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Plated structural elements Case Study svenebbe

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Forside

Table of Contents1 Description...............................................................................................................................................4

1.1 Building and structure.......................................................................................................................4

Figure 1. The building as axonometric view.........................................................................................4

2 Geometri..................................................................................................................................................4

Figure 2.1. Ceiling plane.......................................................................................................................4

Figure 2.2. Elevation of plated frame...................................................................................................5

Figure 2.2. Elevation of tapered frame................................................................................................5

Figure 2.4. Elevation of frame model...................................................................................................6

3 Loads........................................................................................................................................................6

3.1 Dead load...........................................................................................................................................6

3.2 Snow..................................................................................................................................................6

3.3 Wind..................................................................................................................................................6

4 Forces.......................................................................................................................................................9

Figure 4.1. Moment forces in unicross frame......................................................................................9

Figure 4.2. Moment.............................................................................................................................9

Figure 4.3. Shear Forces.....................................................................................................................10

Figure 4.4. Normal Forces..................................................................................................................10

5 Example..............................................................................................................................................11

Figure 1..............................................................................................................................................11

5.1 Cross sections..................................................................................................................................11

5.2 Normal stress...................................................................................................................................12

5.3 Section classes.................................................................................................................................12

5.4 Effective cross-section.....................................................................................................................12

5.5 Bending resistance...........................................................................................................................13

5.6 Normal stress Diagrams...................................................................................................................14

6 Shear resistance..................................................................................................................................14

1


6.2 Contribution from the web..............................................................................................................15

Shear buckling coefficients....................................................................................................................16

6.3 Contribution from flanges...................................................................................................................16

7 Lateral torsional buckling....................................................................................................................17

Torsion constant................................................................................................................................17

Warping constant..............................................................................................................................17

Critical moment.................................................................................................................................18

Interaction between shear force, bending moment and axial force......................................................18

Flange induced buckling........................................................................................................................18

Critical plate buckling stress for plates with one or two stiffeners in the..............................................18

compression zone..................................................................................................................................18

Plastic analysis of beam.........................................................................................................................18

Framed knee..........................................................................................................................................18

Support..................................................................................................................................................18

Finite Element Methods of analysis (FEM)............................................................................................18

Frame model.....................................................................................................................................18

Plate model........................................................................................................................................18

2


Symbols

B Width of structureH Height of structureL Length of structureMi Moment forces from FEM analysisMi Resistance of cross sectionVi Shear forces from FEM analysisVi Resistance of cross sectionNi Normal forces from FEM analysisNi Resistance of cross section

bibshi

hwi Height of cros sectionIy Second moment of inertiagt Deadload from roofps Snow loadPwi Wind loadWi Section Modulus

λw

3


1 Description

1.1 Building and structure

Figure 1. The building as axonometric view

2 Geometri

4


Figure 2.1. Ceiling plane

5


Figure 2.2. Elevation of plated frame

Figure 2.2. Elevation of tapered frame

6


Figure 2.4. Elevation of frame model.

3 Loads

3.1 Dead loadThe deadload is estimated to 0,85kN/m2.

3.2 SnowThe exposure coefficient, Ce, depends on the topography of the surroundings and the size of the structure, and is determined by: Ce = CtopCs, where Ctop is the topography coefficient, and Cs is the size coefficient.Recommended values of Ctop for normal topographies is 1,0. The size coefficient Cs = 1,0 for this building.μw is the shape factor, μw = 1,0.sk is the characteristic value for snowload, sk = 1,0. =>s = μw·Ce·CtSk ·sk=> s = 1.0·1.0·1.0·1.0 => s = 1.0kN/m2.

3.3 Windce = 2,8kN/m2

7


8


Loads

Deadload:gt = 0,85kN/m2·4,2m = 3,57kN/m

Snow:ps = 1kN/m2·4,2m = 4,2kN/m

Wind:Cd = 0,8 => Pwd = ce·Cd·4,2m = 9,4kN/mCe = 0,7 => Pwe = ce·Ce·4,2m = 8,2kN/mCg = 0,7 => Pwg = ce·Cg·4,2m = 4,3kN/mCh = 0,7 => Pwh = ce·Ch·4,2m = 3,1kN/m

9


4 ForcesThe effect of the tapered section is determined:

Figure 4.1. Moment forces in unicross frame

Figure 4.2. Moment

10


Figure 4.3. Shear Forces

Figure 4.4. Normal Forces

11


5 Example

Figure 1

5.1 Cross sections hw2 = 380mm hw1 = 950mm

12


5.2 Normal stressNmin = 120kN, Mmin = 325kN·m.

5.3 Section classes

5.4 Effective cross-section

13


hw1=950mm, tw=4mm

f y=355MPa Nmin=−120 kNMmin=−326 kNm

σ N=N min

AσM=

Mmin

W y

σ 1=σ N+σM

σ 2=σ N−σM

bc=hw 1 ·σ1

σ1−σ2Ψ=

σ2σ1

k σ=7,81−6,29 ·Ψ+9,78·Ψ 2

λw=hw 1

28,4 ·tw ·√ 235f y·√kσ

beff=ρ· bc

be1= ρ·beffbe2=ρ·beff

bh=ρ·(bc−(be1−be 2))

Sw=−bh ·tw ·(hw 1

2−be 1−

bh

2)

ρ=(λw−0,055 · (3+Ψ ))

λw2

5.5 Bending resistance

Cross section Section modulus Moment capacityPlastic 2,8·106·mm3 MpRd = 911·kNmElastic 2,4·106·mm3 MeRd = 785·kNmEffective 2,1·106·mm3 MefRd = 662·kNmFlanges only 1,9·106·mm3 MflRd = 620·kNm

14


5.6 Normal stress Diagrams

15


6 Shear resistance

6.2 Contribution from the web

hw1=950mm, tw=4mm, a=2900mm η =1.2, f y=355MPa

k τ=5,34+4,00 ·( hw 1a )2

+k τsl

ahw 1

>1

k τsl=0

k τsl=9 ·( hw1

a )2

· 4√( I sltw3 ·hw1

)3

But not less than

16


k τ=5,762,1tw

· 3√ I s lhw 1

λw=hw 1

37,4 ·tw ·√ 235f y·√kτ V bw , Rd=

χw · f yw · hw 1 · tw√3 · γM 1

¿>V bw, Rd=166kN

Shear buckling coefficients

17


6.3 Contribution from flanges

18


7 Interaction between shear force, bending moment and axial force

The relative moment capacity depending of the shear utilization/capacity:

19


According to the Eurocode we have this expression:

If these two expressions for the moment capacity is shown together we have.

20


21


8 Lateral torsional buckling

Torsion constant and Warping constant

22


9 Flange induced buckling

23


10 Framed knee

11 Critical plate buckling stress for plates with one or two stiffeners in thecompression zone

12 Plastic analysis of beam

13 Support

14 Finite Element Methods of analysis (FEM)

Plate model

24


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