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Free vibration of plates on Vlasov's foundation
Journal of Sound and Vibration (1977) 54(3), 464-467
FREE VIBRATION
OF PLATES
ON VLASOV’S FOUNDATION
A brief discussion is presented of the free vibration of(i) isosceles right-angled triangular and (ii) equilateral triangular plates, with simply supported edges and resting on a single layered elastic half-space model of Vlasov type. Some numerical results are discussed. 1. INTRODUCTION
The interaction between a structure and its soil foundation is actually a complicated phenomenon, which has led to assumptions of various foundation models. In the Winkler model, the soil is virtually replaced by a series of spring elements having no shear interactions between them. This short-coming has been eliminated by Pasternase and Vlasov, using various assumptions. The free vibration of thin elastic plates within the compass of linear theory has been well studied. But information on the problem under discussion here is not available in the literature. The author thus has made a simple attempt to correlate the natural frequency with nondimensional foundation stiffness parameters. 2. VLASOV’S MODEL Vlasov [l] assumed that the elastic foundation is a compressible layer of thickness H, and does not undergo horizontal movements under the surface load. The vertical displacement w(x, y, z) is assumed to be of the form w(x, y, z) = w(x, u) ICI(z), where $(z) is a function, previously selected according to the nature of the problem, describing the transverse distribution of the displacement and w(x, y) is to be determined from equilibrium considerations. For the case of an elastic half-space, the thickness H tends to infinity. 3. EQUATIONS OF MOTION The equation of motion, in Cartesian model of an elastic half-space is [l]
co-ordinates,
V2 V2 w - 2r2 V2 w
+
s4 w
=
for a plate resting
on a Vlasov-type
-m* a2 wjat=,
(1)
where EO
.y4 =
D(1 - v;)
m* =i[
Here, terial E. = ratio,
E, H r2 ( rC/(z))=dz, - 40(1 + ve) I
H IV
(z))‘dz,
I 0
0
T+F[{i(z)j’dz].
(2)
D is the flexural rigidity of the plate, y and ye are the specific weights of the plate maand soil, respectively, h is the plate thickness, g is the gravitational acceleration, and Es/(1 - vs) and v. = v,/( 1 - v,), where Es and v, are the Young’s modulus and Poisson’s respectively, of the soil material. 464
465
LETTERS TO THE EDITOR
4. ANALYSIS 4.1. Case 1: Simply-supported isosceles right-angledplate (see Figure
1)
Figure 1. Frequency curve for Case 1.
Consider a simply-supported isosceles right-angled triangular plate of equal side lengths a. The axes OX and 0 Y are taken along the equal sides. The boundary conditions are given by
~=a~wp~~=of0r.~=o,
w =
~=a~w/ap~=of0rx+y=a; Consistent with these boundary assumed to be [2]
a2wjay2= 0 for alap = (l/2/2)
conditions,
w(x, y, t) = sinpt n=l: . .
the displacement
,A,(sin(2Ax)
y =
0,
(a/ax + a/+). w(x, y, t) may be
function
sin(Ay) + sir&Ix) sin(2Ay)},
where p is the natural frequency of the plate vibrations and I. = nnja. By using the solution (3) in equation (I), one obtains an expression frequency p in the following form:
for determining
p2 = ( 1/m*)(25A4 + 10r2 A2 + s”). The fundamental
frequency
(3)
the
(4)
is given for n = 1.
of vibration
4.2. Case 2: Simply-supported equilateral triangular plate (see Figure
2)
Consider a simply supported equilateral triangular plate with its centroid as the origin of axes. By using the tri-linear co-ordinate system of Sen [3], the Laplacian operator may be transformed to
a*
~Z~‘,s’,‘______ ap:
ap:
wherep,,p, andp, are perpendiculars sides and are given by p1 = r. + 42 - ~~912,
aps
ap, aP2
a2 ap, ap,
a2 ap, apI’
(5)
drawn from any point p(x, y) inside the triangle to the p2 = r. + xl2 + ~412,
p3 = r. - 4
466
LETTERSTO THEEDITOR
20
40
60
80
100
02
Figure 2. Frequencycurve for Case2. such that p1 +p2 +p3 = 3r, = ufi = K (say). Here r. is the radius of the inscribed circle. The boundary conditions in the new co-ordinate system are w = V2w = 0, forp, = p2 = p3 = 0. For the solution of equation (l), one may assume w(x, y, 2) = sinpt 5 A, (sintp, + sint;p, + sintp3), It=1
with 5 = 2nnJK, which is compatible with the boundary conditions. Substituting the expression (5) in the basic equation (l), one obtains after some algebraic operations, the following expression for determining the natural frequency of the plate : p2 = (l/m*)(t4
+ 29 <’ + s”).
(6)
The fundamental frequency of vibration is given for n = 1. 5. DISCUSSION OFRESULTS Define the non-dimensional quantities D1 = 2a2 r2/n2,
D2 = a4 s4/7c4,
0.1~= m*p2 a4/n4,
where D, is the non-dimensional parameter characterizing shear strains in the soil, D2 is the non-dimensional parameter characterizing compression strains in the soil and, o2 is the nondimensional parameter characterizing the plate frequency. Expressions (4) and (6) can then be written in the general form co2=aon4+
D,a,n2+
D2,
(7)
The curves in Figures 1 and 2 are drawn for w with different variations of D, and D2, for n= 1. It is obvious that the stiffness parameter D, has marked influence on the fundamental frequency. This is very prominent in Case 1 (see Figure 1). It is also clear that as the value of D1 increases, the curve for o tends more and more towards the horizontal. This means that for a foundation having very stiff shear springs the compression springs have no effect on the fundamental frequencies of the two plates considered.
LETTERSTOTHEEDITOR
467
6. CONCLUSIONS Expressions (4) and (6) can be used to obtain natural frequencies of isoceles right-angled triangular plates and equilateral triangular plates, respectively, having simply supported edges, and resting on an elastic half-space of Vlasov type. Structural Engineering Research (Regional) Centre, Adyar, Madras 600020, India
B. BHATTACHARYA
(Received 3 1 May 1977) REFERENCES 1. V. A. VLASOV 1966 IsraelProgramfor
Scientific Translations, Jerusalem. Beams, plates and shells on elastic foundations. 2. M. M. BANERJEE 1975Journalofthe Institute ofEngineers (India) 56,175-179. Thermal buckling of heated plates. 3. B. SEN 1968 Bulletin of the Calcutta Mathematical Society 60, 31-35. Trilinear co-ordinates and boundary value problems.
FREE VIBRATION
OF PLATES
ON VLASOV’S FOUNDATION
A brief discussion is presented of the free vibration of(i) isosceles right-angled triangular and (ii) equilateral triangular plates, with simply supported edges and resting on a single layered elastic half-space model of Vlasov type. Some numerical results are discussed. 1. INTRODUCTION
The interaction between a structure and its soil foundation is actually a complicated phenomenon, which has led to assumptions of various foundation models. In the Winkler model, the soil is virtually replaced by a series of spring elements having no shear interactions between them. This short-coming has been eliminated by Pasternase and Vlasov, using various assumptions. The free vibration of thin elastic plates within the compass of linear theory has been well studied. But information on the problem under discussion here is not available in the literature. The author thus has made a simple attempt to correlate the natural frequency with nondimensional foundation stiffness parameters. 2. VLASOV’S MODEL Vlasov [l] assumed that the elastic foundation is a compressible layer of thickness H, and does not undergo horizontal movements under the surface load. The vertical displacement w(x, y, z) is assumed to be of the form w(x, y, z) = w(x, u) ICI(z), where $(z) is a function, previously selected according to the nature of the problem, describing the transverse distribution of the displacement and w(x, y) is to be determined from equilibrium considerations. For the case of an elastic half-space, the thickness H tends to infinity. 3. EQUATIONS OF MOTION The equation of motion, in Cartesian model of an elastic half-space is [l]
co-ordinates,
V2 V2 w - 2r2 V2 w
+
s4 w
=
for a plate resting
on a Vlasov-type
-m* a2 wjat=,
(1)
where EO
.y4 =
D(1 - v;)
m* =i[
Here, terial E. = ratio,
E, H r2 ( rC/(z))=dz, - 40(1 + ve) I
H IV
(z))‘dz,
I 0
0
T+F[{i(z)j’dz].
(2)
D is the flexural rigidity of the plate, y and ye are the specific weights of the plate maand soil, respectively, h is the plate thickness, g is the gravitational acceleration, and Es/(1 - vs) and v. = v,/( 1 - v,), where Es and v, are the Young’s modulus and Poisson’s respectively, of the soil material. 464
465
LETTERS TO THE EDITOR
4. ANALYSIS 4.1. Case 1: Simply-supported isosceles right-angledplate (see Figure
1)
Figure 1. Frequency curve for Case 1.
Consider a simply-supported isosceles right-angled triangular plate of equal side lengths a. The axes OX and 0 Y are taken along the equal sides. The boundary conditions are given by
~=a~wp~~=of0r.~=o,
w =
~=a~w/ap~=of0rx+y=a; Consistent with these boundary assumed to be [2]
a2wjay2= 0 for alap = (l/2/2)
conditions,
w(x, y, t) = sinpt n=l: . .
the displacement
,A,(sin(2Ax)
y =
0,
(a/ax + a/+). w(x, y, t) may be
function
sin(Ay) + sir&Ix) sin(2Ay)},
where p is the natural frequency of the plate vibrations and I. = nnja. By using the solution (3) in equation (I), one obtains an expression frequency p in the following form:
for determining
p2 = ( 1/m*)(25A4 + 10r2 A2 + s”). The fundamental
frequency
(3)
the
(4)
is given for n = 1.
of vibration
4.2. Case 2: Simply-supported equilateral triangular plate (see Figure
2)
Consider a simply supported equilateral triangular plate with its centroid as the origin of axes. By using the tri-linear co-ordinate system of Sen [3], the Laplacian operator may be transformed to
a*
~Z~‘,s’,‘______ ap:
ap:
wherep,,p, andp, are perpendiculars sides and are given by p1 = r. + 42 - ~~912,
aps
ap, aP2
a2 ap, ap,
a2 ap, apI’
(5)
drawn from any point p(x, y) inside the triangle to the p2 = r. + xl2 + ~412,
p3 = r. - 4
466
LETTERSTO THEEDITOR
20
40
60
80
100
02
Figure 2. Frequencycurve for Case2. such that p1 +p2 +p3 = 3r, = ufi = K (say). Here r. is the radius of the inscribed circle. The boundary conditions in the new co-ordinate system are w = V2w = 0, forp, = p2 = p3 = 0. For the solution of equation (l), one may assume w(x, y, 2) = sinpt 5 A, (sintp, + sint;p, + sintp3), It=1
with 5 = 2nnJK, which is compatible with the boundary conditions. Substituting the expression (5) in the basic equation (l), one obtains after some algebraic operations, the following expression for determining the natural frequency of the plate : p2 = (l/m*)(t4
+ 29 <’ + s”).
(6)
The fundamental frequency of vibration is given for n = 1. 5. DISCUSSION OFRESULTS Define the non-dimensional quantities D1 = 2a2 r2/n2,
D2 = a4 s4/7c4,
0.1~= m*p2 a4/n4,
where D, is the non-dimensional parameter characterizing shear strains in the soil, D2 is the non-dimensional parameter characterizing compression strains in the soil and, o2 is the nondimensional parameter characterizing the plate frequency. Expressions (4) and (6) can then be written in the general form co2=aon4+
D,a,n2+
D2,
(7)
The curves in Figures 1 and 2 are drawn for w with different variations of D, and D2, for n= 1. It is obvious that the stiffness parameter D, has marked influence on the fundamental frequency. This is very prominent in Case 1 (see Figure 1). It is also clear that as the value of D1 increases, the curve for o tends more and more towards the horizontal. This means that for a foundation having very stiff shear springs the compression springs have no effect on the fundamental frequencies of the two plates considered.
LETTERSTOTHEEDITOR
467
6. CONCLUSIONS Expressions (4) and (6) can be used to obtain natural frequencies of isoceles right-angled triangular plates and equilateral triangular plates, respectively, having simply supported edges, and resting on an elastic half-space of Vlasov type. Structural Engineering Research (Regional) Centre, Adyar, Madras 600020, India
B. BHATTACHARYA
(Received 3 1 May 1977) REFERENCES 1. V. A. VLASOV 1966 IsraelProgramfor
Scientific Translations, Jerusalem. Beams, plates and shells on elastic foundations. 2. M. M. BANERJEE 1975Journalofthe Institute ofEngineers (India) 56,175-179. Thermal buckling of heated plates. 3. B. SEN 1968 Bulletin of the Calcutta Mathematical Society 60, 31-35. Trilinear co-ordinates and boundary value problems.