Eulers Formula

The Euler's formula developed during 1759 by Leonard Euler (Swiss mathematician, 1707 to 1783) is used in product designs and also in designs using columns in molds and dies that processes plasdc. Euler's formula assumes that the failure of a column is due solely to the stresses induced by sidewise bending. This assumption is not true for short columns that fail mainly by direct compression, nor is it true for columns of medium length. The failure in such cases is by a combination of direct compression and bending.

Column formulas are found in most machine and tooling hand books as well as strength of materials textbooks. Euler first published this critical-load formula for columns in 1759. For slender columns it is usually expressed in the following form:

where F = Collapsing load on the column in pounds, I = length of the column in inches, A = area of the section in square inches, k = least radius of gyration, which = I/A, E = modulus of elasticity, I = the least moment of inertia of the section, m = a constant depending on the end conditions of the column.

Euler's formula is strictly applicable to long and slender columns, for which the buckling action predominates over the direct compression action and thus makes no allowance for compressive stress. The slenderness ratio is defined as the ratio of length / to the radius of gyration k, represented as l/k.

When the slenderness ratio exceeds a value of 100 for a strong slim column, failure by buckling can be expected. Columns of stiffer and

' Moments of inertia and radii of gyration

MOMENT OF INERTIA I

nD' 128

nr"

"IT

RADIUS OF GYRATION

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more brittle materials will buckle at lower slenderness ratios. The constant factor m in Euler's critical-load formula clearly shows that the failure of a column depends on the configuration of the column ends. The basic four types with their respective m are:

Table 4.3 shows cross sections of the three common slender column configurations. Formulas for each respective moment of inertia I and radius of gyration k are given. With the above formulas buckling force F can be calculated for a column configuration. Table 4.4 lists values of slim ratios (I/k) for small-nominal-diameter column lengths.

Diameter (in.)

Table 4.3 shows cross sections of the three common slender column configurations. Formulas for each respective moment of inertia I and radius of gyration k are given. With the above formulas buckling force F can be calculated for a column configuration. Table 4.4 lists values of slim ratios (I/k) for small-nominal-diameter column lengths.

Diameter (in.)

Column length (in.)

0.031

0.047

0.0625

0.078

0.083

0.125

0.1875

1.0

128

85

64

51

43

32

21

1.5

192

128

96

77

64

48

32

1.75

224

149

112

90

75

56

37

2.0

256

171

128

102

85

64

43

2.25

288

192

144

115

96

72

48

2.5

320

213

160

128

107

80

53

3.0

384

256

192

154

128

96

64

3.25

416

277

206

166

139

104

69

Most failures with the slender columns occur because the slenderness ratio exceeds 100. The prudent designer devises ways to reduce or limit the slenderness ratio.

In the following formula P = axial load; / = length of column; I = least moment of inerda; k = least radius of gyradon; E = modulus of elasticity; y = lateral deflection, at any point along a larger column, that is caused by load P. If a column has round ends, so that the bending is not restrained, the equation of its elastic curve is:

When the origin of the coordinate axes is at the top of the column, the positive direction of x being taken downward and the positive direction of y in the direction of the deflection. Integrating the above expression twice and determining the constants of integration give:

which is Euler's formula for long columns. The factor Q is a constant depending on the condition of the ends. For round ends Q = 1; for fixed ends Q. = 4; for one end round and the other fixed L2 = 2.05. Pis the load at which, if a slight deflection is produced, the column will not return to its original position. If P is decreased, the column will approach its original position, but if P is increased, the deflection will increase until the column fails by bending.

For columns with value of l/k less than about 150, Euler's formula gives results distincdy higher than those observed in tests. Euler's formula is used for long members and as a basis for the analysis of the stresses in some types of structural parts. It always gives an ultimate and never an allowable load.

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