Steel-Concrete Composite Beams

choruspillowUrban and Civil

Nov 29, 2013 (3 years and 10 months ago)

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Steel
-
Concrete Composite Beams

Steel


Concrete Composite Beams

AISC Chapter I

AISC pdf 138




Figure
1



Plastic stress distribution with compression in concrete slab



Figure
2

-

Strain Compatibility

Method

AISC pdf
144




Figure
3

-

Effective Width




Figure
4

-

Negative Moment Strength
-

Rebars in width



Composite Moment Capacity

Fully Composite versus Partially Composite

Steel
-
concrete composite beams are identified as being fully composite or
partially

composite. Lets
focus on positive moment strength. A fully composite steel
-
concrete composite b
eam has its rolled
shape entirely in tension. For rolled shapes with compact webs, the entire shape is at yield in tension. A
beam is fully composite
when

there are a sufficient number of shear connectors provided and the
concrete slab has adequate compres
sion strength.

Positive Moment Strength

For positive moment, the concrete slab carries compression. The steel shape carries tension

(fully
composite) or tension plus some
compression

(partially composite)
. The cross section is shown in
Figure
5
.


Figure
5


There are three cases to consider. The cases correspond to three potential locations of the plastic
neutral axis (PNA).
The PNA is the vertical location, usually expressed as a y
-
coordinate
, of the
transition
from compression stress to tension stress. For plastic analysis, stresses change from a maximum value
in compression to a maximum value in tension.

The three potential locations for the PNA are: 1)
At the top of steel
; 2)

In the top flange of t
he steel
shape, and; 3) In the web of the steel shape. We locate the PNA using equilibrium relations for
horizontal forces.
Total compressions due to bending must equal total tensions due to bending.






Eq.
1

PNA in Slab

A full
y composite beam has its steel shape entirely in tension. A fully composite beam has its PNA
at the
top of steel

(
Figure
6
).

The compression force in the slab is large enough to yield the steel shape in
tension.


Figure
6

The parameter



is the distance from the top of
the

steel falnge to the line of
action

of
the

compression force in the slab.

The thickness of the slab is

.

Moment strength is computed using a moment balance at the top of steel. The compression force


has a momen
t arm equal to


. The tension force


has

a moment arm equal to




when the steel
shape has all

fibers at yield in tension.















Eq.
2


Since


equals

, this becomes















Eq.
3


Tension and compression forces are related as

















Eq.
4


Eq.
3

is re
-
written as



























Eq.
5





























Eq.
6


PNA in Top Flange

The compression force in the concrete slab may not be large enough to yield the steel shape in tension.
How does this happen? Usually
, we make it happen. We choose to install fewer shear connectors, and
so limit the force sent into the concrete slab.

The sum of the strengths of shear connectors is the force sent to the concrete slab.















Eq.
7


We use
Eq.
7

to compute the depth of compression block

.

If the force,



, is less than the yield force





of the steel shape
, then force equilibrium requires
compression stress in a portion
of the steel shape. Let’s say that the portion of steel in compression is
contained in the top flange. The pattern of stresses is shown in
Figure
7
. A parameter



is the distance
from the top fo steel to the PNA.


Figure
7

Force equilibrium is































Eq.
8


Eq.
8

relates
total

compression to net tension in the steel shape. We use this equ
ation to
compute



.




















Eq.
9


The result of
Eq.
9

allows us to check whether the PNA is actually in the top flange. We compute



in
Eq.
9

and then compare









Eq.
10


For PNA in the top flange, moment strength


is































Eq.
11

PNA in Web

If we use even few
er shear connectors, there is less force in the concrete slab, and more of the steel
shape must be in compression to maintain force equilibrium. Note that the end point to this process is
one half of the steel shape in compression and one half in tension:

An
ordinary

steel
beam
.


For PNA in the web of the steel shape, force equilibrium becomes

























































Eq.
12


We use
Eq.
12

to compute

































Eq.
13


We use
Eq.
13

to check that the PNA is in the web.








Eq.
14


The moment strength is













































Eq.
15

Example
-

Composite Beam Flexural Strength
, Positiv
e Moment

Given:

W18 x 40


A992












As

11.8

in2


L

60

ft

d

17.9

in


P at

20

ft

bf

6.02

in


M

2 / 9 PL


tf

0.525

in





tw

0.315

in


f'c

3

ksi





t

6

in





be

10

ft

As Fy

590

k


Shear Connector

0.75

in

N

27.
6

conns


Qn

21
.
4

k

Case 1

P
20 ft
40 ft
28
28
Connectors

Concrete Crushing











1836

k

呥n獩s攠奩敬摩ng






590

k

卨敡爠eonne捴o牳





599

k


呥n獩s攠y楥汤楮g⁣on瑲o汳Ⱐ瑨敲敦e牥r瑨ts⁩猠愠晵aly⁣ompo獩瑥⁳ c瑩tn.

0.85f’c
Fy
Y2
d
a
t

a

1⸹.





5⸰.






619

武k

Case 2

P
20 ft
40 ft
15
15
Connectors

Concrete Crushing

1836

k

Tensile Yielding

590

k

Shear connectors

32
1

k


Shear connectors control. This is a partially composite beam.

0.85f’c
Fy
PNA
Y2
Y1
d
a
t

a

1.05

in

Y2

5.4
8

in

Y1

0.4
4
7

in





3

武k




Cas
e 3

P
20 ft
40 ft
8
8
Connectors

Concrete Crushing

1836

k

Tensile Yielding

590

k

Shear connectors

171

k


Shear connectors control. This is a partially composite beam.

0.85f’c
Fy
PNA
Y2
Y1
d
a
t

a

0.55
9

in

Y2

5.7
2

in

Y1

3.7
9

in





7

武k