Buckling of Column With Two Intermediate Elastic Restraints

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Buckling of Column With Two Intermediate Elastic
Restraints

• Thesis Presentation 15.11.2007

Author Md. Rayhan Chowdhury Mohammad Misbah
Uddin Md. Abu Zaed Khan Md. Monirul Islam
Masud
Supervisor Dr. Mohammad Nazmul Islam
Presidency University
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Introduction

• Background
• In a laterally loaded cross-bracing system, one
  bracing member will be under compression while
  the other member subjected to tension. The
  tension brace may be modeled as a discrete,
  lateral elastic spring attached to the
  compression member. Thus, the prediction of the
  elastic buckling loads of columns with two
  intermediate elastic restraints is therefore of
  practical interest.
• Objectives
• The main objective of this theoretical research
  is to find a set of stability criteria for Euler
  columns with two intermediate elastic restraints.
  
• Scope
• The scope of this thesis is the derivation of
  Euler column buckling theory.

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Model

• Here,
• Column Length, L
• Flexural Rigidity, EI
• Spring 1
• Stiffness, c1
• Located at a1L (a1 lt L)
• Spring 2
• Stiffness, c2
• Located at a2L (a1 a2 L)
• The entire column can be divided into three
  segments as
• Segment-1 0 x a1L
• Segment-1 a1L x a2L
• Segment-1 a2L x L

• Fig. Column with two intermediate elastic
  restraints

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Governing equation for column buckling

• Where i 1, 2, 3 denote the quantity belonging
  to segment 1, segment 2 and segment 3.
• and

General Solution
(1)
for
(2)
for
(3)
for
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Continuity condition at a1
(For deflection, slope, bending moment and shear
force )
(4)
(5)
(6)
(7)
Where,
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Continuity condition at a2
(For deflection, slope, bending moment and shear
force )
(8)
(9)
(10)
(11)
Where,
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Continuation
Substituting Eqs. (1) and (2) into Eqs. (4)-(7),
a set of homogeneous equations is obtained which
may be expressed in forms of Bi in terms of Ai,
i.e.,
(12)
(13)
(14)
(15)
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Continuation
Substituting Eqs. (2) and (3) into Eqs. (8)-(11),
another set of homogeneous equations is obtained
which may be expressed in forms of Ci in terms of
Bi, i.e.,
(16)
(17)
(18)
(19)
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Continuation
Hence, Substituting Eqs. (12) - (15) into Eqs.
(16)-(19), we get Ci in terms of Ai
(20)
(21)
(22)
(23)
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Boundary Condition (fixed free)
At the fixed end
(24)
(25)
At the free end
(26)
(27)
Fig. Boundary condition, fixed - free
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B. C. (fixed free) continuation
a.
b.
Differentiating the boundary equation
Hence, we can develop the Eigen value equation
from the boundary equation in form
(24)
(25)
Where A (A1, A2, A3, A4) and M is the
coefficient matrix of A.
(26)
(27)
Finally the determinant of matrixM yields the
stability criteria.
If we substitute the value of Ci into Eqs. (26)
and (27), the buckling problem involves only four
constants Ai (i 1,2,3,4).
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B. C. (fixed free) continuation
The following tables present the buckling load
parameter for different locations and stiffness
of the intermediate restraints
?120, ?220 (?1c1L3 and ?2c2L3) ?120, ?220 (?1c1L3 and ?2c2L3) ?120, ?220 (?1c1L3 and ?2c2L3) ?120, ?220 (?1c1L3 and ?2c2L3) ?120, ?220 (?1c1L3 and ?2c2L3) ?120, ?220 (?1c1L3 and ?2c2L3) ?120, ?220 (?1c1L3 and ?2c2L3)
  ?2 PL2/(EI) ?2 PL2/(EI) ?2 PL2/(EI) ?2 PL2/(EI) ?2 PL2/(EI) ?2 PL2/(EI)
a2 a 0 a 0.2 a 0.4 a 0.6 a 0.8 a 1
0.1 4.7101 4.7101 4.71 4.7098 4.7091 4.7078
0.3 4.5667 4.5663 4.5609 4.5395 4.4888 4.4004
0.5 4.1476 4.1446 4.107 3.9855 3.794 3.642
0.7 4.1044 4.0962 4.0059 3.807 3.6913 3.7816
0.9 4.4602 4.4396 4.2432 3.9909 4.0748 4.3574
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Conclusions

• Exact stability criteria for columns with two
  intermediate elastic restraints at arbitrary
  location along the column length are derived.
• This stability criteria can be used to determine
  the buckling capacity of compressive member in a
  cross-bracing system.

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Thank you all.
Presidency University