Section properties and capacities

1

CSE 362 Design of Steel StructuresSection properties, section capacities, combined bending and shear, combined compression and bending

Professor K.F. ChungProfessor K.F. Chung

Department of Civil and Structural EngineeringThe Hong Kong Polytechnic UniversityHong Kong SAR, China

2

Design strengths

Section properties

Section capacitiesShear capacityMoment capacity

Section classificationInteraction of forces and moments

Moment capacity under high shearMoment capacity under axial force

Member resistancesColumnBeamCombined compression and bending

Connection design

Design of steel structures

3

Steel grade Flange thickness Design strength

smaller than or equal to py

(mm) (N/mm2)

S275 16 275

40 265

63 255

S355 16 355

40 345

63 335

Design strengths

4

Area, A

Second moment of area, I

Elastic modulus, Z

Plastic modulus, S

Section properties

5

UB 457 x 152 x 52D = 449.8mm B = 152.4mmt = 7.6mm T = 10.9mmr = 10.2mm d = 407.6mm

Cross sectional area

A = +

= (449.8 – 2 x 10.9) x 7.6 + 2 x 152.4 x 10.9= 3253mm2 + 3322mm2

= 6575mm2 (c.f. 66.6cm2 or 6660mm2 from tabulated data)

As Afillet = 4 x 10.22 x = 89.3mm2

A = 6575 + 89.3 = 6664.3mm2

In general, fillets are neglected in most design.

43421

wA

x t2T)(D − 321

fA

T x B 2

⎟⎠⎞

⎜⎝⎛ −

4

π1

d = 428D = 449.8

203.8 219.45

10.9

t = 7.6

B = 152.4

Section propertiesCross-sectional area, A

6

Ix =

= 2 [16.45 x 103 + 80.0 x 106] + 49.66 x 106

= 160.03 x 106 + 49.66 x 106

= 209.69 x 106 mm4 or 20969 cm4

12

7.6 x 428219.45 x 10.9 x 152.4

12

10.9 x 152.4 2

32

3

+⎥⎦

⎤⎢⎣

⎡+

Section propertiesSecond moment of area, I

d = 428D = 449.8

203.8 219.45

10.9

t = 7.6

B = 152.4

7

Z

Zx = = = 932.4 x 103 mm3

= 932cm3

Zw = = 232 x 103mm3 = 232cm3

Zf = 932 – 232 = 700cm3

= = 0.249

= 1 – 0.249 = 0.751

D/2

I449.8/2

10 x 20969 4

6

7.6 x 4282

932

232x

w

Z

Z

x

f

Z

Z

Section propertiesElastic modulus, Z

8

SSx = 2 x [152.4 x 10.9 x 219.45] +

= 729 x 103mm3 + 348 x 103mm3

= 1077 x 103mm3 or 1077cm3 (c.f. 1096 cm3)

= = 0.677 = = 0.323

Shape factor = = 1.156

4

7.6 x 4282

1077

729

x

f

S

S

x

w

S

S

1077

348

932

1077

Section propertiesPlastic modulus, S

9

0.6770.323

729348

0.7510.249

700232

FlangesWeb

-1077-932Total

ratio(cm3)ratio(cm3)

Plastic modulusSx

Elastic modulusZx

Elements

ratio(cm2)

-20969-6575Total

0.7490.232

160034966

0.4980.488

33223253

FlangesWeb

ratio(cm4)

Second moment of area, I

AreaA

Elements

Typical section properties in an I-section

10

PT = Ae x py ≥ P

where P is the applied load

Ae is the effect area allowing for bolt holes, if any

Section capacitiesTension capacity, PT

11

Pc = Ag x py ≥ P

where P is the applied load

Ag is the gross area

Section capacitiesCompression capacity, Pc

12

The shear stress distribution of the I-section under an applied shear force shows that a large proportion of the shear force is taken up by the web, but by the flanges.

wherepv = 0.577 pyAv is the shear area

= D x t rolled section = d x t fabricated section

Pv = Av x pv D

Rolled section Fabricated section

d

Section capacitiesShear capacity, Pv

13

Equivalent stress at failure• σe = von Mises failure criterion

• py = for pure shear

• pv = = 0.577py

2v

2t 3ff +

2v3p

yp3

1

For S275 steel where py = 275.0 N/mm2

pv = 0.577 x 275 = 158.7 N/mm2

For S355 steel where py = 355.0 N/mm2

pv = 0.577 x 355 = 204.8 N/mm2

Section capacitiesShear strength, pv

14

The slenderness of the plate elements,b/T or d/t controls the local buckling behaviour.

The section class of a steel section is always assigned to be the worst classification of its plate elements, i.e. either its compressive flange or its web.

Mc = 1.2 Z py or S py for class 1 plastic or class 2 compact section= Z py for class 3 semi-compact section≤ Z py for class 4 slender section

D

b

t

T

d

B

Section capacitiesMoment capacities and section classification

15

θ

Mc

Class 3 Semi-compact

Class 4 Slender

Class 2 Compact

Class 1 PlasticS py

Z py

py

Class 1 and 2 sections

py

Class 3 sections

≤ py

Class 4 sections


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Web with neutral axis at mid-depth

Outstand element under compression due to bending

(3) Semi-compact(2) Compact(1) Plastic

Class of elementCompression element

Limiting width to thickness ratio

ε 8 T

b≤ ε 9

T

b≤

ε 80 t

b≤ ε 120

t

b≤

ε 13 T

b≤

ε 100 t

b≤

td

T

b

yp

275 ε =


17

As both the compressive flange element and the web element areplastic, the section is also plastic.

Mc = py S or 1.2 py Z= 275 x 1077 x 10-3 = 296.2 kNm

or= 1.2 x 275 x 932 x 10-3 = 307.6 kNm

≤= 6.99T

b

≤= 6.35t

d12010080

1398

Semi-compactCompactPlastic

⇒

Mc = 296.2 kNm

Worked Example 1UB 457 x 152 x 52 S275As T ≤ 16 mm, py = 275 N/mm2 and ε = 1.0


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UB 457 x 152 x 52 S355As T ≤ 16 mm, py = 355 N/mm2 andε = 0.88

As both the compressive flange element and the web element are plastic, the section is also plastic.

Mc = py.S or 1.2 py Z= 355 x 1077 x 10-3 = 382.3 kNm

or= 1.2 x 355 x 932 x 10-3 = 397.0 kNm

≤= 6.99T

b

≤= 6.35t

d105.688.070.4

11.47.97.0


⇒Mc = 382.3 kNm

Worked Example 2


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UB 406 x 178 x 54 S275As T ≤ 16 mm, py = 275 N/mm2, and ε = 1.0

As the compressive flange element is compact while the web element is plastic, the section is hence compact.

Mc = py S or 1.2 py Z= 275 x 1055 x 10-3 = 290.1 kNm

or= 1.2 x 275 x 930 x 10-3 = 306.9 kNm

15.8T

b=

8.46t

d= 12010080

1398


⇒

M = 290.1 kNm

Worked Example 3


20

UB 406 x 178 x 54 S355

As T ≤ 16 mm, py = 355 N/mm2 and ε = 0.88

As the compressive flange element is semi-compact whilethe web element is plastic, the section is semi-compact.

Mc = py Z

= 355 x 930 x 10-3 = 330.2 kNm

15.8T

b=

8.46t

d= 105.688.070.4

11.47.97.0

Semi-compact

Compact

Plastic

⇒

Worked Example 4


21

F

BA

L

BMD

PL

SFDP

At support A, the beam is under co-existing shear force and bending.

Section capacitiesCombined bending and shear

22

SFD ?

BMD ?

SFD ?

BMD ?

Section capacitiesCombined bending and shear

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Mcx = py (S - ρSv) ≤ for class 1 and 2 sections

where

⎟⎟⎠

⎞⎜⎜⎝

⎛ ρ−

5.1

SZp2.1 v

y

2

v

v 1 P

F 2

⎥⎥⎦

⎤

⎢⎢⎣

⎡−⎟⎟

⎠

⎞⎜⎜⎝

⎛=ρ

Shear area to resist shear force

Section to resist bending momentLow shear force

reduced section modulus due to the presence of shear force in the shear area

High shear force

Section capacitiesMoment capacity under shear force

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Mcv = py(Sx – ρSv) ≤ 1.2 py (Z – ρSv/1.5)

where

ρ =

Sv = = 384.4 x 103 mm3

Sx = 1077 x 103 mm3 = = 0.357

Mcx = 275 x 1077 x 103 x 10-6 = 296.2 kNm

Moment capacities of the top and the bottom flangesMf = 275 x 729 x 103 x 10-6 = 200.5 kNm

2

v

v 1P

F2

⎥⎥⎦

⎤

⎢⎢⎣

⎡−⎟⎟

⎠

⎞⎜⎜⎝

⎛

4

x7.6449.82

x

v

S

S1077

4.384

UB 457 x 152 x 52 S275


25

0.986

0.943

0.871

0.771

0.643

292.1

279.3

258.1

228.5

190.5

0.986

0.943

0.871

0.771

0.643

0.04

0.16

0.36

0.64

1.00

0.6

0.7

0.8

0.9

1.0

Mcv/McxMcv (kNm)1 – ρSv/SxΡFv/Pv

UB 457 x 152 x 52 S275


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0

0.2

0.4

0.6

0.8

1.0

1.2

0 0.2 0.4 0.6 0.8 1.0 1.2

Shear force ratio, Fv / Pv

Mo

men

t ra

tio

, Mcv

/ M

cx

Equation A

Equation B

Equation C

UB 457 x 152 x 52 S275


1M

M

P

F

cx

cv

v

v =+

27

For simplicity, assume linear reduction

Mcv = 190.5 + (296.2 – 190.5) x = 190.5 + 52.85= 243.4 kNm ≈ 6% (c.f. 258.1kNm from non-linear reduction)

Linear reduction is always conservative.

Equation A - simple but too conservative

(max. stress for shear and bending does not coincide)

Equation B - reduction due to shear force only when Fv/Pv > 0.6but yet very conservative

Equation C - reduction due to shear force only applied to the web while the flanges are always effective, i.e. Mf

0.61.0

0.60.8

−−

UB 457 x 152 x 52 S275


28

AFD ?

BMD ?

AFD ?

SFD ?

BMD ?

F

F

Section capacitiesCombined compression and bending

29

AFD ?

SFD ?

BMD ?

F

Section capacitiesCombined compression and bending

30

Compression / tension areahn

Low axial force

nA

th

P

F n

c

c ==

2222n Anth =⇒

Mcn = py

= py ( K1 – K2 n2 ) whereK1 = Sx

K2 =

⎟⎟⎠

⎞⎜⎜⎝

⎛−

4

th2n

xS

4t

A2

⇒

Section capacitiesMoment capacity under axial force

31

B

hnD

High axial force

A n = A – DB + hnB

A n – A + DB = hnB

nA

)Bh(DA

P

F n

c

c =−−

=

DB

1)A(nhn +

−=⇒


32

Mcn = py Sxr

Sxr =

=

=

=

Mcn = py K3 (1 - n) (K4 + n)

where K3 = and K4 =

B

hnD

[ ]2n

2 hD4

B−

⎥⎦

⎤⎢⎣

⎡−− 1)(n

B

2AD-D-1)-(n

B

AD

4

B 222

22

⎥⎦⎤

⎢⎣⎡ +−−−

A

2BD1)(n1)(n

4B

A2

⎥⎦⎤

⎢⎣⎡ +− n1

A

2BDn)-(1

4B

A2

4B

A21

A

2BD−


High axial force

33

Low axial force

Mcn = py ( K1 – K2 n2 ) where K1 = Sx and K2 =

As K1 = 1077 cm3 and K2 = = 1460.8cm3

Mcn = 275 x (1077 – 1460.8 n2) x 10-3 kNm

As the maximum value of hn is 428 mm, the limiting value of n is given by

n = = 0.488

4t

A2

7.6x 4

66642

6664

7.6 x 428

High axial forceMcn = py K3 (1 - n) (K4 + n)

As K3 = = = 72.85 cm3

K4 = = = 19.57

Mcn = 275 x 72.85 (1 – n) (19.57 + n) x 10-3 kNm

4B

A2

152.4 x 4

66642

1A

2BD− 1

6664

449.8x 152.4 x 2−

Section capacitiesMoment capacity under axial force for UB 457 x 152 x 52 S275

34

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1 1.2

Axial force ratio, n (Fc / Pc)c

Mo

men

t ra

tio

Mcn

/ Mcx

UB 457 x 152 x 52 S275


35

Linear reduction is always conservative but simple

a)

b)

where α and β= 1 ~ 3 depending on the types of the steel sections

What happens in a section under combined action of axial force, shear force and bending moment?

1M

M

P

F

cx

cn

c

c =+

1M

M

P

F

cx

cn

c

c =⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛βα

UB 457 x 152 x 52 S275


36

Tension members under bi-axial moments

Alternatively, for greater economy in plastic or compact cross sections:

Mx ≤ Mrx

where Mrx or Mcn is the reduced moment capacity about the major axis in the presence of axial load

Mrx = py Sxrwhere Sxr = K1 – K2 n2

K1 = SxK2 = A2 / 4 t for low axial force

= K3 (1 - n) (K4 + n)K3 = A2 / 4 BK4 = (2 BD / A) - 1 for high axial force

1M

M

M

M

P

F

cy

y

cx

x

c

c ≤++

Section capacitiesAxially loaded members under bi-axial moments

37

For plastic or compact cross sections under bi-axial moments in the presence of axial forces

1 M

M

M

M21

Z

ry

yZ

rx

x ≤⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛

1.01.0All other cases

5/35/3Solid and hollow rectangular sections

2.02.0Solid and hollow circular sections

1.02.0I and H sections

Z2Z1


38

Compression members under bi-axial moments

In general

For plastic and compact sections

Refer to ‘Tension members under moments’ for the definition of symbols.

1M

M

M

M

P

F

cy

y

cx

x

c

c ≤++

1 M

M

M

M21

Z

ry

yZ

rx

x ≤⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛


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Section properties and capacities