79

7

Columns and Struts

7.1 BACKGROUND

A column is vertical and supported at both ends. A strut may be inclined or even horizontal and

have a variety of end xings.

The notes below consider that the members are straight and manufactured from a homogeneous

engineering material and are used within the elastic operating range of the material. It is further

considered that the applied load or force is being applied along the centroid of the end features.

The column or strut will remain straight until the end force reaches a certain value and buckling

begins. An increase in force will then result in the column further buckling, but a reduction in this

force will then result in the column or strut returning to its original condition. The value of this

critical force will depend upon the slenderness ratio and the end xing conditions together with the

material of construction.

The slenderness ratio (λ) is dened as:

λ=

l

k

min

(7.1)

where l is the effective length and k

min

is the least radius of gyration of the section.

The principal end xing conditions are as follows:

1. Pinned (hinged) at both ends

2. Fixed (built in) at both ends

3. Fixed at one end and free at the other end

4. Fixed at one end and pinned at the other end

Figure7.1 depicts these end conditions.

The failure of a strut or column is a function of its length and will have a tendency to fail in pure

bending; in this situation the Euler formula is suitable to analyse the condition.

σ=

π

E

L

k

c

2

min

2

(7.2)

Euler’s theory takes no account of the compressive stress in the member. For a material with

a compressive stress less than 300 MPa and a Young’s modulus approximately 200 kPa, the strut

will tend to fail in compression when the slenderness ratio (l/k) is less than 80. It has been found

that Euler’s equation is not reliable for slenderness ratios less than 80 and should be avoided with

slenderness ratios less than 120.

In many practical cases struts may have slenderness ratios below which the Euler formula is

applicable. A number of empirical formulae have been developed to improve the prediction of the

critical stress, and these include, among others:

• Rankine-Gordon

• Perry-Robertson

80 Design Engineer's Case Studies and Examples

• Johnson-Euler

• Euler-Engesser

The rst two theories (Rankine-Gordon and Perry-Robertson) will be considered in this chapter.

7.2 RANKINE-GORDON METHOD

Gordon suggested an empirical formula be used (based on experimental data). Rankine modied

this formula to the one used today.

Formula:

=

σ

+

P

A

1a

2

r

c

l

k

(7.3)

where

P

r

= crushing or crippling load (Rankine-Gordon) (kN)

σ

c

= direct crushing stress (MPa)

A = cross-sectional area of column or strut (m

2

)

l = length of strut or column (m)

k = least radius of gyration of cross section (m)

a = constant depending upon end xing

=

also

I

A

andItheleastsecond moment of area of sectionk

Typical values for σ

c

and a are shown in Table7.1; these will vary dependent upon materials and

type of end xings.

The safe load:

P

P

Factor of safety

r

= (7.4)

L

Pinne

d

Pinned

Fixed

Fixed

Free

Fixed

Pinned

Fixed

FIGURE 7.1 End xing conditions.

81Columns and Struts

For long columns, Euler’s formula applies:

P

EI

l

forhingedorroundedends.

e

2

2

=

π

where

P

e

=

EI

4l

for one endfixed

2

2

π

E = modulus of elasticity (GPa)

P

e

= crippling load (Euler)

For short columns, where buckling effects are absent and hence material is in direct compres-

sion, these equations reduce to:

P

r

= P

e

= Aσ

c

(7.5)

Example 7.1

A strut in a framed structure is manufactured from a steel pipe 150 mm outside diameter and

12.5mm wall thickness. The length is 3.05 m and the pipe is pin-jointed at both ends.

Using a Factor of Safety (FoS) of 5, what is the safe load?

Solution:

c

P

A

1

I

k

r

2

2

=

σ

+

=

π

−A

4

(0.15(2x0.0125

))

22

= 0.00540 m

2

σ

c

= 325 MPa (from Table 7.1)

a

1

7500

(forhinged ends)=

TABLE7.1

Typical Values of a for Use in the Rankine-Gordon Formula

Material σ

c

(MPa)

Value of a

Fixed Ends Hinged Ends

One End Fixed, the

Other End Hinged

Cast iron 560 1/6400 1/1600 1/3600

Low carbon steel 325 1/30,000 1/7500 1/16,875

Note: Since the above values of a are not exactly equal to the theoretical values, the Rankine-Gordon

loads for long columns or struts will not be identical to those estimated by the Euler theory

as estimated.

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