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Dynamic impact of an elastically supported beam—large area contact
0020-7225/85 $3.04 + $6 0 1985 Pewmon Press Ltd.
ht. J. Engns Sci. Vol. 23, No. 10, pi. 987-997, 1985 Printed in Great Britain
DYNAMIC SUPPORTED
IMPACT OF AN ELASTICALLY BEAM-LARGE AREA CONTACT
L. M. KERR and J. C. LEE Department of Civil Engineering, Northwestern University, Evanston, IL 60201, U.S.A. Abstract-The dynamic contact problem of a rigid, smooth striker impacting an elastically supported beam is solved. Use is made of the su~~sition of an elastic layer solution together with an elementary beam theory solution that incorporates the dynamic effects. The problem is formulated in such a manner as to require the solution of a Volterra integral equation for each time increment. Numerical results are presented for various parameters. INTRODUCTION
IN THIS ARTICLE the dynamic impact of a rigid, smooth striker on an elastically supported
beam is studied. The usual studies of such problems make use of the assumptions of Hertz [I], which state that the contact deformation is circular and can be approximated by a parabola having the same curvature as the circle at the origin. By knowing the relationship between the force and deflection and by assuming that wave propagation effects can be neglected, the problem of dynamic impact of two spheres can be solved. The problem of the contact between a sphere and a beam can be solved in a similar manner (see, e.g. Goldsmith [2]), if the local contact is assumed to be Hertzian and the dynamic beam response is taken into consideration. It has been shown by Keer and Miller [3] that for the problem of the indentation of a beam by a cylinder whose radius of curvature is very large, the stress dist~bution in the region of contact will differ from a Hertzian one when the contact length exceeds the thickness of the beam. For this case point contact can no longer be assumed and the Hertzian relations must be corrected. If‘ this is true for the case of static indentation, then it must certainly be true for the dynamic case. In the present investigation we consider the problem in which an indenter with a very large radius of curvature and an initial velocity makes a dynamic impact with an elastically supported beam. For this case the contact area as well as the applied load are unknown and must be forthcoming as a part of the solution. In order to solve the problem two types of solutions are used, First, an elasticity solution for a layer is formulated in order to obtain a load deflection relationship. This solution will be a static solution similar to the one developed by Keer and Miller. It will be included as part of the resistance to deformation of the dynamic beam theory solution (see, e.g. Graff [4]). Using both of these solutions to match the boundary conditions will lead to an integral equation of the Volterra type in terms of the unknown pressure distribution and contact length as functions of time. These quantities will be solved for at each instant of time using a technique developed by Ahmadi, Keer and Mura [5] for the solution of nonHertzian contact problems. Once the contact stress is obtained, then all field quantities internal to the beam can be calculated. FORMULATION
The formulation of the dynamic contact problem will be based on the assumption that local effects can be well approximated by the elastostatic solution for the layer. This assumption implies that the elastodynamic effects occur during such a short interval that their effect is negligible compared with the beam effects and with the effects due to the static deformation of the layer. Thus, two types of solutions are required: First, there is the static layer solution (see, e.g. Keer and Miller [3]), in which the static indentation of a layer of finite length was solved by a global local technique, and second, there is a dynamic solution, which can be obtained from a standard beam analysis (see, e.g. Graff [4]). ESzl:,a* 987
988
L. M. MEER
and
J. 6.
LEE
Static /aye7sohOion The boundary conditions for this portion are for an elastically supported layer of finite thickness h loaded by a smooth symmetric rigid punch (Fig. 4) and can be written as follows: Tyy(X,h) = 0,
1x1-==C 00,
-s_&, iz) = 0,
1x1< co,
62) ‘? {La >
7&,
1x1< a,
(3
7yy(x, 0) = 8,
c < 1x1< co,
(4)
Ty&G 0) = -P(X),
0 < 1x1-=zc,
(5)
0) = 0,
where c is the half length of the contact region. At the ends of the beam the static beam theory boundary conditions will be applied as follows: U, = 0, M-KG=O,
x = 1,
(6)
x = k,
(7)
where I is the half length of the beam. M is the bending moment, K is a spring constant and 8 is the value of the slope averaged through the thickness
The boundary conditions (l)-(7) represent a beam that is loaded over a patch 5r half length c and whose ends are constrained. An elasticity solution that represents loading on the upper surface of an elastic- layer and no loading on its lower surface is given in the following representation by using techniques of Sneddon [6] as
Fig.
1. M is the ratio of the mass of the beam to the mass of the striker
Dynamic impact of an elastically supported beam
989
+ Ph*P + P2 - tyC’y!p + ~h~ch~)lch(~y}~ cos C&W&
(131
where A = p ’ - sh2p3 ,L3= $h and CL,v are the shear modulus and Poisson’s ratio, respectively. It is seen that on y = h the normal and shear stresses automatically vanish and that on y = 0 the shear stress vanishes. The normal stress on y = 0 is given as
7yy(x,0) =
s4) 0
fX3
~0s Wdk
(14)
By substituting rY,,into eqns (4) and (5) and applying a Fourier cosine transform, E(6) is obtained as E(l) = - z &(x)
cos (#$x&r.
(15)
By taking the beam theory solution in the form of a quadratic displacement as in Keer and Miller [3], u, = $AX’ + B
(16)
the bending moment and slope can be written as
s h
MB=
o ~xuy dy = --DA,
D = Eh3/i2(l
- v2),
BB = Ax.
(17) (18)
To determine the constant A we first derive expressions for the moment and average slope due to the Iayer solution. These can be obtained by direct intention by using eqn (13) and partiaf derivative with respect to x of (10). The results are
s h
ML = o 7xxlx=1ydy = - oa F2E(4) ~0s C$ddlr, s
(1%
By using eqns (7), (17), (18), (19) and (20), A is determined to be
(21) where 01 = N/D is a dimensionless parameter determining the support stiffness.
CONTACT LENGTH,
MM
Dynamic impact of an elastically supported beam
991
Fig. 3. Pressure distribution for simply supported beam: M = 1, V = 20 m/s.
11
p - (3 - 2v)shp sin (41) h2a -~ l2 - x2 cos (41) 1+0! ---T-E + 12(1 - V)l P - sh@ t
C
d<.
(23)
As 5 - 0, an asymptotic analysis shows that the integrand in the curly bracket is bounded. As ,$ - co, however, the leading term in the curly bracket does not decay. By adding and subtracting the expression
this difficulty can be removed.
Fig. 4. Pressure distribution for simply supported beam: M = 1 .$, V = 20 m/s.
992
L. WI. KEER and J. C. LEE
Equation (24) can be further simplified to the form: h3
--
s
Bx, _ _i_ l2 - x2 h2a(3 - 2~) 2D 1 + CY 12(8 - Y)l
C
&X’)dK’.
(25)
C - &Y(3 - 2U) l2 - x2 c p(x’)dX + -!pi(X’)K(X:X’)dX’, 24(1-v)D/ I+cx s o ?i-Ds 0
(26)
67rD 0
Thus, eqn (23) becomes the following: -c
u,(x, 0) = - -gj
lx + x’j Ix - X’I ~’ x’l 11- x’l
Ja(-x’)log 11+ o
where
Dynamic beam theory solution For this section we consider the dynamic response to an impulsive ioaded beam using the elementary beam theory. Assuming that the pressure distribution is symmetric with respect to x = 0, only the right half of the beam is considered: i.e. x > 0. By using the Laplace transform and a normal mode expansion (see, e.g. Graff [4]), we find for the case of zero initial conditions that the deflection is given by &~-.-Y&) ph n=l aPf
6’ Yn(u)du
s’p(zl, T) sin afii(t
- 7)d7,
(28)
0
JO
where a = (D/ph)‘12 and p is the mass density. The natural frequency 8, and normal mode Y,(x) satisfy the following equations: Y’,4’(x)- P4Y n n(x) = 0,
O
(29)
Y;(o) = Y;(o) = 0,
(30)
Yn(l) = 0,
(31)
M-K0=0,
x = 1.
Fig. 5. Pressure distribution for simply supported beam: M = 2, V = 20 m/s.
(32)
Dynamic impact of an elastically supported beam
993
250 MPA
Fig. 6. Pressure distribution for simply supported beam: M = 2.25, V = 20 m/s.
The solution for eqn (29) that satisfies the boundary conditions is (331 and the frequency equation becomes (34) The normal modes are then (35)
(36) Fn = {I/2 + sin (2&1)/4@, - 2G,lch(&l) cos (P,E) -I- ti2Gf[i/2 + sJ~(2/3,,1)/4&,]}-‘~~.
Fig. 7. Pressure distribution for simply supported beam: A4 = 1, V = 15 m/s.
L. M. KEER and .I. C. LEE
994 INTEGRAL
EQUATION
FOR
DYNAMIC
From eqns (26) and (28), the total displacement theory solutions is given by
Fig. 8. Simply supported
PROBLEM
from the layer and from the beam
beam subjected to transverse impact: M = I. (a) Force history. (b) Contact area.
Dynamic impact of an elaskally supported beam In the
995
contact region we have z&(.x,0, t) = S(t) - 2,
0 < x < C(l),
139)
where S(t) is the relative approach (a function of time as is the contact region) and R is the radius of curvature of the striker. An additional equation is obtained from Newton’s equation of motion which can be written as
where ( * ) denotes di~erentia~o~ with respect to time and m is the mass of the striker per unit length. Provided that the striker has an initial velocity I’, eqn (40) can be expressed as
where M is the ratio of the mass of the beam to that af the striker,
of
In solving eqns (39) and (41), first the orders the integration and summation of the dynamic parts of eqn (39) are interchanged. For the integration procedure the time domain is poisoned into equally spaced elements. If the number of intervals is even, Filon’s formula is used. We note that ‘in the calculation a ~ornbi~a~o~ of Fil~n’s formula together with a procedure that is similar but where the unknown function is approximated by a third degree polynomial is employed, For each time step the contact area is overestimate and divided into N small patches and a procedure as in Ahmadi et al. IS] is used, where for each patch the pressure is assumed to be constant. We also note that the Volterra integral equation is sensitive to the starting v&e, and in order to reduce the error, the starting value is modified as in Baker [7]. The numerical results are given in Figs. 2-10. For all numerical results the following geometrical and material parameters are used: R = 40 cm, E = 10 cm, 1E1 = 1 cm; E = 206.85 GPa, EJ= 0.29, p = 7833.5 Kg/m’.
996
I_. M. KEER and J. C. LEE
TIME, SECOND (xlo-ag Fig. 10. Clamped beam subjected to transverse impact: M = 1, V = 20 m/s. (a) Force history. (b) Contact area.
Figure 2 shows the contact force and contact area, where the beam is taken to be simply supported and the striker has an impact velocity of 20 m/s. For these cases the mass ratio A4 is varied from 1.0-2.25. With these parameters it is seen that the time history for the force has two peaks. This indicates that after impact the beam is sufficiently flexible that after the first peak the contact will diminish before the beam achieves its maximum deflection. As the maximum deflection is reached, the beam and the striker may have opposite velocities. When the beam releases its stored energy, after achieving maximum deflection, it impacts the striker and a new force maximum is achieved. For the largest value of mass ratio, A4 = 2.25, the beam actually loses contact with the striker and a second impact is noted. The multiple impact which has occurred is different from that shown in Goldsmith [2], where both the spherical striker and the beam (plane stress) are elastic and a second impact occurs in the progressing direction. Furthermore, asM increases, the maximum force transfers from the second peak to the first peak. -The
Dynamic impact of an elastically supported beam
997
maximum half contact lengths, shown in Fig. 2(b), are less than one-fourth of the thickness of the beam, except for the case when M = 1, where c/h > 1. Comparing Fig. 2(b) and Figs. 3-5, which show the pressure distributions corresponding to various values of &I, we note that Fig. 3 shows a distribution that is not Hertzian, while Figs. 4-6 are approximately Hertzian. Figure 3 indicates a situation noted by Keer and Miller [3] in which a contact area greater than the thickness of the beam will tend to produce a nonHertzian contact stress distribution. The appearance of the stress distribution for that case will show peaks nearer to the edge of the contact, while the stresses near the center will be lessened. Note that the non-Hertzian appearance will tend to coincide with a diminished force (Fig. 2). Figures 4-6 show the stress distributions corresponding to the larger mass ratios and Fig. 6 shows the stress distributions corresponding to a loss of contact. Figure 7 shows the pressure distribution which corresponds to M = 1 and I/ = 15 m/s. The force history and contact area are compared with that for V = 20 m/s in Fig. 8. The force histories are similar [Fig. 8(a)], but the contact area becomes much smaller with the decreased impact velocity [Fig. 8(b)]. Figure 9 shows the pressure distribution for a clamped beam impacted by a striker with a mass ratio of M = 1. As in the previous case (Fig. 3) the pressure distribution is non-Hertzian during a portion of the impact time that the maximum half contact length is larger than the thickness of the layer. Figure 10 represents the force history and half contact length for a clamped beam with M = 1 and V = 20 m/s. Since the beam is clamped and hence stiffer, the maximum contact force is much greater than the simply supported case, but the period of impact is much less than the latter. CONCLUSIONS
The problem of the impact of a cylindrical striking an elastically supported beam is solved by combining an elasticity and beam theory solution that allows consideration of the global (beam theory) and local (elasticity) stresses. The impact response appeared to be highly dependent on the geometrical and material parameters. For most of the cases studied the force history for the impact showed two peaks. In the more extreme case a double impact was noted. During the initial stage of the impact the stress distribution was Hertzian, but for relatively small mass ratios a non-Hertzian distribution was observed. During the .non-Hertzian portion of the stress history, the peak stresses showed a diminished magnitude and the load was carried over a larger area. Acknowledgement-Theauthorsare gratefulfor support
from the Air Force Office of Scientific Research under
Grant AFOSR-82-0330. REFERENCES [II S. TIMOSHENKO and J. N. GOODIER, Theory of Elasticity, 3rd Edition. McGraw-Hill, New York (1970). PI W. GOLDSMITH, Impact, The Theory and Physical Behavior of Colliding Solids. Edward Arnold Ltd., London ( 1960). [31 L. M. KEER and G. R. MILLER, Smooth Indentation of a Finite Layer, J. Engng Mech. Div., ASCE, 109, 706 (1983). K. F. GRAFF, Wave Motion in Elastic Solids. Ohio State University Press, Columbus (1975). t:; N. AHMADI, L. M. KEER and T. MURA, Non-He&an Stress Analysis-Normal and Sliding Contact, Int. J. Solids Struct. 19, 357 (1983). WI I. N. SNEDDON, Fourier Transforms. McGraw-Hill, New York (I 95 1). [71 C. T. H. BAKER, The Numerical Treatment of Integral Equations. p. 842. Clarendon Press, Oxford (1977). (Received in production 9 October 1984)
ht. J. Engns Sci. Vol. 23, No. 10, pi. 987-997, 1985 Printed in Great Britain
DYNAMIC SUPPORTED
IMPACT OF AN ELASTICALLY BEAM-LARGE AREA CONTACT
L. M. KERR and J. C. LEE Department of Civil Engineering, Northwestern University, Evanston, IL 60201, U.S.A. Abstract-The dynamic contact problem of a rigid, smooth striker impacting an elastically supported beam is solved. Use is made of the su~~sition of an elastic layer solution together with an elementary beam theory solution that incorporates the dynamic effects. The problem is formulated in such a manner as to require the solution of a Volterra integral equation for each time increment. Numerical results are presented for various parameters. INTRODUCTION
IN THIS ARTICLE the dynamic impact of a rigid, smooth striker on an elastically supported
beam is studied. The usual studies of such problems make use of the assumptions of Hertz [I], which state that the contact deformation is circular and can be approximated by a parabola having the same curvature as the circle at the origin. By knowing the relationship between the force and deflection and by assuming that wave propagation effects can be neglected, the problem of dynamic impact of two spheres can be solved. The problem of the contact between a sphere and a beam can be solved in a similar manner (see, e.g. Goldsmith [2]), if the local contact is assumed to be Hertzian and the dynamic beam response is taken into consideration. It has been shown by Keer and Miller [3] that for the problem of the indentation of a beam by a cylinder whose radius of curvature is very large, the stress dist~bution in the region of contact will differ from a Hertzian one when the contact length exceeds the thickness of the beam. For this case point contact can no longer be assumed and the Hertzian relations must be corrected. If‘ this is true for the case of static indentation, then it must certainly be true for the dynamic case. In the present investigation we consider the problem in which an indenter with a very large radius of curvature and an initial velocity makes a dynamic impact with an elastically supported beam. For this case the contact area as well as the applied load are unknown and must be forthcoming as a part of the solution. In order to solve the problem two types of solutions are used, First, an elasticity solution for a layer is formulated in order to obtain a load deflection relationship. This solution will be a static solution similar to the one developed by Keer and Miller. It will be included as part of the resistance to deformation of the dynamic beam theory solution (see, e.g. Graff [4]). Using both of these solutions to match the boundary conditions will lead to an integral equation of the Volterra type in terms of the unknown pressure distribution and contact length as functions of time. These quantities will be solved for at each instant of time using a technique developed by Ahmadi, Keer and Mura [5] for the solution of nonHertzian contact problems. Once the contact stress is obtained, then all field quantities internal to the beam can be calculated. FORMULATION
The formulation of the dynamic contact problem will be based on the assumption that local effects can be well approximated by the elastostatic solution for the layer. This assumption implies that the elastodynamic effects occur during such a short interval that their effect is negligible compared with the beam effects and with the effects due to the static deformation of the layer. Thus, two types of solutions are required: First, there is the static layer solution (see, e.g. Keer and Miller [3]), in which the static indentation of a layer of finite length was solved by a global local technique, and second, there is a dynamic solution, which can be obtained from a standard beam analysis (see, e.g. Graff [4]). ESzl:,a* 987
988
L. M. MEER
and
J. 6.
LEE
Static /aye7sohOion The boundary conditions for this portion are for an elastically supported layer of finite thickness h loaded by a smooth symmetric rigid punch (Fig. 4) and can be written as follows: Tyy(X,h) = 0,
1x1-==C 00,
-s_&, iz) = 0,
1x1< co,
62) ‘? {La >
7&,
1x1< a,
(3
7yy(x, 0) = 8,
c < 1x1< co,
(4)
Ty&G 0) = -P(X),
0 < 1x1-=zc,
(5)
0) = 0,
where c is the half length of the contact region. At the ends of the beam the static beam theory boundary conditions will be applied as follows: U, = 0, M-KG=O,
x = 1,
(6)
x = k,
(7)
where I is the half length of the beam. M is the bending moment, K is a spring constant and 8 is the value of the slope averaged through the thickness
The boundary conditions (l)-(7) represent a beam that is loaded over a patch 5r half length c and whose ends are constrained. An elasticity solution that represents loading on the upper surface of an elastic- layer and no loading on its lower surface is given in the following representation by using techniques of Sneddon [6] as
Fig.
1. M is the ratio of the mass of the beam to the mass of the striker
Dynamic impact of an elastically supported beam
989
+ Ph*P + P2 - tyC’y!p + ~h~ch~)lch(~y}~ cos C&W&
(131
where A = p ’ - sh2p3 ,L3= $h and CL,v are the shear modulus and Poisson’s ratio, respectively. It is seen that on y = h the normal and shear stresses automatically vanish and that on y = 0 the shear stress vanishes. The normal stress on y = 0 is given as
7yy(x,0) =
s4) 0
fX3
~0s Wdk
(14)
By substituting rY,,into eqns (4) and (5) and applying a Fourier cosine transform, E(6) is obtained as E(l) = - z &(x)
cos (#$x&r.
(15)
By taking the beam theory solution in the form of a quadratic displacement as in Keer and Miller [3], u, = $AX’ + B
(16)
the bending moment and slope can be written as
s h
MB=
o ~xuy dy = --DA,
D = Eh3/i2(l
- v2),
BB = Ax.
(17) (18)
To determine the constant A we first derive expressions for the moment and average slope due to the Iayer solution. These can be obtained by direct intention by using eqn (13) and partiaf derivative with respect to x of (10). The results are
s h
ML = o 7xxlx=1ydy = - oa F2E(4) ~0s C$ddlr, s
(1%
By using eqns (7), (17), (18), (19) and (20), A is determined to be
(21) where 01 = N/D is a dimensionless parameter determining the support stiffness.
CONTACT LENGTH,
MM
Dynamic impact of an elastically supported beam
991
Fig. 3. Pressure distribution for simply supported beam: M = 1, V = 20 m/s.
11
p - (3 - 2v)shp sin (41) h2a -~ l2 - x2 cos (41) 1+0! ---T-E + 12(1 - V)l P - sh@ t
C
d<.
(23)
As 5 - 0, an asymptotic analysis shows that the integrand in the curly bracket is bounded. As ,$ - co, however, the leading term in the curly bracket does not decay. By adding and subtracting the expression
this difficulty can be removed.
Fig. 4. Pressure distribution for simply supported beam: M = 1 .$, V = 20 m/s.
992
L. WI. KEER and J. C. LEE
Equation (24) can be further simplified to the form: h3
--
s
Bx, _ _i_ l2 - x2 h2a(3 - 2~) 2D 1 + CY 12(8 - Y)l
C
&X’)dK’.
(25)
C - &Y(3 - 2U) l2 - x2 c p(x’)dX + -!pi(X’)K(X:X’)dX’, 24(1-v)D/ I+cx s o ?i-Ds 0
(26)
67rD 0
Thus, eqn (23) becomes the following: -c
u,(x, 0) = - -gj
lx + x’j Ix - X’I ~’ x’l 11- x’l
Ja(-x’)log 11+ o
where
Dynamic beam theory solution For this section we consider the dynamic response to an impulsive ioaded beam using the elementary beam theory. Assuming that the pressure distribution is symmetric with respect to x = 0, only the right half of the beam is considered: i.e. x > 0. By using the Laplace transform and a normal mode expansion (see, e.g. Graff [4]), we find for the case of zero initial conditions that the deflection is given by &~-.-Y&) ph n=l aPf
6’ Yn(u)du
s’p(zl, T) sin afii(t
- 7)d7,
(28)
0
JO
where a = (D/ph)‘12 and p is the mass density. The natural frequency 8, and normal mode Y,(x) satisfy the following equations: Y’,4’(x)- P4Y n n(x) = 0,
O
(29)
Y;(o) = Y;(o) = 0,
(30)
Yn(l) = 0,
(31)
M-K0=0,
x = 1.
Fig. 5. Pressure distribution for simply supported beam: M = 2, V = 20 m/s.
(32)
Dynamic impact of an elastically supported beam
993
250 MPA
Fig. 6. Pressure distribution for simply supported beam: M = 2.25, V = 20 m/s.
The solution for eqn (29) that satisfies the boundary conditions is (331 and the frequency equation becomes (34) The normal modes are then (35)
(36) Fn = {I/2 + sin (2&1)/4@, - 2G,lch(&l) cos (P,E) -I- ti2Gf[i/2 + sJ~(2/3,,1)/4&,]}-‘~~.
Fig. 7. Pressure distribution for simply supported beam: A4 = 1, V = 15 m/s.
L. M. KEER and .I. C. LEE
994 INTEGRAL
EQUATION
FOR
DYNAMIC
From eqns (26) and (28), the total displacement theory solutions is given by
Fig. 8. Simply supported
PROBLEM
from the layer and from the beam
beam subjected to transverse impact: M = I. (a) Force history. (b) Contact area.
Dynamic impact of an elaskally supported beam In the
995
contact region we have z&(.x,0, t) = S(t) - 2,
0 < x < C(l),
139)
where S(t) is the relative approach (a function of time as is the contact region) and R is the radius of curvature of the striker. An additional equation is obtained from Newton’s equation of motion which can be written as
where ( * ) denotes di~erentia~o~ with respect to time and m is the mass of the striker per unit length. Provided that the striker has an initial velocity I’, eqn (40) can be expressed as
where M is the ratio of the mass of the beam to that af the striker,
of
In solving eqns (39) and (41), first the orders the integration and summation of the dynamic parts of eqn (39) are interchanged. For the integration procedure the time domain is poisoned into equally spaced elements. If the number of intervals is even, Filon’s formula is used. We note that ‘in the calculation a ~ornbi~a~o~ of Fil~n’s formula together with a procedure that is similar but where the unknown function is approximated by a third degree polynomial is employed, For each time step the contact area is overestimate and divided into N small patches and a procedure as in Ahmadi et al. IS] is used, where for each patch the pressure is assumed to be constant. We also note that the Volterra integral equation is sensitive to the starting v&e, and in order to reduce the error, the starting value is modified as in Baker [7]. The numerical results are given in Figs. 2-10. For all numerical results the following geometrical and material parameters are used: R = 40 cm, E = 10 cm, 1E1 = 1 cm; E = 206.85 GPa, EJ= 0.29, p = 7833.5 Kg/m’.
996
I_. M. KEER and J. C. LEE
TIME, SECOND (xlo-ag Fig. 10. Clamped beam subjected to transverse impact: M = 1, V = 20 m/s. (a) Force history. (b) Contact area.
Figure 2 shows the contact force and contact area, where the beam is taken to be simply supported and the striker has an impact velocity of 20 m/s. For these cases the mass ratio A4 is varied from 1.0-2.25. With these parameters it is seen that the time history for the force has two peaks. This indicates that after impact the beam is sufficiently flexible that after the first peak the contact will diminish before the beam achieves its maximum deflection. As the maximum deflection is reached, the beam and the striker may have opposite velocities. When the beam releases its stored energy, after achieving maximum deflection, it impacts the striker and a new force maximum is achieved. For the largest value of mass ratio, A4 = 2.25, the beam actually loses contact with the striker and a second impact is noted. The multiple impact which has occurred is different from that shown in Goldsmith [2], where both the spherical striker and the beam (plane stress) are elastic and a second impact occurs in the progressing direction. Furthermore, asM increases, the maximum force transfers from the second peak to the first peak. -The
Dynamic impact of an elastically supported beam
997
maximum half contact lengths, shown in Fig. 2(b), are less than one-fourth of the thickness of the beam, except for the case when M = 1, where c/h > 1. Comparing Fig. 2(b) and Figs. 3-5, which show the pressure distributions corresponding to various values of &I, we note that Fig. 3 shows a distribution that is not Hertzian, while Figs. 4-6 are approximately Hertzian. Figure 3 indicates a situation noted by Keer and Miller [3] in which a contact area greater than the thickness of the beam will tend to produce a nonHertzian contact stress distribution. The appearance of the stress distribution for that case will show peaks nearer to the edge of the contact, while the stresses near the center will be lessened. Note that the non-Hertzian appearance will tend to coincide with a diminished force (Fig. 2). Figures 4-6 show the stress distributions corresponding to the larger mass ratios and Fig. 6 shows the stress distributions corresponding to a loss of contact. Figure 7 shows the pressure distribution which corresponds to M = 1 and I/ = 15 m/s. The force history and contact area are compared with that for V = 20 m/s in Fig. 8. The force histories are similar [Fig. 8(a)], but the contact area becomes much smaller with the decreased impact velocity [Fig. 8(b)]. Figure 9 shows the pressure distribution for a clamped beam impacted by a striker with a mass ratio of M = 1. As in the previous case (Fig. 3) the pressure distribution is non-Hertzian during a portion of the impact time that the maximum half contact length is larger than the thickness of the layer. Figure 10 represents the force history and half contact length for a clamped beam with M = 1 and V = 20 m/s. Since the beam is clamped and hence stiffer, the maximum contact force is much greater than the simply supported case, but the period of impact is much less than the latter. CONCLUSIONS
The problem of the impact of a cylindrical striking an elastically supported beam is solved by combining an elasticity and beam theory solution that allows consideration of the global (beam theory) and local (elasticity) stresses. The impact response appeared to be highly dependent on the geometrical and material parameters. For most of the cases studied the force history for the impact showed two peaks. In the more extreme case a double impact was noted. During the initial stage of the impact the stress distribution was Hertzian, but for relatively small mass ratios a non-Hertzian distribution was observed. During the .non-Hertzian portion of the stress history, the peak stresses showed a diminished magnitude and the load was carried over a larger area. Acknowledgement-Theauthorsare gratefulfor support
from the Air Force Office of Scientific Research under
Grant AFOSR-82-0330. REFERENCES [II S. TIMOSHENKO and J. N. GOODIER, Theory of Elasticity, 3rd Edition. McGraw-Hill, New York (1970). PI W. GOLDSMITH, Impact, The Theory and Physical Behavior of Colliding Solids. Edward Arnold Ltd., London ( 1960). [31 L. M. KEER and G. R. MILLER, Smooth Indentation of a Finite Layer, J. Engng Mech. Div., ASCE, 109, 706 (1983). K. F. GRAFF, Wave Motion in Elastic Solids. Ohio State University Press, Columbus (1975). t:; N. AHMADI, L. M. KEER and T. MURA, Non-He&an Stress Analysis-Normal and Sliding Contact, Int. J. Solids Struct. 19, 357 (1983). WI I. N. SNEDDON, Fourier Transforms. McGraw-Hill, New York (I 95 1). [71 C. T. H. BAKER, The Numerical Treatment of Integral Equations. p. 842. Clarendon Press, Oxford (1977). (Received in production 9 October 1984)