Virtual Displacements in
Formulating Framework
Stiffness Equations
This section of the course focuses
on applying the principle of virtual
displacements to construct the
algebraic equations for stiffness
analysis. We will discover that the
principle of virtual displacements
normally produces approximate
solutions and this will be the case
for nonprismatic members and
distributed loading on such
members.
1
element stiffness equations using
the virtual displacements approach
are:
1. Elastic constants relating
stresses and strains of the
material.
2. Relevant differential relationships between strain and
displacements.
3. Descriptions of the real and
virtual displaced states of the
element.
3
Description of the Displaced
State of Framework Elements
As stated in the principle of virtual
displacement notes, we must
assume a function for describing
the real and virtual displacements
over the element. A systematic
procedure will be presented to
describe the axial and bending
displacements over the framework
element.
In linear elastic analysis the only
ingredients needed to construct
2
Elastic constants are known from
experimentation. Differential
relationships between strains and
displacements are basic structural
mechanics relationships, which
have been presented for axial and
bending deformations. Thus, to
form element stiffness equations
we only need to describe the real
displacement variation and take
the virtual displacement variation
to be of same form.
4
1
Q1=Fx1, u1
Axial Force Element
Consider first the axial force element. The governing differential
equation is
Fig. 1: Standard Axial Force Element
Exact solution of (2) is
(1)
where qx(x) = distributed axial load.
For the case where A(x) = constant and qx(x) = 0, (1) becomes
dx
2
0
(2)
which is a homogeneous second5
order differential equation.


N a2 (x) u 2
(5)
where Na1(x) and Na2(x) are known
as shape functions, which describe
in this case the variation of u1 and
u2 over the element, respectively
as shown in Fig. 2.
1
Na1
= 1 – x/L
Na2 = x/L
1
x
0
L
x
0
(3)
Imposing the boundary conditions
u(x=0) = u1 and u(x=L) = u2 (see Fig.
1) leads to
 u1 
 
u2 
1 0  c0 
1 L   c 

  1
or
c0  1  L 0  u1 
  
 
 c1  L  1 1   u 2 
(4)
6
Fig. 2 shows that Na1 = 1 at x = 0 and
0 at x = L whereas Na2 = 0 at x = 0 and
1 at x = L. This is necessary since
u(x=0) = u1 and u(x=L) = u2:
Substituting (4) into (3):
x
u(x)  u1  (u 2  u1 )
L
 x
x
 1  u1    u 2
 L
L
N1a (x) u1
x=L
Node 2
u(x) = c0 + c1 x
d 
du(x) 
  EA(x)
 q x (x)
dx 
dx 
d 2 u(x)
Q4=Fx2, u2
x=0
Node 1
L
7
u(x) = Na1(x) u1 + Na2(x) u2
At x = 0:
u(x=0) = u1 = Na1(x=0) u1 + Na2(x=0) u2
which requires Na1 = 1 and Na2 = 0.
At x = L:
u(x=L) = u2 = Na1(x=L) u1 + Na2(x=L) u2
which requires Na1 = 0 and Na2 = 1.
Such kinematic restraints on the
shape functions satisfy the
Kronecker delta (ij) property: 8
Fig.2: Shape Functions for Axial Force Element
2
Nia (x j )  ij
Shape functions for the elements in
Fig. 3 are:
1 for i  j
ij  
0 for i  j
(6)
You should also note that Na1 + Na2
= 1 anywhere on the element.
A major advantage in using the
principle of virtual displacements is
that you are not limited to the
standard axial force member, see
Fig. 3.
u1
u2
u1
u3
u2 u3
x
x2
N1 (x)  1  3  2 2
L
L
 x x2 
N 2 (x)  4   2 
L L 
N 3 (x)  
(7a)
x
x2
2 2
L
L
u4
(b) Cubic Element
(a) Quadratic Element
9
10
Fig. 3: Higher-Order Axial Force Elements
N1 (x)  1 
11x
x 2 9x 3
9 2  3
2L
L 2L
x 45x 2 27x 3
N 2 (x)  9  2 
L 2L
2L3
N 3 (x)  
2
from 0 to L and the node points
are equally spaced.
(7b)
3
9x
x 27x
 18 2 
2L
L
2L3
N 4 (x)  x 
9x 2
2L2

9x 3
2L3
for the quadratic element (7a) and
cubic element (7b), respectively.
NOTE: The shape functions of (7a)
and (7b) also satisfy the Kronecker
delta property of (6). Also, x goes11
The axial force element shape
functions are known as
Lagrangian polynomials in
numerical analysis. They are
generated using the following
formula:
n
 (x  x j )
Ni (x) 
j1
j i
n
(8)
 (xi  x j )
j1
j i
12
3
where  symbolizes product rule,
i.e., a multiplication sequence and
n = number of element nodes.
Q6=Mz2, 2
Q3=Mz1, 1
Q5=Fy2, v2
Q2=Fy1, v1
x=L
Node 2
x=0
Node 1
Fig. 4: Standard Beam Bending Element
Beam Bending Element
Next, we consider the beam
bending element (Fig. 4). The
governing differential equation is
d 
d v(x) 
EI(x)

  q y (x)
dx 2 
dx 2 
2
2
(9)
13
Evaluating (11) at each dof in Fig.
4 leads to
v x 0  v1   p  x 0 {c}
 1 0 0 0 {c}
x 0
 1 
dp
{c}
dx x 0
 0 1 0 0 {c}
dv
dx
x L
 2 
dx
4
0
(10)
which is a homogeneous fourthorder differential equation.
v(x) = c0 + c1x + c2x2 + c3x3
c0 
or
 
2 3  c1 
v(x)  1 x x x     p(x) {c} (11)
14
c 2 
 c3 
or
 v1  1
  0
e  1  
{}    

 v 2  1
2  0
or
{}e  [P]{c}
0  c 0 
0   c 
 1
2
3  
L L
L c 2 

1 2L 3L2   c3 
0
1
0
0
(12)
Solving (12) leads to
v x  L  v 2   p  x  L {c}
 1 L L2
d 4 v(x)
The exact solution of (10) is
where qy(x) = distributed transverse load. For the case qy(x) = 0
dv
dx
and I(x) = constant, (9) becomes
{c}  [P]1 {}e
L3 {c}
and
dp
{c}
dx x  L
[P]1
 0 1 2L 3L2 {c}
15
1
0

  3
2
L
 23
L
(13)
0
1
0
0
2
L
3
L2
2
L3
1
L2
0
0

1 
L

1

L2 
16
4
Substituting (13) into (11):
 v1 
 
 
v(x)  1 x x 2 x 3 [P]1  1 
v2 
2 
N3b (x)  N 02 (x) 
N2
N3
  N(x) {}e
 v1 
 
 
N4  1 
v2 
2 
where
2x 3
L3
2
2x
x3

L
L2
3x 2
L2
N b4 (x)  N12 (x)  
(14)

L2
N b2 (x)  N11 (x)  x 
e
 N1
3x 2
N1b (x)  N10 (x)  1 
e

2x 3
L3
x 2 x3

L L2
j
Shape function Ni (x) designates
the node i for which the shape
function has been developed and
superscript j designates the order
of differentiation on the nodal
displacement variable, i.e., 0 =
17
function value (v) and 1 = first
derivative function value (). The
shape functions in (14) are known
as cubic Hermitian polynomials;
cubic is the order of the polynomial
and Hermitian designates that
derivative function values are
included in the approximation.
The shape functions of (14) also
satisfy the Kronecker delta
property, except now it needs to be
expanded due to the presence of
dv/dx in the approximation:
19
18
N1b
x 0
dN1b
dx
N3b
 N10
x 0
dN10
dx

x  0,L
x 0
 N 02
dN3b
dx
1; N1b

x 0
x L
 N10
x L
 0;
0
x  0,L
 0; N3b
dN 02
dx
x L
 N 02
x L
1;
0
x 0,L
x  0,L
b
1
N2
 N1
 0;
x  0,L
x  0,L
dN b2
dx
dN b2
dx

x 0
dN11
dx

x L
 1;
x 0
dN11
dx
0
x L
20
5
N b4
x  0,L
dN b4
dx
x 0
 N12
dN12

dx
dN b4
dx
x 0,L

x L
 0;
 0;
x 0
dN12
dx
generate Hermitian interpolation
functions. A plot of the shape
functions are shown in Fig. 5.
1
1
x L
x
0
0
L
(a) Plot of Nb1(x)
The procedure used to generate
the cubic Hermitian shape functions for the standard beam
bending element is known as
generalized coordinate
interpolation. Unfortunately, this
is the only method available to 21
1
0
x
L
0
(b) Plot of Nb3(x)
22
1
0.3
2
0.2
v2, v2’, v2”, …, v2(k)
v1, v1’, v1”, …, v1(k)
0.1
1
0
0
0
0
L
(c) Plot of Nb2(x)
1
L
(a)
x
1
2
k+1
v1, v1’ v2, v2’
x
vk+1, v’k+1
(b)
-0.1
1
-0.2
2
v1, v1’, v1”
-0.3
3
v2
v3, v3’, v3”
(d) Plot of Nb4(x)
(c)
Fig. 5: Plot of the Cubic Hermitian Shape Functions
Figure 6. Example Hermitian Elements
As was the case for the axial force
element, we are not restricted The
shape functions of (14) for beam
bending elements. Fig. 6 shows
some additional choices.
v(x) 
2
2
2
i 1
i 1
i 1
 Ni0 (x) vi   N1i (x) v 'i     Nik (x) vi(k)
v(x) 
v(x) 
k 1
i 1
i 1
 Ni0 (x) vi   N1i (x) v 'i
3
 Ni0 (x) vi  
i 1
23
k 1
i 1,3
N1i (x) v 'i 

i 1,3
Ni2 (x) v"i
24
6
The three cases illustrated in Fig. 6
can collectively be expressed as
v(x) 
 Ni0 (x) vi   N1i (x) v 'i     Nik (x) vi(k)
i0
i1
ik
where the summations over i0, i1, ik
are performed on the nodes with
function value unknowns, first
derivative unknowns, ..., kth
derivative unknowns, respectively.
These shape functions must also
satisfy the Kronecker delta
property, i.e.
Ni0 (x j )  ij
N1i (x j )  0

Nik (x j )  0
for all nodes where the
function value is
unknown
dNi0 (x j ) / dx  0
dN1i (x j ) / dx  ij

dNik (x j ) / dx  0
for all nodes where the
first derivative value is
unknown

k
d Ni0 (x j ) / dx k  0
d k N1i (x j ) / dx k  0

k
d Nik (x j ) / dx k  ij
for all nodes where the
kth derivative value is
unknown
25
Generation of Element Stiffness
Matrices
Now that the real displacement
variations over an element have
been developed:
(15)
  (x)   N  (x) {  }e
where  = framework element
deformation mode ( = a – axial or
= b – bending),  =  mode
displacement variable (u – axial, v
– bending), <N(x)> = vector of
element shape functions, and {}e
= vector of element nodal
displacement parameters; the
27
26
element stiffness matrices can be
expressed as (assuming the virtual
displacement approximation is the
same as the real displacement
approximation):
[k  ] 
L
 {B (x )}c  (x)  B (x )  dx
(16)
0
where <B> = strain displacement
matrix for deformation mode ; c
= constitutive constant for
deformation mode  (c = EA for
axial deformation and = EI for
bending deformation); and [k] =
28
7
element stiffness matrix in the
local coordinate system for
deformation mode . The straindisplacement vectors for axial and
bending deformation modes are
Ba (x) 
Bb (x) 
dN1a
dN a2
dx
dx
d 2 N1b
d 2 N b2
dx 2
dx 2


dN an a
freedom used to approximate the
bending displacement v. Substituting the strain displacement
expressions into (16) leads to
B1B 2
 B1B1

 B 2 B1 B 2 B 2
[k  ]   c 


0

 B  B1 B  B 2
n
 n
L
dx
d 2 N bn b
dx 2
where na = number of nodes used to
approximate the axial displacement
u; and nb = number of degrees of


 B 2 B  
n
 dx (17)



 B B  
n
n 

B1B
n 
Eq. (17) is valid for both axial and
bending deformation and can also
be used for nonprismatic members
as well. For a nonprismatic
member, c will be a function of x.
29
Assuming na = 2 (standard axial
force element) and a prismatic
member, the axial stiffness matrix
from (17) is
a
a
L  dN1 dN1
 dx dx
[k a ]  EA 
 dNa2
 dx
0
L
 EA  

0

dN1a
dx
dN1a dNa2 
dx dx 
dN a2 dN a2 

dx dx 
dx
 L1  L1   L1  L1 
dx
 L1  L1   L1  L1  
L
EA  1 1
EA  1 1
dx 
2  1 1  
L  1 1 
L 
0
(18a)
Similarly, assuming nb = 4 (standard
beam element) and a prismatic
31
member, the bending stiffness
30
matrix coefficients are
L
k ijb 

d 2 Nib (x)
d 2 N bj (x)
EI
dx
2
2
dx
dx
0
for i,j = 1,2,3,4 resulting in the
stiffness matrix
6L 12 6L 
 12

2
2
EI  6L 4L 6L 2L 
b
[k ] 

 (18b)
L3  12 6L 12 6L 
 6L 2L2 6L 4L2 


Obviously, (18a) and (18b) match
the previous results we generated.
32
8

As stated previously, the advantage of using the principle of virtual
displacements is that non-standard
element shape functions can be
used to generate different order
stiffness matrices.
where {Qf }  element fixed-end
force vector for deformation mode
; {N} = element shape functions
used in deformation mode ; and
q(x) = distributed load for
deformation mode .
Element Fixed-End Forces
Element fix-end forces due to initial
strain loading can be expressed as
The element fixed-end forces due
to mechanical loads can be
expressed as
L
 {B (x)}c  (x) E

dx
(20)
0
E
L
{Qf }    {N  (x)}q  (x)dx
{Qf } 
(19)
0
33
Nonprismatic Beam Elements
The shape functions developed
thus far are exact for prismatic
frame elements. Generation of
exact shape functions for nonprismatic elements requires that
the shape functions be developed
using the generalized interpolation
approach for exact solution of (1)
for the axial force element with
A(x) and (9) for beam bending with
I(x). This is not practical unless
you are going to use the same
35
where
= initial strain for deformation mode ; and the other
variables are as previously defined.
34
element geometry repeatedly. The
usual practice is to use shape
functions derived for prismatic
geometry and insert the exact
geometric cross section properties
in (17) or to approximate the cross
section geometry.
Yang (1986) developed axial force
and beam bending stiffness matrices for the standard elements of
Figs. 1 and 4 using:


x 
A(x)  A 0 1 s   
 L  

(21a)
36
9


x 
I(x)  I0 1 r   
 L  

(21b)
The stiffness matrices of Yang
(1986) are
EA 0 
s   1 1
(22a)
[k ] 
1

L    1   1 1 
a
 122 C11

symmetric
 L

6

4C22
EI0  L C21
b
[k ] 
 12
 (22b)
6
12
L  2 C11  L C21
C

2 11
L
L
 6

6C
C
2C
4C


42
44 
L 41
 L 41
inertia at the beginning of the
element; s,  = area interpolation
constants; r,  = moment of inertia
interpolation constants; and
 1 1  4 2  4 3 
2  7  6
C21 1 2r  
1  2 3 
4  12  9
C22 1 r  
1  2  3 
1  5  6
C41 1 2r  
1  2 3 
2  9  9
C42 1 2r  
1  2 3 
1  6  9
C44 1 r  
1  2  3 
C11 1 3r
A0, I0 = area, moment of inertia at
37
38
10