Basic Equations: Differential Form - Mechanics of Structures: Variational and Computational Methods - page 28

functions, e.g., Papkovich 17 Neuber 18 functions, (similar to stress functions) which auto-
matically satisfy the compatibility conditions [Neuber, 1985]. Then Eq. (1.67), the conditions
of equilibrium, can be written in a more tractable form.

1.6.2 Force Formulation

Another alternative formulation of the governing equations entails the development of
force, flexibility, or compatibility formulations. These equations are found by writing the
compatibility equations in terms of stresses. If the constitutive relationship of Eq. (1.32) is
substituted in the compatibility relation of Eq. (1.29), then

D 1 E − 1 σ =

D 1 = 0 (1.68)

These six equations involve the six (σ

x ,

σ

y ,

σ

z ,

τ

xy ,

τ

xz ,

τ ) unknown stresses.

yz

Frequently, the equilibrium condition s a re also included in a force (stress) formulation.
Differentiate Eq. (1.52) to find σ ij, jk =− p Vi,k . Substitution of this into Eq. (1.68) leads to

[Sokolnikoff, 1956].

1 ν

2 2

∇ σ ij +

+ ν σ kk,i j − + ν δ ij ∇

σ kk =− ( p Vi, j + p Vj,i )

1 1

− ν

It can be verified from the compatibility relations of Eq. (1.68) that σ ij,ij =

1

+ ν ∇

2

σ kk .

1

2 1 + ν

Also, from the equilibrium conditions σ ij,ij =− p Vj, j . Then, ∇ σ kk =− p Vj, j and the

1 − ν

compatibility relations in terms of the stress components become

1 ν

2

∇ σ ij +

+ ν σ kk,i j = − ν δ ij p Vk,k − (

p Vi, j + p Vj,i )

(1.69)

1 1

where

2

∇ σ ij = σ ij,kk , δ ij = 1 f i = j and δ ij = 0 f i = j

These are Michell's 19 equations , which together with prescribed static boundary conditions
can be used to find the unknown stresses. If the volume (body) forces are constant, we
obtain Beltrami's 20 equations

1

∇

2

σ ij +

− ν σ kk,i j =

0

(1.70)

1

These can be rewritten as a biharmonic differential equation

2 2

∇ ∇ σ ij = 0 (1.71)

17 Petr Fedorovich Papkovich (1887-1946) was a Russian-Soviet shipbuilding engineer, and was a corresponding
member of the Soviet Academy of Science. His developments in the vibration and strength analysis of ship
structures are considered to be significant contributions to the foundations for the theory of shipbuilding. He
published the displacement functions for the solution of the biharmonic equation in 1932.
18 Heinz Neuber (1906-1989) was a German machanical engineer who received his doctorate under August F oppl
in Munich. After lengthy service in the aircraft industry, he became a faculty member of the Technische Hochschule
in Dresden in 1946. In 1955, he replaced Ludwig F oppl (the son of August) as a chaired professor at the Technische
Hochschule in Munich.
19 John Henry Michell (1863-1943) was a Cambridge-educated Australian mathematician. He was a professor at
the University of Melbourne. His papers were collected into one volume in 1964, along with those of his mechanical
engineer brother Anthony-George-Maldon Michell (1870-1959).
20 Eugenio Beltrami (1835-1900) was a versatile Italian mathematician who served as a professor of mechanics at
several Italian universities.