- Email: [email protected]
Elastic instability of beams subjected to a partially tangential force
Journal of Sound and Vibration (1995) 186(1), 111–123
ELASTIC INSTABILITY OF BEAMS SUBJECTED TO A PARTIALLY TANGENTIAL FORCE S. Y. L Department of Mechanical Engineering, National Cheng Kund University, Tainan, Taiwan 701, Republic of China
K. C. H Department of Precision Machinery, R&D Center, Taichung, Taiwan 400, Republic of China (Received 30 November 1993, and in final form 25 August 1994) The influence of the tangency coefficient and the elastically restrained boundary conditions on the elastic instability of an uniform Bernoulli–Euler beam is investigated. It is shown that at both ends of the beam, if any one of the two elastic spring constants is infinite then the tangency coefficient has no influence on the critical load of the beam, and the coefficient may either increase or reduce the stability of a clamped–translational and rotational elastic spring supported beam. The boundary curves for the flutter and divergence instability of the beam in the tangency coefficient and translational spring constant plane with various values of the rotational spring constant, and in the tangency coefficient and translational spring constant plane with various values of the rotational spring constant are shown. It is found that, in general, the boundary curves can be divided into four sections by three critical points. When the tangency coefficient, the translational elastic spring constant and the rotational elastic spring constant are increased to cross over the boundary curves, except the critical points, the instability mechanism changes and the critical load makes a jump. The jump phenomenon of critical load owing to the change of instability mechanism is explored. 7 1995 Academic Press Limited
1. INTRODUCTION
Studies of stability problems of elastic systems subjected to follower forces have been developed into a new branch of the theory of elastic stability, namely non-conservative elastic instability [1–4]. In general, there are two types of elastic instability mechanisms for these problems: divergence and flutter instability. It is known that if the type of instability is divergence, the critical loads of the system can be determined by a static approach, while for flutter the critical loads should be determined using the dynamic criterion [5]. The elastic instability of an elastic uniform Bernoulli–Euler beam subjected to a tangential force has been examined by many investigators [6–14]. Dzhanelidze [6] found that if a cantilever beam is subjected to a partially tangential force then the instability mechanism for the beam is divergence if the tangency coefficient is less than or equal to 0·5. Smith and Herrmann [7] showed that a uniform Winkler elastic foundation has no influence on the critical flutter load of uniform Bernoulli–Euler cantilever beams subjected to a partially tangential force. Sundararajan [8] found that if a Winkler-type elastic foundation has a modulus distribution geometrically similar to the mass distribution then this foundation will have no influence on the flutter load and will increase the divergence load of the cantilever beams. Sundararajan [9] and Sugiyama et al. [10, 11] found that the end elastic spring 111 0022–460X/95/360111+13 $12.00/0
7 1995 Academic Press Limited
. . . .
112
may reduce or increase the stability of a clamped–translational elastic spring supported Bernoulli–Euler beam. Using the static approach, Kounadis [12] found that the boundary between flutter and divergence instability for a Bernoulli–Euler beam subjected to an end tangential force and with general boundary conditions always passes through a double critical point where the first and second static buckling eigenmodes coincide. Tomski and Przybylski [13] used Kounadis’ conclusion and method to establish the region of divergence instability for an elastically restrained cantilever Bernoulli–Euler beam under a partial follower force. However, Kounadis’ conclusion will not be valid for systems in which the flutter load is lower than the divergence load. Consequently, the region of divergence instability determined by Tomski and Przbylski is only partially correct. An explicit expression for the approximate determination of the static buckling loads of a cantilever beam subjected to an end tangential force was given by Gu¨rgo¨ze [14]. For a non-uniform cantilever column subjected to an end tangential force, the elastic stability was examined by Rao and Rao [15]. The existing literature reveals that, even though the elastic instability of a beam subjected to a tangential force has been intensively studied, the investigation of the instability of a beam subjected to a partially tangential force is insufficient. In particular, there are still no answers to the problem of the determination of the boundary conditions for which the tangency coefficient has no influence on critical load. In this paper the dynamic approach is used to investigate the influence of the tangency coefficient and the boundary condition on the elastic instability of an elastically restrained uniform Bernoulli–Euler beam. The characteristic equation for the buckling loads of the beam is derived in terms of the four normalized fundamental solutions of the system. The boundary conditions for which the tangency coefficient has no influence on the critical load are determined. For a clamped–transitional elastic spring supported beam, the jump phenomenon of its critical load owing to instability mechanism change taking place is explored, and the region of flutter and divergence instability is established. 2. ANALYSIS
Consider an elasticity restrained uniform Bernoulli–Euler beam subjected to an end partially tangential force, as shown in Figure 1. Using the dynamic approach, one assumes W(X, t)=w(X) eivt ,
M*(X, t)=M(X) eivt ,
Q*(X, t)=Q(X) eivt ,
(1)
where W, M* and Q* denote the flexural displacement, the bending moment and the shear force respectively, i=(−1)1/2 and v is the angular frequency of vibration of the beam. In terms of the non-dimensional quantities X x= , L bTL=
V(x)= KTL L 3 , EI
w(X) , L buR=
P= KuR L , EI
NL 2 , EI bTR=
L 2=
mv 2L 4 , EI
KTR L 3 , EI
s=
buL=
KuL L , EI
(1−h)NL 2 , EI
(2)
where E is Young’s modulus, I is the area moment of inertia, N is the partially tangential force, h is the partial tangency coefficient, m is the mass per unit length and L is the length of the beam, KuL , KTL and KuR , KTR , are the rotational and translational spring constants at the left and right ends of the beam respectively, the governing equation of the system can be written as V2(x)+PV0(x)−L 2V(x)=0,
x $ (0, 1),
(3)
113
and the associated boundary conditions are, at x=0, V0(0)−buL V'(0)=0,
V1(0)+sV'(0)+bTL V(0)=0,
(4)
V0(1)+buR V'(1)=0,
V1(1)+sV'(1)−bTR V(1)=0,
(5)
and at x=1,
where primes indicate differentiation with respect to the spatial variable x. Let the general solution of the differential equation (3) be in the form V(x)=C1 V1 (x)+C2 V2 (x)+C3 V3 (x)+C4 V4 (x),
(6)
where C1 , C2 , C3 and C4 are four arbitrary constants to be determined from the specified boundary conditions. To simplify the analysis, the four linearly independent fundamental solutions V1 (x), V2 (x), V3 (x) and V4 (x) are chosen such that they satisfy the normalization condition at the origin of the co-ordinate system [16], and are V1 (x)=
z 2 cosh lx+l 2 cos zx , z 2+l 2
V2 (x)=
cosh lx−cos zx , z 2+l 2
V4 (x)=
V3 (x)=
z 2l−1 sinh lx+l 2z−1 sin zx , z 2+l 2
l−1 sinh lx−z−1 sin zx , z 2+l 2
(7)
where z={ 12 [P+(P 2+4L 2 )1/2 ]}1/2 .
l={ 12 [−P+(P 2+4L 2 )1/2 ]}1/2 ,
(8)
After substituting the general solution (6) and (7) into the boundary conditions (4) and (5), one obtains the characteristic equation of the system: d=a1−sa2+a3 bTL+a4 buL+a5 bTL buL=0,
(9)
X W
KθR
KθL
N
N
ηθ L
ηθ R
K TL
K TR
L Figure 1. Geometry of the system.
114
. . . .
where ai=V0j (1)V1 k (1)−V1 j (1)V0 k (1)−s[V' j (1)V0 k (1)−V0 j (1)V' k (1)] +bqR [V'j (1)V1 k (1)−V' k (1)V1 j (1)]+bTR [Vj (1)V0 k (1)−Vk (1)V0 j (1)] +bqR bTR [Vj (1)V'k (1)−V'j (1)Vk (1)],
(10)
and where the subscripts (i, j, k)=(1, 1, 2), (2, 1, 4), (3, 2, 4), (4, 3, 1) and (5, 3, 4). By root finding of the characteristic equations, the buckling loads can be found. If the tangential force N induces double roots of the frequency parameter L then N is called a flutter load. If N induces a zero L then N is called a divergence load. The lowest of the flutter and divergence load values is the critical load of the system. Now, four special cases are examined, as follows. Case 1. buL and buR are constants; bTL :a and bTR :a. In this case, the characteristic equation (9) becomes V2 (1)V04 (1)−V02 (1)V4 (1)+bqR [V2 (1)V'4 (1)−V'2 (1)V4 (1)] +bqL [V3 (1)V04 (1)−V03 (1)V4 (1)]+bqL bqR [V3 (1)V04 (1)−V03 (1)V4 (1)]=0.
(11)
Case 2. bTL and buR are constants; buL :a and bTR :a. In this case, the characteristic equation (9) becomes V1 (1)V03 (1)−V01 (1)V3 (1)+buR [V1 (1)V'3 (1)−V'1 (1)V3 (1)] +bTL [V3 (1)V04 (1)−V03 (1)V4 (1)]+bTL buR [V3 (1)V04 (1)−V03 (1)V4 (1)]=0.
(12)
Case 3. buL and bTR are constants; bTL :a and buR :a. In this case, the characteristic equation (9) becomes V'2 (1)V1 4 (1)−V1 2 (1)V' 4 (1)+bTR [V2 (1)V' 4 (1)−V' 2 (1)V4 (1)] +buL [V'3 (1)V1 4 (1)−V1 3 (1)V' 4 (1)]+buL bTR [V3 (1)V' 4 (1)−V' 3 (1)V4 (1)]=0.
(13)
Case 4. bTL and bTR are constants; buL :a and buR :a. In this case, the characteristic equation (9) becomes V'1 (1)V1 3 (1)−V1 1 (1)V' 3 (1)+bTR [V1 (1)V' 3 (1)−V' 1 (1)V3 (1)] +bTL [V'3 (1)V1 4 (1)−V' 4 (1)V1 3 (1)]+bTL bTR [V3 (1)V' 4 (1)−V' 3 (1)V4 (1)]=0.
(14)
Because the four characteristic equations (11)–(14) do not contain the tangency coefficient, it follows that if any one of the two elastic spring constants at both ends of the beam is infinite then the tangency coefficient has no influence on the critical load of the beam. For a cantilever beam, the characteristic equation can be obtained by substituting the normalized fundamental solutions (7) into the associated characteristic equation. The characteristic equations for the beam with typical kinds of boundary conditions are given in the Appendix. The results are the same as those reported by Smith and Herrmann [7]. 3. NUMERICAL RESULTS AND DISCUSSION
In the following, the influence of the tangency coefficient and the elastic springs on the elastic instability of a clamped–elastic spring supported beam is examined. The regions of flutter and divergence instability are established. Figure 2 shows the influence of the tangency coefficient on the critical loads of a clamped–translational elastic spring supported beam subjected to a partially tangential
115
Figure 2. Critical load versus tangency coefficient for a clamped-translational elastically spring supported beam: ----, divergence; – – –, flutter.
force. For the beam with non-dimensional translational spring constant bTR=0, the instability mechanism for the beam is divergence, since the tangency coefficient is less than or equal to 0·5. The result is consistent with that given by Dzhanelidze [6]. In addition, it is found that when bTRq28·515, all beams with tangency coefficient equal to 0·18 will have the same critical divergence loads. When bTRq34·821, the instability mechanism from the system is divergence. When bTRQ34·821, there exists a critical value of the tangency coefficient where the instability mechanism changes from divergence to flutter as the tangency coefficient is increased. When the type of instability mechanism changes, there may exist a finite jump for the critical load. When bTRQ11·242 or 30·367QbTRQ34·821, the critical load will jump upward, and when 11·242QbTRQ30·367 the critical load will jump downward as the tangency coefficient is increased to cross over the critical point. Hence the tangency coefficient may either increase or reduce the stability of the system, depending upon the value of the non-dimensional translational spring constant bTR . In Figures 3–5, three different types of instability mechanism change of a clampedtranslational elastic spring supported beam are illustrated. Here iL in the ordinate represents the purely imaginary value of the eigenfrequency. From Figure 3, it can be seen that for a beam with bTR=5, when the tangency coefficient h is equal to or less than the associated critical tangency coefficient hc=0·320, the instability mechanism is divergence. However, when h is greater than the associated critical tangency coefficient, the instability mechanism of the system changes from divergence to flutter and the critical load jumps upward. From Figure 4, it can be seen that for a beam with bTR=20, when the tangency coefficient h is increased to become greater than the associated critical tangency coefficient hc=0·106, the instability mechanism of the system changes from divergence to flutter and the critical load jumps downward. In this case, owing to the coexistence of flutter and divergence loads and the jump in the critical load, the boundary between flutter and divergence instability cannot be determined by the static approach proposed by Kounadis [12]. In Figure 5, it is shown
. . . .
116
that for a beam with bTR=11·242, when the tangency coefficient is equal to the critical tangency coefficient hc=0·208 the first two eigenfrequencies coalesce at a point that lies on a line associated with zero frequency. Hence, when the tangency coefficient h is increased to become greater than the associated critical tangency coefficient, the instability mechanism of the system changes. However, the critical load does not jump. Figure 6 shows the boundary curves for the flutter and divergence instability of a clamped–elastic spring supported beam in the tangency coefficient h and non-dimensional translational spring constant bTR plane, for various values of the non-dimensional rotational spring constant buR . For each value of buR , the region above the associated boundary curve is the region of flutter instability and the other region is that of divergence instability. Each boundary curve is divided into four sections by three critical points: ui1 , ui2 and ui3 . Different values of i denote the boundary curves corresponding to different values of the non-dimensional rotational spring constant buR . The values of uij are given in Table 1. The second and third sections lie between (ui1 , ui2 ) and (ui2 , ui3 ) respectively. The curves above ui1 and ui3 correspond to the first and fourth sections respectively. When the non-dimensional translational spring constant bTR is increased to cross over the first and the third sections of the boundary curve, the type of instability mechanism changes and the critical load jumps downward. On the other hand, the critical load will jump upward as the non-dimensional translational spring constant is increased to cross over the other sections of the boundary curve. When the tangency coefficient h is increased to cross over the first and fourth sections
80 (a)
(b)
(c)
(d)
60
Λ
40
i Λ
20 0 –20 80 60
Λ
40
i Λ
20 0 –20 0
5
10
15 P
20
25
0
5
10
15
20
P
Figure 3. Characteristic curves with bTR=5: (a) h=0·200; (b) 0·273; (c) 0·320; (d) 0·325.
25
117
80 (a)
(b)
(c)
(d)
60
Λ
40
i Λ
20 0 –20 80 60
i Λ
40 20
Λ
0 –20 0
5
10
15 P
20
25
0
5
10
15
20
25
P
Figure 4. Characteristic curves with bTR=20: (a) h=0·050; (b) 0·106; (c) 0·150; (d) 0·200.
of the boundary curve, the type of instability mechanism changes and the critical load jumps upward. On the other hand, the critical load will jump downward as the tangency coefficient is increased to cross over the other sections of the boundary curve. When the physical parameters h and bTR are increased to cross right over the critical points, there will be no jump in the critical load. It can also be seen that when the non-dimensional rotational spring constant buR is increased, the region of flutter instability decreases and the three critical points move closer to one another: i.e., the second and the third sections decrease. From Figure 6, one can qualitatively and quantitatively explain the results revealed in Figure 2. One sees that the tangency coefficient may either increase or reduce the stability of a clamped–translational and rotational elastic spring supported beam. It also generalizes the observation reported by Sugiyama et al. [10, 11] that the end translational elastic spring may either reduce or increase the stability of a clamped–translational elastic spring supported beam. In Figure 7, with an increase in the non-dimensional translational spring constant, the change of the instability mechanism and the jump of the critical load of a beam with buR=0 and various values of tangency coefficient are illustrated to provide a further explanation of the jump phenomena revealed in Figure 6. When h=0·1 and bTR is increased from zero, it will pass through the third section first then cross over the second section in Figure 6. Consequently, the critical load first jumps downward and then jumps upward. At the same time, the type of instability mechanism first changes from divergence to flutter and then
. . . .
118
changes back to divergence. When h=0·3 and bTR is increased from zero, it will cross over the fourth and second sections. Hence the critical load jumps upward twice. When h=1·0 and bTR is increased from zero, it will cross over the first section. Hence the critical load jumps downward and the type of instability mechanism changes from flutter to divergence. This curve is consistent with the one given by Sundararajan [9]. In Figure 8, the boundary curves for the flutter and divergence instability of a beam in the tangency coefficient h and non-dimensional rotational spring constant buR plane with various values of non-dimensional translational spring constant bTR are shown. Similarly to the behavior illustrated in Figure 6, for each value of bTR the region above the associated boundary curve is the region of flutter instability and the other region is of divergence instability. Each boundary curve may be divided into different sections by critical points: Ti1 , Ti2 and Ti3 . Different values of i denote the boundary curves corresponding to different values of the non-dimensional translational spring constant. The values of Tij are given in Table 1. The second and third sections lie between (Ti1 , Ti2 ) and (Ti2 , Ti3 ) respectively. The curves above Ti1 and Ti3 correspond to the first and fourth sections respectively. However, there are not three critical points on all the boundary curves. When bTRq30·175, the boundary curve is divided into four sections by three critical points. When 28·122QbTRQ30·175, the boundary curve is divided into three sections by two critical points Ti1 and Ti2 . When 11·249QbTRQ28·122, the boundary curve is divided into two sections by one critical point Ti1 . When bTRQ11·249, there is no critical point, and the whole
80 (b)
(a) 60
Λ
40
i Λ
20 0 –20
0
5
10
15
20
P
80 (c) 60
Λ
40 20
i Λ
0 –20 0
5
10
15
20
25
P Figure 5. Characteristic curves with bTR=11·242: (a) h=0·200; (b) 0·208; (c) 0·250.
25
119
Figure 6. Boundary curves for the flutter and divergence instability in the tangency coefficient and nondimensional translational spring constant plane with various values of the non-dimensional rotational spring constant.
boundary curve belongs to the first section. When the non-dimensional rotational spring constant buR is increased to cross over the first and third sections of the boundary curve, the type of instability mechanism changes and the critical load jumps downward. On the other hand, the critical load will jump upward as the non-dimensional translational spring constant is increased to cross over the other sections of the boundary curve. When the tangency coefficient h is increased to cross over the first and fourth sections of the boundary curve, the type of instability mechanism changes and the critical load jumps upward. On the other hand, the critical load will jump downward as the tangency coefficient is increased to cross over the other sections of the boundary curve. It is found that when the non-dimensional translational spring constant is increased, the region of flutter instability increases. In Figure 9, with an increase in the non-dimensional rotational spring constant, the change of the instability mechanism and the jump in the critical load of a beam with various values of tangency coefficient and non-dimensional translational spring constant are illustrated to T 1 Critical points in Figure 6 and Figure 8 Figure 6 u11 u12 u13 u21 u22 u23 u31 u32 u33
Figure 8
h
bTR
0·483 0·003 0·208 0·216 0·0003 0·047 0·03 0·0001 0·003
30·367 28·229 11·242 34·138 32·625 29·712 38·412 38·309 38·301
T11 T21 T22 T31 T32 T33 T41 T42 T43
h
buR
0·208 0·058 0·0001 0·046 0·0001 0·470 0·013 0·001 0·134
0·0 8·398 0·0 10·887 4·30 0·0 32·061 27·783 21·963
120
. . . .
Figure 7. Critical load versus the translational spring constant for a beam with various values of the tangency coefficient and buR=0.
provide a further explanation of the jump phenomena revealed in Figure 8. When h=0·550, bTR=36·0 and buR is increased from zero, it will first pass through the fourth section then cross over the first section in Figure 8. Consequently, the critical load first jumps upward and then jumps downward. At the same time, the type of instability mechanism first changes from divergence to flutter and then changes back to divergence. When h=0·3, bTR=30·175 and buR is increased from zero, it will cross over the third section first and then the first section. Hence the critical load jumps downward twice. When h=0·030, bTR=28·122 and buR is
Figure 8. Boundary curves for the flutter and divergence instability in the tangency coefficient and nondimensional rotational spring constant plane with various values of the non-dimensional translational spring constant.
121
Figure 9. Critical load versus rotational spring constant for a beam with various values of the tangency coefficient and translational spring constant.
increased from zero, it will cross over the second section. Hence the critical load jumps upward.
4. CONCLUSIONS
The influence of the tangency coefficient and the elastically restrained boundary conditions on the elastic instability of a uniform Bernoulli–Euler beam has been investigated. The following results have been obtained. (1) At both ends of the beam, if any one of the two elastic spring constants is infinite then the tangency coefficient has no influence on the critical load of the beam. (2) The boundary curves for the flutter and divergence instability of a beam in the tangency coefficient and translational spring constant plane with various values of rotational spring constant, and in the tangency coefficient and translational spring constant plane with various values of rotational spring constant, in general, can be divided into four sections by three critical points. When the translational and the rotational elastic spring constants are increased to cross over the boundary curve of the first and third sections and the other two sections, the instability mechanism changes and the critical load jumps upward and downward respectively. When the tangency coefficient is increased to cross over the first and fourth sections of the boundary curve, the type of instability mechanism changes and the critical load jumps downward. On the other hand, the critical load will jump upward as the tangency coefficient is increased to cross over the other sections of the boundary curve. (3) The tangency coefficient may either increase or reduce the stability of a clamped–translational and rotational elastic springs supported beam. (4) In the tangency coefficient and non-dimensional translational spring constant plane with various values of non-dimensional rotational spring constant, when the nondimensional rotational spring constant is increased, the region of flutter instability decreases and the three critical points move closer; i.e., the second and third sections decrease.
122
. . . .
(5) In the tangency coefficient and non-dimensional rotational spring constant plane with various values of non-dimensional translational spring constant, when the non-dimensional translational spring constant is increased, the region of flutter instability increases. It will be of interest to make a further investigation of the fact revealed in Figure 2; namely, that for a clamped–translational elastic spring supported beam with bTRq28·515, all beams with tangency coefficient equal to 0·18 will have the same critical divergence loads.
ACKNOWLEDGMENTS
This research work was sponsored by the National Science Council of Taiwan, R. O. C. under grant NSC81-0401-E006-16. The authors would like to thank Mr Yin-Re, Pan and Mr Ju-Chun, Lin for helping to prepare the figures.
REFERENCES 1. V. V. B 1979 Non-conservative Problems of the Theory of Elastic Stability. London: Pergamon Press. 2. G. H 1967 Applied Mechanics Reviews 20, 103–108. Stability of equilibrium of elastic systems subjected to non-conservative forces. 3. C. S 1975 The Shock and Vibration Digest 7, 89–105. The vibration and stability of elastic systems subjected follower forces. 4. H. L 1980 Stability of Elastic Systems (first edition). The Netherlands: Sijthoff & Noordhoff. 5. H. Z 1977 Principle of Structural Stability. Basel: Birkhauser-Verlag. 6. G. D 1958 Trudy Leningrad Politeknika Instituta 192, 21–27. On the stability of rods due to the action of follower forces (in Russian). 7. T. E. S and G. H 1972 Transactions of the American Society of Mechanical Engineers, Journal of Applied Mechanics 39, 628–629. Stability of a beam on elastic foundation subjected to a follower force. 8. C. S 1974 Journal of Sound and Vibration 37, 79–85. Stability of column on elastic foundations subjected to conservative and non-conservative forces. 9. C. S 1976 International Journal of Mechanical Science 18, 239–241. Influence of an elastic end support on the vibration and stability of Beck’s column. 10. Y. S, S. M and H. K 1974 Theoretical and Applied Mechanics 22, 33–45. Destabilizing effect of elastic constraint on the stability of non-conservative elastic systems. 11. Y. S, H. K and S. M 1975 Theoretical and Applied Mechanics 23, 125–135. Destabilizing effect of elastic constraint on the stability of non-conservative elastic systems, II: the effect of elastic support. 12. A. N. K 1983 International Journal of Solids and Structures 19, 725–733. The existence of divergence instability for non-conservative systems under follower forces. 13. L. T and J. P 1985 American Institute of Aeronautics and Astronautics Journal 23, 1637–1639. Static instability of an elastically restrained cantilever under a partial follower force. 14. M. G¨¨ 1989 Journal of Sound and Vibration 131, 527–532. On the static buckling of a Petterson–Konig beam in a subtangential range. 15. B. N. R and G. V. R 1988 Journal of Sound and Vibration 122, 188–191. Application of static or dynamic criterion for the stability of a non-uniform cantilever column subjected to a tip-concentrated subtangential follower force. 16. S. Y. L and Y. H. K 1991 Journal of Sound and Vibration 148, 11–24. Elastic stability of non-uniform columns.
123
APPENDIX: CHARACTERISTIC EQUATIONS
Case 1: Clamped–clamped. In this case, bTL :a, buL :a, bTR :a and buR :a, and the characteristic equation is V3 (1)V'4 (1)−V'3 (1)V4 (1)=0. Case 2: Clamped–hinged. In this case, bTL :a, buL :a, bTR :a and buR=0, and the characteristic equation is V3 (1)V04 (1)−V03 (1)V4 (1)=0. Case 3: Clamped–free. In this case, bTL :a, buL :a and bTR=buR=0, and the characteristic equation is V03 (1)V1 4 (1)−V1 3 (1)V0 4 (1)−s[V' 3 (1)V0 4 (1)−V0 3 (1)V' 4 (1)]=0. Case 4: Hinged–hinged. In this case, bTL :a, bTR :a and buL=buR=0, and the characteristic equation is V2 (1)V04 (1)−V02 (1)V4 (1)=0.
ELASTIC INSTABILITY OF BEAMS SUBJECTED TO A PARTIALLY TANGENTIAL FORCE S. Y. L Department of Mechanical Engineering, National Cheng Kund University, Tainan, Taiwan 701, Republic of China
K. C. H Department of Precision Machinery, R&D Center, Taichung, Taiwan 400, Republic of China (Received 30 November 1993, and in final form 25 August 1994) The influence of the tangency coefficient and the elastically restrained boundary conditions on the elastic instability of an uniform Bernoulli–Euler beam is investigated. It is shown that at both ends of the beam, if any one of the two elastic spring constants is infinite then the tangency coefficient has no influence on the critical load of the beam, and the coefficient may either increase or reduce the stability of a clamped–translational and rotational elastic spring supported beam. The boundary curves for the flutter and divergence instability of the beam in the tangency coefficient and translational spring constant plane with various values of the rotational spring constant, and in the tangency coefficient and translational spring constant plane with various values of the rotational spring constant are shown. It is found that, in general, the boundary curves can be divided into four sections by three critical points. When the tangency coefficient, the translational elastic spring constant and the rotational elastic spring constant are increased to cross over the boundary curves, except the critical points, the instability mechanism changes and the critical load makes a jump. The jump phenomenon of critical load owing to the change of instability mechanism is explored. 7 1995 Academic Press Limited
1. INTRODUCTION
Studies of stability problems of elastic systems subjected to follower forces have been developed into a new branch of the theory of elastic stability, namely non-conservative elastic instability [1–4]. In general, there are two types of elastic instability mechanisms for these problems: divergence and flutter instability. It is known that if the type of instability is divergence, the critical loads of the system can be determined by a static approach, while for flutter the critical loads should be determined using the dynamic criterion [5]. The elastic instability of an elastic uniform Bernoulli–Euler beam subjected to a tangential force has been examined by many investigators [6–14]. Dzhanelidze [6] found that if a cantilever beam is subjected to a partially tangential force then the instability mechanism for the beam is divergence if the tangency coefficient is less than or equal to 0·5. Smith and Herrmann [7] showed that a uniform Winkler elastic foundation has no influence on the critical flutter load of uniform Bernoulli–Euler cantilever beams subjected to a partially tangential force. Sundararajan [8] found that if a Winkler-type elastic foundation has a modulus distribution geometrically similar to the mass distribution then this foundation will have no influence on the flutter load and will increase the divergence load of the cantilever beams. Sundararajan [9] and Sugiyama et al. [10, 11] found that the end elastic spring 111 0022–460X/95/360111+13 $12.00/0
7 1995 Academic Press Limited
. . . .
112
may reduce or increase the stability of a clamped–translational elastic spring supported Bernoulli–Euler beam. Using the static approach, Kounadis [12] found that the boundary between flutter and divergence instability for a Bernoulli–Euler beam subjected to an end tangential force and with general boundary conditions always passes through a double critical point where the first and second static buckling eigenmodes coincide. Tomski and Przybylski [13] used Kounadis’ conclusion and method to establish the region of divergence instability for an elastically restrained cantilever Bernoulli–Euler beam under a partial follower force. However, Kounadis’ conclusion will not be valid for systems in which the flutter load is lower than the divergence load. Consequently, the region of divergence instability determined by Tomski and Przbylski is only partially correct. An explicit expression for the approximate determination of the static buckling loads of a cantilever beam subjected to an end tangential force was given by Gu¨rgo¨ze [14]. For a non-uniform cantilever column subjected to an end tangential force, the elastic stability was examined by Rao and Rao [15]. The existing literature reveals that, even though the elastic instability of a beam subjected to a tangential force has been intensively studied, the investigation of the instability of a beam subjected to a partially tangential force is insufficient. In particular, there are still no answers to the problem of the determination of the boundary conditions for which the tangency coefficient has no influence on critical load. In this paper the dynamic approach is used to investigate the influence of the tangency coefficient and the boundary condition on the elastic instability of an elastically restrained uniform Bernoulli–Euler beam. The characteristic equation for the buckling loads of the beam is derived in terms of the four normalized fundamental solutions of the system. The boundary conditions for which the tangency coefficient has no influence on the critical load are determined. For a clamped–transitional elastic spring supported beam, the jump phenomenon of its critical load owing to instability mechanism change taking place is explored, and the region of flutter and divergence instability is established. 2. ANALYSIS
Consider an elasticity restrained uniform Bernoulli–Euler beam subjected to an end partially tangential force, as shown in Figure 1. Using the dynamic approach, one assumes W(X, t)=w(X) eivt ,
M*(X, t)=M(X) eivt ,
Q*(X, t)=Q(X) eivt ,
(1)
where W, M* and Q* denote the flexural displacement, the bending moment and the shear force respectively, i=(−1)1/2 and v is the angular frequency of vibration of the beam. In terms of the non-dimensional quantities X x= , L bTL=
V(x)= KTL L 3 , EI
w(X) , L buR=
P= KuR L , EI
NL 2 , EI bTR=
L 2=
mv 2L 4 , EI
KTR L 3 , EI
s=
buL=
KuL L , EI
(1−h)NL 2 , EI
(2)
where E is Young’s modulus, I is the area moment of inertia, N is the partially tangential force, h is the partial tangency coefficient, m is the mass per unit length and L is the length of the beam, KuL , KTL and KuR , KTR , are the rotational and translational spring constants at the left and right ends of the beam respectively, the governing equation of the system can be written as V2(x)+PV0(x)−L 2V(x)=0,
x $ (0, 1),
(3)
113
and the associated boundary conditions are, at x=0, V0(0)−buL V'(0)=0,
V1(0)+sV'(0)+bTL V(0)=0,
(4)
V0(1)+buR V'(1)=0,
V1(1)+sV'(1)−bTR V(1)=0,
(5)
and at x=1,
where primes indicate differentiation with respect to the spatial variable x. Let the general solution of the differential equation (3) be in the form V(x)=C1 V1 (x)+C2 V2 (x)+C3 V3 (x)+C4 V4 (x),
(6)
where C1 , C2 , C3 and C4 are four arbitrary constants to be determined from the specified boundary conditions. To simplify the analysis, the four linearly independent fundamental solutions V1 (x), V2 (x), V3 (x) and V4 (x) are chosen such that they satisfy the normalization condition at the origin of the co-ordinate system [16], and are V1 (x)=
z 2 cosh lx+l 2 cos zx , z 2+l 2
V2 (x)=
cosh lx−cos zx , z 2+l 2
V4 (x)=
V3 (x)=
z 2l−1 sinh lx+l 2z−1 sin zx , z 2+l 2
l−1 sinh lx−z−1 sin zx , z 2+l 2
(7)
where z={ 12 [P+(P 2+4L 2 )1/2 ]}1/2 .
l={ 12 [−P+(P 2+4L 2 )1/2 ]}1/2 ,
(8)
After substituting the general solution (6) and (7) into the boundary conditions (4) and (5), one obtains the characteristic equation of the system: d=a1−sa2+a3 bTL+a4 buL+a5 bTL buL=0,
(9)
X W
KθR
KθL
N
N
ηθ L
ηθ R
K TL
K TR
L Figure 1. Geometry of the system.
114
. . . .
where ai=V0j (1)V1 k (1)−V1 j (1)V0 k (1)−s[V' j (1)V0 k (1)−V0 j (1)V' k (1)] +bqR [V'j (1)V1 k (1)−V' k (1)V1 j (1)]+bTR [Vj (1)V0 k (1)−Vk (1)V0 j (1)] +bqR bTR [Vj (1)V'k (1)−V'j (1)Vk (1)],
(10)
and where the subscripts (i, j, k)=(1, 1, 2), (2, 1, 4), (3, 2, 4), (4, 3, 1) and (5, 3, 4). By root finding of the characteristic equations, the buckling loads can be found. If the tangential force N induces double roots of the frequency parameter L then N is called a flutter load. If N induces a zero L then N is called a divergence load. The lowest of the flutter and divergence load values is the critical load of the system. Now, four special cases are examined, as follows. Case 1. buL and buR are constants; bTL :a and bTR :a. In this case, the characteristic equation (9) becomes V2 (1)V04 (1)−V02 (1)V4 (1)+bqR [V2 (1)V'4 (1)−V'2 (1)V4 (1)] +bqL [V3 (1)V04 (1)−V03 (1)V4 (1)]+bqL bqR [V3 (1)V04 (1)−V03 (1)V4 (1)]=0.
(11)
Case 2. bTL and buR are constants; buL :a and bTR :a. In this case, the characteristic equation (9) becomes V1 (1)V03 (1)−V01 (1)V3 (1)+buR [V1 (1)V'3 (1)−V'1 (1)V3 (1)] +bTL [V3 (1)V04 (1)−V03 (1)V4 (1)]+bTL buR [V3 (1)V04 (1)−V03 (1)V4 (1)]=0.
(12)
Case 3. buL and bTR are constants; bTL :a and buR :a. In this case, the characteristic equation (9) becomes V'2 (1)V1 4 (1)−V1 2 (1)V' 4 (1)+bTR [V2 (1)V' 4 (1)−V' 2 (1)V4 (1)] +buL [V'3 (1)V1 4 (1)−V1 3 (1)V' 4 (1)]+buL bTR [V3 (1)V' 4 (1)−V' 3 (1)V4 (1)]=0.
(13)
Case 4. bTL and bTR are constants; buL :a and buR :a. In this case, the characteristic equation (9) becomes V'1 (1)V1 3 (1)−V1 1 (1)V' 3 (1)+bTR [V1 (1)V' 3 (1)−V' 1 (1)V3 (1)] +bTL [V'3 (1)V1 4 (1)−V' 4 (1)V1 3 (1)]+bTL bTR [V3 (1)V' 4 (1)−V' 3 (1)V4 (1)]=0.
(14)
Because the four characteristic equations (11)–(14) do not contain the tangency coefficient, it follows that if any one of the two elastic spring constants at both ends of the beam is infinite then the tangency coefficient has no influence on the critical load of the beam. For a cantilever beam, the characteristic equation can be obtained by substituting the normalized fundamental solutions (7) into the associated characteristic equation. The characteristic equations for the beam with typical kinds of boundary conditions are given in the Appendix. The results are the same as those reported by Smith and Herrmann [7]. 3. NUMERICAL RESULTS AND DISCUSSION
In the following, the influence of the tangency coefficient and the elastic springs on the elastic instability of a clamped–elastic spring supported beam is examined. The regions of flutter and divergence instability are established. Figure 2 shows the influence of the tangency coefficient on the critical loads of a clamped–translational elastic spring supported beam subjected to a partially tangential
115
Figure 2. Critical load versus tangency coefficient for a clamped-translational elastically spring supported beam: ----, divergence; – – –, flutter.
force. For the beam with non-dimensional translational spring constant bTR=0, the instability mechanism for the beam is divergence, since the tangency coefficient is less than or equal to 0·5. The result is consistent with that given by Dzhanelidze [6]. In addition, it is found that when bTRq28·515, all beams with tangency coefficient equal to 0·18 will have the same critical divergence loads. When bTRq34·821, the instability mechanism from the system is divergence. When bTRQ34·821, there exists a critical value of the tangency coefficient where the instability mechanism changes from divergence to flutter as the tangency coefficient is increased. When the type of instability mechanism changes, there may exist a finite jump for the critical load. When bTRQ11·242 or 30·367QbTRQ34·821, the critical load will jump upward, and when 11·242QbTRQ30·367 the critical load will jump downward as the tangency coefficient is increased to cross over the critical point. Hence the tangency coefficient may either increase or reduce the stability of the system, depending upon the value of the non-dimensional translational spring constant bTR . In Figures 3–5, three different types of instability mechanism change of a clampedtranslational elastic spring supported beam are illustrated. Here iL in the ordinate represents the purely imaginary value of the eigenfrequency. From Figure 3, it can be seen that for a beam with bTR=5, when the tangency coefficient h is equal to or less than the associated critical tangency coefficient hc=0·320, the instability mechanism is divergence. However, when h is greater than the associated critical tangency coefficient, the instability mechanism of the system changes from divergence to flutter and the critical load jumps upward. From Figure 4, it can be seen that for a beam with bTR=20, when the tangency coefficient h is increased to become greater than the associated critical tangency coefficient hc=0·106, the instability mechanism of the system changes from divergence to flutter and the critical load jumps downward. In this case, owing to the coexistence of flutter and divergence loads and the jump in the critical load, the boundary between flutter and divergence instability cannot be determined by the static approach proposed by Kounadis [12]. In Figure 5, it is shown
. . . .
116
that for a beam with bTR=11·242, when the tangency coefficient is equal to the critical tangency coefficient hc=0·208 the first two eigenfrequencies coalesce at a point that lies on a line associated with zero frequency. Hence, when the tangency coefficient h is increased to become greater than the associated critical tangency coefficient, the instability mechanism of the system changes. However, the critical load does not jump. Figure 6 shows the boundary curves for the flutter and divergence instability of a clamped–elastic spring supported beam in the tangency coefficient h and non-dimensional translational spring constant bTR plane, for various values of the non-dimensional rotational spring constant buR . For each value of buR , the region above the associated boundary curve is the region of flutter instability and the other region is that of divergence instability. Each boundary curve is divided into four sections by three critical points: ui1 , ui2 and ui3 . Different values of i denote the boundary curves corresponding to different values of the non-dimensional rotational spring constant buR . The values of uij are given in Table 1. The second and third sections lie between (ui1 , ui2 ) and (ui2 , ui3 ) respectively. The curves above ui1 and ui3 correspond to the first and fourth sections respectively. When the non-dimensional translational spring constant bTR is increased to cross over the first and the third sections of the boundary curve, the type of instability mechanism changes and the critical load jumps downward. On the other hand, the critical load will jump upward as the non-dimensional translational spring constant is increased to cross over the other sections of the boundary curve. When the tangency coefficient h is increased to cross over the first and fourth sections
80 (a)
(b)
(c)
(d)
60
Λ
40
i Λ
20 0 –20 80 60
Λ
40
i Λ
20 0 –20 0
5
10
15 P
20
25
0
5
10
15
20
P
Figure 3. Characteristic curves with bTR=5: (a) h=0·200; (b) 0·273; (c) 0·320; (d) 0·325.
25
117
80 (a)
(b)
(c)
(d)
60
Λ
40
i Λ
20 0 –20 80 60
i Λ
40 20
Λ
0 –20 0
5
10
15 P
20
25
0
5
10
15
20
25
P
Figure 4. Characteristic curves with bTR=20: (a) h=0·050; (b) 0·106; (c) 0·150; (d) 0·200.
of the boundary curve, the type of instability mechanism changes and the critical load jumps upward. On the other hand, the critical load will jump downward as the tangency coefficient is increased to cross over the other sections of the boundary curve. When the physical parameters h and bTR are increased to cross right over the critical points, there will be no jump in the critical load. It can also be seen that when the non-dimensional rotational spring constant buR is increased, the region of flutter instability decreases and the three critical points move closer to one another: i.e., the second and the third sections decrease. From Figure 6, one can qualitatively and quantitatively explain the results revealed in Figure 2. One sees that the tangency coefficient may either increase or reduce the stability of a clamped–translational and rotational elastic spring supported beam. It also generalizes the observation reported by Sugiyama et al. [10, 11] that the end translational elastic spring may either reduce or increase the stability of a clamped–translational elastic spring supported beam. In Figure 7, with an increase in the non-dimensional translational spring constant, the change of the instability mechanism and the jump of the critical load of a beam with buR=0 and various values of tangency coefficient are illustrated to provide a further explanation of the jump phenomena revealed in Figure 6. When h=0·1 and bTR is increased from zero, it will pass through the third section first then cross over the second section in Figure 6. Consequently, the critical load first jumps downward and then jumps upward. At the same time, the type of instability mechanism first changes from divergence to flutter and then
. . . .
118
changes back to divergence. When h=0·3 and bTR is increased from zero, it will cross over the fourth and second sections. Hence the critical load jumps upward twice. When h=1·0 and bTR is increased from zero, it will cross over the first section. Hence the critical load jumps downward and the type of instability mechanism changes from flutter to divergence. This curve is consistent with the one given by Sundararajan [9]. In Figure 8, the boundary curves for the flutter and divergence instability of a beam in the tangency coefficient h and non-dimensional rotational spring constant buR plane with various values of non-dimensional translational spring constant bTR are shown. Similarly to the behavior illustrated in Figure 6, for each value of bTR the region above the associated boundary curve is the region of flutter instability and the other region is of divergence instability. Each boundary curve may be divided into different sections by critical points: Ti1 , Ti2 and Ti3 . Different values of i denote the boundary curves corresponding to different values of the non-dimensional translational spring constant. The values of Tij are given in Table 1. The second and third sections lie between (Ti1 , Ti2 ) and (Ti2 , Ti3 ) respectively. The curves above Ti1 and Ti3 correspond to the first and fourth sections respectively. However, there are not three critical points on all the boundary curves. When bTRq30·175, the boundary curve is divided into four sections by three critical points. When 28·122QbTRQ30·175, the boundary curve is divided into three sections by two critical points Ti1 and Ti2 . When 11·249QbTRQ28·122, the boundary curve is divided into two sections by one critical point Ti1 . When bTRQ11·249, there is no critical point, and the whole
80 (b)
(a) 60
Λ
40
i Λ
20 0 –20
0
5
10
15
20
P
80 (c) 60
Λ
40 20
i Λ
0 –20 0
5
10
15
20
25
P Figure 5. Characteristic curves with bTR=11·242: (a) h=0·200; (b) 0·208; (c) 0·250.
25
119
Figure 6. Boundary curves for the flutter and divergence instability in the tangency coefficient and nondimensional translational spring constant plane with various values of the non-dimensional rotational spring constant.
boundary curve belongs to the first section. When the non-dimensional rotational spring constant buR is increased to cross over the first and third sections of the boundary curve, the type of instability mechanism changes and the critical load jumps downward. On the other hand, the critical load will jump upward as the non-dimensional translational spring constant is increased to cross over the other sections of the boundary curve. When the tangency coefficient h is increased to cross over the first and fourth sections of the boundary curve, the type of instability mechanism changes and the critical load jumps upward. On the other hand, the critical load will jump downward as the tangency coefficient is increased to cross over the other sections of the boundary curve. It is found that when the non-dimensional translational spring constant is increased, the region of flutter instability increases. In Figure 9, with an increase in the non-dimensional rotational spring constant, the change of the instability mechanism and the jump in the critical load of a beam with various values of tangency coefficient and non-dimensional translational spring constant are illustrated to T 1 Critical points in Figure 6 and Figure 8 Figure 6 u11 u12 u13 u21 u22 u23 u31 u32 u33
Figure 8
h
bTR
0·483 0·003 0·208 0·216 0·0003 0·047 0·03 0·0001 0·003
30·367 28·229 11·242 34·138 32·625 29·712 38·412 38·309 38·301
T11 T21 T22 T31 T32 T33 T41 T42 T43
h
buR
0·208 0·058 0·0001 0·046 0·0001 0·470 0·013 0·001 0·134
0·0 8·398 0·0 10·887 4·30 0·0 32·061 27·783 21·963
120
. . . .
Figure 7. Critical load versus the translational spring constant for a beam with various values of the tangency coefficient and buR=0.
provide a further explanation of the jump phenomena revealed in Figure 8. When h=0·550, bTR=36·0 and buR is increased from zero, it will first pass through the fourth section then cross over the first section in Figure 8. Consequently, the critical load first jumps upward and then jumps downward. At the same time, the type of instability mechanism first changes from divergence to flutter and then changes back to divergence. When h=0·3, bTR=30·175 and buR is increased from zero, it will cross over the third section first and then the first section. Hence the critical load jumps downward twice. When h=0·030, bTR=28·122 and buR is
Figure 8. Boundary curves for the flutter and divergence instability in the tangency coefficient and nondimensional rotational spring constant plane with various values of the non-dimensional translational spring constant.
121
Figure 9. Critical load versus rotational spring constant for a beam with various values of the tangency coefficient and translational spring constant.
increased from zero, it will cross over the second section. Hence the critical load jumps upward.
4. CONCLUSIONS
The influence of the tangency coefficient and the elastically restrained boundary conditions on the elastic instability of a uniform Bernoulli–Euler beam has been investigated. The following results have been obtained. (1) At both ends of the beam, if any one of the two elastic spring constants is infinite then the tangency coefficient has no influence on the critical load of the beam. (2) The boundary curves for the flutter and divergence instability of a beam in the tangency coefficient and translational spring constant plane with various values of rotational spring constant, and in the tangency coefficient and translational spring constant plane with various values of rotational spring constant, in general, can be divided into four sections by three critical points. When the translational and the rotational elastic spring constants are increased to cross over the boundary curve of the first and third sections and the other two sections, the instability mechanism changes and the critical load jumps upward and downward respectively. When the tangency coefficient is increased to cross over the first and fourth sections of the boundary curve, the type of instability mechanism changes and the critical load jumps downward. On the other hand, the critical load will jump upward as the tangency coefficient is increased to cross over the other sections of the boundary curve. (3) The tangency coefficient may either increase or reduce the stability of a clamped–translational and rotational elastic springs supported beam. (4) In the tangency coefficient and non-dimensional translational spring constant plane with various values of non-dimensional rotational spring constant, when the nondimensional rotational spring constant is increased, the region of flutter instability decreases and the three critical points move closer; i.e., the second and third sections decrease.
122
. . . .
(5) In the tangency coefficient and non-dimensional rotational spring constant plane with various values of non-dimensional translational spring constant, when the non-dimensional translational spring constant is increased, the region of flutter instability increases. It will be of interest to make a further investigation of the fact revealed in Figure 2; namely, that for a clamped–translational elastic spring supported beam with bTRq28·515, all beams with tangency coefficient equal to 0·18 will have the same critical divergence loads.
ACKNOWLEDGMENTS
This research work was sponsored by the National Science Council of Taiwan, R. O. C. under grant NSC81-0401-E006-16. The authors would like to thank Mr Yin-Re, Pan and Mr Ju-Chun, Lin for helping to prepare the figures.
REFERENCES 1. V. V. B 1979 Non-conservative Problems of the Theory of Elastic Stability. London: Pergamon Press. 2. G. H 1967 Applied Mechanics Reviews 20, 103–108. Stability of equilibrium of elastic systems subjected to non-conservative forces. 3. C. S 1975 The Shock and Vibration Digest 7, 89–105. The vibration and stability of elastic systems subjected follower forces. 4. H. L 1980 Stability of Elastic Systems (first edition). The Netherlands: Sijthoff & Noordhoff. 5. H. Z 1977 Principle of Structural Stability. Basel: Birkhauser-Verlag. 6. G. D 1958 Trudy Leningrad Politeknika Instituta 192, 21–27. On the stability of rods due to the action of follower forces (in Russian). 7. T. E. S and G. H 1972 Transactions of the American Society of Mechanical Engineers, Journal of Applied Mechanics 39, 628–629. Stability of a beam on elastic foundation subjected to a follower force. 8. C. S 1974 Journal of Sound and Vibration 37, 79–85. Stability of column on elastic foundations subjected to conservative and non-conservative forces. 9. C. S 1976 International Journal of Mechanical Science 18, 239–241. Influence of an elastic end support on the vibration and stability of Beck’s column. 10. Y. S, S. M and H. K 1974 Theoretical and Applied Mechanics 22, 33–45. Destabilizing effect of elastic constraint on the stability of non-conservative elastic systems. 11. Y. S, H. K and S. M 1975 Theoretical and Applied Mechanics 23, 125–135. Destabilizing effect of elastic constraint on the stability of non-conservative elastic systems, II: the effect of elastic support. 12. A. N. K 1983 International Journal of Solids and Structures 19, 725–733. The existence of divergence instability for non-conservative systems under follower forces. 13. L. T and J. P 1985 American Institute of Aeronautics and Astronautics Journal 23, 1637–1639. Static instability of an elastically restrained cantilever under a partial follower force. 14. M. G¨¨ 1989 Journal of Sound and Vibration 131, 527–532. On the static buckling of a Petterson–Konig beam in a subtangential range. 15. B. N. R and G. V. R 1988 Journal of Sound and Vibration 122, 188–191. Application of static or dynamic criterion for the stability of a non-uniform cantilever column subjected to a tip-concentrated subtangential follower force. 16. S. Y. L and Y. H. K 1991 Journal of Sound and Vibration 148, 11–24. Elastic stability of non-uniform columns.
123
APPENDIX: CHARACTERISTIC EQUATIONS
Case 1: Clamped–clamped. In this case, bTL :a, buL :a, bTR :a and buR :a, and the characteristic equation is V3 (1)V'4 (1)−V'3 (1)V4 (1)=0. Case 2: Clamped–hinged. In this case, bTL :a, buL :a, bTR :a and buR=0, and the characteristic equation is V3 (1)V04 (1)−V03 (1)V4 (1)=0. Case 3: Clamped–free. In this case, bTL :a, buL :a and bTR=buR=0, and the characteristic equation is V03 (1)V1 4 (1)−V1 3 (1)V0 4 (1)−s[V' 3 (1)V0 4 (1)−V0 3 (1)V' 4 (1)]=0. Case 4: Hinged–hinged. In this case, bTL :a, bTR :a and buL=buR=0, and the characteristic equation is V2 (1)V04 (1)−V02 (1)V4 (1)=0.