Steel Structures – Formulae 2013/14 - ULisboa Master Degree in Civil Engineering – Steel structures – Formulae – 2013/14 1/20 Steel Structures – Formulae 2013/14 Elastic

Integrated Master Degree in Civil Engineering – Steel structures – Formulae – 2013/14 1/20

Steel Structures – Formulae 2013/14

Elastic stresses formulae

Axial force and bending moment

z

z

y

y yMzMAN

II−+=σ Transverse shear t

SVI

=τ

Torsion moment

Cross-section type Torsional stiffness Shear stress

Radial symmetric cross-sections It =π R4

2 τ = T ρ

Ιt

Open Thin-walled cross-sections ∑= 3

t th31I

t

ii

tTI

max =τ

Closed Thin-walled cross-sections

It =4 Am

2

dst∫

tA2T

m

=τ

Shear flows and forces distribution in connections

Shear flow in a fillet weld due to a “moment”. ρ=p

MFIw

Force in a bolt due to a “moment” ii ρρ

=∑=

n

1j

2j

MF

Virtual Work Principle (table of integrals)

abL 1

2 abL 12 abL

13 abL

13 abL

23 abL

23 abL

12 abL

13 abL

16 abL

14 abL

112 abL

512 abL

14 abL

13 abL

14 abL

112 abL

15 abL

130 abL

310 abL

215 abL

Elastic solutions of Internal forces and displacements

MA = −p L2

12;Mmax =

p L2

24;

VA =VB =p L2;

δmax =pL4

384EI

MA = −p L2

8;Mmax =

p L2

14.2

VA =5p L8

;VB =3p L8

δmax =pL4

185EI

Mmax =p L2

8;

VA =VB =p L2;

δmax =5pL4

384EI

MA = −PL1L2

2

L2;MB = −

PL12L2L2

;

VA =PL2

2 3L1 + L2( )L3

;

VB =PL1

2 L1 +3L2( )L3

;δC =PL1

3L23

3EIL3

MA = −PL1L2 L + L2( )

2L2;VA =

PL2 3L2 − L2

2( )2L3

;

VB =PL1

2 3L − L1( )2L3

;

δC = PL13L22

12EIL34L − L1( )

MC = PL1L2L

;

VA =PL2L;VB =

PL1L;

δC = PL12L2

2

3EIL


Abaqus for evaluating buckling lengths of columns with elastic supports


EC3 – Ultimate limit state of cross-section resistance – sections of class 1 and 2

Separated internal forces

Axial force

tension Nt.Ed/Nt.Rd ≤ 1,0 Nt.Rd =

⎩⎨⎧Npl.Rd = Afy/γM0

Nu.Rd = 0.9Anetfu/γM2

Axial force

compression Nc.Ed/Nc.Rd ≤ 1,0 Nc.Rd = Npl.Rd = Afy/γM0

Bending

moment MEd/Mc.Rd ≤ 1,0 Mc.Rd = Mpl.Rd = Wplfy/γM0

Shear force VEd/Vc.Rd ≤ 1,0 Vc.Rd = Vpl.Rd = Avfy/ 3/γM0 Av ≈ hwtw

Bending and axial force

Rectangular cross-section

MN.Rd = Mpl.Rd⎣⎢

⎡⎦⎥⎤1-

⎝⎜⎛

⎠⎟⎞NEd

Npl.Rd

2

I shaped cross-section

If n≤0,25 and

n≤0,5a MN.y.Rd = Mpl.y.Rd

n = NEdNpl.Rd

a = A - 2btfA ≈ hwtw

A

If n>0,25 or

n>0,5a MN.y.Rd = Mpl.y.Rd

1-n1-0.5a ≤ Mpl.y.Rd

If n≤a MN.z.Rd = Mpl.z.Rd

If n>a MN.z.Rd = Mpl.z.Rd⎣⎢⎡

⎦⎥⎤1-

⎝⎜⎛

⎠⎟⎞n-a

1-a2

Hollow cross-sections

MN.y.Rd = Mpl.y.Rd1-n

1-0.5aw ≤ Mpl.y.Rd

n = NEdNpl.Rd

aw = (A - 2btf)/A≤0,5

MN.z.Rd = Mpl.z.Rd1-n

1-0.5af ≤ Mpl.z.Rd

n = NEdNpl.Rd

af = (A - 2htw)/A≤0,5


EC3 – Ultimate limit state of cross-section resistance – sections of class 1 and 2

Bi-axial flexure

⎣⎢⎡

⎦⎥⎤My.Ed

MN.y.Rd α +

⎣⎢⎡

⎦⎥⎤Mz.Ed

MN.z.Rd β ≤ 1

Cross-section α β Remarks

Ι, H 2 5n β ≥ 1

2 2 -

1.661 - 1.13n2

1.661 - 1.13n2 α, β ≤ 6

Bending and shear interaction

General case fy.red = (1 - ρ) fy ρ = ⎝⎜⎛

⎠⎟⎞2VEd

Vpl.Rd - 1

2

I shaped cross-section

My.V.Rd =

⎝⎜⎛

⎠⎟⎞Wpl.y -

ρAw2

4tw fy

γM0

Short Version of EN1993 Part 1-1 – Materials


Short Version of EN1993 Part 1-1 – Cross-section classification



Short Version of EN1993 Parte 1-1








Coeficiente de redução χ Coeficiente de redução χ

a0 a b c d a0 a b c d

λ α = 0,13 α = 0,21 α = 0,34 α = 0,49 α = 0,76 α = 0,13 α = 0,21 α = 0,34 α = 0,49 α = 0,76

0,2 1,0000 1,0000 1,0000 1,0000 1,0000 1,7 0,3150 0,2994 0,2781 0,2577 0,2289 0,3 0,9859 0,9775 0,9641 0,9491 0,9235 1,8 0,2833 0,2702 0,2521 0,2345 0,2093 0,4 0,9701 0,9528 0,9261 0,8973 0,8504 1,9 0,2559 0,2449 0,2294 0,2141 0,1920 0,5 0,9513 0,9243 0,8842 0,8430 0,7793 2,0 0,2323 0,2229 0,2095 0,1962 0,1766 0,6 0,9276 0,8900 0,8371 0,7854 0,7100 2,1 0,2117 0,2036 0,1920 0,1803 0,1630 0,7 0,8961 0,8477 0,7837 0,7247 0,6431 2,2 0,1937 0,1867 0,1765 0,1662 0,1508 0,8 0,8533 0,7957 0,7245 0,6622 0,5797 2,3 0,1779 0,1717 0,1628 0,1537 0,1399 0,9 0,7961 0,7339 0,6612 0,5998 0,5208 2,4 0,1639 0,1585 0,1506 0,1425 0,1302 1,0 0,7253 0,6656 0,5970 0,5399 0,4671 2,5 0,1515 0,1467 0,1397 0,1325 0,1214 1,1 0,6482 0,5960 0,5352 0,4842 0,4189 2,6 0,1404 0,1362 0,1299 0,1234 0,1134 1,2 0,5732 0,5300 0,4781 0,4338 0,3762 2,7 0,1305 0,1267 0,1211 0,1153 0,1062 1,3 0,5053 0,4703 0,4269 0,3888 0,3385 2,8 0,1216 0,1182 0,1132 0,1079 0,0997 1,4 0,4461 0,4179 0,3817 0,3492 0,3055 2,9 0,1136 0,1105 0,1060 0,1012 0,0937 1,5 0,3953 0,3724 0,3422 0,3145 0,2766 3,0 0,1063 0,1036 0,0994 0,0951 0,0882 1,6 0,3520 0,3332 0,3079 0,2842 0,2512

λ


Short version EN 1993 Part 1-1 6.3.3 – Beam-columns

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

≤

γ

+

γ

+

γχ

≤

γ

+

γ

+

γχ

1MM

kMM

kNN

1MM

kMM

kNN

1M

Rkz

Edzzz

1M

Rky

Edyzy

1M

Rkz

Ed

1M

Rkz

Edzyz

1M

Rky

Edyyy

1M

Rky

Ed

,

,

,

,

,

,

,

,

Annex B


Short version EN 1993 Part 1-8 – Connections





Bolt data

d [mm] do [mm] As [mm2]

12 14 84,3 16 18 157 20 22 245 24 26 353 27 30 459 30 33 561

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Steel Structures – Formulae 2013/14 - ULisboa Master Degree in Civil Engineering – Steel structures – Formulae – 2013/14 1/20 Steel Structures – Formulae 2013/14 Elastic