MODERATE ROTATION THEORY FOR BEAMS AND SHALLOW SHELLS

1. Introduction 2 Moderate Rotation Theory for Beams with Small Initial Curvature 3. Linearized Theory for Beams with Small Initial Curvature 4. Shell Geometry and Kinematics 5. Stress and Moment Resultants and Loads 6. Constitutive Relations for Shallow Shells 7. Equilibrium Equations and Boundary 8. Perturbations relative to a Nominal Stress State 9. The Shallow Shell Equations in Terms of the Airy Stress Function 10. Conclusion

Abstract

The principle of stationary total potential energy is employed in order to derive the equilibrium equations and boundary conditions for: (i) beams with small initial curvature, with the effect of moderate rotation of the neutral axis included; and (ii) shallow shells, with the effect of moderate rotation of the midsurface included. For both the beam and shell, the nonlinear strain terms associated with moderate rotation are retained, which leads to governing equations which are similar to those of von Kármán plate theory. In addition, nonlinear and linearized governing equations are derived for perturbations relative to a given nominal equilibrium state. It is shown that the equilibrium equations for the shallow shell can be written as two coupled biharmonic equations which, for the case of transverse displacements which are small relative to the initial deformation, reduce to the equations of linearized shallow shell theory.

Moderate Rotation Theory for Beams and Shallow Shells

1 Introduction The purpose of this report is to provide a concise derivation of the equations of equilibrium for thin shallow shells and beams with small initial curvature, with the geometric nonlinearity due to moderate rotation included. The resulting equations are therefore similar to the von Kármán equations for the large deflection of plates (see, e.g., Szilard (1974)). The derivations are based on the variational principle of stationary total potential energy. The total potential energy includes the internal strain energy, as well as the potential energy of distributed loads and edge loads. For the beam, this approach yields the equations of axial and transverse equilibrium and the appropriate boundary conditions at each end, in terms of the displacements and of the neutral axis. For the shallow shell, the variational approach yields the three equations of equilibrium and the appropriate boundary conditions along the edges, in terms of the midsurface displacements u, v, and w. To begin, we derive the equilibrium equations for a beam with small initial curvature, with the effect of moderate rotation included. This simpler case serves as a prelude to the shallow shell by clarifying the overall approach and the physical phenomena under consideration. 2 Moderate Rotation Theory for Beams with Small Initial Curvature Consider a beam of length L which has a small amount of initial curvature, as shown in Fig. 1. The displacements of the beam's neutral axis in the x and z directions are denoted by u and w, and the z coordinate of the neutral axis after deformations is given by wtotal ( ) x = w0 ( ) x + ( ) w x (1) where w0( )x

represents the initial undeformed shape of the neutral axis, and ( )w x

is the z displacement of the neutral axis relative to its

undeformed configuration.

Figure 1 .

y

x

b

–b

–a a

Figure 2. Rectangular planform and coordinate system.

z,wp

zp

xQ M

L

Nx,u

0

Beam with initial curvature and reference coordinate system.

The strain ε of the neutral axis in the x direction is given by

ε = u, x + 12 ( ) w0 + w , x2 – w0, x

2 (2)

= u, x + w0, x + 12 w , x2

where the subscript , x denotes differentiation with respect to x. The terms in w in Eq.(2) represent the additional stretching of the neutral axis due to moderate rotation. The internal forces in the beam are the axial stress resultant N, the bending moment M, and the transverse shear Q, with sign conversions defined as in Fig. 1. The resultant N, M, and Q can be expressed in terms of the displacements u and w as follows.

N = E Aε = ( ) E A u, x + w0, x w , x + 12 w , x2 (3a)

M = E I κ = – E I w , x x (3b)

Q = M, x = – ( ) E I w , x x , x (3c)

where E is the elastic modulus, A and I are the area and moment of inertia of the beam cross section, and the quantity κ , defined by

κ = – w , x x (4)

is the curvature change of the neutral axis. The total potential energy Π of the beam is

Π u,w = ∫ ( ) 12 E A ε2 + 12 E I κ2 – pxu – pzw

0

L

d x –

– ( ) N *u + Q*w – M*w , x OL

(5) where ε and κ are given in terms of u and w by Eqs. (2) and (4), and N*, M*, and Q* are the end loads applied at x = 0 or x = L. We can now obtain the equilibrium equations by means of the principle of stationary total potential energy, which can be written in the variational form

δ Π = ∫ ( ) E Aεδε + E I κδκ – pxδu – pzδw d x

0

L

– ( ) N *δu + Q*δw – M*δw , x 0L= 0

(6)

where the operator δ denotes an arbitrary small variation. Equation (6) may be rewritten

δ Π = ∫ ( ) N δε + M δκ – pxδu – pxδw

0

L

dx

– ( ) N *δu + Q*δw – M*δw , x 0L = 0 (7)

Equation (7) also follows from the principle of virtual work, which states that a force distribution which satisfies equilibrium performs zero total work when it acts through any small arbitrary displacement distribution. In order to collect the coefficients of δu and δw the first two terms of the integral in Eq.(7) are integrated by parts:

∫ N δε dx = ∫ N δu, x + ( ) w0 + w , x δw , x dx

0

L

0

L

= ( ) N δu 0L – ∫ N , x δu dx

0

L

+ N ( ) w0 +w , x δw 0L – ∫ N ( ) w0 + w , x , x δw dx

0

L

(8a)

∫ M δκ dx = – ∫ M δw , xx0

L

0

L

= ( ) – M δw , x 0L + ∫ M, xδw , xdx

0

L

= ( ) – M δw , x 0L + ( ) M, xδw 0

L – ∫ M, xx δw dx 0

L

(8b)

Substituting Eqs.(8) into Eqs.(7) and collecting the coefficient of δu and δw yields

δ Π = – ∫ ( ) N , x + px δu dx – ∫ { } M, xx + N ( ) w0 + w , x , x + pz δw dx0

L

0

L

+ ( ) N – N * δu 0L – ( ) M – M* δw , x 0

L (9)

+ { } M, x + N ( ) w0 + w , x – Q* δw 0L = 0

Because δu and δw are arbitrary, each of the terms in the expression (9) must vanish independently, which leads to the equilibrium equations

N , x + px = 0 (10a)

M, xx + N ( ) w0 + w , x , x + pz = 0 (10b) and the boundary conditions at x = 0 and x = L :

N = N * or δ u = 0 ; i.e., u = u* (11a)

M = M * or δ w , x = 0 ; i.e., w , x = w , x * (11b)

M , x + N ( ) w0 + w , x = Q* or δ w = 0 ; i.e., w = w * (11c)

where the superscript* denotes a prescribed value. Note that by substituting the expressions (3) for N and M, Eqs. (10) and (11) can be expressed in terms of the displacements u and w . By substituting from Eq (10a ), the equilibrium equation (10b ) can be written

M, xx + N ( ) w0 + w , xx + p z – px( ) w0 + w , x = 0 (12)

If there are no distributed loads in the x direction, then px = 0 and the equilibrium equations (10) reduce to

N ( )x = N * = Const. (13a)

M, xx + N *( ) w0 + w , xx + pz = 0 (13b) 3 Linearized Theory for Beams with Small Initial curvature Due to the nonlinear terms in the axial strain (2), the total potential given by Eq. (5) contains terms of up quartic order in w , x , and the equilibrium equations (10) and boundary conditions (11a, c) are

nonlinear in u and w. It is frequently advantageous to linearize the equations by considering a solution which is a small perturbation relative to some given nominal solution. For the perturbation analysis, we first assume that the displacements, strain, curvature, and resultants

{ } u, w , ε, κ, N , M, Q (14)

represent a known nominal solution to the nonlinear equations(10), where the overbar symbol denotes that a quantity is associated with this given nominal deformation. Now consider a small perturbation of the loads, i.e.,

px = px + px ; pz = pz + pz (15)

N * = N * + N * ; M* = M* + M * ; Q* = Q* + Q* where the ^ symbol denotes a small perturbation relative to the nominal solution value. These loads will give rise to the perturbed solution

{ } u + u , w + w , ε + ε , κ + κ, N + N , M + M , Q + Q (16)

Substitution of the perturbed force and displacement quantities (15) and (16) into the governing equations (10) and (11) yields the equilibrium equations

N , x + px = 0 (17a)

M, xx + N w , x + N( ) w0 + w + w , x , x + pz = 0 (17b) for the perturbed state, as well as the boundary conditions

N = N * or u = u* (18a)

M = M * or w , x = w , x*

(18b)

M , x + N w , x + N ( ) w0 + w + w , x = Q* or w = w * (18c)

to be applied at x = 0 and x = L. In arriving at Eqs.(17) and (18), several terms which contain barred quantities cancel one another, due to the fact that the nominal (barred) solution satisfies the original equilibrium equations (10) as well as the associated boundary conditions(11). From Eq. (2) for the axial strain,

ε = u, x + 12 ( ) w0 + w , x2 – w0, x

2

(19a)

ε + ε = ( )u +u , x +12 ( )w 0 + w + w , x

2 – w 0, x2 (19b)

thus

ε = u, x + 12 ( )w0 + w + w , x2 – ( )w0 + w , x2

(20)

= u, x + ( ) w0 + w , x w , x + 12 w , x2

and from Eq.(4) for the curvature,

κ = – w , xx (21a)

κ + κ = – ( ) w +w , xx (21b)

thus

κ = – w , x x (22)

The perturbations N and M in Eqs. (17) and (18) are therefore given by

N = E A ε (23a)

M = E I κ

(23b) with ε and κ defined by Eqs. (20) and (22). The axial stress resultant N is nonlinear in w , but can be linearized by

neglecting the higher - order term 12 w , x

2 in the axial strain ε , in order to obtain

N = E Aε ≅ E A u , x + ( ) w0 + w , x w , x (24)

The nonlinear equations (17b) and (18c) can then be linearized by neglecting the higher -

order term N w , x This results in the equations

M , xx + N w , x + N ( ) w0 + w , x , x + p z = 0 (25a)

M , x + N w , x + N ( ) w 0 + w , x = Q * or w = w * (25b) which can be used in place of Eqs. (17b) and (18c), provided that the perturbation is sufficiently small. Substituting the perturbed force and displacement quantities (15) and (16) into the potential energy expression (5) and expanding yields the following expression for the total potential energy of the perturbed state:

Π u ,w = ∫ N ε + 12 E Aε 2 + M κ + 12 E I κ 2 0

L

– ( ) p x + p x u – ( ) p z + p z w dx (26)

– ( ) N * + N * u + ( ) Q* + Q * w – ( ) M * + M * w , x 0L + Const .

where ε and κ are given by Eqs. (20) and (22). The final additive constant represents several terms which depend only on the nominal solution and the loads. These terms are independent of and u and w can therefore be omitted, because they do not affect the results obtained when the principle of minimum potential energy is applied. Further simplification of Eq. (26) is possible by exploiting the fact that the nominal (barred) solution satisfies the original equilibrium condition (7). Equation (26) can be rewritten as

Π u ,w = ∫ ( ) 12 N w , x

2 + 12 E A ε 2 + 12 E I κ 2 – p xu – p zw dx0

L

– ( ) N *u + Q *w – M *w , x 0L

(27)

+ ∫ { } N u , x + ( ) w0 + w , xw , x – M w , xx – p xu – pzw dx

0

L

– ( ) N *u + Q *w – M *w , x 0L

in which the third and fourth lines are the terms which contain barred force quantities and are linear in u and w . The third and fourth lines of Eq.(27) taken together must equal zero, because they are of the same form as the right hand side of Eq. (7), or equivalently, Eq. (9), with u and w playing the roles of δu and δw .

The total potential energy of the beam, in terms of u and w , is therefore given by

Π u ,w = ∫ { 12 N w , x

2 + 12 E A u , x + ( ) w 0 + w , xw , x + 12 w , x 2 2

0

L

+ }12 E I w , xx

2 – p xu – p zw dx – ( ) N *u + Q *w – M *w , x 0

L

(28) By neglecting the terms of Eq. (28) which are higher than quadratic order in u, x and w , x , we obtain the linearized potential energy function

Π u, w ≅ ∫ { 12 N w , x

2 + 12 E A u , x + ( ) w0 + w , xw , x 2

0

L

+ 12 E I w , xx2

– }p xu – p zw dx – ( ) N *u + Q *w – M *w , x 0L

(29) Minimization of the function (29) leads to the linearized equilibrium equations (17a) and (25a), and boundary conditions (18a,b) and (25b), where N is given by Eq. (24). If the barred quantities are equal to zero, Eq. (29) reduces to the linearized potential function for small perturbations relative to the initial undeformed configuration. 4 Shell Geometry and Kinematics

We now consider a shallow shell with undeformed midsurface u0′ ( ) x , y

.

Alternatively, the shell can be viewed as a plate with the initial geometric imperfection w 0( ) x , y

. The plan form is taken as rectangular, with length 2a and width 2b as shown in Fig. 2 The displacements of the shell midsurface in the x,y , and z directions are denoted by u,v, and w. The z coordinate of the midsurface after deformation is given by

w total ( ) x , y = w 0 ( ) x y + ( ) w x , y (30)

where w( x, y ) is the z displacement of the midsurface relative to its undeformed configuration. The midsurface strains, which include the nonlinear terms due to moderate rotation of the midsurface, are given by

εx = u, x + 12 ( ) w0 + w , x2 – w0, x

2

y

x

b

–b

–a a

Figure 2. Rectangular planform and coordinate system.

= u, x + w0, x w , x + 12 w , x2

(31a)

εy = V , y + 12 ( ) w0 + w , y2 – w , y2

= v , y + w0 , y w , y + 12 w , y2 (31b)

γx y = u, y + v , x + ( ) w0 + w , x ( ) w0 + w , y – w0, x w0, y

= u, y + v , x + w0, x w , y + w0, y w , x + w , x w , y

(31c) where εx and εy are the strains in the x and y directions, and εy is the shear strain. For notational convenience in Eqs. (31) and the remainder of this report, partial differentiation with respect to a particular independent variable is denoted by a comma followed by the variable; for example

w , x =

∂w

∂x ; w , xy =

∂2w

∂x ∂y ; εx , yy =

∂2εx

∂y 2

(32) The midsurface curvature changes are defined by

κx = – w , xx (33a)

κ y = – w ,y y (33b)

κ x y = – w ,x y

(33c) where κx and κy are the curvature changes in the x and y directions, and κx y is the change in twist. The kinematic equations (31) and (33) are applicable to deformations for which: (i). The strains are small relative to unity; and (ii). The squares of the slopes of the deformed and undeformed midsurface are small relative to unity. 5 Stress and Moment Resultants and Loads The positive directions of action for the in-plane stress resultants Nx , Ny , and Nx y, which have units of force per unit length, and the distributed loads px and py , which have units of force unit area, are shown in Fig. 3a. The positive directions of action for the bending moment resultants Mx , My , and Mx y, the Kirchhoff shear resultants Vx and Vy , the corner forces F, and the transverse load pz are shown in Figs.3b, c.

NyNxy

Py

Px

Nxy

Nx

y

xMy

Mxy

Mx

Mxy

(a) (b)

F

Vy

F F Corner Force

y

Vy

x

Vy

Pz

F

(c)

Vx

Figure 3. Sign conventions and notation for (a) in-plane stress resultants Nx , Ny , and Nxy and in-plane distributed loads px and py ; (b) bending moment resultants Mx , My , and Mx y; and (c) effective transverse shear resultants Vx and Vy , corner forces F, and transverse load pz .

The external loads which act on the shell consist of the distributed loads px , py ,

and pz, and the edge loads, which are:

Along x = ± a : Nx

* , Nx y * , Vx

*, Mx*

Along y = ± b : Ny * , Nx y

* , Vy * , My

* (34)

At each corner ( ) ± a, ± b : the corner force F*

where the superscript* denotes that the force quantity acts at the edge of the shell.

6 Constitutive Relations for Shallow Shells Provided that the shell's thickness is small compared to it's plan form dimensions and radii of curvature, the constitutive relations are of the same form as for flat plate, and are given by

Nx = E t

1 – ν2( ) εx + ν εy

(35a)

Ny = E t

1 – ν2( ) εy + ν εx

(35b)

Nxy = E t

( )2 1 + ν γx y

(35c)

and

Mx = ( ) D κx + νκy (36a)

My = ( ) D κy + νκx

(36b)

Mx y = ( ) D 1 – ν κx y (36c)

where t is the shell thickness, E is the elastic modulus, ν is Poisson's ratio, and

D = E t3

12 ( ) 1 – ν2 (37)

is the bending stiffness. Note that the appropriate expressions for the strains in Eqs. (35) and the curvature changes in Eqs.(36) are given by Eqs.(31) and (33), which include the effects of initial curvature and nonlinear and nonlinearity.

7 Equilibrium Equations and Boundary Conditions for Shallow Shells The principle of stationary total potential energy can now be used to derive equations of equilibrium and the associated boundary conditions. The total potential energy Π of the shell is given by

Π u, v , w = ∫

– b

b

∫ – a

a

{ E t 2 ( ) 1 – ν2

⎛⎝

⎞⎠ εx

2 + 2 ν εx εy + εy2 + 1 – ν

2 γx y2

+ 12 } D κx2 + 2 ν κx κy + κy

2 + 2 ( ) 1 – ν κx y2 – px u – py u – pz w dx dy

– ∫ ( ) Nx

* u + Nx y* v + Vx

* w – Mx* w , x dy

– b

b

x = –a

x = a

– ∫ ( ) Ny

* v + Nx y* u + Vy

* w – My* wy dx

– a

a

x = – b

y = b

– ( ) F* w x = – ax = a y = – b

y = b (38)

in which the first integral represents the membrane and bending strain energy plus the potential energy of the external distributed loads; the second and third integrals represent the potential energy

of the external loads which act on the edges x = ± a and y = ± b and the final term represents the potential energy of the concentrated loads which act on the shell at the corners. The condition that Π be stationary can be expressed

δ Π ∫ =

– b

b

∫ ( Nx δ εx + Ny δ εy + Nx y δ γx y – a

a

) + Mx δ κx + My δ κ y + 2 Mx yδ κx y – pxδ u – pyδ v – pzδ w dx dy

– ∫ ( ) Nx

*δ u + Nx y* δ v + Vx

*δ w – Mx*δ w , x dy

– b

b

x = – a

x = a

– ∫ ( ) Ny

* δ v + Nx y* δ u + Vy

* δ w – M y * δ w , y dx

– a

a

y = – b

y = b

–( ) F*w x = –ax = a y = –b

y = b (39)

Equation (39) also follows from the principle of virtual work, and is analogous to Eq. (7) for the beam. The midsurface strains and curvature changes in Eq.(39) can be written in terms of the midsurface displacements u, v, and w by means of Eqs. (31) and (33). Equation (39) can then be integrated by parts in the conventional manner in order to collect the coefficients of δu, δv and δ w . The result is

δ Π ∫ = –

– b

b

∫ { ( ) Nx , x + Nxy , y + px δ u + ( ) Ny , y + Nxy , x + py δ v – a

a

+ Mx , xx + 2 Mxy , xy + My , yy + Nx ( ) w0 + w , xx

+ 2 Nx y ( ) w0 + w , xy + Ny ( ) w0 + w , y y + ( ) Nx , x + Nxy , y ( ) w0 + w , x

+ } ( ) Ny , y + Nxy , x ( ) w0 + w , y + pz δ w dx dy

+ ∫ { ( ) Nx – Nx

* δ u + ( ) Nx y – Nx y* δ v – ( ) Mx – Mx

* δ w , x– b

b

+ } Mx , x + 2 Mxy , y + Nx ( ) w0 + w , x – Nx y ( ) w0 + w , y – Vx

* δ w dy x = –a

x = a

+ ∫ {( ) Ny – Ny

* δ v + ( ) Nx y – Nxy * δu – ( ) My – My

* δ w , y– a

a

+ } M y , y + 2 Mxy , x + Ny ( ) w0 + w , y + Nx y ( ) w0 + w , x – Vy * δ w δ x y = – b

y = b

– ( ) 2 Mx y + F* δ w x = – ax = a

y = –b y = b = 0

(40)

Because the coefficients of δu, δv and δw must vanish independently, Eq.(40) provides the equilibrium equations and boundary conditions. The first integral of Eq.(40) yields the x, y , and z equilibrium equations

Nx , x + Nxy , y + px = 0

(41a)

Ny , y + Nxy , x + py = 0 (41b)

Mx , xx + 2Mxy , xy + M y , yy + Nx ( ) w 0 + w , xx + 2Nxy ( ) w 0 + w , xy

E t 1 – ν 2

+ Ny ( ) w0 + w , yy – px ( ) w0 + w ,x – py ( ) w 0 + w , y + pz = 0

(41c) If E , ν , and t are taken to be constant over the shell, the constitutive relations (35) and (36) can be employed in order to express the equilibrium equations (41) as

E t 1 – ν 2

⎛⎝

⎞⎠ εx , x + ν εy , x + 1 – ν

2 γ xy , y + px = 0

(42a)

E t 1 – ν 2

⎛⎝

⎞⎠ εy , y + νεx, y + 1 – ν

2 νxy , x + py = 0

(42b)

DΔΔw = px – px( ) w0 + w , x – py ( ) w0 + w , y

+ E t

1 – ν 2 ( ) εx + νε y ( ) w0 + w , xx + ( ) εy + νεx ( ) w0 + w , yy

+ ( ) 1 – ν γx y ( ) w0 + w , xy (42c)

where

Δ =

∂2

∂x 2 +

∂2

∂y 2

(43)

represents the harmonic differential operator, and the midsurface strains εx , εy and γxy are given in terms of the displacements by Eqs. (31). The remaining terms of Eq. (40) provide the boundary conditions along x = ± a :

E t 2 ( ) 1 + ν 2

( ) εx + νεy = N x *

or u = u * (44a)

E t 2 ( ) 1 + ν

γx y = Nxy *

or v = v * (44b)

– ( ) w , xx + ν w , yy = M x* or w , x = w , x* (44c)

E t 1 – ν2

( ) ε x + ν εy ( ) w0 + w , x + E t 2 ( ) 1 + ν

γ x y ( ) w0 + w , y

– D w , xxx + ( ) 2 – ν w , xyy = V x

* or w = w * (44d) along y = ± b :

E t 1 – ν2

( ) εy + νεx = N y* or v = v * (45a)

E t ( )2 1 + ν

γx y = N xy*

or u = u * (45b)

– ( ) D w , yy + ν w , xx = My* or w , y = w , y * (45c)

E t 1 – ν2

( ) εy + ν εx ( ) w0 + w , y + E t ( )2 1 + ν

γx y ( ) w0 + w , x

– D w , yyy + ( ) 2 – ν w , xxy = Vy

* or w = w * (45d)

and at four corners ( a, b ), ( - a, b ), ( a, - b ), and (- a, - b ):

2 ( )D 1 – ν w , xy = F* or w = w * (46) where the superscripts* denote prescribed edge values.

8 Perturbations Relative to a Nominal Stress Stat Due to nonlinear terms in the midsurface strains (37, the total potential (38) for the shallow shell contains terms of up to quartic order in w ,x and w ,y and the equilibrium equations (42) and boundary conditions ( 44 a, b, d ) are nonlinear in the displacements u,v, and w . As for the case of the beam, it can be useful to rewrite the equations in terms of a perturbation relative to some given nominal solution. A linearized analysis can then be employed for perturbations which are sufficiently small. As for beam, it is assumed that the displacements, strain, curvature, and resultants

{ } u , v , w , εx , κx , Nx , Mx , etc. (47)

represent an initial known nominal solution to the nonlinear equations(42). The distributed loads and edge loads are then perturbed as follows

px = px + px ; py = py + py ; pz = pz + pz

Nx* = Nx

* + Nx* ; Ny

* = Ny* + Ny

* ; Nx y* = Nx y

* + Nx y*

(48)

Mx* = Mx

* + Mx* ; My

* = My* + My

*

Vx* = Vx

* + Vx* ; Vy

* = Vy* + Vy

* ; F* = F* + F * where the ^ symbol again denotes a small perturbation relative to the nominal solution value. The loads (48) give rise to the perturbed solution

{ } u + u, v + v , w + w , εx + εx , κx + κx , Nx + Nx , Mx + Mx , etc. (49) Substituting the perturbed force and displacement quantities (48) and (49) into the equilibrium equations (41) yields a set of perturbed equilibrium equations, which can be simplified by noting that the nominal (barred) solution satisfies the original equilibrium equations, (41). After simplification, the equilibrium equations for the perturbed state are

Nx , x + Nxy , y + px = 0 (50a)

Ny , y + Nxy , x + py = 0 (50b)

– DΔΔw + Nxw , xx + Nx( ) w0 + w + w , xx + Nyw , yy + Ny( ) w0 + w + w , yy + 2Nx yw , xy + 2Nx y( ) w0 + w + w , xy + pz – pxw , x – px( ) w0 + w + w , x

– pyw , y – py( ) w0 + w + w , y = 0 (50c)

where an equivalent form for Eq. (50 c) is

– DΔΔw + Nxw , x + Nx( ) w0 + w + w , x , x + Nyw , y + Ny( ) w0 + w + w , y , y

(51)

+ Nx yw , x + Nx y( ) w0 + w + w , x , y + Nx yw , y + Nx y( ) w0 + w + w , y , x + pz = 0 Analogously to Eq. (20) for the beam, the perturbations of the midsurface strains are given by

εx = u, x + ( ) w0 + w + w , x2 – ( ) w0 + w , x2

= u, x + ( ) w0 + w , xw , x + 12 w , x2

(52a)

≅ u, x + ( ) w0 + w , xw , x (52b)

εy = v, y + 12 ( ) w0 + w + w , y2 – ( ) w0 + w , y2

= v, y + ( ) w0 + w , yw , y + 12 w , y2

(52c)

≅ v, y + ( ) w0 + w , yw , y (52d)

γx y = u, y + v, x + ( ) w0 + w + w , x( ) w0 + w + w , y – ( ) w0 + w , x( ) w0 + w , y

= u, y + v, x + ( ) w0 + w , xw , y + ( ) w0 + w , yw , x + w , xw , y (52e)

≅ u, y + v, x + ( ) w0 + w , xw , y + ( ) w0 + w , yw , x

(52f) where Eqs. (52b, d, f ) provide the linearized strains, for which the quadratic terms in w , x and w , y are omitted. The constitutive relations (35) may now be employed in order to express Eqs. (50) in terms of u, v , and w :

Et1 – ν2

⎛⎝

⎞⎠ εx , x + νεy , x + 1 – ν

2 γxy , y + px = 0

(53a)

Et1 – ν2

⎛⎝

⎞⎠ εy , y + νεx , y + 1 – ν

2 γxy , x + py = 0

(53b) DΔΔw = pz – pxw , x – px( ) w0 + w + w , x – pyw , y – py( ) w0 + w + w , y + Nxw , xx + Nyw , yy + 2Nx yw , xy

+ Et1 – ν2

( )εx + νεy ( ) w0 + w + w , xx + ( ) εy + νεx ( ) w0 + w + w , yy

+ ( ) 1 – ν γx y( ) w0 + w + w , y

(53c)

where εx , εy , and γx y are given by Eqs. (52), and E, ν, and t are taken to be constant over the shell. For perturbations which are sufficiently small, Eq. ( 53c ) can be approximated by the linearized equation

DΔΔw ≅ pz – pxw , x – px( ) w0 + w , x – pyw , y

– py( ) w0 + w , y + Nxw , xx + Nyw , yy + 2Nx yw , xy

(54)

+ Et1 – ν2

( ) εx + νεy ( ) w0 + w , xx + ( ) εy + νεx ( ) w0 + w , yy

+ ( ) 1 – ν γx y ( ) w0 + w , xy in which only the first-order terms are retained. The appropriate equilibrium equations for a linearized analysis are therefore Eqs. ( 53a, b ) and ( 54 ), with the strains given by Eqs. ( 52b, d, f ). Substitution of the perturbed force and displacement quantities (48 ) and ( 49 ) into Eqs. (44), (45 ), and ( 46 ) provides the following boundary conditions along x = ± a :

Et1 – ν2

( ) εx + ν εy = Nx* or u = u* (55a)

Et( )2 1 + ν

γx y = Nx y*

or v = v * (55b)

– ( ) D w , xx + ν w , yy = Mx* or w , x = w , x

* (55c)

Et1 – ν2

( ) εx + ν εy ( ) w0 + w + w , x + Et( )2 1 + ν

γx y ( ) w0 + w + w , y

+ Nxw , y + Nx yw , y – D w , xxx + ( ) 2 – ν w , xyy = Vx* or w = w *

(55d) along y = ± b:

Et1 – ν2

( ) εy + ν εx = Ny* or v = v * (56a)

Et( )2 1 + ν

γ x y = N x y*

or u = u* (56b)

– ( ) D w , yy + ν w , xx = M y* or w , y = w , y

* (56c)

E t1 – ν 2

( ) εy + ν εx ( ) w0 + w + w , y + E t ( )2 1 + ν

γ x y ( ) w0 + w + w , x

+ Nyw , y + Nx yw , x – D w , yyy + ( ) 2 – ν w , xxy = V y* or w = w *

(56d) and at the four corners ( a, b ), ( – a, b ), ( a, – b ), and ( – a, – b ):

2 ( )D 1 – ν w , xy = F * or w = w * (57)

where the superscripts * denote prescribed edge values. For a linearized analysis in which only the first-order terms are retained, the nonlinear conditions ( 55d ) and ( 56d ) can be approximated by the linearized conditions

E t 1 – ν 2

( ) εx + ν εy ( ) w0 + w , x + E t ( )2 1 + ν

γ x y( ) w0 + w , y

+Nxw , x + Nx yw , y – D w , xxx + ( ) 2 – ν w , xyy ≅ Vx* or w = w *

(58a)

E t 1 – ν 2

( ) εy + ν εx ( ) w0 + w , y + E t ( )2 1 + ν

γx y( ) w0 + w , x

+ Nyw , y + Nx yw , x – D w , yyy + ( ) 2 – ν w , xxy ≅ V y* or w = w *

(58b) The appropriate boundary conditions for a linearized analysis are therefore Eqs. ( 55a, b, c ), ( 56a, b, c ), ( 57 ), and ( 58a, b ), with the strains given by Eqs. ( 52b, d, f ). Substituting the perturbed force and displacement quantities ( 48 ) and ( 49 ) into the potential energy expression ( 38 ) and simplifying in a manner analogous to that employed for the beam, we obtain the following expression for the total potential energy of the shell in terms o u, v , and w :

Π u, v , w = ∫

– b

b

∫ – a

a

{ 12 ( ) Nxw , x2 + Nyw , y

2 + 2Nx yw , xw , y

+ E t

( )2 1 – ν 2 ⎛⎝

⎞⎠ εx

2 + 2ν εxεy + εy2 + 1 – ν

2 γ x y2

} + 12D κx2 + 2ν κxκy + κy

2 + ( )2 1 – ν κx y2 – pxu – pyv – pzw dx dy

– ∫ ( )N x

*u + N x y* v + V x

*w – M x*w , x dy

– b

b

x = – a

x = a

– ∫ ( ) N y

*v + N x y* u + V y

*w – M y*w , y dx

– a

a

y = – b

y = b

– ( ) F *w x = – a x = a

y = – b y = b

(59)

where εx ,εy , and γ x y are given by Eqs. ( 52a, c, e ) for the nonlinear case, and by Eqs. ( 52b, d, f ) for the linearized analysis, for which terms of the potential energy (59) which are higher that quadratic order in the displacements are neglected. 9 The Shallow Shell Equations in Terms of the Airy Stress Function In this section it is shown how the three equilibrium equations for shallow shells, which were derived in the preceding two sections, can be reduced to two equilibrium equations in terms of the transverse displacement w and the Airy stress function Φ . For simplicity, we consider the case of normal loading only, with px = py = 0 . Perturbations with respect to a known nominal state are considered, as in the preceding section. The Airy stress function Φ is related to the stress resultants by

Nx = Φ, yy ; Ny = Φ, xx ; Nx y = – Φ, xy (60)

The definitions ( 60 ) guarantee the satisfaction of the conditions ( 41a, b ) for equilibrium in the x and y directions. As in the preceding section, the solution can be viewed as the sum of a known nominal solution and a relative perturbation:

φ = φ + φ (61a)

w = w + w

(61b) The stress resultants associated with the nominal solution and the relative perturbation are consequently given by

Nx = φ, yy ; Ny = φ, xx ; Nx y = – φ, xy (62a)

Nx = φ, yy ; Ny = φ, xx ; Nx y = – φ, xy (62b)

From Eqs. ( 52a, c, e ), it follows that

εx , yy + εy , xx – γxy , xy

= ( ) w0 + w + w , xy2 – ( ) w0 + w + w , xx( ) w0 + w + w , yy

(63) – ( ) w0 + w , xy

2 - ( ) w0 + w , xx( ) w0 + w , yy Equation ( 63 ), which must be satisfied by the strains and the normal displacement w , can be viewed as a compatibility condition which includes the additional stretching of the midsurface due to moderate rotation. The constitutive relations (35 ) yield the identity

εx , yy + εy , xx – γxy , xy

= 1E t

( ) Nx – ν Ny , yy + 1E t

( ) N y – ν Nx , xx – ( )2 1 + ν E t

Nxy , xy (64)

= 1E t

ΔΔφ provided that E, ν, and t are constant over the shell. Equating the right hand sides of Eqs. ( 63 ) and ( 64 ) thus yields the equation

1

E t ΔΔφ = ( ) w0 + w + w , xy

2 – ( ) w0 + w + w , xx( ) w0 + w + w , yy

(65)

– ( ) w0 + w , xy2 – ( ) w0 + w , xx( ) w0 + w , yy

which represents the shell's membrane behavior. Substitution of Eqs. ( 62 ) into the equation ( 50 ) for equilibrium in the z direction yields

DΔΔw = pz + φ, yyw , xx – 2φ, xyw , xy + φ, xxw , yy

+ φ, yy( ) w0 + w + w , xx – 2φ, xy( ) w0 + w + w , xy (66)

+ φ, xx( ) w0 + w + w , yy Equations (65 ) and ( 66 ) are the coupled governing equations for a shallow shell

in terms of φ and w with the effect of moderate rotation included. Note that so far, no

assumption has been made regarding the magnitudes of φ and w relative to the nominal solution. Equations ( 65 ) and ( 66 ) can be linearized by assuming that the derivatives of

φ are small relative to the derivatives of φ , and that the derivatives of w are small relative to the derivatives of w and w0 . Retaining only the terms which are first order

in φ and w then leads to the equations

1

E t ΔΔφ = – ( ) w0 + w , yyw , xx – 2( ) w0 + w , xyw , xy

+ ( ) w0 + w , xxw , yy

(67a)

DΔΔw = pz + φ, yyw , xx – 2φ, xyw , xy + φ, xxw ,y y

+ φ, yy( ) w0 + w , xx – 2φ, xy( ) w0 + w , xy + φ, xx( ) w0 + w , yy (67b)

If the nominal solution is taken to correspond to the undeformed configuration,

then φ = w = 0, and Eqs. ( 65 ) and ( 66 ) reduce to

1E t

ΔΔφ= ( ) w0 + w , xy2 – ( ) w0 + w , xx( ) w0 + w , yy

(68a)

– ( ) w0, xy2 – wo , xxw0,y y

DΔΔw = pz + φ, yy( ) w0 + w , xx – 2φ, xy( ) w0 + w , xy

+ φ,x x( ) w0 + w , yy (68b)

where the ^ accents are removed, because in this case the perturbation is equivalent to the total solution. Equations (68 ) can be linearized by neglecting terms which are higher than first order in φ and w , which leads to the well known equations

1

E t ΔΔφ= – ( ) w0, yyw , xx – 2w0, xyw , xy + w0, xxw , yy

(69a)

DΔΔw = pz + w0, yy φ, xx – 2w0, xyφ, xy + w0, xxφ, yy (69b)

of linearized shallow shell theory. 10 Conclusion The variational principle of stationary total potential energy was employed in order to derive the equilibrium equations and boundary conditions for shallow shells and beams with small initial curvature, with the nonlinear effect of moderate rotation included. The total potential energy for the beam, given by Eq. (5 ), and the total potential energy for the shallow shell, given by Eq. (38 ), contain terms of up quartic order in the displacements, and are suitable for nonlinear analyses by the Ritz method. For the nonlinear analysis of perturbations relative to a given nominal stress state, the potential energy expressions ( 28 ) and ( 59, 52a, c, e ) can be used. The corresponding linearized potential energy expressions (29) and ( 59, 52b, d, f ), which are quadratic in the displacements, can be employed for the analysis of small perturbations. References

Steele, C. R , Stanford University course notes for shell analysis. Szilard, R. ( 1974 ). Theory and Analysis of Plates - Classical and Numerical Methods. ( Prentice-Hall, Englewood Cliffs, NJ. )