INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, VOL. 26, 1709-1728 (1988)

TIME-DOMAIN TRANSIENT ELASTODYNAMIC ANALYSIS OF 3-D SOLIDS BY BEM

S. AHMAD' AND P. K. BANERJEE'

Department of Civil Engineering, State University of New York at Buffalo, Amherst, New York 14260, U.S.A

SUMMARY The numerical implementation of the Direct Boundary Element formulation for time-domain transient analysis of three-dimensional solids is presented in a most general and complete manner. The present formulation employs the space and time dependent fundamental solution (Stokes' solution) and Graffis dynamic reciprocal theorem to derive the boundary integral equations in the time domain. A time-stepping scheme is then used to solve the boundary initial value problem by marching forward in time. Higher order shape functions are used to approximate the field quantities in space as well as in time, and a combination of analytical (time-integration) and numerical (spatial-integration) integration is carried out to form a system of linear equations. At the end of each time step, these equations are solved to obtain the unknown field quantities at that time.

Finally, the accuracy and reliability of this algorithm is demonstrated by solving a number of example problems and comparing the results against the available analytical and numerical solution.

INTRODUCTION

The ability to predict the dynamic response of solid bodies subjected to time and space dependent loads and boundary conditions has gained considerable importance in all engineering fields such as structural dynamics, machine foundation design, aircraft structure design, seismology and soil-structure interaction analysis. For a realistic problem from any of the above engineering fields, it is extremely difficult to obtain a closed-form solution. Therefore, numerical methods such as the Finite Difference Method (FDM), the Finite Element Method (FEM) or the Boundary Element Method (BEM) need to be used. Of the numerical methods, the most versatile and widely used method is the FEM. It has, however, two major deficiencies: (a) it cannot model an infinite or semi-infinite medium properly; and (b) the computational cost (both CPU and data preparation) involved in analysing three-dimensional transient dynamic problems by the FEM is so enormous that only a few researchers can afford it. Similarly, the FDM has not been used frequently, primarily because of the difficulties associated with handling complicated geometries and boundary conditions.

In contrast, it is convincingly demonstrated that accurate and efficient solutions to dynamic problems can be easily obtained by using the BEM'-3 because an infinite or semi-infinite medium can be modelled easily, and for linear problems only the surface of the geometry needs to be discretized. Thus, a considerable reduction in the size of the problem can be achieved for a very wide class of engineering problems.

* Assistant Professor + Professor

0029-598 1/88/08 1709-20s 10.00 0 1988 by John Wiley & Sons, Ltd.

Received 1 1 September 1987 Revised 23 November 1987

1710 S. AHMAD AND P. K. BANERJEE

The boundary element formulation for the time-domain transient dynamic analysis with constant boundary elements has been implemented by Cole et aL4 for the anti-plane strain case (i.e. scalar problem), by Niwa et al.' for the two-dimensional wave scattering problem, by Rice and Sadd6 for the anti-plane strain wave scattering problem, by Spyrakos and Beskos' and Antes* for the strip-footing problem and by Mansur and Brebbiag- lo for the two-dimensional problem.

Three-dimensional problems of transient dynamic by time-domain BEM were not attempted until recently, principally because of the formidable task of numerical implementation and enormous computing requirements. In order to reduce the computation and complications involved, simplifications of the BEM formulation dictated by the nature of the problem to be solved have been developed by Karabalis and Beskos.' ' They have simplified the BEM formul- ation for the special case of transient surface loading on rectangular foundations. A more general numerical implementation of the BEM formulation for time-domain transient dynamic analysis has been attempted by Banerjee et aL2 and Banerjee and Ahmad' using constant temporal and quadratic spatial variations of the field variables.

Most of the above mentioned work other than that of References 1 and 2 suffers from one or more of the following: lack of generality, crude assumption of constant variation of the field variables in space and time, inadequate treatment of singular integrals and unacceptable level of accuracy. For example, Cole et aL4 found the transient dynamic formulation to be unstable, leading to a building up of errors as the time-stepping progresses: Rice and Sadd6 found that dominant errors in the method arise from integrating the Green's function over the singularity and their time-domain formulation when applied to time harmonic problems reveals a large build up of errors; and Niwa et aL5 find considerable errors in their numerical results and suggest that use of higher order interpolation functions for approximating the temporal and spatial variations of field variables may improve the accuracy and the stability of their method. Mansur and BrebbiagV1O re-implemented the scalar dynamic formulation of Cole et al.," as well as the elastodynamic formulation outlined in Niwa et al.,' both for two-dimensional problems of dynamics. They did not appear to have encountered any serious errors in their solutions.

It is not really surprising that a number of the above mentioned authors report some problems of accuracy. Unlike the FEM method which has an energy based error minimization principle embodied in the discretized system, the BEM system in the discretized state does not satisfy any energy principle. Only the original boundary integral equation satisfies the energy principle (namely, the reciprocal work theorem) exactly. Such a system therefore needs to be implemented as accurately as possible.

In References 1 and 2 three-dimensional elastodynamic formulations were implemented using quadratic isoparametric shape functions and a very sophisticated error control in integration. For the transient dynamic analyses, however, only a constant temporal variation was used. While these analyses eliminated most of the inaccuracies in the boundary solution, the interior stresses were found to be somewhat inaccurate. This is primarily due to the presence of a boundary acceleration term in the stress kernel which cannot be satisfactorily handled when the boundary displacements are assumed to be constant over a time step.

Almost all of the above mentioned problems have been eliminated in the present work by using a quadratic shape function in space and a linear shape function in time, taking care of the singular integral in an accurate and elegant manner, using superior and sophisticated integration techniques and implementing the BEM formulation in a complete and general manner. The time- domain transient algorithm developed in this work is capable of producing accurate results for general three-dimensional elastodynamic problems. Furthermore, the use of a linear temporal

TIME-DOMAIN TRANSIENT ELASTODYNAMIC ANALYSIS 1711

shape function makes it possible to extend the present algorithm to non-linear dynamic analysis, which will be the subject of a forthcoming paper by the authors.

TRANSIENT BOUNDARY INTEGRAL FORMULATION

The direct boundary integral formulation for a general, transient, elastodynamic problem can be constructed by combining the fundamental point-force solution (Stoke’s solution) of the govern- ing equation of motion (Navier-Cauchy equations) with Graffis12 dynamic reciprocal theorem. Details of this construction can be found in Banerjee et a1.2 For zero initial conditions and zero body forces, the boundary integral formulation for transient elastodynamics reduces to

r

where

Gij*t i= [:Gij(x, r 5, z) t , (x , T)dr

Fij*ui=j :Fi j (x . T; 5, z)ui(x, r)dr

are Riemann convolution integrals, and 5 and x are the space positions of the receiver (field point) and the source (source point). The fundamental solutions G , and Fij are the displacements ui and tractions t i at a point x at time Tdue to a unit force vector applied at a point 5 at a preceding time T. These functions are listed in a compact and elegant form in Appendix I. The tensor C i j is the discontinuity (or jump) tensor, arising from the singularity of the Fij kernel.

Equation (1) represents an exact formulation involving integration over the surface as well as the time history. It should also be noted that this is,an implicit time-domain formulation because the response at time T is calculated by taking into account the history of surface tractions and displacements up to and including the time T. Furthermore, equation (1) is valid for both regular and unbounded domains.

Once the boundary solution is obtained, the stresses at the boundary can be calculated by combining the constitutive equations, the directional derivatives of the displacement vector and the values of field variables in an accurate matrix formulation. For calculating displacements at interior points equation (1) can be used with ci j=6, , and the interior stresses can be obtained from

O j k ( 5 , TI= CG;k(x, 5, T ) * t i ( x , T) -FGk(x , 5, T)*u i (x , T)I dS(x) (3) I The functions GZk and FGk of the above equation are listed in Appendix 11. These functions are

presented for the first time; they do not exist in any published literature. It can be seen that these functions contain first and second order derivatives with respect to time. Therefore, use of constant temporal variation of the field variables, as done in all previous work, produces unsatisfactory results for interior stress.

NUMERICAL IMPLEMENTATION

Details of the numerical schemes for surface discretization, spatial integration, assembly and solution of system equations can be found in References 1, 2 and 3. In this paper, only the temporal integration and the time-marching scheme used in the present work are presented in

1712 S. AHMAD A N D P. K. BANERJEE

detail. However, for completeness a brief description of the rest of the numerical schemes is also provided.

The surface modelling is done using either six noded triangles or eight or nine noded quadrilaterals. The functional variation over them can be either linear or quadratic or a mixture of linear and quadratic. The spatial integration is carried out to a pre-selected 3 to 5 digit precision. Facilities to include all possible local and global boundary conditions have been provided. The entire analysis can be carried out for an assemblage of up to 15 substructured regions. At the interfaces between sub-regions, facilities to include sliding, bonded or spring loaded connections have been included.

This implementation is a part of BEST3D (Boundary Element Solution Technique, 3- Dimensional) system.

Time-marching scheme

time intervals, i.e. In order to obtain the transient response at a time TN, the time axis is discretized into N equal

N

n = 1 TN= 1 nAT (4)

where AT is the time step. Utilizing equations (4) and (2), equation (1) can be written as

cijui(k, TN)- jTN TN- I j S [ G i j t t - F i j u i ] dSdz= j::' js[Gijti-Fijui] dSdz (5 )

were the integral on the right hand side is the contribution due to past dynamic history. It is of interest that equation (5), like equation (l), still remains an exact formulation of the

problem since no approximation has yet been introduced. However, in order to solve equation (5) , one has to approximate the time variation of the field quantities in addition to the usual approximation of spatial variation. For this purpose a linear interpolation function is used which is described with the resulting time-stepping algorithms as follows.

The field variables (i.e. displacements, tractions or stresses) are assumed to vary linearly during a time step, i.e.

N U i ( X , 7)= c [M;uy-'(x)+M",y(x)] (64

n = 1

N ti(X, T)= 1 [M;t ; - ' (x)+M"Fy(x)]

n = 1

where

N is the total number of time steps; ul(x) and t l (x) represent the spatial variation of ui and t i , respectively, at time T,; M, and MF are the temporal interpolation functions related to local time nodes I and F (see Figure l), and are of the form

TIME-DOMAIN TRANSIENT ELASTODYNAMIC ANALYSIS 1713

Function f ( t )

where

or

Node 'F"

Tn-l T" time (1)

Figure 1. Definition of temporal interpolation nodes

4,(7)= 1 for (n- l )AT<tSnAT, and =O otherwise; and

4" (~ ) = [H { 7 - (n - l )A T } - H { T - nA T } ] H being the Heaviside function.

For illustration purposes, consider the boundary integral equation ( 5 ) for the first time step, i.e.

CijUi ( t9 TI)- j l l [ s [ G i j t i - F i j ~ i ] dSdr=O (8)

The time integration in equation (8) after using equation (6) is done analytically and the spatial integration is performed numerically (see References 1, 2 and 3). This analytical temporal integration will be discussed in the following section.

After the integrations and the usual assembly process, the resulting system equation has the form

where c41 {X') - CBkH yl> + { 4 1 {XO) -CB:l{ YO} = (0) (9)

[A] and [B] are the matrices related to the unknown and known field quantities, respectively; { X} and { Y } are the vectors of unknown and known field quantities, respectively; for {X} and { Y } the superscript refers to the time step; for [A] and [B] the superscript denotes the time step at which they are calculated, and the subscript denotes the local time nodes (I or F) during that time-stepping interval.

1714 S. AHMAD AND P. K. BANERJEE

Since all the unknowns at time T=O are assumed to be zero, equation (9) reduces to

C4l { X 1 } = CBkI { y '1 + D:l{ y O) (10) Now consider the boundary integral equation for the second time step, i.e.

Ci jUi (S9 T2)- 11; Is [ Gijt i - F, ui] dSdz = [I: ] s [ G i j 4 - F i j ~ i ] dSdt (11)

If the time interval ( T, - Tl) is the same as ( T, - To) the resulting coefficient matrices of the left hand sides of equations (8) and (11) become identical. This is due to the time translation properties of the fundamental solutions G , and F,, which contain time functions with arguments (T-z), and therefore the convoluted integral corresponding to the interval T1 < z < Tt with T = T2 is identical to that of the interval To <t< TI with T= Tl.

The right hand side of equation (1 1) is evaluated at time T = T2 with the time integration over the interval To to T, and thus provides the effects of the dynamic history of the first time interval on the current time node (i.e. T2).

Now, the resulting system equation for this time node ( T2) is of the form

[&I {X'} -[BbI{ Y ' } + CAtI {XI} -CB:I { Y1} = - {CA:I {X' } -CGI { Y l }

+ [A:] { X O } - [GI { YO)} (12)

(13)

Equation (12) can be rearranged such that

[Ak] {X'} = [ B k ] { Y '} - [A: +A:] {XI} + [B: +B:] { Y '} + [B:] { Y O} In the above equation, all the quantities on the right hand side are known. Therefore the unknown vector { X 2 } at time Tz can be obtained by solving the above equation.

Thus, for the Nth time step, the boundary integral equation (5) can be written in a discretized form as

N T 1

or

where vector { R"} is the effect of the past dynamic history on the current time node. The above equation can be solved to find the unknown vector { X " ) at time TN. It may appear

at first glance that a prodigious calculation of coefficients is involved. However, a closer examination will reveal that:

(i) if the time step size is constant, the [A;] and [B;] matrices do not change from time step to time step;

(ii) for each time step, a new { R N } vector needs to be formed. This involves the evaluation of a new set of coefficient matrices [A;] , [B;] , [A;] and [B;] involving the effects of the dynamic history of the first time interval on the current time node. Eventually, however, this contribution to (RN} reduces to zero and from that point onwards no new coefficients need to be evaluated.

Finally, it is of interest to note that if the linear temporal shape functions M; and M; are replaced by Mf = MF = 0.5 $"(z), the present time-stepping scheme reduces to a constant tem-

CAkl{xN)=CB;l{ Y N > + { R N 1 (15)

TIME-DOMAIN TRANSIENT ELASTODYNAMIC ANALYSIS 1715

poral variation scheme. In this constant temporal variation scheme, the averaged value of a field variable between the two local time nodes (I and F) is taken as the representative value for the field variable during a time step. This averaging yields more accurate results'. ' than that obtained without averaging. Nevertheless, the latter approach has been invariably used by most of the researchers in past.

Analytical temporal integration

For linear time interpolation, the field variables are expressed as N

n= 1 f = c W!f I- '(x) + M)f"(x)l (16)

where fn(x) represents the spatial variation of the field variablef(x, t) at time Tn( = nAT), and M, and M, are the temporal shape functions expressed by equation (7).

The transient dynamic kernels listed in Appendices I and I1 have one or more of the following time functions embedded in them.

Time function (I): S( T-t-r/c)

Time function (2): Ah( T-r-r/c)dt

Time function (3): 6( T-t-r/c) (19)

Time function (4) : 8( T - t - r l c ) (20)

where 'c' is either the pressure wave velocity c1 or the shear wave velocity c2, and 6 is the Dirac delta function.

Using equation (16), the time integrals related to any of the above listed time functions (equations (17), (18X (19) and (20)) can be expressed as

I:g( T- z - r / c ) f ( x , t) d t = 1 g( T-T-~ /C)~(X, t)dz n = l (n-1AT

nAT

r = [fn-'(x)! M!(t)g( T-.r-r/c)dT

n= 1 (n- 1)AT

1 + f "(x) yT Mt(r)g( T- t - r/c) d t ( n - 1)AT

An important characteristic of the transient dynamic kernels is the time translation pro- perty.' -3Because of this characteristic, at each time step, only the effects of the dynamic history of the first time interval on the current time node need to be evaluated: i.e. at each time step the analytical time integration has to be done only for n = 1. Thus, equation (21) reduces to

AT

g( T-r-r/c)f(x,t)dt=f'(x) 41(t)g( T-t-r/c)dr

+f '(x) 1 AT(t/AT)&(T)g( T- T -r/c)dr 0

1716 S. AHMAD AND P. K. BANERJEE

The time integrals on the right hand side of equation (22) can be rearranged into the following two types of integrals.

Type A:

Type B: loAT r 4 ,(r)g ( T - t - r/c) dz

The convoluted time integrals of types A and B with time functions ( 1 ) and (2) are evaluated analytically as follows.

Time function 1:

Type A J +,(t)s( T-r-r/c)dt=4,( T-r/c)=H( T-r/c)-H( T-AT-r/c) 0

Type B: J:T tq5,(t)6 ( T - t - r/c) = ( T - r/c)& ( T - r/c)

where

[(A2/2)H(T- lr)]$t =O if T< r/cl

if T>r/c2 1 1 -

2c: 2c:

The second term on the right hand side of equation (27) can be obtained in a similar manner by

Type B: IOAT 1”” AS( T-r-lr)t4,(t)dldt=

replacing ‘T’ by ‘T-AT’ in equation (28).

A( T- l r )4 , ( T-b)d l l / C l 1::::

= [:I:TA+l( T-Ar)dl- rA24,(T-Ar)dA i:::: = TC(A2/2)4,( 7.-vr)l;:: - [(n3/3)4 (T- wi :;:: (29)

The terms on the right hand side of equation (29) are evaluated in a similar manner as that of

For the time functions (3) and (4), the time integrals of equations (21) are evaluated as follows. equation (27).

TIME-DOMAIN TRANSIENT ELASTODYNAMIC ANALYSIS 1717

Time function 3

Timefunction 4. The temporal integration involving the time function (4) (i.e. 8( T-z-r lc ) ) is approximated by using a backward finite difference scheme as follows:

f' (a) Mesh I1

3 six-noded triangular elements per octan

2

i' (b) Mesh 12

3 nine-noded quadrilateral elements per octan

i' ( c ) Mesh 13

3 eight-noded quadrilateral elements per octan

x 2

Figure 2. Boundary element models for radial expansion of a cavity

1718 S. AHMAD A N D P. K. BANERJEE

EXAMPLES OF APPLICATION

(a) Spherical cavity

A spherical cavity is embedded in an infinitely extending medium with E = 8-993 x lo6, v = 025 and p = 2 3 x in self-consistent units. The radius of the cavity is a=212. Three different meshes for its surface discretization are shown in Figure 2. Using the built in symmetry capabilities, this problem is modelled by one octant only. The characteristic times required for the

1.4 :::I --

.e A Mesh #I

'r/(u, ) AM * &St

%/Po

1: lpo Loadin; Curve a Mesh 13 o Mesh 12

.e t

.8 -LA,

0

.0 1.0 2 .0 3.0 4 . 8 5.0 6 . 0 7 .0 tc - t - 1

(a ) Time History of Radial Displacement at Cavity Wall

I P ( t ) .9 I

a Mesh 13

o Mesh 12

.8

- . I /!ill--- .0 1.0 2 . 8 3.0 4.0 5 . 8 6.0 7 . 0

a (b) Time Histocy of Tangential Stress at Cavity H a l l

.0

1.0

Figure 3. (a) Time history of radial displacement at cavity wall. (b) Time history of tangential stress at cavity Wall

TIME-DOMAIN TRANSIENT ELASTODYNAMIC ANALYSIS 1719

pressure and shear waves to travel a cavity radius are 0*00102 and 0.00177 secs, respectively. Four cases are considered.

(i) Spherical cavity under sudden radial expansion. A radial pressure P = lo00 is suddenly applied and maintained at the cavity surface. Figure 3(a) shows the time variation of radial displacement at the cavity surface obtained by the time-domain algorithm. Concurrently plotted are the analytical results of Tirno~henko.'~ The dynamic displacements are normalized by the analytical static solution. Figure 3(b) shows the normalized hoop stress at the surface of the cavity

OGIO.

0008.

c z w z 0 0006

Y

2 0-004

a

< 0

0

0 <

0002

OQO

-9002

LOAOING CURVE

TIME

- NUMERICAL (A*=-) . NUMERICAL IAi-00006)

TIME

Figure 4. Radial expansion of a cavity under triangular pulse

LOADING CURVE

TIME

- NUMERICAL t&r00003 SEC) NUMERY:AL~ASDWOO~SEC)

TIME

Figure 5. Radial expansion of a cavity under rectangular pulse

1720 S. AHMAD A N D P. K. BANERJEE

1.4

1.2

1.n

.e

'r .6

(ur ) A n a l . St

compared with the analytical solution. The numerical results of displacements for all three meshes agree well with the analytical solution. The stresses, however, show some oscillations at later times. This is principally due to the abruptness of the applied loading, which results in a shock that cannot be modelled in a numerical scheme. The time step used in this analysis is equal to 0.33t*, where t* is the time required for the pressure wave to travel the cavity radius.

/ *gk6 e x e

1 f,a / -

* a f i r , - I @ fi-8. 8-g 8-1 9-1 -g s-8 - Analytical Sol.for Step

Loadi - I

' A Mesh 8, Linear Var.

- I 4 X Mesh 11, Constant Var. P(t)

0 Mesh 13, Linear Var.

(i i) Spherical cavity subjected to a triangular pulse of radial pressure. A triangular pulse of radial pressure, as shown in Figure 4, is applied at the cavity surface. This example is solved by using the present analysis with two different time steps. The radial displacements at the cavity

I .t

1.n

.e

-s

. 4

.P

P(t)

X X

X

-

I \ - 2t* - I x'Q a x 3

-I , \ @ a x * d _ I t N / -9 5-B 4 - w - m-8 H @-

- I X B S Analytical Sol.for Step

a E%,Linear Var.

x Mesh 13,Linear Var. O Mesh 12,Linear Var.

Mesh 11,constant Var. I

Figure 6. (a) Time history of radial displacement at a distance R=0.2a from the cavity wall. (b) Time history of radial stress at R=0.2a

TIME-DOMAIN TRANSIENT ELASTODYNAMIC ANALYSIS 1721

-5

.4

.3

-2 - 0.3

.1

surface are plotted in Figure 4. The numerical results from both the time steps are almost identical. Thus, this example demonstrates the stability of the present algorithm.

- O 0 0 0 9 0 3 -. / F-@T. n m H II

Analytical Sol.for Step

Mesh #l, Constant Var.

-

- Loading A Mesh #I, Linear Var.

0 Mesh t3, Linear Var.

- k 0 I -

( i i i ) Spherical cavity subjected to a rectangular pulse of radial pressure. A rectangular pulse of radial pressure, as shown in Figure 5, is applied at the cavity surface. This example is also solved by using two different time-increments. Figure 5 shows the time history of the radial displacement of the cavity. By comparing these results with those due to the triangular pulse (i.e. Figure 4), it can be seen that, in general, the displacement at any time interval due to the rectangular pulse is twice of that due to the triangular pulse. This is because the response depends upon the total impulse, and the total impulse due to the rectangular pulses is double that due to the triangular one. In addition, since the duration of the energy input is the same in both problems, the response curves for both cases have the same shape.

-.z

-.3

(iu) Spherical caoity subjected to ramp loading. It was found from the analysis of previous cases that a suddenly applied loading results in a shock which could not be modelled very effectively. This is so because the fundamental solution due to Stoke which has been used in the development of the BEM formulation does not satisfy the kinetic energy requirement at the shock front. The boundary integral representation also does not admit any such discontinuity.

It was therefore decided to consider the case of a ramp loading history, as shown in Figure qa, b). The time t* is defined as the time for pressure wave to travel the radius of the cavity. Figure 6(a) shows the normalized displacement responses from a number of analyses compared with the analytical solution for the suddenly applied loading (since the exact solution is not known for the present case). The results clearly show a delayed response for the ramp loading. All results (obtained by using time step 0*33t*), including the one for a constant time variation for mesh 1, seem satisfactory. Figure qb) shows the normalized radial stress at a distance of 02a (a = radius of the cavity) outside the cavity surface obtained from a series of analyses compared with that of an abruptly loaded cavity. Although the results for all meshes using the linear variation are similar, those obtained from a constant time variation are clearly not satisfactory. This is because of the presence of the boundary acceleration term in the interior stress calculations which cannot be satisfactorily represented in a constant time interpolation scheme. In Figure qb), for constant

Loading Curve

2t* 3 -

- \'

1722 S. AHMAD A N D P. K. BANERJEE

time interpolation, the radial stress obtained at time t = t* is greater than the applied pressure, which is physically impossible.

Figure 7 shows the time history of the hoop stress at a distance 0.2~ from the surface of the cavity, where, once again, all linear time variation analyses give consistent results but the constant time averaging does not, particularly at earlier times. The analytical result shown here once again is for the case of suddenly applied loading.

(b) Circular loaded area on a halfspace

In this example, a circular area of radius R is subjected to suddenly applied vertical loading of Po= 1OOO. The mesh for this problem is shown in Figure 8(a). The necessary details of

I & a .

888.

688.

4 8 8 .

288.

0 .

, 'RI 9R I

(a) Boundary Element Mesh for a Circular Footing on an Elastic Half-space

.0

(b) Time History of Vertical Displacements at the Center and at the Edge of a Circular Footing

Figure 8. (a) Boundary element mesh for a circular footing on an elastic half-space. (b) Time history of vertical displacements at the centre and at the edge of a circular footing

TIME-DOMAIN TRANSIENT ELASTODYNAMIC ANALYSIS 1723

geometry and material properties are radius R = 1.0, v=0-333, density= 1.0, pressure and shear wave velocities of 2.0 and 1.0, respectively, in self-consistent units.

The vertical displacement response (pRu,), where p is the shear modulus, is shown in Figure 8(b) plotted against non-dimensional time t = tc,/R. As expected, the responses of the centre and the edge are different. Whereas the centre responds very quickly and deflects to the full static value, the edge takes nearly twice as long to reach its static value. It is also of interest to note the use of different time steps AT=0-1, 0.15 and 0-2 leads essentially to the same results.

(c) A Jexible square plate foundation on an elastic half space

Karabalis and Beskos,' who described an approximate BEM implementation for half-space problems, examined the transient dynamic response of a 60 x 60 inch square, 11.517 in thick

(a) Footing Mesh

(b) Half-space Mesh

Figure 9. Boundary element model Tor a flexible square footing on an elastic half-space (Mesh # 1)

1724 S. AHMAD AND P. K. BANERJEE

flexible foundation resting on the surface of an elastic half space. The elastic constants for the plate were modulus of elasticity E , = 30.004 x lo6 lbjin', Poisson's ratio vp = 0.3 and mass density pp=7.34 x lb-s2/in4. Those of the half-space were E, =844 x lo6 lb/in2, v s = 0 3 , p,=2.82 x lb-sz/in.4 The above choice of elastic constants led Karabalis and Beskos" to obtain a relative plate stiffness ratio K=0.004, where K is defined by

~ , h 3 ( 1 - v,) K =

12( 1 - v ; ) ~ , b

where h and b are the plate thickness and width, respectively, and ,us is the shear modulus of the half space.

I

(a ) Footing Mesh

I 1 I

(b) Half-space Mesh

Figure 10. Boundary element model for a flexible square footing on an elastic, half-space (Mesh #2)

TIME-DOMAIN TRANSIENT ELASTODYNAMIC ANALYSIS 1725

Karabalis and Beskos” used a finite element idealization (4x4mesh) for the plate and a boundary element idealization (5 x 5 mesh) for the half-space. The boundary elements used were of constant variation over both space and time. This was obviously a very courageous attempt to solve such a difficult transient dynamic problem. Those authors probably gained the necessary confidence by comparing with known static solutions which must have been satisfactory.

In the present attempt at the analyses of the same problem, both the plate and the surface of the half-space were modelled by BEM. The mesh used for one quarter of the plate region is shown in Figure 9(a) for the coarse mesh and in Figure lqa ) for the fine mesh. Similarly two sets of meshes (coarse and fine) for the half-space region are shown in Figures 9(b) and lqb). In all cases, quadratic elements with linear time variations were used.

.* 30.

K-0.004 - (a) Loading Curve and the Dimensions of the Footing

.0 4 . 0 9.0 12.0 16.0 20.0 (sec. x 4.12E-053 time

(b) Time History of Vertical Displacement at the Center of the Footing-

Figure 11. (a) Loading curve and the dimensions of the footing. (b) Time history of vertical displacement at the center of the footing

1726 S. AHMAD AND P. K. BANERJEE

The results of the present analysis for a dynamic pressure pulse shown in Figure ll(a) are plotted in Figure ll(b), where the results of Karabalis and Beskos” are also shown for comparison. The results of Karabalis and Beskos” are obviously incorrect, while both sets of BEM results from the present analysis give almost identical results. It should be noted that in the present model a plate of finite thickness has resulted in a slight delay in the wave front reaching the half-space. This, however, does not account for the very large discrepancies in the response times.

CONCLUDING REMARKS

An advanced algorithm based on the direct boundary element formulation for time-dependent elastodynamic analysis of three-dimensional solids has been presented. The algorithm is an unconditionally stable implicit time-marching scheme and is capable of producing very accurate results. However, for better accuracy, it is recommended that the time step should remain smaller than L/c,, L being the smallest distance measured along the surface between two corner nodes of an element. This algorithm is a viable alternative to that based on the finite element methodology, particularly for soil-structure interaction problems.

ACKNOWLEDGEMENT

The work presented in this paper was made possible by the NASA contract NAS3 23697. The authors are indebted to Dr C. Chamis, the NASA Project Manager, and Dr E. Todd, the Pratt and Whitney Project Manager, for their support and encouragement, and to Dr R. B. Wilson and N. Miller of Pratt and Whitney for valuable discussions.

APPENDIX I

Boundary kernels for transient dynamics

The tensors Gi j and F,, are of the form

Gij(x, T; 6, r ) = A6(u - Ar) d l + aij{ (l/c2)6(u - r/cl)

A6(u-Ar)dA+(12ai j -2bi j ){6(u-r/cz)

- ( C ~ / C ~ ) ~ ~ ( U - ~ / C ~ ) } + 2 r a i j / c z { 6 ’ ( u - r / c 2 ) - ( c z / c l ) 3 6 ’ ( u - r / ~ 1 ) }

-cij(l - 2c~/c~){6(u-r/c,)+(r/cl)~’(u-r/cl)} - d i j { 6 ( u -r/c2)

+(r/cz)Wo - I/.,)> (A2) 1

TIME-DOMAIN TRANSIENT ELASTODYNAMIC ANALYSIS 1727

where v = T - t

dij = yi yj ~mnrn/r' cij = yjni/r3 dij=(y,nj+Gijymnm)/r3 b . . = c . . + d .

IJ IJ li

APPENDIX I1

Interior stress kernels for transient dynamics

The tensors G t k and FGk are of the form

-(12aijk - 2bijk)[6(u -r/c2) - (C2 /C1)2S(U - r / c l ) }

- 2raijk/c2{6'(u- r/c2) - ( c ~ / c , ) ~ ~ ( u - r/c,)f

+ cijk( 1 - 2ci/c:) {S(u - r/cl) + (r/cl)d'(u - r/c ,)}

643) 1 + di jk { S(v - r /~2) + ( ~ / C ~ P ' ( L J - ~ / C ~ ) 1

.2c:(35aijk - 5bijk + - 4ci(45aijk - 6b, + c i j k ) (6 (u - r/c2) -(cz/cl)2S(u - r / c l ) }

- 4 ~ , r ( l O a ~ ~ ~ - b ~ ~ ~ ) ( ~ ' ( u - r / c ~ ) - ( c ~ / c ~ ) ~ d ' ( u - r / c , ) }

+ 2c31 -2c~/c~)(3dijk-2eijk){6(u-r/cl)+(r/cl)b'(u-r/cl)}

+ c:(3fjk - 2 g i j k ) { - r /c2) + ( r / cZ a'(u - r / c 2 ) >

- 4aijkr2{S"(u - r/c2) - (c2/cl)"S"(u - r / c l ) )

+ r2(1 -2c2,/c:){2c$dijk/c:

+(I - 2 c ~ / c ~ ) e i j k } ~ " ( u - r / c l ) + r ~ j k 6 " ( u - r / c Z ) (A41 1

1728 S . A H M A D A N D P. K. BANERJEE

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