1. EULER’S THEORY
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Index
Sr. No. Topic Page. No.
01. Title 02.
02. Introduction 02.
03. Abstract 03.
04. Assumption Made in Euler’s Theory 04.
05. Euler’s Formula 05.
06. Euler’s Quotients 06.
07. Slenderness Ratio 13.
08. Rankine’s Formula For Long Column 14.
09. Reference 16.
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01. Title
Euler’s Theory
02. Introduction
Stability Of long column was first studied by the Swiss Mathematician Leonard Euler in 1757.
He neglected the effect of direct compressive stresses totally and determined critical loads that
would cause failure due to buckling (bending) only. Euler’s analysis based on the following
assumptions.
Euler's Theorem. The generalization of Fermat's theorem is known as Euler's theorem. In
general, Euler's theorem states that, “if p and q are relatively prime, then ”, where φ
is Euler's totient function for integers. That is, is the number of non-negative numbers that are
less than q and relatively prime to q.
In number theory, Euler's theorem (also known as the Fermat–Euler theorem or Euler's totient
theorem) states that if n and a are coprime positive integers, then
where is Euler's totient function. (The notation is explained in the article modular
arithmetic.) In 1736, Leonhard Euler published his proof of Fermat's little
theorem, which Fermat had presented without proof. Subsequently, Euler presented other
proofs of the theorem, culminating with "Euler's theorem" in his paper of 1763, in which
he attempted to find the smallest exponent for which Fermat's little theorem was always
true.
The converse of Euler's theorem is also true: if the above congruence is true, then and must be
coprime.
The theorem is a generalization of Fermat's little theorem, and is further generalized
by Carmichael's theorem.
The theorem may be used to easily reduce large powers modulo . For example, consider finding
the ones place decimal digit of , i.e..The integers 7 and 10 are coprime, and . So Euler's theorem
In general, when reducing a power of modulo (where and are coprime), one needs to work
Euler's theorem is sometimes cited as forming the basis of the RSA encryption system, however
it is insufficient (and unnecessary) to use Euler's theorem to certify the validity of RSA
encryption. In RSA, the net result of first encrypting a plaintext message, then later decrypting it,
amounts to exponentiating a large input number by , for some positive integer . In the case that
the original number is relatively prime to , Euler's theorem then guarantees that the decrypted
output number is equal to the original input number, giving back the plaintext. However,
because is a product of two distinct primes, and , when the number encrypted is a multiple of or ,
Euler's theorem does not apply and it is necessary to use the uniqueness provision of the Chinese
Remainder Theorem. The Chinese Remainder Theorem also suffices in the case where the
number is relatively prime to , and so Euler's theorem is neither sufficient nor necessary.
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03. Abstract
The Euler theory for the buckling of long, slender columns having different end conditions is
presented. The effect of the variation in slenderness ratio is discussed and expressions for the
critical stress in the inelastic buckling of a column are derived. The effects of initial
imperfections are analysed and expressions for the displaced shape and maximum bending
moment obtained. The stability of beams under combined axial and transverse loads is
investigated and the energy method for the calculation of buckling loads in columns is
demonstrated. Finally, the theory for the flexural-torsional buckling of thin-walled columns is
presented.
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04. Assumptions Made in Euler’s Theory
1. The compressive load is exactly i.e. it passes through the centroid of column section.
2. The material of the column is perfectly homogeneous and isotopic.
3. The column is initially straight and of uniform lateral dimensions.
4. The column is long and fails due to buckling only
5. Shortening of the column due to direct compression is neglected
6. The self weight of column is neglected
7. The stresses do not exceed the limit of proportionality
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05. Euler’s Formula
The buckling load for a long column of constant cross-sectional area and length L hinged at both
ends and subjected to axial compression is given by equation
P = (3.14*E*Imin)/(Le)^2
Where
P = Euler’s Buckling Load At Failure
E = Modulus of elasticity of column material
Imin = Minimum M.I of column section
Le = Effective Length of column which depends upon column end condition
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06. Euler’s Quotient
The Euler quotient of an integer a with respect to n is defined as:
The special case of an Euler quotient when n is prime is called a Fermat quotient.
Any odd number n that divides is called a Wieferich number. This is equivalent to saying
that 2φ(n)
≡ 1 (mod n2
). As a generalization, any number n that is coprime to a positive
integer a, and such that n divides , is called a (generalized) Wieferich number to base a. This
is equivalent to saying that aφ(n)
≡ 1 (mod n2
).
a numbers n coprime to a that divide (searched up to 1048576) OEIS sequence
1
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22,
23, 24, 25, 26, 27, 28, 29, 30, ... (all natural numbers)
A000027
2
1, 1093, 3279, 3511, 7651, 10533, 14209, 17555, 22953, 31599,
42627, 45643, 52665, 68859, 94797, 99463, 127881, 136929, 157995,
228215, 298389, 410787, 473985, 684645, 895167, 1232361,
2053935, 2685501, 3697083, 3837523, 6161805, 11512569, ...
A077816
3
1, 11, 22, 44, 55, 110, 220, 440, 880, 1006003, 2012006, 4024012,
11066033, 22132066, 44264132, 55330165, 88528264, 110660330,
221320660, 442641320, 885282640, 1770565280, 56224501667,
112449003334, ...
A242958
4
1, 1093, 3279, 3511, 7651, 10533, 14209, 17555, 22953, 31599,
42627, 45643, 52665, 68859, 94797, 99463, 127881, 136929, 157995,
228215, 298389, 410787, 473985, 684645, 895167, ...
5
1, 2, 20771, 40487, 41542, 80974, 83084, 161948, 643901, 1255097,
1287802, 1391657, 1931703, 2510194, 2575604, 2783314, 3765291,
3863406, 4174971, 5020388, 5151208, 5566628, 7530582, 7726812,
8349942, 10040776, 11133256, 15061164, 15308227, 15453624,
16699884, ...
A242959
6
1, 66161, 330805, 534851, 2674255, 3152573, 10162169, 13371275,
50810845, 54715147, 129255493, 148170931, 254054225,
273575735, 301121113, 383006029, 646277465, ...
A241978
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72105, 129789, 216315, 389367, 648945, ...
29 1, 2, ...
30 1, 7, 160541, ...
The least base b > 1 which n is a Wieferich number are
2, 5, 8, 7, 7, 17, 18, 15, 26, 7, 3, 17, 19, 19, 26, 31, 38, 53, 28, 7, 19, 3, 28, 17, 57, 19, 80,
19, 14, 107, 115, 63, 118, 65, 18, 53, 18, 69, 19, 7, 51, 19, 19, 3, 26, 63, 53, 17, 18, 57, ...
(sequence A250206 in the OEIS)
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For both end hinged:
n=1
For one end fixed and other free:
For both end fixed:
n=2,
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For one end fixed and other hinged:
Effective Length for different End conditions:
Modes of failure of Columns
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07. Slenderness Ratio
( λ) Slenderness ratio of a compression member is defined as the ratio of its effective length
to least radius of gyration.
Buckling Stress:
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08. Rankine’s Formula for Columns:
It is an empirical formula, takes into both crushing PCS and Euler critical load (PR).
PR = Crippling load by Rankine’s formula
Pcs = σcs A = Ultimate crushing load for column
Crippling load obtained by Euler’s formula
Where, A = Cross-section are of column, K = Least radius of gyration, and A = Rankine’s
constant.
Shape of kern in eccentric loading:
To prevent any kid of stress reversal, force applied should be within an area near the cross
section called as CORE or KERN.Shape of kern for rectangular and l-section is Rhombus
and for square section shape is square for circular section shape is circular.
