An elementary derivation of basic equations of the Reissner and Mindlin plate theories

Engineering Structures 32 (2010) 906–909

Contents lists available at ScienceDirect

Engineering Structures journal homepage: www.elsevier.com/locate/engstruct

Short communication

An elementary derivation of basic equations of the Reissner and Mindlin plate theories Milan Batista University of Ljubljana, Faculty of Maritime Studies and Transport, Slovenia

article

info

Article history: Received 18 October 2009 Received in revised form 10 November 2009 Accepted 22 December 2009 Available online 13 January 2010

abstract The basic equations of the Reissner and Mindlin plate theories are derived in an elementary way from the underlying assumptions of the theories about the distribution of in-plane stresses in the Reissner case and the distribution of displacement components in the Mindlin case. The derived equations include a parameter which allows the interpretation of the theories as an approximation of isotropic plates or transversally inextensible plates. Qualitative comportment between the theories is also given. © 2009 Elsevier Ltd. All rights reserved.

Keywords: Elasticity Plate theory Shear deformable plates

1. Introduction This short communication is motivated by the articles of Wang et al. [1] and Batista [2] wherein the authors emphasize the practical importance of the Reissner and Mindlin plate theories resulting from their simplicity and notice that despite the fact that the theories rely on different assumptions they are often treated as a single theory, referred to as the Reissner–Mindlin plate theory. They, among others, qualitatively describe the differences between the theories and state that one difference between them is that the Reissner plate theory was derived from the variational principle of complementary strain energy. This is true for Reissner’s primary derivation [3–5]; however, in his later paper [6] he abandoned variational principles and showed how his equations may be derived directly from elasticity equations by assumptions that the transverse shear stresses are parabolic and that the sum of the inplane normal stresses is linearly distributed over the plate thickness. How Reissner’s plate equations may be derived directly from elasticity equations without using variational principles was, however, previously shown by Green ([7], 224–229), who used Reissner’s weighted displacements [5] and assumed that transverse shear stress components are parabolically distributed across the plate thickness. Variants of such derivation may be found in the books of Timoshenko ([8], 168–171), Girkmann ([9], 583–591) and Panc ([10], 34–41). How Reissner’s equation for an unloaded plate may be derived without any special assumptions was recently

E-mail address: [email protected]. 0141-0296/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.engstruct.2009.12.046

shown by the present author [2]. Nevertheless, we may find statements alluding to an intrinsic connection between Reissner’s plate theory with variational principles even in contemporary books (e.g. [11]). As opposed to Reissner, who considered plate equilibrium, Mindlin, who followed Uflyand [12], considered the vibration of plates [13], and he derived his plate equations directly from elasticity equations without using variational principles. The static case of equations was previously deduced by Bolle [14] and the case with variational principles was deduced by Hencky [15]. In what follows, the theories are visited one more time, showing how the governing equations of the theories may be derived in an elementary way. In the derivation of equations only the basic assumptions of theories are retained: so in this paper the Reissner plate theory means the theory for which the basic equations are derived by assumption that the in-plane stresses are linearly distributed across the plate thickness, while the Mindlin plate theory means the theory where the in-plane displacements are assumed to be linearly distributed across the plate thickness. 2. The shear deformable plate theories Governing elasticity equations. In describing a plate, a rectangular Cartesian coordinate system with coordinates (x, y, z ) is used. The coordinate z is perpendicular to the plane of the plate and the plate faces are at z = ±h/2. For a plate in equilibrium the following stress equilibrium equations must be satisfied:

∂σx ∂τxy ∂τxz + + =0 ∂x ∂y ∂z

(x  y)

∂τxz ∂τyz ∂σz + + = 0 (1) ∂x ∂y ∂z

M. Batista / Engineering Structures 32 (2010) 906–909

where σx , σy , σz , are the normal stress components, and τxy , τxz , τyz are the shear stress components. Here and in what follows the symbol x  y on the right-hand side of an equation means that the equation for the other coordinate is obtained by interchanging x and y. Since we want to consider an elastic isotropic plate as well as an elastic transversally inextensible plate, we write the constitutive equations which connect the stress components with the displacement components u, v , w in the following halfinverted form:


 ∂w ω = σz − ν σx + σy ∂z E   E ∂u ∂v σx = + ν + 1 − ν2 ∂ x ∂y   ∂ u ∂v + τxy = G ∂y ∂x

∂ u ∂w τxz + = (x  y) ∂z ∂x G νω σz (x  y) 1−ν

(2)

(3)

Mindlin plate quantities, respectively. Also, the z coordinate is normalized as follows:

ζ ≡

z h/2

ω=

0 1

transversally inextensible plate isotropic plate

(4)

and where E is the modulus of elasticity, ν is the Poisson ratio, and G ≡ E /2 (1 + ν) is the shear modulus, assumed to be isotropic. It is worthwhile mentioning that the transversally inextensible plate was introduced by Kromm [16], but neither Reissner [3–5] or Mindlin [13] discuss this possibility in their derivations. The boundary conditions on the plate faces are

τxz (x, y, ±h/2) = τyz (x, y, ±h/2) = 0 σz (x, y, ±h/2) = ±

p

(5)

2

where p ≡ σz (x, y, h/2) − σz (x, y, −h/2). The last of these conditions is asymmetric with respect to coordinate z and defines the pure bending of the plate. In this case the transverse displacement w should be a symmetric function of z and, as follows from the governing equations, the stress components σx , σy , σz , τxy and the displacement components u, v are asymmetric functions of z and the stress components τxz , τyz are symmetric functions of z. Consequently, the stress components σx , σy , σz , τxy and the displacement components u, v vanish at the plate middle plane, z = 0. On a high level the elastic state of the plate is described by the stress resultants [8]

Z Mx ≡

−h/2

Z Qx ≡

h/2

h/2

−h/2

σx zdz τxz dz

(x  y)

Z Mxy ≡

h/2

−h/2

τxy zdz

(x  y)

(6)

where Mx , My are the bending moments, Mxy is the twisting moment and Qxz , Qyz are the transverse shear forces, all on one unit of length. By means of the definitions (6) and the face boundary conditions (5), by integration of the stress equilibrium equation (1) across the plate thickness, we obtain the well-known plate equilibrium equations [8]: Qx =

∂ Mx ∂ Mxy + ∂x ∂y

(x  y)

∂ Qx ∂ Qy + = −p. ∂x ∂y

(7)

The task of a shear deformable plate theory is now to find an approximate solution of the elasticity equations (1)–(3) subject to the face boundary conditions (5) which includes enough free functions by which a prescribed plate’s edge boundary conditions may be satisfied. Besides that, it is required that an approximate solution satisfy the plate equilibrium equations (7). In what follows the superscripts R and M will be used to denote the Reissner and

∈ [−1, 1] .

(8)

Note 1. The requirement of the satisfaction of plate equilibrium equations (7) is abandoned in some plate theories where equations are derived from the principle of virtual work [17] Note 2. For the case p = 0 the exact solution of the stated problem exists, which may be found by the method of symbolic integration [18] or by the method of successive approximation [2]. Reissner’s plate theory. As stated in the introduction, the basic assumption of Reissner’s primary theory of plates is that the inplane stresses are linearly distributed across the plate thickness [3,4]. Thus, by using the definition of stress resultants (6), the inplane stress components may be written in the following wellknown form:

where



907

σxR =

6MxR h2

ζ

(x  y)

τxyR =

R 6Mxy

h2

ζ.

(9)

By means of these expressions, by integration of the equilibrium equations (1) with respect to z, use of the equilibrium equation (7)2 and plate face boundary conditions (5), we obtain the transverse stress components:

τxzR =

3QxR 2h

1 − ζ2




(x  y)

σzR =

p 4


ζ 3 − ζ2 .

(10)

The displacement components may now be obtained by the integration of constitutive equations (2) as is shown in [2]. However, Reissner at this point made another approximation:

w = w0R (x, y) .

(11)

By this, the constitutive equation (2)1 may not generally be satisfied unless a plate is transversally inextensible. Now, following integration of the constitutive equation (2)2 with respect to z, by means of (10)1 , (11) and condition u (x, y, 0) = 0 we find the expressions for the in-plane displacement components: uR = −

h ∂w0R 2 ∂x

ζ+

QxR 4G

ζ 3 − ζ2




(x, u  y, v) .

(12)

In this way all the stress components and in-plane displaceR ments are expressed by six unknowns: MxR , MyR , Mxy , QxR , QyR and w0R . For the determination of these unknowns we have at our disposal three plate equilibrium equations (7); hence three more equations must be deduced from the remaining three constitutive equations (3) in such a way that these constitutive equations are satisfied in some approximate way. This may be achieved if we require that the stresses produced by displacement (12) produce the same moments as the stress components (9). So, by substituting (12) into constitutive equation (3), equating the results with (9) and performing integration over the plate thickness the three missing equations are available:

 ∂ 2 w0R ∂ 2 w0R h2 ∂ QxR + = −D + ν ∂ x2 ∂ y2 5 ∂x 2 h ν (2 − ω) − p (x  y) (13) 10 1 − ν ! ∂ QyR h2 ∂ QxR ∂ 2 w0R R Mxy = − (1 − ν) D + + ∂ x∂ y 10 ∂y ∂x
where D = h3 E /12 1 − ν 2 is the plate bending stiffness. For ω = 1 these equations become Reissner’s equations for the MxR



moments [3,4].

908

M. Batista / Engineering Structures 32 (2010) 906–909

The number of equations obtained may now be reduced in the usual way. Introducing the moments from (13) into the equilibrium equation (7)1 and taking (7)2 into account yields the equations for the shear forces: QxR −

h2 10

∂ ∆w0R ∂x h2 1 + ν (1 − ω) ∂ p − 10 1−ν ∂x

(x  y)

(14)

where ∆ ≡ ∂∂x2 + ∂∂y2 is the two-dimensional Laplace operator. By differentiating the above equations with respect to x and y, adding the results and using (7)2 , we arrive at the governing biharmonic equation for deflection: D∆2 w0R = p −

h (2 − ων) 10 (1 − ν)

∆p.

(15)

For ω = 1, Eqs. (14) and (15) become the plate equations obtained by Reissner using the variational procedure [4]. In Reissner’s theory the boundary conditions involving in-plane displacements may not be directly prescribed, since, by (12), the displacements are cubically distributed across the plate thickness. The in-plane displacements must thus be linearized and this may be done by utilizing the method of least squares (LSQ method). So if we require that on the boundary of the plate

Z

h/2

−h/2

uR − φxR z


2

dz = min

(x, u  y, v)

(16)

then we have

φxR =

12

Z

h/2

uzdz = −

h3

−h/2

∂w0R 6Q R + x ∂x 5hG

(x, u  y, v) ,

(17)

(x, u  y, v)

(18)

where φxR is Reissner’s equivalent change of slope of the normal to a plate midplane [5]. If this approximation is taken to be valid over the whole plate then the above plate equations may be put into yet another form. By expressing the shear forces from (17) and then substituting the resulting expressions into the moment’s equations (13), we obtain the following constitutive equations:

∂φyR ∂φxR +ν ∂x ∂y

MxR = D

! −

h2 ν (2 − ω) 10

1−ν

! ∂φyR ∂φxR = D + 2 ∂y ∂x   R 5hG ∂w0 QxR = φxR + (x  y) . 6 ∂x

R Mxy

1−ν

p

uM = φxM (x, y) z

h/2

−h/2

z

Z

wdz 0 − w0R z 0

2

dz = min .

(22)

τ

∂φyM

! ∂φxM + z =G ∂x ∂y   ∂w0M = κ 2 G φxM + (x  y) ∂x

M xy

τxzM

(23)

where κ 2 is Mindlin’s correction factor, which he introduced since the above transverse stress components do not satisfy the face boundary conditions (5)1,2 and its value is out of the scope of the theory. From this, by using the definition of stress resultants (6), we obtain Mindlin’s constitutive equations:

QxM (19)

∂φyM

! ∂φxM D + =− 2 ∂x ∂y   M ∂w0 = κ 2 hG φxM + (x  y) . ∂x 1−ν

(24)

By means of these the shear stress components, (23) may be rewritten in the form

τxyM =

Together with plate equilibrium Eq. (7), these equations form a complete set of equations for the unknown stress resultants, deflection and equivalent change of slope. Again, for ω = 1, we obtain Reissner’s equations, which he derived by using variational principles [5]. It now remains to provide an appropriate interpretation of w0R for the case of an isotropic plate. R z In the derivation of the in-plane displacement (12) we replace 0 w dz 0 by w0R z. This may be justified by utilizing the LSQ method, where we approximate the integral with

Z

v M = φyM (x, y) z wM = w0M (x, y)

where φxM and φyM are the changes of slope of the normal to a plate midplane. Since the stress components are determined by the displacement components the theory includes only three unknowns: φxM , φyM and w0M . Note 3. Mindlin’s derivation [13] is different from the one that will be presented here. He started with the definition of stress resultants, which he called plate-stress components, and defined the corresponding plate-strain components by explicitly ignoring the strain normal to the faces of the plate, Eq. (2)1 . By integration of the constitutive equations over the plate thickness he obtained the plate constitutive relations, wherein he also noted that σz = 0 is R h/2 not assumed, but rather −h/2 σz zdz = 0. Also he explicitly noted that the z-dependence of the stress components is not specified. Now, introducing displacements (22) into the constitutive equations (2)2 and (3) yields the shear stress components:

M Mxy

(x  y)

h

This is Reissner’s weighted deflection, which he obtained by the variational procedure [5], while Green introduced it in a heuristic way [7]. Mindlin plate theory. The starting point of Mindlin’s theory [13] is the assumption that the in-plane components of the displacements are linearly distributed across the plate thickness and that the deflection is independent of the z coordinate [15,13]:

so we may approximate u˜ R = φxR z

R 2 z


0

2h −h/2

∆QxR = −D

2

R h/2 R z
wdz 0 zdz. But = 12 0 h3 −h/2
2 0 w dz zdz = d z 0 w dz 0 − w z 2 , so  Z h/2  3 4z 2 R w0 (x, y) = w 1 − 2 dz . (21) From this we have w0R

Rz

(20)

M 6Mxy

ζ τxzM =

QxM

(25) (x  y) . h2 h From the equilibrium equations (1)2 and (7)2 , and condition σz (x, y, 0) = 0, we may now calculate the transverse normal stress component: p ζ. (26) 2 Introducing (22) and (26) into constitutive equations (3) yields

σzM =

σ

M x

=

hE 2 1 − ν2

+




ων p ζ 2 (1 − ν)

∂φyM ∂φxM +ν ∂x ∂y (x  y) .

! ζ (27)

M. Batista / Engineering Structures 32 (2010) 906–909

By these we establish all the expressions for the stress components. By definition (6), the bending moments corresponding to the in-plane stress components (27) are

∂φyM ∂φxM =D +ν ∂x ∂y

MxM

!

ων h2 + p 12 (1 − ν)

(28)

For ω = 0, these expressions become those obtained by Mindlin [13]. By means of (28) the stress components (27) become

σxM =

6MxM h2

ζ

(x  y) .

(29)

QM ∂w0M + 2x ∂x κ hG

(x  y) ;

(30)

hence the moments (24) and (28)1 may be expressed as

 ∂ 2 w0M h2 ∂ QxM ∂ 2 w0M + ν + = −D ∂ x2 ∂ y2 6κ 2 ∂ x
2 2 h ν 2 − ωκ − p (x  y) 12κ 2 (1 − ν) h2 ∂ 2 w0M + = − (1 − ν) D ∂ x∂ y 12κ 2

M Mxy

(31)

∂ QyM ∂ QxM + ∂y ∂x

! .

Now, similar to Reissner’s case, we may now deduce the equations for the shear forces: QxM

−

h2 12κ

∆QxM 2

∂ ∆w0M = −D ∂x
2 h 1 + ν − ωνκ 2 ∂ p − 12κ 2 (1 − ν) ∂x

(x  y) ,

(32)

and the equation for deflection: D∆ w 2

M 0

−h/2 h/2

Z

=p−

h2 2 − ωνκ 2




12κ 2 (1 − ν)

∆p.

u − φxM z


2

dz = min

w − w0M


2

dz = min

(x, u  y, v)

(34)

so

φ = M x

12 h3

h/2

Z

uzdz −h/2

(x, u  y, v)

w = M 0

1

Z

h/2

h −h/2

wdz .

(35)

Consequently, for any of the discussed material models φxM and φ differ since φxM is defined by the true displacement u while φxR is defined in terms of uR . The deflection differs only for an isotropic plate: w0M represents an average deflection while w0R represents a weighted average deflection. On the stress resultants level we may establish the following relationship between the theories. Comparing the deflection equations (15) and (33) we may see that these equations become equal when R x

κ2 =

10 12 − ων

.

(36)

However, the equations for the shear forces (14) and (32) and for the calculation of the moments (13) and (31) become equal only for



MxM

h/2

−h/2

Now, the in-plane stress components (23)1 and (29) do not satisfy the stress equilibrium conditions (1), but the theory demands that the plate equilibrium equations (7) should be satisfied. So, by substituting the expressions for the moments and the shear forces (28)2 into the plate equilibrium equations (7) we obtain three equations for the unknowns φxM , φyM and w0M . However, in this article, in order to compare Mindlin’s theory with Reissner’s, equations (24)1 and (28) will now be transformed into the form in which the shear forces Qx and Qy configure as unknowns instead of as kinematical variables φxM and φyM . Thus, from the plate constitutive equation (24)2 we have

φxM = −

The theories differ also in the interpretation of kinematical variables. Utilizing the LSQ method, in Mindlin’s case, we could obtain the following interpretation of (22):

Z (x  y) .

909

(33)

3. Discussion On the basis of the derived equations we may establish the following difference between the theories. It is evident that on the stress level they can never yield the same results, since in the Reissner theory the transverse shear stresses are distributed parabolically and thus satisfy the zero free plate face boundary condition, while in Mindlin’s approach the transverse shear stresses are constant across the plate thickness so the zero shear stress condition on the plate face is not satisfied. Also, σz in Reissner’s case is distributed cubically while in Mindlin’s case the distribution is linear over the plate thickness. To that we add that Reissner’s in-plane stress components satisfy the stress equilibrium equations, while Mindlin’s do not. Both stress approximations, however, satisfy the transverse stress equilibrium equation.

ω = 0 κ2 =

5 6

.

(37)

Thus only for the case of transversally inextensible plates and for κ 2 = 5/6 do both plate theories give the same results on the stress resultant level, but as already noted the interpretation of kinematical variables differs. References [1] Wang CM, Lim GT, Reddy JN, Lee KH. Relationships between bending solutions of Reissner and Mindlin plate theories. Eng Struct 2001;23:838–49. [2] Batista M. The derivation of the equations of moderately thick plates by the method of successive approximations. Acta Mech, Online, 2009. [3] Reissner E. On the theory of bending of elastic plates. J Math Phys 1944;23: 184–91. [4] Reissner E. The effect of transverse shear deformation on the bending of elastic plates. J Appl Mech 1945;12: A-69–A-77, Trans ASME 67. [5] Reissner E. On bending of elastic plates. Quart Appl Math 1947;5:55–68. [6] Reissner E. On transverse bending of plates including the effect of transverse shear deformation. Int J Solids Struct 1975;11:569–73. [7] Green AE, Zerna W. Theoretical elasticity. Oxford press; 1954. [8] Timoshenko S, Woinowsky-Krieger S. Theory of plates and shells. McGrawHill; 1959. [9] Girkmann K. Flächentragwerke. Springer Verlag; 1963. [10] Panc V. Theories of elastic plates. Nordhoff; 1975. [11] Ventsel E, Krauthammer T. Thin plates and shells: Theory, analysis, and application. CRC Press; 2001. [12] Uflyand YaS. The propagation of waves in transverse vibrations of bar and plates. Appl Math Mech 1948;21:287–300. (in Russian). [13] Mindlin RD. Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates. Trans ASME J Appl Mech 1951;18:31–8. [14] Bolle L. Contribution au problème linéaire de flexion d’une plaque élastique. Bull Tech De La Suisse Romande 1947;73(21):281–5. [15] Hencky H. Über die Berücksichtigung der Schubverzerrung in ebenen Platten. Ingenieur-Archiv 1947;16:72–6. [16] Kromm A. Verallgemeinerte Theorie der Plattenstatik. Ingenieur-Archiv 1953; 21:266–86. [17] Wang CM, Reddy JN, Lee KH. Shear deformable beams and pates: Relationships with classical solutions. Oxford, U.K: Elsevier; 2000. [18] Cheng S. Elasticity theory of plates and a refined theory. Trans ASME 1979;46: 644–9.