Short Notes: Euler's Theory of Columns | Short Notes for Civil Engineering - Civil Engineering (CE) PDF Download

 Page 1


Euler's Theory of Columns 
Columns and Struts:
• A structural member subjected to an axial compressive force is called strut. 
As per definition strut may be horizontal, inclined or even vertical.
• The vertical strut is called a column.
Euler's Column Theory
This theory has the following assumptions.
• Perfectly straight column and the axial load applied.
• Uniform cross-section of the column throughout its length.
• Perfectly elastic, homogeneous and isotropic material.
• The length of the column is large as compared to its cross-sectional 
dimensions.
• The shortening of the column due to direct compression is neglected.
• The failure of the column occurs due to buckling alone.
Limitation of Euler's Formula
• There is always crookedness in the column and the load may not be exactly 
axial.
• This formula does not take into account the axial stress and the buckling load 
is given by this formula may be much more than the actual buckling load.
Euler's Buckling (or crippling load)
Page 2


Euler's Theory of Columns 
Columns and Struts:
• A structural member subjected to an axial compressive force is called strut. 
As per definition strut may be horizontal, inclined or even vertical.
• The vertical strut is called a column.
Euler's Column Theory
This theory has the following assumptions.
• Perfectly straight column and the axial load applied.
• Uniform cross-section of the column throughout its length.
• Perfectly elastic, homogeneous and isotropic material.
• The length of the column is large as compared to its cross-sectional 
dimensions.
• The shortening of the column due to direct compression is neglected.
• The failure of the column occurs due to buckling alone.
Limitation of Euler's Formula
• There is always crookedness in the column and the load may not be exactly 
axial.
• This formula does not take into account the axial stress and the buckling load 
is given by this formula may be much more than the actual buckling load.
Euler's Buckling (or crippling load)
• The maximum load at which the column tends to have lateral displacement oc 
tends to buckle is known as buckling or crippling load. Load columns can be 
analysed with the Euler's column formulas can be given as
P£ = (« = 1 . 2. 3,....)
or Pr =
t t 2 EI
where, E = Modulus of elasticity, / = Effective Length of column, and / = Moment of 
inertia of column section.
1. For both end hinged:
n=1
P e =¦
n'El
l1
Both end hinaed
2. For one end fixed and other free:
3. For both end fixed:
n=2,
Pe =
Air1 El 
El
Page 3


Euler's Theory of Columns 
Columns and Struts:
• A structural member subjected to an axial compressive force is called strut. 
As per definition strut may be horizontal, inclined or even vertical.
• The vertical strut is called a column.
Euler's Column Theory
This theory has the following assumptions.
• Perfectly straight column and the axial load applied.
• Uniform cross-section of the column throughout its length.
• Perfectly elastic, homogeneous and isotropic material.
• The length of the column is large as compared to its cross-sectional 
dimensions.
• The shortening of the column due to direct compression is neglected.
• The failure of the column occurs due to buckling alone.
Limitation of Euler's Formula
• There is always crookedness in the column and the load may not be exactly 
axial.
• This formula does not take into account the axial stress and the buckling load 
is given by this formula may be much more than the actual buckling load.
Euler's Buckling (or crippling load)
• The maximum load at which the column tends to have lateral displacement oc 
tends to buckle is known as buckling or crippling load. Load columns can be 
analysed with the Euler's column formulas can be given as
P£ = (« = 1 . 2. 3,....)
or Pr =
t t 2 EI
where, E = Modulus of elasticity, / = Effective Length of column, and / = Moment of 
inertia of column section.
1. For both end hinged:
n=1
P e =¦
n'El
l1
Both end hinaed
2. For one end fixed and other free:
3. For both end fixed:
n=2,
Pe =
Air1 El 
El
Both end fixed
4. For one end fixed and other hinged:
One end fixed and other hinged
Effective Length for different End conditions
End condition Both end 
hinged
One end fixed 
o th er free
Both end fixed One end fixed
Effective
le n g th ^ ;/)
L 2L 1/2 1/V2
Modes of failure of Columns
Type o f Column M od e of Failure
Short column Crushing
Long column Buckling
Interm ediate column Combined Crushing and Buckling
Slenderness Ratio ( A)
• Slenderness ratio of a compression member is defined as the ratio of its 
effective length to least radius of gyration.
Tm in
Page 4


Euler's Theory of Columns 
Columns and Struts:
• A structural member subjected to an axial compressive force is called strut. 
As per definition strut may be horizontal, inclined or even vertical.
• The vertical strut is called a column.
Euler's Column Theory
This theory has the following assumptions.
• Perfectly straight column and the axial load applied.
• Uniform cross-section of the column throughout its length.
• Perfectly elastic, homogeneous and isotropic material.
• The length of the column is large as compared to its cross-sectional 
dimensions.
• The shortening of the column due to direct compression is neglected.
• The failure of the column occurs due to buckling alone.
Limitation of Euler's Formula
• There is always crookedness in the column and the load may not be exactly 
axial.
• This formula does not take into account the axial stress and the buckling load 
is given by this formula may be much more than the actual buckling load.
Euler's Buckling (or crippling load)
• The maximum load at which the column tends to have lateral displacement oc 
tends to buckle is known as buckling or crippling load. Load columns can be 
analysed with the Euler's column formulas can be given as
P£ = (« = 1 . 2. 3,....)
or Pr =
t t 2 EI
where, E = Modulus of elasticity, / = Effective Length of column, and / = Moment of 
inertia of column section.
1. For both end hinged:
n=1
P e =¦
n'El
l1
Both end hinaed
2. For one end fixed and other free:
3. For both end fixed:
n=2,
Pe =
Air1 El 
El
Both end fixed
4. For one end fixed and other hinged:
One end fixed and other hinged
Effective Length for different End conditions
End condition Both end 
hinged
One end fixed 
o th er free
Both end fixed One end fixed
Effective
le n g th ^ ;/)
L 2L 1/2 1/V2
Modes of failure of Columns
Type o f Column M od e of Failure
Short column Crushing
Long column Buckling
Interm ediate column Combined Crushing and Buckling
Slenderness Ratio ( A)
• Slenderness ratio of a compression member is defined as the ratio of its 
effective length to least radius of gyration.
Tm in
Lg = E ffe c tiv e len g th
rm in= A *m in/A )
rmin ~ Least radius of gyration
Buckling Stress:
Pe _ n 2E
A A2
Rankine’s Formula for Columns
• It is an empirical formula, takes into both crushing Pcs and Euler critical load
(Pr)-
1 - 1 _ 1 
~ P z
• Pr = Crippling load by Rankine’s formula
• Pcs = °c s A = Ultimate crushing load for column
P; =
it‘El
Crippling load obtained by Euler’s formula
P* =
^ ± ^ 7 l= A k 2
l + a | i
Where, A = Cross-section is of the column, K = Least radius of gyration, and A = 
Rankine’s constant.
The shape of Kern in eccentric loading
• To prevent any kind of stress reversal, the force applied should be within an 
area near the cross section called as CORE or KERN.
• The shape of Kern for rectangular and l-section is Rhombus and for the 
square section, the shape is square for circular section shape is circular.
Rectangular Column
Page 5


Euler's Theory of Columns 
Columns and Struts:
• A structural member subjected to an axial compressive force is called strut. 
As per definition strut may be horizontal, inclined or even vertical.
• The vertical strut is called a column.
Euler's Column Theory
This theory has the following assumptions.
• Perfectly straight column and the axial load applied.
• Uniform cross-section of the column throughout its length.
• Perfectly elastic, homogeneous and isotropic material.
• The length of the column is large as compared to its cross-sectional 
dimensions.
• The shortening of the column due to direct compression is neglected.
• The failure of the column occurs due to buckling alone.
Limitation of Euler's Formula
• There is always crookedness in the column and the load may not be exactly 
axial.
• This formula does not take into account the axial stress and the buckling load 
is given by this formula may be much more than the actual buckling load.
Euler's Buckling (or crippling load)
• The maximum load at which the column tends to have lateral displacement oc 
tends to buckle is known as buckling or crippling load. Load columns can be 
analysed with the Euler's column formulas can be given as
P£ = (« = 1 . 2. 3,....)
or Pr =
t t 2 EI
where, E = Modulus of elasticity, / = Effective Length of column, and / = Moment of 
inertia of column section.
1. For both end hinged:
n=1
P e =¦
n'El
l1
Both end hinaed
2. For one end fixed and other free:
3. For both end fixed:
n=2,
Pe =
Air1 El 
El
Both end fixed
4. For one end fixed and other hinged:
One end fixed and other hinged
Effective Length for different End conditions
End condition Both end 
hinged
One end fixed 
o th er free
Both end fixed One end fixed
Effective
le n g th ^ ;/)
L 2L 1/2 1/V2
Modes of failure of Columns
Type o f Column M od e of Failure
Short column Crushing
Long column Buckling
Interm ediate column Combined Crushing and Buckling
Slenderness Ratio ( A)
• Slenderness ratio of a compression member is defined as the ratio of its 
effective length to least radius of gyration.
Tm in
Lg = E ffe c tiv e len g th
rm in= A *m in/A )
rmin ~ Least radius of gyration
Buckling Stress:
Pe _ n 2E
A A2
Rankine’s Formula for Columns
• It is an empirical formula, takes into both crushing Pcs and Euler critical load
(Pr)-
1 - 1 _ 1 
~ P z
• Pr = Crippling load by Rankine’s formula
• Pcs = °c s A = Ultimate crushing load for column
P; =
it‘El
Crippling load obtained by Euler’s formula
P* =
^ ± ^ 7 l= A k 2
l + a | i
Where, A = Cross-section is of the column, K = Least radius of gyration, and A = 
Rankine’s constant.
The shape of Kern in eccentric loading
• To prevent any kind of stress reversal, the force applied should be within an 
area near the cross section called as CORE or KERN.
• The shape of Kern for rectangular and l-section is Rhombus and for the 
square section, the shape is square for circular section shape is circular.
Rectangular Column
Kern/core
Circular Column