30 Chapter Two cylindrical bodies in contact Figure 2.6 R I ,R2 = radii of cylinders (positive when convex and negative when concave) -=- 1 +g E, El El, E2 = modulus of elasticity for the two materials Case 7: General Case of Contact Between Elastic Bodies with Continuous and Smooth Surfaces at the Contact Zone Analysis of this case by Hertz can be found in Refs and A diagrammatic representation of this problem is shown in Fig 2.7 and the contact area is expected to assume an ellipitcal shape Assuming that (RI, and (R2, R;) R;) are the principal radii of curvature at the point of contact for the two bodies respectively, and $ is the angle between the planes of principle curvature for the two surfaces containing the curvatures l/R1and l/R2, the curvature consants A and B can be calculated from: These expressions can be used to calculate the contact parameter P from the relationship: The Contact Between Smooth Surfaces Figure 2.7 General case of contact B-A cos0 = A+B The semi-axes of the elliptical area are: where m, n = functions of the parameter as given in Fig 2.8 P = total load 1-U : kl =- k2 - u2 =- nEl nE2 u l , u2 = Poisson’s ratios for the two materials E l , E2 = corresponding modulii of elasticity Case 8: Beams on Elastic Foundation The general equation describing the elastic curve of the beam is: 31 Chapter 32 degrees Figure 2.8 Elliptical contact coefficients where k = foundation stiffness per unit length E = modulus of elasticity of beam material = moment of inertia of the beam With the notation: the general sohtion for beam deflection can be represented by: where A , B, C and D are integration constants which must be determined from boundary conditions For relatively short beams with length smaller than (0.6/8), the beam can be considered rigid because the deflection from bending is negligible compared to the deflection of the foundation In this case the deflection will be constant and is: s=- P kL and the maximum bending moment = P L / 33 The Contact Between Smooth Surfaces For relatively long beams with length greater than ( / / ? ) the deflection will have a wave form with gradually diminishing amplitudes The general solution can be found in texts on advanced strength of materials Table 2.2 lists expressions for deflection y , slope 8, bending moment A and shearing force V for long beams loaded at the center Case 9: Pressure Distribution Between Rectangular Elastic Bars in Contact The determination of the pressure distribution between two bars subjected to concentrated transverse loads on their free boundaries is a common problem in the design of mechanical assemblies This section presents an approximate solution with an empirical linear model for the local surface contact deformation The solution is based on an analytical and a photoelastic study [18] A diagrammatic representation of this problem is shown Table 2.2 Condition Beam on Flexible Supports Governing equations" Chapter 34 in Fig 2.9a The problem is approximately treated as two beams on an elastic foundation, as shown in Fig 2.9b The equations describing the system are: where qX = load intensity distribution at the interface (lb/in.) ZI, Z2 = moments of inertia of beam cross-sections E l , E2 = modulii of elasticity z l , z2 = local surface deformations y I , y 2= beam deflections k l , k2 = empirical linear contact stiffness for the two bars respectively calculated as: Et Et k2 =kl =0.544 0.544 P Figure 2.9a Two rectangular bars in contact The Contact Between Smooth Surfaces 35 YI A 1 I I1 I I I 11' I I kl I t Y2 Figure 2.9b Simplified model for two beams in contact The criterion for contact requires, in the absence of initial separations, the total elastic deflection to be equal to the rigid body approach at all points of contact, therefore: where a is the rigid body approach defining the compliance of the entire joint between the points where the loads are applied: The continuity of force at the interface yields: Equations (2.1)-(2.4) are a system of four equations in four unknowns This system is now reduced to a single differential equation as follows Adding Eqs (2.1) and (2.2) gives: Chapter 36 Substituting Eq (2.4) into Eq (2.5) yields: The combination of Eqs (2.3) and (2.4) gives: Substituting Eq (2.7) into Eq (2.6) yields the governing differential equation: where Kc, is an effective stiffness: With the following notation: (2.10) Eq (2.10) may now be rewritten as: d" -+ 4$" d,r4 = 4$a (2.1 I ) The following are the boundary conditions which the solution of (2.1 I ) must satisfy, provided L C, where L is half the length of the pressure zone: The beam deflections are zero at the center location The slope is zero at the center The summation of the interface pressure equals the applied load The pressure at the end of the pressure zone is zero The moment at the end of the pressure zone is zero The shear force at the end of the pressure zone is zero The six unknowns to be determined by the above boundary conditions are the four arbitrary constants of the complementary solution, the rigid body approach a, and the effective half-length of contact C The four constants 37 The Contact Between Smooth Surfaces and the rigid body approach are determined as a function of the parameter R = /3t.A plot of the rigid body approach versus A is shown in Fig 2.10 At A = n/2, the slope of the curve is zero For values of R greater than n/2, the values of z1 and z2 become negative, which is not permitted The maximum permitted values of i is then n/2 and the effective half-length of contact is l t = n/(2/3) n / ( p ) is greater than L, the effective length is then 2L If The expression for the load intensity at the interface is: PB (cosh 2A + cos 2E + 2) cosh , cos @U ! ? U q = k1z1 = - [ sinh 2A + sin 21 (cosh i - COS 2A) sinh /!?x /!?.U sin (sinh 2;1+ sin 2A) - sinh B x cos Px + cosh Bx sin Bx (2.12) where A = ge q = the load intensity (lb/in.) x = restricted to be x e The normalized load intensity versus position is shown in Fig 2.1 for ge = n/2 This figure represents a generalized dimensionless pressure distribution for cases where t < L < 4.0 a % 3.0 s U e a 2.0 c ;1.0 v) v) v) n CI Figure 2.10 1.2 1.6 d2 Dimensionless approach of the two beams versus the parameter ge 38 Chapter Figure 2.1 Normalized load intensity over contact region The assumption of a constant contact stiffness can be considered adequate as long as the theoretical contact length is far from the ends of the beams For cases where t approaches L, it is expected that the compliance as well as the stress distribution would be influenced by the free boundary As a result, it is expected that the actual pressure distribution would deviate from the theoretical distribution based on constant contact stiffness A proposed model for treating such conditions is given in the following The approximate model gives a relatively simple general method for determining the contact pressure distributions between beams of different depths which is in general agreement with experimental t oelast ic investigations In the model the contact half-length t is calculated from the geometry of the beam according to the formula: When t < L, the true half-length of contact is equal to t and the corre< sponding pressure is directly calculated from Eq (2.12) or directly evaluated from the dimensionless plot of Fig 2.11 As t approaches L, the effect of the free boundary comes into play and the constant stiffness model can no longer be justified An empirical method 39 The Contact Between Smooth Surfaces to deal with the boundary effect for such cases is explained in the following The method can be extended for the cases where C L Because of the increase in compliance at the boundaries of a finite beam as the stressed zone approaches it, a fictitious theoretical contact length l ’ ( e ’ > e) can be assumed to describe a hypothetical contact condition for equivalent beams with L‘ > e (according to the empirical relationship given > in Fig 2.12 The pressure distribution for this hypothetical contact condition is then calculated Because the actual half-length of the beam is L, it would be expected that the pressure between e’ and L would have to be carried over the actual length L for equilibrium The redistribution of the pressure outside the physical boundaries of the beam is assumed to follow a mirror image, as shown in Fig 2.13 The superposition of this reflected pressure on the pressure within the boundaries of the beam gives the total pressure distribution The general procedure cam be summarized as follows: Calculate /3 from geometry and the material of the contacting bars according to Eq (2.10) Calculate l from the equation t = n/(2/3) Using e and L, find f?’ from Fig 2.12 Notice that for e < L, < e’ = e 1.8 1.6 h $ 1.4 1.2 1.o (L f Figure 2.1 1.2 Empirical relationship for determining t! ’ 1.6 2.0 40 Chapter i- Boundary Figure 2.13 The “mirror image” procedure Evaluate the pressure distribution over l ’ from the normalized graph, Fig 2.1 For l < L, the pressure distribution as calculated in step is the < true contact pressure When t ’ is greater than L, the distribution calculated by step is modified by reflection (as a mirror image of the pressure outside the physical boundaries defined by the length t) A MATHEMATICAL PROGRAMMING METHOD FOR ANALYSIS A N D DESIGN OF ELASTIC BODIES IN CONTACT The general contact problem can be divided into two categories: Situations where the interest is the evaluation of the contact area, the pressure distribution, and rigid body approach when the system configuration, materials and applied loads are known; Systems which are to be designed with appropriate surface geometry for the objective of obtaining the best possible distribution of pressure over the contact region In this section a general formulation is discussed for treating this class of problems using a modified linear programming approach A simplex-type algorithm is utilized for the solution of both the analysis and design situations A detailed treatment of this problem can be found in Refs 18 and 19 41 The Contact Between Smooth Surfaces The Formulation of the Contact Problem 2.3.1 The contact problems which are analyzed here are restricted to normal surface loading conditions Discrete forces are used to represent distributed pressures over finite areas The following assumptions are made: The deformations are small The two bodies obey the laws of linear elasticity The surfaces are smooth and have continuous first derivatives Problem formulation and geometric approximations can therefore be made within the limits of elasticity theory 2.3.2 Condition of Geometric Compatibility At any point k in the proposed zone of contact (Fig 2.14), the sum of the elastic deformations and any initial separations must be greater than or equal to the rigid body approach This condition is represented as: where &k \ l ’ k ( l ) , kt’k(2) = initial separation at point k = elastic deformations of the two bodies respectively at point k a = rigid body approach Figure 2.14 Zone of contact 42 2.3.3 Chapter Condition of Equilibrium The sum of all the forces F acting at the discrete points (k = 1, , N where ’ N is the number of candidate points for contact) must balance the applied load (P) normal to the surface The equilibrium condition can therefore be written as: (2.14) 2.3.4 The Criterion for Contact At any point k , the left-hand side of the inequality constraint in Eq (2.13) may be strictly positive or identically zero Defining a slack variable Yk representing a final separation, the contact problem can be formulated as follows Find a solution ( F , a,Y ) which satisfies the following constraints: -SF+cre+IY=e eTF = P (2.15) Either where skj = a k j , ( l ) -k a k j ( ) akj(l), akj(2) influence coefticients = skj for the deflection of the two bodies respectively = N x N matrix of influence coefticients F = N x vector of forces Y = N x vector of slack variables (or final separation) e = N x vector of 1’s E = N x vector of initial separations (Y = rigid body approach, a scalar The Contact Between Smooth Suflaces 2.4 43 A GENERAL METHOD O SOLUTION BY A SIMPLEX-TYPE F ALGORITHM The problem as formulated in Eq (2.15) can be solved using a modification of the simplex algorithm used in linear programming The changes required for the modification are minor and are similar to those given by Wolfe [S] When Eq (2.15) is represented in a tableau form in Table 2.3, the condition for the solution can be stated as: Find the set of column vectors corresponding to ( F , a,Y ) subject to the conditions, either F k = or Y k = 0, such that the right-hand side is a nonnegative linear combination of these column vectors These column vectors are called a basis For a problem with N discrete points, the number of possible combinations of these column vectors taken ( N 1) at a time is: + Because of the very large number of combinations, an efficient method is required for finding the unique, feasible solution The following algorithm proved to be effective for the problem under investigation The original problem as formulated in Eq (2.15) can be rewritten as: N+l Minimize X Z , j= I such that -SF Table 23 Fl F2 + ae + IY + Iz = E eTF + ZN+I = P Representation of Eq (2.15) FN YI y +1 +I (2.16) + +1 YN = 62 =P 44 Chapter Subject to the conditions that Either where Z j = artificial variables which are required to be nonnegative (j= 1, , N = an N x vector of artificial variables with components Z , , , Z N + 1) The above problem can be classified as a linear programming problem [ 131 if it were not for the condition that either Fk = or Yk = The simplex algorithm for linear programming can, however, be utilized to solve by making a modification of the entry rules The conditions of Eq (2.15) require some restrictions on the entering variables Suppose the entering variable is chosen as F, A check must be made to see if the Y, is not in the basis, F, is free to enter the basis The actual replacement of variables is accomplished by an operation called pivoting This pivot operation consists of N elementary operations which replace a system by an equivalent system in which a specified variable has a coefficient of unity in one equation and zero elsewhere [ 131 A flow diagram of the modified simplex algorithm is shown in Fig 2.15 Computational experience has shown the simplex-type algorithm to converge to the unique feasible point in at most (3/2)(N+ ) cycles, the majrity of cases converge in N cycles The simplex-type algorithm for the solution of the contact problem requires less computer storage space when compared to available solution algorithms such as Rosen's gradient projection method [14] or the FrankWolfe algorithm [ 151 Only minor modifications of the well-known simplex algorithm are required This algorithm is also readily adaptable to the design problem which is discussed later in this section + + EXAMPLE The classical problem of two spheres in contact is considered as an example In this case the influence coefficient matrix S in Eq (2.15) is calculated according to a Boussinesq model as discussed earlier in this chapter: 45 The Contact Between Smooth Surfaces I S a t with tr standard equations I Chooserby If s corresponds to Ff is V, in the Basis? If scorrespondsto Y, A Yes N+l c r= j-I z j Make canonical relative to the artificial variables and ' I I r t , Define J=(jllIjIZN+I} Basic feasible solution * No Would Fr replace or V, replace Remove sfo J rm , No Choose s by P Replace the r'* basic variable by x, by pivoting on the term a,x, d, = mindj j d Test 4) is d, O? Is 6) O? > I Define J = (Jll I I2N + 1) j STOP No feasible solution Figure 2.1 Flow diagram for the simplex-type algorithm where U = Poisson's ratio dk, = distance from point k to pointj in the contact zone Figure 2.16 shows a comparison between the classical Hertzian pressure distribution and that obtained by the described technique The spheres considered are steel with radii of in and loin., respectively and the applied load is 1001b The algorithm solution gave a value of 0.000281 in for the rigid body approach which compares favorably with 0.000283 in for the classical Hertz solution Chapter 46 r( h 180,000 - g 120,000 & m - 12 v -Computer Based Model v) with 1 Points Across the Diameter c , U 60,000- 0 0150 Classical Theory 0100 0050 0050 0100 0150 Radial Distance from the Applied Load (in) Figure 2.1 Pressure distribution between two spheres THE DESIGN PROCEDURE FOR UNIFORM LOAD DISTRIBUTION The design system discussed in this section automatically produces initial separations which produce the best possible distribution of load based on a selected function for surface modification (initial separation) A secondorder curve is selected for the initial separation since it can be readily generated The equation for such curve is given by: y = ox2 + b x + c (2.17) where y = initial separation profile and is required to be Y = axial position along the face The correction profile can be attained by modifying one or both of the contacting surfaces The objective of the design system is to evaluate the constants (a, b, c) for the optimal corrections corresponding to the distribution giving the minimum possible value for the maximum load intensity In the formulation of the design system the compatibility condition given in Eq (2.15) is used with E being replaced by Eq (2.17) Accordingly: -SF + aye + IY - aX2 - bX - c = (2.18) 47 The Contact Between Smooth Surfaces where -2 X = N x vector whose kth element is x i X = N x vector whose kth element is xk xk = position of the kth point The condition of equilibrium and the criterion for contact are the same as in Eqs (2.14) and (2.15) The initial separations are required to be nonnegative, therefore: a y 2+ b y + c2 where a governs the sign of the second derivative If we define Ak as the length of the line segment at the kth point, the average load intensity over that segment is Fk/Ak The value of pmax must be greater than the average load intensities at all the candidate points This constraint is written as follows: where D is a diagonal matrix whose kth element is l/Ak The design system is now stated in a concise form as: Minimize pmax such that (2.19) e'F=P F , Y , a,c Subject to the condition that either Yk = or Fk = It should be noted that an upper bound must be given to c to keep the values of c and a finite in Eq (2.19) The algorithm for solving the design problem (Fig 2.17) is divided into two parts The first part finds a feasible solution for the load distribution 48 Chapter while the initial separations are constrained to be zero The second part minimizes the maximum load intensity using the parameters (a, b, c ) as design variables The simplex-type algorithm is used in both parts The minimization of the maximum load intensity is a nonlinear programming problem, the objective function is linear but the constraints are nonlinear [ 171 Since all the constraints are linear except for the criterion for contact, the basic simplex algorithm can again be used with the modified entry rules as discussed previously EXAMPLE The case of a steel beam on an elastic foundation is considered here as an illustration of the design system It is required in this case to calculate the necessary initial separations which produce, as closely as possible, a uniform pressure Given in this example are: L , t , and d = length, width, and depth of beam = 8.9 in., 1.0 in., and 4.0 in., respectively k = foundation modulus = 10’ lb/in./in The results from the solution algorithm with a quadratic modification are given in Figs 2.18 and 2.19 and the pressure distribution without initial separation is shown in Fig 2.18 for comparison The initial separation as calculated from the analysis program for a uniform pressure distribution is also shown in Fig 2.19 It can be seen that the quadratic modification, although it does not provide an exactly uniform pressure distribution, represents the best practical initial separation for the stated objective An approximation for evaluation of the surface modification can be obtained by assuming a uniform load distribution, computing the necessary initial separations and then fitting these data to a curve with the stated form of surface modification In the process of curve fitting, the main objective is to approximate the computed initial separations without regard to the resulting load distribution The same approach can be readily applied to the surface modification of bolted joints to produce uniform pressure in the joint and consequently minimize the tendency for leakage or fretting depending on the application EXAMPLE In this example, the same approach is applied for determining the initial separation necessary to produce uniform pressure at the interface between multiple-layered beams The case considered for illustration is shown in Fig 2.20 where three cantilever beams are subjected to 49 The Contact Between Smooth Surfaces S m with rhc Standard Equations (5.18) I Add the artificial , variables Z Z2, ,Z Add objective function z , = I f s cornponds t F, o i s Y in the basis? , or Ifs cormponds to Y , is F, in thc basis? z zj - j=1 Yes No r Make canonical relative to the artificial variables, I ,.J o + Define J = the s t of all variabks e e x a p a E Choose s by d; - i”’ Ji Would F, Test Z , YC.5 No A V Remove s fivrnJ STOP No Fusible Solution Is > O ? , v v Replace the r* basic variable by xs by pivoting on the term ursxs Phase I No A/ Yes ’ I Choosesby d; =J“‘ dJ Test P Is d.; O? n p l m r, OT r, replace F,? Is D s 2O? \L STOP Best Basic Feasible Solution Phase I1 * - allvllriablcs all v a h b l u except I Basic Feasible Solution -_ Figure 2.1 - stan Phase I1 Design Phasc Flow diagram for the design algorithm I Use the pmPx row I 50 Chapter Figure 2.1 2.0 Pressure distribution of beam on elastic foundation -Optimum Quadratic Correction Pressure I i i LI c ) (d ' d C n I D s a c From Center of Beam (in.) itne Fiaure 2.1 Initial separation for uniform pressure distribution and optimal quadratic correction 51 The Contact Between Smooth Surfaces P Figure 2.20 Multiple cantilever beams an end load The applied load P is assumed to be 60001b, the length of the beam is 12in and width is in and the thicknesses are 5in., 2in and 5in., respectively The beams are made of steel with modulus of elasticity equal to 30 x 106psi The interface areas are divided into 24 segments and the force on each segment is found to be 70.621b for both interfaces which is equivalent to 141.24psi The calculated initial separations are given in Fig 2.21 Interface (F = 70.62 Ib at each point) o 0 0 no 0 0 0Q t l -0- - Figure 2.21 -0 " O O U Interface (F= 70.62 Ib at each point) 10 Distance From the Fixed End (in.) H l = , HZzO.1, H3=5 Initial separations 12 52 Chapter EXAMPLE In this case a steel cantilever beam with length 12in., width I in and thickness 3in is subjected to an end load of 60001b The beam is supported by another steel cantilever beam with the same length and width and different thickness H2 as shown in Fig 2.22 The same algorithm is used with 24 segments at the interface to determine the maximum attainable uniform pressure at the interface and the corresponding initial separation Smaxat the free end for different values of the thickness H z The results are given in Fig 2.23 and show that Smax will reach an asymptotic limit when H2 is either very large or very small The uniform load on each segment F,,, is shown to reach a limit value when H2 is very large F I , F2, FN X Figure 2.22 Discrete forces 103 102 01 H,=5 in - - 008 1 - - OM2 h " 01 W K K 10-2 t , - 004 10-3 104 - 002 10-5 10-6 I I I11111 I I I111111 I 1 Ill11 I I I 11111 I I I L U0 L ration The Contact Between Smooth Surfaces 53 Numerous illustrative examples for simulated bolted joints, multiple layer beams and elastic solids with finite dimensions are given in Ref 20 A list of some of the publications dealing with different aspects of the contact problem is given in Refs 21-39 REFERENCES 10 11 12 13 14 15 16 17 Timoshenko, S.P., Theory of Elasticity, McGraw Hill Book Company, New York, 1951 Love, A E H., A Treatise on the Mathematical Theory of Elasticity, Dover, New York, 1944 Timoshenko, S P., Strength of Materials, Part 11, D Van Nostrand, New York, 1950 Galin, L A., Contact Problems in the Theory of Elasticity, Translation by H Moss, North Carolina State College, 1961 Keer, L M., “The Contact Stress Problem for an Elastic Sphere Indenting an Elastic Layer,” Trans ASME, Journal of Applied Mechanics, 1964, Vol 86, pp 143-145 Tu, Y , “A Numerical Solution for an Axially Symmetric Contact Problem,” Trans ASME, Journal of Applied Mechanics, 1967, Vol 34, pp 283 Tsai, N., and Westmann, R A., “Beam on Tensionless Foundation,” Proc ASCE, J Struct Div., April 1966, Vol 93, pp 1-12 Wolfe, P., “The Simplex Method for Quadratic Programming,” Econometrica, 1959, Vol 27, pp 328-398 Dorn, W S., “Self-Dual Quadratic Programs,” SIAM J Appl Math., 1961, Vol 9, pp 51-54 Cottle, R W., “Nonlinear Programs with Positively Bounded Jacobians,” JSIAM Appl Math., 1966, Vol 14(1), pp 147-158 Kortanek, K., and Jeroslow, R., “A Note on Some Classical Methods in Constrained Optimization and Positively Bounded Jacobians,” Operat Res., 1967, Vol 15(5), pp 964-969 Cottle, R W., “Comments on the Note by Kortanek and Jeroslow,” Operat Res., 1967, Vol 15(5), pp 964-969 Dantzig, G W., Linear Programming and Extensions, Princeton University Press, Princeton, NJ, 1963 Rosen, J B., “The Gradient Projection Method for Non-Linear Programming,” SIAM J Appl Math., 1960, Vol 8, pp 181-217; 1961, Vol 9, pp 514-553 Frank, M., and Wolfe, P., “An Algorithm for Quadratic Programming,” Naval Research Logist Q., March-June 1956, Vol 3(1 & 2), pp 95-1 10 Kerr, A D., “Elastic and Viscoelastic Foundation Models,” Trans ASME, J Appl Mech., 1964, Vol 86, pp 491-498 Mangasarian, L., Nonlinear Programming, McGraw Hill Book Company, New York, NY, 1969 54 Chapter 18 Conry, T F., “The Use of Mathematical Programming in Design for Uniform Load Distribution in Nonlinear Elastic Systems,” Ph.D Thesis, The University of Wisconsin, 1970 19 Conry, T F., and Seireg, A., “A Mathematical Programming Method for Design of Elastic Bodies in Contact,,’ Trans ASME, J Appl Mech., 1971, Vol 38, pp 387-392 20 Ni, Yen-Yih, “Analysis of Pressure Distribution Between Elastic Bodies with Discrete Geometry,” M.Sc Thesis, University of Florida, Gainesville, 1993 21 Johnson, K L., Contact Mechanics, Cambridge University Press, New York, NY, 1985 22 Ahmadi, N., Keer, L M., and Mura, T., “Non-Hertzian Contact Stress Analysis - Normal and Sliding Contact,” Int J Solids Struct., 1983, Vol 19, p 357 23 Alblas, J B., and Kuipers, M., “On the Two-Dimensional Problem of a Cylindrical Stamp Pressed into a Thin Elastic Layer,” Acta Mech., 1970, Vol 9, p 292 24 Aleksandrov, V M., “Asymptotic Methods in Contact Problems,’’ PMM, 1968, Vol 32, pp 691 25 Andersson, T., Fredriksson, B., and Persson, B G A., “The Boundary Element Method Applied to 2-Dimensional Contact Problems,” New Developments in Boundary Element Methods CML Publishers, Southampton, England, 1980 26 Barovich, D., Kingsley, S C., and Ku, T C., “Stresses on a Thin Strip or Slab with Different Elastic Properties from that of the Substrate,” Int J Eng Sci., 1964, Vol 2, p 253 27 Beale, E M L., “On Quadratic Programming,’, Naval Res Logist Q., 1959, Vol 6, p 74 28 Bentall, R H., and Johnson, K L., “An Elastic Strip in Plane Rolling Contact,,’ Int J Mech Sci., 1968, Vol 10, p 637 29 Calladine, C R.,and Greenwood, J A., “Line and Point Loads on a NonHomogeneous Incompressible Elastic Half-Space,” Quarterly Journal of Mechanics and Applied Mathematics, 1978, Vol 1, p 507 30 Comniou, M., “Stress Singularities at a Sharp Edge in Contact Problems with Friction,” ZAMP, 1976, Vol 27, p 493 31 Dundurs, J., Properties of Elastic Bodies in Contact, Mechanics of Contact between Deformable Bodies, University Press, Delft, Netherlands, 1975 32 Greenwood, J A., and Johnson, K L., “The Mechanics of Adhesion of Viscoelastic Solids,” Philosphical Magazine, 1981, Vol 43, p 697 33 Matthewson, M J., “Axi-Symmetric Contact on Thin Compliant Coatings,” Journal of Mechanics and Physics of Solids, 1981, Vol 29, p 89 34 Maugis, D., and Barquins, M., “Fracture Mechanics and the Adherence of Viscoelastic Bodies,’, Journal of Physics D (Applied Physics), 1978, Vol 1 35 Meijers, P., “The Contact Problems of a Rigid Cylinder on an Elastic Layer,” Applied Sciences Research, 1968, Vol 18, p 353  ... Figure 2.22 Discrete forces 10 3 10 2 01 H,=5 in - - 008 1 - - OM2 h " 01 W K K 10 -2 t , - 004 10 -3 10 4 - 002 10 -5 10 -6 I I I 111 11 I I I 111 111 I 1 Ill 11 I I I 11 111 I I I L U0 L ration The Contact... of the contact problem is given in Refs 21- 39 REFERENCES 10 11 12 13 14 15 16 17 Timoshenko, S.P., Theory of Elasticity, McGraw Hill Book Company, New York, 19 51 Love, A E H., A Treatise on the... 60001b, the length of the beam is 1 2in and width is in and the thicknesses are 5in. , 2in and 5in. , respectively The beams are made of steel with modulus of elasticity equal to 30 x 10 6psi The interface