17 FLEXURE OF THICK COMPOSITE PLATES The mechanical behavior of a laminated plate as studied in Chapter 12 involves the definition of stress resultants NN.T and moment resultants MM M These resultants are obtained from the membrane stresses o.T The other stress components,o,t have not been taken into account until now. In this chapter we note how these stresses exist and have influence on the mechanical behavior of the laminate.We will also examine the configurations of plates for which the influence of these stresses are significant (for example,plates with relatively high thicknesses,justifying the title given for this chapter).This study is based on the previous definition of the displacement parameters based on integral displacement forms,and constitutes an approach analogous (and also original)to that used in Chapter 15 for the description of the bending of composite beams. 17.1 PRELIMINARY REMARKS 17.1.1 Transverse Normal Stress o, The plate is situated as in Chapter 12 from which the name of transverse normal stress o..Such stress appears due to the application of a transverse load (con- centrated or distributed)which will cause bending of the plate. A very local concentration of load in a very small zone cannot be examined with the theory of plates,which is not able to provide spatial distribution of the stresses in the neighborhood of the point of load application.This phenomenon is complex even in three-dimensional numerical modeling. Therefore,what will be presented will not be valid in the immediate surround- ings of a very local transverse load (for example,an insert). A distributed load gives rise to the stresses o.with small amplitude as compared with the stresses ox and o.This is the reason why o.is often neglected. 17.1.2 Transverse Shear Stresses tz and tz Due to the assumption of perfect bonding between the plies,the stress vector remains continuous across an interfacial element with normal vector=2,between two consecutive plies of the laminate.Then tz and z remain continuous at the 2003 by CRC Press LLC
17 FLEXURE OF THICK COMPOSITE PLATES The mechanical behavior of a laminated plate as studied in Chapter 12 involves the definition of stress resultants Nx, Ny, Txy and moment resultants Mx, My, Mxy. These resultants are obtained from the membrane stresses sx, sy, txy. The other stress components, sz, txz, tyz have not been taken into account until now. In this chapter we note how these stresses exist and have influence on the mechanical behavior of the laminate. We will also examine the configurations of plates for which the influence of these stresses are significant (for example, plates with relatively high thicknesses, justifying the title given for this chapter). This study is based on the previous definition of the displacement parameters based on integral displacement forms, and constitutes an approach analogous (and also original) to that used in Chapter 15 for the description of the bending of composite beams. 17.1 PRELIMINARY REMARKS 17.1.1 Transverse Normal Stress sz The plate is situated as in Chapter 12 from which the name of transverse normal stress sz. Such stress appears due to the application of a transverse load (concentrated or distributed) which will cause bending of the plate. A very local concentration of load in a very small zone cannot be examined with the theory of plates, which is not able to provide spatial distribution of the stresses in the neighborhood of the point of load application. This phenomenon is complex even in three-dimensional numerical modeling. Therefore, what will be presented will not be valid in the immediate surroundings of a very local transverse load (for example, an insert). A distributed load gives rise to the stresses sz with small amplitude as compared with the stresses sx and sy. This is the reason why sz is often neglected. 17.1.2 Transverse Shear Stresses txz and tyz Due to the assumption of perfect bonding between the plies, the stress vector remains continuous across an interfacial element with normal vector = , between two consecutive plies of the laminate. Then txz and tyz remain continuous at the n z TX846_Frame_C17 Page 317 Monday, November 18, 2002 12:33 PM © 2003 by CRC Press LLC
interfaces between plies (see Section 15.1.2).In addition,the upper face and lower face of the laminate are assumed to be free of tangential forces.The thickness of the laminate is denoted as b.One then has: Tz=Tz=0forz=±h/2 Assume stress and moment resultants to be constant in a given zone of the laminate: NN,To My,Me My constants (x,y) Then,by inversion of Equation 12.20,for example,one notes that the following global strain Eae Ean Yo d'woldx,woldy,20woldxdy are constant in the zone under consideration.The local strains of Equation 12.12 then depend only on the coordinate z of the laminate.This is the same for the membrane stresses ox,o t With the above consideration,local equilibrium can be written as (in the absence of body forces): g+2+=0 ++ (17.1) 荣+努+0 The transverse shear stresses then appear to be constant across the thickness of a ply.Being continuous at the interface and nil at the location z =tb/2,they are nil in all the thickness of the laminate. From this,these stresses do not play an important role in all cases:they do not always exist,their existence being related to variable stresses and moment resultants.When they exist and depending on the composition of the laminate, they can have influence on the deformation in bending,and on the interlaminar adhesion (between layers). We will assume that these stresses exist,associated with the hypotheses of the following paragraph. 17.1.3 Hypotheses The plate has midplane symmetry. The plies are orthotropic,the orthotropic axes coinciding with the x,y,z axes of the laminate. The stress o,is negligible. For example,this is the case for a laminate made of layers of balanced fabric at,,or 45o,-45°，for unidirectional layers at 0°and9o°，or for mats.Instead of this hypothesis,,one can also adopt the less restrictive hypothesis of a balanced laminate.In this case the following calculations are much more involved,without appreciable gain on the enlargement of the field of applications examined in Section 17.6.3. 2003 by CRC Press LLC
interfaces between plies (see Section 15.1.2). In addition, the upper face and lower face of the laminate are assumed to be free of tangential forces. The thickness of the laminate is denoted as h. One then has: txz = tyz = 0 for z = ±h/2 Assume stress and moment resultants to be constant in a given zone of the laminate: Nx, Ny, Txy, My, Mx, Mxy constants "(x, y) Then, by inversion of Equation 12.20, for example, one notes that the following global strain eox, eoy, goxy, ∂2 w0/∂x2 , ∂2 w0/∂y 2 , 2∂2 w0/∂x∂y are constant in the zone under consideration. The local strains of Equation 12.12 then depend only on the coordinate z of the laminate. This is the same for the membrane stresses sx, sy, txy. With the above consideration, local equilibrium can be written as (in the absence of body forces): (17.1) The transverse shear stresses then appear to be constant across the thickness of a ply. Being continuous at the interface and nil at the location z = ±h/2, they are nil in all the thickness of the laminate. From this, these stresses do not play an important role in all cases: they do not always exist, their existence being related to variable stresses and moment resultants. When they exist and depending on the composition of the laminate, they can have influence on the deformation in bending, and on the interlaminar adhesion (between layers). We will assume that these stresses exist, associated with the hypotheses of the following paragraph. 17.1.3 Hypotheses The plate has midplane symmetry. The plies are orthotropic, the orthotropic axes coinciding with the x, y, z axes of the laminate.1 The stress sz is negligible. 1 For example, this is the case for a laminate made of layers of balanced fabric at 0∞, 90∞, or 45∞, –45∞, for unidirectional layers at 0∞ and 90∞, or for mats. Instead of this hypothesis, one can also adopt the less restrictive hypothesis of a balanced laminate. In this case the following calculations are much more involved, without appreciable gain on the enlargement of the field of applications examined in Section 17.6.3. ∂sx ∂x -------- ∂txy ∂y --------- ∂txz ∂z + + --------- = 0 ∂txy ∂x --------- ∂sy ∂y -------- ∂t yz ∂z + + --------- = 0 TX846_Frame_C17 Page 318 Monday, November 18, 2002 12:33 PM © 2003 by CRC Press LLC
Remarks: For each ply having orthotropic axes x,y,z,the constitutive Equation 13.3 can be written as,taking into account the simplification o,0: Ey 0 0 Ey 1马 0 0 E 0 y x 0 0 0 0 Yxz 0 0 0 Txz 0 0 0 0 1 or under inverse form Ox [E E12 0 0 0 9 E22 0 0 0 S 人分 0 0 Es=Gxy 0 0 (17.2) 0 0 0 E4=G知 0 Yz 0 0 0 Ess =Gy Yyz where: Eu=1-VxVxs Ey E22=1-VxyVyx The transverse shear is at the origin of distortions as illustrated in Figure 17.1 for the shear stress T As a consequence,the displacements due to flexion discussed in Section 12.2.1 can be adapted as shown in Figure 17.2. yz orthotropic plate sandwich plate laminated plate in x,y,z axes Figure 17.1 Distortion of Section due to Transverse Shear Tyz 2003 by CRC Press LLC
Remarks: For each ply having orthotropic axes x, y, z, the constitutive Equation 13.3 can be written as, taking into account the simplification sz # 0: or under inverse form (17.2) where: The transverse shear is at the origin of distortions as illustrated in Figure 17.1 for the shear stress tyz. As a consequence, the displacements due to flexion discussed in Section 12.2.1 can be adapted as shown in Figure 17.2. Figure 17.1 Distortion of Section due to Transverse Shear tyz ex e y g xy g xz g yz Ó ˛ Ô Ô Ô Ô Ô Ô Ô Ô Ô Ô Ì ˝ Ô Ô Ô Ô Ô Ô Ô Ô Ô Ô Ï ¸ 1 Ex --- nyx Ey –----- 00 0 n xy Ex –----- 1 Ey --- 00 0 0 0 1 Gxy ------ 0 0 0 00 1 Gxz ------ 0 0 000 1 Gyz ------ sx sy txy txz t yz Ó ˛ Ô Ô Ô Ô Ô Ô Ô Ô Ô Ô Ì ˝ Ô Ô Ô Ô Ô Ô Ô Ô Ô Ô Ï ¸ = sx sy txy txz Ót yz ˛ Ô Ô Ô Ô Ô Ô Ì ˝ Ô Ô Ô Ô Ô Ô Ï ¸ E11 E21 0 0 0 E12 E22 0 0 0 0 0 E33 = Gxy 0 0 0 0 0 E44 = Gxz 0 0 0 0 0 E55 = Gyz ex e y g xy g xz Óg yz ˛ Ô Ô Ô Ô Ô Ô Ì ˝ Ô Ô Ô Ô Ô Ô Ï ¸ = E11 Ex 1 – nxynyx = -----------------------; E12 nyxEx 1 – nxynyx = -----------------------; E22 Ey 1 – nxynyx = ----------------------- TX846_Frame_C17 Page 319 Monday, November 18, 2002 12:33 PM © 2003 by CRC Press LLC
Wo (a) (b) Figure 17.2 Flexural Displacements In this figure,(a)represents a section before and after bending and (b)shows the evolution of the section as a rigid displacement(parameters vo,wo,and e) to which are associated increments ny and n in the plane y,z.Note:Due to the existence of midplane symmetry,the antisymmetric manner with respect to z, these increments are small,but they cannot be neglected a priori.(At this stage, we do not have a definition for the equivalent rotation,noted as e in b). This justifies the interest in the definition of the displacement field relating these increments.A supplementary interest rests in the possibility,during the study,to look after the necessary approximations more closely to lead to a useful technical formulation. 17.2 DISPLACEMENT FIELD The elastic displacement at each point of the laminate has the components u(x,y,z),v(,y,z)and w(x,y,z).With simplified description of Paragraph 12.2.1, one sees in Figure 17.2 (b),average translations denoted as vo and wo.and a rotation of the section denoted as e,to which one superimposes the supple- mentary displacements n,and n.We will define these averages in integral forms as follows: Translation along x direction:By definition,this is uo such that: 1r2 uo =p5 u(x,y,z)dz b/2 Rotation about the y axis:By definition,this is (x,y)such that': .b2 E12 u(x,y,z)x zdz -b/2E1 Approximations that do not appear neatly in the specialized literature. 3 Such a definition of the"average rotation"will be fundamental in the following to ensure the energetic coherence of the formulation for the transverse shears (see Section 17.6.6). 2003 by CRC Press LLC
In this figure, (a) represents a section before and after bending and (b) shows the evolution of the section as a rigid displacement (parameters v0, w0, and qx) to which are associated increments hy and hz in the plane y,z. Note: Due to the existence of midplane symmetry, the antisymmetric manner with respect to z, these increments are small, but they cannot be neglected a priori. (At this stage, we do not have a definition for the equivalent rotation, noted as qx in b). This justifies the interest in the definition of the displacement field relating these increments. A supplementary interest rests in the possibility, during the study, to look after the necessary approximations more closely to lead to a useful technical formulation. 2 17.2 DISPLACEMENT FIELD The elastic displacement at each point of the laminate has the components u(x, y, z), v (x, y, z) and w (x, y, z). With simplified description of Paragraph 12.2.1, one sees in Figure 17.2 (b), average translations denoted as v0 and w0, and a rotation of the section denoted as qx, to which one superimposes the supplementary displacements hy and hz. We will define these averages in integral forms as follows: Translation along x direction: By definition, this is u0 such that: Rotation about the y axis: By definition, this is qy(x,y) such that 3 : Figure 17.2 Flexural Displacements 2 Approximations that do not appear neatly in the specialized literature. 3 Such a definition of the “average rotation” qy will be fundamental in the following to ensure the energetic coherence of the formulation for the transverse shears (see Section 17.6.6). u0 1 h -- u x( ) , y, z dz –h/2 h/2 Ú = qy E11 EI11 --------- E12 EI12 + --------- Ë ¯ Ê ˆ u x( ) , y, z ¥ zdz –h/2 h/2 Ú = TX846_Frame_C17 Page 320 Monday, November 18, 2002 12:33 PM © 2003 by CRC Press LLC
where one has reused the notations of Section 12.1.6 for the terms _.4 The longitudinal displacement u(x,y,z)then takes the form: u(x,y,z)=uo(x,y)+20,(x,y)+nx(x,y,2) with: In effect,one can obtain starting from this expression: the integrals disappearing due to antisymmetry in 2: (y型）nzd …+,+面面2 In the second member,the first integral disappears due to midplane symmetry.In addition,taking into account the definition of 0,written above,the second integral is also nil. Translation along the y direction:This is vo(,y)such that: 】bM2 。=J(x,八2) -b12 Rotation about the x axis:This is e such that: w(x,y,z)×zdz The longitudinal displacement u(,y,z)then takes the form: v(x,y)=vo(x,y)-z0(x,y)+n(x,y,z) Recall that (Section 12.1.6): ply k=l ply 5 The coefficient of 0,is 1 because one can note that: 原+)+ 品面：Gnca-GGca-G1 2003 by CRC Press LLC
where one has reused the notations of Section 12.1.6 for the terms .4 The longitudinal displacement u(x, y, z) then takes the form: with: In effect, one can obtain starting from this expression: the integrals disappearing due to antisymmetry in z 5 : In the second member, the first integral disappears due to midplane symmetry. In addition, taking into account the definition of qy written above, the second integral is also nil. Translation along the y direction: This is v0(x,y) such that: Rotation about the x axis: This is qx such that: The longitudinal displacement v(x,y,z) then takes the form: 4 Recall that (Section 12.1.6): 5 The coefficient of qy is 1 because one can note that: 1 EIij ----- 1 EI ----- [ ] C –1 , where Cij E ij k zk 3 zk-1 3 – 3 -------------------- Ë ¯ Ê ˆ k=I st ply nth ply = = Â u x( ) , y, z = u0( ) x, y + zqy( ) x, y + hx( ) x, y, z E11 EI11 --------- E12 EI12 + --------- Ë ¯ Ê ˆ hx z zd –h/2 h/2 Ú = 0 u zd –h/2 h/2 Ú h u¥ o + qy z z hx dz –h/2 h/2 Ú d + –h/2 h/2 Ú = E11 EI11 -------- E12 EI12 + -------- Ë ¯ Ê ˆ z2 dz –h/2 h/2 Ú C11 EI11 -------- C12 EI12 + -------- C11C22 C11C22 C12 2 – ---------------------------- C12 2 C11C22 C12 2 – == = – ---------------------------- 1 E11 EI11 --------- E12 EI12 + --------- Ë ¯ Ê ˆ uz zd –h/2 h/2 Ú uo E11 EI11 --------- E12 EI12 + --------- Ë ¯ Ê ˆ z zd º –h/2 h/2 Ú = … qy E11 EI11 --------- E12 EI12 + --------- Ë ¯ Ê ˆ hx z zd –h/2 h/2 Ú + + v0 1 h -- v x( ) , y, z dz –h/2 h/2 Ú = qx E22 EI22 --------- E12 EI12 + --------- Ë ¯ Ê ˆ v x( ) , y, z ¥ z dz –h/2 h/2 Ú = – v x( ) , y = v0( ) x, y – zqx( ) x, y + hy( ) x, y, z TX846_Frame_C17 Page 321 Monday, November 18, 2002 12:33 PM © 2003 by CRC Press LLC
with: (b/2) E2+ ny z dz =0 -b/2 Translation along the z direction:This is wo(x,y)such that: 1(b/2) o(x,y）= w(x,y,z)dz The vertical displacement takes the form: w(x,y,z)=wo(x,y)+n2(x,y,z) In summary,one obtains for the elastic displacement field: w=lo+z6,+刀x(x,y,z) v=vo-zex +n(x,y,z) (17.3) w Wo nz(x,y,z) nx,ny,n antisymmetric in z. (17.4) E+ (17.5) 17.3 STRAINS One deduces from the previous displacements the strains: dey onx e=ex+派+0派 + ∂8x,any 6=0y-z+历 + ∂， a0x Y=g+苏- + (17.6) d +8，+梁+架 dx wo-0x+ ny+ z Yy= dz dy 17.4 CONSTITUTIVE RELATIONS 17.4.1 Membrane Equations Recall the method that was already used in Section 12.1.1. 2003 by CRC Press LLC
with: Translation along the z direction: This is w0(x,y) such that: The vertical displacement takes the form: In summary, one obtains for the elastic displacement field: 17.3 STRAINS One deduces from the previous displacements the strains: (17.6) 17.4 CONSTITUTIVE RELATIONS 17.4.1 Membrane Equations Recall the method that was already used in Section 12.1.1. E22 EI22 --------- E12 EI12 + --------- Ë ¯ Ê ˆ hy z zd –h/2 ( ) h/2 Ú = 0 w0( ) x, y 1 h -- w x( ) , y, z dz –h/2 ( ) h/2 Ú = w x( ) , y, z = w0( ) x, y + hz( ) x, y, z u u = 0 + + zqy hx( ) x, y, z v v = 0 – zqx + hy( ) x, y, z (17.3) w w = 0 + hz( ) x, y, z hx, hy, hz antisymmetric in z. (17.4) E11 EI11 --------- E12 EI12 + --------- Ë ¯ Ê ˆ hx z zd –h/2 h/2 Ú E22 EI22 --------- E12 EI12 + --------- Ë ¯ Ê ˆ hy z zd –h/2 h/2 Ú = = 0 (17.5) ex e 0x z ∂qy ∂x -------- ∂hx ∂x = + + -------- e y e 0y – z ∂qx ∂y -------- ∂hy ∂y = + -------- g xy g 0xy z ∂qy ∂y -------- ∂qx ∂x – -------- Ë ¯ Ê ˆ ∂hx ∂y -------- ∂hy ∂x = + + + -------- g xz ∂w0 ∂x --------- qy ∂hx ∂z -------- ∂hz ∂x = ++ + -------- g yz ∂w0 ∂y --------- – qx ∂hy ∂z -------- ∂hz ∂y = + + -------- TX846_Frame_C17 Page 322 Monday, November 18, 2002 12:33 PM © 2003 by CRC Press LLC
stress resultant Ns =dx:from [17.2]et [17.6] b/2 u6+A+d+斯 stress resultant N=dz: Ny A2Eox+A22E0y stress resultant T=dz: =+(-++梁 Txy =A3sYox In summary,one finds again the relations already established in Chapter 12 (Equations 12.5)as: Nx A11 A12 0 Eox Ny A21 A22 0 0 0 Yoxy or,in inverse form,by using the notations in Equation 12.9: Eox N 1/Eg -Vjax/Ey N 1 Eoy b[A] N -Vxy/Ex 1/E, 0 N (17.7) T 0 0 1/Go 17.4.2 Bending Behavior One has again the already known moment resultants (see Section 12.2.1). Moment resultant M,=dz: with[17.2and[17.5: any dz +(-+ 6/2 dy dy a+C2×- My=Cu dx + 历+axJb2 + E2nyzdz The simplifications are due to the antisymmetry of the integrated functions (midplane symmetry). 2003 by CRC Press LLC
stress resultant 6 : stress resultant stress resultant In summary, one finds again the relations already established in Chapter 12 (Equations 12.5) as: or, in inverse form, by using the notations in Equation 12.9: (17.7) 17.4.2 Bending Behavior One has again the already known moment resultants (see Section 12.2.1). Moment resultant with [17.2] and [17.5]: 6 The simplifications are due to the antisymmetry of the integrated functions (midplane symmetry). Nx Ú–h/2 h/2 = sxdx: from 17.2 [ ] et 17.6 [ ] Nx E11 e 0x z ∂qy ∂x -------- ∂hx ∂x + + -------- Ë ¯ Ê ˆ z E 12 e 0y – z ∂qx ∂y -------- ∂hy ∂y + -------- Ë ¯ Ê ˆ dz –h/2 h/2 Ú d + –h/2 h/2 Ú = Nx A11e 0x A12e 0y ∂ ∂x ------ E11hx dz ∂ ∂y ----- E12hy dz –h/2 h/2 Ú + –h/2 h/2 Ú = + + Ny Ú–h/2 h/2 = sy dz: Ny = A21e 0x + A22e 0y Txy Ú–h/2 h/2 = txy dz: Txy E33 g oxy z ∂qy ∂y -------- ∂qx ∂x – -------- Ë ¯ Ê ˆ ∂hx ∂y -------- ∂hy ∂x + + + -------- Ë ¯ Ê ˆ dz –h/2 h/2 Ú = Txy = A33g oxy Nx Ny ÓTxy ˛ Ô Ô Ì ˝ Ô Ô Ï ¸ A11 A12 0 A21 A22 0 0 0 A33 e ox e oy Óg oxy ˛ Ô Ô Ì ˝ Ô Ô Ï ¸ = e ox e oy Óg oxy ˛ Ô Ô Ì ˝ Ô Ô Ï ¸ h A[ ]–1 1 h ¥ -- Nx Ny ÓTxy ˛ Ô Ô Ì ˝ Ô Ô Ï ¸ 1 h -- 1/Ex nyx – /Ey 0 nxy – /Ex 1/Ey 0 0 01/Gxy Nx Ny ÓTxy ˛ Ô Ô Ì ˝ Ô Ô Ï ¸ = = My Ú–h/2 h/2 = sx z zd : My E11 ze ox z2 ∂qy ∂x -------- z ∂hx ∂x + + -------- Ë ¯ Ê ˆ dz… –h/2 h/2 Ú = º E12 ze oy z2 – ∂qx ∂y -------- z ∂hy ∂y + -------- Ë ¯ Ê ˆ dz –h/2 h/2 Ú + My C11 ∂qy ∂x -------- C12 ∂qx ∂y – -------- ∂ ∂x ¥ + ------ E11hx z z ∂ ∂y ----- E12hy z zd –h/2 h/2 Ú d + –h/2 h/2 Ú = + TX846_Frame_C17 Page 323 Monday, November 18, 2002 12:33 PM © 2003 by CRC Press LLC
In the last two terms there appear the nonzero integrals of even functions.If one neglects the contribution of the rates of variation along the x and y direction,respectively,of these terms,the previous equation is reduced to' a0+C× My =Cu ax d0. dy Moment resultantM b/2 Oz dz b/2 dz.. 6M2 -b12 which is reduced to: -Ms=Gi2dx and in neglecting the contribution of the last two terms': a8+C2×- -Ms Cdx aθ dy 。Moment resutantM=-」2g2tk b/2 which is reduced to: Essnyzdz and in neglecting the contribution of the variations of the differences nx and ny: -M=C- The existence of such approximation does not appear if one neglects apriori the increments nx nn:in Equation 17.3. 2003 by CRC Press LLC
In the last two terms there appear the nonzero integrals of even functions. If one neglects the contribution of the rates of variation along the x and y direction, respectively, of these terms, the previous equation is reduced to7 Moment resultant : which is reduced to: and in neglecting the contribution of the last two terms7 : Moment resultant which is reduced to: and in neglecting the contribution of the variations of the differences hx and hy 7 : 7 The existence of such approximation does not appear if one neglects a priori the increments hx, hy, hz in Equation 17.3. My C11 ∂qy ∂x -------- C12 ∂qx ∂y = + ¥ –-------- Mx sy z zd –h/2 h/2 Ú = – –Mx E12 ze ox z2 ∂qy ∂x -------- z ∂hx ∂x + + -------- Ë ¯ Ê ˆ dzº –h/2 h/2 Ú = º E22 ze oy z2 – ∂qx ∂y -------- z ∂hy ∂y + -------- Ë ¯ Ê ˆ dz –h/2 h/2 Ú + –Mx C12 ∂qy ∂x -------- C22 ∂qx ∂y ¥ –-------- ∂ ∂x ------ E12hx z z ∂ ∂y ----- E22hy z zd –h/2 h/2 Ú d + –h/2 h/2 Ú = + + –Mx C12 ∂qy ∂x -------- C22 ∂qx ∂y = + ¥ – -------- Mxy txy z zd –h/2 h/2 Ú = – –Mxy E33 zg oxy z2 ∂qy ∂y -------- ∂qx ∂x – -------- Ë ¯ Ê ˆ z ∂hx ∂y -------- z ∂hy ∂x + + + -------- Ë ¯ Ê ˆ dz –h/2 h/2 Ú = –Mxy C33 ∂qy ∂y -------- ∂qx ∂x – -------- Ë ¯ Ê ˆ ∂ ∂y ----- E33hx z z ∂ ∂x ------ E33hy z zd –h/2 h/2 Ú d + –h/2 h/2 Ú = + –Mxy C33 ∂qy ∂y -------- ∂qx ∂x – -------- Ë ¯ Ê ˆ = TX846_Frame_C17 Page 324 Monday, November 18, 2002 12:33 PM © 2003 by CRC Press LLC
In summary,one finds again a form analogous to Equation 12.16(with Cis= C23=0 due to the orientation of the plies (see Hypotheses in Section 17.1.3): [C1 C12 0 e -Mx C22 0 a (17.8) 0 0 Or in the inverse form,by reusing the notations of Section 12.1.6. 品 赢 0 My dy 0 -Mx (17.9) 学-) 0 0 -Mxy 17.4.3 Transverse Shear Equation We define here new stress resultants starting from the transverse shear stresses, which are denoted as transverse shear stress resultants: Shear stress resultantdz Using Equations 17.2 and 17.6: in setting b/2 (bG〉=Gek -b/2 yields (17.10) where one can note the presence of the integral of an even function: Shear stress resultant= 2003 by CRC Press LLC
In summary, one finds again a form analogous to Equation 12.16 (with C13 = C23 = 0 due to the orientation of the plies (see Hypotheses in Section 17.1.3): (17.8) Or in the inverse form, by reusing the notations of Section 12.1.6. (17.9) 17.4.3 Transverse Shear Equation We define here new stress resultants starting from the transverse shear stresses, which are denoted as transverse shear stress resultants: Shear stress resultant Using Equations 17.2 and 17.6: in setting yields (17.10) where one can note the presence of the integral of an even function: Shear stress resultant My –Mx Ó–Mxy ˛ Ô Ô Ô Ô Ì ˝ Ô Ô Ô Ô Ï ¸ C11 C12 0 C21 C22 0 0 0 C33 ∂qy ∂x -------- ∂q x ∂y –-------- ∂qy ∂y -------- ∂q x ∂x – -------- Ë ¯ Ê ˆ Ó ˛ Ô Ô Ô Ô Ì ˝ Ô Ô Ô Ô Ï ¸ = ∂qy ∂x -------- ∂q x ∂y –-------- ∂qy ∂y -------- ∂q x ∂x – -------- Ë ¯ Ê ˆ Ó ˛ Ô Ô Ô Ô Ì ˝ Ô Ô Ô Ô Ï ¸ 1 EI11 --------- 1 EI12 --------- 0 1 EI21 --------- 1 EI22 --------- 0 0 0 1 C33 ------- My –Mx Ó–Mxy ˛ Ô Ô Ô Ô Ì ˝ Ô Ô Ô Ô Ï ¸ = Qx Ú–h/2 h/2 = txz dz Qx Gxz ∂w0 ∂x --------- qy ∂hx ∂z -------- ∂hz ∂x ++ + -------- Ë ¯ Ê ˆ dz –h/2 h/2 Ú = hGxz · Ò Gxz dz –h/2 h/2 Ú = Qx hGxz · Ò ∂w0 ∂x --------- + qy Ë ¯ Ê ˆ Gxz dhx ∂z --------- dz –h/2 h/2 Ú = + Qy Ú–h/2 h/2 = t yz dz Qy Gyz ∂w0 ∂y --------- – qx ∂hy ∂z -------- ∂hz ∂y + + -------- Ë ¯ Ê ˆ dz –h/2 h/2 Ú = TX846_Frame_C17 Page 325 Monday, November 18, 2002 12:33 PM © 2003 by CRC Press LLC
in setting 012 yields ∂nyd (17.11) 17.5 EQUILIBRIUM EQUATIONS These are the same for the plates in general,no matter what are their compositions and are,therefore,classical. One recalls here only the equilibrium equations for bending. 17.5.1 Transverse Equilibrium 。1 cequbrum reion空+安+架+f:0 In integrating across the thickness,one reveals the transverse shear stresses Qr and +9+o+∫k=0 b12 dx ∂y Then denoted by p,the transverse stress density: a2+9+p.=0 dx dy 17.5.2 Equilibrium in Bending ■local qurium relation装+安+=+f=0 dy After multiplication with z,the integration over the thickness leads to 装-警+小[吴-+e=0 dx dy .a-Q.+x+m水=0 dx dy In neglecting the moment density: M:OMs-Qx =0 dx dy (17.12) ■local equilibrium relation安+g+爱+f=0 An analogous calculation leads to aM+M+9,=0 (17.13) dx dy 2003 by CRC Press LLC
in setting yields (17.11) 17.5 EQUILIBRIUM EQUATIONS These are the same for the plates in general, no matter what are their compositions and are, therefore, classical. One recalls here only the equilibrium equations for bending. 17.5.1 Transverse Equilibrium local equilibrium relation In integrating across the thickness, one reveals the transverse shear stresses Qx and Qy: Then denoted by pz, the transverse stress density: 17.5.2 Equilibrium in Bending local equilibrium relation After multiplication with z, the integration over the thickness leads to In neglecting the moment density: (17.12) local equilibrium relation An analogous calculation leads to (17.13) hGyz · Ò Gyz dz –h/2 h/2 Ú = Qy hGyz · Ò ∂w0 ∂y --------- – qx Ë ¯ Ê ˆ Gyz ∂hy ∂z -------- dz –h/2 h/2 Ú = + ∂t zx ∂x ---------- ∂t zy ∂y --------- ∂sz ∂z -------- f + ++ z = 0 ∂Qx ∂x --------- ∂Qy ∂y --------- sz [ ]–h/2 h/2 fz dz –h/2 h/2 Ú ++ + = 0 ∂Qx ∂x --------- ∂Qy ∂y + + --------- pz = 0 ∂sx ∂x -------- ∂t xy ∂y --------- ∂t xz ∂z ---------- f +++ x = 0 ∂My ∂x --------- ∂Mxy ∂y – ----------- ∂ ∂z ----- ztxz ( ) – txz z zfx dz –h/2 h/2 Ú d + –h/2 h/2 Ú + = 0 ∂My ∂x --------- ∂Mxy ∂y – ----------- – Qx ztxz [ ]–h/2 h/2 zfx dz –h/2 h/2 Ú + + = 0 ∂My ∂x --------- ∂Mxy ∂y – ----------- – Qx = 0 ∂t yx ∂x --------- ∂sy ∂y -------- ∂t yz ∂z --------- f ++ + y = 0 ∂Mxy ∂x ----------- ∂Mx ∂y + + --------- Qy = 0 TX846_Frame_C17 Page 326 Monday, November 18, 2002 12:33 PM © 2003 by CRC Press LLC
