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


Short-term composite section


Permanent loads applied before the slab has hardened, are
carried by the steel 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)
The procedure shown in this
picture is only valid if the neutral
axis is not in the concrete.
Use iterations otherwise.
b
My t
f ct 
nI tr
t
b/n
t
yt
yb
Original section
Transformed section
f sb 
My b
I tr
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
12  b (3   3 )
Rh 
12  2b

Where, b=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
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:
Fcrw 
0.9 E k
D
 
 tw 
2
where, k  bend buckling coefficient 
9
 Dc / D 2
where, Dc  depth of web in compression in elastic range

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
Web Bend-Buckling Reduction (6.10.1.10)

If the previous conditions are not met then:

 2 Dc

awc
Rb  1  



rw   1.0
 1200  300 awc  tw

where, rw  5.7
and
awc 
E
Fyc
2 Dc tw
b fc t fc
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
Dc 

f DC1  f DC 2  fW S  f LL  IM
tf
f DC1 f DC 2  fW S f LL  IM


csteel
c3n
cn
In lieu, you can use
  fc 
Dc  
d  t fc  0
 f c  ft 


where, d  depth of steel sec tion
f c and ft 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:
'


F
A

F
A

0
.
85
f
D yt t
yc c
c As  Fyr Ar
Dcp  
 1
2 
Fyw Aw

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:
bf  D / 6
bf
2t f
 12.0
t f  1.1 tw
0.1 
I yc
I yt
 10
Section Behavior
Moment
Mp
My
Compact
Noncompact
Slender
Curvature
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
2 Dcp
tw
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




E
 3.76
Fyc
6.10.7 Flexural Resistance
Composite Sections in Positive Flexure
Compact sections
 At the strength limit state, the section must satisfy
1
M u  fl S xt   f M n
3
 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.
6.10.7 Flexural Resistance
Composite Sections in Positive Flexure
Non-Compact sections (6.10.7.2)
 At the strength limit state:

The compression flange must satisfy fbu ≤ f Fnc

The tension flange must satisfy fbu + fl/3 ≤ f 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
6.10.7 Flexural Resistance
Composite Sections in Positive Flexure

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 Noncomposite Sections (6.10.6.2.2)
Sections with Fyf ≤ 70 ksi
Web satisfies the non-compact slenderness limit
2 Dc
E
 5.7
tw
Fyc



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
6.10.8 Flexural Resistance Composite Sections in
Negative Flexure and Non-Composite Section

Discretely braced flanges in compression
fbu 

Discretely braced flanges in tension
fbu 

1
fl   f Fnc
3
1
fl   f Fnt
3
Continuously braced flanges: fbu≤ f 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
Flange Local buckling or Lateral Torsional
Buckling Resistance
Fn or Mn
Fmax or Mmax
Inelastic Buckling
(Compact)
Inelastic Buckling
(non-compact)
Fyr or Mr
Elastic Buckling
(Slender)
Lp
Lr
pf
rf
Lb
f
6.10.8 Flexural Resistance Composite Sections in
Negative Flexure and Non-Composite Section

Fnc Compression flange flexural resistance – local buckling
f 
b fc
2 t fc
 pf  0.38
E
Fyc
rf  0.56
E
Fyr
When,  f   pf
Fnc  Rb Rh Fyc
When,  f  rf
 
Fyr    f   pf
Fnc  1  1 

  Rh Fyc   rf   pf
Fyr  0.7 Fyc

 Rb Rh Fyc

Fnc Compression flange flexural resistance
Lateral torsional buckling
E
Fyc
L p  1.0 rt
Lb
When, Lb  L p
rf   rt
Fnc  Rb Rh Fyc
 
Fyr   Lb  L p 
Fnc  Cb 1  1 

 Rb Rh Fyc  Rb Rh Fyc
  Rh Fyc   Lr  L p 
Fnc  Fcr  Rb Rh Fyc
When, Lb  Lr
When, Lb  Lr
Where,
2
2
 f 
 f 
Cb  1.75  1.05  1   0.3  1   2.3
 f2 
 f2 
Fcr 
rt 
E
Fyr
Cb Rb  2 E
2
 Lb 
 
 rt 
b fc

Dt
12 1  c w
 3 b fc t fc





Lateral Torsional Buckling
Unstiffened Web Buckling in Shear
Web plastification in shear
Vn  V p  0.58Fyw D.t w
Inelastic web buckling
Vn  1.48tw2 EFyw
Elastic web buckling
3
4.55t w E
Vn 
D
2.46
E
Fy
3.07
E
Fy
D/tw
6.10.9 Shear Resistance – Unstiffened webs



At the strength limit state, the webs must satisfy:
Vu ≤ v 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
D
Ek
If ,
 1.12
;
tw
Fyw
If ,
If ,
1.12
then C  1.0
Ek
D
Ek
1.12

 1.40
; then C 
D
Fyw tw
Fyw
tw
D
Ek
 1.40
;
tw
Fyw
then C 
1.57
D
 
tw
2
Ek
Fyw

Ek
Fyw
Tension Field Action
Vn  Vcr  VTFA
Beam Action
Tension Field Action
D
g
d0
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
If the sec tion is proportioned such that :
2 D tw
 b fct fc  b ft t ft 





0.87 (1  C ) 
Vn  0.58 Fyw D tw C 

2

 do  
1   

 D  

where, d o  transverse stiffener spacing
k  shear buckling coefficient  5 
If not , then Vn  0.58 Fyw
5
 do 
D
 
2



0.87 (1  C )
D t w C 
2

do
 do 
1



D
D
 








 2.5
Transverse Stiffener Spacing
D
Interior panel
End
panel
d o  1.5D
d o  3D
do  1.5D
Types of Stiffeners
Transverse
Intermediate
Stiffener
Longitudinal
Stiffener
D
do  1.5D
do  1.5D
Bearing
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
Single Plate
Angle
Double Plate
Less than 4 tw or more than 6tw
6.10.11 Design of Stiffeners



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:
 bt  D 
I t   
 Il
 bl  3.0 do 
6.10.11 Design of Stiffeners

The stiffener strength must be greater than that required for
TFA to develop. Therefore, the area requirement is:
 Fyw 2

 Vu 
D
tw
As  0.15 B (1  C ) 
  18
F
V

t
 crs

w
 v n
where, Fcrs  elastic local buckling stress
Fcrs 
0.31E
2
 Fys
 bt 
 
 tp 
 
and , B  1.0 for stiffener pairs
B  1.8 for sin gle angle stiffener
B  2.4 for sin gle plate stiffener

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
6.10.11 Design of Stiffeners

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.
6.10.11 Design of Stiffeners

To prevent local buckling before yielding, the following
should be satisfied.
E
bt  0.48t p
Fys

The factored bearing resistance for the fitted ends of
bearing stiffeners shall be taken as:
 Rsb n  1.4 Apn Fys

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.
6.10.11 Design of Stiffeners
D
Interior panel
d o  3D
End
panel
d o  1.5D
tp
bt
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
Qr  scQn
0.85
Cross-sectional are of the stud in square inches
Qn  0.5 Asc
'
f c Ec
 Asc Fu
Minimum tensile strength of the stud (usually 60 ksi)
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”