The resistance of cross sections subject
to shear and bending – theoretical
analysis and practical design rules
Sections subject to both bending and shear have a reduced bending resistance where the
shear force is greater than half the shear resistance. Richard Henderson of the SCI discusses the
background and design rules.
NSC 27
April 18
Work carried out between 1930 and 1965 on the resistance
of cross sections capable of being designed plastically was
presented by Baker, Horne and Heyman1. Theoretical treatments
of the effect of shear force on the resistance moment of sections
were developed and were subsequently compared with tests.
The design rules presented in BS 5950-1:2000 and subsequently
in BS EN 1993-1-1 were based on this work.
Horne2 examined rectangular and I sections and developed
expressions for the reduction in the bending resistance of
cross sections where the sections are subject to both bending
and shear. In the examination, the sections are assumed to
be capable of carrying their full plastic moment: sections are
assumed to be restrained from global buckling and I sections are
either class 1 or class 2 according to EC3.
Rectangular Section
A rectangular section will carry a bending moment equal to
its elastic moment of resistance where only the extreme fibres
reach yield stress. The remainder of the cross section is able to
resist a shear force. The shear stress distribution is parabolic
over the depth of the section and is zero at the extreme fibres
with a maximum value at the neutral axis. The average shear
stress is two thirds of the maximum value. If the bending
moment is increased above the elastic moment of resistance,
the area of the section available to resist shear is reduced until
it vanishes when the plastic moment of resistance is reached. At
this point, the whole section reaches its yield stress. The plastic
resistance moment of the section is Mp = (bh2/4)f. and its plastic
yshear resistance is Vy = bhτy if the bending and shear are each
considered on their own.
When the bending moment is between the elastic and plastic
moment of resistance, the elastic core of the section has a depth
yo above and below the neutral axis and yo < h/2 where h is the
depth of the section. The resistance moment is given by the sum
of the plastic moment of resistance of the outer portion and the
elastic moment of resistance of the core:
M = b/4(h2 – 4y2)σy + 2/3by2σo
o
y
and the shear resistance is provided by the core and given by
V = 4/3byoτy.
Eliminating y0 and using the expressions for Mp and Vp gives:
Mpr/Mp = 1 – 3/4(V/Vp)2 (1)
Mpr is the reduced plastic moment of resistance in the presence
of shear. The expression is valid for values of V up to that for
which yo = h/2 ie V/Vp ≤ 2/3.
Horne showed that using the Tresca yield criterion, a less
conservative estimate is given by Mpr/Mp = 1 – 0.444(V/Vp)2
provided V/Vp ≤ 0.792.
The interaction between shear and bending according to this
expression is shown in Figure 1
Figure 1: Interaction of shear and bending – Rectangular section
According to the less conservative estimate, the bending
resistance of the section is about 89% of the plastic resistance
moment when the shear force is half the shear resistance.
I Section
A similar analysis can be made of an I section, if the shear
stresses are assumed only to be in the web. The plastic resistance
moment of the web is denoted by Mpw = (d2t/4)σy and the
w
wshear resistance by Vpw = dtτw , where dw and tw are the depth
wwand thickness of the web. Using equation 1, the reduced plastic
moment is given by:
Mpr = Mp – 3/4(V/V)2 M.
pwpwThis equation is valid provided V/Vpw ≤ 2/3 which means that
the plastic zones in the section extend beyond the flanges and
into the web.
Horne and Morris3 discussed the effect of shear force on the
plastic moment, assuming the web of the I section provides all
the shear resistance and the shear stress τw is assumed to be
uniform over the depth of the web. The longitudinal bending
stress in the web is reduced because of the presence of the shear
stress to a value which can be determined using the Von Mises
yield criterion: σw = f2 − 3τ20.5. The reduction in longitudinal
y
w
bending stress in the web results in a reduced bending resistance
given by:
Mpr = Mp – Mpw 1 – {1 – (V/Vpw)2}0.5
The interaction between the bending moments and the
ratio of the applied shear force and shear resistance is shown in
Figure 2.
28
Technical
/Design#BS_5950
/Design_codes_and_standards#General_rules_and_rules_for_buildings
/Design_codes_and_standards#Eurocode_3_-_Steel_structures
/Steel_material_properties#Yield_strength
/Member_design#Bending
/Member_design#Bending_and_shear
/Member_design#Shear
