ELASTIC MODULI OF MATERIALS IN PAVEMENT DETERMINING BY SURFACE DEFLECTION DATA, SYSTEMS A. William Research Celik Ozyildirim H. Engineer Research Highway & Transportation Research Council Virginia Cooperative Organization Sponsored Jointly by the Virginia (A 1975 September 76R-I0 VHTRC A FEAS IBI LITY STUDY by Engineer Carpenter and K. Vaswani Nari Research Scientist Senior )epartment of Highways & Transportation and the University of Virginia) Charlottesville, Virginia
determination of the elastic, or Young's, modulus, E, The the materials in each layer in an n-layered pavement system of the number, order, thi knesses, and Poisson's ratios of given layers, and the surface load and deflection data, is not the using the classical theory of elasticity alone. This possible develops some assumptions and techniques, based on the report modulus concept, Burmister's deflection equation, the effective element method, and the concepts of beams and plates on finite foundations, which yield mathematical solutions for such elastic moduli. ABSTRACT iii
determination of the elastic, The Young's, modulus, E, or the materials in each layer in of n-layered pavement system an determining deterioration in pavement systems i. re- as in changes in moduli, and hence the need for flected determining the structural behavior of 2. materials pavement pavement systems for the purpose of opti- and mizing pavement designs; and establishing quality control techniques during 3. construction. preliminary investigation of n-layered A systems by the pavement has shown that given the number, order, thick- authors and Poisson's ratios of the layers, and the surface load nesses, the dynaflect deflection data it is not possible and utilize to classical theory of elasticity alone to determine the elastic moduli the the materials in each layer. Therefore other methods of be employed to determine the elastic moduli of the materials must multi-layer systems. in objective of this research The to investigate the pos- was of determining the elastic moduli of the materials in sibility multi- layer pavement systems from dynaflect deflection data. following concepts and procedures The investigated as were their individual and combined potentials- to I. the effective moduli of pavement systems, 3. the finite element method, and 4. the concepts of beams and plates on elastic foundations. INTRODUCTION is desirable for-- rehabilitation; OBJECTIVE SCOPE 2. Burmister's equation,
concept of an effective modulus of a pavement system is The on a spring analogy extended to columns and on Boussinesq's based a simple two-layer pavement system. If it is as- Consider that, Poisson's ratio, is zero for each layer, and that sumed in the original problem), which may be analyzed as noted (layers reference i. in k X 6 k X 6 a 1 a 2' Xl 62 d8 and and d2 are the deflections at the upper boundaries of 61 1 and 2, respectively, layers and 68 are the deflections within the first and second a respectively, layers, 1 and X2 are the external loads appiied to the upper bound- X of layers 1 and 2, respectively, and aries k and k8 are the a respectively. layers, spring constants of the first and second EFFECTIVE MODULUS OF A PAVEMENT SYSTEM settlement equation. Spring Analogy layers are of finite depth, the pavement system reduces to both spring system composed of a connected column of two subsprings a Given the system in Figure I, one may write (i) X 2 -k X 6 + (k + k 8) X 2' (2) (3) + 6 a ' I (4) (5) X 2 0 (no external force), where 2
the two-layer spring system, if the external load In X2 and the stiffnesses k and k are known, the two unknown and i and 2, can be determined using equations 1 deflections, 2. and the inverse problem, only X 1 and l are given. Rewriting In 1 through 5, one obtains equations X k8 i 62 X X k 1 (ks' X X I) (k Rigid Boundary Figure I. A two-layer spring system. X ke 1 -, and.6-1-_ (6) (7) the solution for k and k8 involves three unkn[,wns, Therefore, ks, and 2, in only two equations, equations 6 and 7. Thus, ke, there are an infinity of solutions of the form i e
there is one other experimentally measurable para- However, kef f, which is the effective stiffness of the system. This meter, k X 6 eff i' (8) Xl k determine and kb. However, equation 9 may be derived from s 1 and 2, by rewriting them as. equations 7 k 1 2 e + k k 1 k + k n + k X I keff X mentioned in the previous section, kef f for a spring sys- As is an experimentally measurable quantity. To extend the con- tem h, cross sectional area A, and modulus E, for such a column height, a compressive force P, the deflection at the top is under parameter is defined by which implies that (9) eff m this concept appears to give one additional equation Intuitively, may be used in conjunction with equations 6 and 7 to fully which (i0) and k 2 k X k - i X Xl k X 8 (ii) equation 9 does not increase the row dimension of the Therefore, matrix. coefficient Extension of Spring... Analogy to Columns of kef f to a three-dimensional problem, one needs to determine cept equivalent of k in the layered system Consider a column of the P X h A X E' (12) or A p x_e X 6, (13) h which is reminiscent of the spring relation P k X d- (14),, AE one can see that the form q is the stiffness" of a column. Thus, this reasoning to an n-±ayered system, one may write Extending
1 =E (E/h)i' (E/h) eff [ eff.th is the thickness of the i layer, and h h. 1 validity of the above approach must be demonstrated using The known data or the Chevron(2, 3) technique in combination either X (I Z 17) 2 X P X r' E one can see that treating an n-layered system as a one-layered Thus, under the assumption that nonhomogeniety dies not radically system, equation 17, will yield affect 2 X (I X p X r 2 6 eff Eef f has been empirically related to the E i's of the layers as Vaswani (5) However, equation 16 is a potentially more reward- by relationship between E and the E. 's. eff ing (15) or (16) where =Eh.. eff with Boussinesq's Settlement Equation (which is described below) Settlement Equation Boussinesq's settlement equation (4) for the deflection under Boussinesq's a flexible plate is 2 6 where p is the load intensity, and r is the radius of the bearing area. E (18) where Eef f is the effective modulus of the entire system. E eff E hi (19) l
equation (an extension of Boussinesq's settle- Burmister's equation) for deflections under a flexible bearing area for ment E 2 the settlement coefficient, is a function of r/h i and EI/E 2 F w, F (charts. are given in reference 4) for w equation (when the dynaflect data are known) yields a This for E, when E 2 is known and solutions for E 1 and E 2 when solution El/E2 is known. finite element method can yield a complete solution for The n Ei's and n @i's in an n-layered pavement system, if, in the to the number, order, thicknesses and Poisson's ratios addition the layers, and the external load, n of the 2 X n Ei's and of give the design Es, subgrade modulus (6)). However, this would becomes progressively r ore difficult to achieve as the solution of unknown E i's increases. Thus knowing n-i of the E i's number 1 of the 6i's the solution is much simpler than that when, and n-3 of the Ei's and 3 of the @i's are known. Furthermore, say, Ei's and 6i's are not directly available for analysis. these Yx 2 X k X e X (cossx + sinsx) (21) BURMISTER'S DEFLECTION EQUATION a two-layer elastic system (4) is 2 =2X (i- )XPX rxf (20) w where p is the load intensity, r is the radius of the bearing area, and FINITE ELEMENT METHOD are known. (The dynaflect deflection data, of course, yields i's value of i- Also, Vaswani's soil classification scheme the Thus, auxiliary methods must be employed to obtain them. BEAMS AND PLATES ON ELASTIC FOUNDATIONS a two-layer system composed of an infinitely long Given supported on an elastic foundation (spring foundation), beam a point load, the theory of beams on elastic foundations(7,8) and states-
2 X 8 -Sx 8x P X e sinsx (22) k k 1/4 equals 4 X E X I > is the moment of inertia (second moment of area) of the I and, beam P is the point load at x o. values for y and @x are determined from dynaflect deflection When equations 2 and 22 may be used to determine E and k. data, application of these results to pavement deflections The the theory of plates on elastic foundations (9)) requires (really the rigidity of a plate be used in place of the rigidity of that beam. This is accomplished by simply substituting Eh3/(12(1 a EI in the expression for 8.(8) Thus an approximation for for deflections may be obtained by using equations 21 and 22 ment X (i x_k 3 X h3 E this manner, dynaflect data may be employed to determine E of In top layer of a pavement system and the combined k of the re- the theory of plates on elastic foundations would, of course, The better solutions than this extension of the theory of beams yield the equivalent system of equations for plates on elastic whereas do not. Solutions to the plate equations require iter- foundations improvement techniques and they are solvable in only certain ative Thus, the authors feel that equations 21, 22, and 23 instances. an acceptable engineering approximation to the problem constitute plates on elastic foundations. of and where is the deflection at point x (x o directly under the load), Yx is the slope of the deflection curve at x, 8 x is. the spring modulus of the foundation, E is the elastic modulus of the beam, 2 pave- where 8 (23) maining layers. elastic foundations would yield for E and k in a two-layer sys- on However, equations 21, 22, and 23 have analytical solutions, tem.
three methods discussed above can be used for determining The elastic moduli of the materials in a pavement system. Based the these methods, five possible algorithms have been prepared for on of two-layer systems, and nineteen possible combinations solution algorithms and subalgorithms have been prepared for solution of three-layer systems. These algorithms of given in the Appen- are 1 and E 2 for two-layer pavement systems can be determined E various combinations of Burmister's procedure, the finite from method, and the Eef f concept. The requirements for sol- element are that either E 1 be known from the theories of beams ution plates on elastic foundations or that E 2 Es be known from and soil classification scheme. Vaswani's E2, and E 3 for three-layer pavement systems can be deter- El, from various combinations of Burmister's procedure, the mined element method, and Eeff concept, and the treatment of finite comof layers as single layers. The requirements for solu- binations are that both E 1 and E3 Es be known from the theories of tion and plates on elastic foundations and Vaswani's soil clas- beams report has demonstrated that two and three-layer prob- This are theoretically solvable. Thus, the authors recommend lems the techniques presented in this report be systematically that evaluated, and, if necessary, modified based employed, field on The authors further recommend that the most appropriate data. as determined from such evaluations be presented to techniques SOLUTIONS dix. CONCLUS IONS sification scheme, respectively. RECOMMENDATIONS Department in implementable forms such the computer programs as sets of graphs. or
Martin, H. C., Introduction to Matrix Methods of Structural McGraw-Hill Book Company, New York 1966. thods, Me a Circular Area", California Research Corporation, Richmond, on September 24, 1963. California, H., and W. L. Dieckmann, "Numerical Computation of Warren and Strains in a Multiple-Layered Asphalt Pavement Stresses Runways" HRB Proceedings., Vol 23, Washington, D C 1943 N. K., Method for Separately Evaluating the Vaswani, Performance of Subgrades and Overlying Flexible tural N. K., "Evaluation of Subgrade Moduli Vaswani, Working Plan, Virginia Highway and Pavements", M., Beams on Elastic Foundation, the Hetenyi, Press, Ann Arbor, Michigan, 1946. Michigan Struc- Pave- R., Theory_ and Analysis of Plates Classical and Szilard, Methods, Prentice-Hall, Inc., New Jersey, 1974. Numerical REFERENCES io J., "Analysis of Stresses and Displacements in an Michelow, Elastic System Under a Load Uniformly Distributed n-layered California Research Corporation, Richmond, California, System", 24, 1963. September D. M., "The Theory of Stresses and Displacements Burmister, Layered Systems and Application to the Design of Airport in ments" HRR 362 Washington, D C 1971 Flexible for Transportation Research Council, May 1975. of University Van Nostrand D. i956. S., Strength of Materials, Part II, Timoshenko, Inc., Princeton, N.J., Third Edition, Company,
Algorithm 1 Estimate E2 1. scheme(5) Algori thin 2 Algorithm 3 Algorithm 4 Algorithm 5 Algorithm 6 Vaswani's soil using (vscs). Determine E 2 and 62 using the finite element method, (FEM). APPENDIX S OLUT I ON AL GO RI THMS and A-3 illustrate the notation used in the A-l, A-2, Figures algorithms. solution Two-Layer Systems classification 2. Determine E 1 using Burmister's equation. (4) Determine E 1 using equations 21, 22, and 23. Determine E 1 using equations 21, 22, and 23. 2. Determine Eef f using equation 18. 3. Determine E 2 using equation 16. i. Estimate E2 us ing VSCS. 2. Determine Eef f using equation 18. 3. Determine El using equation 16. Estimate E2 using VSCS. Determine E 1 and 62 using the FEM. Three-Layer Systems i. Determine E 1 using equations 21, and Estimate E3 2. Determine E 3. us ing VSCS. using equation 18. ff 4. Determine E 2 using equation 16. A-I
g%rithm Determine i. 3. De te rmi n e Subalgorithm A Subalgorithm B E l, E23, and E Given 3. E Determine using 2 E23 E 2 E 3 E l, El2, and E Given 3. E Determine using 2 h h12 h 2 I E E 12 E 2 1 Algorithms 8, 9, I0 3. Determine E (9) using steps 1 and 2 of any Algorithm 17 1 20. through 3. Determine E (i0) using steps 1 and 2 of 1 24. through Apply Subalgorithm B. Algorithms Ii, 12, 13 (Ii) 3. (13) 3 2. Apply Algorithm 4 to determine El2 and E 3. E 1 using equations 21, 22, and 2. Estimate us ing VSCS. E3 @2' and @3 using the FEM. E2' h 3 (AI) h2 3 h2 + (A2) i. Treat the top two layers as a single layer. Apply Algorithm 1 to determine El2 and E 2. 3. 3. Determine E (8) using equations 21, 22, and 23. 1 any Algorithm 21 i. Treat the top two layers as a single layer. (12) 3 Same as Algorithms 8, 9, i0, respecti.vely. 4. Apply subalgorithm B.
Algorithms 14, 15, 16 (16) 3. 2. Apply Algorithm 5 to determine El2 and E 3. Apply Subalgorithm B. Algorithms 17, 18, 19, 20 Determine E 3 E Determine 3 i0. through Determine E (19) 3 13. through 3. Determine E3 (20) 16. through Apply Subalgorithm A. Algorithms 21, 22, 23, 24 (21) 3. (22) 3 (23) 3 (24) 3 Apply Subalgorithm A. using steps 1 and 2 of any Algorithm 8 using steps 1 and 2 of any Algorithm ll using steps 1 and 2 of any Algorithm 14 i. Treat the top two layers as a single layer. (15) 3. Same as Algorithms 8, 9, 10, respectively. i. Treat the bottom two layers as a single layer. 2. Apply Algorithm 2 to determine E 1 and E23. using (17) 3. VSCS. (18) 3. i. Treat the bottom two layers as a single layer. Apply Algorithm 3 to determine E 1 and E23. as Algorithms 17, 18, 19, 20, re spe ctiw:_ ly. Same
E 2 E 1 E 3 P h 62 R±g±d Boundary Figure AI. Two-layer system. I i h h Rigid Boundary Figure A2. Three-layer system.
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