Lecture 4
Lecture annotation
Variational principles of linear elasticity. Fundamental relations, extreme of functionals,
classical variational principles (Lagrange, Castiligliano, Reisssner, Hu-Washizu), modified
variational principles. Hybrid models of FEM.
Principle of virtual work, virtual displacements, and virtual stresses
Consider a mixed boundary value problem of the theory of deformable bodies, where
there are prescribed displacements g , i.e. u  g on the part of the boundary u , and on the
remaining part,  , there are prescribed surface tractions p , i.e.  ij n j  pi . All displacement
functions and stress functions are assumed to be sufficiently smooth so that Gauss’ integral
theorem holds and differential relations and boundary conditions have sense.
Definition 1 A tensorial function σ (x) is called the statically admissible stress field if
 ij   ji and the components  ij satisfy the equilibrium equations in a domain 
 ij
x j
 fi  0
(*)
(fi denotes the components of the vector of body forces) and on the part of the boundary 
they satisfy the force boundary conditions
 ij n j  pi .
(**)
A vectorial function u(x) is called the geometrically admissible field of displacement,
if on the part of the boundary u it satisfies the boundary condition (back to Lecture 9-10)
(***)
ui  gi .
Principle of virtual work. Let σ(x) be a statically admissible stress field and u(x)
geometrically admissible displacement field. Denote
1  u u 
 ij (u)   i  j  .
(****)
2  x j xi 
Then it holds
(1)
 ij ij (u)dV   fiui dV   piui dS   ij n j gi dS .



u
P r o o f immediately follows from Gauss’ integral theorem and the use of both the
equilibrium equations and boundary conditions.
Remark 1. The principle of virtual work is often formulated as follows: The virtual work of internal forces (lefthand side of the equation) is equal to the virtual work of external forces (the virtual work of body forces, surface
tractions and the work exerted by “yielding of support”).
Let us point out that the principle of virtual work holds for mutually independent stress
fields and displacement fields, i.e. for an arbitrary type of generalized Hooke’s law (both
linear and nonlinear).
Principle of virtual displacements. We know, according to the definition of classical
solution u 0 , that u 0 is a geometrically admissible displacement field. Let the real stress field
σ 0 is statically admissible. Substitute into the principle of virtual work both the real stress
field σ 0 and the real displacement field u 0 , and then also an arbitrary, but geometrically
1
admissible displacement field u0  u . Subtracting so obtained equalities we obtain the
principle of virtual displacement:
(2)
  0ij ij dV   fi ui dV   pi ui dS ,



where
ij   ij (u) ,
whereas u is so-called virtual displacement (also „variation“ of u ).
From the definition 1 it follows, that ui  0 on the part of the boundary u , but
otherwise it is an arbitrary (sufficiently smooth) vectorial field. Let us point out again that the
equation (2) holds for an arbitrary type of Hooke’s law. The principle of virtual
displacements serves as a basis of so-called v a r i a t i o n a l f o r m u l a t i o n o f t h e
s o l u t i o n o f b a s i c b o u n d a r y v a l u e p r o b l e m s o f e l a s t i c i t y (weak or
generalized solution).
Note that the principle of virtual displacements is often called the principle of virtual work!
Remark 2. If the integral on the left-hand side of Eq. (2) is integrated by parts, we get
  0ij

 fi   ui dV    pi   0ij n j   ui dS  0 ,


  x

j



(3)
wherefrom the equilibrium equations and the boundary condition (natural boundary condition) on the part of the
boundary  follow. The principle of virtual displacement (2) is thus equivalent in a certain sense to the
assertion that σ 0 is a statically admissible stress field. Observe that the equation (3) is a global (integral) form of
the conditions of equilibrium and the force boundary conditions. According to the fundamental lemma of the
calculus of variations the integral relation (3) is equivalent to the local conditions (*) and (**).
In engineering practice the principle of virtual displacements (2) represents the basis of so-called
displacement
method
for
the
solution
of
statically
indeterminate
s t r u c t u r e s .
Principle of virtual stresses. Substitute in the principle of virtual work both the real
displacement field u 0 and the real stress field σ 0 , and then also an arbitrary, but statically
admissible stress field σ 0  σ . Subtracting so obtained equalities we obtain the principle of
virtual stresses:
(4)
  ij  u0   ij dV   gi ij n j dS ,

u
where ij are so-called virtual stress components („variation“ of σ ). It follows from the
definition 1, that ij   ji , ij x j  0 , and ij n j  0 on the part of the boundary  .
Eq. (4) also holds for an arbitrary type of generalized Hooke’s law.
In engineering practice the principle of virtual stresses (2) represents the basis of socalled f o r c e m e t h o d f o r t h e s o l u t i o n o f s t a t i c a l l y i n d e t e r m i n a t e
structures.
Note that the principle of virtual stresses is often called the principle of complementary
virtual work!
Principle of the potential energy minimum in the theory of elasticity (Lagrange
principle)
Consider again a mixed boundary value problem of the theory of deformable bodies,
but now assume a specific form of generalized Hooke’s law, namely that one concerning
anisotropic bodies, i.e.
 ij  Cijkl  kl ,
(5)
2
where Cijkl are bounded functions of the variable x   . In practice, they are most often
constant or piecewise constant. It is well-known the following symmetry relations hold:
Cijkl  Cklij  C jikl  Cijlk .
Presume that the related quadratic form is uniformly positive definite in a given domain, i.e.
there exists such a constant C0  0 , so that
2W (ε)  Cijkl (x) ij  kl  Co ij  ij
(6)
holds for all symmetric tensors  ij and for almost all x   .
In the case of an isotropic body we have only two independent elastic constants, e.g.
Lame’s coefficients  , and G . (   E 1  1  2   , where E is the Young modulus and
 is Poisson’s ratio.). If   0 , G(x)  G0  0 almost everywhere in  , we get
2W (ε)   2  2G ij ij  2G0 ij ij ,
(7)
(where =11+22+33) hence the presumption (6) is satisfied. (The extension of the
presumption (6) to anisotropic materials is consistent with energetic considerations
concerning the stable equilibrium.) (back to Lecture 9-10)
Assume that the real stress field σ 0 is statically admissible. If we substitute into the
principle of virtual displacements the relation (5), we arrive at
(8)
 Cijkl kl (u0 ) ij dV   fi ui dV   pi ui dS  0 .



Since the variation of W((u0)) – so-called strain energy density – with respect to u is
d
 W  ε  u0    W ε  u0  t u   Cijkl  kl  u0   ij  u  ,
dt
t 0
the condition (8) can be written in the form of variation:

 W ε(u ) dV  

0


fi u0i dV   pi u0i dS  0 .

(9)
The expression in compound parentheses is denoted by   u0  and is called the potential
energy of elastic body. The solution of the problem u 0 is a critical point of the functional
 (u) and renders this functional stationary.
Compute for geometrically admissible displacement field, u0  u , the difference
  u0   u     u0  
  W  ε  u 0   u    W  ε  u 0    dV   fi ui dV   pi ui dS 



(10)
1
C


d
V

f

u
d
V

p

u
d
S

ijkl
ij
kl
i
i
i
i



2 
1
1
  Cijkl ij kl dV  c   ij ij dV  0


2
2
The inequality follows from (6) and (8). Hence,  (u) attains for u 0 a minimum. We arrive to
the result:
  Cijkl  kl (u 0 ) ij dV 
Principle of minimum potential energy. The solution u 0 of a mixed boundary
problem of elasticity renders the functional of potential energy
1
 (u)   Cijkl  ij (u) kl (u)dV   fi ui dV   pi ui dS
(11)


2 
a minimum value on the set of geometrically admissible displacement fields.
3
Remark 3. Under the assumption that generalized Hooke’s law (5) holds, the principle of virtual
displacements expressing that   u0   0 (see (9)) is a consequence of the principle of minimum potential
energy.
Principle of minimum complementary energy
The principle of minimum potential energy will be transformed to the variational
problem of maximum using Friedrichs’ method:
Consider the problem u  min , and assume that the strain-displacements relations
1  u
u 
 ij   i  j  ,
2  x j xi 
and the boundary conditions ui  gi on the part of the boundary u are subsidiary conditions,
i.e. solve the problem  (e, u)  min with subsidiary conditions (so called constrained
extreme).
Using the method of Lagrange multipliers construct a new functional
H (ε, u, λ , μ) 
 1
 1  u u j 
 
   Cijkl  ij kl  ij   i 
  ij   dV 



 2

 2  x j xi 
 
(12)
  fi ui dV   pi ui dS   i ( gi  ui )dS ,


u
where ij (x)   ji (x) and i (x) are new unknown functions (Lagrange multipliers) and seek a
stationary value of the functional H with respect to all independent arguments without any
subsidiary conditions.
Carry out the variation of H and integrate by part. Obtain

 

 H    Cijkl  kl  ij   ij   ik  fi   ui 

 xk



 1  u u 

   i  j    ij  ij  dV 



 2  x j xi 


  n

 p   n   u dS  0 .
u

ij
i
j
(13)

 i   ui   gi  ui  i dS 
ij
j
i
Among the conditions (Euler’s equations) (13) select only those conditions that are
complementary to the original subsidiary conditions, i.e. take into consideration
ij  Cijkl  kl in 
(14)
i   ij n j on u ,
ik
 fi  0 in  ,
xk
 ij n j  pi on   .
(15)
(16)
(17)
The relations (14) - (17) indicate that the functions ij have the physical meaning of
stress tensor components, and  i have the physical meaning of surface tractions. Eq. (16)
then stands for the conditions of equilibrium in the body and Eq. (17) is the force boundary
condition.
Generalized Hooke’s law is invertible, i.e. we can write
4
 ij  Dijkl kl ,
(18)
where Dijkl (x) are the compliance coefficients having the same symmetry properties as the
moduli Cijkl have. The related quadratic form, with  ij as variables, is also uniformly positive
definite in  , because the moduli Cijkl (x) are bounded in  .
Integrating by parts we obtain
 ui u j 
ij
dV  

ui dV  ij n j ui dS .
(19)
 x j xi 

 x

j


If we substitute into the functional H (ε, u, λ , μ) firstly the relation (17), then the conditions
(15)- (17), and finally we write  ij instead of ij , we get the functional
1
2

ij 


1
Dijkl ij kl dV   gi ij n j dS .
2
u
Lemma 1. If there exists (u0 )  min(u) on the set of geometrically admissible
S1 (σ)   
displacements, then there exists also max S1 (σ)  S1 (σ0 ) for a certain tensor σ 0 on the set of
statically admissible stresses and it holds S1 (σ 0 )  (u0 ) .
P r o o f can be based e.g. on tracing the heuristic construction of the functional S1 (σ) .
The functional S (σ)  S1 (σ) is called the complementary energy, and it holds:
Principle of minimum complementary energy (Castigliano–Menabre principle). If
the real stress field σ 0 of a mixed boundary value problem of elasticity is statically
admissible, then it renders the functional of complementary energy
1
S (σ )   Dijkl ij kl dV   gi ij n j dS
(20)
u
2 
minimum on the set of all statically admissible stress fields.
Remark 4. In the case gi  0 , or u   the functional S (σ) is reduced to the first integral in (20) (strain
energy). The principle is then called the principle of minimum strain energy.
1. If generalized Hooke’s law (5) is invertible, then the principle of virtual stresses follows as a
consequence of the principle of minim complementary energy. The principle of virtual stresses expresses that the
variation  S (σ 0 )  0 .
2. Remark 6 explains the meaning of Euler conditions (Euler’s equations) for the problem  S (σ )  0 ,
namely the represent the compatibility of the minimalizing tensor σ 0 .
3. The principle of complementary energy can be derived from the principle of virtual stresses using
inverse Hooke’s law.
Hybrid principles in the theory of elasticity, Hellinger-Reissner principle
In 1914 E. Hellinger proposed mixed (hybrid) variational formulation of elastic
boundary value problems, which involves two “dual” fields such as displacements and forces
simultaneously as unkowns. E. Reissner (1950, 1961) extended this formulation by
incorporating the boundary conditions. K. Washizu and Hu Hai Chang (1955) generalized the
classical variational principles (minimum potential energy and complementary energy,
respectively) by employing variations of the strain fields in addition to the variations of the
displacements and forces.
Hybrid principles can be derived using the method of Lagrange multipliers, already
demonstrated at Friedrichs’ transformation (12). Hence, set up the functional H (ε, u, λ, μ )
5
from Eq. (12), and employ variation and integration by parts. Exclude  i using the condition
(15) and write  ik instead of ik on the basis of (14). Thus, derive the functional
I (ε, u, σ) 
1

1  u u 
   Cijkl  ij  kl   ij  ij   ij  i  j   fi ui  dV 
 2
2  x j xi 


(21)
  pi ui dS    ij n j ( gi  ui )dS ,

u
which was proposed by Hu Hai Chang and Washizu. From the condition  J (ε,u,σ )  0 there
follow all fundamental relations of a given problem, i.e. the equilibrium equations, the straindisplacements relations, generalised Hooke’s law and boundary conditions.
If we exclude the strain tensor from the functional J using generalized Hooke’s law
(18), we get the functional
R (u, σ ) 
 1

1  u u 
    Dijkl ij kl   ij  i  j   fi ui  dV 

2  x j xi 
 2

(22)
  pi ui dS    ij n j ( gi  ui )dS ,

u
which was proposed by E. Reissner (it is often referred to as the Hellinger-Reissner functional
and the Hellinger-Reissner principle). It is ease to show that from the condition  R (u,σ )  0
there follow the equilibrium equations, further the relations
1  ui u j 

(23)

  Dijkl kl
2  x j xi 
and boundary conditions. (back to Lecture 8)
Remark 5. The Hellinger-Reissner principle can also be derived using the method of Lagrange multipliers from
the principle of minimum complementary energy, if we append the equilibrium equations and the boundary
conditions on the part of the boundary  as subsidiary conditions to the functional S (σ ) . Thus we come to the
functional
R 1 (σ, λ,μ) 
1
  ij

   Dijkl ij kl  i 
 fi   dV 


 2
 x j


(24)
   ij n j gi dS   ( pi   ij n j ) i dS ,
u

where all variables σ , λ , μ are mutually independent. Employing variation and integration by part we get
R1 
 
 
 
1    j  
    Dijkl kl   i 
 ij   ij  fi  i  dV 


 x


2  x j xi  
 
 j
 
(25)
  (i  gi ) ij n j dS 
u


(   )
i
i
ij
n j  ( pi   ij n j )i  dS .
From the condition  R 1  0 there follow, as Euler’s equations, the equilibrium equations, the force boundary
conditions on  , and
i  gi on u ,
i  i on  ,
1    
Dijkl kl   i  j  in  .
2  x j xi 
6
(26)
(27)
From (26) and (37) it is apparent that the multipliers i ,  i have the meaning of displacements. If we substitute
into R 1 for i  ui , and i  ui , change the sign and integrate by parts, we obtain the Hellinger-Reissner
functional R (u,σ ) .
Remark 6. In the Remark 5 we have simultaneously shown that the necessary (Euler) conditions following from
the principle of minimum complementary energy are the equations (26) and (27). These conditions are
equivalent to the assertion that there exists a vectorial displacement field, from which the real stress field can be
derived using generalized Hooke’s law. The conditions (26) and (27) can be understood as conditions of
compatibility of the stress tensor which minimalizes the functional S (σ ) .
Remark 7. In essence, the Hellinger-Reissner principle represents a canonical form of the principle of minimum
potential energy and the related Euler conditions are so-called Hamilton’s canonical equations (in our case the
equilibrium equations and the relations (23).
The critical point (u0 , σ 0 ) of the functional R (u,σ ) is not an extreme point, but a
saddle point, as the following lemma shows:
Lemma 2. Let U be the set of all geometrically admissible displacement fields u , Q
be the set of all symmetric stress tensors, u 0 be the solution of a given mixed boundary value
problem and σ 0 is the related stress tensor. Then
(28)
R (u0 , σ0 )  min max R (u,σ)  (u0 )  max min R (u,σ) ,
uU
σQ
σQ
uU
where  is the functional of potential energy.
P r o o f . Let the displacement u  uˆ U is chosen fixed. Then R (uˆ , σ ) attains its
maximum on the set Q just when it holds
Dijklˆ kl   ij (uˆ ) .
(29)
Indeed, the last integral in (20) falls off owing to uˆ U and
 R (uˆ , σ)    Dijkl kl   ij (uˆ )   ij dV  0 .

ˆ ) attains a stationary value, just when σ satisfies (20). Because
Hence, the functional R (u,σ
the inequality
1
ˆ ˆ   σ )  R (u,σ
ˆ ˆ )    Dijkl ij kl dV  0 ,
R (u,σ
 2
holds, it follows that there is a maximum at the point σ̂ .
ˆ ) , we arrive to the equation
If we substitute (29) into R (u,σ
1

R (uˆ , σˆ )    Cijkl  ij (uˆ ) kl (uˆ )  fi uˆi  dV   pi uˆi dS   (uˆ ) .
 2



According to the principle of minimum potential energy it holds
ˆ ˆ (uˆ ))  min  (uˆ )   (u 0 ) .
min R (u,σ
U
U
Consider that σ  σˆ  Q is a fixed stress field. Then the functional R (u, σˆ ) is linear in
the variable u . Let u  u0  v , where u0 U , v is an arbitrary, geometrically admissible
field satisfying the homogenous boundary condition, i.e. v  0 on the part of the boundary u
and   (, ) is a real parameter. Then u U for all coefficients  and it holds
R (u0  v,σˆ )  A  B ,
where A and B do not depend on  , and
B   ˆ ij ij ( v)  f i vi  dV   pi vi dS .


Hence, if we search
max min R (u,σ) ,
σQ
uU
7
we can confine ourselves to statically admissible stress fields σˆ  , for which the expression
B is equal to zero for all fields v (see Remark 2). Indeed, for σ  Q \  there exists such a
field v , so that B  B( v )  0 , and thus
min R (u,σ)   .
uU
Therefore, it is
max min R (u,σ)  max min R (u,σˆ )  max(S (σˆ ))  (u0 ) ,
σQ
uU
σˆ 
uU
σˆ 
because for σˆ  both lemma 1 and the principle of virtual work (1) hold; hence
R (u,σˆ )  S (σˆ ) .
In conclusion we present the overview of f o r m a i n v a r i a t i o n a l p r i n c i p l e s
in the Table 1. By „a“ we denote the relations which are required to hold in a specific
variational formulation „a priori“, and by „E“ we denote the relations which follow from a
specific principle as necessary Euler conditions.
For the Hellinger-Reissner principle, and for the principle of minimum of
complementary energy, there are indicated two alternatives in the first two columns. Namely,
when interpreting the relation (23), we can assume a priori, that Hooke’s law holds. Then the
right-hand side is  ij and we obtain the strain-displacements relations as the Euler conditions.
If we choose the strain-displacements relations to hold a priori, then the left-hand side is  ij
and generalized Hooke’s law is considered as the Euler conditions.
H y b r i d p r i n c i p l e s are of great importance in the following situations:
1. They are useful even in the situation when e.g. the principle of minimum of
potential energy cannot be used for an approximate variational solution because of numerical
instability (e.g. for “nearly incompressible” elastic materials) .
2. They are used to the derivation of boundary conditions in the shell theory.
3. They form the basis of the so-called h y b r i d m e t h o d s o f f i n i t e
e l e m e n t s a n a l y s i s . It is sometimes difficult to find basis (shape) functions that satisfy
the required conditions. Thus is particularly true for the force method. This shortcoming can
be corrected with the use of hybrid variational principles, in which those conditions causing
difficulties are attached to the variational expressions using Lagrange multipliers.
Table 1
Principle
Hu, Washizu
Hellinger-Reissner
minimum
potential energy
minimum
complementary
energy
Relations in a domain
Boundary conditions
geometrical
force
generalized strainequilibrium conditions on conditions
Hooke’s
equations
on 
u
law
displacements
E
E
E
E
E
a
E
E
E
E
E
a
a
a
a
E
E
a
E
a
E
a
E
a
It is instructive to show the connection between the condition (8), following from the
principle of virtual displacements or from the principle of minimum potential energy,
respectively, and the theory of generalized (weak) solution exposed in the Lecture 2.
8
Consider for simplicity an isotropic material characterized by Lame’s coefficients, which are
generally functions of location. Hooke’s law can be written as follows
(30)
 ij  ij div u  2G ij  u  ,
where ij is Kronecker’s symbol. If we substitute (30) into the equations of equilibrium, we
obtain so-called Lame’s equations of equilibrium


(31)

  div u   2  G ij  u    fi ,
xi
x j
or, in vectorial form
(32)
 grad   div u   2divGε  u   f
(back to Lecture 5-6) (back to Lecture 9-10). Write for short the vectorial equation (32) in the
form
(33)
Au  f ,
which is actually a vectorial analogue of the operator equation (1) in Lecture 2, and the
operator of boundary value problem of elasticity A follows by comparison of (33) and (31).
Consider a mixed boundary value problem of the theory of elasticity, where there are
prescribed displacements g , i.e. u  g on the part of the boundary u , and on the remaining
part  there are prescribed surface tractions p , i.e.  ij n j  pi . The boundary condition
prescribed on u is the essential boundary condition in the sense of definition in Lecture 2.
The energetic space HA of the operator A is formed by a set of functions with metric defined
by the scalar product (u,v)A. Further, we define the linear set

V=   C 1    supp    

(34)
(supp is the compact support of ) The set V is completed by adding limit functions
3
(elements) in the metric of the energetic space HA and denoted byV . The space V  V  is
called the space of testing functions v, which are 1-times continuously differentiable in 
and equal to zero on u. As a matter of fact, the testing functions v correspond to the virtual
displacements u, if their norm v  1, see the Remark 2 of Lecture 1.
Definition 2 The function u0  HA is called the generalized (weak) solution of a mixed
boundary value problem of elasticity, if it holds
u0  g V ,
(35)


  div u 0  div v   2G ij  u 0   ij  v   dV   fi vi dV   pi vi dS .


(36)
The definition of the weak solution can be substantiated in this way: The condition (35)
means that u0 = g on u in the sense of traces. Eq. (36) is, in fact, Eq. (8) written down for an
isotropic material using the testing functions. Conceive that the weak solution u0 is
sufficiently smooth. Then using Gauss’s theorem we obtain from (36)
 


 vi  xi   div u0   2 x j  G ij  u0    fi  d    vi ni  div u0  2Gn j ij  u0   pi  d S  0 .(37)



The expressions in brackets of the first integral are Lame’s equations of equilibrium (31). The
expressions in brackets of the second integral are the force boundary conditions expressed in
terms of displacements using (30) and these expressions are equal to zero. If (37) holds for all
functions vV, we say that the differential equations of equilibrium are satisfied in the weak
sense.
Remark 8. The difference between the classical Ritz method and FEM consists in that the testing (basis)
functions in the classical Ritz method are nonzero in a whole investigated domain which makes their selection
difficult or even impossible for domains that have a complicated shape and for complex boundary conditions,
while in the FEM the testing (basis) functions are very simple and nonzero only in a nearest surroundings of
9
individual nodes, see Fig. 1, see also Lecture 3. In mathematical language we say that the compact support of
basis function in FEM is a subdomain formed by neighbouring elements. If we wish to improve accuracy of the
classical Ritz method, we have to add another linearly independent basis functions. In FEM we proceed in a
similar way. We add new basis functions so that we divide the investigated domain into larger number of
elements. Thus, in the same domain there appear more “pyramids”, which corresponds to the increased number
of (global) basis functions. The accuracy of FEM depends on:
a) the method of division of investigated domain (number of elements and the approximation of boundary)
b) approximation of solution on individual elements.
Fig. 1
Convergence to the correct solution is critical to the proper use of a finite element analysis. To achieve
monotonic convergence, the element must be
a) compatible (or conforming)
b) complete.
Compatibility assures that no gaps occur within the elements and between the elements when the system of
elements is assembled and loaded. To satisfy the condition of compatibility, the approximate functions
(displacements, temperature etc) should be chosen such that (1) they are continuous within the element, and (2)
at the element interfaces at least the first r derivatives are continuous, where r +1 is the highest derivative
appearing in the functional of the principle of virtual work. For linear elastic elements where r =0, the
compatibility condition requires that the approximate function be continuous both inside the element and on the
inter-element boundaries– so-called C0 continuity. For bending elements of the sort needed for beams and plates
where r = 1, compatibility means that the slope of the approximate function must be continuous inside the
element and on its boundaries– so-called C1 continuity. The requirement of Cr continuity will ensure that no
contribution is made from the element interface to the total functional of variational principle. This condition is
satisfied by a complete polynomial of degree r +1. Elements that do not satisfy the compatibility requirement are
called incompatible or nonconforming. If the incompatibility disappears with increasing mesh refinement, the
elements can still be acceptable as they may lead to convergence to the correct solution.
The completeness will be defined with regard to the solution of elasticity problems. Approximate functions
satisfy the condition of completeness, if
1. they are able to represent displacements that the element undergoes as a rigid body without developing
stress. For example, consider a cantilevered beam with a concentrated force acting at the midpoint.
Since stresses will not be generated beyond the location of load application, the approximate functions
for the elements at the free end must be able to permit the elements to translate and rotate stress free.
2. The displacement functions of an element must be such that the strain in each element approaches a
constant value in the limit as the element approaches a very small size. Then a complex variation of
strain within the structure can be approximated. For this constant strain representation, the displacement
function must contain those terms that can eventually result in constant strain states. If the structure is
actually in a constant strain state, the functions must be able to represent this constant strain.
The condition of completeness (1) can be easily explained in the following way: The necessary condition for the
existence of the solution of the first basic boundary value problem of elasticity (i.e. on the boundary there are
prescribed only surface tractions, hence u = ) is that the overall equilibrium condition must be satisfied:
(38)
fi d V  pi d S  0,
 x  f  dV   x  p dV  0 ,





i



i
(zero resultant force and zero resultant moment). Substitute for the testing function v in the condition (36) such
a displacement that corresponds to the displacement and rotation of a rigid body. Then ij(v) = 0 and the lefthand side of (36) is equal to zero If there exists a weak solution, then also the right-hand side of (36) must be
equal to zero. i.e.
(39)
fi vi dV  pi vi dS  0 for every v representing displacement and rotation of a rigid body .




The function v representing the displacement and rotation of a rigid body has the form v = a +bx for arbitrary
vectors a and b. If we set b=0, we obtain
10


fi ai dV   pi ai dS  0

pro a  R3
(40)
wherefrom (38)1 follows. If we set a=0, then (39) reduces to
 f  b  x  dV   p  b  x  d S  0
pro  b  R3 .
(41)

Because f.(bx) = b.(xf) and similarly p.(bx) = b.(xp), we obtain (34)2.
Modified variational principles
Assume that a body  consists of several parts of simple geometrical shape, which will be
called elements. The elements are denoted by (i) and corresponding displacements by u(i) and
stresses by ij(i). The body in Fig. 2 consists of two elements, (1) and (2). According to the
character of a variational principle, there is necessary to require on the inter-element boundary
(12) either the displacement continuity (so-called compatible model)
u(1)  u(2)  0 .
(42)
or the traction continuity (so-called equilibrium model)
Fig. 2
 ij n j   ij2nj2  0 .
1 1
(43)
If we wish to weak the requirements posed on the field u(i), and ij(i), respectively, we have to
incorporate the subsidiary conditions (42) and/or (43) into a variational principle. In such a
way, modified variational principles are set up.
Modified Hellinger-Reissner principle will be presented in two versions:
1. In the first case, rewrite the condition (42) to the form
u (1)  μ  0, u (1)  μ  0 on 12 ,
(44)
The new variable – interfacial displacement between elements  - can be characterized as a
“prescribed“ displacement on 12. If we now extend the functional (22) also on the interelement boundary 12, we obtain a modified functional
1
1
1
2
2
2
R m (u, σ, μ)  R (u, σ )    ij  nj  ( i  ui  )dS    ij  nj  ( i  ui  )dS 
12
12
(45)
 R (u, σ )   ( ij 2 nj2ui 2   ij1 nj1ui1 )dS   i ( ij1 nj1   ij 2  nj2  )dS ,
12
12
The functional R (u,σ ) can be convert using Gauss’-theorem, in which we put  = 1 +2, 
= 1 +2 + 12. As a result of this operation we obtain
1
1
2
2
R m (u, σ, μ)  R  (u, σ)   i ( ij  nj    ij  nj  )dS
(46)
12
2. In the second (dual) case, rewrite (43) to the form
 ij1 nj1  i  0,  ij 2 nj2  i  0 .
(47)
The new variable represents interfacial forces related to the boundary surface of the element
1. As a result of operation analogical to that one leading to (45) we obtain the following
modified functional
11
R m (u, σ, λ )  R m (u, σ)   ui  ( ij  nj   i )dS   ui  ( ij  nj   i )dS 
1
1
1
2
12
2
2
12
 R m (u, σ )   (ui1 ij1 nj1  ui 2 ij 2 nj2 )dS   i (ui1  ui 2 )dS ,
12
(48)
12
which can be converted using Gauss’ theorem to the form
1
2
R m (u, σ, λ )  R m (u, σ)   i (ui   ui  )dS .
12
(49)
Using the relations (42)-(44) and (47) we can convince ourselves that the functionals (46) and
(48), or (45) and (49) are identical. In addition to variations of fields u(i) and jk(i), i=1,2, is the
variation also employed to the multipliers  and .
The modified Hellinger-Reissner principle forms the basis of so-called hybrid models of
structural analysis.
Modified Lagrange principle will be derived from Eq. (49). If we require that only the straindisplacements relations (****) are satisfied a priori, but not the geometrical boundary
conditions (***), we come to the expression
1
2
m     i (ui   ui  )dS    ij n j (ui  gi )dS
(50)
12
u
(i)
In addition to variations of displacements u , i=1,2, is the variation also employed to the
multipliers i  ij1 nj1  ij 2 nj2 on 12 and ij on u. The modified Lagrange principle forms
the basis so-called compatible hybrid model of structural analysis.
Modified Castigliano principle will be derived from Eq. (46). If we require that only the
equilibrium equations (*) are satisfied a priori, but not the force boundary conditions, we
come to the expression (beware of the change of sign, see Lemma 1)
S m  S -  i ( ij1 nj1   ij 2 nj2 )dS   ui  ij n j  pi  d S
(51)
12

In addition to variations of stresses ij , i=1,2 is the variation also employed to the
multipliers  = u(1)=u(2) on 12 and u on . The modified Castigliano principle forms the
basis so-called equilibrium hybrid models of structural analysis.
The compatible model is based on the approximation of the displacement field and, therefore
it is, by nature, connected with the displacement method of solution. It leads to the stiffness
matrix. The equilibrium model is based on the approximation of the stress field, and, therefore
it is, by nature, connected with the force method of solution. It leads to the compliance matrix.
(i)
Hybrid models of FEM
It is advantageous to formulate FEM models in compact matrix notation. Therefore, we will
first present matrix notation of displacement field, strain and stress fields and also the notation
of some matrix operators.
1. displacement field u = {u,v,w}T
2. strain field presented in the form of 6 dimensional vector {} = {xx, yy, zz, 2yz, 2xz,
2xy}T
3. stress field presented in the form of 6 dimensional vector {} = {xx, yy, zz, yz, xz,
xy}T (Compound parentheses distinguish the 6 dimensional vector from the
corresponding tensor of 2. order. The sequence of the components of stress and/or
strain tensor in the 6 dimensional vector is determined according to the following rule:
First we assign numbers to the indices in usual manner, i.e. x 1, y2, z3. Then
the couple of indices 11 correspond to the 1. component of the 6 dimensional vector,
222, 333. In the case of different indices, the order of precedence is determined
form the missing number increased by 3, i.e.. 234, 135, 126)
4. body forces field f = {fx, fy, fz}T
12
5. Matrix differential operator
 x 0 0 0 z y 


   0 y 0 z 0 x 
(52)





 0 0 z y x 0 
5. Matrix of direction cosines of outer normal to the 
 nx 0 0 0 nz n y 


(53)
n   0 ny 0 nz 0 nx 
 0 0 nz n y nx 0 


3 equations of equilibrium and 6 strain-displacements relations are then written in the form
(54)
 σ  f  0, ε  T u ,
and Hooke’s law
(55)
σ  Cε resp. ε  Dσ ,
where {C}, or {D} respectively, is symmetric 6x6 matrix of elastic moduli, or symmetric
matrix of compliances of a material, respectively. (back to Lecture 5-6) (back to Lecture 8) 3
force boundary conditions on  can be written in matrix form as follows
(56)
nσ  p .
The geometrical boundary condition on u u=g remains unchanged. Using preceding relations
e.g.
Dijklijkl switches to {}T{D}{}, ijui/xj switches to {}TTu etc.
(57)
In the case of the displacement model of FEM, the approximate displacements ui in the ith
element are related to the unknown displacements ri at the nodes by
ui  Ni r i ,
(58)
where N i is the matrix of the basis or shape functions whose compact support consists of the
elements in the neighbourhood of individual nodes. Similarly in the case of the force method,
approximate stresses i in the ith element are related to the unknown parameters (generalized
stresses) si by
σ i  N isi ,
(59)
For the displacement and force methods, it is sometimes difficult to find approximate
solutions that satisfy the required conditions. This shortcoming can be corrected with the use
of hybrid variational principles, in which those conditions causing difficulties are attached to
the variational expressions using Lagrange multipliers. The hybrid FEM has features of the
displacement and force method. Typically, the hybrid variational principle is discretized using
two set of approximate solutions, one for the interior of the element and the other especially
for the boundaries.
We choose to introduce the hybrid method by treating it as an extended force method. We
start with the Hellinger-Reissner principle (22), which we rewrite using Gauss’ theorem
applied to the second term. We obtain
1
R (u, σ)    Dijkl ij kl dV    ij n j  pi  ui dS    ij n j gi dS .
(60)


u
2
For the derivation of (60) we assumed that the field ij satisfies the equations of equilibrium
inside the domain, but it need not satisfy the force boundary condition on.. Let us now
divide the domain into elements k, k= 1,….M and rewrite the Hellinger-Reissner functional
(60) using the matrix notation mentioned above. We have
N1
M
1
T
T
T

R      σ Dσ dV   uT nσ  p  dS    σ n gdS , (61)
k


k
uj
2
k 1 
j 1

13
where N1 denotes the number of elements where boundary displacements are prescribed.
Notice the second integral in Eq. (61). In the case when k stands for the inter-element
boundary, this term represents the virtual work of a traction jump across the interface
Apparently, this virtual work is equal to zero if the tractions are continuous on inter-element
boundaries. In the case when k stands for the part of the domain boundary , the second
term in (61) expresses a measure of fulfilment of the force boundary conditions. The
boundary displacements u serve then as Lagrange multipliers.
For the boundary displacements, choose an approximate solution such as
(62)
u  N Br ,
where r are the nodal displacements, and the matrix of the basis functions NB defines the
assumed boundary displacements. An approximate solution for stresses is assumed in the
form (59). Note that when the structure is discretized into elements, the vector {n}{} in (61)
contains the reaction forces on the boundaries of the elements. Also, u is the boundary
displacement which can occur on all elements. Hence, the first term in the second integral
should be applied to all elements. Substitute now approximate solutions (62) and (59) into the
Hellinger-Reissner functional (61) and. Obtain
N2
M
M
1 T T
R   
s N D N sdV    rT NTB n N sdS    rT NTBpdS 
k 2
 k
 i
k 1
k 1
i 1
(63)
N1
T
   sT NT n gdS ,
u j
j 1
where N2 denotes the number of elements where boundary tractions are prescribed. It can be
seen that the functional R turns to be a function of parameters r and s The conditions for
R to attain its stationary value then read
R
R
 0 and
 0.
(64)
r
s
Detailing these relations we obtain the following system of linear algebraic equations
M


k 1
 k
N
T
B
n N dS

 
k 1
k

N D N dV s 
T
global compliance matrix of the system


s 
vector of generalized (transformed) tractions
M
N2
i 1
 i
NTB pdS

vector of generalized (transformed) loading
M


k 1
 k

N n N B dS r
T
T
vector of generalized (transformed) displacements

N1

j 1
u j
N n g .
T
T
(65)
vector of generalized (transformed)
prescribed displacements
Example 1. For a beam element extending from x = a to x = b derive the matrix of the basis functions N.
Assume the deflection can be approximated by a polynomial
(66)
w  c1  c2 x  c3 x 2  c4 x 3  ...wˆ1  wˆ 2 x  wˆ 3 x 2  wˆ 4 x 3  ... ,
where ci  w
ˆ i , i  1, 2,.. are the unknown constants of the assumed series that are chosen so as to obtain a good
approximation. Often the variable representation by a trial series will be supplemented with a superscript tilde,
e.g. w , to indicate that it is being approximated. For this beam element, we choose to retain only the first four
terms. Write the polynomial in the form
 wˆ1 
 wˆ 
(66)
ˆ w
ˆ T NTu
w  1, x, x 2 , x3   2   Nu w
 wˆ 3 
 
 wˆ 4 
where N u  1, x, x 2 , x 3  and w
ˆ   wˆ1 , wˆ 2 , wˆ 3 , wˆ 4  .
It is) convenient to rewrite the assumed series in terms of physically meaningful parameters (unknown
displacements at the ends of the beam element, rather than in terms of the unknown constants wˆ i . Thus, the
T
vector of unknowns ŵ can be transformed into the unknown nodal displacement vector
14
 wa 
   w 
(67)
a.
r a
 wb 


b   wb 
The assumed series is then referred to as an interpolation function. The derivative of w , which is needed to form
Eq. (67), is given by
 wˆ1 
 wˆ 
(68)
ˆ w
ˆ T NuT
w  0,1, x, x 2   2   Nu w
 wˆ 3 
 
 wˆ 4 
Now, evaluate w and   w at x = a and x = b, giving
0
0   wˆ 1 
 wa   w  0   1 0
     w 0  0 1 0
0   wˆ 2  .
(69)
  
 a

2
3
 wb   w  l   1 l
l
l   wˆ 3 
 
  

2
 b    w  l   0 1 2l 3l   wˆ 4 
r
ˆu
N
ˆ
w
ˆ ,
The constants ŵ are found in terms of the (unknown) displacements at a and b by using the inverse of N
u
1
ˆ
(70)
ˆ  Nu r  Gr
w
where
0
0
0 
 1
 0


1
0
0
(71)
ˆ 1  
.
GN
u
 3 l 2
2l
3 l2
1l 


3
1 l 2  2 l 3 1 l 2 
2l
The relationship between w and r, i.e. between the deflection w and the values of displacements w and  at the
ends of the element, is
(72)
ˆ  Nu Gr  Nr
w  Nu w
or


 x2
 x 2 x3 
x2
x3 
x 2 x3 
x3 
(73)
w  x   1  3 2  2 3  wa    x  2  2 a   3 2  2 3  wb    2  b
l
l
l
l
l
l
l
l








where N = NuG. This expression is the desired from of the assumed series, where The matrix N, which is often
called a shape function matrix, characterizes the “interpolation“ or ”shape” between the nodes.
Recommended literature for further reading
Washizu, K. Variational Methods in Elasticity and Plasticity, 3rd ed., Pergamon Press, Oxford,
1982.
Problems
1. An elastic string of length L, modulus E, and cross-sectional area A is fixed at its ends
and subjected to a transverse loading pz. The strain is given by
1
 xx  x   u   x   w2  x  ,
2
where u is the axial component of displacement and w is the transverse component.
(compare with Problem 2, in Appendix 1). Use the principle of minimum potential
energy to show that the governing equations are
d  
1

EA  u  w2    0 and u x 0  u x  L  0,

dx 
2

d  
1
 
EA  u  w2  w  pz  0 and w x 0  w x  L  0

dx 
2
 
15
Hint:
2
L
L
L
1

1
  1 2 
  A  xx xx d x   pz w d x    EA  u  w   p z w  d x
2 0
2


2
0
0 

and  = 0 leads to the governing equations.
2. The principle of virtual displacement (work) expression for a flat member of thickness
t with in-plane loading is
  xx xx   yy yy  2 xy xy  t d A    px ux  p y u y  d 

A
(a) Derive the equation of equilibrium in .
(b) Derive the force boundary conditions on .
3. Use the Hellinger-Reissner functional to derive the governing equations for a beam.
Include the effect of shear deformation.
Hint: Employ Eq. (22). There are only two stresses or strains of significance in
engineering beam theory. These are xx, xz, xx,, xz. Then
 xx2 2 1   2  xx2  xz2
1
Dijkl ij kl 

 xz 

2
2E
2E
2 E 2G
The integrand of the second term in (22) becomes
 ij  ij   xx xx  2 xz xz
Suppose that only applied loading is the distributed transverse loading pz. Then
R (u,σ ) becomes
L
  2  2 
u
 u w  
R (u, σ)      xx  xz    xx
  xz  
d
V


0 pz w d x .

x
 z x  
  2 E 2G 
To proceed, introduce the kinematical relations for engineering beam theory. If the
extension of the centreline is ignored, the axial displacement of a point on a cross
sectional plane is u(x,z)=z (x), where  = tan  is the slope of the beam axis. Hence
u

1  u w  1 
w 
 xx 
z
,  xz   
   

x
x
2  z x  2 
x 
The stresses and net resultant forces are related by
Mz
T
 xx 
,  xz   average  ,
I
A
where M or T is the bending moment or shear force respectively. Obtain
L
L
  1  Mz 2 1  T  2  Mz 2  T 
w  
R (u, σ )      





d
A
d
x





 


0 pz w d x 
A
I x A 
x  
  2 E  I  2G  A  
0
L
L
 M2
T2

w  

  

M
 T  
d
x

pz w d x


2
EI
2
Gk
A

x

x


s

0 
0
where the area A has been modified by the shear factor ks. Now, set  R (u,σ )  0 ,
where variations are taken independently with respect to the forces (M,T) and
displacements (w,). Thus,
 R (u, σ) 
 M  M T T



w 
w 


  

 M
 M
  T  
  T    
  pz w d x  0.
EI
Gks A
x
x
x 
x 



0
Integrate by parts the integral containing /x and w/x. Obtain
L
16
 M  
 T
w 

 
 T 
 M  
EI x 
x 
 Gks A
0 
L
 R (u, σ)   M   T  w0    
L
(P3.1)
 M

 T
 

 T     
 pz   w d x  0.
 x

 x
 
In matrix form, with z = [w,,T,M]T (so-called state vector) this appears as
 0
0
 x
0   w   pz  



L
0
1
 x      0  
T  0
0  z  x 1 1 ksGA 0   T    0   d x   Boundary terms  0 (P3.2)


    
  0
 x
0
1 EI   M   0  
Euler’s equations stemming from (P3.1) are
Equilibrium conditions
M
T
 T,
  pz
x
x
Kinematical equations including the material law

w 

M  EI
, T  Gk s A   

x
x 

Boundary conditions at x = 0 and x = L
M  0 or   0, T  0 or  w  0
In summary, by utilizing the kinematical conditions initially (assuming they hold a
priori, see Table 1), the remaining governing equations have been derived by using a
variational theorem in which both displacements and forces are varied independently.
4. Use the Hellinger-Reissner functional to derive the governing differential equations
for a bar under extension.
5. Use the Hellinger-Reissner functional and find approximate solution for a simply
supported beam loaded by constant transverse loading pz.
Hint: Let the whole beam be represented by a single element of length l, beginning at x
= -l/2 and ending at x = +l/2. To express the axial coordinate in non-dimensional form,
define  = 2x/l. Then the element is defined in the range -1    1. Also, note that
 x   2 l    . Moreover, if the boundary terms are assumed to be satisfied at the
outset, Eq. (P3.2) becomes

0
0
  2 l   
0
  w   pz  



1
0
0
1
 2 l         0   l
T 

d   0 .(P5.1)
1 z  2 l   
  T   0  2
1

1
k
GA
0
s

    


0
2
l



0

1
EI


  M   0  

To justify ignoring the boundary terms in (P3.2), choose the basis functions for both
displacements and forces that satisfy the boundary conditions (w= M = 0 at   1)
w  N w wˆ  1  2  wˆ ,   N ˆ  ˆ , T  NT Tˆ  Tˆ , M  N M Mˆ  1   2  Mˆ , (P5.2)
for the used notation see Example 1. (P5.2) can be written in matrix form as
0
0   wˆ  1  2 0 0
0   wˆ 
 w   Nw 0



  ˆ 
 ˆ
  0 N
0
0
0

0
0



  .

 



 0
T   0
0 NT
0  Tˆ
0 
0   Tˆ 


 


  
2
0
0 N M   Mˆ   0
0
0
1



M   0
  Mˆ 
 
z
zˆ
Insert (P5.3) into (P5.1), integrate, and obtain
17
(P5.3)
 0

0
T 
 zˆ 
4

  2 l  3
 0

  wˆ   43 pz  

 
2
0
 2 l  43   ˆ   0   l
3

(P5.4)
  0.
1
1 k s GA
0   Tˆ   0   2
  

1
0
 16
 2 l  43
  Mˆ   0  
15 EI 
Solve this set of equations for wˆ , ˆ , Tˆ , Mˆ . Substitute the results into (P.2), giving
0
 2 l  43
0
pz l 2  l 2
pz l 3
1 
2
w

,

 1  ,  
8  10 EI k s GA 
20 EI


pl
p l2
T   z , M  z 1   2
2
8
These, of course, are approximate relationships. Solution using Maple.

18

(P5.5)