25

Kirchhoff Plates:
BCs and
Variational Forms

25–1
Chapter 25: KIRCHHOFF PLATES: BCS AND VARIATIONAL FORMS 25–2

TABLE OF CONTENTS

Page
§25.1. INTRODUCTION 25–3
§25.2. BOUNDARY CONDITIONS FOR KIRCHHOFF PLATE 25–3
§25.2.1. Conjugate Quantities . . . . . . . . . . . . . . . 25–3
§25.2.2. The Modified Shear . . . . . . . . . . . . . . . 25–4
§25.2.3. Corner Forces . . . . . . . . . . . . . . . . . 25–5
§25.2.4. Common Boundary Conditions . . . . . . . . . . . 25–6
§25.2.5. Strong Form Diagram . . . . . . . . . . . . . . 25–7
§25.3. THE TOTAL POTENTIAL ENERGY PRINCIPLE 25–7
§25.3.1. The TPE Functional . . . . . . . . . . . . . . . 25–7
§25.3.2. Finite Element Conditions . . . . . . . . . . . . . 25–8
§25.4. THE HELLINGER-REISSNER PRINCIPLE 25–8
§25.4.1. Finite Element Conditions . . . . . . . . . . . . 25–9
§25.5. THE CURVATURE-DISPLACEMENT DE VEUBEKE PRINCIPLE 25–10
§25.5.1. The dV Functional . . . . . . . . . . . . . . . 25–10
§25.5.2. Finite Element Conditions . . . . . . . . . . . . 25–10

25–2
25–3 §25.2 BOUNDARY CONDITIONS FOR KIRCHHOFF PLATE

§25.1. INTRODUCTION
In this Chapter we continue the discussion of the governing equations of Kirchhoff plates with the
consideration of boundary conditions (BCs).
When plates and shells are numerically idealized by finite elements the proper modeling of boundary
conditions can be a difficult subject. Two factors contribute to this. First, displacement derivatives in
the form of rotations are now involved in the kinematic boundary conditions. Second, the correlation
between physical support conditions and mathematical B.C. can be tenous. Some mathematical
BC used in practice are nearly impossible to reproduce in the laboratory, let alone on an actual
structure.

§25.2. BOUNDARY CONDITIONS FOR KIRCHHOFF PLATE

One of the mathematical difficulties associated with this plate model is the “Poisson paradox”
resolved by Kirchhoff:
• The plate deflection satisfies a fourth order partial differential equation (PDE), which is the
biharmonic equation ∇ 2 w = q/D for an isotropic homogeneous plate.
• A fourth order PDE can only have two boundary conditions at each boundary point.
• But three conjugate quantities: normal moment, twist moment and transverse shear appear
naturally there.
The reduction from three to two requires variational methods. But it is not necessary to look at a
complete functional. The procedure can be explained directly through the external boundary work.

§25.2.1. Conjugate Quantities

Conside a Kirchhoff plate of general shape as in Figure 25.1(a). Assume that the boundary  is
smooth, that is, contains no corners. Under those assumptions the exterior normal n and tangential
direction s at each boundary point B are unique, and form a system of local Cartesian axes.
The kinematic quantities referred to these local axes are

∂w ∂w
w, = −θs , = θn , (25.1)
∂n ∂s
where θn and θs denote the rotations of the midsurface at B about axes n and t, respectively, see
Figure 25.1(b). The work conjugate static quantities, shown in Figure 25.1(c), are

Qn , Mnn , Mns , (25.2)

respectively. By conjugate it is meant that the boundary work can be expressed as the line integral
   
∂w ∂w  
WB = Q n w + Mns + Mnn ds = Q n w + Mns θn − Mnn θs ds (25.3)
 ∂s ∂n 

where ds ≡ d denotes the differential boundary arclength. This integral appears naturally in the
process of forming the energy functionals of the plate. Given the configuration of W B , it appears at

25–3
Chapter 25: KIRCHHOFF PLATES: BCS AND VARIATIONAL FORMS 25–4

dx θs
(b)
y s
(a) dy
ds
B n
θn

Γ
x
Qx
Ω B Mxy
A (c) y
Qy
Mxx x
Myx Qn
s
B
Myy n
Mns
Mnn

Figure 25.1. BCs at a smooth boundary point B of a Kirchhoff plate: n = external normal,
s = tangential direction. (a) boundary traversed in the counterclockwise
sense (looking down from +z) leaving the plate proper on the left;
(b) shows kinematic quantities θs and θn ; (c) shows force-moment
quantities Q n , Mnn and Mns on the boundary face.

first sight as if three boundary conditions can be assigned at each boundary point, taken from the
conjugate sets (25.1) and (25.2). For example:

Simply supported edge w=0 θn = 0 Mnn = 0,

(25.4)
Free edge Qn = 0 Mns = 0 Mnn = 0.

The boundary conditions for a free edge were indeed expressed by Poisson in this form.1 As noted
above, this is inconsistent with the order of the PDE. Kirchhoff showed2 that three conditions are
too many and in fact only two are independent.

§25.2.2. The Modified Shear

The reduction to two independent conjugate pairs may be demonstrated through integration of
(25.3) by parts with respect to s, along a segment AB of the boundary :
  
B
∂ Mns  ∂w
W B | BA = Qn − w + Mnn dt + Mns w| BA . (25.5)
A ∂s ∂n

Introducing the modified shear3

∂ Mns
Vn = Q n − , (25.6)
∂s

1 See e.g., I. Todhunter and K. Pierson, History of Theory of Elasticity, Vol. I.

2 G. Kirchhoff, publications cited in §24.2.
3 Also called Kirchhoff equivalent force, or Kirchhoffische Ersatzkräfte.

25–4
25–5 §25.2 BOUNDARY CONDITIONS FOR KIRCHHOFF PLATE

s
s
Rc

_ +
Mns Mns

Figure 25.2. Effect of of modified shear (Kirchhoff shear) at a plate corner:

the force-pairs do not cancel, producing a corner force Rc .

we may rewrite (25.5) as

  
B
∂w
W B | BA = Vn w + Mnn dt + Mns w| BA . (25.7)
A ∂n
This transformation reduces the conjugate quantities to two work pairs:
∂w
Vn , w and Mnn , = −θs . (25.8)
∂n

§25.2.3. Corner Forces

The last term in (25.7) deserves analysis. First consider a plate with smooth boundary as in Figure
25.1. Assuming that Mns is continuous over , and we go completely “around” the plate so that
A ≡ B,
Mns w| BA = 0. (25.9)
Next consider the case of a plate with a corner C as in Figure 25.2. At C the twisting moment jumps
− +
from, say, Mns to Mns . The transverse displacement w must be continuous (if fact, C 1 continuous).
Place A and B to each side of C, so that A → C from the minus side while C ← B from the plus
side. Then
+ − + −
Mns | BA = Rc w = (Mns − Mns )w, with Rc = Mns − Mns . (25.10)
This jump in the twisting moment is called the corner force Rc ; see Figure 25.3. Note that Rc has
the physical dimension of force, because the twisting moment is a moment (force times length) per
unit length.

Simple Support,
No Edge Anchorage

Figure 25.3. Manifestation of modified shear as corner lifting forces.

25–5
Chapter 25: KIRCHHOFF PLATES: BCS AND VARIATIONAL FORMS 25–6

Free edge

;; water

;;;;
;;;;
;; Clamped
(fixed) edges

REMARK 25.1
;
"Point support" for
reinforced concrete slab

Figure 25.4. Boundary condition examples..

The physical interpretation of modified shears and of corner forces is well covered in Timoshenko and
Woinowsky-Krieger.4 Suffices to say that if a plate corner is constrained not to move laterally, a concentrated
force Rc called the corner reaction, appears. If the corner is not held down the reaction cannot physically
manifest and the plate will have a tendency to move away from the support. This is the source of the well known
“corner lifting” phenomenon that may be observed on a laterally loaded square plate with simply supported
edges that do not prevent lifting. See Figure 25.3.
This effect does not appear if the edges meeting at C are free or clamped, because if so the twist moment Mns
on both sides of the corner point are zero.

§25.2.4. Common Boundary Conditions

Below we state homogeneous boundary conditions frequently encountered in Kirchhoff plates as
selected combinations of the conjugate quantities (25.8). Some BCs are illustrated in the structures
depicted in Figure 25.4.
Clamped or Fixed Edge (with s along edge):
∂w
w = 0, = −θs = 0. (25.11)
∂n
Simply Supported Edge (with s along edge):

w = 0, Mnn = 0. (25.12)

Free Edge (with s along edge):

Vn = 0, Mnn = 0. (25.13)

4 Theory of Plates and Shells monograph cited in previous Chapter

25–6
25–7 §25.3 THE TOTAL POTENTIAL ENERGY PRINCIPLE

w=w ^
Prescribed
deflections θs = ^θs Deflection Lateral
& rotations load
^ θ
w, ^s Displacement w q Γ
BCs Ω

Kinematic κ=Pw Equilibrium PT M = q

in Ω in Ω

Constitutive Mnn= M^
nn Prescribed
Curvatures M=Dκ Bending Vn = V^n moments
moments & shears
κ in Ω M Force BCs ^ , V^
M nn n

Figure 25.5. The Strong Form diagram for the Kirchhoff plate,
including boundary conditions.

Symmetry Line (with s along line):

∂w
Vn = 0, = −θs = 0. (25.14)
∂n
Point Support:
w=0 (25.15)

Non-homogeneous boundary conditions of force type involving prescribed normal moment M̂nn or
prescribed modified shear V̂n , are also quite common in practice. Non-homogeneous B.C. involving
prescribed nonzero transverse displacements or rotations are less common.

§25.2.5. Strong Form Diagram

The Strong Form diagram of the governing equations, including boundary conditions, for the
Kirchhoff plate model is shown in Figure 25.5.
We are now ready to present several energy functionals of the Kirchhoff plate that have been used
in the construction of finite elements.

§25.3. THE TOTAL POTENTIAL ENERGY PRINCIPLE

The only master field is the transverse displacement w. The departure Weak Form is shown in Figure
25.5. The weak links are the internal equilibrium equations and the force boundary conditions.

§25.3.1. The TPE Functional

Proceeding as explained in previous chapters one arrives at the functional The TPE functional with
the conventional forcing potential is

TPE [w] = UTPE [w] − WTPE [w] (25.16)

The internal energy is

  
UTPE [w] = 2 (Mw )T κw =
1 1
2
w T
(κ ) D κ d =w 1
2
(wPT )D (Pw) d, (25.17)
  

25–7
Chapter 25: KIRCHHOFF PLATES: BCS AND VARIATIONAL FORMS 25–8

w=w ^
Prescribed
deflections θs = ^θs Deflection w Master
Lateral
& rotations w load
^ θ
w, ^s Displacement Rotations θ q Γ
BCs Ω

Kinematic κw = P w Equilibrium
in Ω

Constitutive Slave Prescribed

Slave Curvatures Mw= D κw Bending moments
moments & shears
κw in Ω Mw Force BCs ^ , V^
M nn n

Figure 25.6. The Weak Form departure point to derive the TPE
variational principle for a Kirchhoff plate.

where P = [ ∂ 2 /∂ x 2 ∂ 2 /∂ y 2 2 ∂ 2 /∂ x∂ y ]T is the curvature-displacement operator. The groupings

Pw in the last of (25.17) emphasize that P is to be applied to w to form the slave curvatures κw .
The external work WTPE [w] is more complicated than in plane stress and solids. It is best presented
as the sum of three components. These are due to apply lateral loads, applied edge moments and
transverse shears, and to corner loads, respectively:

WTPE [w] = Wq [w] + W B [w] + WC [w], (25.18)

The first two terms apply to all plate geometries and are
 
Wq [w] = q w d, W B [w] = (V̂n w − M̂nn θsw ) d. (25.19)
 V M

The last term: WC arises if the plate has j = 1, 2, . . . , n c corners at which the displacement w j is
not prescribed. If so,
nc nc
+ −
WC = Rcj w j = ( M̂ns − M̂ns ) w. (25.20)
j=1 j=1

§25.3.2. Finite Element Conditions

The variational index of the TPE functional is m = 2, since second derivatives of w (the curvatures)
appear in UTPE . The convergence conditions for finite elements derived using this principle are:
Completeness. The assumed w over each element should reproduce exactly all {x, y} polynomials
of order ≤ 2.
Continuity. The assumed w should be C 1 continuous over the FEM mesh.
The second condition is not easy to satisfy using standard polynomial assumptions. These difficul-
ties have motivated the development of various techniques to alleviate the continuity requirement.

25–8
25–9 §25.4 THE HELLINGER-REISSNER PRINCIPLE

w=w ^
Prescribed
deflections θs = ^θs Deflection w Master
Lateral
& rotations w load
^ θ
w, ^s Displacement Rotations θ q
BCs
Γ
Kinematic κ =Pw
w
Equilibrium Ω
in Ω

Slave Curvatures
κw

Constitutive
Master Prescribed
Curvatures Mw= D κw Bending moments
Slave moments & shears
κM in Ω M Force BCs ^ , V^
M nn n

Figure 25.7. The Weak Form departure point to derive the HR

variational principle for a Kirchhoff plate.

§25.4. THE HELLINGER-REISSNER PRINCIPLE

The Weak Form useful as departure point for the HR principle is shown in Figure 25.7. Both the
transverse displacement w and the bending moment field M are chosen as master fields. The weak
links are the internal equilibrium equations,
The HR functional with the conventional forcing potential is

HR [w, M] = UHR [w, M] − WHR [w, M]. (25.21)

The internal energy is

 
UHR [w, M] = (M κ −
T w 1
2
MT D−1 M) d = (MT κw − U ∗ ) d. (25.22)
 

Here U ∗ = 12 MT D−1 M) is the complementary energy density (per unit of plate area) written in
terms of the bending moments. Integration of this over  gives the total complementary energy
U ∗.
The external work WHR is the same as for the TPE principle treated in the previous section.

§25.4.1. Finite Element Conditions

The variational indices of the HR functional are m w = 2 for the transverse deflection and m M = 0
for the bending moments. Consequently the completeness and continuity conditions for w are the
same as for the TPE, and nothing is gained by going to the more complicated functional.

25–9
Chapter 25: KIRCHHOFF PLATES: BCS AND VARIATIONAL FORMS 25–10

w=w ^
Prescribed
deflections θs = ^θs Deflection w Master
Lateral
& rotations w load
^ θ
w, ^s Displacement Rotations θ q
BCs
Γ
Kinematic κ =Pw
w Ω
in Ω

Constitutive Slave
Curvatures M =Dκ
w w Bending Equilibrium
Slave moments
κw in Ω
Mw

Constitutive Slave
Mκ = D κ
Prescribed
Bending moments
Master Curvatures moments & shears
κ in Ω Mκ Force BCs ^ , V^
M nn n

Figure 25.8. The Weak Form departure point to derive the curvature-displacement
de Veubeke variational principle for a Kirchhoff plate.

It is possible to balance the variational indices so that m w = m M = 1 by integrating the previous

form by parts once. The resulting principle was exploited by Herrmann5 to construct a plate element
with linearly varying w, Mx x , M yy , Mx y . This element, however, was disappointing in accuracy.
Furthermore enforcing moment continuity can be physically wrong. Progress in the construction
of elements of this type was achieved later using hybrid principles.

§25.5. THE CURVATURE-DISPLACEMENT DE VEUBEKE PRINCIPLE

This kind of principle (for elastic solids) was introduced by de Veubeke, and functionals will be
accordingly identified by a dV subscript.

§25.5.1. The dV Functional

In this case both the transverse displacement w and the curvatures field κ are chosen as master
fields. The departure Weak Form is shown in Figure 25.8.

dV [w, κ] = UdV [w, κ] − WdV [w, κ]. (25.23)

The internal functional is

dV [w, κ] = (κT Mw − 12 κT Dκ) d. (25.24)


The external work is the same as for TPE.

5 L. R. Herrmann, A bending analysis for plates, in Proceedings 1st Conference on Matrix Methods in Structural Mechanics,
AFFDL-TR-66-80, Air Force Institute of Technology, Dayton, Ohio, pp. 577-604, 1966.

25–10
25–11 §25.5 THE CURVATURE-DISPLACEMENT DE VEUBEKE PRINCIPLE

§25.5.2. Finite Element Conditions

The variational indices of the dV functional is m w = 2 for the transverse displacement and m κ = 0
for the curvatures. Consequently the completeness and continuity conditions for w are the same as
for the TPE, as nothing is gained by going to the more complicated functional. To get a practical
scheme that reduces the continuity order it is necessary to proceed to hybrid principles.

25–11