

































IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                               I.U. Udo-Akpan1  * 

https://ijojournals.com/                                              Volume 07 Issue 02 || February, 2024 || 

ON A-TWO-PARAMETER DYNAMIC BUCKLING OF A VISCOUSLY DAMPED BUT CLAMPED COLUMN STRESSED BY A STEP LOAD 
 

 

 

ON A-TWO-PARAMETER DYNAMIC BUCKLING OF A VISCOUSLY DAMPED 

BUT CLAMPED COLUMN STRESSED BY A STEP LOAD 
 

I.U. Udo-Akpan1  and J. U. Chukwuchekwa2 
1Department of Mathematics and Statistics, University of Port Harcourt, Port Harcourt, Rivers 
State, Nigeria. 
2Department of Mathematics, Federal University of Technology, Owerri, Imo State, Nigeria. 

 

Abstract 

In this investigation, we extend our search on the dynamic buckling loads of some elastic 

structures to that of a clamped column lying on a nonlinear (cubic) elastic foundation but 

impacted upon axially by a step load. In order to ensure a uniformly valid solution, we employ 

multi–scaling two–timing regular perturbation procedures in asymptotic expansions of the 

variables. It is shown that (a) clamped columns buckle at higher buckling loads than columns 

with simply–supported ends irrespective of whether the columns are loaded statically or 

dynamically and whether damped or undamped, (b) Specifically, the inequalities satisfied by 

the static buckling load �� and dynamic buckling load�� in the clamped case are respectively 

given as 1 < �� < 2.125 and 1 < �� < 2.125 as against 0 < �� < 1 and 0 < �� < 1 for simply–

supported end conditions, (c) At low values of the static buckling load ��, there is no 

appreciable change in the values of the dynamic buckling load �� but at higher values of ��, �� 

increases sharply with increased static buckling load ��. However, the increase seems to 

decrease with increased damping. We are able to mathematically relate the dynamic buckling 

load �� to the static buckling load �� and thereby by–passing the labour of repeating the entire 

process for different imperfection parameters. Thus, given either �� or ��, we can predict either 

value without the actual knowledge of the size of the small imperfection parameter. 

 

Keywords: Dynamic buckling, viscously damped clamped column, elastic structures, 

step load, two-timing perturbation 2010 Mathematical Subject Classification: 74B20, 

74H10, 34E10 

 

1.        INTRODUCTION 

Investigations into the static or dynamic stability (or otherwise) of columns (finite or 

infinite) are age old problems that have been embarked upon by researchers for some 

years now. As observed by [1] and [2], buckling of structures is one of the structural 

instabilities that have been known for centuries ever since the equation to derive the 

critical buckling of a column was derived by Leonhard Euler [3]. As a result of this, 

previous studies on the subject matter are indeed enormous and varied, and include 

investigations by [4]-[7], among others. We must stress that columns, are in themselves, 

indispensable structural materials and their utility cuts across all cultures in human 

history. 

This investigation is concerned with analytical determination of the dynamic buckling 

load of a viscously damped but clamped imperfect column that rests on a nonlinear 

(cubic) elastic foundation, where the column is struck by a step load. The viscous 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                               I.U. Udo-Akpan1  * 

https://ijojournals.com/                                              Volume 07 Issue 02 || February, 2024 || 

ON A-TWO-PARAMETER DYNAMIC BUCKLING OF A VISCOUSLY DAMPED BUT CLAMPED COLUMN STRESSED BY A STEP LOAD 
 

 

 

damping, though small in magnitude, is not in any way related either physically or 

mathematically, to an equally small imperfection that is stress-free and twice-

differentiable. The formulation therefore contains two small but dimensionally 

independent parameters upon which asymptotic expansions are initiated using a two-

timing multi-scaling perturbation procedure.  

The analysis contained here is an extension of a similar study espoused by [8], where, in 

that study, damping was taken to be related in some way, to the imperfection �� (�), so 

that once the imperfection was fixed, the damping was equally fixed. However, we note, 

from physical reasoning, that damping need not in any way, be related to the 

imperfection in all probabilities. Similar studies were done by [9]-[12], among others. 

Apart from addressing the phenomenon of viscous damping in a special fashion, this 

investigation is related, in spirit, to similar studies by [13]-[25]. 

 

2.        FORMULATION OF THE PROBLEM  

As in [6], the dimensional differential equation satisfied by the deflection �(�, �) of a 

finite viscously damped column lying on a nonlinear (cubic) elastic foundation but 

struck by a load �(�) is 

���,�� + ��,� + ���,���� + 2�(�)�,�� + ��� −  ����� 

      =  −2�(�)
����

��� , � > 0,      (2.1a)  

�(�, 0) =  �,�(�, 0) = 0,       (2.1b) 

� =  �,� = 0, �� � = 0, π,      � > 0      (2.1c) 

Here,  �� is the mass per unit length, � is the damping coefficient, EI is the bending 

stiffness, where E and I are the Young’s modulus and the moment of inertia respectively. 

The nonlinear elastic foundation exerts a force per unit length given by ��� −  ����� 

on the column, where �� and �� are constants such that ��>0, ��>0, and � is the 

imperfection-sensitivity parameter which is such that for � = 1, the nonlinear elastic 

foundation is said to “softening”, whereas for � = -1, the foundation is said to be 

“hardening”. We have here excluded all nonlinearities of �(�, �) higher than the cubic 

and have also excluded all nonlinear derivatives of�(�, �). 

 

3.     NON-DIMENSIONALIZATION OF THE GOVERNING EQUATION 

We shall non-dimensionalize the equations (2.1a - c) by using the following non-

dimensional quantities 

� = �
��

��
�

�

�

�,    � =  �
��

��
�

�

�

�,     ��(�) =  
�(�)

2(����)
�

�

 ,      ��� =  �
��

��
�

�

�

�� ,      2� =
�

(����)
�

�

 , 

  � =  �
��

��
�

�

�

�,    0 <  � << 1,   0 < δ ≪ 1;   0 <  � < 2.125;  

 

 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                               I.U. Udo-Akpan1  * 

https://ijojournals.com/                                              Volume 07 Issue 02 || February, 2024 || 

ON A-TWO-PARAMETER DYNAMIC BUCKLING OF A VISCOUSLY DAMPED BUT CLAMPED COLUMN STRESSED BY A STEP LOAD 
 

 

 

On introducing these non-dimensional quantities into (2.1a-c) and simplifying, we get 

   �,��  + 2��,�  + �,���� +  2��(�)�,�� +  � −  ��� =  −2���(�)
����

���
,   � > 0,    

0 < � < π        (3.1a) 

 �(�, 0) =  �,�(�, 0) = 0,   0 < � < π    (3.1b)                     

 � =  �,� = 0, �� � = 0, π,      � > 0           (3.1c) 

Here, a subscript following a comma indicates partial differentiation and �(�) indicates 

the actual time dependence of the load having its magnitude as λ.  In our case, �(�) is a 

step load characterized by 

  �(�) = �
1, � > 0
0, � < 0         

�      (3.2)  

 

4.  CLASSICAL BUCKLING LOAD, �� 

The classical buckling load �� is the load that the associated linear perfect column 

buckles statically. The equations required are obtained from (3.1a) as 

  �,���� +  2��,�� +  � = 0,     0 < � <  �       (4.1a) 

� =  �,� = 0, ��  � = 0, π.           (4.1b) 

We note that the deflection � at this stage depends only on�. To solve (4.1a,b), we let 

                �(�) =  ∑ (1 − ���2��)��
�
���       (4.2) 

On substituting (4.2) into (4.1a),multiplying by ���2�� for fixed� and integrating form 

0 to �, we see that for � = �, we get 

(16�� −  8��� + 1)�� = 0         (4.3) 

where, m is a fixed value of n. 

According to [18], the condition for static buckling is 
��

��
= 0          (4.4) 

where, � is the displacement (or deflection). This gives the classical buckling load �� as   

  λ� =  
����� �

���        (4.5a) 

The least value of �� is when � = 1 and in this case, we get 

  λ� =  
��

�
  = 2.125        (4.5b) 

In retrospect, a similar column with simply-supported end conditions that satisfy the 

same equation as (4.1a) but, instead of (4.1b), it would satisfy the conditions  

� =  �,�� = 0, �� � = 0, π      (4.5c) 

has the classical buckling load as �� = 1 which is different from (4.5a,b). Thus, a 

clamped column has a higher classical buckling load than the same column with simply 

– supported end conditions. 

 

 

 

 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                               I.U. Udo-Akpan1  * 

https://ijojournals.com/                                              Volume 07 Issue 02 || February, 2024 || 

ON A-TWO-PARAMETER DYNAMIC BUCKLING OF A VISCOUSLY DAMPED BUT CLAMPED COLUMN STRESSED BY A STEP LOAD 
 

 

 

5.    STATIC BUCKLING LOAD,  �� 

This is the load that the column would require to buckle statically. The required 

differential equation is 

 �,���� +  2��,�� +  � −  ��� =  −2��
����

���
,     0 < � <  �    (5.1a) 

� =  �,� = 0, �� � = 0, π,          (5.1b) 

To determine the displacement (or deflection) �(�) in (5.1a,b), we set �(�)  ≡  1and let 

��  = ���(1 − ���2��),     │���│ ≪ 1         

Next, we let 

∑ �(�)�
��� �� ;      �(�)  =  �(�)(�)     (5.2) 

By substituting (5.2) into (5.1a) and equating the coefficients of powers of ��, � =

1, 2, 3, …,  we get 

O(ϵ):��(�)  ≡  �,����
(�)

+  2��,��
(�)

+  �(�) = −8���������2��     (5.3)  

O(��):��(�) = 0           (5.4) 

O(��):        ��(�) =  ���(�)�
�

       (5.5) 

 etc.  

�(�)(�) = �,�
(�)

   �� � = 0, π      (5.6) 

We seek for the solutions of (5.3) – (5.6) by letting 

�(�)(�) = 2 ∑ ��
(�)�

��� sin� �� =  ∑ (1 − ���2��)��
(�)�

���    (5.7) 

On substituting (5.7) into (5.3), we get 

∑ ���
(�)(8��� − 16��)���2�� + ��

(�)(1 − ���2��)��
��� =  −8���������2�� (5.8) 

On multiplying (5.8) through by Cos2mx and integrating from 0 to π, we see that for 

� = �, 

(16�� −  8��� + 1)��
(�)

=  8������ 

This gives 

��
(�)

=  
�������

����� ������
 = �        (5.9a) 

∴ �(�) =  ��
(�)(1 − ���2��)       (5.9b) 

On substituting (5.7) into (5.4), we easily get 

��
(�)

 =   0             (5.10) 

Next, we substitute (5.7) in (5.5) and use (5.9a,b) to get 

����
(�)(8��� − 16��)���2�� +  ��

(�)(1 − ���2��)�  =  ���(�)�
�

(1 − ���2��)�

�

���

 

 =  ���(�)�
�

�
�

�
−  

��

�
���2�� +  

�

�
���4�� −  

�

�
���6���(5.11) 

We multiply (5.11) through by Cos2mx and integrate from 0 to π, to get 

��
(�)

=  
�����

�(����� ������)
       (5.12) 

 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                               I.U. Udo-Akpan1  * 

https://ijojournals.com/                                              Volume 07 Issue 02 || February, 2024 || 

ON A-TWO-PARAMETER DYNAMIC BUCKLING OF A VISCOUSLY DAMPED BUT CLAMPED COLUMN STRESSED BY A STEP LOAD 
 

 

 

 

Next, we multiply (5.11) by Cos4mx and integrate from 0 to π to get 

���
(�)

=  
�����

�(������ �������)
       (5.13) 

Lastly, we multiply (5.11) by Cos6mx and integrate from 0 to π and get,  

���
(�)

 =  �
���

��������������
�       (5.14) 

Thus, we get 

 �(�) =  ��
(�)(1 − ���2��) + ���

(�)(1 − ���4��) +  ���
(�)(1 − ���6��)(5.15) 

So that 

            �(�) =  ���
(�)(1 − ���2��) + �����

(�)(1 − ���2��)� +  ���
(�)(1 − ���4��) 

�+  ���
(�)(1 − ���6��)� + …        (5.16) 

Before determining the static buckling load, we need to evaluate (5.16) at � =  
�

��
. 

This is informed by the fact that eventually, we shall need to determine the associated 

dynamic problem at  � =  
�

��
 (which also means finding the maximum of (5.16)). Such 

evaluation yields 

  � = 2���
(�)

+  2�����
(�)

−  ���
(�)

� +  …     (5.17) 

We can write (5.17) as 

 � =  ��� + ���� + …       (5.18) 

where, 

�� =  2��
(�)

,       �� =   2���
(�)

− ���
(�)

�     (5.19) 

As in [18], the static buckling load is obtained by first reversing the series (5.18) in the 

form,  

� =  ��� +  ���� +  …       (5.20) 

On substituting for w from (5.18) in (5.20) and equating the coefficients of powers of ϵ, 

we get 

�� =  
�

��
,        �� =  

���

��
�        (5.21) 

The maximization (4.4) is now easily executed from (5.20) to yield 

�� +  3����
�  = 0         (5.22) 

where, �� is the value of � evaluated at static buckling. From (5.18), we get 

�� =  �
���

���
  =  

�

√�
�

��
�

��
�

�

�
      (5.23)               

Next, we determine (5.20) at static buckling and get 

� =  ��� +  ���� +  … =  ��(�� +  ����
�) =  

�

�√�
�

��

��
�

�

�
   (5.24) 

On substituting in (5.24) for �� and ��, we get 

(16�� −  8���� + 1)
�

� = 18√5���
�

������� �1 + 
�

��
�

����� �������

���������������
��

�

�
 (5.25) 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                               I.U. Udo-Akpan1  * 

https://ijojournals.com/                                              Volume 07 Issue 02 || February, 2024 || 

ON A-TWO-PARAMETER DYNAMIC BUCKLING OF A VISCOUSLY DAMPED BUT CLAMPED COLUMN STRESSED BY A STEP LOAD 
 

 

 

where �� is the static buckling load. The least value of �� is obtained when � = 1, and 

for this, we get 

(17 −  8��)
�

� = 18√5���
�

����� �1 +  
�

��
�

��� ���

����� ���
��

�

�
   (5.26) 

If it is required that the buckling mode be strictly in the shape of imperfection, then the 

results corresponding to (5.25) and (5.26) respectively become 

(16�� −  8���� + 1)
�

� = 18√5�����
�

�����     (5.27) 

and 

(17 −  8��)
�

� = 18√5���
�

�����         (5.28) 

By way of comparison, we can perform a similar analysis on the same column but with 

simply-supported end conditions and the results corresponding to those of (5.25), 

(5.26), (5.27) and (5.28) are respectively given by 

(�� −  2���� + 1)
�

� =
�

�
���������

�

� �1 + �
��� �������

�������������
��

�

�
  (5.29) 

(1 −  ��)
�

� =
�

�√�
�������

�

� �1 +  �
����

������
��

�

�
    (5.30) 

(�� −  2���� + 1)
�

� =
�

�
���������

�

�     (5.31)  

and 

(1 −  ��)
�

� =
�

�√�
�������

�

�      (5.32) 

The result (5.32) was first obtained by [6]. So far, we conclude that the static buckling 

load of structures largely depends, among other things, on the type of end constraints of 

the structures. 

 

6. THE DYNAMIC PROBLEM 

The associated dynamic problem follows from (3.1a) which we now recast as 

 �,��  + 2��,�  +  �,���� +  2��(�)�,�� +  � −  ��� 

    =  −2���(�)
����

���  , � > 0, 0 < � < π    (6.1a) 

�(�, 0) =  �,�(�, 0) = 0,   0 < � < π     (6.1b)  

� =  �,� = 0, �� � = 0, π,      � > 0       (6.1c) 

Henceforth, we shall substitute for the step load �(�) as in (3.2). 

We let 

  � =  ��,          (6.2a) 

 �̂ = � +  �
��(�)���  ��(�)���⋯

�
�      (6.2b) 

where, ��(0) = 0,        i = 1, 2, 3, …,   �� =  ��(�) 

We stress that ϵ and δ are small and unrelated parameters. Thus, from (6.2a,b), we get 

��

��
=  

��

��̂

��̂

��
+  

��

��̂

��̂

��

��

��
+ 

��

��

��

��
 

 =  (1 + ��
� �� + ��

� �� + ⋯ )�,�� +  ��,�    (6.3a)  

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(ISSN: 2992-4421 )                                                                                                               I.U. Udo-Akpan1  * 

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ON A-TWO-PARAMETER DYNAMIC BUCKLING OF A VISCOUSLY DAMPED BUT CLAMPED COLUMN STRESSED BY A STEP LOAD 
 

 

 

where,(… )� =  
�(… )

��
and a subscript after a comma indicates partial differentiation. 

∴
���

���
=  (1 +  ��

� �� + ��
� �� + ⋯ )��,���� +  ���,�� + 2�(1 +  ��

� �� +  ��
� �� + ⋯ )�,��� 

+ �(��
���� +  ��

���� + ⋯ )�,��    (6.3b)  

On substituting (6.3a,b) into (6.1a) for �(�) = 1, we get 

�(1 +  ��
� �� +  ��

� �� + ⋯ )��,����
� +  ���,�� + 2�(1 + ��

� �� +  ��
� �� + ⋯ )�,��� 

+��(��
���� +  ��

���� + ⋯ )�,��� +  2��(1 +  ��
� �� +  ��

� �� + ⋯ )�,�� +  ��,�� 

+ �,���� +  2��(�)�,�� +  � −  ��� =  −2���(�)
����

���
   (6.4) 

Next, we adopt the asymptotic series 

  � =  ∑ ∑ �(��)(�, �̂�
���

�
��� , �)����       (6.5) 

and substitute same into (6.4), and afterwards, equate powers of ���� to get 

�(�):      ��(��) =  �,����
(��)

+  �,����
(��)

+  2��,��
(��)

+  �(��) =  −2��(�)
����

���    (6.6)  

�(��):     ��(��) =  −2 ��,���
(��)

+  �,��
(��)

�       (6.7) 

�(���):   ��(��) =  −2 ��,���
(��)

+  �,��
(��)

� −  �,��
(��)

      (6.8) 

�(��):      ��(��) = 0              (6.9) 

�(���):    ��(��) = −2 ��,���
(��)

+  �,��
(��)

�      (6.10) 

�(����):   ��(��) = −2 ��,���
(��)

+  �,��
(��)

� −  �,��
(��)

    (6.11) 

�(��):       ��(��) =   ���(��)�
�

− 2��
� �,����

(��)
      (6.12) 

�(���):    ��(��) = 3���(��)�
�

�(��) − 2 ��,���
�� + �,��

(��)
� − ��

���,��
(��)

− 2��
� �,��

(��)
(6.13) 

�(����):   ��(��) = 3� ��(��)��(��)�
�

+ ��(��)�
�

�(��)� − �,��
(��)

 

−2 ��,���

(��)
+ ��

� �,���

(��)
� −  ��

���,��
(��)

−  2 ��,��
(��)

+ ��
� �,��

(��)
+  �,�

(��)
�       (6.14) 

The initial conditions, which are evaluated at �̂ = 0 =  � are 

  �(��)(�, 0, 0) =  0,       � = 1, 2, 3, … ;   � = 0, 1, 2, 3, …   (6.15) 

Ο(ϵ):        �
,��
(��)(x, 0, 0) =  0         (6.16) 

Ο(ϵδ):       �
,��
(��)(x, 0, 0) +  �,�

(��)(x, 0, 0) = 0      (6.17) 

Ο(ϵδ�):     �
,��
(��)(x, 0, 0) +  �,�

(��)(x, 0, 0) = 0      (6.18) 

In general, we have 

 �
,��
(��)(x, 0, 0) +  �,�

�(���)(x, 0, 0) = 0,   k = 1, 2, 3, …     (6.19)

 Ο(ϵ�):     �
,��
(��)(x, 0, 0) = 0        

 (6.20) 

 

 

 

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(ISSN: 2992-4421 )                                                                                                               I.U. Udo-Akpan1  * 

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ON A-TWO-PARAMETER DYNAMIC BUCKLING OF A VISCOUSLY DAMPED BUT CLAMPED COLUMN STRESSED BY A STEP LOAD 
 

 

 

In general, we have 

 �
,��
(��)(x, 0, 0) + �,�

�(���)(x, 0, 0) = 0,   k = 1, 2, 3, …        (6.21) 

Ο(ϵ�):        �
,��
(��)(x, 0, 0) +  ��

� (0)�
,��
(��)(x, 0, 0) = 0         (6.22)  

O(ϵ��):      �
,��
(��)(x, 0, 0) +  ��

� (0)�
,��
(��)(x, 0, 0) +  �,�

(��)(x, 0, 0) = 0       (6.23) 

Ο(ϵ���):    �,��
��(x, 0, 0)  +   ��

� (0)�
,��
(��)(x, 0, 0) +   �,�

(��)(0, 0)  = 0       (6.24) 

Generally, we have 

 �,��
(��)

(x, 0, 0) +  ��
� (0)�,��

(��)
(x, 0, 0) +   �,�

��(���)�
(x, 0, 0) = 0 , k = 1, 2, 3, …     (6.25) 

The boundary conditions are 

 �(��) =  U,�
(��)

= 0 �� � = 0, �.  

For solution to all the systems of equation involved here, we set 

 �(��)(x, t̂, τ) =  2 ∑ U�
(��)(t̂, τ)sin�nx  =  ∑ U�

(��)
(t̂, τ)(1 − cos 2��)�

���
�
���      (6.26a) 

We shall assume 

��(�) =  ���(1 − cos 2��)      (6.26b)  

for �, fixed. 

On substituting (6.26a,b)  into (6.6) and simplifying, we get 

� �U�,����
(��)

+ U�
(��)

� (1 − cos 2��)

�

���

+ �(8��� − 16��)U�
(��)

cos 2��

�

���

 

= −8���������2��  (6.27a)   

Next, we multiply (6.27a) by ���2�� and integrate from 0 to π , and note that for � =

�, we get 

 U�,����
(��)

+  ��U�
(��)

=   8������      (6.27b)  

U�
(��)(0,0) = 0,      U�,��

(��)
(0,0) = 0      (6.27c) where,  

�� =  (16�� −  8��� + 1) > 0,   ∀ �      (6.27d) 

The solution of (6.27b-d) is 

U�
(��)

(t̂, τ) =  ��(�) cos ��̂ + ��(�) sin ��̂ + �     (6.28a) 

� =  
�������

��  =  
�������

����� ������
       (6.28b) 

where, 

 ��(0) =  −�,         ��(0) = 0       (6.28c) 

This means that 

 �(��) =  U�
(��)

(1 − cos 2��)       (6.28d) 

We next substitute (6.28d) into (6.7), using (6.25), and thereafter, multiply by 

cos 2�� and note that for � = 2�, we get 

 U
�,����
(��)

+  ��U�
(��)

=  −2 � U�,���

(��)
+ U�,��

(��)
�      (6.29a) 

 U�
(��)(0,0) = 0,          U�,��

(��)(0,0) +  U�,�
(��)(0,0)    (6.29b) 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                               I.U. Udo-Akpan1  * 

https://ijojournals.com/                                              Volume 07 Issue 02 || February, 2024 || 

ON A-TWO-PARAMETER DYNAMIC BUCKLING OF A VISCOUSLY DAMPED BUT CLAMPED COLUMN STRESSED BY A STEP LOAD 
 

 

 

On substituting for U�
(��)

 in (6.29) from (6.28a), we ensure a uniformly valid solution in 

�̂ by equating to zero the coefficients of cos ��̂  and sin ��̂ and getting 

 ��
� +  �� = 0,      ��

� +  �� = 0       (6.30a) 

The solution of (6.30a) using (6.28c) yields 

 ��(�) = 0,           ��(�) = −����     (6.30b) 

The remaining equation in (6.29a,b) is solved to get 

 U�
(��)

=  ��(�) cos ��̂ +  ��(�) sin ��̂                                                                  (6.31a) 

 ��(0) = 0,     ��(0) =  
�

�
                                                                                          (6.31b) 

We however note that  

 ��
�(0) = B,    ��

��(0) =  −B       (6.31c) 

We equally note at this stage that       (6.32) 

�(��) =  U�
(��)

(1 − cos 2��) 

On substituting from (6.32) and (6.28d) in (6.8), multiplying thereafter by cos 2�� and 

integrating from 0 to π, we get, (for � = �) 

U
�,����
(��)

+  ��U�
(��)

=  −2 � U�,���

(��)
+  U�,��

(��)
� −  U�,��

(��)
    (6.33a) 

U�
(��)(0,0) = 0,       U�,��

(��)
(0,0) +  U�,�

(��)(0,0)    (6.33b) 

Next, we substitute in (6.33a) for U�
(��)

 and U�
(��)

 from (6.31a) and (6.28a) and ensure a 

uniformly valid solution in �̂ by equating to zero the coefficients of cos ��̂ and sin ��̂ and 

so, get, respectively 

 ��
� +  �� =

���
��

��
     and     ��

� +  �� = 0     (6.33c) 

On solving (6.33c), we get 

 ��(τ) =  ��� �∫
���

��(�)

��

�

�
�� +  ��(0)�,     ��(τ) = 0    (6.33d) 

The remaining equation in the substitution into (6.33a) is solved to yield 

 U�
(��)(t̂, τ) =   ��(�) cos ��̂ +  ��(�) sin ��̂     (6.34a) 

where, 

 ��(0) = 0,       ��(0) = 0        (6.34b) 

In passing, we note from (6.33c) that  

 ��
� (0) = −��(0)

���
��(�)

��
 =  

���

��
 ,      ��

��(0) =   
�

�
    (6.34c) 

We conclude that 

 �(��) =  U�
(��)

(1 − cos 2��)      (6.35) 

On solving equations (6.9) to (6.11), using the appropriate initial and boundary 

conditions, we get 

 �(��)(�, �̂, �) =   �(��)(�, �̂, �) =  �(��)(�, �̂, �) = 0    (6.36) 

 

 

 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                               I.U. Udo-Akpan1  * 

https://ijojournals.com/                                              Volume 07 Issue 02 || February, 2024 || 

ON A-TWO-PARAMETER DYNAMIC BUCKLING OF A VISCOUSLY DAMPED BUT CLAMPED COLUMN STRESSED BY A STEP LOAD 
 

 

 

We next substitute on the right hand side of (6.12), using (6.25) and simplify to get 

��(��) =  � ���� +
3��

��

2
� + 3 ����� +  

��
�

4
� cos � �̂ + 

3���
� cos 2��̂

2
� 

+ �
��

� cos 3��̂

4
� �

5

2
− 

15 cos 2��

4
+  

3

2
cos 4�� −

1

4
cos 6��� 

−2��
�  U�,����

(��)
(1 −  cos 2��)            (6.37)  

Next, we assume 

 �(��) =   ∑ U�
(��)

(1 − cos 2��) �
���      (6.38) 

On substituting (6.38) into (6.37) and first multiplying through by cos 2��, and 

thereafter integrating from 0 to π, we see that, for � = �, we get 

 U�,����
(��)

+  ��U�
(��)

=  
15�

4
���� +

3��
��

2
� + 3 ����� + 

��
�

4
� cos � �̂ +

3���
� cos 2��̂

2
� 

+ �
��

� ��� ����

�
� − 2��

�  U�,����
(��)

    (6.39a)  

 U�
(��)(0,0) = 0,    �

�,��
(��)(0, 0) + ��

� (0)�
�,��
(��)(0, 0) = 0    (6.39b) 

Next, if in the substitution in (6.12), we multiply through by ���4�� and thereafter 

integrate from 0 to π, we get (for� = 2�) 

 U��,����
(��)

+  ���
� U��

(��)
=  

−3�

2
���� +

3��
��

2
� + 3 ����� +  

��
�

4
� cos � �̂ +

3���
� cos 2��̂

2
� 

+ �
��

� ��� ����

�
�    (6.40a) 

 U��
(��)(0,0) = 0,       �

��,��
(��)(0, 0) = 0                                                                            (6.40b) 

where, 

���
� =  (256�� −  32��� + 1) > 0,   ∀ �                                                       (6.40�) 

Lastly, if in the substitution into (6.12), we multiply through by cos 6��, and integrate 

from 0 to π, we see that, for � = 3�, we get 

U��,����
(��)

+  ���
� U��

(��)
=  

�

4
���� +

3��
��

2
� + 3 ����� +  

��
�

4
� cos � �̂ +

3���
� cos 2��̂

2
� 

+ ���
� ��� ����

�
�     (6.41a) 

U��
(��)(0,0) = 0,    �

��,��
(��) (0, 0) = 0                                                                  (6.41b) 

where, 

 ���
� =  (1296�� −  72��� + 1) > 0,   ∀ �                                                 (6.41�) 

To solve (6.39a,b), we substitute for U�
(��)

 from (6.28a), (noting (6.28c)), and ensure a 

uniformly valid solution in �̂ by equating to zero, the coefficient of cos � �̂ and get 

 ��
� (�) =  

����

��
��� +  

��
�

�
�,     ��

� (0) =   
�������

����     (6.42a) 

 ��
��(0) =   

�����

����           (6.42b) 

 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                               I.U. Udo-Akpan1  * 

https://ijojournals.com/                                              Volume 07 Issue 02 || February, 2024 || 

ON A-TWO-PARAMETER DYNAMIC BUCKLING OF A VISCOUSLY DAMPED BUT CLAMPED COLUMN STRESSED BY A STEP LOAD 
 

 

 

The remaining equation in (6.39a) is now solved to get 

U�
(��)(�̂, �) = �� cos � �̂ + �� sin � �̂ +

15�

4
�

��

��
−

3���
� cos 2��̂

2��
�

−  ���
� cos 3��̂

32��
�             (6.43�) 

where  

��(�) = ��� +
���

��

�
�        (6.43b)  

  ��(0) =   
����

�����
,      (0) = 0        (6.43c) 

 ��(0) =  
���

�
 ,     ��

�(0) =  −3��,     ��
��(0) =   6��    (6.43d) 

Turning to (6.40a,b), we solve to get 

U��
(��)

(�̂, �) =  �� cos ��� �̂ +  �� sin ��� �̂ − 
3�

2
�

��

���
� +   

�� cos ��̂

���
� − ��

+
3���

� cos 2��̂

2(���
� − 4��)

� 

+ �
��

� ��� ����

�����
� �����

�     (6.44a) 

where, 

��(�) = 3 ����� + 
��

�

�
�,   ��(0) =

�����

�
,  ��

�(0) =
����

�
,  ��

��(0) =  
�����

�
 (6.44b) 

 ��(0) =  
������

�
,       ��(0) = 0              (6.44c) 

�� =
�

����
� +  

��

�����
� �����

−
�

���
� ����

+
�

�����
� �����

          (6.44d) 

We next solve (6.41a) and get 

U��
(��)(�̂, �) =  �� cos ��� �̂ +  �� sin ��� �̂ +  

�

4
�

��

���
� +  

�� cos ��̂

���
� − ��

+
3���

� cos 2��̂

2(���
� − 4��)� 

  + �
��

� ��� ����

�����
� �����

�            (6.45a) 

where, 

 ��(0) =
�����

4
,    �� = �

1

���
� +   

15

���
� − ��

−
3

���
� − 4��

� + � 1

2(���
� − 9��)

�,   ��(0)

= 0       (6.45�) 

Thus, we have 

 �(��) =  U�
(��)(1 − cos 2��) + U��

(��)(1 − cos 4��) +  U��
(��)(1 − cos 6��)        (6.45�) 

 

 

 

 

 

 

 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                               I.U. Udo-Akpan1  * 

https://ijojournals.com/                                              Volume 07 Issue 02 || February, 2024 || 

ON A-TWO-PARAMETER DYNAMIC BUCKLING OF A VISCOUSLY DAMPED BUT CLAMPED COLUMN STRESSED BY A STEP LOAD 
 

 

 

We next substitute on the right hand side of (6.13) and get 

��(��) =  
3�

4
��� ���� +  

��
�

4
� sin � �̂ + ��� sin2 � �̂ −

��
�

4
sin3 � �̂�� �

5

2
−

15 cos 2��

4
� 

��+ 
3 cos 4��

2
−  

cos 6��

4
�� − 2 �U�,���

(��)
(1 − cos 2��) +  U��,���

(��)
(1 − cos 4��)� 

+ U��,���

(��)
(1 − cos 6��) +  U�,��

(��)
(1 − cos 2��) +  U��,��

(��)
(1 − cos 4��) 

�+ U��,��
(��)

(1 − cos 6��)� − ��
�� U�,��

(��)
(1 − cos 2��) 

−2��
�  U�,��

(��)
(1 − cos 2��)      (6.46) 

Let   

 �(��) =   � U�
(��)

(1 − cos 2��)                                                                                     (6.47)

�

���

 

On substituting (6.47) into (6.46), first multiplying through by cos 2�� and integrating 

from 0 to π, we get, for � = � 

U�,����
(��)

+  ��U�
(��)

=  
45�

4
��� ���� + 

��
�

4
� sin � �̂ + ��� sin2 � �̂ −  

��
�

4
sin3 � �̂�� 

− ��
�� U�,��

(��)
−  2��

�  U�,��
(��)

−  2 �U��,���

(��)
+  U��,��

(��)
� (6.48a) 

U�
(��)(0,0) = 0,    �

�,��
(��)(0, 0) + ��

� (0)�
�,��
(��)(0, 0) +  ��,�

(��)(0, 0) = 0                     (6.48b)  

Next, we multiply (6.46) by cos 4��, integrate from 0 to π, we get, for � = 2� 

U��,����
(��)

+  ���
� U��

(��)
=  

−9���

8
����� +  

��
�

4
� sin � �̂ + ��� sin2 � �̂ −  

��
�

4
sin3 � �̂�� 

− 2 �U��,���

(��)
+  U��,��

(��)
�    (6.49a) 

 U��
(��)(0,0) = 0,      �

��,��
(��)(0, 0) +  ���,�

(��) (0, 0) = 0                                                          (6.49b) 

Similarly, we multiply (6.46) by cos 6��, integrate from 0 to π, we get, for � = 3� 

U��,����
(��)

+  ���
� U��

(��)
=  

3���

16
����� + 

��
�

4
� sin � �̂ + ��� sin2 � �̂ −  

��
�

4
sin3 � �̂�� 

− 2 �U��,���

(��)
+  U��,��

(��)
�    (6.50a) 

 U��
(��)(0,0) = 0,     �

��,��
(��) (0, 0) + ���,�

(��) (0, 0) = 0     (6.50b) 

 

 

 

 

 

 

 

 

 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                               I.U. Udo-Akpan1  * 

https://ijojournals.com/                                              Volume 07 Issue 02 || February, 2024 || 

ON A-TWO-PARAMETER DYNAMIC BUCKLING OF A VISCOUSLY DAMPED BUT CLAMPED COLUMN STRESSED BY A STEP LOAD 
 

 

 

To solve (6.48a), we substitute for terms there and after, ensure a uniformly valid 

solution by equating to zero the coefficients of cos � �̂ and sin � �̂. This gives, for cos � �̂ 

 ��
� +  �� =

0                                                                                                                     (6.51�)  

Forsin � �̂: 

 ��
� + �� =  

��

2
(��

�� + 2��
� ) −

45�

32�
(�� + ��

�)��                                                         (6.51�) 

On solving (6.51a,b), we get  

 �� = 0;    �� =  ��� ���(0)

− � �� �
��

2
(��

�� + 2��
� ) +

45�

32�
(�� + ��

�)���

�

�

���             (6.51�)    

��
�(0) =  

��������

�����
        (6.52d) 

The remaining equations, having ensured a uniformly valid solution in (6.48a) are 

U�,����
(��)

+  ��U�
(��)

=  �� sin2 � �̂ + ��sin3 � �̂                                                                 (6.52�) 

U�
(��)(0,0) = 0,    �

�,��
(��)(0, 0) + ��

� (0)�
�,��
(��)(0, 0) +  ��,�

(��)(0, 0) = 0                     (6.52b) 

where, 

 �� =  
��������

��
− 

����

��
{(��

�)� +  (��
�)}      (6.52c) 

 �� =   
−45������

�

64
− 

15�

32�
{(��

�)� +  (��
�)}                                                             (6.52�)  

(0) =   
75���

16�
,      ��(0) =  

−225���

64�
                                                                       (6.52�) 

We note the following 

  r�
� (0) =  

−345���

128�
,    r�

� (0) =  
465���

32�
,     r�

� (0) =  
−45���

128�
                                      (6.52�) 

On solving (6.52a,b), we get 

 U�
(��)

=  �� cos � �̂ +  �� sin � �̂  −
�� sin2 � �̂

3��
−  

��sin3 � �̂

8��
                                        (6.53�) 

where, 

                ��(0)

= 0                                                                                                                         (6.53�) 

���(0) −  
2��(0)

3�
−   

3��(0)

8�
+  ��(0)���(0) +  

15�

4
�

��
�

��
−  

���
���

��
− 

3��
���

�

��
� │��� = 0 

This yields 

                    ��(0)

=   
−5695��

512��
                                                                                                    (6.53�) 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                               I.U. Udo-Akpan1  * 

https://ijojournals.com/                                              Volume 07 Issue 02 || February, 2024 || 

ON A-TWO-PARAMETER DYNAMIC BUCKLING OF A VISCOUSLY DAMPED BUT CLAMPED COLUMN STRESSED BY A STEP LOAD 
 

 

 

 

To ensure a uniformly valid solution in terms of �̂ in (6.49a), we equate to zero the 

coefficients of cos2 ��� �̂and sin2 ��� �̂ and respectively get 

                  ��
� +  �� = 0                                                                                                                 (6.53�) 

and 

  ��
� + �� =  

�

����
�

� ��

�
��� + 

��
�

�
� +  

�����
�� ���

���
� � ��

�          (6.53e) 

On solving (6.53d, e), we get 

 �� = 0,      �� =  ������(0) +  ∫ ����
�

�
���          (6.53f) 

where, 

 �� =   
�

����
�

� ��

�
��� +  

��
�

�
� + 

�����
�� ���

���
� � ��

�      (6.53g) 

From (6.53e), we get 

 ��
� (0) =  �����,    �� =  

�

����
�

��

���
+  

��

�����
� � ���

� −  
���

�
   (6.53h) 

The remaining equation in (6.49a) is now written as 

 U
��,����
(��)

+  ���
� U��

(��)
=  �� sin � �̂ + �� sin2 � �̂ + ��sin3 � �̂  (6.54a) 

 U��
(��)(0,0) = 0,     �

��,��
(��) (0, 0) + ���,�

(��) (0, 0) = 0     (6.54b) 

where, 

 �� =   
�����

�
��� +  

��
�

�
� +  

������
�� ���

���
� � ��

     (6.54c) 

 �� =   
��������

�
−  

��������
��

�
� ���

���

���
� � ���       (6.54d) 

  �� =  
����

���

��
−  

�������
��

�
� ���

���

�����
� � ����

       (6.54e) 

where, 

 ��(0) =  �����,           �� =  
���

���
+ 

��

�����
� � ���

    (6.54f) 

��(0) =  �����,            �� =    9 �
�

��
+  

�

���
� � ���

�    (6.54g) 

��(0) =  ������,        ��� =    
�

���
−  

��

�����
� � ����

    (6.54h) 

 

 

 

 

 

 

 

 

 

 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                               I.U. Udo-Akpan1  * 

https://ijojournals.com/                                              Volume 07 Issue 02 || February, 2024 || 

ON A-TWO-PARAMETER DYNAMIC BUCKLING OF A VISCOUSLY DAMPED BUT CLAMPED COLUMN STRESSED BY A STEP LOAD 
 

 

 

On solving (6.54a,b), we get 

U��
(��)

= �� cos ����̂ + �� sin ����̂ + 
�� ��� ���

���
� � ��

+
�� ��� ����

���
� � ���

+
�� ��� ����

���
� � ���

 (6.55a) 

��(0) = 0,    ��(0) = −
1

2���
�

���

���
� −  ��

+ 
2���

���
� −  4��

+  
3���

���
� −  9��

+  ��
� (0)� 

= ��− 
��

�
�

��
�

���
� + 

��
�

���
� � ��

+
����

� ��

���
� � ���

+
���

���

�����
� � ����

���
���

(6.55b) 

That is 

 ��(0) =  ������,         (6.55c) 

��� =   −
1

2���
�� �

��

���
� − ��

+  
2��

���
� −  4��

+ 
3���

���
� − 9��

�� 

+ ��
�

�
�

���
� −  

��

�����
� � ���

+  
�

���
� � ��� − 

�

���
� � �����         (6.55d) 

Next we ensure a uniformly valid solution in �̂ in (6.50a) by first substituting for the 

relevant terms there, and equating to zero the coefficients of cos ����̂ and sin ����̂ to 

get 

 ��
� +  �� = 0,      ��

� + �� = 0      (6.56a) 

On solving (6.56a), we get 

 ��(�) = 0,      ��(�) =   ��(0)���     (6.56b) 

where,  

 ��
� (0) = −��(0),       ��

��(0) = ��(0)     (6.56c) 

The remaining equation in (6.50a) is now written as 

 U��,����
(��)

+  ���
� U��

(��)
=  �� sin � �̂ + �� sin2 � �̂ +  ���sin3 � �̂                                     (6.57�) 

  U��
(��)(0,0) = 0,          �

��,��
(��)( 0, 0) +  ���,�

(��) (0, 0) = 0                                                  (6.57b) 

where, 

  �� =   
����

��
��� +  

��
�

�
� + 

��������
� � 

���
���

�

�
�

�����
� � ���

         (6.57c) 

 �� =   
�������

��
+ 

�������
� ��� ��

��

�����
� � ����

           (6.57d) 

 ��� =  
�����

���

��
+  

���

�
(��

���
� + ��

�)          (6.57e) 

��(0) =  �����,            �� =   
��

���
+ 

��

�����
� � ���

         (6.57f) 

 ��(0) =  
�����

���
,          ���(0) =   

�����

��
          (6.57g) 

 ��
�(0) =  

������

����
−  

�����

�����
� � ���

           (6.57h) 

 ��
�(0) =  ��� �

��

���
−  

��

���
� � ���           (6.57i) 

 ���
� (0) =  

����

���
         (6.57j) 

 

 

https://zenodo.org/doi/10.5281/zenodo.10726874 IJO JOURNALS

Volume 07 | Issue 02 | February 2024 | https://ijojournals.com/index.php/m/index 15



IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                               I.U. Udo-Akpan1  * 

https://ijojournals.com/                                              Volume 07 Issue 02 || February, 2024 || 

ON A-TWO-PARAMETER DYNAMIC BUCKLING OF A VISCOUSLY DAMPED BUT CLAMPED COLUMN STRESSED BY A STEP LOAD 
 

 

 

On solving (6.57a,b), we get 

U��
(��)

= �� cos ����̂ + �� sin ����̂ + 
�� ��� ���

���
� � ��

+
�� ��� ����

���
� � ���

+  
��� ��� ����

���
� � ���

 (6.58a) 

with 

 ��(0) = 0,           (6.58b) 

��(0) =  −
1

���
�
� �

��

���
� −  ��

+ 
2��

���
� −  4��

+  
3���

���
� −  9��

� + ��
�
� 

+ ���
�

�
��

�

���
� +  

��
�

���
� � ��

+  
�����

��
�
��

�����
� � ����

+  
���

���
�

�����
� � ����

���
���

 (6.58c) 

We note from (6.56a) that  

 ��
� (0) = −��(0) =

������

�
       (6.58d) 

On simplifying (6.58c), we get 

��(0) =   �����,             (6.58e) 

�� =  −
1

���
�� �

��

���
� −  ��

− 
3

8(���
� −  4��)

−  
9

56(���
� −  9��)

�� −
��

8
 

+ ��
�

�
�

����
� +  

��

�����
� � ���

−  
�

���
� � ��� +  

�

�����
� � ����

��       (6.58f) 

We next substitute on the right hand side of (6.14) and get 

��(��) = 3� ���
(��)

���
(��)

�
�

+  ���
(��)

�
�

��
(��)

� (1 − cos 2��) 

− ���,��
(��)(1 − cos 2��) +  ���,��

(��) (1 − cos 4��) +  ���,��
(��) (1 − cos 6��)� 

−2 ���,���

(��)
(1 − cos 2��) +  ���,���

(��)
(1 − cos 4��) + ���,���

(��)
(1 − cos 6��)� 

+ ���,��
(��)

(1 − cos 2��) + ���,��
(��)

(1 − cos 4��) + ���,��
(��)

(1 − cos 6��)� 

 + ��
� ��,���

(��)
(1 − cos 2��) −  ��

����,��
(��)

(1 − cos 2��) 

−2 ���
� ��,��

(��)
(1 − cos 2��) + ��,�

(��)(1 − cos 2��) + ���,�
(��) (1 − cos 4��)� 

+����,�
(��) (1 − cos 6��)�        (6.59a) 

By letting  

 �(��) =  ∑ U�
(��)

(1 − cos 2��) �
���        (6.59b) 

and substituting same into (6.59a), multiplying the resultant equation by cos 2�� and 

integrating from 0 to π, we see that, for  � = �, we get 

U�,����
(��)

+ ��U�
(��)

=  
45�

4
���

(��)
���

(��)
�

�

+ ���
(��)

�
�

��
(��)

� − ��,��
(��)

 

 − 2 ���,���

(��)
+ ��,��

(��)
+ ��

� ��,���

(��)
� − ��

����,��
(��)

− 2��
� ��,��

(��)
− 2��,�

(��)
      (6.60a) 

 U�
(��)(0,0) = 0,      �

�,��
(��)(0, 0) + ��,�

(��)(0, 0)  + ��
� ��,��

(��)
= 0       (6.60b) 

 

 

https://zenodo.org/doi/10.5281/zenodo.10726874 IJO JOURNALS

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                               I.U. Udo-Akpan1  * 

https://ijojournals.com/                                              Volume 07 Issue 02 || February, 2024 || 

ON A-TWO-PARAMETER DYNAMIC BUCKLING OF A VISCOUSLY DAMPED BUT CLAMPED COLUMN STRESSED BY A STEP LOAD 
 

 

 

For � = 2� in the substitution into (6.59a), we get 

 U��,����
(��)

+  ���
� U��

(��)
=  

−9�

4
���

(��)
���

(��)
�

�

+ ���
(��)

�
�

��
(��)

� − ���,��
(��)

 

   − 2 ����,���

(��)
+  ���,��

(��)
� − 2���,�

(��)
        (6.61a) 

U��
(��)(0,0) = 0,          �

��,��
(��)(0, 0) + ���,�

(��) (0, 0)        (6.61b) 

For � = 3� in the substitution into (6.59a), we get 

       U��,����
(��)

+  ���
� U��

(��)
=  

3�

4
���

(��)
���

(��)
�

�

+ ���
(��)

�
�

��
(��)

� − ���,��
(��)

 

    −2 ����,���
(��)

+ ���,��
(��)

� − 2���,�
(��)

      (6.62a) 

 U��
(��)(0,0) = 0,          �

��,��
(��) (0, 0) +  ���,�

(��) (0, 0)                                                      (6.62�) 

If we substitute for terms on the right hand side of (6.60a), we get 

 U
�,����
(��)

+ ��U�
(��)

=  
45�

4
���

� �
�� cos ��̂

4
+  

�

2
(1 − cos 2��̂) − 

��

4
cos 3��̂�� 

  + ������ +  �� ��� + 
���

�

�
� cos ��̂ +  �� ��� + 

��
�

�
� ��� ��̂� 

  + ������� cos 2��̂ + �������� 2��̂ + 
����

�

�
cos 3��̂ + 

����
�

�
sin 3��̂�� 

  − ���
�� cos ��̂ +  

���

�
�

��
��

��
−  

����
��

��

���
cos 2��̂ − 

���
��

��

����
cos 3��̂�� 

−2 �����
� cos ��̂ − ���

� sin ��̂ −
2��

�

3�
cos 2��̂ −

3��
�

8�
cos 3��̂�� 

−���
� ��

� � cos ��̂] − ��
��� ��cos ��̂ − 2��

� ��� cos ��̂ 

  −2 ���
� cos ��̂ +   

���

�
�

��
�

��
−  

����
��

�

���
cos 2��̂ − 

���
��

�

����
cos 3��̂��   (6.63a) 

To ensure a uniformly valid solution in �̂ as far as (6.63a) is concerned, we equate to 

zero the coefficients of cos ��̂ and sin ��̂ and respectively get 

  ��
� + �� = −ℎ�(�) = −

� 

��
�� ��� +

��
�

�
�     (6.63b) 

and 

  ��
� +  �� =  ℎ�(�) 

 = −
� 

��
�

���

�
�

����
�

�
+ �� ��� +

���
�

�
�� − 2��

� − ��
�� − ��

�����
� 

  −���
� �(2�� +  ��

� )]      (6.63c) 

On solving (6.63b,c), we get 

 �� =  ��� ���(0) − � ℎ�(�)����
�

�

� 

 �� =  ��� ���(0) +  � ℎ�(�)����
�

�

� 

 

 

 

https://zenodo.org/doi/10.5281/zenodo.10726874 IJO JOURNALS

Volume 07 | Issue 02 | February 2024 | https://ijojournals.com/index.php/m/index 17



IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                               I.U. Udo-Akpan1  * 

https://ijojournals.com/                                              Volume 07 Issue 02 || February, 2024 || 

ON A-TWO-PARAMETER DYNAMIC BUCKLING OF A VISCOUSLY DAMPED BUT CLAMPED COLUMN STRESSED BY A STEP LOAD 
 

 

 

The remaining equation in (6.60a) is 

U
�,����
(��)

+ ��U�
(��)

= ��� + ��� cos 2��̂ + ��� sin 2��̂ + ��� cos 3��̂ + ��� sin 3��̂      (6.64�) 

 U�
(��)(0,0) = 0,    �

�,��
(��)(0, 0) +  ��,�

(��)(0, 0) + ��
� ��,��

(��)
= 0   (6.64b) 

where, 

 ��� =  
���

�
������ −

���
�

�
� −  

����
��

���
−  

����
���

���
    (6.64c) 

��� =  
���

�
������ −

���
�

�
� +  

������
��

��

���
+ 

����� ��
��

��
+ 

�������
��

�

���
  (6.64d) 

��� =  
��������

�
        (6.64e) 

��� =  
���

��
(��

��� − ��
���) +  

������
��

��

�����
+  

����
�����

��
    (6.64f) 

��� =
�������

�

��
+  

������
��

�

����
       (6.64g) 

���(0) =
������

��� ,          ���(0) =
������

����  ,    ���(0) =  
������

��
,   (6.64h) 

���(0) =  
�������

�����
,       ���(0) =  

������

����
     (6.64i) 

 

On solving (6.64a), using (6.64b) we get 

U�
(��)(�̂, �) = ��� cos ��̂ + ��� sin ��̂ +

���

��
−  

1

3��
(��� cos 2��̂ +  ��� sin 2��̂) 

− 
�

���
(��� cos 3��̂ +  ��� sin 3��̂)     (6.65a) 

where, 

  ���(0) =   
−���(0)

��
+  

���(0)

3��
+  

���(0)

8��
 

  ���(0) =   
���������

�����
,               ���(0) = 0      (6.65b) 

As was in the case of U�
(��)

 in (6.60a) which eventually led to (6.65a), we now substitute 

for terms into (6.61a) and to ensure a uniformly valid solution in �̂, we equate to zero 

the coefficients of cos ����̂ and sin ����̂ and respectively get 

 ��
� +  �� =  

��

��
(��

�� + 2��
� )       (6.66a) 

and 

 ��
� + �� =  

��

��
(��

�� + 2��
� )       (6.66b) 

On solving, we get 

 �� =  ��� ���(0) −
�

��
∫ ��(��

�� + 2��
� )��

�

�
�    (6.66b) 

 �� =  ��� ���(0) −  
�

��
∫ ��(��

�� + 2��
� )��

�

�
�    (6.66c) 

 

 

 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                               I.U. Udo-Akpan1  * 

https://ijojournals.com/                                              Volume 07 Issue 02 || February, 2024 || 

ON A-TWO-PARAMETER DYNAMIC BUCKLING OF A VISCOUSLY DAMPED BUT CLAMPED COLUMN STRESSED BY A STEP LOAD 
 

 

 

The remaining equation in the substitution into (6.61a) is 

U��,����
(��)

+ ���
� U��

(��)
= ��� + ��� cos � �̂ + ��� sin � �̂ + ���cos 2� �̂ + ��� sin 2� �̂ 

   + ���cos 3� �̂ +  ���sin 3� �̂    (6.67a) 

 U��
(��)(0,0) = 0,          �

��,��
(��)(0, 0) + ���,�

(��) (0, 0)    (6.67b) 

where, 

 ��� =  
���

�
(���

� + �����) + 
���

��

���
� + 

���
���

����
�      (6.67c) 

��� =  
���

�
�

����
�

�
+  �� �

���
�

�
+  ���� +  

����
��

�����
� ����

+  
����

�

�����
� ����

−  
�����

�����

���
� ���

 (6.67d)  

��� =  
�����

�
�

���
�

�
+  ���        (6.67e) 

��� =  −9� �
���

�

�
+ ������ +  

������
��

��

�����
� �����

+   
������

��
�

�����
� �����

− 
�����

�����

���
� ����

  (6.67f) 

��� =
�������

�
         (6.67g) 

��� = −9� �
��

���

�
−

����

�
� + 

�����
��

��

�����
� �����

+  
�����

��
�

�����
� �����

− 
�����

�����

���
� ����

  (6.67h) 

 ��� =  
�������

�

�
         (6.67i) 

On solving (6.67a,b), we get 

U��
(��)(�̂, �) = ��� cos ����̂ + ��� sin ����̂ +

���

���
� + �

1

���
� −  ��

� (��� cos ��̂ + ��� sin ��̂) 

  + �
�

���
� � ���� (��� cos 2��̂ + ��� sin 2��̂) 

  + �
�

���
� � ���� (��� cos 3��̂ + ��� sin 3��̂)    (6.68a) 

We may not need ���(0) and ���(0). We next substitute into (6.62a) and to ensure a 

uniformly valid solution in �̂, equate to zero, the coefficients of cos ����̂ and sin ����̂ 

and respectively get 

  ��
� +  �� =  

��

����
(��

�� + 2��
� )      (6.68b) 

and 

  ��
� +  �� =  

�

����
(��

�� + 2��
� )      (6.68c) 

On solving, we get 

 �� =  ��� ���(0) −
�

����
∫ ��(��

�� + 2��
� )��

�

�
�   (6.68d) 

  �� =  ��� ���(0) +  
��

����
∫ ��(��

�� + 2��
� )��

�

�
�    (6.68e) 

The remaining equation in the substitution into (6.62a) is  

 U��,����
(��)

+  ���
� U��

(��)
=  ��� +  ��� cos � �̂ + ��� sin � �̂ + ���cos 2� �̂ +  ��� sin 2� �̂ 

   + ���cos 3� �̂ +  ���sin 3� �̂   (6.69a) 

  U��
(��)(0,0) = 0,     �

��,��
(��) (0, 0) + ���,�

(��) (0, 0)    (6.69b) 

 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                               I.U. Udo-Akpan1  * 

https://ijojournals.com/                                              Volume 07 Issue 02 || February, 2024 || 

ON A-TWO-PARAMETER DYNAMIC BUCKLING OF A VISCOUSLY DAMPED BUT CLAMPED COLUMN STRESSED BY A STEP LOAD 
 

 

 

where, 

 ��� =  
�����

�

�
+  

�������

�
−  �

���
��� ��

��

����
�       (6.69c) 

��� =  
����

���

��
+  

����

�
�

���
�

�
+  ��� −  

���
��

�����
� ����

−
�����

�� ���

���
� ���

−  
����

�

�����
� ����

 (6.69d) 

��� =  3��� ��� −
��

�

�
�        (6.69e) 

��� =  
������

�

�
 +  

�������

�
−

������
��

��

�����
� �����

−   
������

��
�

���
� ����

− 
�����

�����

���
� ����

   (6.69f) 

��� =
�������

�
         (6.69g) 

��� = −
�������

�

�
 +  

���
����

��
−

����
��

��

�����
� �����

−   
����

��
�

�����
� �����

−  
������

� �����

���
� ����

  (6.69h) 

 ��� =  
������

�

��
         (6.69i) 

 ���(0) =
����

���
,      ���(0) =   3������      (6.69j) 

 ��� =  
��

���� +  
��

������
� ����

+  
���

���
� ��� �

��

����
+  

��

�����
� �����

� − 
��

�����
� ����

 (6.69k) 

���(0) = 0,        ���(0) =  3������       (6.69l) 

 ��� =   
��

��� −  
�

�����
� ����

−  
��

���
� ��� �

��

���
−  

��

���
� ���� +  

�

���
� ���   (6.69m) 

���(0) = 0,       ���(0) =  3������,        ���(0) = 0     (6.69n) 

 ��� =   
�

���� +  
��

������
� �����

       (6.69o) 

On solving (6.69a,b), we get 

���
�� (�̂, �) =  ��� cos ����̂ +  ��� sin ����̂ +  

���

���
� + �

1

���
� − ��

� (��� cos ��̂ + ��� sin ��̂) 

  + �
�

���
� � ���� (��� cos 2��̂ + ��� sin 2��̂) 

  + �
�

���
� � ���� (��� cos 3��̂ + ��� sin 3��̂)    (6.70a) 

where, 

 ���(0) =  3������,         (6.70b) 

 ��� =  − �
�

������
� + 

���

���
� � �� + 

���

���
� � ��� +  

���

���
� � ����   (6.70c) 

���(0) = 0          (6.70d) 

So far, we write the deflection as 

�(�, �̂, �) =  ����
(��)

+  ���
(��)

+  ����
(��)

+ ⋯ �(1 − cos 2��) 

+�����
(��)(1 − cos 2��) +  ���

(��)(1 − cos 4��) +  ���
(��)(1 − cos 6��)� 

 +����
(��)(1 − cos 2��) + ���

(��)(1 − cos 4��) + ���
(��)(1 − cos 6��)� 

 +�����
(��)(1 − cos 2��) +  ���

(��)(1 − cos 4��) + ���
(��)(1 − cos 6��)� 

 �+ ⋯ ] + ⋯         (6.71) 

 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                               I.U. Udo-Akpan1  * 

https://ijojournals.com/                                              Volume 07 Issue 02 || February, 2024 || 

ON A-TWO-PARAMETER DYNAMIC BUCKLING OF A VISCOUSLY DAMPED BUT CLAMPED COLUMN STRESSED BY A STEP LOAD 
 

 

 

7. CRITICAL VALUES OF THE DEPENDENT VARIABLES AT MAXIMUM DEFLECTION 

As in [18, 19], the dynamic buckling load ��, which is defined as the largest load 

parameter for the solution of the problem to be bounded, is obtained from the 

maximization 

  
��

���
= 0         (7.1) 

where, �� is the maximum deflection whose maximum values of the dependent 

variables we shall now determine. 

Let ��, �̂�, �� be the values, at maximum displacement of �, �̂and � respectively and let us 

now assume the following asymptotic series 

�̂� =  �̂� +  ��̂�� +   ���̂�� + ⋯ +  ��(�̂�� +  ��̂�� +  ���̂�� + ⋯ ) + ⋯  (7.2a) 

�� =  �� +  ���� +  ����� + ⋯ +  ��(��� +  ���� +  ����� + ⋯ ) + ⋯ 

�� = ��� = �[�� + ���� + ����� + ⋯ + ��(��� + ���� + ����� + ⋯ ) + ⋯ ] (7.2b) 

As a function of �, �̂, �, the conditions for �(�, �̂, �) to have a maximum are 

 �,� = 0,       (1 +  ��
� �� +  ��

� �� + ⋯ )�,�� +  ��,� = 0   (7.3) 

On substituting (6.71) into the first of (7.3), we get the value of � at maximum 

deflection, namely ��, as 

 �� =   
�

��
,      � = 1, 2, 3, …      (7.4) 

On evaluating (6.71) at � =  �� =   
�

��
, we get 

� =  2����
(��)

+ ���
(��)

+ ����
(��)

+ ⋯ � +  2�� ���
(��)

+ ���
(��)

+ ����
(��)

+  ���
(��)

�� 

 +�� ����
(��)

+   ���
(��)

� + ⋯ �       (7.5) 

We shall now expand each of the terms of (7.3), (which is evaluated at (��, �̂�, ��)) by 

using (7.2a-c), as well as (7.4) and (7.5). Thus, we get 

���,��
(��)

=  � ���,��
(��)

+ {��̂�� + ���̂�� + ⋯ +  ��(�̂�� + ��̂�� + ���̂�� + ⋯ ) + ⋯ }��,����
(��)� 

+�{�� +  ���� +  ����� + ⋯ +  ��(��� +  ���� +  ����� + ⋯ ) + ⋯ }�
�,���

(��)
 

+
1

2
�{��̂�� + ���̂�� + ⋯ +  ��(�̂�� +  ��̂�� + ���̂�� + ⋯ ) + ⋯ }��

�,������
(��) � 

+2�{��̂�� + ⋯ + ��(�̂�� + ��̂�� + ⋯ ) + ⋯ } 

× {�� + ���� + ⋯ + ��(��� + ���� + ⋯ )}�
�,�����

(��)
 

 +��{�� + ���� + ⋯ + ��(��� +  ���� + ⋯ ) + ⋯ } �����,����

(��)
���

(���,�)
 (7.6a) 

���
�,��
(��)

=  ����
�,��
(��)

+  {��̂�� + ⋯ +  ��(�̂�� +  ��̂�� +  … )}�
�,����
(��)� 

+�{�� + ⋯ +  ��(��� +  ���� + ⋯ ) + ⋯ }�
�,���

(��)
 

+
1

2
�{��̂�� +   … +  ��(�̂�� +  ��̂�� + ⋯ ) + ⋯ }���,������

(��) � 

+2�{��̂�� + ⋯ + ��(�̂�� + ��̂�� + ⋯ ) + ⋯ } 

× {�� + ���� + ⋯ + ��(��� + ���� + ⋯ )}�
�,�����
(��)

 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                               I.U. Udo-Akpan1  * 

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ON A-TWO-PARAMETER DYNAMIC BUCKLING OF A VISCOUSLY DAMPED BUT CLAMPED COLUMN STRESSED BY A STEP LOAD 
 

 

 

 +��{�� + ���� + ⋯ + ��(��� +  ���� + ⋯ ) + ⋯ } �����,����
(��)

���
(���,�)

 (7.6b) 

�����,��
(��)

=  ��� ���,��
(��)

+  {��̂�� + ⋯ + ��(�̂�� +  ��̂�� +  … )}��,����
(��)� 

  + �{�� + ⋯ +  ��(��� +  ���� + ⋯ ) + ⋯ } ���
�,���

(��)
+ ⋯ ��

(���,�)
 (7.6c) 

�� ���,��
(��)

+ ���,��
(��)

� = �� ����,��
(��)

+ ���,��
(��)

� + { ��̂�� + ���̂�� + ⋯ } ���,����
(��)

+ ���,����
(��)

�� 

   +  �{�� +  ���� + ⋯ } ����
�,���

(��)
+  �

��,���

(��)
� + ⋯ ��

(���,�)
 (7.6d) 

�����
�,��
(��)

+ �
��,��
(��)

� = ��� ����,��
(��)

+ ���,��
(��)

� + { ��̂�� + ⋯ } ���,����
(��)

+ ���,����
(��)

�� 

   +  �{�� +  ���� + ⋯ } ����
�,���

(��)
+  �

��,���

(��)
� + ⋯ ��

(���,�)
 (7.6e) 

������
�,��
(��)

+ �
��,��
(��)

� =  ���� ����
�,��
(��)

+ �
��,��
(��)

� + ⋯ ��
(���,�)

   (7.6f) 

����
� �

�,��
(��)

= �� ���
� �

�,��
(��)

+ ��
� {��̂�� + ���̂�� + ⋯ }�

�,����
(��)

+ �{�� + ���� + ⋯ }���
� �

�,��
(��)

�
,�

� 

 +
�

�
{�̂�� + ⋯ }���

� �
�,����
(��)

+ �{�� + ���� + ⋯ }{��̂�� + ���̂�� + ⋯ }���
� �

�,����
(��)

�
,�

 

 +  
�

�
��(�� + ⋯ )� �����

� �
�,��
(��)

�
,��

+ ⋯ ��
(���,�)

    (7.6g) 

�����
� ��,��

(��)
=  ��� ���

� ��,��
(��)

+ (��̂�� + ⋯ )��
� ��,����

(��)� 

  + ���(�� + ⋯ )���
� �

�,����
(��)

�
,�

+ ⋯ ��
(���,�)

     (7.6h) 

������
� �

�,��
(��)

=  ���� ����
� �

�,��
(��)

+ ⋯ ��
(���,�)

        (7.6i) 

����,�
(��)

= �� ���,�
(��)

+  {��̂�� + ⋯ +  ��(�̂�� +  ��̂�� +  … )}��,���

(��) � 

 +
�

�
�{��̂�� +   … +  ��(�̂�� +  ��̂�� + ⋯ ) + ⋯ }��

�,�����
(��) � 

 +2�{��̂�� + ⋯ +  ��(�̂�� +  … ) + ⋯ } 

 × {�� +  ���� + ⋯ +  ��(��� +  ���� + ⋯ )} ����
�,����
(��)

+ ⋯ ���
(���,�)

 (7.6j) 

�����,�
(��)

= ������,�
(��)

+ ⋯ + {��̂�� + ⋯ + ��(�̂�� + ��̂�� + ⋯ )}� ����,���

(��)
+ ⋯ ��

(���,�)
(7.6k) 

������,�
(��)

+  ���,�
(��)

� = ��� ����,�
(��)

+  ���,�
(��)

� + �(��̂�� +  … ) ���,���

(��)
+ ���,���

(��)
� + ⋯ �� 

   ��+�(�� +   … )���,��
(��)

+  ���,��
(��)

� + ⋯ ��
(���,�)

   (7.6l) 

�������,�
(��)

+ ���,�
(��)

� =  ������,�
(��)

+  ���,�
(��)

�│(���,�) + ⋯      (7.6m) 

 

 

 

 

 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                               I.U. Udo-Akpan1  * 

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ON A-TWO-PARAMETER DYNAMIC BUCKLING OF A VISCOUSLY DAMPED BUT CLAMPED COLUMN STRESSED BY A STEP LOAD 
 

 

 

By substituting (7.6a – m) into the second equation of (7.3) and equating the 

coefficients of ����, we get 

�(�):         ��,��
(��)

(�̂�, 0) = 0         (7.7a) 

�(��):      �̂���
�,����
(��)

 + ���
�,���

(��)
+   ��,�

(��)
= 0      (7.7b) 

�(���):    �̂����,����
(��)

 + �����,���

(��)
+  

(�̂��)�

2
��,����

(��)
+  �̂������,�����

(��)
+ 

(��)�

2
��,����

(��)
 

  +�̂����,����
(��)

+ ����,���
(��)

+  ��,��
(��)

+ �̂����,���
(��)

+ ����,��
(��)

+  ��,�
(��)

 (7.7c) 

�(��):      �̂���
�,����
(��)

 +  ��
�,����
(��)

+  �
��,����
(��)

� +  ��
� (0)�

�,��
(��)

= 0   (7.7d) 

�(���):    �̂����,����
(��)

 +  �����,���
(��)

+   
1

2
�2�̂���̂����,������

(��)
+  2�̂������,�����

(��)
� +  �̂����,����

(��)
 

+ �̂�� ���,����
(��)

+  ���,����
(��)

� + �� ���,���

(��)
+  ���,���

(��)
� + ���,��

(��)
+ ���,��

(��)
� 

  + �̂���
�,���

(��)
+  ���,�

(��)
+  ���,�

(��)
� = 0     (7.7e) 

etc., where equations (7.7a) to (7.7e) are evaluated at (�̂�, 0). 

From (7.7a), we get  

  ��̂� =  ��,        � = 1, 2, 3, … 

 ∴                  �̂� =  
�

�
,       (� = 1)       (7.8a) 

where we have taken  � = 1. 

On substituting (7.8) in (7.7b) and simplifying, we get 

 �̂�� = �
���

�,��
(��)

� ��,�
(��)

� �
�,���

(��)
�

�
�,����
(��) �

(���,�)

= 0      (7.8b) 

On substituting for terms in (7.7c) and simplifying, we get 

 �̂�� =  �������,��
(��)

�
�,����
(��) �

(���,�)

=  
���

�� =  
�

��      (7.8c) 

We next substitute into (7.7d) and get 

�̂�� = �
���

�,����
(��)

� �
��,����
(��)

�

�
�,����
(��) �

(���,�)

      (7.8d) 

Now, 

��,����
(��)

(�̂�, 0) =  
�����

��
,      ���,����

(��)
(�̂�, 0) =  ������,     (7.8e) 

��� =
�

�
��

�

����
� −  

��

�����
� � ���

+  
�

�����
� � ����

− 
�

�����
� � ����

��   

− ��
����

�����
� � ���

+  
����

����
� � ����

−  
���

�����
� � ����

��   (7.8f) 

 �
�,����
(��)(�̂�, 0) =  −���         (7.8g) 

On substituting for terms in (7.8d), we get 

 �̂�� =  ������,       ��� =  ���� +  
��

��
�     (7.8h) 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                               I.U. Udo-Akpan1  * 

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ON A-TWO-PARAMETER DYNAMIC BUCKLING OF A VISCOUSLY DAMPED BUT CLAMPED COLUMN STRESSED BY A STEP LOAD 
 

 

 

After substituting for terms in (7.7e) and simplifying, many terms vanish and the 

remaining terms give 

 �̂�� = ��
��

�
�,����
(��) ��̂���� +  �

�,�����

(��)
+ �

�,��
(��)

+ �
��,��
(��)

+  ��,�
(��)

+  ���,�
(��)

���
(���,�)

 (7.8i) 

We now evaluate each of the terms in (7.8i) as follows: 

  ��,��
(��)

(�̂�, 0) =
�������

���
       (7.9a) 

 ���,��
(��)

(�̂�, 0) =  ������       (7.9b)  

 ��� =   �� cos ����̂� − 
���

�����
� � ���

−  
�

������
� � ����

−  
�

������
� � ����

  (7.9c) 

  ��,�
(��)(�̂�, 0) =  

������

����
       (7.9d) 

  ���,�
(��) (�̂�, 0) =  ������       (7.9e) 

��� =  −
�� ��� ������

�
− 

�

�
�

�

���
� +  

��

�����
� � ���

+  
�

������
� � ����

+ 
�

������
� � ����

� (7.9f) 

  �
�,�����
(��) (�̂�, 0) =  ����

�(0) =  ���       (7.9g) 

On substituting for terms in (7.8i), we get 

�̂�� =  ������,         ��� =
�

�� �������� + 
����

���
+  ��� −  

�����

����� +  ����  (7.10) 

Later, we shall also need terms like ��, ���, ���, ��� ��� ��� which we now evaluate 

directly from (6.2b) (evaluated at maximum values of the variables). Thus, at maximum 

deflection, (6.2b) becomes 

 �̂� = �� +  �
��(��)���  ��(��)���⋯

�
�      (7.11) 

By using (7.2a – c), we can write (7.11) as 

�̂� +  ��̂�� +   ���̂�� + ⋯ + ��(�̂�� +  ��̂�� +  ���̂�� + ⋯ ) + ⋯ 

=   �� +  ���� +   ����� + ⋯ +  ��(��� +  ���� +  ����� + ⋯ ) 

+ �� ���(0) + ����
� (0) +

���
�

2
��

��(0)� 

  +�� ���(0) + ����
� (0) +

���
�

�
��

��(0) + ⋯ � + ⋯    (7.12) 

 

 

 

 

 

 

 

 

 

 

 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                               I.U. Udo-Akpan1  * 

https://ijojournals.com/                                              Volume 07 Issue 02 || February, 2024 || 

ON A-TWO-PARAMETER DYNAMIC BUCKLING OF A VISCOUSLY DAMPED BUT CLAMPED COLUMN STRESSED BY A STEP LOAD 
 

 

 

We note the fact that ��(0) = 0 and equally note that �� can be expanded using (7.2b). 

By equating coefficients of  ���� in (7.12), we get 

�(�):         �̂� =  
�

�
=   �� 

 ∴               ��   =    
�

�
          (7.13a) 

�(��):       �̂�� = 0 =   ��� 

 ∴                ��� = 0          (7.13b) 

�(���):         �̂�� =   
��

�
=  

�

��
=  ��� 

 ∴                 ��� =
�

��
         (7.13c) 

�(��):         �̂�� =  ������ =  ��� + ��
� (0)�� 

∴                   ��� =  ������ − ��
� (0)�� =  ������

(�)
     (7.13d) 

  ���
(�)

=   ��� +  
����

����
       (7.14) 

�(���):      �̂�� =  ������ =  ��� +  �����
� (0) +  

��
�

2
��

��(0) 

∴ ��� =  ������ −  �����
� (0) − 

��
�

�
��

��(0) =   ������
(�)

   (7.15a) 

  ���
(�)

=  ��� −  
����

����       (7.15b) 

 

8.  MAXIMUM DEFLECTION, �� 

To determine the maximum deflection w�, we evaluate (7.5) at �̂�,  �� ��� ��. Thus, we 

get 

w� = 2�����
(��)

+ ����
(��)

+ �����
(��)

� + 2�� �����
(��)

+  ����
(��)

� +  �����
(��)

+  ����
(��)

�� 

  + �������
(��)

+  ����
(��)

� + ⋯ � + ⋯     (8.1) 

where,���
(��)

=  ��
(��)

(�̂�, ��). 

We now expand each term of (8.1) using (7.2a – c). Thus, we have 

����
(��)

= ���
(��)(�̂�, ��, ��) = ����

(��)
+ {�̂��� + ���̂�� + ⋯ + ��(�̂�� + ��̂�� + ���̂�� + ⋯ )}� 

 +
�

�
{{�̂��� + ⋯ +  ��(�̂�� +  ��̂�� + ���̂�� +  … )}�� 

 +  �{�� +  ���� + ⋯ + ��(��� +  ���� + ⋯ )} 

 × {�̂��� + ⋯ + ��(�̂�� + ��̂�� + ���̂�� +  … )}��,��
(��)

 

 ��+��{�� +  ���� + ⋯ +  ��(��� +  ���� + ⋯ )}� + ⋯ ]|(���,�) (8.2) 

�����
(��)

=  �� ���
(��)

+ {�̂��� +  ���̂�� + ⋯ + ��(�̂�� +  ��̂�� + ���̂�� +  … )}��,��
(��)� 

  + �{�� +  ���� + ⋯ + ��(��� +  ���� + ⋯ )} 

  +
�

�
{{�̂��� + ⋯ +  ��(�̂�� +  ��̂�� + ���̂�� +  … )}�� 

  +  2�{�� +  ���� + ⋯ +  ��(��� +  ���� + ⋯ )} 

  × {�̂��� + ⋯ + ��(�̂�� + ��̂�� + ���̂�� +  … )}��,���

(��)
 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                               I.U. Udo-Akpan1  * 

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ON A-TWO-PARAMETER DYNAMIC BUCKLING OF A VISCOUSLY DAMPED BUT CLAMPED COLUMN STRESSED BY A STEP LOAD 
 

 

 

  ��+��{�� +  ���� + ⋯ +  ��(��� +  ���� + ⋯ )}� + ⋯ ]|(���,�)  (8.3) 

������
(��)

=  ��� ���
(��)

+ {�̂��� + ⋯ + ��(�̂�� + ⋯ )}��,��
(��)� 

  ��+�{�� + ⋯ + ��(��� + ���� + ⋯ )} + ⋯ ]|(���,�)    (8.4) 

�����
(��)

+ ���
(��)

� = �� ����
(��)

+  ���
(��)

� +  {�̂��� + ⋯ }���
(��)

+  ���
(��)

�
,��

� 

  ��+�{�� + ⋯ + ��(��� +  ���� + ⋯ )}���,�
(��)

+  ���,�
(��)

���
(���,�)

 (8.5) 

������
(��)

+ ���
(��)

� = ��� ����
(��)

+  ���
(��)

� + {�̂��� + ⋯ }���
(��)

+  ���
(��)

�
,��

� 

   ��+ �(�� + ⋯ )���,�
(��)

+  ���,�
(��)

���
(���,�)

   (8.6) 

�������
(��)

+ ���
(��)

� =  ���������,�
(��)

+  ���,�
(��)

���
(���,�)

    (8.7) 

On substituting (8.2) – (8.7) into (8.1), we observe that most terms vanish and the 

remaining ones are given as 

w� = 2� �����
(��)

+ �����,�
(��)

+
����

�

2
��,��

(��)
���

(���,�)

 

 +2�� ����
(��)

+ ���
(��)

� +  � ������,�
(��)

+ �̂����,��
(��)

+ �����,�
(��)

+ ���,�
(��)

��� 

 +��������,�
(��)

+ �̂���̂���
�,����
(��)

+ �������,��
(��)

+ �̂���
�,��
(��)

+ ���̂���
�,���

(��) � 

 ���+
��

�

�
���,�

(��)
+ ���,�

(��)
� + ���

(��)
+ ���

(��)
����

(���,�)
+ ⋯   (8.8) 

We however note that 

��
(��)(�̂�, 0) =  

�����

�� ,        (8.9a) 

  ���
(��)(�̂�, 0) =  �����       (8.9b) 

�� =  
�

�
�

�(����� ������)

��� +
�(����� ������)

�����
� ����

+
�(����� ������)

�����
� �����

+
(����� ������)

�����
� �����

�  (8.9c) 

 ��
(��)(�̂�, 0) =  

������

��� ,       (8.10a) 

  ���
(��)(�̂�, 0) =  ������       (8.10b) 

��� =  �
�(����� ������)

����
� �� −  

���(����� ������)

���
� ��� +  

���(����� ������)

���
� ���� −  

���(����� ������)

�����
� �����

� (8.10c) 

On substituting (8.9a) – (8.10c) into (8.8) and simplifying, we get 

w� = 4�� �1 −  
��

��
+ 

�

�
�

��

�
�

�
� +  

�������

��
��1 +  

����

��
� + ���� +  ������ + ⋯ (8.11a) 

where, 

��� =  
��

��
�

�

�
�

���

���� +  ���� − ���
(�)

−  ����      (8.11b) 

 ��� =  
��

��
��

�

�
� ���

(�)
− ���

(�)
− �

�

�
� ��� − ��� +

�

�
�

�

��� �
���

���� + ���� −
��

��� + ���� (8.11c) 

 

 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                               I.U. Udo-Akpan1  * 

https://ijojournals.com/                                              Volume 07 Issue 02 || February, 2024 || 

ON A-TWO-PARAMETER DYNAMIC BUCKLING OF A VISCOUSLY DAMPED BUT CLAMPED COLUMN STRESSED BY A STEP LOAD 
 

 

 

9. DYNAMIC BUCKLING LOAD, �� 

Having determined the maximum deflection, w� as in (8.11a – c), we shall now 

determine the dynamic buckling load �� from the maximization (7.1).  We shall first 

reverse the series (8.10a) which we now write as 

   w� =  ��� + ���� + ⋯      (9.1a) 

where, 

   �� = 4 �1 −  
��

��
+  

�

�
�

��

�
�

�

�      (9.1b) 

   �� =
�����

��
��1 +  

����

��
� +  ���� +  ������    (9.1c) 

Thus, for the reversal, we write 

  � =  ��w� +  ����
� + ⋯      (9.2) 

Upon substituting in (8.13) for w� from (8.12a), and equating the coefficients of �, we 

get 

   �� =  
�

��
,      �� =  − �

��

��
��      (9.3a) 

The maximization (7.1) is now easily executed from (9.2) to yield 

   �� +  3�����
� = 0,        (9.3b) 

where w�� is the maximum value of the deflection at dynamic buckling. Thus, we have 

   w�� =  �
���

���
        (9.4) 

On substituting from (9.3a) in (9.4) we get 

   w�� =  
�

√�
�

��
�

��
�       (9.5) 

If we next evaluate (9.2) at dynamic buckling, we get 

 � =  ��w�� +  �����
� + ⋯  =    w��( �� + 3�����

� )    (9.6) 

On substituting in (9.6) for ��, �� ��� w��, we get 

 � =  
�

�√�
�

��

��
�

�

�
        (9.7) 

On substituting in (9.7), we get the equation for determining the dynamic buckling load 

λ� as 

(16�� − 8��λ� + 1)
�

� = 18√10(����)��λ��
�

� �
���� 

����
��

�� ����� ������

�� 
��

��
� 

�

�
�

��

�
�

� �

�

�

 (9.9) 

Equation (9.9) gives an implicit formula for determining the dynamic buckling load λ�. 

By using equation (5.25), we can eliminate the imperfection parameter ϵ and hence 

relate λ� to λ�. This gives 

 �
������������

������������
�

�

�
=  √2 �

��

��
�

⎣
⎢
⎢
⎢
⎢
⎡ ���� 

����
��

�� ����� ������

�� 
��
��

� 
�
�

�
��
�

�
�

�� 
�

��
�

����� �������

���������������
�

⎦
⎥
⎥
⎥
⎥
⎤

�

�

   (9.10) 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                               I.U. Udo-Akpan1  * 

https://ijojournals.com/                                              Volume 07 Issue 02 || February, 2024 || 

ON A-TWO-PARAMETER DYNAMIC BUCKLING OF A VISCOUSLY DAMPED BUT CLAMPED COLUMN STRESSED BY A STEP LOAD 
 

 

 

The least value of the dynamic buckling load λ� is obtained when � = 1 and for this 

value, equations (9.9) and (9.10) respectively become 

  (17 − 8λ�)
�

� = 18√10(����)λ��
�

� �
���� 

����
��

�� ����� ������

�� 
��

��
� 

�

�
�

��

�
�

� �

�

�

 (9.11) 

and 

  �
������

������
�

�

�
=  √2 �

��

��
�

⎣
⎢
⎢
⎢
⎢
⎡ ���� 

����
��

�� ����� ������

�� 
��
��

� 
�
�

�
��
�

�
�

�� 
�

��
�

������
���������

�

⎦
⎥
⎥
⎥
⎥
⎤

�

�

   (9.12) 

where the right hand sides of (9.11) and (9.12) are to be evaluated at � = 1. 

 

10. ANALYSIS OF RESULTS 

The graphical plots of the results were done using Q-Basic codes and the results are 

hereby presented in Fig. 1, Fig. 2 and Fig. 3.  

 

 
Fig. 1: Relationship between the Static buckling load, �� and Imperfection factor, ϵ using Eqn. 

(5.26), Eqn.(5.28), Eqn.(5.30) and Eqn.(5.32). 

 

 

0

0.5

1

1.5

2

2.5

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.2

ST
A

TI
C

 B
U

C
K

LI
N

G
 L

O
A

D

IMPERFECTION FACTOR

STATIC BUCKLING LOAD (EQN 5.28)

STATIC BUCKLING LOAD (EQN 5.30)

STATIC BUCKLING LOAD (EQN 5.32)

STATIC BUCKLING LOAD (EQN. 5.26)

〖λ�〗^

ϵ

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                               I.U. Udo-Akpan1  * 

https://ijojournals.com/                                              Volume 07 Issue 02 || February, 2024 || 

ON A-TWO-PARAMETER DYNAMIC BUCKLING OF A VISCOUSLY DAMPED BUT CLAMPED COLUMN STRESSED BY A STEP LOAD 
 

 

 

 
Fig. 2: Relationship between the Dynamic buckling loads and Imperfection parameters at some 

fixed values of the damping factor, δ, using Eqn. (9.11). 

 
Fig. 3: Relationship between the Dynamic buckling loads and the Static buckling loads at some 

fixed values of the damping factor, δ, using Eqn. (9.12). 

 

From Fig. 1, we observe that the static buckling load of a clamped column is always 

higher than that of the same column with simply–supported boundary conditions. In 

general, the static buckling load of a column with either clamped or simply-supported 

boundary conditions and whose deflections are strictly in the shape of imperfection 

always has the least static buckling load. However, while the static buckling load of a 

clamped column satisfies the inequality, 1 < �� < 2.125, a similar column with simply–

supported boundary supports satisfies the inequality 0 < �� < 1.  

Fig. 2 shows that the dynamic buckling load �� decreases with increased imperfection 

for any value of the damping parameter. 

From Fig. 3, we observe that at low values of the static buckling load �� (precisely for 

1 < �� < 1.45), there is no significant difference in the value of the dynamic buckling 

load �� of the column, whether damped or undamped. However, at higher values of �� 

(i.e,  1.45 < �� < 2.125), the dynamic buckling load  �� rises with �� and the highest of 

0

0.2

0.4

0.6

0.8

1

1.2

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.2

D
Y

N
A

M
IC

 B
U

C
K

LI
N

G
 L

O
A

D

IMPERFECTION FACTOR

DYNAMIC BUCKLING LOAD, δ = 0
DYNAMIC BUCKLING LOAD, δ = 0.01
DYNAMIC BUCKLING LOAD, δ = 0.02
DYNAMIC BUCKLING LOAD, δ = 0.03

λD

ϵ

0

0.5

1

1.5

2

2.5

1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1D
Y

N
A

M
IC

 B
U

C
K

LI
N

G
 L

O
A

D

STATIC BUCKLING LOAD

DYNAMIC BUCKLING LOAD , δ = 0

DYNAMIC BUCKLING LOAD , δ = 0.01

DYNAMIC BUCKLING LOAD , δ = 0.02

DYNAMIC BUCKLING LOAD , δ = 0.03

λD

λS

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                               I.U. Udo-Akpan1  * 

https://ijojournals.com/                                              Volume 07 Issue 02 || February, 2024 || 

ON A-TWO-PARAMETER DYNAMIC BUCKLING OF A VISCOUSLY DAMPED BUT CLAMPED COLUMN STRESSED BY A STEP LOAD 
 

 

 

such rise is the undamped case. It is not clear whether such a result is specific to 

clamped boundary conditions or whether it is general. As in the static loading case, the 

inequality satisfied by the clamped column is 1 < �� < 2.125. We thus observe that 

generally, whether in the static or dynamic loading cases, clamped columns buckle at 

much more higher loads than similar columns with simply–supported boundary 

supports. For reasons attributed to nonlinearity and imperfection, clamped columns on 

nonlinear elastic foundations buckle at lower values of buckling loads than the 

corresponding classical buckling load of the column. 

We observe that while the buckling modes split into three distinct modes proportional 

to (1 − cos 2��), (1 − cos 4��) and (1 − cos 6��), it is only the buckling modes in the 

shapes of (1 − cos 2��) and (1 − cos 6��) that eventually contribute to dynamic 

buckling. The mode in the shape of (1 − cos 4��) does not contribute. 

Lastly, as seen in equations (8.20) and (8.22), we are able to directly relate �� to �� 

even without the knowledge of the size of the associated imperfection. In this way, we 

have circumvented the process of repeating the arduous manipulations for different 

imperfection parameters. 

 

11. CONCLUSION 

We have carried out an analytical investigation of the buckling of an imperfect clamped 

column lying on a nonlinear elastic foundation but struck axially by a step load. It is our 

contention that similar enquiries can be extended to other loading conditions apart 

from step load and to other structures apart from columns.  

 

 

ACKNOWLEDGEMENTS:  

Authors acknowledge the valuable contribution their mentor, Prof. A.M. Ette (FNMS), 

for his guardian and support on this work. 

 

REFERENCES 

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(ISSN: 2992-4421 )                                                                                                               I.U. Udo-Akpan1  * 

https://ijojournals.com/                                              Volume 07 Issue 02 || February, 2024 || 

ON A-TWO-PARAMETER DYNAMIC BUCKLING OF A VISCOUSLY DAMPED BUT CLAMPED COLUMN STRESSED BY A STEP LOAD 
 

 

 

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(ISSN: 2992-4421 )                                                                                                               I.U. Udo-Akpan1  * 

https://ijojournals.com/                                              Volume 07 Issue 02 || February, 2024 || 

ON A-TWO-PARAMETER DYNAMIC BUCKLING OF A VISCOUSLY DAMPED BUT CLAMPED COLUMN STRESSED BY A STEP LOAD 
 

 

 

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