Surya N. Patnaik

Ohio Aerospace Institute, Brook Park, Ohio

Dale A. Hopkins

Glenn Research Center, Cleveland, Ohio

Completed Beltrami-Michell Formulation in

Polar Coordinates

NASA/TM—2005-213634

October 2005
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Surya N. Patnaik

Ohio Aerospace Institute, Brook Park, Ohio

Dale A. Hopkins

Glenn Research Center, Cleveland, Ohio

Completed Beltrami-Michell Formulation in

Polar Coordinates

NASA/TM—2005-213634

October 2005

National Aeronautics and

Space Administration

Glenn Research Center
Available from

NASA Center for Aerospace Information

7121 Standard Drive

Hanover, MD 21076

National Technical Information Service

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Springfield, VA 22100

This report is a formal draft or working

paper, intended to solicit comments and

ideas from a technical peer group.

This report contains preliminary

findings, subject to revision as

analysis proceeds.

Available electronically at http://gltrs.grc.nasa.gov

This work was sponsored by the Low Emissions Alternative

Power Project of the Vehicle Systems Program at the

NASA Glenn Research Center.
NASA/TM—2005-213634 1 

Completed Beltrami-Michell Formulation in Polar Coordinates 

Surya N. Patnaik 

Ohio Aerospace Institute 

Brook Park, Ohio 44142 

Dale A. Hopkins 

National Aeronautics and Space Administration 

Glenn Research Center 

Cleveland, Ohio 44135 

Summary 

 A set of conditions had not been formulated on the boundary of an elastic continuum since the time of 

Saint-Venant. This limitation prevented the formulation of a direct stress calculation method in elasticity 

for a continuum with a displacement boundary condition. The missed condition, referred to as the 

boundary compatibility condition, is now formulated in polar coordinates. The augmentation of the new 

condition completes the Beltrami-Michell formulation in polar coordinates. The completed formulation 

that includes equilibrium equations and a compatibility condition in the field as well as the traction and 

boundary compatibility condition is derived from the stationary condition of the variational functional of 

the integrated force method. The new method is illustrated by solving an example of a mixed boundary 

value problem for mechanical as well as thermal loads. 

Introduction 

 The stress-strain law, the equilibrium equation (EE), and the compatibility condition (CC) are the 

three fundamental relations in elasticity. The material law was formulated in the mid-seventeenth century 

by Hooke (1635–1703). The equilibrium equation or the stress formulation is credited to Cauchy (1789–

1857). Saint-Venant (1797–1886) developed the CC, or the strain formulation. It is a general belief that 

the fundamental elasticity relations were known for over a century. The thrust, therefore, was to develop 

approximate solution techniques because a closed-form solution cannot be generated for the vast majority 

of the solid mechanics problems. Such techniques included Airy’s method (ref. 1), Ritz’s method (ref. 2), 

the moment distribution technique (ref. 3), Kani’s method (ref. 4), the finite element technique (ref. 5), 

and others. 

 It is surprising that the strain formulation was not known on the boundary of an elastic continuum, 

even though Cauchy’s stress formulation explicitly contained the boundary conditions also known as the 

traction conditions. Because of this deficiency, problems with displacement boundary conditions could 

not be solved using the direct stress calculation method, popularly referred to as the Beltrami-Michell 

formulation (BMF) (ref. 6). The strain formulation that was missed on an elastic boundary is referred to 

as the boundary compatibility condition (BCC). The BCC has been derived. Now the stress and strain 

formulations are parallel in form; each contain field equations and boundary conditions. Earlier, we 

derived the BCC for two-dimensional (ref. 7) and three-dimensional (ref. 8) problems in elasticity in 

Cartesian coordinates. The BMF was completed by adding the new BCC to the classical method. The 

completed Beltrami-Michell formulation (CBMF) can be used to solve displacement as well as mixed 

boundary value problems in elasticity. The CBMF stress formulation is as versatile as the Navier 
NASA/TM—2005-213634 2 

displacement method, yet its equation structure is simpler. Solutions to plate and shell problems via the 

CBMF are discussed in references 9 to 11. 

 A conservative elastician, believing the set of existing equations to be sufficient, may be reluctant to 

accept the new BCCs. However, it should be realized that some formulae and equations of the solid 

mechanics discipline were not completed in the first attempt, but were perfected eventually. For example, 

perfecting the flexure formulae required a century between Galileo, Bernoulli, and Coulomb. SaintVenant completed the shear stress formula that was initiated by Navier. Cauchy formulated the stress 

equilibrium equation that was also developed by Navier in terms of displacement, but it contained only a 

single material constant instead of two. 

 The formulation of the BCC in polar coordinates is the primary contribution of this report. The use of 

the new condition is also illustrated through the solution of a mixed boundary value problem for 

thermomechanical loads. The CBMF containing the BCC is obtained from the stationary condition of the 

variational functional πs of the integrated force method (IFM) (ref. 8). The variational calculation is 

performed in two distinct steps: 

 (1) The terms of the functional πs are transformed to obtain integrands, whose coefficients are either 

displacement variables, stress function, or reactions. 

 (2) The stationary condition (ref. 12) of the functional δπs with respect to displacement, stress 

function, and reaction yield all the expressions of the CBMF. 

The BCC is the coefficient of the variational stress function in the line integral term. Variational calculus 

in polar coordinates is more difficult than that in the Cartesian system (ref. 7) because the coefficients of 

the terms in the functional are functions of the r-coordinate. Also the Jacobian (J = r) has to be used. A 

nonvariational approach or carelessness can easily miss an expression because of the tensorial nature of 

stress and strain. The accuracy of CBMF derivation is essential because solution of elasticity problems in 

polar coordinates is very popular. Many existing elasticity solutions can be verified by back-substituting 

into the CBMF. To demonstrate the use of the new condition, two mixed boundary value problems are 

solved. The first example is for mechanical load, while the second is for thermal load. 

 This report is organized as follows: First, a variational derivation is given for the CBMF. Green’s 

theorem is used for a quick validation of the new boundary condition. Then the CBMF is used to solve a 

problem with stress and displacement boundary conditions. This is followed by discussion and 

conclusions. Appendix A is a listing of symbols and acronyms found in this report. Appendix B presents 

the major steps of the variational derivation, which can be used by the reader to verify the BCC. 

Appendix C presents the solution of a structure using the IFM, which is the discrete analogue of the 

CBMF. 

Completed Beltrami-Michell Formulation in Polar Coordinates 

 The CBMF in polar coordinates is obtained from the stationary condition of the variational functional 

(ref. 7) of the IFM. The functional πs has three terms (eq. (1a)). The first term A(σ, u) represents the strain 

energy, expressed in terms of stress σ and displacement u. The second term B(ε, φ) is the complementary 

strain energy written in terms of the strain ε and the stress function φ. The third term W is the potential of 

the work done. Basic steps of the derivation are given in appendix B. The functional is transformed into 

integrals with integrands whose coefficients are either displacement, stress function, or reaction variables. 

Symbolically it can be represented as follows: 

πs =+− ABW (1a) 
NASA/TM—2005-213634 3 

( ){ } ( ) ( ){ } ( )

( ){ }

1

2

field EE field CC boundary EE boundary CC

continuity condition reaction 0

s

D

u ds u d d

d

π= + φ + + φ ⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎣ ⎦ ⎣ ⎦⎣ ⎦

+ =

∫∫ ∫ ∫

∫

A A

A

A A

A

 (1b) 

where the two displacement components 

u

v

⎧ ⎫ ⎨ ⎬ ⎩ ⎭

are represented as {u}. Likewise, {reaction} represents the 

two reactions reaction along . reaction along 

⎧ ⎫r ⎨ ⎬ ⎩ ⎭θ Also, D is the plate domain; 1A and 2 A are boundary segments where 

traction is prescribed and reaction is induced, respectively; and A is the line segment where stress is 

indeterminate. For a two-dimensional problem, A = 2 A . The stationary condition of the functional in 

equation (1b) with respect to displacement, stress function, and reaction can be represented by the 

following symbolic expression: 

( ) {} ( ) ( ) {}

( ) ( ){ }

1

2

field EE field CC boundary EE

boundary CC continuity condition reaction 0

s

D

u ds u d

d d

δπ = δ + δφ + δ ⎡ ⎤⎡ ⎤ ⎣ ⎦⎣ ⎦

+ δφ + δ = ⎡ ⎤ ⎣ ⎦

∫∫ ∫

∫ ∫

A

A A

A

A A

 (1c) 

 The field EE and field CC are the coefficients of the variational displacement and stress function, 

respectively, in the surface integral terms of the functional (see also appendix B). Likewise the 

boundary EE and boundary CC are the coefficients of the variational displacement and stress function, 

respectively, in the line integral terms. The continuity conditions are the coefficients of the variational 

reactions. The field equations and boundary conditions of the CBMF recovered from the stationary 

condition of the variational functional in equation (1c) are as follows: 

Equilibrium equations 

Field: 

1 ( ) 0 r r

r b

rr r

∂σ ∂τ σ −σθ + + +=

∂ ∂θ (2a) 

1 2 b 0

rr r

θ

θ

∂τ τ ∂σ

+ + +=

∂ ∂θ (2b) 

Boundary: 

rr r n nP σ + τ= θ (3a) 

r nn P θθ θ τ+ σ = (3b) 

Compatibility conditions 

Field: 

( )

( )

( ) ( )

2 2

22 2

2 2

2 22

2

2

1 1 1 2

1 2

1 1

0

rr r

E r r r r

r r r r

rr r r

θ θ θ

⎛ ⎞⎡ ∂ σ ∂ σ ∂σ + υ ⎢ −υ − ⎜ ⎟ ⎝ ⎠⎢ ∂θ ∂ ∂ ⎣

∂ σ ∂ σ ∂σ υ + υ

+− +

∂ ∂θ ∂

+υ +υ ∂ τ ∂τ ⎤ − −=⎥ ∂ ∂θ ∂ ⎥⎦

 (4) 

. 

Boundary: 

( ) ( )

( ) ( ) ( )

1

2 1

11 0

r rr n

r r

n

r r

θ θ

θ

θ

⎡ ⎤ ∂ ∂τ + υ ⎛ ⎞ ⎢ ⎥ σ − υσ − + σ − σ ⎜ ⎟ ∂ ∂θ ⎝ ⎠ ⎣ ⎦

⎡ ⎤ ∂τ ∂σ + υ

+ ⎢ ⎥ +υ − +υ + τ =

∂ ∂θ ⎣ ⎦

(5) 
NASA/TM—2005-213634 4 

The kinematics boundary conditions are 

u u − = 0 (6a) 

υ −υ= 0 (6b) 

and 

r and θ are the polar coordinates 

σr, σθ, and τ are the stress components 

εr, εθ, and γ are the strain components 

u and υ are the displacements 

br and bθ are body forces 

Pr and Pθ are tractions applied along boundary segment 1 A

u and υ are initial displacements along boundary segment 2 A

φ is the stress function 

 

The CBMF in equations (2) to (6) contains the following: 

 (1) The stress formulation of Cauchy as coefficients of variational displacements δu and δ υ in the 

surface integral. It consists of two EEs in the field (eqs. (2a) and (2b)) and two on the boundary (eqs. (3a) 

and (3b)) that are popularly known as the traction conditions. The stress formulation has two distinct 

components: the field equations and the boundary conditions. 

 (2) Strain formulation as the coefficient of variational stress function δφ. It is written in terms of 

stress for an isotropic material with Young’s modulus E and Poisson’s ratio υ. It consists of the single 

field CC (eq. (4)), along with one new BCC (eq. (5)). Saint-Venant, unlike Cauchy, formulated only the 

field condition. He missed the boundary conditions that we have completed. Now, both the stress and 

strain formulations are consistent, containing the field equations (eqs. (2a), (2b), and (4)) as well as the 

boundary conditions (eqs. (3a), (3b), and (5)). In appendix B, the strain formulation is derived in terms of 

the strains. 

 (3) Displacement boundary conditions. Two kinematics displacement boundary conditions (eqs. (6a) 

and (6b)) are obtained as coefficients of the variational reactions. A rigorous derivation of the continuity 

condition is more difficult than the stress and the strain formulations, which are straightforward. 

 The three-component stress tensor (σr, σθ, and τ) is indeterminate in the field and on the boundary 

because the state of equilibrium provides only two equations. To achieve determinacy of the stress state, 

we must add one CC in the field as well as one on the boundary. Saint-Venant has given us the field CC. 

We have formulated the BCC. For the derivation of elasticity equation, the variational technique is an 

elegant method because of the tensorial nature of stress and strain. A nonvariational approach may miss 

an equation. 

 The CC should be imposed only when the domain is indeterminate, whether it is the field or the 

boundary. The CC has no relevance for a determinate domain or a determinate boundary. A plane stress 

problem is one degree indeterminate in the field because there are three stresses and two displacements. It 

has one field CC. A BCC should not be imposed on a free or a determinate boundary, where at least one 

stress component is zero. A clamped boundary is typically indeterminate; thus, one BCC is imposed. 

 The solution of an elasticity problem using the CBMF has two distinct steps: 

 (1) The stress state is calculated first using the equations (1) to (5) for an elastic continuum with stress 

and displacement boundary conditions. The displacement boundary conditions (eq. (6)) are not used, but 

the BCC is used. 
NASA/TM—2005-213634 5 

 (2) Displacements are back-calculated by integrating the strain field. The kinematics boundary 

conditions given by equation (6) are used to evaluate the constants of integration in the displacement 

function. 

The CBMF recognizes that displacement does not induce stress. The derivative of displacement, which 

becomes the strain that induces stress, is accounted for through the BCC. Displacement conditions are 

used to eliminate rigid body movement as explained in step (2). 

 The BCC expressed in strain, stress, and displacement is as follows: 

Expressed in strain: 

( ) 0

2 2

r r r rn n

r rr r r r r

θ θ

⎡ ⎤⎡ ⎤ ε ∂ε ∂ ∂γ ∂γ γ − ε+ + − − = ⎢ ⎥⎢ ⎥ ⎣ ⎦⎣ ⎦ ∂ ∂θ ∂θ ∂ (7a) 

Written in stress for an isotropic material with Poisson’s ratio υ: 

( ) ( ) ( ) ( ) 1 21( ) 1 1 0 r rr n n

rr r r

θ

θθ θ

⎡ ⎤⎡ ⎤ ∂ ∂τ ∂τ +υ +υ ⎛ ⎞ ∂σ ⎢ ⎥⎢ ⎥ σ − υσ − + σ − σ + + υ − + υ + τ = ⎜ ⎟ ∂ ∂θ ∂ ∂θ ⎝ ⎠ ⎣ ⎦⎣ ⎦

 (7b) 

In terms of displacements u and v: 

22 2 2

2

2 2 0 r

u uu

r nr r r rn

rr r r

θ

⎡ ⎤⎡ ⎤ ∂ ∂ υ ∂υ ∂ ∂ υ ∂υ ∂ ⎢ ⎥⎢ ⎥ − − + +υ− − − =

∂θ ∂ ∂ ∂θ ∂θ ∂ ∂θ ∂ ∂θ ⎣ ⎦⎣ ⎦

 (7c) 

The BCC expression contains either all three strains, or three stresses, or it contains derivatives of two 

displacement components. The BCC is not a continuity condition in displacement, stress, or strain; 

however, it is a function of the variables. As such, the BCC is expressed in the derivatives of stress, strain, 

and displacement, but it is not a component of rotation. The BCC is an independent condition. It forms a 

new elasticity expression that was missed since the time of Saint-Venant. The field CC is a second-order 

differential equation, while the boundary counterpart is a first-order equation. This characteristic is 

applicable to the stress formulation. The field EEs are first-order equations, while the boundary (or 

traction) conditions are algebraic equations. 

Annular Plate Subjected to Thermomechanical Load 

 We will now illustrate the CBMF calculation strategy through the solution of a radially symmetrical 

annular plate with mixed boundary conditions for mechanical and thermal loads. Consider a plate made of 

an isotropic material with Young’s modulus E and Poisson’s ratio υ. It has thickness h (considered unity) 

with outer and inner radii of a and b, respectively, as shown in figure 1(a). The mechanical load case 

consists of a uniform radial load of intensity p applied at the outer boundary r = a. The inner boundary is 

restrained: u = 0 at r = b. The CBMF for the mixed boundary value problem is generated from a special 

case of the variational functional. It is obtained using the condition of symmetry or by setting the shear 

stress τ and transverse displacement υ to zero (τ = 0, υ = 0) as well as by neglecting variation with 

respect to the angle θ: 

f 0 ⎛ ⎞ ∂ = ⎜ ⎟ ⎝ ⎠ ∂θ . Also, a simpler stress function ψ is used. 
NASA/TM—2005-213634 6 

σr =ψ−U (8a) 

d r U

dr r θ

⎛ ⎞ ψ ψ σ = +− ⎜ ⎟ ⎝ ⎠ (8b) 

where U = 0 for this example. 

Solution for Mechanical Load Only 

 The equations of CBMF for a symmetrical annular plate subjected to a uniform mechanical load of 

intensity p are listed in equations (9) to (11): 
NASA/TM—2005-213634 7 

Equilibrium equations 

Field: 

( ) 0 r r

r r

∂σ σ −σθ + =

∂ (9a) 

Boundary: 

 at r σ= = p r a (9b) 

Compatibility conditions 

Field: 

( ) ( )( ) 1

0 r r r r

θ θ

∂ + υ

σ − υσ + σ − σ =

∂ (10a) 

Boundary: 

 0 at r r b σθ − υσ = = (10b) 

where the displacement boundary condition is 

u rb = 0 at = (11) 

 There are two stresses σr and σθ and one displacement u. In the field, there is one EE (eq. (9a)) and one 

CC (eq. (10a)). On the boundary there is one traction condition (eq. (9b)) and a single BCC (eq. (10b)). 

Also, at the inner boundary there is one kinematics condition (eq. (11)). Solution to equations (9a), (9b), 

(10a), and (10b) yield the stress response. The single displacement u is back-calculated by integrating the 

strain and evaluating the integration constant using the kinematics condition (eq. (11)). 

 The EE and CC in the field are arranged to obtain the following two simpler uncoupled equations: 

( ) 0 r

d

dr

σ +σ = θ (12a) 

( ) 0 d r r

dr r

σ σ −σθ + = (12b) 

Integration of the first equation (12a) yields the sum of the stresses σr + σθ to be a constant. The second 

equation (12b) is uncoupled and solved. The two constants in the stress variables are determined from the 

traction condition (eq. (9b)) and the new BCC (eq. (10b)). The stress solution follows: 

() ()

() ()

2 2 2

2 22

1 1

1 1 r

r b a

p

a br

⎡ ⎤ +υ + −υ ⎛ ⎞ ⎣ ⎦ σ = ⎜ ⎟ ⎜ ⎟ +υ + −υ ⎝ ⎠

 (13a) 

() ()

() ()

2 2 2

2 22

1 1

1 1

r b a

p

a br θ

⎡ ⎤ +υ − −υ ⎛ ⎞ ⎣ ⎦ σ = ⎜ ⎟ ⎜ ⎟ +υ + −υ ⎝ ⎠

 (13b) 

( )

( )

2

2

1

1

1

r

p

b

a

σ +σ = θ

− υ ⎛ ⎞ + ⎜ ⎟ + υ ⎝ ⎠

 (13c) 

2

2 for 0.3

1 0.54

r

p

b

a

σ +σ = υ= θ

⎛ ⎞ + ⎜ ⎟ ⎝ ⎠

 (13d) 
NASA/TM—2005-213634 8 

For a plate with E = 30×106

 psi, υ = 0.3, p = 1 psi, a = 20 in., and b = 10 in., the stresses at the outer 

boundary are σr = 1.0, while σθ = 0.763 psi and σr + σθ = 1.763 psi. At the inner boundary these are 

σr = 1.356 psi, σθ = 0.407 psi, and σr + σθ = 1.763 psi. The sum of the stresses σr + σθ = 1.763 psi is 

independent of the r coordinate of the plate. The BCC 2

62 0.62 r r

σ − υσ = − θ has an inverse quadratic 

variation with respect to the radius, with a minimum value of zero at the restrained boundary (r = 10 in.) 

and a maximum value of 0.46 psi at the outer free boundary (r = 20 in.). The BCC should not be imposed 

on the free boundary at r = a = 20 in. The stress state in the mixed boundary value problem is obtained 

without any use of the prescribed displacement boundary condition. The displacement function u is 

obtained following the standard elasticity solution strategy. Stress is changed into strain using Hooke’s 

law. It is integrated to obtain the displacement function that contained a constant c. 

( )( )

() ()

22 2 2

2 2

1

1 1

r b a

u pc

Ea b r

−υ − ⎛ ⎞

= + ⎜ ⎟ ⎜ ⎟ +υ + −υ ⎝ ⎠

 (14a) 

The constant c does not affect the stress state. For the problem, the constant is calculated to be zero 

(c = 0) from the homogeneous kinematics boundary condition u = 0 at r = b (eq. (11)): 

( )( )

() ()

22 2 2

2 2

1

1 1

r b a

u p

Ea b r

−υ − ⎛ ⎞ = ⎜ ⎟ ⎜ ⎟ +υ + −υ ⎝ ⎠

 (14b) 

The values of displacements are u = 0 at the inner boundary, r = b, and u = 3.5×10–7 in. at the outer 

boundary, r = a. 

 The CBMF produced the solution to the mixed boundary value problem in two steps: First the stress 

state was calculated using the field EE and CC, along with the traction condition as well as the BCC. 

Then the displacement function was back-calculated. Solution to the mixed boundary value problem 

could not have been obtained by the classical BMF stress formulation. Solution to the mixed boundary 

value problem is not available in standard textbooks in elasticity (refs. 6, 13, and 14). 

Solution for Thermal Load Only 

 The CBMF solution for the annular plate is obtained for a temperature distribution given as 

( )

( ) ( ) a b

b

T T

TT rb

a b

− = + − − (15) 
NASA/TM—2005-213634 9 

The temperature distribution is shown in figure 1(b). It has a linear variation with values Ta and Tb at 

r = a and r = b, respectively. The coefficient of thermal expansion is α. The CBMF equations for the 

annular plate subjected to a thermal load are given below: 

Equilibrium equations 

Field: 

( ) 0 r r

r r

∂σ σ −σθ + =

∂ (16a) 

Traction or boundary: 

 0 at r σ= =r a (16b) 

Compatibility conditions 

Field: 

( ) ( )( ) 1

r r r r

dT E

dr

θ θ

∂ + υ

σ − υσ + σ − σ

∂

= −α

 (17a) 

Boundary: 

 at r ET rb σθ − υσ = −α = (17b)

where the displacement boundary condition is 

u rb = 0 at = (18) 

Both the field and the boundary CCs (eqs. (10a) and (10b)) for the mechanical load are modified for the 

temperature load to obtain equations (17a) and (17b). The field EE is not changed. The mechanical load is 

set to zero (p = 0) in the traction equation (16b). The EE and CC in the field are rearranged to obtain the 

following two simpler working equations: 

( ) r

d dT E

dr dr

σ + σ = −α θ (19a) 

( ) 0 d r r

dr r

σ σ −σθ + = (19b) 

The field equations are solved for the boundary conditions to obtain the response, consisting of σr, σθ, and 

u: 

( )

( ) ( )

{ ( ) ( )( ) ( )( ) }

{ ( ) ( ) ( )( ) }

2 22 22

22 2 2 2 2 2

22 2 2 2 2 2

3

12 12

2 (1 1 2

r

a

b

E ar

r aba b a b

T r a b a b rb a b ab a b

T r a b a b rb a b ab b a

⎧ ⎫ ⎪ ⎪ α −

σ = ⎨ ⎬

− + +υ − ⎡ ⎤ ⎪ ⎪ ⎩ ⎭ ⎣ ⎦

⎛ ⎡ ⎤ × + +υ − + −υ − −υ + −υ − −υ ⎜ ⎡ ⎤⎡ ⎤ ⎣ ⎦⎣ ⎦ ⎝ ⎣ ⎦

⎡ ⎤ ⎞ + − + +υ − + +υ − +υ + − +υ + +υ ⎡ ⎤⎡ ⎤ ⎟ ⎣ ⎦⎣ ⎦ ⎣ ⎦ ⎠

 (20a) 
NASA/TM—2005-213634 10 

( ) ( )

{ ( ) () ( )( ) }

{ ( ) () ( )( ) }

2 22 22

3 2 2 2 2 2 3 3 3 3 22

3 2 2 2 2 2 3 3 2 3 3 22

3

2 2 12

2 3 21

a

b

E

rbaa b a b

T r a b a b r a b a b ab a b

T r a b a b r a b ab a b a b a b

θ

⎧ ⎫ ⎪ ⎪ α

σ = ⎨ ⎬

− + +υ − ⎡ ⎤ ⎪ ⎪ ⎩ ⎭ ⎣ ⎦

⎛ ⎡ ⎤⎡ ⎤ × + +υ − − + +υ − + −υ − −υ ⎜ ⎡ ⎤ ⎣ ⎦ ⎝ ⎣ ⎦⎣ ⎦

⎡ ⎤⎡ ⎤ ⎞ + − + +υ − + − + +υ − + +υ − +υ ⎡ ⎤ ⎟ ⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎠

 (20b) 

( )( )

( ) ( )

{ ( ) ( ) ( )( ) }

{ ( ) ( ) ( )( ) }

2 22 22

22 2 2 2 2 2

22 2 2 2 2 2

1

3

2 12

2 21

a

b

b r

u

rbaa b a b

T r a b a b ra a b a b a b a b

T r a b a b ra a b a b a b a b

⎧ ⎫ ⎪ ⎪ α +υ − = ⎨ ⎬

− + +υ − ⎡ ⎤ ⎪ ⎪ ⎩ ⎭ ⎣ ⎦

⎛ ⎡ ⎤ × + +υ − + − −υ − + −υ − −υ ⎜ ⎡ ⎤⎡ ⎤ ⎣ ⎦⎣ ⎦ ⎝ ⎣ ⎦

⎡ ⎤ ⎞ + − + +υ − + − +υ − + +υ − +υ ⎡ ⎤⎡ ⎤ ⎟ ⎣ ⎦⎣ ⎦ ⎣ ⎦ ⎠

 (20c) 

The numerical values of the response parameters for Ta = 100 °C, Tb = 50 °C, and α = 12×10–6/°C are 

 (1) at r = a: σr = 0 ksi, σθ = –17.5 ksi, and u = 0.012 in. 

 (2) at r = b: σr = 14.2 ksi, σθ = –13.7 ksi, and u = 0 in. 

The sum of the stresses σr + σθ = 18508 – 1800r has a linear variation with respect to the r coordinate 

because of a similar distribution of temperature (see eq. (15)). The BCC 

2

1427.797 6.478 0.780 ksi r ET r

r

σ − υσ + α = + − θ is zero at the inner boundary. 

 The CBMF solved the thermal load problem for a mixed boundary value problem. Superposition of 

solutions for mechanical and thermal loads yields the result for thermomechanical combined load. 

Discussions 

 This section examines the CBMF concept. Attributes of the CC are also given. The annular plate 

example is supplemented with an eight-bar discrete truss structure. The solution to the truss problem 

using the integrated force method (IFM), which is the discrete analogue of CBMF, is given in 

appendix C. 

Completed Beltrami-Michell Formulation 

 Hooke’s law, which is common to all analysis methods, relates stress to strain through the material 

matrix [G]: 

{σ} = ε [G]{ } (21) 
NASA/TM—2005-213634 11 

Stress σ must satisfy the state of equilibrium in the field as well as on the boundary of an elastic 

continuum. Likewise, strain has to comply with the condition of compatibility in the domain as well as on 

the boundary. The stress and strain formulations are sufficient for the determination of the stress state in 

an elastic continuum with stress and displacement boundary conditions. The equations that are required to 

calculate the stress state can be conceptualized in the following symbolic expression: 

{ } Equilibrium equations Mechanical load

stress Compatibility conditions Initial deformation

⎡ ⎤⎧ ⎫ ⎪ ⎪ ⎢ ⎥ = ⎨ ⎬ ⎣ ⎦⎩ ⎭ ⎪ ⎪

 (22) 

The state of equilibrium and compatibility is sufficient for the determination of the stress state. 

Displacement is not required to calculate stress. An elastic body can undergo rigid body displacement and 

rotation that does not induce stress. Total displacement can be decomposed into an elastic component and 

a rigid body component: disp = dispelastic + disprigid. The stress calculation in the CBMF accounts for the 

elastic component via the strain in the field and on the boundary. Recovery of the displacement from the 

stress state uses the kinematics or the rigid body displacement component. 

 Calculating stress by combining the equilibrium and compatibility was envisioned by Michell, and it 

is described by Love in the following quotation (ref. 1): 

 

“It is possible by taking account of these relations [compatibility conditions] to obtain a complete 

system of equations [Equation 19] which must be satisfied by stress components, and thus the 

way is open for a direct determination of stress without the intermediate steps of forming and 

solving differential equations to determine the components of displacements.” 

The proposition of Beltrami and Michell can be realized now with the availability of the new BCC. A 

direct method is now available to calculate the stress state in a general elastic continuum with 

displacement as well as stress boundary conditions. The stress state is obtained without any recourse to 

displacement, which is back-calculated by integrating the strains. In the quotation, “intermediate steps” 

refers to Navier’s displacement method that contains higher order differential equations. For the annular 

plate example, the CBMF required the solution of two uncoupled differential equations. Navier’s method 

in contrast would have required the solution of a third-order differential equation. 

 An IFM of structural analysis has previously been formulated (ref. 8). IFM is the discrete analogue of 

the CBMF in elasticity. In IFM, forces are calculated from a set of equations [S]{F} = {P*} that include 

the EE and the CC. Displacements are back-calculated. The IFM solution to a truss problem is given in 

appendix C. 

Nature of Compatibility Condition 

 The CC is a controller type of relation. Strains are controlled, f(εr, εθ, γ) = 0, in elasticity (or the 

strain formulation); likewise the deformations β are balanced, f(β1, β2, . . . , βn) = 0, in a discrete 

structural system. The controller type of relation cannot be derived from an application of the standard 

concepts of mechanics, like “action equal to reaction” (leading to the EE), or the “cause effect relation” 

(that has given us Hooke’s law), or the “displacement continuity concept” (the “strain continuity” is 

conceptually incorrect). This is probably one important reason for the late development of these CCs. In 

elasticity, the field CC (or Saint-Venant's “strain formulation”) can be derived by simply eliminating 

the displacements from the strain displacement relations. However, the derivation of the BCC requires the 

use of variational calculus. For structures, a direct application of Saint-Venant's “strain formulation” 

would have been sufficient for the derivation of the CC (ref. 8). No calculus would have been required 
NASA/TM—2005-213634 12 

because, like EE, the CC is also an algebraic equation. But such a procedure was not adopted, and the CC 

was not developed as a deformation balance concept. Variational calculus is the right tool to derive the 

BCC because of the tensorial nature of stress and strain. 

Nontriviality Property of Boundary Compatibility Condition 

 The field CCs fCC are satisfied automatically when expressed in continuous displacement functions u

and υ : fCC(u, υ ) = ξ(u, υ ) – ξ(u, υ ) = 0. However, the BCC, when expressed in terms of displacements, 

produces a nontrivial condition: 

22 2 2

2

2 2 0 r

u uu

r nr r r rn

rr r r

θ

⎛ ⎞⎛ ⎞ ∂ ∂ υ ∂υ ∂ ∂ υ ∂υ ∂ ⎜ ⎟⎜ ⎟ − − + +υ− − − =

∂θ ∂ ∂ ∂θ ∂θ ∂ ∂θ ∂ ∂θ ⎝ ⎠⎝ ⎠

 (23) 

The Navier displacement method should account for the BCC because this is not a trivial condition in 

displacement. The BCC should be enforced along the interelement boundaries in a finite element model. 

The role of BCC should be investigated further in the Navier displacement method. A two-span plate 

made of two different materials supported on an elastic foundation may be an ideal example for the 

investigation. 

Rotation and Compatibility Condition 

 The BCC should not be confused with rotation. An elastic body under load moves from its initial 

position to occupy the final form by undergoing strain, an x translation, and a θ rotation (see fig. 1(c)). 

Only strain (not translation or rotation) induces stress. Strain is zero when the body is rigid. Rotation and 

strain are independent of each other, even though both quantities are defined in terms of the derivatives of 

displacement. For example, the BCC, which is a function of the strains is defined in polar coordinates as 

( ) 0

2 2

r r r rn n

r rr r r r r

θ θ

⎡ ⎤⎛ ⎞ ε ∂ε ∂ ∂γ ∂γ γ − ε+ + − − = ⎢ ⎥ ⎜ ⎟ ⎣ ⎦⎝ ⎠ ∂ ∂θ ∂θ ∂ (24) 

The BCC enforces an equality constraint on the strain components; it imposes no restriction on either 

translation or rotation. The annular disk requires the BCC for analysis even while it is undergoing 

translation and rotation on a flat surface (with z = 0), as shown in figure 1(c). 
NASA/TM—2005-213634 13 

Stability of Structure 

 Consider the discrete truss shown in figure 2. Its analysis is given in appendix C. The truss has one 

field CC and one boundary CC as follows: 

Field CC: 

12 3 456 β +β − β − β +β +β = 2 2 0 (expressed in bar deformation, β) (25a) 

12 3 456 σ +σ − σ − σ +σ +σ 2 2 = 0 (expressed in bar stress, σ) (25b) 

Boundary CC: 

2 7 β +β = 0 (in deformation) (25c) 

2 7 σ +σ = 0 (in stress) (25d) 
NASA/TM—2005-213634 14 

The field CC in equation (25a) restrains the six bar stresses. This is the discrete analogue of 

Saint-Venant’s strain formulation (see eq. (4)). On the boundary, two member stresses are related 

(eq. (25c)). The two stresses σ2 and σ7 cannot assume independent values in the lower boundary 

chord in figure 2(a). The situation is similar to the BCC of the annular plate. The stresses σr 

and σθ cannot assume independent values along the inner boundary because of the BCC 

0 0.4068 0.3 1.356 0, or . σ − υσ = ⇒ − × = σ = υσ θ θ r r Compatibility conditions are required for the 

analysis of indeterminate structures, which are more stable than their determinate counterpart. Stability of 

a structure may concern the state of equilibrium. The CC does not degrade the stability of a structure. The 

original truss shown in figure 2(a) is displaced in two steps. The axial constraint at node 5 is released first. 

This process eliminates the BCC but the structure is still stable. It is one degree indeterminate and the 

support at node 5 can move along the x-coordinate direction. If the transverse restraint is released at that 

node, then the structure rotates. The truss undergoes an x translation and a θ rotation as depicted in 

figure 2(b). It can be analyzed as a mechanism by accounting for the field CC (given by eq. (25a)). 

Number of Boundary Compatibility Conditions 

 There are two displacement boundary conditions in a plane elasticity problem. The question is 

“should there be two BCCs?” The answer is “no.” The number of BCCs is equal to the number of field 

CCs, which is equal to the indeterminacy r, defined as the difference in the number of stress n and 

displacement m variables (r = n – m). The elasticity problem in polar coordinates has three stresses and 

two displacements. It is one degree indeterminate, and it has one field CC and one boundary CC. The 

number of displacement boundary conditions in the Navier displacement method is not equal to the 

number of BCCs. Consider the displaced position for the annular plate and the truss shown in figures 1(c) 

and 2(b), respectively. The plate undergoes translation and rotation, referred to as the kinematics 

conditions. The plate is restrained at the inner boundary, which is the elastic condition. The calculation of 

the displacement function in the Navier displacement method requires the simultaneous compliance of 

both types of conditions: kinematics as well as the elastic conditions. In the CBMF, the kinematics and 

the elastic conditions are enforced in two steps. First, the elastic condition, which essentially is the BCC, 

is used to calculate the stress response. In the annular plate example, the BCC is used to calculate stress 

in the CBMF. The kinematics condition u = 0 is then used to evaluate the integration constant in the 

displacement function. 

Concluding Remarks 

 The boundary compatibility condition (BCC) for an elastic continuum has been derived in polar 

coordinates using a variational approach. The BCC in essence is a constraint that is imposed on the strain 

or the stress state. The new boundary condition completes the stress formulation in elasticity. The 

completed Beltrami-Michell stress formulation can be used to calculate the stress state in a general elastic 

continuum without any reference to the displacement in the field or on the boundary. The displacement is 

back-calculated from the stress state. The BCC when expressed in displacements yields a nontrivial 

condition. 
NASA/TM—2005-213634 15 

Appendix A 

Symbols and Acronyms 

Symbols: 

A strain energy 

a, b plate outer and inner radii, respectively 

B complementary strain energy 

br, bθ body forces 

c displacement function constant 

D plate domain 

E Young’s modulus 

f function 

{F} member force vector 

Gr, Gθ Green’s functions 

[G] material matrix 

h plate thickness 

J Jacobian 

A domain boundary 

1 2 A A, boundary segments 

nr, nθ direction cosines 

{P*} load vector 

P P r , θ prescribed loads or tractions 

p mechanical load 

Rr, Rθ reactions 

r, z, θ polar coordinates 

[S] IFM governing matrix 

T temperature distribution 
NASA/TM—2005-213634 16 

U potential function 

u, v displacements 

u v, prescribed displacements 

V body force potential 

W potential of work done 

x translation 

α coefficient of thermal expansion 

β deformation 

εr, εθ, γ plain strain components 

θ rotation 

ξ function of displacement 

πs variational functional of the integrated force method 

σi bar stress 

σr, σθ, τ plain stress components 

υ Poisson’s ratio 

ψ simple stress function 

φ stress function 

Acronyms: 

BCC boundary compatibility condition 

BMF Beltrami-Michell formulation 

CBMF completed Beltrami-Michell formulation 

CC compatibility condition 

EE equilibrium equation 

IFM integrated force method 
NASA/TM—2005-213634 17 

Appendix B 

Variational Formulation for the Completed Beltrami-Michell Formulation 

 This appendix provides the variational derivation of the completed Beltrami-Michell formulation in 

polar coordinates that includes the new boundary compatibility condition (BCC). The equations are 

obtained from the stationary condition of the integrated force method (IFM) functional πs, defined 

previously in equation (1a) as 

πs =+− A B W (B1) 

where 

12 4 3 56 r

D

u u A u hrdrd

rr rr r

θ

θ

⎛ ⎞ σ ∂ τ ∂ τ∂υ τ υ σ ∂υ = + +σ + + − θ ⎜ ⎟ ∂ ∂θ ∂ ∂θ ⎝ ⎠

w∫∫ (B2a) 

7 8 9 10 2 2 11 2

22 2 2

r r

D

B hrdrd

r rr rr rr

θ ⎛ ⎞ ε ∂ φ ε ∂φ ε∂φ γ∂φ γ∂ φ = ++ + − θ ⎜ ⎟

∂θ ∂ ∂θ ∂ ∂ ∂θ ⎝ ⎠

w∫∫ (B2b) 

( )( )

1 2

12 14 13 15 16

rrr 1 2

D

W h uP P d uR R d b u b rdrd θ θθ

⎡ ⎤ ⎛ ⎞⎛ ⎞

= + υ + + υ + +υ θ ⎢ ⎥ ⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠ ⎣ ⎦

∫ ∫ ∫∫

A A

A A vv w (B2c) 

The plate domain D has boundary A, that is separated into segments 1 A and 2 A ; A = 1 A + 2 A . Body 

forces are br and bθ. Along the boundary segment 1 A , loads Pr and Pθ are prescribed, and displacements 

u and υ are free. The segment 2 A has prescribed displacements u and υ that can induce reactions Rx

and Rr. The derivation sets the uniform plate thickness to unity (h = 1) without any consequence. 

 The term A represents the strain energy, and it is expressed in stress and displacement, which are 

considered independent of each other. The strain energy term B is expressed in strain and stress function 

φ, which are also considered independent of each other. The potential of the work done is W. Body force 

potential V is defined as and r

V V b b

r r

θ

∂ ∂ = = ∂ ∂θ

. The stress function φ is defined as 

2

2 2 r V

r r r

∂φ ∂ φ σ =+ −

∂ ∂θ

 (B3a) 

2

2 V

r

θ

∂ φ σ = −

∂

 (B3b) 

r r

∂ ⎛ ⎞ ∂φ τ=− ⎜ ⎟ ∂ ⎝ ⎠ ∂θ (B3c) 
NASA/TM—2005-213634 18 

 Each term of the functional is reduced to obtain new terms that contain two factors. The second factor 

can be displacement, a stress function, or reaction. The first factor is an expression in terms of stress, 

strain, and load. The stationary condition of the functional with respect to displacement, stress function, 

and reactions will yield the following expressions: 

 (1) Field equilibrium equations (EEs) in stress. They are the coefficients of the variational 

displacements δu and δ υ in the surface integral terms. 

 (2) Boundary EEs, or traction conditions. They are the coefficients of δu and δ υ in the line integral 

terms. 

 (3) Field CC in strains. It is the coefficients of the variational stress function ( ) δφ in the surface 

integral term. 

 (4) Boundary CC. It is the coefficient of (δφ) in the line integral term. 

 (5) The displacement continuity condition. It is the coefficient of the variational reactions in the line 

integral term. 

Derivation of equations stated in items (1) to (4) listed above (see eq. (B4) below) is straightforward. The 

derivation of the continuity condition (item (5)) required back-calculation (see eqs. (B10) to (B12) 

below). 

 The first 11 terms of the functional reduced using techniques of calculus are given in equation (B4). 

The other five terms (12 through 16) are retained without any operation. 

[ ] [ ]

1 r r r rr

u ds n u d u ds

r rr

⎛ ⎞ ∂ ⎛ ⎞ ∂σ σ

σ =σ − + ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ∂ ∂ ⎝ ⎠ ∫∫ ∫ ∫∫ wvw A

[ ] [ ]

2 u ds n u d u ds

r r

θ

⎛⎞ ⎛⎞ τ∂ ∂τ

=τ − ⎜⎟ ⎜⎟ ⎝⎠ ⎝⎠ ∂θ ∂θ ∫∫ ∫ ∫∫ wvw A

[ ]

3 u ds u ds

r r

⎛ ⎞ ⎛⎞ σ σ θ θ = ⎜ ⎟ ⎜⎟ ⎝ ⎠ ⎝⎠ w w ∫∫ ∫∫

[ ] [ ]

4

r ds n d ds

r rr

⎛⎞ ⎛ ⎞ τ∂υ ∂τ τ ⎜⎟ ⎜ ⎟ = τυ − + υ ⎝⎠ ⎝ ⎠ ∂ ∂ ∫∫ ∫∫ ∫∫ ww w A

[ ] [ ]

5

ds n d ds

r r

θ θ

θ θ

⎛ ⎞ ⎛⎞ σ ∂υ ∂σ

⎜ ⎟ ⎜⎟ =σ υ − υ

⎝ ⎠ ⎝⎠ ∂θ ∂θ ∫∫ ∫ ∫∫ wv w A

[ ]

6

ds ds

r r

⎛ ⎞ ⎛⎞ τυ τ

− =− υ ⎜ ⎟ ⎜⎟ ⎝ ⎠ ⎝⎠ w w ∫∫ ∫∫

( ) ( ) [ ] ( ) [ ]

7 2 2

2 2 22

r r 1 1 r r ds n d n d ds

r r r r

θ θ

ε ε ∂ φ ∂φ ∂ ∂ ⎛ ⎞ ⎛⎞ ⎛⎞ ⎡ ⎤ =− + ε φ + ε φ ⎜ ⎟ ⎜⎟ ⎜⎟ ⎢ ⎥ ⎝⎠ ⎝⎠ ∂θ ∂θ ⎣ ⎦ ∂θ ∂θ ⎝ ⎠ ∫∫ ∫ ∫ ∫∫ wv v w A A

[ ] [ ]

8 r rr r n ds d ds

rr r rr

ε ∂φ ε ∂ε = φ− φ ∂ ∂ ∫∫ ∫ ∫∫ wv w A
NASA/TM—2005-213634 19 

( ) ( )

[ ] ( )[ ]

( ) ( )[ ] ( ) [ ]

2 2 9

2 2

2

2

2

r

r

r

r

nr r

ds n d d ds

r rr r rr

n r

n d d ds

r rr r r r

θ θ θ

θ

θ θ θ

θ

⎛ ⎞ ε∂φ ⎡ ⎤ ∂φ ∂ε ∂ ε ⎜ ⎟ =ε − φ + φ ⎜ ⎟ ⎢ ⎥ ∂ ∂ ⎣ ⎦ ∂ ∂ ⎝ ⎠

∂ε ∂ε ⎛ ⎞ ⎡ ⎤ ∂φ ∂ε

=ε − φ + + φ ⎜ ⎟ ⎢ ⎥ ⎣ ⎦ ∂∂ ∂ ∂ ⎝ ⎠

∫∫ ∫ ∫ ∫∫

∫ ∫ ∫∫

A A

A A

wv v w

vv w

[ ] [ ]

10

2 2 ds n d ds

r r r

θ

⎛ ⎞ γ∂φ γ ∂γ ⎛ ⎞ ⎜ ⎟ =− φ − φ ⎜ ⎟ ⎝ ⎠ ∂θ ∂θ ⎝ ⎠ ∫∫ ∫ ∫∫ wv w A

[ ]

[ ]

11 2

2

1

2

2 2

r

r

ds n n d

rr r r

n d n d ds

r r rr

θ

θ

⎛ ⎞ γ ∂ φ ∂γ ∂γ ⎛ ⎞ − = − + δφ ⎜ ⎟ ⎜ ⎟ ∂ ∂θ ∂ ∂θ ⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎡ ⎤ ⎛⎞ ⎡ ⎤ γ ∂φ γ ∂φ ∂ γ − + −φ ⎜ ⎟ ⎜⎟ ⎢⎥ ⎢⎥ ⎝ ⎠ ⎣ ⎦ ⎝⎠ ⎣ ⎦ ∂θ ∂ ∂ ∂θ

∫∫ ∫

∫ ∫ ∫∫

A

A A

w v

v v w

 

 (B4) 

All 11 terms are combined to obtain the following form of the functional: 

[ ] [ ]

[ ]

( ) ( )[ ] ( )[ ]

( )

1

2 2 2

22 2 2

2

2

2

r r s r

D

r r

D

rr r r

r r

b u b ds

r r r rr r

ds

r rr rr rr rr

n n Pu n n P d

r

n

r rr r

θ θ

θ

θ θ

θ θθ θ

θ

⎧ ⎫ ⎛ ⎞⎛ ⎞ ∂σ ∂τ ∂τ τ σ − σ ∂σ

π =− + − + + + + + υ ⎨ ⎬ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠ ∂ ∂θ ∂ ∂θ ⎩ ⎭

⎛ ⎞ ∂ ε ∂ε ∂ ε ∂ε ∂γ ∂ γ + −+ + − − φ ⎜ ⎟

∂θ ∂ ∂θ ∂ ∂ ∂ ∂θ ⎝ ⎠

+ σ +τ − + τ +σ − υ

⎧ ⎫ ⎪ ⎪ ε ∂ε ∂ ε ∂γ +−+ + ⎨ ⎬ ⎪ ⎪ ∂ ∂θ ⎩ ⎭

∫∫

∫∫

∫

A

A

w

w

v

[ ]

( )

2 2

2

2

2 2

r

r rr r

n d

r rr

n n n n d uR R d

r rr

θ

θθ θ θ

⎛ ⎞ ⎧ ⎫ ∂γ γ ⎛ ⎞ ⎜ ⎟ ⎨ ⎬ −− φ ⎜ ⎟ ⎩ ⎭ ∂θ ∂ ⎝ ⎠ ⎝ ⎠

⎪ ⎪ ⎧ ⎫ ⎛ ⎞⎡ ⎤ ⎡ ⎤ γ ∂φ γ ∂φ ⎛ ⎞ ε

+ ε + − + + +υ ⎨ ⎬ ⎜ ⎟⎢⎥ ⎢⎥ ⎜ ⎟ ⎪ ⎪ ⎩ ⎭ ⎝ ⎠⎣ ⎦ ⎣ ⎦ ∂ ∂θ ⎝ ⎠

∫

∫ ∫

A

A A

A

A A

v

v v

 (B5) 

The variation of the functional with respect to displacements δu and δ υ yields the field EEs 

0 r r r b

rr r

σ − σθ ∂σ ∂τ

+ − +=

∂ ∂θ (B6a) 

2 b 0

rr r

θ

θ

∂τ τ ∂σ

+ + +=

∂ ∂θ (B6b) 
NASA/TM—2005-213634 20 

Likewise, the field CC is obtained as the coefficient of the variation of the stress function δφ: 

2 2 2

22 2 2

2

0 r r

r rr rr rr rr

θ θ ∂ ε ∂ε ∂ ε ∂ε ∂γ ∂ γ −+ + − − =

∂θ ∂ ∂θ ∂ ∂ ∂ ∂θ (B7) 

Along the boundary segment 1 A , the variation of the displacements δu and δ υ yields the EEs or the 

traction conditions 

rr r σ n nP +τ = θ (B8a) 

r n nP θ θ θ τ +σ = (B8b) 

Along an indeterminate boundary, the BCC is obtained as the coefficient of the variation of the stress 

function δφ: 

( ) 0

2 2

r r r

r

n n

r rr r r r r

θ

θ

⎛ ⎞ ε ∂ε ∂ ε ∂γ ∂γ γ ⎛ ⎞ ⎜ ⎟ − + + −− = ⎜ ⎟ ∂ ∂θ ∂θ ∂ ⎝ ⎠ ⎝ ⎠

 (B9) 

 

 In summary, stress equilibrium is enforced in the field (eq. (B6)) and on the boundary (eq. (B8)). 

Likewise strain compliance is achieved in the field (eq. (B7)) and on the boundary (eq. (B9)). 

Displacement Continuity 

 The displacement boundary conditions u u − = 0 and υ −υ= 0 are routinely used in analysis. Their 

derivations are shown through back-calculation. This strategy is followed to avoid artificiality in a direct 

derivation process. The expression (uu P n n P n n − δ =σ +τ υ−υ δ =τ +σ = ) ( r rr r θ θ θθ ) + ( ) 0 ( ) yields 

the continuity conditions. Because u and υ are contained in terms 14 and 15 in equation (B2c), we have 

to prove the following formula along boundary segment 2 A :

( )( )

2 2

2 2

r r r rr r n n n n d u n n n nd

r rr

θ θ θ θ θθ

⎡ ⎤ ⎛ ⎞⎛ ⎞ ⎛ ⎞ γ ∂φ γ ∂φ ⎛ ⎞ ε ⎢ ⎥ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ε + − + = σ +τ +υ τ +σ ⎜ ⎟ ⎣ ⎦ ⎝ ⎠⎝ ⎠ ⎝ ⎠ ∂ ∂θ ⎝ ⎠ ∫ ∫

A A

A A v v (B10a) 

The variational form of equation (B10a) can be written as 

( )( )

2 2

2 2

r r r rr r n n n n d u n n n nd

r rr

θ θ θ θ θθ

⎡ ⎤ ⎛ ⎞⎛ ⎞ ⎛ ⎞ γ ∂φ γ ∂φ ⎛ ⎞ ε ⎢ ⎥ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ε + δ − + δ = δ σ + τ + υδ τ + σ ⎜ ⎟ ⎣ ⎦ ⎝ ⎠⎝ ⎠ ⎝ ⎠ ∂ ∂θ ⎝ ⎠ ∫ ∫

A A

A A v v

 (B10b) 

Consider the reduction of the first of the two right-hand terms in equation B10a: 

( )

2

2 2

a b c

rr r r u n nd u n n n d

rr r r r

θ θ

⎧ ⎫ ⎪ ⎛ ⎞ ⎛⎞ ∂φ ∂ φ ∂ ∂φ ⎛ ⎞ ⎡ ⎤ ⎪ σ +τ = + − ⎨ ⎜ ⎟ ⎜⎟ ⎜ ⎟ ⎢ ⎥ ⎬ ⎪ ⎝ ⎠ ⎝⎠ ∂ ∂ ∂θ ⎝ ⎠ ∂θ ⎣ ⎦ ⎪ ⎩ ⎭

∫ ∫ A A v v (B11a) 
NASA/TM—2005-213634 21 

The terms b and c that contain higher derivatives of the stress function are reduced to obtain terms in the 

first derivative of the stress function: 

( )

2

22 2

b b

r r

u u n ud n d un d n d

r r rr r r

θ θ

⎛ ⎞ ∂ φ ∂φ ∂ ∂ ∂φ ∂φ ∂ ⎡ ⎤ ⎛⎞ ⎛ ⎞ ⎛ ⎞ ⎜ ⎟ =− − = ⎢ ⎥ ⎜⎟ ⎜ ⎟ ⎜ ⎟ ∂θ ∂θ ⎣ ⎦ ⎝⎠ ⎝ ⎠ ⎝ ⎠ ∂θ ∂ ∂θ ∂θ ∂ ⎝ ⎠ ∫ ∫ ∫∫ A AAA v v vv

(B11b) 

The variation of the first right-hand term of equation (B10a) becomes 

( ) 2

 

rr r

uu u u n nd n n d

rr rr r

θ θ

⎧ ⎫ ⎪ ⎪ ⎡ ⎤ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎛ ⎞ δ∂φ ∂ ∂φ ∂ ∂φ ⎛ ⎞ δ σ +τ = − δ + δ ⎨ ⎬ ⎢ ⎥ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎪ ⎪ ⎩ ⎭ ⎣ ⎦ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠ ∂ ∂θ ∂ ∂θ ⎝ ⎠ ∂θ ∫ ∫ A A v v

 

 (B11c) 

Likewise, the second right-hand term is reduced: 

( ) 2 r r n n d n nd

r r r rr

θθ θ

⎧⎪⎡ ⎤ ⎛ ⎞ υδ∂φ ∂υ δ∂φ ∂υ δ∂φ ⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎫⎪ δ τ +σ υ = + − ⎨⎢ ⎥ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎬ ⎪⎩ ⎭ ⎣ ⎦ ⎝ ⎠ ∂θ ∂θ ∂ ∂ ∂ ⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎪ ∫ ∫ A A v v (B11d) 

Verification of the formula given by equation (B10a) is obtained by combining the two equations (B11c) 

and (B11d): 

( ( ) ( ))

( )

2

2

2 2

rr r

r r

r rr

u n n n nd

u uu n n nn d

rr r r r r r r

n n nn

r r

θ θθ

θ θ

θθ θ

δ σ + τ + υδ τ + σ

⎧ ⎫ ⎪ ⎪ ⎡ ⎤⎡ ⎤ ⎛ ⎞ ⎛⎞ ⎛⎞⎛ ⎞ ⎛ ⎞ ∂υ ∂υ ∂φ ∂ υ ∂ ∂φ = + − δ− − − δ ⎨ ⎬ ⎢ ⎥⎢ ⎥ ⎜ ⎟ ⎜⎟ ⎜⎟⎜ ⎟ ⎜ ⎟ ⎪ ⎪ ⎣ ⎦⎣ ⎦ ⎝ ⎠ ⎝⎠ ⎝⎠⎝ ⎠ ⎝ ⎠ ∂θ ∂ ∂ ∂θ ∂ ∂θ ⎩ ⎭

⎡ ⎤⎡ ⎤ ⎛⎞ ⎛ ⎞ ⎛⎞ ⎛ ⎞ γ ∂φ γ ∂φ = ε + δ − −ε δ ⎢ ⎥⎢ ⎥ ⎜⎟ ⎜ ⎟ ⎜⎟ ⎜ ⎟ ⎣ ⎦⎣ ⎦ ⎝⎠ ⎝ ⎠ ⎝⎠ ⎝ ⎠ ∂ ∂θ

∫

∫

A

A

A

A

v

v

2

d ⎧ ⎫ ⎪ ⎪ ⎨ ⎬ ⎪ ⎪ ⎩ ⎭ ∫

A

A v

 (B12) 

Verification of Boundary Compatibility Condition 

 Green’s theorem is used for a quick verification of the BCC. The BCC is inserted in the line integral 

coefficient to recover the well known field CC in the surface integral term. The integral theorem in polar 

coordinates can be written as 

( ) ( ) 1 r rr

G rG ds G n G n d

rr r

θ

θ θ

⎛ ⎞ ∂ ∂

−=+ ⎜ ⎟ ⎝ ⎠ ∂ ∂θ ∫∫ ∫ w v A (B13) 

where Gr and Gθ are the coefficients of direction cosines nr and nθ in equation (B9), respectively: 

( )

2

r r

r

G

r rr r

θ ε ∂ ε ∂γ =− +

∂ ∂θ (B14a) 
NASA/TM—2005-213634 22 

2

r

r rr Gθ

∂ε ∂γ γ − − ∂θ ∂ = (B14b) 

The surface integral terms are generated as 

( ) 2 2

2

2

2

r r rG

rr rr rr rr r

θ θ ∂ ⎛ ⎞ ∂ε ⎛ ⎞ ∂ε ∂ ε ∂ γ =−++ ⎜ ⎟ ⎜ ⎟ ∂ ∂ ∂ ∂ ∂θ ⎝ ⎠ ∂ ⎝ ⎠

 (B15a) 

2 2

22 2 2

r

r r r r

G

r

θ ∂ ε ∂ γ ∂γ − −

∂θ ∂θ ∂ ∂θ

∂ = ∂θ (B15b) 

( ) 2 2 2

22 2 2

r 2 r r rG G ds ds

rr r rr rr rr r rr

θ θθ ⎧ ⎫ ⎪ ⎪ ∂ ∂ ∂ ε ∂ε ⎛ ⎞ ∂ ε ∂ε ∂γ ∂ γ ⎨ ⎬ − =− − + + − − ⎜ ⎟ ⎪ ⎪ ⎩ ⎭ ∂ ∂θ ∂ ∂ ∂ ∂θ ∂θ ∂ ∂θ ⎝ ⎠

w w ∫∫ ∫∫ (B16)

The coefficient within the bracket is the field CC. The compatibility concept applies to the field as well as 

to the boundary. The nature of the compatibility expression changes in compliance with the domain and 

the boundary. The same interpretation is true for Cauchy’s field EEs. 
NASA/TM—2005-213634 23 

Appendix C 

 Solution to an Eight-Bar Truss 

 The solution to the eight-bar truss shown in figure 2(a) is obtained using the integrated force method 

(IFM) (ref. 13), which is the discrete analogue of the completed Beltrami-Michell formulation (CBMF). 

The IFM, like the CBMF, generates the force solution by coupling the equilibrium equations (EEs) to the 

compatibility conditions. Displacements are back-calculated from the force solution. The truss is made of 

steel with Young’s modulus E = 30 000 ksi. Each of the eight bars has an area of 1 in2

. Nodes 1 and 5 are 

fully restrained. It is subjected to a gravity load of magnitude P = –10 kip at the midspan location. The 

problem is to calculate the force and displacement response. 

 The six EEs of the structure can be written in terms of bar forces F as 

1

2

3

4

5

6

7

8

1 00 0 1 0 0 0

2

1 10 0 0 0 0 0

2 0

1 0

01 0 0 0 1 0

2 0

1 10 00 0 0 1 0 0

2 0

1 1 0 00 0 1 0 0

2 2

1 1 00 0 0 1 0

2 2

F

F

F

F

F

F

F

F

⎡ ⎤ − − ⎢ ⎥

⎧ ⎫

⎪ ⎪ ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ − ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪ ⎨ ⎬⎨ ⎬ = − − − ⎪ ⎪⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪⎪ ⎪ − ⎪ ⎪ ⎩ ⎭ ⎪ ⎪ ⎢ ⎥ ⎪ ⎪ ⎩ ⎭

⎣ ⎦

 (C1) 

The two compatibility conditions (CCs) in bar deformations β can be written as 

1

2

3

4

5

6

7

8

11 2 2 1100 0

01 0 0 0010 0

⎧ ⎫ β

⎪ ⎪ β⎪ ⎪

⎪ ⎪ β

⎪ ⎪ ⎡ ⎤ − − ⎪ ⎪ β ⎧ ⎫ ⎢ ⎥ ⎨ ⎬ ⎨⎬ =

⎣ ⎦⎪ ⎪ β ⎩ ⎭

⎪ ⎪ β

⎪ ⎪

⎪ ⎪ β

⎪ ⎪ β⎩ ⎭

 (C2) 

The CC is rewritten in member forces using the flexibility relation FL

AE

⎛ ⎞ β = ⎜ ⎟ ⎝ ⎠ for bar length L, area A, 

modulus E, and member force F: 
NASA/TM—2005-213634 24 

1

2

3

4

5

6

7

8

11 2 21100 0

01 0 0 0010 0

F

F

F

F

F

F

F

F

⎧ ⎫

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪ ⎡ ⎤ ⎧⎫ − − ⎪ ⎪ ⎨ ⎬ ⎨⎬ = ⎢ ⎥ ⎣ ⎦ ⎩⎭ ⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪ ⎩ ⎭

 (C3) 

Simultaneous solution of the six EEs and the two CCs yields the eight member forces {F}. The six 

displacements {X} are back-calculated: 

1

2 1

3 2

4 3

5 4

6 5

7 6 in.

8 kip

2.6

1.3 0.021

3.3 0.009

3.7 0.004

and

2.6 0.059

7.4 0.012

1.3 0.034

7.1

F

F X

F X

F X

F X

F X

F X

F

⎧ ⎫ ⎧ ⎫ − ⎪ ⎪ ⎪ ⎪ − ⎧ ⎫ ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ − − ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪ ⎪− ⎪ ⎨ ⎬⎨ ⎬ ⎨ ⎬⎨ ⎬ = = − − ⎪ ⎪⎪ ⎪ ⎪ ⎪⎪ ⎪

⎪ ⎪⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎩ ⎭ ⎪− ⎪

⎪ ⎪⎪ ⎪ ⎩ ⎭ ⎩ ⎭ −

 (C4) 
NASA/TM—2005-213634 25 

References 

 1. Love, A.E.H.: A Treatise on the Mathematical Theory of Elasticity. Cambridge University Press, 

Cambridge University Press, England, 1927. 

 2. Ritz, W.: Über eine neue Methode zur Lösung gewisser Variationsprobleme der mathematischen 

Physik. J. Reine Angew Math, vol. 135, 1909, pp. 1−61. 

 3. Cross, H.: Analysis of Continuous Frames by Distributing Fixed-End Moments. Am. Soc. Civ. Eng. 

Trans., paper 1793, 1932. 

 4. Thadani, Bhagwan Nebhraj: Modern Methods in Structural Mechanics. Asia Pub. House, New York, 

NY, 1964. 

 5. Gallagher, Richard H.: Finite Element Analysis: Fundamentals. Prentice-Hall, Englewood Cliffs, NJ, 

1974. 

 6. Sokolnikoff, Ivan Stephen: Mathematical Theory of Elasticity. McGraw-Hill, New York, NY, 1956. 

 7. Patnaik, S.N.: Variational Energy Formulation for the Integrated Force Method. AIAA J., vol. 24, 

no. 1, 1986, pp. 129−137. 

 8. Patnaik, Surya N.; and Hopkins, Dale A.: Strength of Materials: A New Unified Theory for the 21st 

Century. Elsevier, Boston, MA, 2004. 

 9. Patnaik, S.N.; and Nagaraj, M.S.: Analysis of Continuum by the Integrated Force Method. Comput. 

Struct., vol. 26, no. 6, 1987, pp. 899−905. 

10. Patnaik, S.N.; and Satish, H.G.: Analysis of Continuum Using Boundary Compatibility Conditions of 

Integrated Force Method. Comput. Struct., vol. 34, no. 2, 1990, pp. 287−295. 

11. Patnaik, Surya N., et al.: Completed Beltrami⎯Michell Formulation for Analyzing Mixed Boundary 

Value Problems in Elasticity. AIAA J., vol. 34, no. 1, 1996, pp. 143−148. 

12. Washizu, K.: Variational Methods in Elasticity and Plasticity. Pergamon Press, Oxford, England, 

1968. 

13. Timoshenko, S.P.; and Goodier, J.N.: Theory of Elasticity. McGraw-Hill, New York, NY, 1969. 

14. Saada, Adel S.: Elasticity Theory and Applications. R.E. Krieger Publishing Co., Malabar, FL, 1983. 
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NASA TM—2005-213634

E–15117

WBS–22–708–24–05

31

Completed Beltrami-Michell Formulation in Polar Coordinates

Surya N. Patnaik and Dale A. Hopkins

Boundary; Compatibility; Derivation; Elasticity; Variational

Unclassified -Unlimited

Subject Category: 39

A set of conditions had not been formulated on the boundary of an elastic continuum since the time of Saint-Venant.

This limitation prevented the formulation of a direct stress calculation method in elasticity for a continuum with a

displacement boundary condition. The missed condition, referred to as the boundary compatibility condition, is now

formulated in polar coordinates. The augmentation of the new condition completes the Beltrami-Michell formulation

in polar coordinates. The completed formulation that includes equilibrium equations and a compatibility condition in

the field as well as the traction and boundary compatibility condition is derived from the stationary condition of the

variational functional of the integrated force method. The new method is illustrated by solving an example of a mixed

boundary value problem for mechanical as well as thermal loads.

Surya N. Patnaik, Ohio Aerospace Institute, 22800 Cedar Point Road, Brook Park, Ohio 44142; and Dale A. Hopkins,

NASA Glenn Research Center. Responsible person, Surya N. Patnaik, organization code RSS, 216–433–5916.