elastic stresses in unshored composite sectionlibvolume3.xyz/civil/btech/semester7/designand... ·...

Elastic Stresses in Unshored Composite Section

� The elastic stresses at any location shall be the sum of

stresses caused by appropriate loads applied separately

� Steel beam

� Permanent loads applied before the slab has hardened, are

carried by the steel section.

� Short-term composite section

� Transient loads (such as live loads) are assumed to be carried

by short-term composite action. The short-term modular ratio,

n, should be used.

� Long-term composite section.

� Permanent loads applied after the slab has been hardened are

carried by the long-term composite section. The long-term

modular ratio, 3n, should be used.

Elastic Stresses (6.10.1.1)

Original section Transformed section

The procedure shown in this

picture is only valid if the neutral

axis is not in the concrete.

Use iterations otherwise.

Elastic Stresses (6.10.1.1)

Effective Width (Interior)

� According to AASHTO-LRFD 4.6.2.6.1, the effective width

for interior girders is to be taken as the smallest of:

� One quarter of the effective span length (span length in

simply supported beams and distance between permanent load inflection points in continuous beams).

� Average center-to-center spacing.

� Twelve times the slab thickness plus the top flange width.

Hybrid Sections 6.10.3, 6.10.1.10

� The web yield strength must be:

� 1.20 fyf ≥ fyw≥ 0.70 fyf and fyw≥ 36 ksi

� The hybrid girder reduction factor = Rh

� Where, β=2 Dn tw / Afn

� Dn = larger of distance from elastic NA to inside flange face

� Afn = flange area on the side of NA corresponding to Dn

� fn = yield stress corresponding to Afn

βρρβ

)3(12 3

Additional sections

� 6.10.1.4 – Variable web depth members

� 6.10.1.5 – Stiffness

� 6.10.1.6 – Flange stresses and bending moments

� 6.10.1.7 – Minimum negative flexure concrete deck rft.

� 6.10.1.8 – Net section fracture

Web Bend-Buckling Resistance (6.10.1.9)

� For webs without longitudinal stiffeners, the nominal bend buckling

resistance shall be taken as:

� When the section is composite and in positive flexure Rb=1.0

� When the section has one or more longitudinal stiffeners,

and D/tw≤ 0.95 (E k /Fyc)0.5 then Rb = 1.0

� When 2Dc/tw ≤ 5.7 (E / Fyc)0.5 then Rb = 1.0

where k bend buckling coefficientD D

where D depth of web in compression in elastic range

Web Bend-Buckling Reduction (6.10.1.10)

� If the previous conditions are not met then:

21 1.0

1200 300

wc cb rw

Ewhere

D tand a

= − − ≤

Calculating the depth Dc and Dcp (App. D6.3)

� For composite sections in positive flexure, the depth of the

web in compression in the elastic range Dc, shall be the

depth over which the algebraic sum of the stresses in the

steel, the long-term composite and short term composite

section is compressive

� In lieu, you can use

IMLLWSDCDCc

ffffD −

fD d t

where d depth of steel tion

f and f are the compression and tension flange stresses

−= − ≥ +

Calculating the depth Dc and Dcp (App. D6.3)

� For composite sections in positive flexure, the depth of the

web in compression at the plastic moment Dcp shall be

taken as follows for the case of PNA in the web:

−−−= 1

ryrsccyctyt

AFAfAFAFDD

6.10 I-shaped Steel Girder Design

Proportioning the section (6.10.2)

� Webs without longitudinal stiffeners must be limited to

D/tw ≤ 150

� Webs with longitudinal stiffeners must be limited to

D/tw≤ 300

� Compression and tension flanges must be proportioned

such that:

0.1 10

≤ ≤

Slender

Noncompact

Compact

Moment

Curvature

Section Behavior

6.10 I-Shaped Steel Girder Design

� Strength limit state 6.10.6

� Composite sections in positive flexure (6.10.6.2.2)

� Classified as compact section if:

� Flange yield stress (Fyf ) ≤ 70 ksi

� where, Dcp is the depth of the web in compression at the

plastic moment

� Classified as non-compact section if requirement not met

� Compact section designed using Section 6.10.7.1

� Non-compact section designed using Section 6.10.7.2

t F≤

6.10.7 Flexural Resistance

Composite Sections in Positive Flexure

Compact sections

� At the strength limit state, the section must satisfy

� If Dp≤ 0.1 Dt , then Mn = Mp

� Otherwise, Mn = Mp(1.07 – 0.7 Dp/Dt)

� Where, Dp = distance from top of deck to the N.A. of the

composite section at the plastic moment.

� Dt = total depth of composite section

� For continuous spans, Mn = 1.3 My. This limit allows for

better design with respect to moment redistributions.

3 nu l xt fM f S Mφ+ ≤

Non-Compact sections (6.10.7.2)

� At the strength limit state:

� The compression flange must satisfy fbu ≤ φf Fnc

� The tension flange must satisfy fbu + fl/3 ≤ φf Fnt

� Nominal flexural resistance Fnc = Rb Rh Fyc

� Nominal flexural resistance Fnt= Rh Fyt

� Where,

� Rb = web bend buckling reduction factor

� Rh = hybrid section reduction factor

� Ductility requirement. Compact and non-compact sections

shall satisfy Dp ≤ 0.42 Dt

� This requirement intends to protect the concrete deck

from premature crushing. The Dp/Dt ratio is lowered to

0.42 to ensure significant yielding of the bottom flange when the crushing strain is reached at the top of deck.

6.10 I-Shaped Steel Girder Design

� Composite Sections in Negative Flexure and Non-

composite Sections (6.10.6.2.2)

� Sections with Fyf ≤ 70 ksi

� Web satisfies the non-compact slenderness limit

� Where, Dc = depth of web in compression in elastic range.

� Designed using provisions for compact or non-compact web

section specified in App. A.

� Can be designed conservatively using Section 6.8

� If you use 6.8, moment capacity limited to My

� If use App. A., get greater moment capacity than My

t F≤

6.10.8 Flexural Resistance Composite Sections in

Negative Flexure and Non-Composite Section

� Discretely braced flanges in compression

� Discretely braced flanges in tension

� Continuously braced flanges: fbu≤ φf Rh Fyf

� Compression flange flexural resistance = Fnc shall be taken

as the smaller of the local buckling resistance and the lateral torsional buckling resistance.

� Tension flange flexural resistance = Fnt = Rh Fyt

3 ncbu l ff f Fφ+ ≤

3 ntbu l ff f Fφ+ ≤

Flange Local buckling or Lateral Torsional

Buckling Resistance

Fn or Mn

Inelastic Buckling

(non-compact)

Elastic Buckling

(Slender)

Fmax or Mmax

Inelastic Buckling

(Compact)

λrf λf

Fyr or Mr

6.10.8 Flexural Resistance Composite Sections in

Negative Flexure and Non-Composite Section

� Fnc Compression flange flexural resistance – local buckling

0.38 0.562

f pf rffc yc yr

f pf nc b h yc

yr f pff rf nc b h yc

h yc rf pf

When F R R F

FWhen F R R F

λ λ λ

λ λλ λ

− ≤ = − − −

Fnc Compression flange flexural resistance

Lateral torsional buckling

1.75 1.05 0.3 2.3

b p t rf t

b p nc b h yc

yr b p

b r nc b b h yc b h ych yc r p

b r nc cr b h yc

E EL L r r

When L L F R R F

F L LWhen L L F C R R F R R F

R F L L

When L L F F R R F

λ π= =

− ≤ = − − ≤ −

≥ = ≤

= − + ≤

C R EF

Lateral Torsional Buckling

Lateral support

Unstiffened Web Buckling in Shear

E46.2 D/tw

Web plastification in shear

Inelastic web buckling

Elastic web buckling

wywpntDFVV .58.0==

21.48n w ywV t EF=

EtV wn

6.10.9 Shear Resistance – Unstiffened webs

� At the strength limit state, the webs must satisfy:

Vu ≤ φv Vn

� Nominal resistance of unstiffened webs:

Vn = Vcr = C Vp

where, Vp = 0.58 Fyw D tw

� C = ratio of the shear buckling resistance to shear yield strength

k = 5 for unstiffened webs

, 1.12 ; 1.0

1.12, 1.12 1.40 ;

1.57, 1.40 ;

yw w yw yw

w yw yw

D E kIf then C

E k D E k E kIf then C

DF t F F

D E k E kIf then C

t F FD

< ≤ =

> = ×

Tension Field Action

n cr TFAV V V= +

Beam Action

Tension Field Action

6.10.9 Shear resistance – Stiffened Webs

� Members with stiffened webs have interior and end panels.

� The interior panels must be such that

� Without longitudinal stiffeners and with a transverse

stiffener spacing (do) < 3D

� With one or more longitudinal stiffeners and transverse

stiffener spacing (do) < 1.5 D

� The transverse stiffener distance for end panels with or

without longitudinal stiffeners must be do < 1.5 D

� The nominal shear resistance of end panel is

Vn = C (0.58 Fyw D tw)

� For this case – k is obtained using equation shown on next

page and do = distance to stiffener

Shear Resistance of Interior Panels of Stiffened Webs

2sec : 2.5

0.87 (1 )0.58

, 0.58

fc fc ft ft

n yw w

D tIf the tion is proportioned such that

b t b t

CV F D t C

where d transverse stiffener spacing

k shear buckling coefficientd

If not then V

0.87 (1 )

CF D t C

Transverse Stiffener Spacing

Interior panel End

3≤ Ddo 5.1≤

1.5od D≤

Types of Stiffeners

1.5od D≤1.5od D≤

Bearing

Stiffener

Transverse

Intermediate

Stiffener

Longitudinal

Stiffener

6.10.11 Design of Stiffeners

� Transverse Intermediate Stiffeners

� Consist of plates of angles bolted or welded to either one or

both sides of the web

� Transverse stiffeners may be used as connection plates for

diaphragms or cross-frames

� When they are not used as connection plates, then they shall

tight fit the compression flange, but need not be in bearing

with tension flange

� When they are used as connection plates, they should be

welded or bolted to both top and bottom flanges

� The distance between the end of the web-to-stiffener weld

and the near edge of the adjacent web-to-flange weld shall

not be less than 4 tw or more than 6 tw.

Transverse Intermediate Stiffeners

Less than 4 tw or more than 6tw

Single Plate

Double Plate

� Projecting width of transverse stiffeners must satisfy:

bt ≥ 2.0 + d/30

and bf/4 ≤ bt ≤ 16 tp

� The transverse stiffener’s moment of inertia must satisfy:

It ≥ do tw3 J

where, J = required ratio of the rigidity of one transverse

stiffener to that of the web plate = 2.5 (D/do)2 – 2.0 ≥ 2.5

It = stiffener m.o.i. about edge in contact with web for

single stiffeners and about mid thickness for pairs.

� Transverse stiffeners in web panels with longitudinal

stiffeners must also satisfy:

b DI I

0.15 (1 ) 18

1.8 sin

2.4 sin

ywus w

w v n crs

crs ys

FVDA B C t

where F elastic local buckling stress

and B for stiffener pairs

B for gle angle stiffener

B for gle plate stiffener

≥ − −

� The stiffener strength must be greater than that required for

TFA to develop. Therefore, the area requirement is:

� If this equation gives As negative, it means that the web alone

is strong enough to develop the TFA forces. The stiffener

must be proportions for m.o.i. and width alone

� Bearing Stiffeners must be placed on the web of built-up

sections at all bearing locations. Either bearing stiffeners will

be provided or the web will be checked for the limit states of:

� Web yielding – Art. D6.5.2

� Web crippling – Art. D6.5.3

� Bearing stiffeners will consist of one or more plates or

angles welded or bolted to both sides of the web. The stiffeners will extend the full depth of the web and as closely

as practical to the outer edges of the flanges.

� The stiffeners shall be either mille to bear against the flange

or attached by full penetration welds.

� To prevent local buckling before yielding, the following

should be satisfied.

� The factored bearing resistance for the fitted ends of

bearing stiffeners shall be taken as:

� The axial resistance shall be determined per column

provisions. The effective column length is 0.75D

� It is not D because of the restraint offered by the top and bottom flanges.

Etb 48.0≤

( ) 1.4sb pn ysnR A F=

Interior panel End

3≤ Ddo 5.1≤

9tw 9tw 9tw

General Considerations

� Shear studs are needed to transfer the horizontal shear that is developed between the concrete slab and steel beam.

� AASHTO-LRFD requires that full transfer (i.e. full composite action) must be achieved.

� Shear studs are placed throughout both simple and continuous spans.

� Two limit states must be considered: fatigue and shear. Fatigue is discussed later.

Strength of Shear Studs

uscccscn FAEfAQ ≤= '5.0

Cross-sectional are of the stud in square inches

Minimum tensile strength of the stud (usually 60 ksi)

nscr QQ φ=

Placement

� A sufficient number of shear studs should be placed

between a point of zero moment and adjacent points of

maximum moment.

� It is permissible to evenly distribute the shear studs along the length they are needed in (between point of inflection

and point of maximum moment), since the studs have the

necessary ductility to accommodate the redistribution that

will take place.

Miscellaneous Rules

� Minimum length = 4 x stud diameter � Minimum longitudinal spacing = 4 x stud diameter

� Minimum transverse spacing = 4 x stud diameter

� Maximum longitudinal spacing = 8 x slab thickness

� Minimum lateral cover = 1".

� Minimum vertical cover = 2”. � Minimum penetration into deck = 2”

elastic stresses in unshored composite sectionlibvolume3.xyz/civil/btech/semester7/designand... ·...

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