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Stability analysis of structural frames subjected to circulatory loading by displacement method
Jounml of Sound and Vibration (1992) 153(2), 359-367
9 S T A B I L I T Y A N A L Y S I S O F S T R U C T U R A L F R A M E S SUBJECTED T O C I R C U L A T O R Y L O A D I N G BY D I S P L A C E M E N T M E T t t O D G. M. A R w
Department of Fire Safeo' Enghwerhzg, Lund bzstitute of Technology, Box 118, S-221 00, Lund, Sweden (Received 24 May 1991)
I. INTRODUCTION
Problems involving non-conservative loading are frequently encountered in aeromechanics, and there is a considerable amount of literature Covering the effects of various parameters upon the critical loading of flutter type systems; see, for example, a recent review by Bogacz and Janiszewski [1]. The instability of structural frames loaded by circulatory forces has been investigated by Kounadis et al. [2, 3] using both dynamic and static equilibrium methods. In this letter the method of displacement adopted by Kornoukhov [4] and Chudnovskii [5] in the stability analysis of frames under conservative loading is extended to be applied for a stability investigation of frames subjected to follower forces. 2. STATEMENT OF PROBLEM
The static part of the differential equation of motion of the beam illustrated in Figure 1. is
d ' y / d ~ " + v 2 d 2),/d~ 2 - t,'t, = 0,
(1)
v: = N I 2 / E I ,
(2)
with u4 = mco214/EL
tlere y denotes, the transverse deflection of the beam, ~ = x / l with x the longitudinal coordinate, E1 is the flexural rigidity of the beam, i is the beam length; N is the applied axial load, m is the mass density per unit length and co is the frequency of vibration. The boundary conditions of the beam are
y(O)=-,L, Nr
A
y(I)=6h,
y'(O) = ~'a,
m
Elo~
y'(l)=eb.
B
(3)
No, *4
;o.o
1,.
|
X
)7 bo
iObo
Figure I. Member end displacements and forces of prismatic beam.
359 0022-460X/92/050359 + 09 $03.00/0
~.') 1992 Academic Press Limited
360
LE'I-I'ERS TO THE EDITOR
The general solution of equation (!) has the form y ( ~ ) =At cos Z,~+A2 sin ~.l~ +A3 cosh Z2~+A4 sinh Z2~,
(4)
where Z2.t = ( - l ) " ( v 2 / 2 ) + ~ / ( v ' / 4 ) + u
n= 1, 2.
4,
(5)
By using the method of initial parameters, Chudnovskii derived the following expressions for the bending moments and shear forces at the ends A and B of the bar shown in Figure 1: 31~b = i.b[ ao~9. + flohgb
-
~'.o( 3./l.t,)
~.~( Sdlat,)].
-
(6)
Q.t, = - ( i~t,/l~b) [ ?'~t,cp.,+ ri.bgb -- p.b( 6 J l.o) -- f~t,( Sb/Lb) ].
For a hinge at the end B, the expressions become: 9
o
eot,=
9
(7)
/at,
Here aab = v a , / b , tan v,
fl~b= v(v--sin v ) / b l sin v,
i~h = El~b/iab,
7~h = rl.b = a.b + flab = V 2 tan ( v / 2 ) / b , , /tab =
v3
~a~,--
o _ ~tah o = rlab o Gab --
2 tan ( v / 2 ) - v' 0 0 r3 p~b=~'ab=~ /a,,
aj=tan
--
v2
tan v
-
-
-
-
,
al
bl=2tan(v/2)-v.
v - v ,
(8)
For Mba and Qb,,, the indices a and b are interchanged in equations (6) and (7). 3.
APPLICATIONS A N D N U M E R I C A L RESULTS
The equations derived in the previous section have been used to find the critical loading and frequency for some types of one- and two-storey frames, subjected to compressive follower forces; (F) denotes flutter and (D) divergence instability, respectively. Case 1 : one-storey frame fixed at the base and having two equal concentrated masses at the joints 2 and 3 (see Figure 2). The frame is subjected to follower forcesp at the joints (o)
p
~'//f.
(b)
~
. ]p
"///P
[o
I '~
bl
OPzo
N21
:L PP2,
rnw 2 S
~9 M$4 N54
Figure 2. One-storey frame.
361
LE"I"TERS TO TIlE EDITOR
35
I I I
30
'0.75
25 20
15 / 10
5 0-0
1
I
I
)
I
0"2
0"4
06 k
0-8
1-0
Figure 3. Critical buckling load v~2 versus k for different values of k- for the frame shown in Figure 2; n=2.
2 and 3 with a direction of action determined by the angle cpp=kq~+k62j/h.The equations of equilibrium of moments and shearing forces, with account taken of the antisymmetry of the frame, are
Q21..ba34-2mco2621-2pcpp=O.
~,t"21 q- M34 = 0,
(9, 10)
Substituting the expressions for M2,, M34, and Q2, and Q3~, and rearranging, one obtains a pair of simultaneous homogenous algebraic equations. For a non-trivial solution of
(a)
~m
"z
J12 J23 ti4. L I; (b) . PP2p
m~,,2821
3!
~ M
9 Q34
021 I
M34
NZl
Figure 4. One-siorey frame with one concentrated load.
LETTERS TO TIlE EDITOR
362
35 ~=0-75
30
0'5
,=# 25 20
15 '~
I
0'0
0-2
I
I
0-4
0-6
k
1
!
0-8
1-0
Figure 5. Critical buckling load v~2 versus k for different values of/~ for the frame shown in Figure 4;
P:,
n=2
~z
ikm2
rnz
Iv3 i34 4t ! i23 i4 P! 2
P!
d hml
rn 1
I il2
I
i5
hi
I
I.
h3
l, I
Figure 6. Two-storey frame.
o.~ 0-0 I
0-0
I
I
I
i
I
0.2
0-4
0'6 k
0-8
1"0
Figure 7. Critical buckling load v~z versus k for various values of/~ for the frame in Case 4;
n=
2-0.
363
LETTERS TO TIlE EDITOR
and 812/h the determinant of the coefficients of ~P2and 8 u / h must vanish: i.e., 6n+a2,
-Y
I
P2, + v~2~:-u 4 =0,
- ~ ' + v~2k
(i 1)
where y = a21 +~21 and n = i2/il. The critical buckling load for n = 2 , k = 1.0 and ks=0.0 is 1 !.775. The relation between the critical loading and the direction parameters are shown in Figure 3. The critical buckling load under conservative loading (k--ks=0.0) is 8-434. Case 2: one-storey frame fixed at the base and having a concentrated mass only at the joint 2, as shown in Figure 4. The frequency equation for the same action of follower forces as in Case 1 is given in the following determinant form: a2t + 4 n
2t!
-Y2,
2n
4n+4
-6
l-y2, + v~2k
-6
P2, + 12+ v~zks-u 4
A =l
=0.
(12)
The influence of the parameters k and kson the divergence buckling load is shown in Figure 5. Case 3: a two-storey frame fixed at the base, having two concentrated masses m, and m2 and subjected to follower forces p~ and p2 at the joints, as shown in Figure 6. The direction of actions of the. forces are ~ p = kt 92 + ks, ~52,/hl and ~03e= k:~p3+ ks2($32+ $ 2 1 ) / h 2 . The critical buckling load for the frame with m, =m2, h~ =h2 = 2h3 and k, =kl = k 2 = 0 and k2 = 1-0 is 8.611, with a flutter frequency of 1.328. The critical buckling load for the same frame with conservative loading is 6.75. In Tables l(a) and l(b) the critical buckling loads and frequencies are given for various values of the parameters k,,/7.~, k2 and ks2. Case 4: the frame shown in Figure 2 is re-analyzed for stability with a hinged base. The frequency determinant for this case is
d=
a~ + 6n
--a2j
- a 2 , + v22k p o + v~2ks_u.S =0.
(13)
For the effect of the directional parameters (k, kS) of the forces on the critical buckling, see Figure 7. Case 5: The two-storey frame of Case 3 is analyzed with a hinged base. The dependency of the critical loading on various parameters is presented in Table 2. Case 6: the one-storey frame of Figure 4 is studied with a hinged base. The results are shown in Figure 8 for two values of n. Case 7 : the partial frame shown in Figure 9 is subjected to a compressive follower force p and carries a concentrated mass m at its joint. The distributed masses of the bars mj, and m2 and the moment of inertia of the concentrated mass i.~have bee,i taken into account. The following determinant has to be solved for critical buckling load and frequency of the frame: A = /5/21+Zla23-1112(Z2+Z3Z4)
I
o
4
-~2,+v~Z2k
-Y2,
2 ,7- - r o- u t42 v12k
--P2|
[
. =0. - k,l,I
(14)
l lere zl = i2.~/i2t, t~l = m2h/m,I, z2 = i.a/m,h 3, z3 = m / m l h , z4 = (a/h) 2 and k,e is a non-dimensional spring constant (in case of lateral bracing at joint 3). The critical loading and frequencies for various parameter values, are given in Table 3. The critical divergence loads v22 against the parameter k are shown in Figure 10 for two values ofti'i (ff~= I, 2). The influence of the mass m and its moment of inertia is a stabilizing one; at the value of i~1= 1, the critical loading v22 is 23.92 and the frequency u~2 is 15-28 with z3 = 1.0, a = 0.047 and/3=0"002119.
364
LE'I'TERS T O T H E
EDITOR
TABLE 1
Critical buckling loads andfiequencies for various values of k2 (kl=k2=kl=O, m l = m 2 = 1, h i = h2--2h3); (b) divergence buckling loads for various values of k2 and at/~2=0.25 (El = k t = 0 , h i = h 2 = (a)
2h3)
(a) k2
v2,3
u"
0"0 0"25 0-35 0-4 0.475 0.5 0-55 0.6 0.8 1-0
6"765 7.347 7.542 8"345 8.671 8.777 8.735 8-703 8.628 8.611
0"0 0"0 00 00 0.0 0.389 0-526 0.647 1.037 1.328
(D) (D) (D) (D) (D) (F) (F) (F) (F) (F)
(b) k~ 0.0 0.1 0.2 0-3 0.4 0-5 0-6 0.7 0.8 0-9 I-0
/~
v~
0-25 0.25 0.25 0.25 0.25 0-25 0.25 0.25 0.25 0-25 0-25
8.996 9.081 9.155 9.218 9.273 9.321 9.36 9.398 9.43 9.458 9.483
TABLE 2
Critical bucklhzg loads I.,z 23 for various values of k2 (kl =/'S'z=/~l = 0"0 k~ v~3 0"0 0. I 0"2 0-3 0.4 0"5 0-6 0-7 0-8 0-9 1.0
7"23 7.44 7"67 7-92 8-175 8-453 8-75 9-1 9-42 9-80 9"87
45r
LETTERS TO T H E EDITOR
365
4.3
4-1 3'9
:3'7 3-5
0"0
I
I
0"2
0'4
t
0"6 k
,
0-8
i
1'0
Figure 8. Critical buckling load v~2 versus parameter k for the frame in Case 6 at two different values o f n . p
1
..L
7 77.. Figure 9. Partial frame. 17 15
13
0.0
~ = 1 - 0 / /
I
I
1
I
0.2
0.4
0.6 k
0-8
I
1.0
Figure I0. Critical load ~'~ versus the parz;meler k for the partial frame shown in Figure 9 for 61= 1.0 and 2.0.
Case 8: the partial frame shown in Figure 10 is re-examined with the column hinged at the base. The divergence buckling loads are given in Table 4. The values in brackets are divergence buckling loads from the dynamic analysis. The value v~2=2-467 for k = l coincides with the value found in reference [6] by a non-linear stability analysis.
366
LETTERS TO T I l E E D I T O R TABLE 3
Critical buckling load vl2 and frequencies ul2for various values o f k ( h = l = 1.0, m = 0 . 0 , k,p=0-0) k
Vt2
ut2
0'5 0'6 0-7 0.8 I-0
4-975 4.942 4.914 4.893 4.862
1"663 1.8165 1.925 2.01 2-135
(F) (F) (F) (F) (F)
TABLE 4 Critical buckling loads v22 for various vahtes of k with IJ1= 1-0 and k,p = 0 . 0 k
vI22
0.0 0. I 0-2 0-3 0-4 0.5 0.6 0-7 0.8 0-9 !-0
1.821 1.873 i-92 1-982 2.041 2.104 2.169 2.238 2-387 2.387(I-591) 2.467(!.571)
4. CONCLUSIONS The displacement method has been used to investigate the flutter and divergence instability of a number of plane frames subjected to compressive follower forces. The method is very effective for complex systems without inclined members; it minimizes convergence problems and has a wide range of applications. ACKNOWLEI)GMENT
This study has been financed by the Department of Fire Safety Engineering, Lund University. REFER ENCES
1. R. BOGACZand R. JANISZEWSKI 1986 lnstytut Podstawowych Problemow Techniki Polskiej,4kademii Nauk, 1-92. Zagadnienie analizy i syntezy kolumn obciazonych silami sledzacymi ze wzgledu na statecznosc (in Polish). 2. A. N. KOUNADtS and T. N. AX~RAM 1981 American Institute of,4eronautics and Astronautics Journal 19, 761-765. Linear and nonlinear analysis of a nonconservative frame of divergence instability. 3. A. N. KOUNADIS 1983dcta Mechaniea 47, 247-262. Interaction of the joint and of the lateral bracing stiffness for the optimum design of unbraced frames. 4. N. V. KORNOUKIIOV 1949 Strength and Stability of Beam and Frame Systems. Moscow: Stroiizdat (in Russian).
LETTERS TO TIlE EDITOR
367
5. V. G. CHUDNOVSKH 1952 Methods of Calculath~g Vibrations and Stability of Beam and Frame Systems. Kiev: A N Ukr SSR (in Russian). 6. A . N . KOONADtS 1979 Journalde M~canique Applique~3, 173-185. The effects ofsome parameters on the nonlinear divergence buckling of a nonconservative simple frame.
9 S T A B I L I T Y A N A L Y S I S O F S T R U C T U R A L F R A M E S SUBJECTED T O C I R C U L A T O R Y L O A D I N G BY D I S P L A C E M E N T M E T t t O D G. M. A R w
Department of Fire Safeo' Enghwerhzg, Lund bzstitute of Technology, Box 118, S-221 00, Lund, Sweden (Received 24 May 1991)
I. INTRODUCTION
Problems involving non-conservative loading are frequently encountered in aeromechanics, and there is a considerable amount of literature Covering the effects of various parameters upon the critical loading of flutter type systems; see, for example, a recent review by Bogacz and Janiszewski [1]. The instability of structural frames loaded by circulatory forces has been investigated by Kounadis et al. [2, 3] using both dynamic and static equilibrium methods. In this letter the method of displacement adopted by Kornoukhov [4] and Chudnovskii [5] in the stability analysis of frames under conservative loading is extended to be applied for a stability investigation of frames subjected to follower forces. 2. STATEMENT OF PROBLEM
The static part of the differential equation of motion of the beam illustrated in Figure 1. is
d ' y / d ~ " + v 2 d 2),/d~ 2 - t,'t, = 0,
(1)
v: = N I 2 / E I ,
(2)
with u4 = mco214/EL
tlere y denotes, the transverse deflection of the beam, ~ = x / l with x the longitudinal coordinate, E1 is the flexural rigidity of the beam, i is the beam length; N is the applied axial load, m is the mass density per unit length and co is the frequency of vibration. The boundary conditions of the beam are
y(O)=-,L, Nr
A
y(I)=6h,
y'(O) = ~'a,
m
Elo~
y'(l)=eb.
B
(3)
No, *4
;o.o
1,.
|
X
)7 bo
iObo
Figure I. Member end displacements and forces of prismatic beam.
359 0022-460X/92/050359 + 09 $03.00/0
~.') 1992 Academic Press Limited
360
LE'I-I'ERS TO THE EDITOR
The general solution of equation (!) has the form y ( ~ ) =At cos Z,~+A2 sin ~.l~ +A3 cosh Z2~+A4 sinh Z2~,
(4)
where Z2.t = ( - l ) " ( v 2 / 2 ) + ~ / ( v ' / 4 ) + u
n= 1, 2.
4,
(5)
By using the method of initial parameters, Chudnovskii derived the following expressions for the bending moments and shear forces at the ends A and B of the bar shown in Figure 1: 31~b = i.b[ ao~9. + flohgb
-
~'.o( 3./l.t,)
~.~( Sdlat,)].
-
(6)
Q.t, = - ( i~t,/l~b) [ ?'~t,cp.,+ ri.bgb -- p.b( 6 J l.o) -- f~t,( Sb/Lb) ].
For a hinge at the end B, the expressions become: 9
o
eot,=
9
(7)
/at,
Here aab = v a , / b , tan v,
fl~b= v(v--sin v ) / b l sin v,
i~h = El~b/iab,
7~h = rl.b = a.b + flab = V 2 tan ( v / 2 ) / b , , /tab =
v3
~a~,--
o _ ~tah o = rlab o Gab --
2 tan ( v / 2 ) - v' 0 0 r3 p~b=~'ab=~ /a,,
aj=tan
--
v2
tan v
-
-
-
-
,
al
bl=2tan(v/2)-v.
v - v ,
(8)
For Mba and Qb,,, the indices a and b are interchanged in equations (6) and (7). 3.
APPLICATIONS A N D N U M E R I C A L RESULTS
The equations derived in the previous section have been used to find the critical loading and frequency for some types of one- and two-storey frames, subjected to compressive follower forces; (F) denotes flutter and (D) divergence instability, respectively. Case 1 : one-storey frame fixed at the base and having two equal concentrated masses at the joints 2 and 3 (see Figure 2). The frame is subjected to follower forcesp at the joints (o)
p
~'//f.
(b)
~
. ]p
"///P
[o
I '~
bl
OPzo
N21
:L PP2,
rnw 2 S
~9 M$4 N54
Figure 2. One-storey frame.
361
LE"I"TERS TO TIlE EDITOR
35
I I I
30
'0.75
25 20
15 / 10
5 0-0
1
I
I
)
I
0"2
0"4
06 k
0-8
1-0
Figure 3. Critical buckling load v~2 versus k for different values of k- for the frame shown in Figure 2; n=2.
2 and 3 with a direction of action determined by the angle cpp=kq~+k62j/h.The equations of equilibrium of moments and shearing forces, with account taken of the antisymmetry of the frame, are
Q21..ba34-2mco2621-2pcpp=O.
~,t"21 q- M34 = 0,
(9, 10)
Substituting the expressions for M2,, M34, and Q2, and Q3~, and rearranging, one obtains a pair of simultaneous homogenous algebraic equations. For a non-trivial solution of
(a)
~m
"z
J12 J23 ti4. L I; (b) . PP2p
m~,,2821
3!
~ M
9 Q34
021 I
M34
NZl
Figure 4. One-siorey frame with one concentrated load.
LETTERS TO TIlE EDITOR
362
35 ~=0-75
30
0'5
,=# 25 20
15 '~
I
0'0
0-2
I
I
0-4
0-6
k
1
!
0-8
1-0
Figure 5. Critical buckling load v~2 versus k for different values of/~ for the frame shown in Figure 4;
P:,
n=2
~z
ikm2
rnz
Iv3 i34 4t ! i23 i4 P! 2
P!
d hml
rn 1
I il2
I
i5
hi
I
I.
h3
l, I
Figure 6. Two-storey frame.
o.~ 0-0 I
0-0
I
I
I
i
I
0.2
0-4
0'6 k
0-8
1"0
Figure 7. Critical buckling load v~z versus k for various values of/~ for the frame in Case 4;
n=
2-0.
363
LETTERS TO TIlE EDITOR
and 812/h the determinant of the coefficients of ~P2and 8 u / h must vanish: i.e., 6n+a2,
-Y
I
P2, + v~2~:-u 4 =0,
- ~ ' + v~2k
(i 1)
where y = a21 +~21 and n = i2/il. The critical buckling load for n = 2 , k = 1.0 and ks=0.0 is 1 !.775. The relation between the critical loading and the direction parameters are shown in Figure 3. The critical buckling load under conservative loading (k--ks=0.0) is 8-434. Case 2: one-storey frame fixed at the base and having a concentrated mass only at the joint 2, as shown in Figure 4. The frequency equation for the same action of follower forces as in Case 1 is given in the following determinant form: a2t + 4 n
2t!
-Y2,
2n
4n+4
-6
l-y2, + v~2k
-6
P2, + 12+ v~zks-u 4
A =l
=0.
(12)
The influence of the parameters k and kson the divergence buckling load is shown in Figure 5. Case 3: a two-storey frame fixed at the base, having two concentrated masses m, and m2 and subjected to follower forces p~ and p2 at the joints, as shown in Figure 6. The direction of actions of the. forces are ~ p = kt 92 + ks, ~52,/hl and ~03e= k:~p3+ ks2($32+ $ 2 1 ) / h 2 . The critical buckling load for the frame with m, =m2, h~ =h2 = 2h3 and k, =kl = k 2 = 0 and k2 = 1-0 is 8.611, with a flutter frequency of 1.328. The critical buckling load for the same frame with conservative loading is 6.75. In Tables l(a) and l(b) the critical buckling loads and frequencies are given for various values of the parameters k,,/7.~, k2 and ks2. Case 4: the frame shown in Figure 2 is re-analyzed for stability with a hinged base. The frequency determinant for this case is
d=
a~ + 6n
--a2j
- a 2 , + v22k p o + v~2ks_u.S =0.
(13)
For the effect of the directional parameters (k, kS) of the forces on the critical buckling, see Figure 7. Case 5: The two-storey frame of Case 3 is analyzed with a hinged base. The dependency of the critical loading on various parameters is presented in Table 2. Case 6: the one-storey frame of Figure 4 is studied with a hinged base. The results are shown in Figure 8 for two values of n. Case 7 : the partial frame shown in Figure 9 is subjected to a compressive follower force p and carries a concentrated mass m at its joint. The distributed masses of the bars mj, and m2 and the moment of inertia of the concentrated mass i.~have bee,i taken into account. The following determinant has to be solved for critical buckling load and frequency of the frame: A = /5/21+Zla23-1112(Z2+Z3Z4)
I
o
4
-~2,+v~Z2k
-Y2,
2 ,7- - r o- u t42 v12k
--P2|
[
. =0. - k,l,I
(14)
l lere zl = i2.~/i2t, t~l = m2h/m,I, z2 = i.a/m,h 3, z3 = m / m l h , z4 = (a/h) 2 and k,e is a non-dimensional spring constant (in case of lateral bracing at joint 3). The critical loading and frequencies for various parameter values, are given in Table 3. The critical divergence loads v22 against the parameter k are shown in Figure 10 for two values ofti'i (ff~= I, 2). The influence of the mass m and its moment of inertia is a stabilizing one; at the value of i~1= 1, the critical loading v22 is 23.92 and the frequency u~2 is 15-28 with z3 = 1.0, a = 0.047 and/3=0"002119.
364
LE'I'TERS T O T H E
EDITOR
TABLE 1
Critical buckling loads andfiequencies for various values of k2 (kl=k2=kl=O, m l = m 2 = 1, h i = h2--2h3); (b) divergence buckling loads for various values of k2 and at/~2=0.25 (El = k t = 0 , h i = h 2 = (a)
2h3)
(a) k2
v2,3
u"
0"0 0"25 0-35 0-4 0.475 0.5 0-55 0.6 0.8 1-0
6"765 7.347 7.542 8"345 8.671 8.777 8.735 8-703 8.628 8.611
0"0 0"0 00 00 0.0 0.389 0-526 0.647 1.037 1.328
(D) (D) (D) (D) (D) (F) (F) (F) (F) (F)
(b) k~ 0.0 0.1 0.2 0-3 0.4 0-5 0-6 0.7 0.8 0-9 I-0
/~
v~
0-25 0.25 0.25 0.25 0.25 0-25 0.25 0.25 0.25 0-25 0-25
8.996 9.081 9.155 9.218 9.273 9.321 9.36 9.398 9.43 9.458 9.483
TABLE 2
Critical bucklhzg loads I.,z 23 for various values of k2 (kl =/'S'z=/~l = 0"0 k~ v~3 0"0 0. I 0"2 0-3 0.4 0"5 0-6 0-7 0-8 0-9 1.0
7"23 7.44 7"67 7-92 8-175 8-453 8-75 9-1 9-42 9-80 9"87
45r
LETTERS TO T H E EDITOR
365
4.3
4-1 3'9
:3'7 3-5
0"0
I
I
0"2
0'4
t
0"6 k
,
0-8
i
1'0
Figure 8. Critical buckling load v~2 versus parameter k for the frame in Case 6 at two different values o f n . p
1
..L
7 77.. Figure 9. Partial frame. 17 15
13
0.0
~ = 1 - 0 / /
I
I
1
I
0.2
0.4
0.6 k
0-8
I
1.0
Figure I0. Critical load ~'~ versus the parz;meler k for the partial frame shown in Figure 9 for 61= 1.0 and 2.0.
Case 8: the partial frame shown in Figure 10 is re-examined with the column hinged at the base. The divergence buckling loads are given in Table 4. The values in brackets are divergence buckling loads from the dynamic analysis. The value v~2=2-467 for k = l coincides with the value found in reference [6] by a non-linear stability analysis.
366
LETTERS TO T I l E E D I T O R TABLE 3
Critical buckling load vl2 and frequencies ul2for various values o f k ( h = l = 1.0, m = 0 . 0 , k,p=0-0) k
Vt2
ut2
0'5 0'6 0-7 0.8 I-0
4-975 4.942 4.914 4.893 4.862
1"663 1.8165 1.925 2.01 2-135
(F) (F) (F) (F) (F)
TABLE 4 Critical buckling loads v22 for various vahtes of k with IJ1= 1-0 and k,p = 0 . 0 k
vI22
0.0 0. I 0-2 0-3 0-4 0.5 0.6 0-7 0.8 0-9 !-0
1.821 1.873 i-92 1-982 2.041 2.104 2.169 2.238 2-387 2.387(I-591) 2.467(!.571)
4. CONCLUSIONS The displacement method has been used to investigate the flutter and divergence instability of a number of plane frames subjected to compressive follower forces. The method is very effective for complex systems without inclined members; it minimizes convergence problems and has a wide range of applications. ACKNOWLEI)GMENT
This study has been financed by the Department of Fire Safety Engineering, Lund University. REFER ENCES
1. R. BOGACZand R. JANISZEWSKI 1986 lnstytut Podstawowych Problemow Techniki Polskiej,4kademii Nauk, 1-92. Zagadnienie analizy i syntezy kolumn obciazonych silami sledzacymi ze wzgledu na statecznosc (in Polish). 2. A. N. KOUNADtS and T. N. AX~RAM 1981 American Institute of,4eronautics and Astronautics Journal 19, 761-765. Linear and nonlinear analysis of a nonconservative frame of divergence instability. 3. A. N. KOUNADIS 1983dcta Mechaniea 47, 247-262. Interaction of the joint and of the lateral bracing stiffness for the optimum design of unbraced frames. 4. N. V. KORNOUKIIOV 1949 Strength and Stability of Beam and Frame Systems. Moscow: Stroiizdat (in Russian).
LETTERS TO TIlE EDITOR
367
5. V. G. CHUDNOVSKH 1952 Methods of Calculath~g Vibrations and Stability of Beam and Frame Systems. Kiev: A N Ukr SSR (in Russian). 6. A . N . KOONADtS 1979 Journalde M~canique Applique~3, 173-185. The effects ofsome parameters on the nonlinear divergence buckling of a nonconservative simple frame.