16
Linear elastic theory of elastic plates - Mindlin-Reissner theory
In this theory we make the first step to account for the 3D nature of a plate by making allowance in
an approximate way for shear deformation through the thickness of a plate in addition to bending
and twisting deformations. The “fibres” that are initially straight and normal to the mid-surface are
still assumed to remain straight, but they are given independent degrees of freedom to rotate about
axes in the plane of the plate. Thus each fibre is given 3 degrees of freedom: transverse deflection
w, and two rotations ψx and ψy.
Curvatures are again defined by the first derivatives of the rotations in Figure 14, and the moment
curvature relations as a form of Hooke’s law are unchanged, but remember that now we do not
express curvatures as second derivatives of deflections. The additional aspect of Hooke’s law
concerns the relations between shear forces and shear strains.















Figure 14: rotations of fibres through the thickness of a plate.
Transverse shearing of plate
The distinction between Mindlin-Reissner and Kirchhoff plate theories rests on the inclusion or
exclusion respectively of transverse shear deformation through the plate thickness. Both theories
assume that normals to the midsurface remain straight. However Mindlin allows for a shear strain,
i.e. a change of angle between the normal and the tangent to the midsurface, whereas Kirchhoff
neglects such shear strains so that the normals remain perpendicular to the midsurface. The shear
strain









 xzx
x
w
 , (8)

indicated by the symbol in Figure 18, is treated as constant throughout the plate thickness. In
Kirchhoff theory it is assumed that xzx
x
w
 


 that i.e. , 0 . In Mindlin-Reissner theory the shear
strain zx is accounted for, but note that there is an inconsistency between the assumption of a
constant shear strain and the parabolic distribution of shear stress zx as illustrated in Figure 15. The
latter is required to satisfy equilibrium of stress according to engineer’s beam theory.






















2
24
1
2
32
1
2
1
2
3
t
z
t
Q
t
z
t
z
t
Q xx
zx (9)


mid-surface plane
z
y
x
z z
y
x
u = -z . x
v = -z . y
17





















Figure 15: assumptions for shear stress and shear strain in Mindlin-Reissner theory
This inconsistency is resolved in a “weak” way by equating two expressions for virtual work
developed in terms of the shear force stress-resultant Qx per unit width of a beam strip and the shear
stresses zx:
   dx
t
Q
G
dxdz
G
dxQ x
t
t
zx
zxx 
















 


22/
2/
2
5
61

where G is the shear modulus defined by G = E/2(1 + ν).
Thus zxx GtQ 
6
5
(10)
as indicated in Cook et al [1] Chapter 15, Equation (15.1-5) when k = 5/6.
Simple supported boundary conditions
Now that we have 3 independent degrees of freedom for each transverse fibre, the number of
boundary conditions that can be specified at each point increases to 3. Thus at a loaded boundary,
we can specify both components of moment (bending and twisting) and a shear force intensity.
Furthermore at simple supports we specify zero transverse deflection w and bending moment Mn
about the line of support, with a choice for the third condition. There are two alternative forms
which result from using Mindlin-Reissner theory: termed “soft” and “hard”. “Soft” specifies zero
twisting moment Mns, whereas “hard” specifies zero rotation s about the normal to the boundary in
the xy plane. Which type of support best represents the physical problem requires your decision –
Cook has words of wisdom in Section 15.5 [1]!
It is important to appreciate that linear theory only works so long as the vertical deflections are very
small. Otherwise in-plane forces can develop and plates tend to transmit loads in ways you may not
have imagined! Figure 16 refers to the non-linear elastic behaviour of a 2m square vertical glass
plate with 6mm thickness, simple supports and subjected to a uniform pressure (kN/m2). In this
context “soft” implies that the edges have freedom to rotate in the torsional sense and freedom to
zx
Qx
Qx
zx
x
z
x
x
w



dx
18
move in the plane of the plate; whereas “hard” implies in this case that freedom of the edges to
rotate is maintained but in-plane movement is fully constrained.



Figure 16: load deflection curves for non-linear behaviour due to large deflections; straight line for linear
behaviour; solid curved line for nonlinear elastic analysis with “soft” boundary conditions to imply edge
deflections, e.g. midside pull-in is unconstrained; dashed line for nonlinear elastic analysis with “hard”
boundary conditions;
References
[1]. Concepts and applications of finite element analysis, R D Cook, D S Malkus, M E Plesha, R
J Witt. 4th ed Wiley 2002, Chapter 15, various sections.

[2]. Mechanics of Structures, Variational and Computational Methods, W. Wunderlich & W.D.
Pilkey, CRC Press, 2nd ed., 2003. Sections 13.4 for shear deformation effects.