# 4 PLASTIC DESIGN METHOD

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26 Νοε 2013 (πριν από 4 χρόνια και 6 μήνες)

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1

4

PLASTIC DESIGN METHOD

4.1 Positive Bending Moment

The ultimate load resistance of a simply supported beam is determined by the moment of
resistance of the critical cross
-
section [5]. The determination of the moment of resistance of
the cross
-
section is b
ased on the following assumptions:

a. The shear connectors are able to transfer the forces occurring between the steel and the
concrete at failure (full shear connection).

b. No slip occurs between the steel and the concrete (complete interaction).

c. Tens
ion in concrete is neglected.

d. The strains caused by bending are directly proportional to the distance from the neutral
axis; in other words, plane cross
-
sections remain plane after bending, even at failure.

e. The relationship between the stress

a
, and

the strain

a

of steel is schematically represented
by the diagram shown in Figure 9a.

f. The relation between the stress

c

and the strain

c

of concrete is schematically represented
by the diagram shown in Figure 9b.

Both materials are assumed to behave in a perfectly plastic manner, and therefore, the strains
are not limited. This assumption is similar to that made when calculating the plas
tic moment

2

resistance for Class 1 steel sections used independently. The idealised diagram for steel is
shown in Figure 9a. The deviation between the real and the idealised diagram is much smaller
than for concrete as shown in Figure 9b. The use of f
ck

for

the maximum stress in the concrete
will clearly result in an unconservative design although in practice the overestimate does not
appear to be very significant. To allow for this overestimate a conservative approximation for
concrete strength (kf
ck
) is us
ed in design.

Experimental research has proved that the plastic method with k = 0,85, leads to a safe value
for the moment of resistance. This is only true if the upper flange cross
-
section is less than or
equal to that of the lower flange, as will usually

be the case.

Application of these assumptions leads to the stress distributions shown in Figures 10
-

12.
Clearly, the calculation of the moment of resistance M
c

is dependent on the position of the
neutral axis, which is determined by the relationship bet
ween the cross
-
section of the concrete
slab and the cross
-
section of the steel beam. Two cases can be identified as follows:

a. the neutral axis is situated in the concrete slab:

1. in the solid part of the composite slab (R
s

< R
c
; see Figure 10)

2. in the

rib of the composite slab (R
s
= R
c
)

b. the neutral axis is situated in the steel beam:

1. in the flange of the steel section (R
s

> R
c

> R
w
;

see Figure 11)

2. in the web of the steel section (R
s

> R
c

< R
w
; see Figure 13)

3

4

The plastic moment resistance, assuming full shear connection and a symmetric steel section,
is expressed in terms of the resistance of various elements of the beam as follows:

Resistance of concrete

flange : R
c

= b
eff

h
c

0,85 f
ck

/

c

Resistance of steel flange : R
f

= b t
f

f
y

/

a

Resistance of shear connection : R
q

= N Q

Resistance of steel beam : R
s

= A f
y

/

a

5

Resistance of clear web depth : R
v

= d t
w

f
y

/

a

Resistance of overall web depth : R
w

= R
s

-

2 R
f

where

A is the area of steel beam

b is the breadth of steel flange

b
eff

is the effective breadth of concrete flange

h is the overall depth of the steel beam

h
p

is the depth of profiled steel sheet

h
c

is the depth of concrete flange above upper flan
ge of profiled steel sheet

d is the clear depth of web between fillets

f
ck

is the characteristic cylinder compressive strength of the concrete

M
pl

is the plastic moment resistance of steel beam

N is the number of shear connectors in shear span length betwe
en two critical cross
-
sections

Q is the resistance of one shear connector

t
f

is the thickness of steel flange

t
w

is the thickness of web

is

Full shear connection applies when R
q

is greater than (or equal to) the lesser of R
c

and R
s
.

The concrete flange is assumed to be a solid concrete slab, or a composite slab with profiled
steel sheets running perpendicular to the beam. The Equations are con
servative for a
composite slab where the profiled steel sheets run parallel to the beam because in the
resistance R
c
, the concrete in the ribs is neglected.

For a composite section with full shear connection, where the steel beam has equal flanges,
the pla
stic moment resistance M
c

for positive moments is given by the following:

Case a1
:

If the neutral axis is situated in the concrete flange as shown in Figure 10, R
s

< R
c

and the
positive bending moment of resistance is:

6

M
pl.Rd

= R
s

z

where: z = h/2 + h
p

+ h
c

-

x/2

x = (Af
y

/

a
) / (b
eff

kf
ck

/

c
).h
c

=

(R
s
/R
c

).h
c

M
pl.Rd

= R
c

(h/2 + h
p

+ h
c

-

R
s
.h
c
/2R
c
)

(3)

Case a2
:

If the neutral axis is situated in the rib of the composite slab, R
s

= R
c

and Equation (3) can be
rewritten as:

M
pl.Rd

= R
s

(h + 2h
p
+ h
c
)/2

or, M
pl.Rd

= R
s
.h/2+ R
c
.(h
c
/2 + h
p
)

(4)

Case b1
:

If the neutral axis is situated in the steel flange, R
s

> R
c

> R
w
. From equilibrium of normal
forces it can be shown that the axial compression force R in th
e steel flange (see Figure 11) is:

R
c

+ R = R
s

-

2 R + R

2 R = R
s

-

R
c

R = (R
s

-

R
c
)/2

This axial force R is located in the middle of the upper part of the flange, with a depth equal
to: (Rt
f
)/R
f
= (R
s

-

R
c
).t
f
/2R
f

. Therefore, the moment of resistance is equal to the resistance
expressed by the Equation (
4) minus (2R)½(R
s

-

R
c
).t
f
/2R
f

equal to (R
s
-
R
c
)
2
.t
f
/4R
f

as
illustrated in Figure 11.

This can be written as:

M
pl.Rd

= R
s
.h/2+ R
c

(h
c
/2 + h
p
)
-

(R
s

-

R
c
)
2
.t
f
/4R
f

(5)

Case b2
:

If the neutral axis is in the web of the steel section, R
s

> R
c

<
R
w
. In this case, a part of the
web is in compression and, as already discussed, this could influence the classification of the
web.

Webs not fully effective ("non
-
compact webs") are not treated in this lecture. If the depth to
thickness ratio of the web o
f a steel section is less than or equal to 83

/(1
-
R
c
/R
v
) where

=
, it is considered as a compact web and the total depth is effective. The p
ositive
bending resistance is as illustrated in Figure 12:

7

M
pl.Rd

= R
c

z + M
pl,N
-
red.Rd

= R
c
.(h + 2 h
p
+ h
c
)/2+ M
pl,N
-
red.Rd

(6)

where:

M
pl,N
-
red.Rd

= plastic moment resistance of the steel beam reduced by a normal force R
c
.

According to Eurocode 3 [2] the plastic moment reduced by a normal force for standard rolled
I and H steel sections, can be approximated by:

M
pl,N
-
red.Rd

= 1,11 M
pl.a.Rd

(1

-

R
c
/R
s
)

M
pl.Rd

(7)

So the resistance can be written as:

M
pl.Rd

= R
c
.(h + 2 h
p
+ h
c
)/2 + 1,11 M
pl.a.Rd

(1
-

R
c
/R
s
) (8)

8

M
pl,N
-
red.Rd

can also be written as M
pl.a.Rd

-

(R
c
2
/R
v
2
)(d/4)

In this case the moment of resistance is:

M
pl.Rd

= R
c
.(h + 2 h
p
+ h
c
)/2

+ M
pl.a.Rd

-

(R
c
2
/R
v
2
)(d/4) (9)

The formulae for the positive moment of resistance values are summarised in Table 2.