Critical Stress ocritical

In each case described in Sections 6.2.1 to 6.2.3 the critical load Pc (i.e. critical stressx cross-sectional area) must be estimated for design purposes. Since the critical stress depends on the slenderness it is convenient to quantify slenderness in mathematical terms as:

r where:

Le is the effective buckling length, r is the radius of gyration= ^^ and

I and A are the second moment of area about the axis of bending and the cross-sectional area of the section as before.

6.3.1 Critical Stress for Short Columns

Short columns fail by yielding/crushing of the material and ccritical=Py, the yield stress of the material. If, as stated before, columns can be assumed short when the length is not greater than (10xthe least horizontal length) then for a typical rectangular column of cross-section (bxd) and length L~10b, a limit of slenderness can be determined as follows:

ractius of gyration

r bf 2V3

From this we can consider that short columns correspond with a value of slenderness less than or equal to approximately 30 to 35.

6.3.2 Critical Stress for Slender Columns

Slender columns fail by buckling and the applied compressive stress ocritical<<Py.

The critical load in this case is governed by the bending effects induced by the lateral deformation.

6.3.3 Euler Equation

In 1757 the Swiss engineer/mathematician Leonhard Euler developed a theoretical analysis of premature failure due to buckling. The theory is based on the differential equation of the elastic bending of a pin-ended column which relates the applied bending moment to the curvature along the length of the column, i.e.

Bending Moment = EÎ


dx approximates to the curvature of the deformed column.

Since this expression for bending moments only applies to linearly elastic materials, it is only valid for stress levels equal to and below the elastic limit of proportionality. This therefore defines an upper limit of stress for which the Euler analysis is applicable. Consider the deformed shape of the assumed centre-line of a column in equilibrium under the action of its critical load Pc as shown in Figure 6.8.

Column Analysis With Initial Curvature

Figure 6.8

The bending moment at position x along the column is equal to * J-')] *

and hence Bending Moment

This is a 2" Order Differential Equation of the form: dx

The solution of this equation can be shown to be: where:

This expression for Pc defines the Euler Critical Load (PE) for a pin-ended column. The value of n=0 is meaningless since it corresponds to a value of Pc=0. All other values of n correspond to the 1st, 2nd, 3rd.. .etc. harmonics (i.e. buckling mode shapes) for the sinusoidal curve represented by the differential equation. The first three harmonics are indicated in Figure 6.9.

Figure 6.9—Buckling mode-shapes for pin-ended columns

The higher level harmonics are only possible if columns are restrained at the appropriate levels, e.g. mid-height point in the case of the 2nd harmonic and the third-height points in the case of the 3rd harmonic.

The fundamental critical load (i.e. n=1) for a pin-ended column is therefore given by:

Euler Critical Load Pe -

This fundamental case can be modified to determine the critical load for a column with different end-support conditions by defining an effective buckling length equivalent to that of a pin-ended column.

6.3.4 Effective Buckling Length (LE)

The Euler Critical Load for the fundamental buckling mode is dependent on the buckling length between pins and/or points of contra-flexure as indicated in Figure 6.9. In the case of columns which are not pin-ended, a modification to the boundary conditions when solving the differential equation of bending given previously yields different mode shapes and critical loads as shown in Figure 6.10.

Figure 6.10—Effective Buckling Lengths for Different End Conditions

The Euler stress corresponding to the Euler Buckling Load for a pin-ended column is given by:

and l=Ar

& Kulcf

where (L/r) is the slenderness X as before m

Note: In practical design it is very difficult to achieve full fixity as assumed for the end conditions. This is allowed for by modifying the effective length coefficients e.g. increasing the value of 0.5L to 0.7L and 0.7L to 0.85L.

A lower limit to the slenderness for which the Euler Equation is applicable can be found by substituting the stress at the proportional limit g e for g Euler as shown in the following example with a steel column.

Assume that g e=200 N/mm2 and that E=205 kN/mm2

ir x 205x103


jt2 x 205xi0i

s too

In this case the Euler load is only applicable for values of slenderness>~100 and can be represented on a stress/slenderness curve in addition to that determined in Section 6.3.1 for short columns as shown in Figure 6.11.

The Euler Buckling Load has very limited direct application in terms of practical design because of the following assumptions and limiting conditions:

• the column is subjected to a perfectly concentric axial load only,

• the column is pin-jointed at each end and restrained against lateral loading,

• the material is perfectly elastic,

• the maximum stress does not exceed the elastic limit of the material,

• there is no initial curvature and the column is of uniform cross-section along its length,

• lateral deflections of the column are small when compared to the overall length,

• there are no residual stresses in the column,

• there is no strain hardening of the material in the case of steel columns,

• the material is assumed to be homogeneous.

30 slflO

Increasing Slendcmess (Lfr) -

Figure 6.11

Practical columns do not satisfy these criteria, and in addition in most cases are considered to be intermediate in terms of slenderness.

6.3.5 Critical Stress for Intermediate Columns

Figure 6.12

Since the Euler Curve is unsuitable for values of stress greater than the elastic limit it is necessary to develop an analysis which overcomes the limitations outlined above and which can be applied between the previously established slenderness limits (see Figure

6.11) as shown in Figure 6.12.

6.3.6 Tangent Modulus Theorem

Early attempts to develop a relationship for intermediate columns included the Tangent Modulus Theorem. Using this method a modified version of the Euler Equation is adopted to determine the stress/slenderness relationship in which the value of the modulus of elasticity at any given level of stress is obtained from the stress/strain curve for the material and used to evaluate the corresponding slenderness. Consider a column manufactured from a material which has a stress/strain curve as shown in Figure 6.13(a).

Figure 6.13

The slope of the tangent to the stress/strain curve at a value of stress equal to c is equal to the value of the tangent modulus of elasticity Et (Note: this is different from the value of

E at the elastic limit). The value of Et can be used in the Euler Equation to obtain a modified slenderness corresponding to the value of stress c as shown at position 'x' in

: Slcndcmcss A at position = (Lfr)

x Et

If successive values of X for values of stress between c e and c y are calculated and plotted as shown, then a curve representing the intermediate elements can be developed. This solution still has many of the deficiencies of the original Euler equation.

6.4 Perry-Robertson Formula

The Perry-Robertson Formula was developed to take into account the deficiencies of the Euler equation and other techniques such as the Tangent Modulus Method. This formula evolved from the assumption that all practical imperfections could be represented by a hypothetical initial curvature of the column.

As with the Euler analysis a 2nd Order Differential Equation is established and solved using known boundary conditions, and the extreme fibre stress in the cross-section at mid-height (the assumed critical location) is evaluated. The extreme fibre stress, which includes both axial and bending effects, is then equated to the yield value. Clearly the final result is dependent on the initial hypothetical curvature.

Consider the deformed shape of the assumed centre-line of a column in equilibrium under the action of its critical load Pc and an assumed initial curvature as shown in Figure 6.14.

Critical Buckling Load Formula

Figure 6.14

The bending moment at position x along the column is equal to=-Pc(y+y0)

and hence the bending moment = Ei d y dx2

v — —

f p "I * t




If the initial curvature is assumed to be sinusoidal, then k E

where a is the amplitude of the initial displacement and the equation becomes:

The solution to this differential equation is:

(P }

(p ^

A Cos

+ B Sin —je


The constants A and B are determined by considering the boundary values at the pinned ends, i.e. when x=0 y=0 and when x=L y=0.

Substitution of the boundary conditions in the equation gives:

a Sin




The value of the stress at mid-height is the critical value since the maximum eccentricity of the load (and hence maximum bending moment) occurs at this position;



i J






(Note: yo at mid-height is equal to the amplitude a of the assumed initial curvature). The maximum bending moment

The maximum combined stress at this point is given by:

<7i f axial load bending moment x c { A 1

where c is the distance from the neutral axis of the cross-section to the extreme fibres.

The maximum stress is equal to the yield value, i.e. a, maximum


The average stress over the cross-section is the load divided by the area, i.e. (Pc/A)

The (ac/r) term is dependent upon the assumed initial curvature and is normally given

^average J

This equation can be rewritten as a quadratic equation in terms of the average stress: Oy (<TE — = Gvmgt [0 + ~ "^average]

average e« - Oavmet + (i + /?)cFj:] + o"j <TE - 0

The solution of this equation in terms of caVerase is:

+1 0

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