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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 based 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. Tension 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 plastic moment
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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 fck 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 (kfck) is used 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 Mc is dependent on the position of the
neutral axis, which is determined by the relationship between 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 (Rs < Rc; see Figure 10)
2. in the rib of the composite slab (Rs= Rc)
b. the neutral axis is situated in the steel beam:
1. in the flange of the steel section (Rs > Rc > Rw; see Figure 11)
2. in the web of the steel section (Rs > Rc < Rw; see Figure 13)
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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 : Rc = beff hc 0,85 fck /c
Resistance of steel flange : Rf = b tf fy /a
Resistance of shear connection : Rq = N Q
Resistance of steel beam : Rs = A fy /a
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Resistance of clear web depth : Rv = d tw fy /a
Resistance of overall web depth : Rw = Rs - 2 Rf
where
A is the area of steel beam
b is the breadth of steel flange
beff is the effective breadth of concrete flange
h is the overall depth of the steel beam
hp is the depth of profiled steel sheet
hc is the depth of concrete flange above upper flange of profiled steel sheet
d is the clear depth of web between fillets
fck is the characteristic cylinder compressive strength of the concrete
Mpl is the plastic moment resistance of steel beam
N is the number of shear connectors in shear span length between two critical cross-sections
Q is the resistance of one shear connector
tf is the thickness of steel flange
tw is the thickness of web
 is
Full shear connection applies when Rq is greater than (or equal to) the lesser of Rc and Rs.
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 conservative for a
composite slab where the profiled steel sheets run parallel to the beam because in the
resistance Rc, the concrete in the ribs is neglected.
For a composite section with full shear connection, where the steel beam has equal flanges,
the plastic moment resistance Mc 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, Rs < Rc and the
positive bending moment of resistance is:
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Mpl.Rd = Rs z
where: z = h/2 + hp + hc - x/2
x = (Afy /a) / (beff kfck /c).hc = (Rs/Rc ).hc
Mpl.Rd = Rc (h/2 + hp + hc - Rs.hc/2Rc)
(3)
Case a2:
If the neutral axis is situated in the rib of the composite slab, Rs = Rc and Equation (3) can be
rewritten as:
Mpl.Rd = Rs (h + 2hp + hc)/2
or, Mpl.Rd = Rs.h/2+ Rc.(hc/2 + hp)
(4)
Case b1:
If the neutral axis is situated in the steel flange, Rs > Rc > Rw. From equilibrium of normal
forces it can be shown that the axial compression force R in the steel flange (see Figure 11) is:
Rc + R = Rs - 2 R + R  2 R = Rs - Rc  R = (Rs - Rc)/2
This axial force R is located in the middle of the upper part of the flange, with a depth equal
to: (Rtf)/Rf = (Rs - Rc).tf/2Rf . Therefore, the moment of resistance is equal to the resistance
expressed by the Equation (4) minus (2R)½(Rs - Rc).tf/2Rf equal to (Rs-Rc)2.tf/4Rf as
illustrated in Figure 11.
This can be written as:
Mpl.Rd = Rs.h/2+ Rc (hc/2 + hp) - (Rs - Rc)2.tf/4Rf
(5)
Case b2:
If the neutral axis is in the web of the steel section, Rs > Rc < Rw. 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 of a steel section is less than or equal to 83/(1-Rc/Rv) where  =
, it is considered as a compact web and the total depth is effective. The positive
bending resistance is as illustrated in Figure 12:
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Mpl.Rd = Rc z + Mpl,N-red.Rd = Rc.(h + 2 hp + hc)/2+ Mpl,N-red.Rd (6)
where:
Mpl,N-red.Rd = plastic moment resistance of the steel beam reduced by a normal force Rc.
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:
Mpl,N-red.Rd = 1,11 Mpl.a.Rd (1 - Rc/Rs)  Mpl.Rd (7)
So the resistance can be written as:
Mpl.Rd = Rc.(h + 2 hp + hc)/2 + 1,11 Mpl.a.Rd (1 - Rc/Rs) (8)
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Mpl,N-red.Rd can also be written as Mpl.a.Rd - (Rc2/Rv2)(d/4)
In this case the moment of resistance is:
Mpl.Rd = Rc.(h + 2 hp + hc)/2 + Mpl.a.Rd - (Rc2/Rv2)(d/4) (9)
The formulae for the positive moment of resistance values are summarised in Table 2.