FULL SHEAR DEFORMATION FOR ANALYSIS OF THICK PLATE

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1 FULL SHAR DFORMATION FOR ANALYSIS OF THICK PLAT 1 IBARUGBULM, OWUS M., 2 NJOKU, KLCHI O., 3 ZIFULA, UCHCHI G. 1,2,3 Civil ngineering Department, Federal University of Tecnology, Owerri, Nigeria -mail: 1 ibeowums@gmail.com, 2 njoku@yaoomail.om.com, 3 uceci.eziefula@yaoo.com Abstract- Tis paper presents full sear deformation for analysis of tick plate. Te main assumption ere is tat te vertical sear strain is not zero and sear deformation is not divided into classical and sear deformation components. Wit tese assumptions and oter assumptions of traditional sear deformation teories, total potential energy for tick plate was derived in strict compliance wit te principles of teory of elasticity. Te paper also derived te vertical sear stress profile from matematical principles. Te total potential energy was subjected to direct variation by differentiating it, in turn, wit te coefficients of deflection, rotation in x direction and rotation in y direction. Tis variation resulted into tree simultaneous direct governing equations. Numerical example for a case of a plate wit all te edges simply supported was used to test te new metod. It was observed tat te values of non dimensional forms of displacements and stresses from te present stu agree wit te values from previous studies. Also observed is tat te values of te in-plane quantities did not vary wit span-dept ratio (). Tey are all equal to te values from classical plate teory (CPT) for all te values of. However, te out-of-plane quantities varied wit span-dept ratio from equal to 4 up to equal to 20, after wic tey become constant and approximately equal to values from CPT. Tis sows tat te present metod is reliable and sufficient for tick plate analysis. Keywords- sear deformation, vertical sear strain, stress, deflection, rotation, potential energy I. INTRODUCTION Te callenge encountered in classical plate teory (CPT) is te assumption tat vertical sear strains are zeros: = du = 0 ; = dv = 0 Tis assumption leads to erroneous expressions for in-plane displacements: u = z dw dw ; v = z u = z dw instead of ; v = z dw Te erroneous assumption is referred to as one of te Kircoff ypotesis and stated as: " te vertical line, wic is initially straigt and normal to middle surface before bending remains straigt and normal to te middle surface after bending" (Ibearugbulem et al., 2014). Tis assumption is reasonable for tin plates (span-dept ratio is more tan 20). For tick plates, tis assumption will not be reliable. To improve on CPT, refined plate teories (RPT) evolved. some of te refined plate teories include first order sear deformation teory, FSDT (Reissner, 1945) and Mindlin, 1951), second order sear deformation teory - SSDT, iger order sear deformation teory - HSDT, trigonometric sear deformation teory - TSDT, ypabolic sear deformation teory - HPSDT, exponential sear deformation teory - SDT etc. (Sayyad and Gugal, 2012 Sayyad, 2011; Sayyad and Gugal, 2012; Kruszewski, 1949; Cowper, 1966 and 1968; Bickford, 1982, Krisna Murty, 1984, Heyliger and Red, 1988; Bimaraddi and Candrasekara, 1993, Gugal and Simpi, 2002; Gugal, 2006; Gugal and Sarma, 2009). However, it is important to note tat all tese RPT make a common erroneous assumption. Tey divided te plate deformation into two components - CPT deformation (c, uc and vc) and sear deformation (s, us and vs). To get te total deformation, tey ad to add te CPT deformations and sear deformations ( = c + s, u = uc + us and v = vc + vs). By making tis assumption, te erroneous Kircoff ypotesis was reintroduced in te RPT as: u = z and v = z. In te end erroneous total in-plane displacements were used in te RPTs as: = dw + du dw + dz = du dz = dw + dv dw + dz = dv dz In order to avoid tis erroneous assumption, tis present stu decided to use full in-displacements and rotations (u, v, x and y) witout dividing tem into CPT in-plane displacements and rotations (uc, vc, xc and yc) and sear deformation in-plane displacements and rotations (us, vs, xs and ys). In tis case, single displacements and rotations (u, v, x and y) were used. In te formulation te following assumptions were made: (i) Te displacements, u, v and w are small wen compared wit plate tickness; (ii) Te deformations are woles (not divided into CPT and sear deformations components) see figure 1; (iii) Te in-plane displacements, u and v are differentiable in x, y and z axes, wile te out-ofplane displacement (deflection), w is only 1

2 differentiable in x and y axes. Tis means tat te derivatives of w wit respect to z are zeros. Consequently, z = 0; (iv) Te effect of te out-ofplane normal stress on te gross response of te plate is small wen compared wit oter stresses. Tus, it can be neglected. Tat is z = 0.; (v) Te vertical sear stress (pxz or pyz) distributed troug te plate tickness is te product of ordinary vertical stress ( xz or yz ) and sape factor, G. Tat is: τ = τ G ; τ = () γ τ = τ G ; τ = γ () ; (vi) Te vertical line tat is initially normal to te middle surface of te plate before bending remains straigt but no longer normal to te middle surface after bending. Te specific objectives of te present stu include: (i) To formulate sape factor, G(z) for vertical sear stress matematically in line wit works of Timosenko (1921).; (ii) To develop a direct governing simultaneous equations for tick plate analysis using direct variational calculus.; (iii) To use polynomial displacement functions in te simultaneous equations for pure bending analysis of tick plate. II. DFORMATION AND DISPLACMNT FILD Full Sear Deformation For Analysis of Tick Plate Tat s = z d [A + A ] (8) = du = d [A + A ] (9) = dv = d [A + A ] (10) From te assumptions, we ave tree displacements, u(x,y,z), v(x,yz) and w(x,y). Ibearugbulem (2015) in is lecture note defined deflection and rotations, w, x and y as: w = A (1) d = A (2) d = A (3) By using figure 1 and Pytagoras teory (bearing in mind small angles are considered ere), Ibearugbulem (2015) wrote: u = z = za d v = z = za d (4) (5) III. NGINRING STRAIN COMPONNTS From assumptions erein, te strain normal to z axis is zero. Tis left us wit only five engineering strain components x, y, xy, xz, and yz. Using equations (1), (4) and (5) tese strain components are define as: = du = za d (6) = dv = za d (7) = du + dv = za d + za d Figure 1: Deformation of a section of a tick plate IV. CONSTITUTIV QUATIONS Te constitutive equations for five stress components are: = 1 + = 1. z A d + A d (11) = 1 + = 1. z A d + A d (12) (1 ) = 2(1 ). (1 ) = 1. z[a + A ] d (13) (1 ) = 2(1 ). (1 ) = 2(1 ). [A + A ] d (14) (1 ) = 2(1 ). 2

3 = (1 ) 2(1 ). [A + A ] d Full Sear Deformation For Analysis of Tick Plate (15) V. TOTAL POTNTIAL NRGY Total potential energy is given as: = U + V (16) Were U is te strain energy given as: U =. dz = ) dz (17) V is te external work given as: Substituting equation (18), (19) and (26) into equation (16) and using equations (20) to (24) we oain: = D {[ A 2 d A + 2 A A + A d +A d ] +[ A A A 2 A 2 ] d 6 (1 ) a [A + 2 A A + A ] d 6 (1 ) a [A + 2 A A + A ] d } V = FF (18) Substituting equations (6) to (15) into equation (17) gives: U = 2(1 ) +B A A d d ] {[ B A d +[B A A d d + B A d B (1 ) 2 B (1 ) 2 B (1 ) A A A + A d [A + 2 A A + A ] d [A + 2 A A + A ] d Were B = z dz B = 1 dz = t 12 ] } (19) (20) = t = 12B (21) t Let te span-dept ratio, be define as: = a t t = a (22) Let te flexural rigidity of te plate be defined as D = 1. B t = 12(1 (23) ) Substituting equation (22) into equation (21) gives: B = t = 12 B a (24) Matematically, d d = d (26) FF (27) VI. DIRCT VARIATION OF TOTAL POTNTIAL NRGY QUATIONS Tis total potential energy (equation 27) contains tree unknown coefficients of deflection, rotation in x axis and rotation in y axis (A 1, A 2 and A 3 respectively). Differentiating tis total potential energy equation wit respect to A 1, A 2 and A 3 in turn will give tree simultaneous equations in non dimensional forms of axes R, Q and S (for x, y and z axes): A r D r r a r r r A = C r r r A C (28) Were C = d ab FF (29) da C = d ab FF da (30) C = d ab FF da (31) R = x x = ar (32) a Q = y y = bq (33) b C Aspect ratio, P = b b = ap (34) a S = z z = St (35) t r = 6 (1 ) k + 1 P k (36) r = 6 (1 ) k (37) r = 6 (1 ) k (38) r = k + 1 2P k + 6 (1 ) k (39) 1 + r = 2P k (40) r = 1 P k + 1 2P k + 6 (1 ) P k (41) 3

4 Were k = k = d drdq dr dq k = d dq dr dq k = d dr dr dq k = d dq dr dq dr dq; VII. VRTICAL SHARING STRSS From strengt of materials, te equation sear stress across te dept of a section is given as: τ = V H Ib or τ = V H (42) Ib V, H, I and b are designations for transverse sear force, first moment of area, second moment of inertia and breadt of te section respectively. From figure 2 and using matematical principle te first moment of area is oained as: H = b 2 t 4 z (43) Based on common knowledge te second moment of inertia for a rectangular section is given as: I = 12 Full Sear Deformation For Analysis of Tick Plate (44) Figure 2: A rectangular cross section Substituting equations (43) and (44) into equation (42) we oain: τ = V z t = V G(z) (45) τ = V z V = t G(z) (46) τ = V (47) τ = V (48) G(z) = z t (49) 4 VIII. DFINITION OF SOM QUANTITIS Let us define non-dimensional form of te displacements and stress components according to Sayyad et al. (2012) as: w = 100w u (50); u = (51) v = qt v qt qt (52); σ = σ (53) q σ = σ q (54); τ = τ (55) q τ = τ (56); τ q = τ (57) q IX. NUMRICAL PROBLM Determine te deflection at te center of ssss tick plate. Determine also te in-plane stresses and te vertical sear stresses at te edges of te plate. Polynomial displacement function sall be used. Te polynomial displacement function, is given as: = (R -2R 3 + R 4 )(Q - 2Q 3 + Q 4 ). Te values of ki and Frq erein are given as: k1= ; k2 = ; k3 = k4 = ; k5 = ; Frq = 0.04 X. RSULTS AND DISCUSSIONS A close look at tables 1 to 3 reveals tat te in-plane quantities, u, x, y, and xy are not affected by te span-dept ratio (). Te values of tese quantities for all te span-dept ratios are equal and are equal to te values from CPT. However, tese quantities are all functions of te plate tickness and tey vary linearly wit te plate tickness. On te oter and, te out-of-plane quantities, w and xzp vary wit te span-dept ratio. Teir values decrease as te spandept ratio increases. It is observed tat tis variation of te out-of-plane quantities decreases as te spandept ratio increases and becomes insignificant from span-dept ratio of 20. From tis point, te values of te out-of-plane quantities become equal to te value oained from te classical plate analysis. Anoter spectacular observations were made on tables 4 to 6. Here it is observed tat non dimensional in-plane quantities do not vary wit te span-dept ratio as teir values are same as te values from te CPT. Similarly, te values for vertical sear stress are te same but not equal to te values from CPT. It is worty of note tat it is te deflection tat vary wit te span-dept ratio. As in te case of dimensional quantities (tables 1 to 3) te variation wit span-dept ratio stopped at span-dept ratio of 20. Te values of non dimensional form of deflection for span-dept ratios of 100 and above are equal to te value from CPT. Te result from te present stu was compared wit te values from previous studies. Tis is sown on tables 7 and 8. It is observed tat it is te only te values from te present stu tat maintained te

5 caracteristics of same values for te non dimensional forms of te in-plane quantities and te vertical sear stress for span-dept ratios of 4 and 10. However, it sall be seen tat te values from te present stu vary little wit te values from previous studies. It is worty of note tat te values from te previous studies do not tally wit one oter as te values from te present stu did not tally wit tem irrespective of te closeness. It is te believe of tis present stu tat te erroneous assumption regarding te CPT component of te vertical sear strain (wic is assumed to be zero contrary to te governing assumption of non-zero vertical sear strain) is responsible for uncaracteristic beavior of te values from te previous studies. xcept te non dimension form of deflection, te values of oter non dimensional quantities sould be te same for different span-dept ratios. Table 1: Displacements and stresses for P =1 Full Sear Deformation For Analysis of Tick Plate Table 3: Displacements and stresses for P =2 Table 4: Non dimensional forms of Displacements and stresses for P =1 Table 5: Non dimensional forms of Displacements and stresses for P =1.5 Table 2: Displacements and stresses for P =1.5 Table 6: Non dimensional forms of Displacements and stresses for P =2 5

6 Table 7: Non dimensional forms of Displacements and stresses for P =1 and = 4 Table 8: Non dimensional forms of Displacements and stresses for P =1 and = 10 RFRNC [1]. Bimaraddi A., and Candrasekara K., (1993), Observations on iger-order beam teory, Journal of Aerospace ngineering Proceeding of ASC, Tecnical Note., 6, pp [2]. Bickford W. B., (1982), A consistent iger order beam teory, Development in Teoretical Applied Mecanics, SCTAM, 11, pp [3]. Cikaltankar, S. B., Sayyad, I.I., Nandedkar, V. M. (2013). Analysis of Ortotropic Plate By Refined Plate Teory. International Journal of ngineering and Advanced Tecnology (IJAT) ISSN: , Volume-2, Issue-6, pp [4]. Cowper G. R., (1966), te sear coefficients in Timosenko beam teory, ASM Journal of Applied Mecanics, 33, pp [5]. Cowper G. R., (1968), On te accuracy of Timosenko s beam teory, ASC Journal of ngineering Mecanics Division, 94(6), pp Full Sear Deformation For Analysis of Tick Plate [6]. Gugal Y. M., (2006), A simple iger order teory for beam wit transverse sear and transverse normal effect, Departmental Report 4, Applied mecanics Department, Government college of ngineering, Aurangabad, India, pp [7]. Gugal Y. M., and Sarma R., (2009), Hyperbolic sear deformation teory for flexure and vibration of tick isotropic beams, International Journal of Computational Metods, 6(4), pp [8]. Gugal Y. M., and Simpi R. P., (2002), A review of refined sear deformation teories for isotropic and anisotropic laminated beams, Journal of Reinforced Plastics and Composites, 21, pp [9]. Heyliger P. R., and Red J. N., (1988), A iger order beam finite element for bending and vibration problems, Journal of Sound and vibration, 126(2), pp [10]. Hildebrand F. B., and Reissner. C., (1942), Distribution of stress in built-in beam of narrow rectangular cross section ASM Journal of Applied Mecanics, 64, pp [11]. Ibearugbulem, O. M., ze, J. C. and ttu, L. O. (2014). nergy metods in teory of rectangular plates (use of polynomial sape functions). Liu House of xcellence Ventures, Owerri: ISBN [12]. Ibearugbulem, O. M. (2015). Advanced teory of elasticity Lecture note. Unpublised: Federal University of Tecnology, Owerri, Nigeria [13]. Krisna Murty A. V., (1984), Toward a consistent beam teory, AIAA Journal, 22, pp [14]. Mindlin, R.D. and Deresiewicz, H. (1954). "Tickness-sear and flexural vibrations of a circular disk," Journal of Applied Pysics, 25, [15]. Red, J. N., A simple iger order teory for laminated composite plates, ASM Journal of Applied Mecanics 51 (1984) [16]. Reissner,. and Stavsky, Y. (1961) Bending and stretcing of certain type of eterogeneous aelotropic elastic plates, Journal of App. Mec., 28, pp [17]. Sayyad, A. S. & Gugal, Y. M. (2012). Bending and free vibration analysis of tick isotropic plates by using exponential sear deformation teory. Applied and Computational Mecanics 6 (2012) 65 8 [18]. Sayyad, Attesamuddin S. (2011). Comparison of various sear deformation teories for te free vibration of tick isotropic beams. International journal of civil and structural engineering Volume 2, No 1, pp [19]. Timosenko S. P., (1921), on te correction for sear of te differential equation for transverse vibrations of prismatic bars, pilosopical magazine, series 6, 41, pp