Elastic Stability of Columns | Civil Engineering Optional Notes for UPSC PDF Download

 Page 1


 
Introduction: 
Structural members which carry compressive loads may be divided into two broad categories depending on their 
relative lengths and cross-sectional dimensions. 
Columns: 
Short, thick members are generally termed columns and these usually fail by crushing when the yield stress of the 
material in compression is exceeded. 
Struts: 
Long, slender columns are generally termed as struts, they fail by buckling some time before the yield stress in 
compression is reached. The buckling occurs owing to one the following reasons. 
(a). the strut may not be perfectly straight initially. 
(b). the load may not be applied exactly along the axis of the Strut. 
(c). one part of the material may yield in compression more readily than others owing to some lack of uniformity in the 
material properties through out the strut. 
In all the problems considered so far we have assumed that the deformation to be both progressive with increasing 
load and simple in form i.e. we assumed that a member in simple tension or compression becomes progressively 
longer or shorter but remains straight. Under some circumstances however, our assumptions of progressive and 
simple deformation may no longer hold good and the member become unstable. The term strut and column are 
widely used, often interchangeably in the context of buckling of slender members.] 
At values of load below the buckling load a strut will be in stable equilibrium where the displacement caused by any 
lateral disturbance will be totally recovered when the disturbance is removed. At the buckling load the strut is said to 
be in a state of neutral equilibrium, and theoretically it should than be possible to gently deflect the strut into a simple 
sine wave provided that the amplitude of wave is kept small. 
Theoretically, it is possible for struts to achieve a condition of unstable equilibrium with loads exceeding the buckling 
load, any slight lateral disturbance then causing failure by buckling, this condition is never achieved in practice under 
static load conditions. Buckling occurs immediately at the point where the buckling load is reached, owing to the 
reasons stated earlier. 
The resistance of any member to bending is determined by its flexural rigidity EI and is The quantity I may be written 
as I = Ak
2
, 
Where I = area of moment of inertia 
A = area of the cross-section 
k = radius of gyration. 
The load per unit area which the member can withstand is therefore related to k. There will be two principal moments 
of inertia, if the least of these is taken then the ratio 
 
Page 2


 
Introduction: 
Structural members which carry compressive loads may be divided into two broad categories depending on their 
relative lengths and cross-sectional dimensions. 
Columns: 
Short, thick members are generally termed columns and these usually fail by crushing when the yield stress of the 
material in compression is exceeded. 
Struts: 
Long, slender columns are generally termed as struts, they fail by buckling some time before the yield stress in 
compression is reached. The buckling occurs owing to one the following reasons. 
(a). the strut may not be perfectly straight initially. 
(b). the load may not be applied exactly along the axis of the Strut. 
(c). one part of the material may yield in compression more readily than others owing to some lack of uniformity in the 
material properties through out the strut. 
In all the problems considered so far we have assumed that the deformation to be both progressive with increasing 
load and simple in form i.e. we assumed that a member in simple tension or compression becomes progressively 
longer or shorter but remains straight. Under some circumstances however, our assumptions of progressive and 
simple deformation may no longer hold good and the member become unstable. The term strut and column are 
widely used, often interchangeably in the context of buckling of slender members.] 
At values of load below the buckling load a strut will be in stable equilibrium where the displacement caused by any 
lateral disturbance will be totally recovered when the disturbance is removed. At the buckling load the strut is said to 
be in a state of neutral equilibrium, and theoretically it should than be possible to gently deflect the strut into a simple 
sine wave provided that the amplitude of wave is kept small. 
Theoretically, it is possible for struts to achieve a condition of unstable equilibrium with loads exceeding the buckling 
load, any slight lateral disturbance then causing failure by buckling, this condition is never achieved in practice under 
static load conditions. Buckling occurs immediately at the point where the buckling load is reached, owing to the 
reasons stated earlier. 
The resistance of any member to bending is determined by its flexural rigidity EI and is The quantity I may be written 
as I = Ak
2
, 
Where I = area of moment of inertia 
A = area of the cross-section 
k = radius of gyration. 
The load per unit area which the member can withstand is therefore related to k. There will be two principal moments 
of inertia, if the least of these is taken then the ratio 
 
Is called the slenderness ratio. It's numerical value indicates whether the member falls into the class of columns or 
struts. 
Euler's Theory : The struts which fail by buckling can be analyzed by Euler's theory. In the following sections, 
different cases of the struts have been analyzed. 
Case A: Strut with pinned ends: 
Consider an axially loaded strut, shown below, and is subjected to an axial load „P' this load „P' produces a deflection 
„y' at a distance „x' from one end. 
Assume that the ends are either pin jointed or rounded so that there is no moment at either end. 
 
Assumption: 
The strut is assumed to be initially straight, the end load being applied axially through centroid. 
 
 
In this equation „M' is not a function „x'. Therefore this equation can not be integrated directly as has been done in the 
case of deflection of beams by integration method. 
 
Though this equation is in „y' but we can't say at this stage where the deflection would be maximum or minimum. 
Page 3


 
Introduction: 
Structural members which carry compressive loads may be divided into two broad categories depending on their 
relative lengths and cross-sectional dimensions. 
Columns: 
Short, thick members are generally termed columns and these usually fail by crushing when the yield stress of the 
material in compression is exceeded. 
Struts: 
Long, slender columns are generally termed as struts, they fail by buckling some time before the yield stress in 
compression is reached. The buckling occurs owing to one the following reasons. 
(a). the strut may not be perfectly straight initially. 
(b). the load may not be applied exactly along the axis of the Strut. 
(c). one part of the material may yield in compression more readily than others owing to some lack of uniformity in the 
material properties through out the strut. 
In all the problems considered so far we have assumed that the deformation to be both progressive with increasing 
load and simple in form i.e. we assumed that a member in simple tension or compression becomes progressively 
longer or shorter but remains straight. Under some circumstances however, our assumptions of progressive and 
simple deformation may no longer hold good and the member become unstable. The term strut and column are 
widely used, often interchangeably in the context of buckling of slender members.] 
At values of load below the buckling load a strut will be in stable equilibrium where the displacement caused by any 
lateral disturbance will be totally recovered when the disturbance is removed. At the buckling load the strut is said to 
be in a state of neutral equilibrium, and theoretically it should than be possible to gently deflect the strut into a simple 
sine wave provided that the amplitude of wave is kept small. 
Theoretically, it is possible for struts to achieve a condition of unstable equilibrium with loads exceeding the buckling 
load, any slight lateral disturbance then causing failure by buckling, this condition is never achieved in practice under 
static load conditions. Buckling occurs immediately at the point where the buckling load is reached, owing to the 
reasons stated earlier. 
The resistance of any member to bending is determined by its flexural rigidity EI and is The quantity I may be written 
as I = Ak
2
, 
Where I = area of moment of inertia 
A = area of the cross-section 
k = radius of gyration. 
The load per unit area which the member can withstand is therefore related to k. There will be two principal moments 
of inertia, if the least of these is taken then the ratio 
 
Is called the slenderness ratio. It's numerical value indicates whether the member falls into the class of columns or 
struts. 
Euler's Theory : The struts which fail by buckling can be analyzed by Euler's theory. In the following sections, 
different cases of the struts have been analyzed. 
Case A: Strut with pinned ends: 
Consider an axially loaded strut, shown below, and is subjected to an axial load „P' this load „P' produces a deflection 
„y' at a distance „x' from one end. 
Assume that the ends are either pin jointed or rounded so that there is no moment at either end. 
 
Assumption: 
The strut is assumed to be initially straight, the end load being applied axially through centroid. 
 
 
In this equation „M' is not a function „x'. Therefore this equation can not be integrated directly as has been done in the 
case of deflection of beams by integration method. 
 
Though this equation is in „y' but we can't say at this stage where the deflection would be maximum or minimum. 
So the above differential equation can be arranged in the following form  
Let us define a operator 
D = d/dx 
(D
2
 + n
2
) y =0 where n
2
 = P/EI 
This is a second order differential equation which has a solution of the form consisting of complimentary function and 
particular integral but for the time being we are interested in the complementary solution only[in this P.I = 0; since the 
R.H.S of Diff. equation = 0] 
Thus y = A cos (nx) + B sin (nx) 
Where A and B are some constants. 
Therefore  
In order to evaluate the constants A and B let us apply the boundary conditions, 
(i) at x = 0; y = 0 
(ii) at x = L ; y = 0 
Applying the first boundary condition yields A = 0. 
Applying the second boundary condition gives 
 
From the above relationship the least value of P which will cause the strut to buckle, and it is called the “ Euler 
Crippling Load ” P e from which w obtain. 
Page 4


 
Introduction: 
Structural members which carry compressive loads may be divided into two broad categories depending on their 
relative lengths and cross-sectional dimensions. 
Columns: 
Short, thick members are generally termed columns and these usually fail by crushing when the yield stress of the 
material in compression is exceeded. 
Struts: 
Long, slender columns are generally termed as struts, they fail by buckling some time before the yield stress in 
compression is reached. The buckling occurs owing to one the following reasons. 
(a). the strut may not be perfectly straight initially. 
(b). the load may not be applied exactly along the axis of the Strut. 
(c). one part of the material may yield in compression more readily than others owing to some lack of uniformity in the 
material properties through out the strut. 
In all the problems considered so far we have assumed that the deformation to be both progressive with increasing 
load and simple in form i.e. we assumed that a member in simple tension or compression becomes progressively 
longer or shorter but remains straight. Under some circumstances however, our assumptions of progressive and 
simple deformation may no longer hold good and the member become unstable. The term strut and column are 
widely used, often interchangeably in the context of buckling of slender members.] 
At values of load below the buckling load a strut will be in stable equilibrium where the displacement caused by any 
lateral disturbance will be totally recovered when the disturbance is removed. At the buckling load the strut is said to 
be in a state of neutral equilibrium, and theoretically it should than be possible to gently deflect the strut into a simple 
sine wave provided that the amplitude of wave is kept small. 
Theoretically, it is possible for struts to achieve a condition of unstable equilibrium with loads exceeding the buckling 
load, any slight lateral disturbance then causing failure by buckling, this condition is never achieved in practice under 
static load conditions. Buckling occurs immediately at the point where the buckling load is reached, owing to the 
reasons stated earlier. 
The resistance of any member to bending is determined by its flexural rigidity EI and is The quantity I may be written 
as I = Ak
2
, 
Where I = area of moment of inertia 
A = area of the cross-section 
k = radius of gyration. 
The load per unit area which the member can withstand is therefore related to k. There will be two principal moments 
of inertia, if the least of these is taken then the ratio 
 
Is called the slenderness ratio. It's numerical value indicates whether the member falls into the class of columns or 
struts. 
Euler's Theory : The struts which fail by buckling can be analyzed by Euler's theory. In the following sections, 
different cases of the struts have been analyzed. 
Case A: Strut with pinned ends: 
Consider an axially loaded strut, shown below, and is subjected to an axial load „P' this load „P' produces a deflection 
„y' at a distance „x' from one end. 
Assume that the ends are either pin jointed or rounded so that there is no moment at either end. 
 
Assumption: 
The strut is assumed to be initially straight, the end load being applied axially through centroid. 
 
 
In this equation „M' is not a function „x'. Therefore this equation can not be integrated directly as has been done in the 
case of deflection of beams by integration method. 
 
Though this equation is in „y' but we can't say at this stage where the deflection would be maximum or minimum. 
So the above differential equation can be arranged in the following form  
Let us define a operator 
D = d/dx 
(D
2
 + n
2
) y =0 where n
2
 = P/EI 
This is a second order differential equation which has a solution of the form consisting of complimentary function and 
particular integral but for the time being we are interested in the complementary solution only[in this P.I = 0; since the 
R.H.S of Diff. equation = 0] 
Thus y = A cos (nx) + B sin (nx) 
Where A and B are some constants. 
Therefore  
In order to evaluate the constants A and B let us apply the boundary conditions, 
(i) at x = 0; y = 0 
(ii) at x = L ; y = 0 
Applying the first boundary condition yields A = 0. 
Applying the second boundary condition gives 
 
From the above relationship the least value of P which will cause the strut to buckle, and it is called the “ Euler 
Crippling Load ” P e from which w obtain. 
 
The interpretation of the above analysis is that for all the values of the load P, other than those which make sin nL = 
0; the strut will remain perfectly straight since 
y = B sin nL = 0 
For the particular value of 
 
Then we say that the strut is in a state of neutral equilibrium, and theoretically any deflection which it suffers will be 
maintained. This is subjected to the limitation that „L' remains sensibly constant and in practice slight increase in load 
at the critical value will cause the deflection to increase appreciably until the material fails by yielding. 
Further it should be noted that the deflection is not proportional to load, and this applies to all strut problems; like wise 
it will be found that the maximum stress is not proportional to load. 
The solution chosen of nL = ? is just one particular solution; the solutions nL= 2 ?, 3 ?, 5 ? etc are equally valid 
mathematically and they do, infact, produce values of „P e' which are equally valid for modes of buckling of strut 
different from that of a simple bow. Theoretically therefore, there are an infinite number of values of P e , each 
corresponding with a different mode of buckling. 
The value selected above is so called the fundamental mode value and is the lowest critical load producing the single 
bow buckling condition. 
The solution nL = 2 ? produces buckling in two half – waves, 3 ? in three half-waves etc. 
Page 5


 
Introduction: 
Structural members which carry compressive loads may be divided into two broad categories depending on their 
relative lengths and cross-sectional dimensions. 
Columns: 
Short, thick members are generally termed columns and these usually fail by crushing when the yield stress of the 
material in compression is exceeded. 
Struts: 
Long, slender columns are generally termed as struts, they fail by buckling some time before the yield stress in 
compression is reached. The buckling occurs owing to one the following reasons. 
(a). the strut may not be perfectly straight initially. 
(b). the load may not be applied exactly along the axis of the Strut. 
(c). one part of the material may yield in compression more readily than others owing to some lack of uniformity in the 
material properties through out the strut. 
In all the problems considered so far we have assumed that the deformation to be both progressive with increasing 
load and simple in form i.e. we assumed that a member in simple tension or compression becomes progressively 
longer or shorter but remains straight. Under some circumstances however, our assumptions of progressive and 
simple deformation may no longer hold good and the member become unstable. The term strut and column are 
widely used, often interchangeably in the context of buckling of slender members.] 
At values of load below the buckling load a strut will be in stable equilibrium where the displacement caused by any 
lateral disturbance will be totally recovered when the disturbance is removed. At the buckling load the strut is said to 
be in a state of neutral equilibrium, and theoretically it should than be possible to gently deflect the strut into a simple 
sine wave provided that the amplitude of wave is kept small. 
Theoretically, it is possible for struts to achieve a condition of unstable equilibrium with loads exceeding the buckling 
load, any slight lateral disturbance then causing failure by buckling, this condition is never achieved in practice under 
static load conditions. Buckling occurs immediately at the point where the buckling load is reached, owing to the 
reasons stated earlier. 
The resistance of any member to bending is determined by its flexural rigidity EI and is The quantity I may be written 
as I = Ak
2
, 
Where I = area of moment of inertia 
A = area of the cross-section 
k = radius of gyration. 
The load per unit area which the member can withstand is therefore related to k. There will be two principal moments 
of inertia, if the least of these is taken then the ratio 
 
Is called the slenderness ratio. It's numerical value indicates whether the member falls into the class of columns or 
struts. 
Euler's Theory : The struts which fail by buckling can be analyzed by Euler's theory. In the following sections, 
different cases of the struts have been analyzed. 
Case A: Strut with pinned ends: 
Consider an axially loaded strut, shown below, and is subjected to an axial load „P' this load „P' produces a deflection 
„y' at a distance „x' from one end. 
Assume that the ends are either pin jointed or rounded so that there is no moment at either end. 
 
Assumption: 
The strut is assumed to be initially straight, the end load being applied axially through centroid. 
 
 
In this equation „M' is not a function „x'. Therefore this equation can not be integrated directly as has been done in the 
case of deflection of beams by integration method. 
 
Though this equation is in „y' but we can't say at this stage where the deflection would be maximum or minimum. 
So the above differential equation can be arranged in the following form  
Let us define a operator 
D = d/dx 
(D
2
 + n
2
) y =0 where n
2
 = P/EI 
This is a second order differential equation which has a solution of the form consisting of complimentary function and 
particular integral but for the time being we are interested in the complementary solution only[in this P.I = 0; since the 
R.H.S of Diff. equation = 0] 
Thus y = A cos (nx) + B sin (nx) 
Where A and B are some constants. 
Therefore  
In order to evaluate the constants A and B let us apply the boundary conditions, 
(i) at x = 0; y = 0 
(ii) at x = L ; y = 0 
Applying the first boundary condition yields A = 0. 
Applying the second boundary condition gives 
 
From the above relationship the least value of P which will cause the strut to buckle, and it is called the “ Euler 
Crippling Load ” P e from which w obtain. 
 
The interpretation of the above analysis is that for all the values of the load P, other than those which make sin nL = 
0; the strut will remain perfectly straight since 
y = B sin nL = 0 
For the particular value of 
 
Then we say that the strut is in a state of neutral equilibrium, and theoretically any deflection which it suffers will be 
maintained. This is subjected to the limitation that „L' remains sensibly constant and in practice slight increase in load 
at the critical value will cause the deflection to increase appreciably until the material fails by yielding. 
Further it should be noted that the deflection is not proportional to load, and this applies to all strut problems; like wise 
it will be found that the maximum stress is not proportional to load. 
The solution chosen of nL = ? is just one particular solution; the solutions nL= 2 ?, 3 ?, 5 ? etc are equally valid 
mathematically and they do, infact, produce values of „P e' which are equally valid for modes of buckling of strut 
different from that of a simple bow. Theoretically therefore, there are an infinite number of values of P e , each 
corresponding with a different mode of buckling. 
The value selected above is so called the fundamental mode value and is the lowest critical load producing the single 
bow buckling condition. 
The solution nL = 2 ? produces buckling in two half – waves, 3 ? in three half-waves etc. 
 
 
If load is applied sufficiently quickly to the strut, then it is possible to pass through the fundamental mode and to 
achieve at least one of the other modes which are theoretically possible. In practical loading situations, however, this 
is rarely achieved since the high stress associated with the first critical condition generally ensures immediate 
collapse. 
struts and columns with other end conditions: Let us consider the struts and columns having different end 
conditions 
Case b: One end fixed and the other free: 
 
writing down the value of bending moment at the point C