Steel Structures – Formulae 2013/14 - ULisboa Master Degree in Civil Engineering – Steel...
Transcript of Steel Structures – Formulae 2013/14 - ULisboa Master Degree in Civil Engineering – Steel...
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
Integrated Master Degree in Civil Engineering – Steel structures – Formulae – 2013/14 2/20
Abaqus for evaluating buckling lengths of columns with elastic supports
Integrated Master Degree in Civil Engineering – Steel structures – Formulae – 2013/14 3/20
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
Integrated Master Degree in Civil Engineering – Steel structures – Formulae – 2013/14 4/20
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
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Short Version of EN1993 Part 1-1 – Cross-section classification
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Integrated Master Degree in Civil Engineering – Steel structures – Formulae – 2013/14 7/20
Short Version of EN1993 Parte 1-1
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Integrated Master Degree in Civil Engineering – Steel structures – Formulae – 2013/14 9/20
Integrated Master Degree in Civil Engineering – Steel structures – Formulae – 2013/14 10/20
Integrated Master Degree in Civil Engineering – Steel structures – Formulae – 2013/14 11/20
Integrated Master Degree in Civil Engineering – Steel structures – Formulae – 2013/14 12/20
Integrated Master Degree in Civil Engineering – Steel structures – Formulae – 2013/14 13/20
Integrated Master Degree in Civil Engineering – Steel structures – Formulae – 2013/14 14/20
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
Integrated Master Degree in Civil Engineering – Steel structures – Formulae – 2013/14 15/20
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
Integrated Master Degree in Civil Engineering – Steel structures – Formulae – 2013/14 16/20
Short version EN 1993 Part 1-8 – Connections
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Integrated Master Degree in Civil Engineering – Steel structures – Formulae – 2013/14 18/20
Integrated Master Degree in Civil Engineering – Steel structures – Formulae – 2013/14 19/20
Integrated Master Degree in Civil Engineering – Steel structures – Formulae – 2013/14 20/20
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