Theory of Elasticity Formulation of the Mindlin Plate Equations

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1 Theor of lasticit Formulation of the indlin Plate quations Nwoji, C.U. #1, Onah H.N. #, ama B.O. #3, Ike, C.C. * 4 #1, #, #3 Dept of Civil ngineering, Universit of Nigeria, Nsukka, nugu State, Nigeria. *4 Dept of Civil ngineering nugu State Universit of Science & Technolog, nugu State, Nigeria. 1 clifford.nwoji@unn.edu.ng hginus.onah@unn.edu.ng 3 benjamin.mama@unn.edu.ng 4 ikecc007@ahoo.com Abstract-In this work, the mathematical theor of elasticit has been used to formulate and derive from fundamental principles, the first order shear deformation theor originall presented b indlin using variational calculus. A relaation of the Kirchhoff s normalit hpothesis was used to account for the effect of the transverse shear strains in the behaviour of the plate. This made the resulting theor appropriate for use for moderatel thick plates. A simultaneous use of the strain-displacement relations for small-deformation elasticit, stress-strain laws and the stress differential equations of equilibrium was used to obtain the differential equations of static fleure for indlin plates in terms of three unknown generalised displacements. The equations were found to be coupled in the unknown displacements; but reducible to the Kirchhoff plate equations when the removed Kirchhoff normalit hpothesis was introduced. This showed the Kirchhoff plate theor to be a specialization of the indlin plate theor. The theor of elasticit foundations of the indlin and Kirchhoff plate theories are thus highlighted. Kewords: indlin plate theor, theor of elasticit, shear deformation. BACKGROUND I. INTRODUCTION Plates are three dimensional structural members bounded b two surfaces: z h/, called top and bottom surfaces, and the inplane region on the domain 0 a, 0 b, where the origin is assumed at a corner point ( = 0, = 0, z = 0). The distance, h, between the top and bottom surfaces, is called the thickness, and this dimension is much smaller than the inplane dimensions a, and b [1, ]. Generall, plates are categorized into three, namel: thin plates, moderatel thick plates and thick plates, depending on the ratio of the thickness to the least inplane dimension (h/a) where a b [3, 4]. The can also be classified based on the nature of the deformation as small-deformation plates, large deformation plates, or according to their material properties as homogeneous, heterogeneous plates or isotropic, anisotropic plates etc. Plate problems are elasticit problems that are formulated b simultaneous satisfaction of kinematic, material constitutive laws and the differential equations of equilibrium subject to the geometrical and loading conditions [5]. The complete closed form formulation is often difficult, and approimations are commonl emploed to reduce the three dimensional elasticit problem to two dimensional approimations. II. PLAT THORIS AND ODLS Several theories and models have been formulated to describe plate behaviour. The include Kirchhoff theor, indlin [6] theor, Reissner [7] theor, Shimpi [8] refined theor, Redd [9] theor, Levinson [10] theor, Osadebe [11] plate model, etc. Kirchhoff plate theor, also called the classical small deflection theor of thin plates is the simplest and most commonl used theor of plates. It is a linear infinitesimal theor, suitable for thin isotropic, and orthotropic plates. The equation for static bending of thin istropic plates with variable fleural rigidit is given as: D w D w D w D (, ) w(1 ) q(, ) z (1) where D(, ) is the fleural rigidit, w is the transverse deflection, is the Laplacian () q z (, ) is the distribution of transverse load on the plate, is the Poisson s ratio, and and are the inplane Cartesian coordinates. DOI: /ijet/017/v9i6/ Vol 9 No 6 Dec 017-Jan

2 The governing equation is a fourth order linear partial differential equation with variable coefficients in w, the unknown dependent variable of the equation. For homogeneous plates, the governing equation simplifies to the fourth order equation with constant coefficients given as: w w w 4 D D w q (, ) 4 4 z (3) where 4 (4) 4 is the biharmonic operator. Von Karman presented the large deflection thin plate theor as the sstem of equilibrium and compatibilit equations given b: D 4 q w (, ) Lw (, ) z h h (5) (, ) L( w, w) (6) where w w w Lw (, ) (7) (, ) is the Air s stress function, is the Young s modulus of elasticit, h is the plate thickness and 4 w w Lww (, ) (8) Von Karman s large deflection thin plate theor considers the longitudinal displacement of the middle plane cause b the deflection and the associated stress. The governing equations are coupled, non-linear partial differential equations of fourth order in the dependent variables w(, ) and (, ). Kirchhoff classical plate theor, also called the Kirchhoff-Love plate theor was formulated based on the following kinematic assumptions also called the Kirchhoff hpothesis [1-3] (i) straight lines normal to the middle surface before bending deformation remain straight after bending deformation (ii) straight lines normal to the mid surface before bending deformation remain normal to the middle surface after bending deformation (iii) the thickness of the plate does not change during the bending deformation. The classical Kirchhoff plate theor has been found to give satisfactor results for thin plates, and despite the obvious mathematical simplicit of the governing equations, the theor has the following limitations [1] (i) the accurac of the Kirchhoff plate theor decreases with increase in plate thickness, with rapidl increasing or localized forces and in problems of stress concentration around openings in the plate. (ii) the Kirchhoff plate theor assumes a displacement field for which the strain field is derived. The plate is assumed to be in plane stress state and the stress field derived from the strain field and the stress-strain laws for plane stress conditions. The stress field does not account for the transverse shear stress under the transversel applied load. However, the consideration of equilibrium confirms that the shear stresses in the transverse direction eist. (iii) since the Kirchhoff plate theor disregards the transverse shear deformation, it results in considerable errors when applied to moderatel thick plates for which the transverse shear deformation plas an important role in the behaviour. (iv) the Kirchhoff plate theor uses an incomplete set of boundar conditions resulting in errors especiall around the plate boundaries, as well as in support reactions. These limitations of the Kirchhoff plate theor have led to the development of other plate theories in order to address some of the lapses of the Kirchhoff plate theor. Some of these plate theories developed in answer to the limitations of the Kirchhoff plate theor are Reissner plate theor, indlin plate theor, Redd plate theor, Levinson s plate theor, Shimpi s refined plate theor, etc. In this paper, a sstematic formulation of the indlin plate theor is done using the methods of the theor of elasticit. The formulation of the indlin plate equations uses a displacement based approach. DOI: /ijet/017/v9i6/ Vol 9 No 6 Dec 017-Jan

3 III. RSARCH AI AND OBJCTIVS The research aim is to present a theor of elasticit formulation of the indlin plate equations for static fleure using the equilibrium approach. The objectives are: (i) to obtain the displacement field, the strain field, and the stress fields over the plate due to transverse load; (ii) to obtain the internal force resultants; (iii) to use the differential equations of equilibrium and obtain the internal force resultants in terms of the generalized displacements; (iv) to obtain the governing partial differential equations from a consideration of the differential equations of equilibrium. IV. THORTICAL FRAWORK Plate problems of static fleure and dnamic fleure are elasticit problems, and are formulated b simultaneousl considering the strain-displacement (kinematic) relations, the material constitutive (stress-strain) laws, and the differential equations of equilibrium. KINATIC RLATIONS The kinematic relations for small-displacement assumptions are: u v w zz z u v v w z z u w z (14) z where,, zz are normal strains,, z, and z are shear strains, u(,, z), v(,, z) and w(,, z) are components of the displacement field in the,, and z Cartesian coordinate directions. STRSS-STRAIN LAWS The stress-strain laws for homogeneous isotropic materials are: 1 ( ) (15) zz 1 ( ) (16) zz 1 ( ) (17) zz zz (1 ) (18) G z (1 ) z z (19) G z (1 ) z z (0) G where,, zz are normal stresses,, z, z are shear stresses, is the Young s modulus of elasticit, is the Poisson s ratio and G is the shear modulus. (9) (10) (11) (1) (13) DOI: /ijet/017/v9i6/ Vol 9 No 6 Dec 017-Jan

4 DIFFRNTIAL QUATIONS OF STATIC QUILIBRIU The differential equations of static equilibrium of an infinitesimal plate element; in the absence of bod forces are given in terms of stresses as: z z 0 (1) z z 0 () z z zz z 0 (3) In terms of internal force resultants, the equations are: Q Q qz (, ) 0 (4) Q (5) Q (6) where Q, and Q are shear force resultants,, are the bending moments and is the twisting moment. V. THORY OF LASTICITY FORULATION OF INDLIN PLAT QUATIONS FUNDANTAL ASSUPTIONS (i) (ii) (iii) (iv) (v) The following assumptions are made: The plate deflection under transverse loads are small and finite, hence small displacement relations of elasticit theor are valid. Normals to the plate undeformed middle surface before deformation remain straight and unstretched in length but the are not necessaril normal to the plate middle surface after deformation. The implication of the relaation of the normalit rule is that the shear strains z and z are not zero, and the indlin theor of plates considers the effect of shear deformation. Stresses normal to the plate middle surface are negligible, thus zz 0 (7) There is no deformation in the middle surface of the plate, which remains a neutral surface. The state of deformation is described completel b the transverse displacement in the z direction of the middle surface w(,, z = 0), and (, ), (, ) where, are the rotations about the and aes of lines normal to the middle surface before deformation. THORY OF LASTICITY FORULATION OF TH INDLIN PLAT QUATIONS The theor of elasticit formulation of indlin plate equations for static fleure is based on the kinematic relations, the material constitutive relations, and the differential equations of static equilibrium. DISPLACNT FILD The displacement field at a generic point (,, z) of the plate is given b the deflections of the plate middle surface as follows: wz (,, ) wz (,, 0) w (, ) (8) uz (,, ) uz (,, 0) z (, ) (9) vz (,, ) vz (,, 0) z (, ) (30) where u, v, and w are the components of the displacement field in the,, and z coordinate directions, respectivel. DOI: /ijet/017/v9i6/ Vol 9 No 6 Dec 017-Jan

5 STRAIN FILDS The strain fields are obtained using the kinematic relations as follows: z (31) z (3) 0 (33) zz w w z z z STRSS-STRAIN LAWS The stress-strain laws for the case when zz 0 are; for isotropic, homogeneous materials, ( ) 1 ( ) 1 G (1 ) z Gz z (1 ) z Gz z (1 ) STRSS FILDS (OR STRSS DISPLACNT QUATIONS) The strain fields are substituted into the stress-strain laws to obtain the stress fields as follows: z 1 z 1 z (1 ) w z G w z G INTRNAL FORC RSULTANTS The internal force resultants,,, Q and Q are obtained as the following integrals over the plate thickness: h/ zdz (47) h/ (34) (35) (36) (37) (38) (39) (40) (41) (4) (43) (44) (45) (46) DOI: /ijet/017/v9i6/ Vol 9 No 6 Dec 017-Jan

6 Q Q h/ zdz (48) h/ h/ zdz (49) h/ h/ dz (50) z h/ h/ dz (51) z h/ where, are the bending moments, is the twisting moment, Q and Q are shear forces. B substitution of the strain fields, and integration of the resulting epressions with respect to the z coordinate, we obtain the internal force resultants as follows: h/ z dz (5) / 1 h D (53) h/ z / 1 dz h (54) D (55) h/ z (1 ) dz h/ (56) D(1 ) (57) Shear force correction coefficients k are introduced into the definition for the shear force epressions in order to consider the parabolic distribution of shear stresses in the z direction. Appling the shear correction coefficient (factor), h/ h/ w Q k zdz k G dz h/ h/ (58) w w Q kgh DQ (59) h/ h/ w Q k zdz k G dz (60) h/ h/ w w Q kgh DQ (61) DIFFRNTIAL QUATIONS OF STATIC QUILIBRIU The differential equations of static equilibrium for a differential element of the plate when bod forces are absent are quations (1) (3). In terms of internal force resultants, the differential equations of equilibrium are quations (4) (6). The three differential equations of equilibrium can be epressed as a single equation as follows: q (, ) 0 z (6) B substitution of epressions obtained for,, and, the differential equation of equilibrium become: DOI: /ijet/017/v9i6/ Vol 9 No 6 Dec 017-Jan

7 D(1 ) D D 3 3 q (, ) 0 z Simplifing, D qz (, ) (64) Also, from quation (4), w w kgh kgh q (65) Simplifing, q q w kgh DQ (66) From quation (5), we obtain upon substitution, 1 w D kgh (67) After simplification, D w (1 ) (1 ) Q kgh (68) Similarl, from quation (6), we obtain, upon substitution, D(1 ) w D Q kgh (69) After simplification, D w (1 ) (1 ) Q kgh (70) Let s D (71) where s, the arcus moment, is defined as: s 1 (7) Then, the governing partial differential equations of indlin plates are: D w (1 ) (1 ) DQ 0 (73) D w (1 ) (1 ) DQ 0 (74) DQ( w ) qz(, ) 0 (75) When z 0 (76) i.e. w (77) and z 0 (78) w (79) (63) DOI: /ijet/017/v9i6/ Vol 9 No 6 Dec 017-Jan

8 The indlin plate equations become: w w D (80) w w D (81) w D(1 ) (8) 4 qz w w (83) D Q = 0 (84) Q = 0 (85) VI. DISCUSSION This stud has successfull presented the first principles formulation and derivation of the indlin first order shear deformation plate theor using the methods of the theor of elasticit. The kinematics assumptions of the classical Kirchhoff plate theor was etended b removing the normalit restrictions, thereb including the transverse shear strains. The transverse shear strains were found from quations (45, 46) to be constant through the plate thickness, whereas the stress-equilibrium equations predict a quadratic (parabolic) variation of the transverse shear stresses over the plate thickness. In order to make the shear forces (Q, Q ) computed in the present stud to be equal to those obtained using the parabolic variation of transverse shear stresses obtained from the stress equilibrium equations, shear correction factors were introduced in the definition of the internal shear stress resultants (quation (58) and (60)). In the derivation of the stress-strain equations of the plate, plane stress state assumption is made, similar to the classical Kirchhoff zeroth order shear deformation plate theor. The plane stress form of the stress-strain law quations (37-41) is used. In both the classical Kirchhoff zeroth order shear deformation plate theor and the indlin first order shear deformation plate theor, the assumption of inetensibilit of the middle plane ielding zz 0, and/or straightness of transverse normals can be removed. Such etensions of the assumptions of the present work can ield second order and higher order shear deformation models of plates under static, and/or dnamic fleure. When the transverse shear stresses are made to be zero, i.e. w and w, the equations of the indlin plate theor reduce to the equations of the classical Kirchhoff zeroth order shear deformation plate theor. Bending moment resultants were obtained from a simultaneous consideration of kinematics and stress-strain laws as quations (53) and (55). The twisting moment resultant was similarl obtained as quation (57). The shear force resultants were found, similarl as quations (59) and (61). The sstem of differential equations of static equilibrium epressed in terms of stress resultants were then used, together with the internal force resultants obtained from the simultaneous consideration of the kinematic assumptions, and the stress-strain laws to obtain the governing differential equations of equilibrium of indlin plates as quations (64), (68) and (70) or alternativel as quations (73) (75). The equations are a sstem of three coupled partial differential equations in the three unknown variables w(, ), (, ) and (, ). Despite the man obvious merits of the indlin plate theor, it suffers the following limitations: (i) the theor ields a constant shear strain distribution over the plate thickness, in violation of the known parabolic distribution of shear strain obtained in a rigorous application of theor of elasticit. The introduction of the shear stress correction factor merel ensures that the resultant shear stress agrees with the predictions of the theor of elasticit, but the distributions of the shear stress over the plate thickness are in violation of the theor of elasticit. (ii) the shear free boundar condition on the plate surface is violated (iii) the theor is two-dimensional and cannot correctl predict practical three dimensional plates. VII. CONCLUSIONS The conclusions of the stud are as follows: (i) the governing partial differential equations of equilibrium of indlin plates have been successfull derived using a displacement approach in the theor of elasticit. DOI: /ijet/017/v9i6/ Vol 9 No 6 Dec 017-Jan

9 (ii) the governing equations of indlin plates, were derived from a simultaneous consideration of the kinematic equations, stress-strain laws and differential equations of equilibrium and thus simultaneousl satisf the kinematic, stress-strain laws, as well as the equilibrium equations of elasticit theor. (iii) the governing equations are a sstem of three coupled partial differential equations in terms of three unknown generalized displacements, and can be solved mathematicall. (iv) the indlin plate equations simplif to the classical Kirchhoff zeroth order shear deformation plate equations when the normalit conditions are strictl imposed in the kinematic equations; making the Kirchhoff plate theor a special case of the indlin plate theor. (v) the Kirchhoff plate theor is easil derivable from the indlin plate theor when w, w. RFRNCS [1] J.N. Redd Theor and Analsis of lastic Plates, Talor and Francis, London, [] K. Chandrashekhara Theor of Plates, Universities Press (India) Limited Hderabad, 011. [3] R. Szilard Theories and Applications of Plate Analsis: Classical, Numerical and ngineering ethods, John Wile and Sons Inc, New Jerse, 004. [4] Ashwin Balasubramanian Plate Analsis with Different Geometries and Arbitrar Boundar Conditions, Sc Thesis, echanical ngineering, Facult of Graduate School, the Universit of Teas at Arlington, December 011. [5] V.K. Sebastian An lastic Solution for Simpl Supported Rectangular Plates, Nigerian Journal of Technolog (NIJOTCH) Vol.7 No1, pp 11-16, September, 1983 [6] R.D. indlin Influence of Rotar Inertia on Fleural otion of Isotropic, lastic Plates Journal of Applied echanics Vol. 18 No 1, pp , [7]. Reissner The effect of shear deformation on the bending of elastic plates, Journal of Applied echanics, Vol 1 pp A69, [8] R.P. Shimpi Refined Plate Theor and its Variants, AIAA Journal, Vol 40, pp [9] J.N. Redd A refined non-linear theor of plates with transverse shear deformation, International Journal of Solids and Structures, Vol. 0 pp , [10]. Levinson An accurate simple theor of statics and dnamics of elastic plates, echanics Research Communications, Vol.7, pp , [11] N.N. Osadebe Differential equations of small deflection analsis of thin elastic plates possessing etensible middle surface, Universit of Nigeria Virtual Librar, [1] C.C. Ike quilibrium Approach in the Derivation of Differential quations for Homogeneous Isotropic indlin Plates, Nigerian Journal of Technolog (NIJOTCH) Vol 36, No., pp , April 017. AUTHOR PROFIL ngr Dr. Clifford U. Nwoji obtained his civil engineering education from Universit of Nigeria Nsukka. He is a member of Nigerian Societ of ngineers and also a registered engineer with Council for Regulation of ngineering in Nigeria. He has substantial industrial eperience. Since obtaining his higher degrees in engineering he has been teaching structural engineering courses at both undergraduate and postgraduate levels. He has successfull supervised higher degree theses. He has to his credit a good number of published research papers in reputable international journals. Hginus Nwankwo Onah (B.ng,.ng, PhD) Born in Nigeria on a 4 th 1959, I obtained m B.ng. (Civil ngineering) in After brief work in Lagos Nigeria, I travelled to France in 1987 where I obtained.ng (Civil ngineering) at INSA de Lon and Ph.D (Structural echanics) at CNA Paris in Then I worked as a researcher at Research and Development (R&D) unit of RNAULT otors Reuil almaison France, Department of ngineering Universit of anchester ngland and Becchis Osiride of FIAT otors Torino Ital for one ear each. From Januar 1996 to date as a lecturer in Civil ngineering Department of Universit of Nigeria, Nsukka, I teach and supervise both undergraduate and Postgraduate students and carr out research activities in mechanics and behavior of structures ngr Dr Benjamin Okwudili ama I got m PhD in the ear 009 in the Universit of Nigeria, Nsukka. master s degree in civil Structural ngineering in 199 in the same School. first degree in Civil ngineering in then Anambra State Universit of Technolog, nugu. I have been lecturing in the Department of Civil ngineering, Universit of Nigeria, Nsukka from 1994 till date. ngr Dr. Charles C. Ike is a lecturer at the nugu State Universit of Science and Technolog, nugu, Nigeria. He holds a PhD in structural engineering, and is a member of the Nigerian Societ of ngineers. He is registered with the Council for the Regulation of ngineering in Nigeria (CORN). He has numerous publications in reputable international journals. DOI: /ijet/017/v9i6/ Vol 9 No 6 Dec 017-Jan