Design of Thin-Walled Members

Various shapes of cold-formed sections

Advantages• Lightness• High strength and stiffness• Ease of prefabrication and mass production• Fast and easy erection and installation• Substantial elimination of delays due to weather• More accurate detailing• Non shrinking and non creeping at ambient temperatures• Formwork unneeded• Termite-proof and rot proof• Uniform quality• Economy in transportation and handling• Non combustibility• Recyclable material

Characteristic Behavior• relatively high width to thickness ratios.

• Un-stiffened or incompletely restrained parts of sections.

• singly symmetrical or unsymmetrical shapes.

• geometrical imperfections of the same order as or exceeding the thickness of the section.

• structural imperfections caused by the cold-forming process.

Characteristic Behavior• buckling within the range of large deflections.

• effects of local buckling on overall stability.

• combined torsional and flexural buckling.

• shear lag and curling effects.

• effects of varying residual stresses over the section.

Materials

• For cold-formed sections and sheeting it is preferable to use cold-rolled continuously galvanized steel with yield stresses in the range of 280-320-350N/mm2, and with a total elongation of at least 10%

ASD (Allowable Stress Design) Method

LRFD (Load and Resistance Factor Design) Method

Safety Factors and Resistance Factors

Nominal Loads

Effects of Cold Forming• As cold forming involves work hardening effects, the

yield stress, the ultimate strength and the ductility are all locally influenced by an amount which depends on the bending radius, the thickness of the sheet, the type of steel and the forming process.

• The average yield stress of the section then depends on the number of corners and the width of the flat elements

Effects of Cold Forming• The average yield

stress can be estimated by approximate expressions given in the appropriate codes. In the example, the average yield stress ratio fya/fyb» 1,05 and the corner yield stress ratio fyc/fyb» 1,4.

Effects of Cold Formingfull-section tensile yield strength;

The effective width concept

Effective Cross-sections• evaluate the effective width of the compression

elements of the section• calculate the geometric properties of the

effective section• the design procedure is the same as for thick-

walled sections• the resistance of a thin-walled effective cross-

section is limited by the design yield stress at any part of the section, based on an elastic analysis.

• the design resistance is based on the value fy/M, where M is a partial safety factor for resistance (normally M=1,1).

Validity of the effective width concept

Effective Section

k = the buckling factor = 4 for stiffened Elements = 0.43 for unstiffened element

Uniformly compressed stiffened and unstiffened elements

w = the flat width;ρ = the reduction factorλ = a slenderness factorf = the stress in the element

Effective SectionUniformly compressed stiffened and unstiffened elements

Effective Section

w = the flat width;ρ = the reduction factorλ = a slenderness factorf = the stress in the element

k = the buckling factor = 4 for stiffened Elements = 0.43 for unstiffened element

checking the serviceability state of uniformly compressed elements supported by webs on both edges:

Effective Section

Ψ = f2/f1

b = the effective width (uniformlycompressed stiffened elements) with f1 substituted by f and with k determined fromk=4+2(1–ψ)3+2(1–ψ)

Webs and stiffened elements with stress gradient

Doubly supported element

Singly supported element

Effective SectionUnstiffened elements and edge stiffened with stress gradient

In this case, the code suggests evaluating the effective widths in accordance with Section 5.4.2 (uniformly compressed unstiffened elements) using a uniform distribution with the maximum value of stress f.

Effective Section

Uniformly compressed elements with an edge stiffener

Uniformly compressed elements with one intermediate stiffener

Limits for b/t ratios• The design rules

give limits for b/t ratios as.

• These maximum width-to-thickness ratios depend partly on limited experimental evidence, and partly on experience from manufacturing and handling sections

Resistance vs. geometrical properties.

RESISTANCE OF CROSS-SECTIONSLRFD Factored resistance фRn

Resistance under axial tension

Resistance under axial compression

Resistance under bending moment

Combined axial tension and bending

Combined axial compression and bending

Combined shear force and bending moment

Buckling resistance under axial compression

Buckling resistance under axial compression and bending

Resistance under axial tension - LRFD

An = the net area of the cross-section;

Fy = the design yield stress, which takes into account the increase in basic yield strength due to strain-hardening

Фt = resistance reduction factor

The nominal tensile strength Tn shall be determined as follows

Resistance under axial compression - LRFD

Ae = the effective area at the stress Fy;

Fy = the design yield stress

Фt = resistance reduction factor

Assuming that the member is not subject to the risk of axial buckling,the nominal compression strength Pn shall be determined as follows

Resistance under bending moment - LRFD

Se = the elastic section modulus of the effective section, calculated for extreme compression or tension fiber at Fy;

Fy = the design yield stress—for the section to be fully effective it should take into account the increase of the basic yield strength due to strain-hardening.

Фb = resistance reduction factor

The nominal flexural strength Mn shall be determined as follows:

for sections with unstiffened compression flange

Combined axial tension and bending - LRFD

T = the required tensile axial strength;Tn = the nominal tensile axial strengthMx, My = the required flexural strengths about the centroidal axes;Mnx, Mny = the nominal flexural strengths about the centroidal;Mnxt = SfxtFy Mnyt = SfytFy

Sfxt = the elastic section modulus of the full unreduced section for the extreme tension fibre about the x axis;Sfyt=the elastic section modulus of the full unreduced section for the extreme tension fibre about the y axis;фb, фt=are the appropriate safety factors.

The required strengths T, Mx, My shall satisfy the following equations

Combined axial compression and bending - LRFD

P = the required compressive axial strength;

Pn = the nominal axial strength

Mx, My = the required flexural strengths with respect to the centroidal axes of the effective section determined for the required cornpressive axial strength alone;

Mnx, Mny = the nominal flexural strengths about the centroidal axes;

фb, фc = the appropriate safety factors.

If the member is not subject to the risk of axial buckling, the required strengths P, Mx, My shall satisfy the following equations,

Buckling resistance under axial compression - LRFD

Ae = the effective area at the stress Fn;

Fe = the smallest value for the elastic flexural, torsional and torsional-flexural bucklingstress. (the case here is for elastic flexural buckling)

E is the modulus of elasticity,K is the effective length factorL is the unbraced lengthr is the radius of gyration of the full, unreduced section.

Фt = resistance reduction factor

The nominal axial strength Pn shall be calculated as follows

for λc≤1.5

for λc>1.5

Buckling resistance under axial compression - LRFD

Buckling resistance under axial compression and bending - LRFD

P = the required compressive axial strength;Pn = the nominal axial strength;Pno = the nominal axial strength (i.e. with Fn=Fy);Mx, My = the required flexural strengths with respect to the centroidal axes of the effective section determined for the required compressive axial strength alone;Mnx, Mny = the nominal flexural strengths about the centroidal axes фb, фc = are the appropriate safety factors

The required strengths P, Mx, My shall satisfy the following equations

Buckling resistance under axial compression and bending - LRFD

Ix, Iy=the moments of inertia of the full unreduced section about the x axis and y axis,respectively;Lx, Ly=the actual unbraced lengths for bending about the x axis and y axis,respectively;

The reduction coefficient α is given by:

Buckling resistance under axial compression and bending - LRFD

Kx, Ky = the effective length factors for buckling about the x and y axes,Cmx, Cmy=coefficients defined as follows:Cm=0.85 for compression member in frames subject to joint translation (sideways);Cm=0.6–0.4M1/M2 for restrained compression members in frames braced against joint translation and not subjected to transverse loading between their supports in the plane of bending (M1/M2 is the ratio of the smaller to the larger moment at the end of themember);Cm= 0.85 for compression members in frames braced against joint translation and subjected to transverse loading between their supports in the plane of bending, when their ends are restrained;Cm=1.00 for compression members in frames braced against joint translation and subjected to transverse loading between their supports in the plane of bending, when their ends are not restrained.