- Email: [email protected]
Dynamic stability of cantilever columns resting on an elastic foundation
c.hprrrr Printed
d smcrures in Gmr Britain.
Vol.
2s.
ho.
I, pp.
151-158.
0045.7949,87 53.00 + 0.w Pergamon Joumnlr Ltd.
1987
TECHNICAL NOTE DYNAMIC
STABILITY OF CANTILEVER COLUMNS ON AN ELASTIC FOUNDATION
RESTING
B. P. SHASTRY and G. VE%ATF.SWARA RACJ Department of Space, Vikram Sarabhai Space Centre, Trivandrum 695022, India (Received 4 Murch 1986)
Abstract-Instability boundaries for cantilever columns resting on an elastic foundation are determined using finite element method for various foundation modulii. The study reveals that dynamic stability boundaries for such columns can be represented in terms of three separate dynamic stability curves corresponding to low foundation moduli, high foundation moduli and the transition regions of foundation
INTRODUCTION
The periodic load acting on the cohrmn is represented by
The dynamic stability of structural elements is well discussed in [I]. In [2] and [3] the dynamic stability of slender and short columns, respectively, subjected to a periodic concentrated load at the free end, are investigated using the finite element method. The effect of elastic foundation on the dynamic stability of columns has been investigated [2,4]. It has been shown that in the case of hinged columns [5] the stability mode shape abruptly changes when the foundation modulus exceeds a critical value. However, in the case of cantilever columns the stabifity mode shape remains constant for lower foundation moduli and higher foundation moduli even though the two mode shapes are different and show a gradual change when the foundation modulus changes from lower values to higher values. However, the vibration mode shape remains constant whatever may be the value of foundation modulus. It has been shown [2] that the dynamic stability behaviour of columns has a definite dependency on the stability and vibration mode shapes, but this aspect is not studied in detail; even though the effect of elastic foundation on the dynamic stability behaviour of cantilever columns is shown in[4], the effect of dependency of stability and vibration mode shape is not discussed. The aim of the present study is to bring out in detail the dynamic stability behaviour of cantilever columns resting on elastic foundation over a wide range of foundation moduli. Dynamic stability curves are obtained for various values of the foundation moduli, showing clearly the interaction of stability and vibration mode shapes. FINITE ELEMENT
FORMULATION
Following [I] and [2], with a concentrated periodic load acting on the column at the free end, the instability boundaries are obtained, as a first appro~mation, by the solution of the equation
where [K], [C] and [M] are the assembled elastic stiffness matrix, geometric stiffness matrix and mass matrix respectively. These matrices are obtained using standard proceduresf61. It is to be noted here that the matrix [KJ contains a cont~bution from the stiffness of the elastic foundation also.
P(r) = p*+ P,sb(i),
(2)
where (3) with #(t) = cos er.
(4)
In eqn (3). P, is the critical load and in eqn (4) 9 is the axial frequency in rad/sec. The solution of eqn (1) provides (a) the static
j=e=o;
buckling
load of the column
when
(b) the lateral natural frequency w in rad/sec when a=/?=0
and
o=-;
8 2
(c) regions of unstable solutions with u and b varying.
NUMERICAL The
RESULTS
fundamental lateral frequency parameter ;., (defined as, .+- mo*L’/EI, where m is the mass per unit length, L is the length of the cohunn, E is the Young’s modulus and I is the moment of inertia) and the stability parameter Lb (defined as, 1, = P,L*/EQ and the associated mode shapes and the instability regions when the column is subjected to a periodic axial load for a = 0.5 are obtained for various values of the foundation parameter r (defined as r = KL’IEI, where K is the foundation modulus) for a cantilever column resting on an elastic foundation. An eight-element solution is used, based on convergence study, to obtain the numerical results. Table I gives the fundamental frequency parameter and the stability parameter 5 for several values of r. It is seen from Table 1 that as r increases both &and A, increase. The increase in I, for a given r is equal to the sum of the values of A, for I = 0 and the corresponding r. A similar trend is not seen for 2,. In Fig. 1, the mode shapes of stability and vibration are given. For values of r =O, 0.1, 1 and 10 the stability mode shape is the same and similar to that of the
157
158
Technical Note Table 1. Fundamental frequency parameter j./ and stability parameter i., of a cantilever column resting on an elastic foundation I
i.,
i.,
0.0 0.1 1.0 10.0 50.0 100.0 250.0 500.0 750.0 1000.0
12.36 12.46 13.36 22.36 62.36 112.4 262.4 512.4 762.4 1012
2.467 2.486 2.650 4.178 8.861 11.99 17.04 22.8 1 27.60 31.79
r = 2%.
vibration mode shapes. For values of r = 250, 500, 750 and 1000 the stability mode shapes are same and are different from the vibration mode shape. A transition in the stability mode shape is taking place, and is seen for values of I = 50 and 100. Figure 2 gives the instability regions (dynamic stability curves) plotted in terms of the nondimensional parameters p [defined as p =/l/2(1 -a)] and 0/o,, where o, is the fundamental lateral frequency in rad/sec. This figure indicates that there is strong relationship between the dynamic stability curves and the mode shapes of stability and vibration. For r = 0. 0.1, 1 and 10 the dynamic stability curve is the same, and for r = 250, 500. 750 and 1000 we obtain a different dynamic stability curve as in these cases the mode shapes of stability and vibration are not similar. For r = 50 and 100 two different dynamic stability curves are obtained, because the stability mode shape is undergoing a transition between one fixed mode shape to another. CONCLUDING
REMARKS
Stability, vibration mode shapes and curves for instability boundaries are presented for a cantilever column resting on
MO. 750.1000
r=o-1000
Fig. 1. Stability and vibration mode shapes of a cantilever column resting on an elastic foundation. an elastic foundation as a function of foundation modulus. It is Seen that the stability mode shapes change from one fixed mode shape with lower foundation moduli to another fixed mode shape with higher foundation moduli rather gradually, while the vibration mode shapes remain the same. Instability boundaries for these columns are strongly dependent on the stability mode shapes. They are to be represented as separate dynamic stability curves for low foundation moduli, high foundation moduli and transition curves for the in-between range of foundation moduli.
a=05 05
09
I I
17
I3
I9
21
Eq
Fig. 2. Dynamic stability curves of a cantilever column resting on an elastic foundation.
REFERENCES
1. V. V. Bolotin, Dynamic Stability of Elastic Systems. Holden-Day, San Fransisco (1964). 2. J. E. Brown, J. M. Hutt and A. E. Salama, Finite element solution to dynamic stability of bars. AIAA Jnl 6, 1423-1425 (1968). 3. B. P. Shastry and G. Venkateswara Rao, Dynamic
stability of bars considering shear deformation and rotatory inertia. Compur. Srruct. 19, 823-827 (1984). 4. B. A. H. Abbas and J. Thomas, Dynamic stability of Timoshenko beams resting on an elastic foundation. J. Sound Vibr. 60, 33-44 (1978). 5. S. P. Timoshenko and J. M. Gere, Theory of Elastic Stability. McGraw-Hill, New York (1961). 6. 0. C. Zienkiewicz, Finiie Element Method in Engineering Science. McGraw-Hill, New York (1971).
d smcrures in Gmr Britain.
Vol.
2s.
ho.
I, pp.
151-158.
0045.7949,87 53.00 + 0.w Pergamon Joumnlr Ltd.
1987
TECHNICAL NOTE DYNAMIC
STABILITY OF CANTILEVER COLUMNS ON AN ELASTIC FOUNDATION
RESTING
B. P. SHASTRY and G. VE%ATF.SWARA RACJ Department of Space, Vikram Sarabhai Space Centre, Trivandrum 695022, India (Received 4 Murch 1986)
Abstract-Instability boundaries for cantilever columns resting on an elastic foundation are determined using finite element method for various foundation modulii. The study reveals that dynamic stability boundaries for such columns can be represented in terms of three separate dynamic stability curves corresponding to low foundation moduli, high foundation moduli and the transition regions of foundation
INTRODUCTION
The periodic load acting on the cohrmn is represented by
The dynamic stability of structural elements is well discussed in [I]. In [2] and [3] the dynamic stability of slender and short columns, respectively, subjected to a periodic concentrated load at the free end, are investigated using the finite element method. The effect of elastic foundation on the dynamic stability of columns has been investigated [2,4]. It has been shown that in the case of hinged columns [5] the stability mode shape abruptly changes when the foundation modulus exceeds a critical value. However, in the case of cantilever columns the stabifity mode shape remains constant for lower foundation moduli and higher foundation moduli even though the two mode shapes are different and show a gradual change when the foundation modulus changes from lower values to higher values. However, the vibration mode shape remains constant whatever may be the value of foundation modulus. It has been shown [2] that the dynamic stability behaviour of columns has a definite dependency on the stability and vibration mode shapes, but this aspect is not studied in detail; even though the effect of elastic foundation on the dynamic stability behaviour of cantilever columns is shown in[4], the effect of dependency of stability and vibration mode shape is not discussed. The aim of the present study is to bring out in detail the dynamic stability behaviour of cantilever columns resting on elastic foundation over a wide range of foundation moduli. Dynamic stability curves are obtained for various values of the foundation moduli, showing clearly the interaction of stability and vibration mode shapes. FINITE ELEMENT
FORMULATION
Following [I] and [2], with a concentrated periodic load acting on the column at the free end, the instability boundaries are obtained, as a first appro~mation, by the solution of the equation
where [K], [C] and [M] are the assembled elastic stiffness matrix, geometric stiffness matrix and mass matrix respectively. These matrices are obtained using standard proceduresf61. It is to be noted here that the matrix [KJ contains a cont~bution from the stiffness of the elastic foundation also.
P(r) = p*+ P,sb(i),
(2)
where (3) with #(t) = cos er.
(4)
In eqn (3). P, is the critical load and in eqn (4) 9 is the axial frequency in rad/sec. The solution of eqn (1) provides (a) the static
j=e=o;
buckling
load of the column
when
(b) the lateral natural frequency w in rad/sec when a=/?=0
and
o=-;
8 2
(c) regions of unstable solutions with u and b varying.
NUMERICAL The
RESULTS
fundamental lateral frequency parameter ;., (defined as, .+- mo*L’/EI, where m is the mass per unit length, L is the length of the cohunn, E is the Young’s modulus and I is the moment of inertia) and the stability parameter Lb (defined as, 1, = P,L*/EQ and the associated mode shapes and the instability regions when the column is subjected to a periodic axial load for a = 0.5 are obtained for various values of the foundation parameter r (defined as r = KL’IEI, where K is the foundation modulus) for a cantilever column resting on an elastic foundation. An eight-element solution is used, based on convergence study, to obtain the numerical results. Table I gives the fundamental frequency parameter and the stability parameter 5 for several values of r. It is seen from Table 1 that as r increases both &and A, increase. The increase in I, for a given r is equal to the sum of the values of A, for I = 0 and the corresponding r. A similar trend is not seen for 2,. In Fig. 1, the mode shapes of stability and vibration are given. For values of r =O, 0.1, 1 and 10 the stability mode shape is the same and similar to that of the
157
158
Technical Note Table 1. Fundamental frequency parameter j./ and stability parameter i., of a cantilever column resting on an elastic foundation I
i.,
i.,
0.0 0.1 1.0 10.0 50.0 100.0 250.0 500.0 750.0 1000.0
12.36 12.46 13.36 22.36 62.36 112.4 262.4 512.4 762.4 1012
2.467 2.486 2.650 4.178 8.861 11.99 17.04 22.8 1 27.60 31.79
r = 2%.
vibration mode shapes. For values of r = 250, 500, 750 and 1000 the stability mode shapes are same and are different from the vibration mode shape. A transition in the stability mode shape is taking place, and is seen for values of I = 50 and 100. Figure 2 gives the instability regions (dynamic stability curves) plotted in terms of the nondimensional parameters p [defined as p =/l/2(1 -a)] and 0/o,, where o, is the fundamental lateral frequency in rad/sec. This figure indicates that there is strong relationship between the dynamic stability curves and the mode shapes of stability and vibration. For r = 0. 0.1, 1 and 10 the dynamic stability curve is the same, and for r = 250, 500. 750 and 1000 we obtain a different dynamic stability curve as in these cases the mode shapes of stability and vibration are not similar. For r = 50 and 100 two different dynamic stability curves are obtained, because the stability mode shape is undergoing a transition between one fixed mode shape to another. CONCLUDING
REMARKS
Stability, vibration mode shapes and curves for instability boundaries are presented for a cantilever column resting on
MO. 750.1000
r=o-1000
Fig. 1. Stability and vibration mode shapes of a cantilever column resting on an elastic foundation. an elastic foundation as a function of foundation modulus. It is Seen that the stability mode shapes change from one fixed mode shape with lower foundation moduli to another fixed mode shape with higher foundation moduli rather gradually, while the vibration mode shapes remain the same. Instability boundaries for these columns are strongly dependent on the stability mode shapes. They are to be represented as separate dynamic stability curves for low foundation moduli, high foundation moduli and transition curves for the in-between range of foundation moduli.
a=05 05
09
I I
17
I3
I9
21
Eq
Fig. 2. Dynamic stability curves of a cantilever column resting on an elastic foundation.
REFERENCES
1. V. V. Bolotin, Dynamic Stability of Elastic Systems. Holden-Day, San Fransisco (1964). 2. J. E. Brown, J. M. Hutt and A. E. Salama, Finite element solution to dynamic stability of bars. AIAA Jnl 6, 1423-1425 (1968). 3. B. P. Shastry and G. Venkateswara Rao, Dynamic
stability of bars considering shear deformation and rotatory inertia. Compur. Srruct. 19, 823-827 (1984). 4. B. A. H. Abbas and J. Thomas, Dynamic stability of Timoshenko beams resting on an elastic foundation. J. Sound Vibr. 60, 33-44 (1978). 5. S. P. Timoshenko and J. M. Gere, Theory of Elastic Stability. McGraw-Hill, New York (1961). 6. 0. C. Zienkiewicz, Finiie Element Method in Engineering Science. McGraw-Hill, New York (1971).