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Buckling of Column With Two Intermediate Elastic Restraints
Thesis Presentation 15.11.2007
Author: Md. Rayhan ChowdhuryMohammad Misbah UddinMd. Abu Zaed KhanMd. Monirul Islam Masud
Supervisor: Dr. Mohammad Nazmul Islam
PRESIDENCY UNIVERSITY
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Department of Civil Engineering Undergraduate Thesis Presentation 2007-11-152 PRESIDENCY UNIVERSITY
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
Fig. Column with two intermediate elastic restraints
P
z, w
x
c1
c2L
a2L
a1L
Spring 1
Spring 2
Here,
Column Length, L Flexural Rigidity, EI Spring 1
– Stiffness, c1
– Located at a1L (a1 < 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
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Governing equation for column buckling
4 22
4 20i id w d w
dx dx
Where i = 1, 2, 3 denote the quantity belonging to segment 1, segment 2 and segment 3.
and/ ,x x L / ,w w L 2 2 /( )PL EI
1 1 2 3 4sin cosw A x A x A x A 10 x a
2 1 2 3 4sin cosw B x B x B x B 1 2a x a
3 1 2 3 4sin cosw C x C x C x C 2 1a x
(1)for
(2)for
(3)for
General Solution:
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Department of Civil Engineering Undergraduate Thesis Presentation 2007-11-155 PRESIDENCY UNIVERSITY
Continuity condition at a1
31 1 /( )c L EI
(4)
(5)
Where,
(6)
(For deflection, slope, bending moment and shear force )
11
1 2 0x a x aw w
11
1 2 0x ax a
dw dw
dx dx
1
1
2 21 2
2 20x a
x a
d w d w
dx dx
1
1 1
3 22 21 1 2 2
1 13 30x a
x a x a
d w dw d w dww
dx dxdx dx
(7)
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Department of Civil Engineering Undergraduate Thesis Presentation 2007-11-156 PRESIDENCY UNIVERSITY
Continuity condition at a2
32 2 /( )c L EI
(8)
(9)
Where,
(10)
(For deflection, slope, bending moment and shear force )
(11)
22
2 3 0x a x aw w
22
32 0x ax a
dwdw
dx dx
2
2
2232
2 20x a
x a
d wd w
dx dx
2
2 2
232 23 32 2
2 23 30x a
x a x a
d w dwd w dww
dx dxdx dx
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Continuation…
(12)
(13)
(14)
(15)
11 1 1 1 2 1 3 1 4 13
cos( )[ sin( ) cos( ) ]B a A a A a A a A A
12 1 1 1 2 1 3 1 4 23
sin( )[ sin( ) cos( ) ]B a A a A a A a A A
21
3 1 1 2 1 3 1 422
[ sin( ) cos( ) ( ) ]B A a A a A a A
21 1
4 1 1 2 1 3 1 421 1
[ sin( ) cos( ) (1 )a
B A a A a A a Aa
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.,
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Continuation…
(16)
(17)
(18)
(19)
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.,
21 2 1 2 2 2 3 2 4 13
cos( )[ sin( ) cos( ) ]C a B a B a B a B B
22 2 1 2 2 2 3 2 4 23
sin( )[ sin( ) cos( ) ]C a B a B a B a B B
22
3 1 2 2 2 3 2 422
[ sin( ) cos( ) ( ) ]C B a B a B a B
22 2
4 1 2 2 2 3 2 422 2
[ sin( ) cos( ) (1 )a
C B a B a B a Ba
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Continuation…
(20)
(21)
(22)
(23)
Hence,
Substituting Eqs. (12) - (15) into Eqs. (16)-(19), we get Ci in terms of Ai
1 1 2 3 4( , , , )C A A A A
1 1 2 3 4( , , , )C A A A A
1 1 2 3 4( , , , )C A A A A
1 1 2 3 4( , , , )C A A A A
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Boundary Condition (fixed – free)
(24)
(25)
(26)
(27)
At the fixed end:
Fig. Boundary condition, fixed - free
P
z, w
x
c1
c2L
a2L
a1L
Spring 1
Spring 2
1 00
xw
1
0
0x
dw
dx
23
21
0x
d w
dx
223 3
31
0x
d w dw
dxdx
At the free end:
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B. C. (fixed – free) continuation
(24)
(25)
(26)
(27)
Differentiating the boundary equation:
2 4 0A A
1 3 0A A
21 2( sin( ) cos( )) 0C C
23 0C
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).
[ ]{ } {0}M A
a.
Hence, we can develop the Eigen value equation from the boundary equation in form
b.
Where {A} = (A1, A2, A3, A4) and [M] is the coefficient matrix of {A}.
Finally the determinant of matrix[M] yields the stability criteria.
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B. C. (fixed – free) continuation
ξξ11=20, ξ2=20: (ξ=20, ξ2=20: (ξ11=c=c11LL33 and ξ and ξ22=c=c22LL33) )
λλ22= PL= PL22/(EI)/(EI)
a2a2 α = 0α = 0 α = 0.2α = 0.2 α = 0.4α = 0.4 α = 0.6α = 0.6 α = 0.8α = 0.8 α = 1α = 1
0.10.1 4.71014.7101 4.71014.7101 4.714.71 4.70984.7098 4.70914.7091 4.70784.7078
0.30.3 4.56674.5667 4.56634.5663 4.56094.5609 4.53954.5395 4.48884.4888 4.40044.4004
0.50.5 4.14764.1476 4.14464.1446 4.1074.107 3.98553.9855 3.7943.794 3.6423.642
0.70.7 4.10444.1044 4.09624.0962 4.00594.0059 3.8073.807 3.69133.6913 3.78163.7816
0.90.9 4.46024.4602 4.43964.4396 4.24324.2432 3.99093.9909 4.07484.0748 4.35744.3574
The following tables present the buckling load parameter for different locations and stiffness of the intermediate restraints:
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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