg: 2 4 1 Fig 2 13]11

11

b[Beg: 2.4.1, Fig. 2.13]


Plate-like buckling and effective widthEffective widthp for thin plate

The effective width of a thin plate depends on B d t diti t th t id Boundary support conditions at the two sides, Geometry of the plate (a, b, and thickness), The loading conditions => Direct stress distribution along the edges The loading conditions Direct stress distribution along the edges. Obviously, no buckling for any part of the plate which is under tensile

stress.

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Stephen P. Timoshenko (1878-1972)


Plate-like buckling and effective widthEffective widthp for thin plateEffective widthp for thin plate

Classification of Sections: [EC3 Part 1-1, Table 5.2]. Calculations of beff: [EC3 Part 1-5, Tables 4.1] (Internal compression part) Calculations of beff: [EC3 Part 1 5, Tables 4.1] (Internal compression part)

and 4.2 (Outstand flanges) with key parameter defining the stress ratio.

1313


Plate-like buckling and effective widthEffective widthp for thin plateEffective widthp for thin plate

The parameter defines the effective widthp, beff The reduction factor for the web is defined as [EC3 1-5: 4 4(2)] The reduction factor for the web is defined as [EC3 1 5: 4.4(2)]

+

)0 055(3 0.0550.0850.5for 1.0 p

+>+= 0.0550.0850.5for )0.055(3

p2p

p

T bt iT bt i ff tiff ti idthidthpp bbk28.4

b/t=p To obtain To obtain effective effective widthwidthpp bbeffeff we we

need b, t, need b, t, (stress ratio) only.(stress ratio) only. The buckling factor k() for web (internal compression element) is

defined in the last row of [EC3 1-5: 4.4, Table 4.1]

Note: Compressive stress +ve Tensile stress -ve

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Note: Compressive stress +ve, Tensile stress -ve


Calculations of Class 4 slender section propertiesPrinciple of section properties calculation by EC3Principle of section properties calculation by EC3

The section is first broken down to individual plates with lateral supports. Each of these plates is classified based on its geometry and stress Each of these plates is classified based on its geometry and stress

distribution. The effective widthp, beff of the plates are calculated.

C f G Combine all the plates back, calculate the location of the new centroid G, Aeff and Weff and any other section properties needed.

MEd

NEd

[Beg: 2.4.2.2, Fig. 2.14]

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Calculations of Class 4 slender section propertiesPrinciple of section properties calculation by EC3Principle of section properties calculation by EC3

The reduced, effective section (with all non-effective areas removed) is then treated as a Class 3 section with linear strain distribution.

The ultimate resistance of the section is reached when the centre of the compressive plate located furthest from the new centrod G is yield.

So why Class 4 sections are zebras? If both axial force and bending moment act simultaneously, the calculation

of b should based on the combined stress distributionof beff should based on the combined stress distribution. For a non-symmetrical section under NEd, the shift in centroid GG for a

distance eN will generate an additional moment M=NEd eN and it should be considered for the updated stress distribution calculation, which in turns further change beff and eN and so on ..

GGGeN

1616


Calculations of Class 4 slender section propertiesS h Cl 4 ti b ?So why Class 4 sections are zebras?

For a symmetrical section under MEd, the reduction of effective area in the compression part also shift the centroid GG.compression part also shift the centroid GG .

While there is no change in MEd, to balance the moment, there must be a change in the stress distribution and the stress ratio which changes beff

d i ff i G dand in turns affecting G and so on fyfy fy

GG NA: Gross

Class 4 web


Calculations of Class 4 slender section propertiesClass 4 sections under EC3 Part 1-5

While Class 4 sections are zebras, in somecases they are easier to handle

EC3 1 5 ll h f ll i i lifi i h EC3 1-5 allows the following simplifications whencalculating the section properties of Class 4 sections[EC3 1-5: 4.3]. In general, if both NEd and MEd are present, then9 Aeff could be calculated from the stresses due to compression only,9 W ld b l l t d f th t d t b di l9 Weff could be calculated from the stresses due to bending only.9 However, iterations may still be needed until the position G (or

values of Aeff and Weff) converged. For I-sections (e.g. plate girder) and box sections in bending only, only

one iteration step is required (e.g. stop after the first G is computed). These two simplifications allow hand calculations affordable for many These two simplifications allow hand calculations affordable for many

Class 4 sections.

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Comparison with BS 5950Class 4 sections under EC3 Part 1-5

How Class 4 sections are handed in BS5950? Class 1 to Class 4 sections defined accordingly as in EC3. Similar concept in BS 5950: Aeff,Zeff and addition moments generated. However, for doubly symmetric slender sections, the Aeff are given

explicitly: No iteration is needed!explicitly: No iteration is needed! In many cases, most of the centre

parts of the plates are removed. Effective parts have lengths

limited to at most 20t form ends with lateral support => Perhapswith lateral support > Perhaps may be more conservative when comparing with EC3.

A li i f l f b i An explicit formula for beff are given when the web is Class 4.

2020


E l f ti tiExamples of section properties calculations

A plate girder with Class 4 web A Class 4 box section A Class 4 box section Implications in buckling strength

l l ticalculations

2121


A plate girder with Class 4 webExamples of section properties calculations

g A plate girder (I section) under pure sagging

moment with Class 4 web about Class 1 flanges C l l i f i i

10mm Calculation steps for section properties According to EC3 1-5, only one iteration is needed

to calculate its section properties.

hw=1920

yy G

Agross=51200mm2 Igross=3663530.67cm4 I th 1st it ti th h l b i 40 mm

fy=355MPa=0.814

In the 1st iteration, assume the whole web is effective => G at centre of web => compressive stress = tensile stress => =-1.

400mm40mm

Thus, from EC3 Part 1-5, Tables 4.1, k=23.9.08.1699.1

9.23814.04.2810/1920

4.28/ >===

k

th wwp

550.0699.1

)13(055.0699.1)3(055.022 =

=+=p

p

22

p

22



g Calculation steps for section properties In the first iteration bc=1920/2=960mm

fy

be1 Again from, from EC3 Part 1-5, Tables 4.1beff=0.55960=528mmbe1=0.4528=211.2mm yy G

Compressivestress

dc bc (=hw/2inthetwostepmethod)

e1

be2

x

r

be1 0.4 528 211.2mmbe2=0.6528=316.8mmx= 960-211.2-316.8=432mmA=10432=4320mm2

yy G yy G

tensilestressdt

G

btA=10432=4320mm2Aeff=51200-4320=46880mm2

r=1920/2-211.2-432/2=532.8mm G=Centroid ofeffectivesectionG=Centroid ofgrosssection=fy



g Calculation steps for section

properties If i h d ld ti t it t

fy

Compressived bc (=hw/2in

be1

If wished, we could continue to iteratek=21.51 => => =0.522bc=1920/2+49.10=1009.1mm

yy G

stressdc c ( w/

thetwostepmethod)

be2

x

G

r791.=pbeff=0.5221009.1=526.8mm(c.f. in first iteration: beff=528mm)

But EC3 does not require us to do so!

yy G yy G

tensilestressdt bt

Proceed to calculate the Minimum effective modulus: Weff=Ieff/dc G=Centroid ofeffectivesectionG=Centroid ofgrosssection

=fy42=Class 4

tw=10mm

G

hw=600mm

tf2=20mmrT=233.8mm

yy

9 Top flange: bf1/tf1=580/10=58>42=Class 49 Bottom flange: bf2/tf1=29< 33 = Class 19 Webs (compression): hw/tw=570/10=57>42=Class 4

bf2=600mm

z( p ) w w9 Webs (pure bending, =-1): hw/tw=570/10=57Iterations needed to get section properties!

2626


A Class 4 box section under axial loading Examples of section properties calculations

g Calculation steps for section properties Stress distributions of section components

z

9Top flange: uniform compressive stress9Bottom flange: uniform compressive stress9W b i b di

Gyy eN9Webs: compression + bending =>

Linear compressive stress with stress ratio depends on the location of G

rTGeN

z

rT,eff

N at G M = N e (sagging)depends on the location of G However, remember that EC3 allows the following simplifications:

If both NEd and MEd are present then

NEd at G M = NEdeN (sagging)

If both NEd and MEd are present, then9 Aeff could be calculated from the stresses due to compression only,9 Weff could be calculated from the stresses due to bending only.

These simplifications eventually allow us to compute A and W without These simplifications eventually allow us to compute Aeff and Weff without any iteration!

2727



zg Calculation steps for Aeff Only NEd is considered, rT=233.8mm. All plates (top and bottom flanges and two webs) All plates (top and bottom flanges and two webs)

=> Uniform compression => =1 k=4.0 for both flanges and webs

G

rTyy

GeN

rT,effftop=0.725, beff,ftop=420.5mmweb=0.734, beff,web=418.7mm zNEd at G M = NEdeN (sagging)

Eventually, eN=30.1mm and rT,eff=203.7mm, Aeff=24778.1mm2 Since only NEd is considered the shift of G to G does not

further change , no more iteration is needed!28

28

g ,



zg Calculation steps for Weff Only M is considered, rT=233.8mm. The section is under pure bending The section is under pure bending. Only top flange is Class 4 =1 and k=4.0 again for top flange

G

rTyy

GeM

rT Mftop=0.725, beff,ftop=420.5mmAeff,M=27804.9mm2 (effective area under bending)e =20 1mm r =213 6mm

z

rT,M

NEd at G M = NEdeN (sagging)eM=20.1mm, rT,eff=213.6mmWeff,ftop=4200cm3, Weff,fbottom=7558cm3

Note that we need to recheck to ensure the webs are not Class 4 under

Effective modulus at top edge of bottom flange

Note that we need to recheck to ensure the webs are not Class 4 under the bending action of M as the section is now non-symmetrical.

Stress ratio 1= Weff,ftop/Weff,fbottom=-0.56, from EC3 1-1, Table 5.2, Class 3 limit 42 /(0 67+0 33 ) 79 8>57 >Web is at least Class 3 and elasticlimit=42/(0.67+0.331)=79.8>57 =>Web is at least Class 3 and elastic.

Since only M is considered, no more iteration is needed!

2929



g Checking the adequacy of the section under

NEd = 5500kN W ff for bending resistance is defined as the

z

Weff for bending resistance is defined as the effective section modulus at the centroid of the top flange, which is slightly less than Weff,ftop G

yyand is calculated asrT

yyG

eM

z

rT,M3

f2f1

effeff 4144cm

rtth

IW =++=

NEd at G M = NEdeN (sagging)MT,f2f1

w r2h + Cross section resistance check according to [EC3, 1-1: 6.2.9.3 Eqn. 6.44]

30 15500kN5500kNeNN

OK1.00.95275MPa/1.04144cm

30.1mm5500kN275MPa/1.024778.1mm

5500kN/fWeN

/fAN

32yeff

NEd

yeff

Ed

=>


Implications in buckling strength calculationsExamples of section properties calculations

The previous examples considered the cross section resistance only. In general, even when only axial force NEd is applied to a Class 4

section addition moments are almost always generated due to the shiftsection, addition moments are almost always generated due to the shift of the section centroid.

Such moments should be considered in both section resistance and buckling resistance checks [EC3 1-1: 6.3.3 Eqns. 6.11 and 6.12].

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RSA Symposium, 5 August 2013, NTU Summary and conclusions

Summary and conclusions Remember that Class 1 and 2 sections are horses, Class 3

sections are mustangs and Class 4 sections are zebras! While zebras are rare (at least in Singapore) Class 4 sections While zebras are rare (at least in Singapore), Class 4 sections

are more common! (e.g. plate and box girders). In EC3, the section properties of a Class 4 section depends on

b th it t d th l d li dboth its geometry and the loads applied. To calculate the section properties of a Class 4 section, an

engineer needs knowledge related to plate-like buckling, elastic g g p g,bending theory, centriod, first and second moments of area and elastic modulus calculations.

In general iterations are needed to calculate the section properties In general, iterations are needed to calculate the section properties of Class 4 sections and hand calculations could be tedious.

Some simple calculation tools (e.g. spreadsheet programmes) may p ( g p p g ) ybe helpful.

3232


End of presentationfThanks for your attentions!

All questions are welcome!All questions are welcome!

3333

Eurocode 3 Stress Ratio Presentation 5 by Lee CK

Documents

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