CAN/CSA-S157-05/S157.1-05 April 2007
8 Local buckling of flat elements .1 Buckling stress
8.4 Elements supported at one edge with a lip at the other edge .1 General shapes
For the general case of a flange element attached at one longitudinal edge to a web element with a lip at the other longitudinal edge (see Figure C8 of the Commentary), subjected to uniform compressive stress, the slenderness, λ, used to obtain the buckling stress shall be given by
where
Ip = polar moment of inertia of flange and stiffener about the supported edge J = St. Venant torsion constant for flange and stiffener
Cw= warping constant for rotation of the flange and stiffener about the supported edge
= Is w 2 where
Is = moment of inertia of the stiffener about the inside surface of the flange to which it is attached;
this applies to all types of stiffener, including inclined lips and bulbs
w = flange width measured from the intersection of the median lines of the flange and web k = the spring constant for the restraint provided by the connection between flange and web
= 3 t13/16 (a + 0.5w) for channel and Z sections
= 1.5 t13/16 (a + 0.5w) for I-sections where
t1 = web thickness
a = breadth of the web element to which the flange is connected
8.4.2 Uniform thickness with simple lips 8.4.2.1
For a shape of uniform thickness, with simple lips on the flanges (see Figure C8 of the Commentary), the slenderness, λ, shall be given by
Δ
© Canadian Standards Association Strength design in aluminum
April 2007
9.7.3.2 Members with biaxial moments, not subject to lateral-torsional buckling
For members subjected to axial force combined with bending about both principal axes, where the same extreme fibre carries the maximum stress from both moments and the member fails in flexure, the limiting condition shall be given by
where
Cf = factored compressive force, not greater than the value of Cr given by Clause 9.4.1 Mxf = moment in the member due to the factored lateral load, about the strong axis Myf = moment in the member due to the factored lateral load, about the weak axis φy = resistance factor on the yield strength
Fo = limiting stress (see Clause 9.3.2) A = gross area
Sx = section modulus of the gross section about the strong axis Sy = section modulus of the gross section about the weak axis Cex = Aπ2 E/λx2
rx = radius of gyration of the gross cross-section about the strong axis Cey = Aπ 2 E/λy2
where λy = L/ry
where
ry = radius of gyration of the gross section about the weak axis
9.7.3.3 Members subject to lateral-torsional buckling
For combined axial force and bending about the strong axis, when lateral buckling can occur, the combined factored axial load, Cf , and bending moment, Mf , shall satisfy
where
Cf = applied compressive force due to the factored loads
Mf = moment due to the factored lateral load, or as calculated in Clause 9.5.3.3 Cry = factored resistance for failure about the weak axis, obtained from Clause 9.4.1 Mr = factored moment resistance obtained from Clause 9.5.3.1 or 9.5.3.2, as applicable Cex = elastic buckling force for bending about the strong axis
= Aπ 2 E/λx2
(Replaces p. 33, February 2005)
Licensed for/Autorisé à Reid Costley, Cascade Engineering Group, Sold by/vendu par CSA on/le 10/27/2009. Single user license only. Storage, distribution or use on network prohibited./Permis d'utilisateur simple seulement. Le stockage, la distribution ou l'utilisation sur le réseau est interdit.
S157-05 © Canadian Standards Association
E = elastic modulus λx = KL/rx
where
K = effective length factor (see Table 4 for typical values) L = unbraced length
rx = radius of gyration of the gross cross-section about the strong axis
9.7.4 Eccentric compression 9.7.4.1 General case
For general cases of eccentric compression, the following requirements shall apply:
(a) For failure in the plane of bending, Clause 9.7.3.1(a) or (b) shall be used with a factored moment, Mf , given by
Mf = 1.2eCf
(b) For lateral-torsional buckling, Clause 9.7.3.3 shall be used with a moment, Mf , given by Mf = eCf
where
e = eccentricity
Cf = factored axial force
9.7.4.2 Single angle bracing members
For single angle bracing members,
(a) the factored compressive resistance, Cr , of discontinuous single angles connected through one leg shall be given by Clause 9.4.1 using
where λv = KL/rv
where
K = effective length factor (see Table 4 for typical values) L = unbraced length of member
rv = minimum radius of gyration λt= 5 w/t
where
w = width of longer leg, see Clause 9.4.3.1.2 t = thickness of longer leg
(b) the factored resistance, Cr , shall not exceed (i) φc 0.5AFy for single bolt connections; or
(ii) φc 0.67AFy for double bolt or welded connections where
A = gross cross-sectional area Fy = yield strength
Δ
λ =
(
λv2 + λt2 1/2)
© Canadian Standards Association Strength design in aluminum
April 2007
m = number of shear planes
A = nominal cross-sectional area of the fastener Fu = ultimate strength of the fastener material
If the bolt threads are in a shear plane, the value of Vr shall be multiplied by 0.75.
11.2.3.2 Tensile resistance
The factored tensile resistance of a bolt, Tr , shall be the lesser of the values given by (a) Tr = φf 0.75 A Fu ; and
(b) Tr = φf A Fy where
φf = fastener resistance factor
A = cross-sectional area of the bolt based on the nominal diameter Fu = ultimate strength of the bolt material
Fy = yield strength of the bolt material Rivets are not commonly used in tension.
11.2.3.3 Combined shear and tension
For a bolt subject to both shear and tension, exclusive of tension due to tightening, the reduced factored tensile resistance, T’r , shall be given by the following:
where
Tr = factored tensile resistance given in Clause 11.2.3.2
k = 1.8, or 1.4 when the bolt thread is excluded from the shear plane Vf = factored shear load on the bolt
11.2.4 Bolts and rivets in bearing 11.2.4.1 Bearing strength
The factored bearing resistance, Br , of the connected material for each loaded fastener shall be the lesser of the values given by the following formulas:
(a) Br = φu etFu; and (b) Br = φu 2dtFu where
φu = ultimate resistance factor
e = perpendicular distance from the hole centre to the end edge in the direction of the loading (not less than 1.5d )
t = plate thickness
Fu = ultimate strength of the connected material d = fastener diameter
11.2.4.2 Lap joints
For unrestrained lap joints in tension, the factored bearing resistance, Br , shall be the lesser of the values given by the following formulas:
(a) Br = φu(t1 + t2) e Fu /4; and
(Replaces p. 45, February 2005)
Licensed for/Autorisé à Reid Costley, Cascade Engineering Group, Sold by/vendu par CSA on/le 10/27/2009. Single user license only. Storage, distribution or use on network prohibited./Permis d'utilisateur simple seulement. Le stockage, la distribution ou l'utilisation sur le réseau est interdit.
S157-05 © Canadian Standards Association
t1, t2 = thicknesses of the plates, t1 < t2
e = distance from the hole centre to the end edge, but not less than 1.5d Fu = ultimate tensile strength of the connected material
d = fastener diameter
11.2.4.3 Oblique end edges
Where the end edge is oblique to the line of action of the tension force (see Figure C23 of the
Commentary), the factored bearing resistance, Br , for a single bolt shall be the lesser of the values given by the following formulas:
(a) Br = φu [e + (e – do) cos2 θ] tFu ; and (b) Br = φu 2 dtFu
where
φu = ultimate resistance factor
e = perpendicular distance from the hole centre to the end edge do = hole diameter
d = fastener diameter
θ = angle made by the end edge with the direction of the force t = plate thickness
Fu = ultimate strength of the connected material
11.2.5 Tear-out of bolt and rivet groups (block shear) 11.2.5.1 Tension: Rectangular patterns
For a group of two or more fasteners in a rectangular pattern (see Figure C24(a) of the Commentary) resisting a force directed towards the edge, the factored bearing resistance, Rb , of the group of fasteners shall be the lesser of the values given by the following formulas:
(a) Rb = φu [(m – 1)(g – do ) + (n – 1)(s – do ) + e]tFu; and (b) Rb = φu 2 NdtFu
where
φu = ultimate resistance factor
m = number of fasteners in the first transverse row
g = fastener spacing measured perpendicular to the direction of the force do = hole diameter
n = number of transverse rows of fasteners
s = fastener spacing measured in the direction of the force
e = edge distance in the direction of force for the first row, but not less than 1.5d
= 2d, when e > 2d t = plate thickness
Fu = ultimate strength of the connected material N = total number of fasteners
d = fastener diameter
11.2.5.2 Tension: Trapezoidal patterns
For a triangular or trapezoidal group of fasteners in a staggered pattern (see Figure C24(b) of the
Commentary) resisting a force directed towards the edge, the factored bearing resistance, Rb , of the group shall be the lesser of the values given by the following formulas:
(a) Rb = φu [2 (m – 1)(g – do + s2/4g) + e] tFu; and (b) Rb = φu 2NdtFu
where
φu = ultimate resistance factor
m = number of fasteners in the first transverse row Δ
© Canadian Standards Association Strength design in aluminum
April 2007