Design of Compression Members

Theoretical background

The resistance of a steel member subject to axial compression depends on the cross section resistance or the

occurrence of instability phenomena, such as flexural buckling, torsional buckling or flexural-torsional

buckling. In general, the design for compression is governed by the second condition (instability phenomena)

as steel members are usually of medium to high slenderness.

The cross section resistance to axial compression should be based on the plastic capacity (plastic axial force)

in compact sections (class 1, 2 or 3), but taking into account the local buckling resistance through an effective

elastic capacity in class 4 sections. The buckling resistance should be evaluated according to the relevant

buckling mode and relevant imperfections of real members, as described in the following sections.

Elastic critical load

Buckling is an instability phenomenon that is characterized by the occurrence of transverse deformations in

members under compression forces. The elastic critical load (Euler’s critical load) is derived from the theory

of elastic stability, as the value of the axial force at which an initially-perfect elastic member may start

exhibiting deformations that are not exclusively axial. This phenomenon, rigorously called buckling due to

compression in a bending mode, will be simply referred as flexural buckling.

The elastic critical load of a pinned column, with uniform cross section and subjected to constant axial force,

is evaluated as shown below. In this formulation, the following conditions (ideal design conditions) are

assumed:

material with linear elastic behaviour;

member free from geometric imperfections and from residual stresses;

perfectly centred load;

small displacement theory.

From the elastic column theory,

For other support conditions, the critical load is obtained replacing the real length L by the buckling length

LE. The buckling length LE of a member is defined as the length of a fictitious equivalent pinned member

with the same critical load Figure 3.43 illustrates the buckling lengths for isolated members, for several support

conditions.

By dividing Euler’s critical load by the area of the cross section (A), the critical stress is obtained:

Where , slenderness coefficient is 𝜆 = 𝐿𝐸

𝑖 , radius of gyration, 𝑖 = √

𝐼

𝐴, and 𝑖2 =

𝐼

𝐴

Case 1: Member without imperfections, composed of material with elastic-perfectly plastic behavior, failure

occurs by buckling in the elastic region is Euler’s critical stress is lower than the yield stress fy.

Case 2: For a short member (with a low slenderness coefficient, failure will only occur by buckling in the

elastic range if Euler’s critical, failure occurs by yielding of the cross section, when the applied stress equals

the yield stress, that is, when

𝜎 = 𝑁⁄𝐴 = 𝑓𝑦

The limit between the two types of behaviour is defined by a value of the slenderness coefficient, denoted as

𝜆1, given by:

Based on the slenderness coefficient 𝜆1, the non-dimensional slenderness coefficient �̅�is defined as:

The behaviour of a compressed member, without imperfections, for the full slenderness range, is represented in Figure

3.44.

In compressed members of thin-walled open cross section (such as channels, L sections or cruciform cross

sections) (and hence low torsional stiffness), other instability phenomena may also occur:

torsional buckling

flexural-torsional buckling.

Torsional buckling is due to the rotation of cross sections around the axis of the member, as illustrated in

Figure 3.45a; flexural-torsional buckling consists of the simultaneous occurrence of torsional and bending

deformations along the axis of the member (Figure 3.45b).

For compressed members constituted by I or H sections, the most critical instability mode is usually flexural

buckling.

Effect of imperfections and plasticity

In real structures, imperfections are unavoidable and result in deviations from the theoretical behaviour

previously described; under these circumstances, the critical load, in general, is not reached. Imperfections

can be divided into two types:

1- Geometrical imperfections (lack of linearity, lack of verticality, eccentricity of the loads) and.

2- Material imperfections (residual stresses). differential cooling after hot rolling and any other kind of

process involving heat (like welding and flame cutting, for example), shearing and cold-forming and

cold-bending.

The resistance of compressed members is based on the “European design buckling curves” (ECCS, 1977)

that relate the ratio 𝜒 = 𝜎𝑓𝑦

⁄ with the non-dimensional slenderness �̅�. These (five) curves were the result

of an extensive experimental and numerical research programme (ECCS, 1976) that accounted for all

imperfections in real compressed members (initial out- of -straightness, eccentricity of the loads, residual

stresses).

Design for compression

According to clause 6.2.4(1), the cross section resistance of axially compressed members is verified by the

following condition:

where NEd is the design value of the axial compression force and Nc,Rd is the design resistance of the cross

section for uniform compression, given by (clause 6.2.4(2)):

Class 1, 2 or 3 cross sections

Class 4 cross section

where A is the gross area of the cross section, Aeff is the effective area of a class 4 cross section, fy is the

yield strength of steel and 𝜆𝑀0 is a partial safety factor.

In evaluating Nc,Rd , holes for fasteners can be neglected, provided they are filled by fasteners and are not

oversize (clause 6.2.4(3)).

In compression members it must also be verified that:

where Nb,Rd is the design buckling resistance of the compression member (clause 6.3.1.1(1)) and this

generally controls design. The design flexural buckling resistance of prismatic members is given by:

Class 1, 2 or 3 cross sections

Class 4 cross sections

Where 𝝌 is the reduction factor for the relevant buckling mode and 𝜆𝑀1 is a partial safety factor (clause

6.3.1.1(3)). The reduction factor 𝝌 is obtained from the following equation:

where

Where �̅� is the non dimensional slenderness coefficient, given by:

Class 1, 2 or 3 cross sections

Class 4 cross sections

where, 𝜶 is the imperfection factor;

Ncr is the elastic critical load (Euler’s critical load) for the relevant buckling mode;

ET is the tangent modulus

Lcr is the length of the corresponding buckling mode;

i is the radius of gyration of the cross section;

The effect of imperfections is included by the imperfection factor , which assumes values of 0.13,

0.21, 0.34, 0.49 and 0.76 for curves a0, a, b, c and d (European design buckling curves), respectively.

These curves are illustrated in Figure 3.50. The imperfection factor and the associated buckling curve

to be adopted in design of a given member depends (see Table 3.4):

- the geometry of the cross sections,

- the steel grade,

- the fabrication process

- the relevant buckling plane,

According to clause 6.3.1.2(4), for values of the non-dimensional slenderness 𝜆

�̅� ≤ 0.2 or if 𝑁𝐸𝑑

𝑁𝑐𝑟 ⁄ ≤ 0.04 ,

The effect of buckling can be neglected, and members are designed based only on the cross section

resistance.

In compression members with open cross sections, then according to clause 6.3.1.4(1), account should be

taken of the possibility that resistance to torsional or flexural-torsional buckling could be less than the

resistance to flexural buckling. The design process for these members is very similar to that for flexural

buckling, the non-dimensional slenderness coefficient 𝜆 being replaced by the non-dimensional slenderness

coefficient , �̅�𝑇 evaluated by the following equations (clause 6.3.1.4(2)):

Class 1, 2 or 3 cross sections

Class 4 cross sections

where Ncr = the lower of the values Ncr,T and Ncr,TF.

Ncr,T = Elastic critical load for torsional buckling =

Ncr,TF = elastic critical load for flexural-torsional buckling =

Example 3.9

Design the column BD of the steel structure represented in Figure 3.52, using a UC cross section in S 355 steel, according to EC3-1-1. The column is fixed at the base and hinged at section B (with respect to the two principal axis of the cross section). Cross section B is fixed in both horizontal directions, in the plane of the structure (due to the beam itself) and in the perpendicular plane (because of secondary bracing members).

1- Design value of the applied compressive axial force NEd

ii) Preliminary design – Assuming class 1, 2 or 3 cross sections, yields:

Use UC 203x203x52 section with A=66.3 cm2, h=206.2 mm, b=204.3 mm, tw= 7.9 mm, tf=12.5 mm

Iy-y = 5260 cm4 Iz-z =1780 cm4 iy-y = 8.91 cm iz-z= 5.18 cm. iii) Buckling lengths – According to the support conditions, the buckling lengths are equal in both planes, given by:

Buckling in the plane of the structure (plane x-z) - LEy = 0.85 ×8.0 = 6.8 m. Buckling in the plane of the structure (plane x-y) - LEz = 0.85 ×8.0 = 6.8 m

2- Determination of the slenderness coefficients

= 6.8 /(8.91*10-2) = 76.32

ఒఒభ

= 76.32/76.4 = 1.0

= 6.8 /(5.18*10-2) = 131.27 ఒఒభ

= 131.27/76.4= 1.72

3- Calculation of the reduction factor min h/b = 206.2/204.3 = 1.01 < 1.2 , tf =12.5 mm < 100 mm

= 0.5[1+0.49×(1.72 – 0.2) + 1.722] = 2.35 min =z = [ 1/(2.35+(2.352 – 1.722)0.5)] = 0.253 4- Safety verification

= 0.253 × 66.3 × 10-4 × 355 × 103 /1.0 = 595.5kN

As NEd = 1376 kN > Nb,Rd = 595.5 kN Safety is not verified Try heavier section such 203×203×86 (A= 110 cm2) or 254×254×89 (A=113 cm2)

Example 3.10:

Consider the lattice beam of example 3.3, in S 275 steel, with the internal forces represented in

Figure 3.53. In order to complete its design, design the compressed members, considering the

same types of cross sections, that is:

a) Square hollow sections (SHS), with welded connections between the members of the structure.

b) HEA sections in the chords (horizontal members) and sections built up from 2 channels in the

diagonals, bolted to gusset plates welded to the HEA profiles in the upper and lower chords.

Based on the axial force diagrams, the most compressed chord member is under an axial force of

742.6 kN and it is simultaneously one of the longest members, with L = 3.00 m; concerning the

diagonals, the most compressed member, with a length L = 2.12 m, is under an axial force of

350.2 kN. For the definition of the buckling lengths of the members, it is assumed that all the

nodes of the truss are braced in the direction perpendicular to the plane of the structure.

a) Design with square hollow section (SHS) profiles

i) Preliminary design, assuming a class 1, 2 or 3 cross section, gives:

Upper chord,

Diagonals,

Based on a table of commercial profiles of square hollow sections (SHS), a SHS 120x120x8 mm

(A = 35.5 cm2) is proposed for the upper chord and a SHS 80x80x6.3 mm (A = 18.4 cm2).The

following additional geometrical properties apply to the proposed sections:

SHS 120x120x8 mm: I = 738 cm4, i = 4.56 cm;

SHS 80x80x6.3 mm: I = 165 cm4, i = 3.00 cm.

b) Design with HEA profiles for the chords and sections built up from 2 channels for the

diagonals.

Look at the solution at the text book Page 194