Final Al-khwarizmi Engineering Journal Al-Khwarizmi Engineering Journal, vol. 1, no.1, pp 64-75 (2005) ٥٢ Al-khwarizmi Engineering Journal Al-Khwarizmi Engineering Journal, Vol.1, No.2,pp 52-63, (2005) The Timoshenko Three-Beams Technique To Estimate The Main Elastic Moduli Of Orthotropic Homogeneous Materials Dr. Kamal K.M. Saify * Dr. Adnan N.J. Al-Temimy** Dr. Muhsin J.J. *** *The Technical College-Baghdad / Department of Dies & Tools. ** Mechanical Engineering Dept./College of Engineering./ University of Baghdad *** Mechanical Engineering Dept./ College of Engineering/ University of Al-Nahrain (Received 11 December 2004, accepted 11 February 2005) Abstract:- A New developed technique to estimate the necessary six elastic constants of homogeneous laminate of special orthotropic properties are presented in this paper for the first time. The new approach utilizes the elasto-static deflection behavior of composite cantilever beam employing the famous theory of Timoshenko. Three extracted strips of the composite plate are tested for measuring the bending deflection at two locations. Each strip is associated to a preferred principal axis and the deflection is measured in two orthogonal planes of the beam domain. A total of five trails of testing is accomplished and the numerical results of the stiffness coefficients are evaluated correctly under the contribution of the macromechanics and the approximate bending theory. To insure the validity of the new approach, separate individual tensile tests are performed, and the corresponding results are compared. Excellent agreements are obtained between the different approaches. The ease, simple and accurate predictions are well confident by the new technique. Keywords: Timoshenko ,Beam, composite beam. 1. Introduction The development of composite materials offers great potential in advanced civilian and non-civilian structural applications since the late thirties of the last century [1,2] and still now in rapid progress and evolution [3,4]. The recent century began with a new technological development of the “ smart” composite structures [5,6,7] where a large strength-to-weight ratio is achieved, besides the ability to react actively to disturbance forces while maintaining structural integrity. The assignment of the mechanical engineering properties, of such materials, are strongly demanded for design and behavior analysis. The orthotropic elastic constants (total nine in number) represent an important set of those properties. Starting with the familiar Young, shear moduli and Poisson’s ratio, the traditional static tensile test satisfies, to some level, the mentioned objective but involves uncertainty of the results (due to the localized deformation near the end fixture of the tensile sample (see Ref.[8], chap. Micromechanics) as well as the weak and simple base theory it adopts where the transverse shear effects are ignored as usual [8,9]), and also the relative cost of the test requirements (the available of tensile M/C and minimum three Kamal K.M. Saify./ Al-khwarizmi Engineering Journal ,Vol.1, No. 2,PP 52-63 (2005) ٥٣ samples to be distracted later). Moreover, the traditional test is not able to determine more than four elastic constants in its best conditions (refer to [8]). The theoretical and experimental attempts of Tsai[9] and nextly Halpin and Tsai[10] in the static micromechanics of composites, was found satisfactory if the pre-limitations in analysis were released. Their formulations require that the physical and geometrical properties of the composite constituents as well as a suggestion of two new factors insurt in the formulae, all ought to be prepared in advance. Again only four elastic constants could be obtained by these approaches. Dynamic tests, were firstly conducted by Goens[11] to determine the shear modulus of an isotropic bar under torsion. Later on, the two elastic constants (Young and shear moduli) were obtained by Pickett[12], Hasselmen [13] and Spinner & Teff[14] using two independent tests, the bending vibration and torsional ones. Rubben & Scharr[15] applied excellently both the static tensile test and torsional vibration test to estimate the nine elastic constants of composite using three chosen samples for the two tests, one of which its fabrication procedure was seemed difficult to be achieved and required much care and accuracy. Deobald & Gibson [16] used the classic orthotropic plate theory of Kerishoff to compute the four elastic constants employing the new modal analysis technique (MAT) [17]. Saify & Al-Temimi[18] were the first who succeeded to obtain the two “ effective” elastic moduli by one test of flexural vibrations of prismatic bar. Recently reference [19] presented a developed “three theories technique (TTT)” to determine all the elastic constants set of anistropic material employing the MAT and basing theoretically upon his “ exact” orthotropic simply-supported plate theory from Levinson[20] first concept of the exact isotropic plate theory. The conditions, required to make this approach successful, are: a test rig for sample boundary supporting and the set-up instrumentation of the applied MAT. It seems, generally, that a chosen approach, to determine the orthotropic elastic constants of a composite material, is often incorporating some technical (and/or theoretical) limitations in the employment. The need of (i) inexpensive test, (ii) acceptable base theory, (iii) few tested samples and (v) many estimated elastic constants, is the most preferable thing to put forwards for achieving such aim. Too many demands against so humble abilities!. The present paper looked for accomplishing most of these demands through the adoption of Timoshenko beam theory [21] which is still found as an acceptable engineering theory. A non-destructive static deflection test (instead of the destructive tensile test) of a composite cantilever strip may be sufficed to obtain the two elastic constants associated with the principal axes of the testing sample. Utilizing the familiar configuration of the samples, originally used in the tensile test, the present approach would be able to determine a maximum of six independable elastic constants from three samples preserving the same simple test set-up. The new approach had been examined, for validity assessment, by a resonant frequency test and the comparison of experimental results were made among all mentioned approaches. The present T3BT reflected very obviously its reliability and success in the achievement comparing with other techniques in literature till the time of submitting this report. 2. Theoretical Analysis Referring to Fig.1, the composite sample, under consideration, is modeled as a rectangular beam (strip) with its length, thickness and breadth Kamal K.M. Saify./ Al-khwarizmi Engineering Journal ,Vol.1, No. 2,PP 52-63 (2005) ٥٤ are denoted by L1,L2 and L3 respectively. The beam is supposed to be bent statically in the plane (1-2) due to an arbitrary distributing load P12 (per unit breadth) on the beam upper surface. Generally, the load may be a function of location η along the major axis-1. In accordance to Timoshenko beam theory the constitutive- displacement relationships and the force-moment equilibrium conditions state that: ......(3) ).( 12 1 I , .A :where ......(2) 0Q- , )( ......(1) EM , 6 5 3 233321 12 31112 LLLL d dM P d dQ d d I d d AGQ == == =      += η η η η ψ η δ ψ The quantity (5/6G12), in Q-expression, is the “ effective” shear modulus of the strip material [20], in which the factor (5/6) refers to Reissner’s shear coefficient, while G12 is simply the actual shear modulus associated with existed plane of deformation (1-2). The item E1, in M-expression, is commonly the Young modulus of the cantilever material in the major direction-1. Solving of the ordinary differential eq.(2) for the loading condition of concentrated force P0 at the beam tip (η=L1), and using the results into eq.(1) yields to the general expressions of the displacement components δ (the local deflection) and ψ (the section rotation) as followings: ......(4) 6 1 2 1 2 1 5 6 01 32 1 31 0 1 2 1 31 0 112 0 CCL IE P CL IE P AG P ++      −= +      −−= ηηηδ ηηψ Applying the beam condition at the clamped end 0) 0( ==⇒= δψη , gives the exciplict formula for δ as varied with η, in the form: ( ) 5 3 6 112 032 1 31 0 ηηηδ AG p L IE p +−= ….(5) The two elastic constants (E1 and G12), appeared in above equation, can be calculated whenever the deflection δ is precisely measured at two locations, say the strip tip (η=L1) and the mid- length (η=L1/2), resulting in: 5 3 48 5 5 6 3 112 10 31 3 10 112 10 31 3 10 AG LP IE LP AG LP IE LP M T += += δ δ ……(6) where δT and δM represent the localized deflections at the beam tip and mid-length positions respectively. Eq.(6) suffices now to compute the two elastic constants of the strip from: ( ) ( ) 516 18 5 2 3 2 1 2 3 0 12 3 1 2 3 0 1 TM MT L L L P G L L L P E δδ δδ −            = −            = …(7) The benefit of above formulae, comes from that its mathematical scheme can be held correctly for general beam rotation of the coordinate axes system. It does not enforce any preferable choice of the directions (1,2,3) to be adjusted for any sides of the strip (length, thickness or breadth), i.e. eq.(7) can be utilized for any axis Kamal K.M. Saify./ Al-khwarizmi Engineering Journal ,Vol.1, No. 2,PP 52-63 (2005) ٥٥ rotation through (900) about its plane. This will serve to compute another set of two elastic constants (Young and shear moduli) corresponding to the new axes system. In order to estimate a maximum number of these sets of moduli values, the job was devoted to conduct the deflection tests, firstly on a beam-A whose major axis coinciding with the fiber axis (i.e. parallel), secondly on a beam-B whose major axis is perpendicular to the fiber axis (i.e. normal) and thirdly on a beam-C whose major axis is at 450 with the fiber axis (i.e. inclined). These three beams are actually cut from the composite laminate as illustrated by Fig.(2). Each beam is then tested independently one or two times. In each time the deflection axis is altered by (900) rotation about the major axis. Denoting the Cartesian plane (xy) as the mid-plane of the composite laminate, where the fibers are along x-axis, and choosing z-axis to be orthogonal with (xy) through out the laminate thickness, then the complete deflection tests may be organized as followings: (i) Beam-A (parallel): (1) Test-1: The major axis-1 is x-axis and the deflection axis-2 is z- axis, from which Ex and Gxz can be estimated. (2) Test-2: The major axis-1 is x-axis and the deflection axis-2 is y-axis, from which Ex and Gxy would be then computed. (ii) Beam-B (normal): (1)Test-1: The major axis-1 is y-axis and the deflection axis-2 is z- axis, from which Ey and Gyz can be estimated. (2)Test-2: The major axis-1 is y-axis and the deflection axis-2 is x- axis, from which Ey and Gxy would be then computed. (iii) Beam-C (inclined): (1) Test-1: The major axis is 1-axis and the deflection axis is 2- axis, from which E1 and G12 can be evaluated. Transforming the results to the actual laminate axes (xyz), then the two elastic constants (Gxy and νxy) can be estimated, from this test, using (see Ref.[8]): 11 4 12 11 2 4 141 12 121         +−      =         +−      += yxx xy yxxy EEGE EEGEG ν …(8) By now, the present 3-beam samples are non-destructively tested by simple deflection tests to estimate the six elastic constants (Ex, Ey, Gxy, νxy, Gxz and Gyz) of the given orthotropic material. Table(1) summarizes the total five tests procedure, previously explained . Note that the constants Ex and Ey would be averaged from the test results of beam-A and B respectively. The same thing might be done for Gxy from all the three beam tests, whereas no averaging is there for Gxz, Gyz and νxy since they are computed one time only. 3. Numerical results, discussions and comparison The ever best method to check for the validity of the present and relevant techniques to estimate the different elastic constants of an orthotropic material, is the adoption of a reference sample whose material elastic moduli had been precisely obtained and verified frequently by some reliable technique, other than these mentioned here, and see whether the present approaches retain the same elastic constants values. Unfortunately, this trail failed due to the absence of such material. However, the present aim can be achieved alternatively by adoption of the same experimental and theoretical results of the different approaches. The “ best” approach is that which maintaining the minimum deviations of the results throughout all cycles of the tests. It is a simple sort of “ optimization” of the different four Kamal K.M. Saify./ Al-khwarizmi Engineering Journal ,Vol.1, No. 2,PP 52-63 (2005) ٥٦ approaches: The classical tensile test, The strength of material approach, The elasticity approach and finally the present T3BT. The present manufactured composite plate was firstly well prepared and the three strips(A,B,C) were perfectly cut along the corresponding directions as clarified by Fig.2. Appendices(A,B) display the main formulations to calculate the corresponding four elastic constants (Ex, Ey, Gxy and νxy) in the light of the approaches respectively. Table(2) show the entire collection of the experimental readings of the strips static deflections (using electrical resistance strain gauges) corresponded to given concentrated load at the tip end and for all strips configurations and test trails as proposed by the T3BT. Table(3) presents the experimental acquired readings of the classical tensile test procedure made on the three strips and for all test trails as familiarly performed by this treatment. From these tests readings, the orthotropic elastic constants were computed and organized as shown by Tables(4,5). The T3BT gives the results of six elastic modulii, whereas the classical tensile test gives the results of four elastic constants. In closing, Table(5) illustrates the overall final values of the material elastic moduli as obtained by the current four approaches, mentioned before. In this table the results of the T3BT and the tensile tests are commonly averaged, from which the final standard deviations are computed easily. A little consideration into the last argument of the standard deviations in Table(5) gives definitely that present T3BT estimates the accurate results in respect to the familiar tensile test approach, in addition to its ability of obtaining two further constants upon the common four ones. The mean value of these deviations, among the total six values from the T3BT, is no more than (0.053), while from the tensile test approach (with total four values) reaches to (0.108). It is very obvious that the present T3BT estimates the results two times accurate than the classical approach. Henceforth, the Tsai approach is more reliable in results than the strength of material approach which seems to be the worse one. The most beneficial thing regarding the present T3BT is its success in estimating the orthotropic shear moduli Gxz and Gyz that no other technique had achieved in similar proposition of the present work. 4. References: 1. Lubin G., “Handbook of composites.”, Van Nostrand Reinhold Co., 1982. 2. Rosato D. and Grove C., “Filament Winding: Its Development Manufacture, Application and Design.”, 1964, J. Wiley & Sons Inc., New York. 3. CompositPro., Peak Composite Innovations, 11372 W, Parkhill Dr., Littleton Colorado, 80127 USA, 2003. 4. FiberSimTM , Composite Design Technologies, Inc., 235 Wyman St., Suite 100, Waltham, MA 02451-1219, USA, 2003. 5. Zhon X., Chattopadhyay A. and Thornburg R., “Analysis of piezoelectric smart composites using a coupled piezoelectric- mechanical model.”, J. Intell. Mat. Sys. & Strucs., Vol. 11, 2000, pp. 169-179. 6. Heng Soo Kim, Aditi Chattopadhyay and Xu Zhou, “Stress analysis of smart composite structures using piezoelectric patch using thermal- peozoelectric- mechanical loading.”, AIAA, Vol. 52, 2002, pp. 1-13. Kamal K.M. Saify./ Al-khwarizmi Engineering Journal ,Vol.1, No. 2,PP 52-63 (2005) ٥٧ 7. Henug Soo etal, “Dynamic response of smart composite shell using a Coupled thermo- piezoelectric-mechanical model.”, AIAA, 1-11, 2002. 8. Jones R.M., ”Mechanics of composites.”, McGraw Hill Book Co., Washington D.C., 1975. 9. Tsai S.W. and Spinner G.S., “The determination of the moduli of anistropic plates.”, ASME Transc., J. Appl. Mechs., Vol. 30, 1963, pp. 467-468. 10. Halpin J.C. and Tsai S.W., “Effects of environmental factors on composite materials.”, J. Composite Mat., Vol. 1, No. 1, 1969, pp. 4-10. 11. Joens E., “On the determination of the dynamic modulus of uniform bar under torsional vibration.”, J. Modern Physics (West Germany), Vol. 11, 1931, pp. 649-678. 12. Pickett G., “Equations for computing elastic constants from flexural and torsional frequencies of vibration of prisms and cylinders.”, Procc. Am. Soci. Testing Mats., Vol. 45, 1954, pp. 846-865. 13. Hasselmen D.P., “Tables for the computation of the shear modulus and Young modulus of elasticity from the resonant frequencies of rectangular prisms.”, The Carborundum Co., New York, Niagara Falls, 1961. 14. Spinner S. and Tefft W., “A method for determining mechanical resonance frequencies and for calculating elastic moduli from these frequencies.”, Procc. Am. Soci. Testing Mats., Vol. 61, 1961, pp. 1221-1238. 15. Rubben A. and Scharr G., “Method of determination The complete three- dimensional elastic compliance matrix of composite material.”, J. Comp. Structs., Vol. 27, 1987, pp. 760-773. 16. Deobald L.R. and Gibson R.F., “Determination of the elastic constants of orthotropic plates by a modal analysis/Rayliegh- Ritz technique.”, J. Soun. Vibr., Vol. 124, No. 2, 1988, pp. 269- 283. 17. Dossing O., “Structural testing. Part-II: Modal analysis and simulation.”,Bruel & Kjaer publishings, 1988, pp. 26-27. 18. Saify K.M. and Al-Temimi A.N., “A new proposal to compute the dynamic elastic moduli and Timoshenko shear coefficient of isotropic prismatic bars.”, Procc. 4th Sci. Engg. Conf., Univ. Baghdad, 1997, 2ME33. 19. Saify K.M., “Elasto-static &- dynamic investigation into composite plates & shells with new approaches for estimation of the elastic moduli.”, Ph.D. Thesis, Univ. Baghdad, 2000. 20. Levinson M., “A new rectangular beam theory.”, J. Soun. Vibr., Vol. 74, No. 1, 1981, pp. 81-87. 21. Timoshenko S., “On the correction for shear of the differential equation of transverse vibration of prismatic bars.”, Phil. Magz., Vol. 41, Series 6, 1921, pp. 125-127. Kamal K.M. Saify./ Al-khwarizmi Engineering Journal ,Vol.1, No. 2,PP 52-63 (2005) ٥٨ Given the physical & mechanical properties of the fiber (E-glass) and matrix (epoxy) constituents of the composite material (the present fabricated laminate) as listed below: The apparent elastic constants and mass density of the orthotropic laminate may be computed as followings (see Ref.[8,9] where deep details on the chemical compositions are discussed): 1 1 mmff mmffxy m m f f xy m m f f y mmffx VV VV G V G V G E V E V E EVEVE ρρρ ννν += += += += += ……(A -1) with the notations (f, m) refer to the fiber and matrix constituents respectively Appendix(B): The elasticity approach. Referring to the theoretical concepts of Tsai & Halpin in the micromechanics of composite material of two constituents, discussed earliarly, the four apparent elastic constants were driven in the form of: { } { } { } )1(G , )1( )1(E , )( 65xy43 210y KcKcKcKc KcKcKVEVEkE xy ffmmx +−=+−= +−=+= ν ……(B-1) where c and k are the effective “fudge” factor and the misalignment factor respectively. Their magnitudes are actually taken to be in the range (0.85-1.00) for the first factor and (0.0-0.4) for the second one, as proposed by the authors above. The six K’s coefficients in eq.(B-1) are computed from: Specification Fiber (E-glass), Vf=45% Matrix (epoxy) Vm=55% Young modulus 72.40Gpa 3.40Gpa Shear modulus 29.67Gpa 1.27Gpa Poisson’s ratio 0.220 0.34 Mass density 2.54x10-6 kg/mm3 1.22x10-6kg/mm3 APPENDIXES Appendix(A): Strength of material approach. Kamal K.M. Saify./ Al-khwarizmi Engineering Journal ,Vol.1, No. 2,PP 52-63 (2005) ٥٩ )()( )()( )(2 )(2 )()2( )2()2( )()2( )2()2( )(2)2( )()2( )(2)2( )()2( )1(2 6 5 4 3 2 1 0 mmfmf mmfmf f mmfm mmff m mfmffmf fmmfmmffmf mmfmmmf mmfmmfmmff mfmfm mmfmfmf mmfmm mmfmmmf mmff VGGGG VGGGG GK VGGG VGGG GK VKKGGKK VGKKVGKK K VKKGGKK VGKKVGKK K VKKGK VKKGGKK K VKKGK VKKGGKK K VVK −++ −−+ = ++ −− = −−+ +++ = −−+ +−+ = −−+ −−+ = −++ +−+ = −−= νν νν νν ……(B-2) with all other notations are as being defined in Appendix(A) Kamal K.M. Saify./ Al-khwarizmi Engineering Journal ,Vol.1, No. 2,PP 52-63 (2005) ٦٠ Table(2). The present T3BT readings of the composite cantilever strips under the proposed static deflection tests.(refer to Fig.(2)). Scheme of test Test-1 Test-2 Reading Items P0 (kg) δM (mm) δT (mm) P0 (kg) δM (mm) δT (mm) Strip-A Trial(1) 1.00 1.5875 5.0763 50.00 0.4571 1.2955 Trial(2) 1.50 2.3526 7.5231 75.00 0.6950 1.9544 Trial(3) 2.00 3.1274 10.0007 90.00 0.8338 2.3428 Strip-B Trial(1) 0.50 2.1473 6.8676 50.00 0.9374 2.8374 Trial(2) 0.75 3.0527 9.7637 75.00 1.3575 4.0835 Trial(3) 1.00 4.1033 13.1237 90.00 1.6415 4.9385 Strip-C Trial(1) 25.0 0.8230 2.1363 - - - Trial(2) 35.0 1.1516 2.9673 - - - Trial(3) 45.0 1.4879 3.8349 - - - Table(3). The present simple tensile test readings of the composite cantilever strips. Reading Items P0 (kg) ∆L* (mm) ∆b** (mm) Strip-A Trial(1) 300.0 0.800 0.098 Trial(2) 350.0 0.964 0.078 Trial(3) 400.0 1.096 0.072 Strip-B Trial(1) 300.0 1.943 0.089 Trial(2) 350.0 2.094 0.078 Trial(3) 400.0 2.430 0.072 Strip-C Trial(1) 300.0 0.710 - Trial(2) 350.0 1.312 - Trial(3) 400.0 1.571 - (*) Longitudinal elongation of the tested strip. (**) Lateral contraction of the strip. Table(4). Computations of the elastic constants of the composite strip from two present theoretical/experimental approaches. Approach The estimated elastic moduli of the present orthotropic material Ex (Gpa) Ey (Gpa) Gxy (Gpa) νxy Gxz (Gpa) Gyz (Gpa) T3BT (*) Trial(1) 6.613 2.361 1.985 0.221 1.779 0.879 Trial(2) 6.693 2.491 1.865 0.371 1.869 0.949 Trial(3) 6.713 2.471 1.855 0.371 1.899 0.929 Tensile test (**) Trial(1) 7.380 3.030 2.141 0.444 - - Trial(2) 7.120 3.280 2.441 0.324 - - Trial(3) 7.160 3.230 2.351 0.264 - - (*) refer to eqs.(5,6,7). (**) Ei=P0.Li/(Lj.Lk).∆Li, νij=∆Lj.Li/Lj. ∆Li (i,j,k=x,y,z or 1,2,3) with the help of eq.(7). Kamal K.M. Saify./ Al-khwarizmi Engineering Journal ,Vol.1, No. 2,PP 52-63 (2005) ٦١ Table(5). Comparison of the estimated results of the orthotropic elastic moduli of the present composite material according to variety of current approaches. Approach The main elastic constants Ex (Gpa) Ey (Gpa) Gxy (Gpa) νxy Gxz (Gpa) Gyz (Gpa) T3BT Average 6.673 2.441 1.905 0.321 1.849 0.919 σ(*) 0.043 0.057 0.064 0.071 0.051 0.029 Tensile test Average 7.220 3.180 2.311 0.344 - - σ(*) 0.123 0.108 0.126 0.075 - - Strength of material($) 7.081 4.003 1.502 0.294 - - Elasticity($$) 6.727 2.340 1.822 0.302 - - (*) Standard deviation ( ) 3 3 1 2∑ = − = i iaveragevalue ($) refer to Appendix(A). ($$) refer to Appendix(B). Kamal K.M. Saify./ Al-khwarizmi Engineering Journal ,Vol.1, No. 2,PP 52-63 (2005) ٦٢ Figure (2) Figure (1) Kamal K.M. Saify./ Al-khwarizmi Engineering Journal ,Vol.1, No. 2,PP 52-63 (2005) ٦٣ تقنية توموشينكو ثالثية العتبة لتقدير معامل المرونة الرئيسي للمواد )المتجانسة ثالثية البعد( جويج محسن جبر.د عدنان ناجي جميل التميمي.د كمال مصطفى كمال محمود سيفي.د كلیة الھندسة /قسم المیكانیك كلیة الھندسة /قسم المیكانیك بغداد-الكلية التقنية/قسم القوالب والعدد النهرينجامعة جامعة بغداد :ةخالصال طريقة مطورة جديدة ، لحساب ثوابت المرونة الستة والضرورية لتحليل التصرف الميكانيكي والداينماكي دة الصفات الهندسية، قد قدمت في هذه الورقة للمرة األولى من نوعها للشرائح المركبة المتجانسة متعام للقضبان المركبة الناتئة " تيموشنكو"اعتمدت الطريقة على نظرية . في األساس النظري وإجراءات العمل يتطرق الجانب العملي الى استخدام ثالث شرائح مركبة من المادة على طول المحاور . والمنحنية سكونيا سية الثالث وإيجاد االزاحات السكونية المناظرة لكل شريحة وبمستويين متعامدين من منظومة المحاور األسا تم إجراء خمسة محاوالت تجريبية بواقع اختبارين لكل محاولة وحساب معامالت . األساسية للتركيب إقرار الموثوقية للنتائج لغرض . ونظرية الميكانيك الدقيق للمواد المركبة" تيموشنكو"الصالبة وفق معادالت المختلفة، تم إجراء ثالث اختبارات كالسيكية للشد لجميع نماذج التجربة ومقارنة القراءات النهائية للصالبة لقد أثبتت الطريقة المقدمة كفاءتها وصحة نتائجها بإعطائها أقل االنحرافات العددية . في االتجاهات الرئيسية .متازت ببساطة ودقة الشكل الرياضي للحلمقارنةً بالطرق السابقة، كما ا