Elastic Stresses in Unshored Composite Section

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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, 3
n
, should be used.


Elastic Stresses
(6.10.1.1)

Original section
Transformed section
b
b/n
y
t
y
b
t
t
tr
b
sb
I
My
f

tr
t
ct
nI
My
f

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 f
yf


fyw
≥ 0.70 f
yf

and
f
yw
≥ 36 ksi



The hybrid girder reduction factor = R
h






Where,
b
=2 D
n

t
w

/ A
fn



D
n

= larger of distance from elastic NA to inside flange face



A
fn

= flange area on the side of NA corresponding to D
n



f
n

= yield stress corresponding to A
fn



b


b
2
12
)
3
(
12
3




h
R
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 R
b
=1.0



When the section has one or more longitudinal stiffeners,

and D/t
w
≤ 0.95 (E k /F
yc
)
0.5

then R
b

= 1.0



When 2D
c
/t
w
≤ 5.7 (E / F
yc
)
0.5

then R
b

= 1.0




2
2
0.9
9
,
/
,
crw
w
c
c
Ek
F
D
t
where k bend buckling coefficient
D D
where D depth of web in compression in elastic ra
nge

 
 
 
 

Web Bend
-
Buckling Reduction
(6.10.1.10)


If the previous conditions are not met then:

2
1 1.0
1200 300
,5.7
2
wc c
b rw
wc w
rw
yc
c w
wc
fc fc
a D
R
a t
E
where
F
D t
and a
b t


  
   
  

  


Calculating the depth D
c

and D
cp

(App. D6.3)


For composite sections in positive flexure, the depth of the
web in compression in the elastic range D
c
, 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

f
n
IM
LL
n
WS
DC
steel
DC
IM
LL
WS
DC
DC
c
t
c
f
c
f
f
c
f
f
f
f
f
D










3
2
1
2
1
0
,sec
c
c fc
c t
c t
f
D d t
f f
where d depth of steel tion
f and f are the compression and tension flange st
resses
 

  
 
 

 

Calculating the depth D
c

and D
cp

(App. D6.3)


For composite sections in positive flexure, the depth of the
web in compression at the plastic moment D
cp

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














1
85
.
0
2
'
w
yw
r
yr
s
c
c
yc
t
yt
cp
A
F
A
F
A
f
A
F
A
F
D
D
6.10 I
-
shaped Steel Girder Design

Proportioning the section (6.10.2)


Webs without longitudinal stiffeners must be limited to


D/t
w

≤ 150


Webs with longitudinal stiffeners must be limited to


D/t
w
≤ 300


Compression and tension flanges must be proportioned
such that:




/6
12.0
2
1.1
0.1 10
f
f
f
f w
yc
yt
b D
b
t
t t
I
I



 
Slender

Noncompact

Compact

Moment

Curvature

M
p

M
y

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 (F
yf
)
≤ 70 ksi





where, D
cp

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

2
3.76
cp
w yc
D
E
t F

6.10.7 Flexural Resistance

Composite Sections in Positive Flexure

Compact sections


At the strength limit state, the section must satisfy




If D
p
≤ 0.1 D
t

, then M
n

= M
p



Otherwise, M
n

= M
p
(1.07


0.7 D
p
/D
t
)



Where, D
p

= distance from top of deck to the N.A. of the
composite section at the plastic moment.


D
t

= total depth of composite section


For continuous spans, M
n

= 1.3 M
y
. This limit allows for
better design with respect to moment redistributions.


1
3
n
u l xt f
M f S M

 
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
f
bu



f

F
nc



The tension flange must satisfy

f
bu

+ f
l
/3


f

F
nt



Nominal flexural resistance
F
nc

= R
b

R
h

F
yc



Nominal flexural resistance
F
nt
= R
h

F
yt



Where,


R
b

= web bend buckling reduction factor



R
h

= hybrid section reduction factor




Ductility requirement.

Compact and non
-
compact sections
shall satisfy D
p
≤ 0.42 D
t



This requirement intends to protect the concrete deck
from premature crushing. The D
p
/D
t

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.

6.10.7 Flexural Resistance

Composite Sections in Positive Flexure

6.10 I
-
Shaped Steel Girder Design


Composite Sections in Negative Flexure and Non
-
composite Sections (6.10.6.2.2)


Sections with F
yf
≤ 70 ksi


Web satisfies the non
-
compact slenderness limit





Where, D
c

= 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 M
y


If use App. A., get greater moment capacity than M
y



2
5.7
c
w yc
D
E
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: f
bu


f
R
h
F
yf



Compression flange flexural resistance = F
nc

shall be taken
as the smaller of the local buckling resistance and the
lateral torsional buckling resistance.



Tension flange flexural resistance = F
nt

= R
h

F
yt


1
3
nc
bu l f
f f F

 
1
3
nt
bu l f
f f F

 
Flange Local buckling or Lateral Torsional
Buckling Resistance

F
n

or M
n

L
b

Inelastic Buckling

(non
-
compact)

Elastic Buckling

(Slender)

L
p

F
max

or M
max

Inelastic Buckling

(Compact)


pf

L
r


rf


f

F
yr

or M
r

6.10.8 Flexural Resistance Composite Sections in
Negative Flexure and Non
-
Composite Section


F
nc

Compression flange flexural resistance


local buckling


0.38 0.56
2
,
,1 1
0.7
fc
f pf rf
fc yc yr
f pf nc b h yc
yr f pf
f rf nc b h yc
h yc rf pf
yr yc
b
E E
t F F
When F R R F
F
When F R R F
R F
F F
  
 
 
 
 
  
 
 
  

  
 
   
  

 
  
  
 

F
nc

Compression flange flexural resistance

Lateral torsional buckling

2 2
1 1
2 2
1.0
,
,1 1
,
,
1.75 1.05 0.3 2.3
b p t rf t
yc yr
b p nc b h yc
yr b p
b r nc b b h yc b h yc
h yc r p
b r nc cr b h yc
b
E E
L L r r
F F
When L L F R R F
F L L
When L L F C R R F R R F
R F L L
When L L F F R R F
Where
f f
C
f f
 
 
 
 
  

  
 
    
  

 
  
  
 
  
   
   
   
   
2
2
12 1
3
b b
cr
b
t
fc
t
c w
fc fc
C R E
F
L
r
b
r
D t
b t


 
 
 

 

 
 
 
Lateral Torsional Buckling

Unstiffened Web Buckling in Shear

y
F
E
46
.
2
D/t
w

Web plastification in shear

Inelastic web buckling

Elastic web buckling

y
F
E
07
.
3
w
yw
p
n
t
D
F
V
V
.
58
.
0


2
1.48
n w yw
V t EF

D
E
t
V
w
n
3
55
.
4

6.10.9 Shear Resistance


Unstiffened webs


At the strength limit state, the webs must satisfy:



V
u



v

V
n


Nominal resistance of unstiffened webs:


V
n

= V
cr

= C V
p


where, V
p

= 0.58 F
yw

D t
w


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


k = 5 for unstiffened webs

2
,1.12;1.0
1.12
,1.12 1.40;
1.57
,1.40;
w yw
yw w yw yw
w
w yw yw
w
D Ek
If then C
t F
Ek D Ek E k
If then C
D
F t F F
t
D Ek E k
If then C
t F F
D
t
 
  
  
 
 
 
Tension Field Action

d
0

D

g

n cr TFA
V 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 (d
o
) < 3D


With one or more longitudinal stiffeners and transverse
stiffener spacing (d
o
) < 1.5 D



The transverse stiffener distance for end panels with or
without longitudinal stiffeners must be d
o

< 1.5 D


The nominal shear resistance of end panel is


V
n

= C (0.58 F
yw

D t
w
)


For this case


k is obtained using equation shown on next
page and d
o

= distance to stiffener



Shear Resistance of Interior Panels of Stiffened Webs



2
2
2
sec:2.5
0.87 (1 )
0.58
1
,
5
5
,0.58
w
fc fc ft ft
n yw w
o
o
o
n
Dt
If the tion is proportioned such that
b t b t
C
V F Dt C
d
D
where d transverse stiffener spacing
k shear buckling coefficient
d
D
If not thenV


 
 
 

 
 
 
 

 
 
 
 
 

  
 
 
 

2
0.87 (1 )
1
yw w
o o
C
F Dt C
d d
D D
 
 
 


 
 
 
 
 
 
 
 
 
Transverse Stiffener Spacing

Interior panel

End

panel

D
d
o
3

D
d
o
5
.
1

D

1.5
o
d D

Types of Stiffeners

D

1.5
o
d D

1.5
o
d 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 t
w

or more than 6 tw.

Transverse Intermediate Stiffeners

Less than 4

t
w

or more than 6
t
w


Single Plate

Double Plate

Angle

6.10.11 Design of Stiffeners


Projecting width of transverse stiffeners must satisfy:



b
t

≥ 2.0 + d/30


and b
f
/4 ≤ b
t

≤ 16 t
p



The transverse stiffener’s moment of inertia must satisfy:


I
t

≥ d
o

t
w
3

J


where, J = required ratio of the rigidity of one transverse
stiffener to that of the web plate = 2.5 (D/d
o
)
2



2.0 ≥ 2.5


I
t

= 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:


3.0
t
t l
l o
b
D
I I
b d
  

  
  
6.10.11 Design of Stiffeners

2
2
0.15 (1 ) 18
,
0.31
,1.0
1.8 sin
2.4 sin
yw
u
s w
w v n crs
crs
crs ys
t
p
F
V
D
A B C t
t V F
where F elastic local buckling stress
E
F F
b
t
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 A
s

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.




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.


ys
p
t
F
E
t
b
48
.
0



1.4
sb pn ys
n
R A F

6.10.11 Design of Stiffeners

Interior panel

End

panel

D
d
o
3

D
d
o
5
.
1

D

b
t

t
p

9
t
w

9
t
w

9
t
w

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

u
sc
c
c
sc
n
F
A
E
f
A
Q


'
5
.
0
Cross
-
sectional are of the stud in square inches

Minimum tensile strength of the stud (usually 60 ksi)

n
sc
r
Q
Q


0.85

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”