9. Resistance to transverse forces tables (web bearing and buckling)

9.1 UB sections, UC sections and parallel flange channels

(a)  The design shear resistance

The design shear resistance V_c,Rd is given by:

where:

Av	is the shear area
	for rolled I sections Av = A – 2btf + (tw + 2r) tf but not less than ηhwtw,
	for parallel flange channels Av = A – 2btf + (tw + r) tf
fy	is the yield strength
γM0	is the partial factor for resistance of cross sections (γM0 = 1.0 as given in the National Annex).

(b)  Design resistance to local buckling

The design resistance of an unstiffened web, F_Rd, to local buckling under transverse forces is given by:

where:

fy	is the yield strength
tw	is the thickness of the web
Leff	is the effective length for resistance to transverse force ( = χF ly )
χF	=  ≤ 1.0
ly	is the effective loaded length
γM1	is the partial factor for resistance of members (γM1 = 1.0 as given in the National Annex)
	=

Fcr	=
kF	=   but kF ≤ 6
hw	is the depth between flanges = h – 2tf
ss	is the length of stiff bearing (see figure 6.2 of BS EN 1993-1-5)
c	is the distance from the end of the stiff bearing to the end of the section.

The effective loaded length l_y has been calculated as the least value given by the following three expressions:

where:

le	=  ≤ ss + c	Eq (6.13)
m1	=
m2	=	if
m2	= 0	if

(Note that, at present, BS EN 1993-1-5, 6.5(3) erroneously refers to Equations 6.11, 6.12 and 6.13. This error is due to be corrected by CEN.)

Values of F_Rd have been calculated for two values of c, where c is the distance from the end of the member to the adjacent edge of the stiff bearing length, as shown in Figure 6.1 of EN 1993-1-5. The values are:

c        = 0      for a stiff bearing positioned at the end of the section

c        = c_lim   for a stiff bearing positioned at a distance c_lim from the end of the section. This position represents the minimum value of c at which the maximum resistance of the web for a given stiff bearing length, s_s is attained.

where:

clim
cbuckling
cbearing

For the case where the stiff bearing is positioned at an intermediate distance (i.e. c < c_lim) the resistance given for c = 0 is conservative.

9.2 Square and rectangular hollow sections

BS EN 1993-1-5 does not cover the resistance to transverse forces for hollow sections. Therefore in this website the approach previously presented in P202 Volume 1 Section properties member capacities^[25] has been adopted and is presented below in terminology consistent with BS EN 1993-1-1.

(a)  Bearing

The bearing resistance, F_Rd,bearing, of the unstiffened web may be calculated using the factors C₁ and C₂ in the tables, using:

where:

b1	is the effective bearing length (see figures below)
n	= 5, for continuous web over bearing
	= 2, for end bearing
k	= t, for hollow sections
t	is the wall thickness
fy	is the yield strength.

The bearing factor C₁ represents the contribution from the flanges, adjacent to both webs, and is given by:

C1	= 2 n k t fy / γM0	Generally
C1	= 2 x 5 t2 fy / γM0	For a section continuous over the bearing
C1	= 2 x 2 t2 fy / γM0	For end bearing

The bearing factor C₂ is equal to 2tf_y / γ_M0 representing the stiff bearing contribution for both webs.

(b)  Buckling resistance

The buckling resistance F_Rd,buckling of the two unstiffened webs is given by:

where:

b1	is the stiff bearing length
n1	is the length obtained by dispersion at 45° through half the depth of the section
t	is the wall thickness
χ	is the reduction factor for buckling resistance, based on the slenderness  as given in Section 6.1
fy	is the yield strength of the hollow section.

Unless loads or reactions are applied through welded flange plates, the additional effects of moments in the web due to eccentric loading must be taken into account, which will result in lower buckling values.

The buckling factor C₂ is the stiff bearing component factor and is equal to C₁/h

The buckling factor C₁ is the portion of (n₁ t χ f_y /γ_M1) due to the beam alone.

where:

F	is the limiting force in each web (derived below).

The factor of 4 allows for two webs and dispersion of load in two directions and applies to a member that is continuous over a bearing or an end bearing member with a continuously welded sealing plate.

For non–welded flange plates, the limiting force F depends on the equivalent eccentricity of loading from the centreline of the web given by:

This expression has been derived from research^[13] and is also applicable to cold‑formed hollow sections ^[18]

If the flange is considered as a fixed-ended beam of length b - t, the two forces F create a fixed end moment M;

and thus the moment at mid-height of the web M_z can be found as follows ^[26]:

where:

a

= h/b

Using the interaction criterion 

The limiting value of F is given when the left hand side of this criterion = 1.

If the length of the wall resisting F and M is h/2, given by a 45° dispersal in one direction, A = ht/2 and W_el,z = ht²/12. Substituting these values, introducing k = M_z /F, and rearranging, the limiting value becomes:

where:

nb

=

and k is given by the expression for M_z/F above.

(c)  Shear resistance

The shear resistance is given by:

where:

Av	is the shear area ( = A h / ( b + h ))
A	is the area of the cross section
h	is the overall depth
b	is the overall width.