Chapter 3
BEHAVIOUR AND RESISTANCE OF CROSS SECTION 3.1 GENERAL The individual components of cold-formed steel members are usually so thin with respect to their widths that they buckle at stress levels less than the yield point when subjected to compression, shear, bending or bearing. Local buckling of such elements is therefore one of the major considerations in cold-formed steel design. It is well known that, compared to other kinds of structures, thin plates are characterised by a stable post-critical behaviour (see Figure 3.1). Consequently, cold-formed steel sections, which can be regarded as an assembly of thin plates along the corner lines, will not necessarily fail when their local buckling stress is reached and they may continue to carry increasing loads in excess of that at which local buckling occurs. As already shown in Figure 1.16 of Chapter 1, Figure 3.2 summarises the difference in behaviour of a tick-walled slender bar in compression (see Figure 3.2a), and a thin-walled one (see Figure 3.2b). Both cases of ideal perfect bar and imperfect bar are presented. Figure 3.3 illustrates examples of local buckling patterns of beams and columns (Yu, 2000). In order to account for local buckling, when the resistance of cross sections and members is evaluated and checked for design purposes, the effective section properties have to be used. However, for members in tension the full section properties are used. The appropriate use of full and Design of Cold-formed Steel Structures, 1st Edition. Dan Dubina, Viorel Ungureanu and Raffaele Landolfo. © 2012 ECCS – European Convention for Constructional Steelwork. Published 2012 by ECCS – European Convention for Constructional Steelwork.
_____ 97
3. BEHAVIOUR AND RESISTANCE OF CROSS SECTION
effective cross section properties is explained in this section. Local buckling of individual walls of cold-formed steel sections is a major design criterion and, consequently, the design of such members should provide sufficient safety against failure by local instability with due consideration given to the post-buckling strength of structural components (e.g. walls).
P W
Perfect bar
Perfect bar
Pcr A
P u
Imperfect bar
lenght (unloaded)
W0
a) P
P
Perfect cylinder
w
length (unloaded)
_____ 98
Perfect cylindrical shell
Pcr
u
W
Imperfect cylindrical shell
W0
W
P
b)
P
c) length (unloaded)
w
P
Perfect plate
Pcr Imperfect plate
P u
Perfect plate
W0
W
Figure 3.1 – Post-critical behaviour of elastic structures: a) columns: indifferent post-critical path; b) cylinders: unstable post-critical path; c) plates: stable post-critical path
3.1 GENERAL
Figure 3.2 – Behaviour of (a) slender tick-walled and (b) thin-walled compression bar
_____ 99
a) Beams A-A A A
b) Columns Figure 3.3 – Local buckling of compression walls of cold-formed steel section
The present section provides details of the calculation procedure for the evaluation of cross section resistance for the following design actions: - axial tension; - axial compression;
3. BEHAVIOUR AND RESISTANCE OF CROSS SECTION
-
bending moment; combined bending and axial tension; combined bending and axial compression; torsional moment; shear force; local transverse forces; combined bending moment and shear force; combined bending moment and local transverse force.
Both local buckling and distortional buckling (sectional instability modes) effects on the cross section strength will be examined.
3.2 PROPERTIES OF GROSS CROSS SECTION 3.2.1 Nominal dimensions and idealisation of cross section The properties of the gross cross section shall be determined using specified nominal dimensions (width and thickness) of the walls and stiffeners composing the cross section (see Figure 3.4).
c
bs
h 1,3
y
d
y
C
h 1,1
y
t
h h 1,2
C
h
h
C
y v
t y
b
r
v y
r
t
c
r
_____ 100
z
z b c
z b1u
b2 u
z
z
z
Figure 3.4 – Nominal dimensions of gross cross section
Due to the manufacturing process, cold-formed steel sections have rounded corners. The notional flat widths of plane elements shall be measured from the midpoints of the adjacent corner elements as indicated in Figure 3.5 (CEN, 2006a).
3.2 PROPERTIES OF GROSS CROSS SECTION bp
gr
a) midpoint of corner or bend X
X is intersection of midlines P is midpoint of corner rm r t / 2
P r
t
I
I 2
§ §I¡ § I ¡¡ rm ¨ tan ¨ ¸ sin ¨ ¸ ¸ 2 Š 2 šš Š Š š
gr
2
b bp
sw
I
bp = sw
hw
h
c) notional flat width bp for a web (bp = slant height sw)
b bp
bp
bp,c c bp
bp,c c
d) notional flat width bp of plane parts adjacent to web stiffener bp
bp
bp,d d
e) notional flat width bp of flat parts b) notional flat width bp of plane adjacent to flange stiffener parts of flanges Figure 3.5 – Notional widths of plane cross section parts bp allowing for corner radii
Since many cold-formed steel sections have thin walls and small radii then in many cases of practical design, the determination of section properties can be simplified by assuming that the material is concentrated at the mid-line of the section and the corners are replaced by the intersections of flat elements. The accuracy of these assumptions may be illustrated by considering the lipped channel section shown in Figure 3.6. Figure 3.7a and 3.7b (Rhodes, 1991) illustrate the effects of using the simplified “mid-line�
_____ 101
3. BEHAVIOUR AND RESISTANCE OF CROSS SECTION
cross section on the geometrical properties of the gross cross section. In these figures, Ag/Ag,m-l denotes the ratio of the area calculated considering the round corners versus the “mid-line” one, while Ig/Ig,m-l correspondingly refers to second order moment of the area. As can be seen, the errors are small if radius and thickness are small, but they are significant and lead to an overestimation of the geometric properties if both thickness and radii are large. 150
150
t
25
25
t
50
50
All inside radii = t
25
25
Figure 3.6 – Lipped channel section 1.00
r=0 r=t r = 2t r = 3t r = 4t r = 5t
0.95 Ag/Ag,m-l
0.90
0.80 0.75 0.70
0
1
2 3 4 Thickness t (mm)
1.00
5
r=0 r=t r = 2t
0.95 0.90 Ig/Ig,m-l
_____ 102
0.85
r = 3t
0.85 0.80
r = 4t
0.75
r = 5t
0.70
0
1
2 3 4 Thickness t (mm)
5
Figure 3.7 – Effect of inside radius on calculated geometrical properties of lipped channel section
3.2 PROPERTIES OF GROSS CROSS SECTION
Generally, in order to avoid the overestimation of area and second moment of area the influence of rounded corners on section properties shall be taken into account. This may be done (CEN, 2006a) with sufficient accuracy by reducing the properties calculated for the same cross section with sharp corners (see Figure 3.8) using following approximations:
Ag | Ag , sh (1 G )
(3.1a)
I g | I g , sh (1 2G )
(3.1b)
I w | I w, sh (1 4G )
(3.1c)
with:
Ij
n
G
0.43
¦r j 1 m
j
90$
¦b
(3.1d)
p ,i
i 1
bp,i
_____ 103
Actual cross section Idealized cross section Figure 3.8 – Approximate allowance for rounded corners
where Ag Ag,sh bp,i Ig Ig,sh Iw Iw,sh
is the area of the gross cross section; is the value of Ag for a cross section with sharp corners; is the notional flat width of plane element i for a cross section with sharp corners; is the second moment of area of the gross cross section; is the value of Ig for a cross section with sharp corners; is the warping constant of the gross cross section; is the value of Iw for a cross section with sharp corners;
3. BEHAVIOUR AND RESISTANCE OF CROSS SECTION
I m n rj
is the angle between two plane elements, in degrees; is the number of plane elements; is the number of curved elements; is the internal radius of curved element j.
The reductions given by eqn. (3.1) may also be applied in calculating the geometrical properties of the effective cross section, provided that the notional flat widths of the plane walls are measured to the points of intersection of their midlines. As stated in EN1993–1–3, when the internal radius r ! 0.04t E / f y , then the resistance of the cross section should be determined by tests. However, for practical design purposes, EN1993–1–3 in §§5.1, allows the influence of rounded corners to be ignored if the following conditions are fulfilled: r d 5 t
(3.2a)
r d 0.10 bp
(3.2b)
and
3.2.2 Net geometric properties of perforated sections _____ 104
The net geometrical properties of a cross section, or of an element of a cross section, shall be taken as its gross cross section properties minus the appropriate deduction for all fastener holes and other openings. In the evaluation of the section properties of members in bending or compression, holes made specifically for fasteners such as screws, bolts, etc., may be neglected when the hole is filled with material and able to transfer load. However, for openings or holes in general, the reduction in cross sectional area and cross sectional properties caused by such holes or openings should be taken into account. This may be accomplished either by testing or by analysis. If the section properties are to be evaluated analytically, they should be calculated considering the net cross section, which has the most detrimental arrangement of holes, which is not necessarily the same cross section for bending analysis and compression analysis. This is illustrated in Figure 3.9, where for the channel section shown the net cross section A–A has a smaller area than the net cross section
3.2 PROPERTIES OF GROSS CROSS SECTION
B–B, and is therefore critical with regard to axial loading. For loading about the x–x axis, the second moment of area and the section modulus of cross section B–B however are less than those of section A–A, and so for bending strength section B–B is critical. In the case of tension members, fasteners do not effectively resist the tension loading, which is tending to open the fastener holes, and holes made for fasteners must also be taken into consideration. A B
x
A
One hole diameter d each side
One hole diameter d
x B
Figure 3.9 – Channel section with holes (Rhodes, 1991)
In determining the net area of tension members, the cross section having the largest area of holes should be considered. The area that should be deducted from the gross cross sectional area is the total cross sectional areas of all holes in the cross section. In determining the area of fastener holes the nominal hole diameter should be used. In the case of countersunk holes, the countersunk area should be deducted. In case of a member in tension, which has staggered holes, the weakening effects of holes which are not in the same cross section, but close enough to interact with the holes in a given cross section should be taken into account. If two lines of holes are far apart from another line of holes, they have not any effect on the strength of the section at the position of the other line of holes. If the lines are close, then each line of holes affects the other one. In such a case (see Figure 3.10), the area that should be deducted from the gross cross sectional area is the greater of: a) the deduction of non-staggered holes, along the section 1–1 in Figure 3.10, i.e. Anet ,1 1
bp t 2 d t
(3.3a)
b) the sum of the sectional area of all holes in any diagonal or zigzag line extending progressively across the member or element (i.e. along the
_____ 105
3. BEHAVIOUR AND RESISTANCE OF CROSS SECTION
section 2–2 in Figure 3.10), minus an allowance for each gauge space p in the chain of holes. This allowance shall be taken as 0.25s2t/p, but not more than 0.6s t, where: p
s
t
is the gauge space, i.e. the distance measured perpendicular to the direction of load transfer, between the centres of two consecutive holes in the chain; is the staggered pitch, i.e. the distance, measured parallel to the direction of the load transfer, between the centres of the same two holes; is the thickness of the material. 1
2
d p bp
Direction of load transfer
p d s
s
Figure 3.10 – Staggered holes and appropriate net cross section
Anet ,2 2
t s2 bp t (2 d t 2 0.25 s 2 ) bp t 2 (d 0.5 ) t p p
(3.3b)
In case of cross sections such as angles with holes in more than one plane, the spacing p shall be measured along the centre line of thickness of the material, as shown in Figure 3.11 for the case of staggered holes in the two legs. 2
1
d
Direction of load transfer
d
p
2b-t
b t p
_____ 106
Therefore, the net area for section 2–2 in Figure 3.10 is calculated as follows:
1
2
Figure 3.11 – Angles with staggered holes in both legs
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