Download Analysis of Symmetric Structures SYMMETRIC STRUCTURES

Analysis of
Symmetric Structures
SYMMETRIC
STRUCTURES
Many structures, because of
aesthetic and/or functional
considerations, are arranged in
symmetric patterns. Recognition of
such symmetry will be identified
and the use of this symmetry will
be used to reduce the computational effort in analyzing such a
structure.
Definition of symmetry is expedited
by using the concept of reflection,
or mirror image. A plane structure
is symmetric with respect to an
axis of symmetry in its plane if the
reflection of the structure about the
axis is identical in geometry,
supports, and material properties
to the structure itself. Examples in
Fig. 10.3 where the s-axis defines
the axis of symmetry.
1
2
Figure 10.3 – Example Symmetric
Structures
NOTE: Most symmetric structures
can be identified by inspection –
simply compare the geometry,
supports and material properties of
the two halves of the structure on
each side of the axis of symmetry.
3
4
1
When examining structural
symmetry for the purpose of
analysis, it is necessary to
consider symmetry of only those
structural properties that influence
the results of the analysis. For
example, the truss structure of Fig.
10.4 can be considered symmetric
when subjected to vertical loads
because under such loads the
horizontal reaction is zero.
However, this truss cannot be
considered symmetric when
subjected to any horizontal loads.
s
P1
P3
h
0
L
L
P2
Truss
Structure
E = constant
A = constant
Figure 10.4 – Symmetric Analysis:
Horizontal Reaction = 0
5
6
SYMMETRIC AND
ANTISYMMETRIC
COMPONENTS OF LOADINGS
The reflection of a system of forces
and displacements about an axis
can be obtained by rotating the
force and displacement system
through 180 about the axis as
shown in Fig. 10.8.
7
y
Fy, vy
Fy, vy
Fx, vx
M, 
M, 
A(-x,y)
(b) Reflection about y-Axis
A(x,y)
Fx, vx
(a) Force/Displacement
System
x
M, 
A(x,-y)
Fx, vx
Fy, vy
(c) Reflection about x-Axis
Figure 10.8 – Force and Displacement
Reflections
8
2
Symmetric Loadings
P
A loading is considered to be
symmetric with respect to an axis
in its plane if the reflection of the
loading about the axis is identical
to the loading itself.
A
s
P
a
2
P
B
a
Antisymmetric Loadings
a
2
s
B
a
a
w
w
C
B
C
s
a
a
s
C
w
a
2
P
A
s
D
D
P
a
Loading
w
B
A
Reflection
Loading
A loading is considered to be
antisymmetric with respect to an
axis in its plane if the negative of
the reflection of the loading about
the axis is identical to the loading
itself.
9
A
Reflection
a
2
A
P
a
Loading
w
s
a
2
B
B
a
2
P
B
w
C
P
a
A
Reflection
Figure 10.9 – Examples of Symmetric
10
Loadings
General Load Decomposition
into Symmetric and
Antisymmetric Components
Any general loading can be
decomposed into symmetric and
antisymmetric components with
respect to a symmetry axis by
applying the following procedure:
1. Divide the magnitude of the
forces and/or moments of the
given loading by two (e.g., Fig.
10.11(b)).
Figure 10.10 – Examples of
Antisymmetric Loadings
11
2. Draw a reflection of the half
loading about the specified axis
12
(Fig. 10.11(c)).
3
2P
3. Determine the symmetric
component of the given loading
by adding the half loading to its
reflection (Fig. 10.11(d)). Divide
the magnitude of the forces
and/or moments of the given
loading by two (e.g., Fig.
10.11(b)).
4. Determine the antisymmetric
component of the loading by
subtracting the symmetric
loading component from the
given loading (Fig. 10.11(e)).
V
2H
=
b
s
P
w
a
b
(b) Half Loading
s
P
w
b
a
(c) Reflection of Half Loading
P
P
s
w
w
a
a
b
(d) Symmetric Loading
Figure 10.11 – General Load Decomposition
14
BEHAVIOR OF SYMMETRIC
STRUCTURES UNDER
SYMMETRIC AND
ANTISYMMETRIC LOADING
V
H
a
(a) Given Loading
b
13
2V
2w
s
H
(b) Symmetric
Loading
When a symmetric structure is
subjected to a loading with
respect to the structure’s axis of
symmetry, the response of the
structure is also symmetric.
+
(a) Given Loading
V
H
V
H
(c) Antisymmetric
Loading
Frame Example of General
Load Decomposition Superposition
15
16
4
Displacement behavior along the
axis of symmetry for symmetric
loading results in no rotation
(unless there is a hinge at such a
point) nor any deflection
perpendicular to the axis of
symmetry.
M≠ 0
w
w
Fx≠ 0
Fy= 0
h
 collar
support
L
2
L
2
L
2
s
(a) Symmetric Frame and
Loading
P
2P
P
w
h
s
(b) Half Frame w/ Symmetric
Boundary Conditions
A,I
B
Force behavior along the axis of
symmetry for symmetric loading
results in zero force along the axis
of symmetry.
17
If support B in Fig. 10.16(c) was a
roller rather than a hinge support,
would the boundary condition at B in
Fig. 10.16(d) change?
s
P
L
L
P
L
(a) Symmetric Truss and Loading
L
2
Figure – Example Symmetric Truss Structure
What BC at the point indicated for
the symmetric truss structure? 19
L
L
s
(c) Symmetric Frame and
Loading
18
A symmetric structure is subjected to a loading that is antisymmetric with respect to the
structure’s axis of symmetry, the
response of the structure is also
antisymmetric.
Displacement behavior along the
axis of symmetry for antisymmetric
loading results in no displacement
along the axis of symmetry.
Force behavior along the axis of
symmetry for antisymmetric loading results in zero force normal to
the axis of symmetry and zero 20
bending moment.
5
Symmetric Beam
P
P
s
P
Half Beam w/ Loading
s
s
P
s
P
Symmetric Frame
P
Half Frame w/ Loading
Figure 10.17 – Symmetric Structures
w/ Antisymmetric Loadings
21
For general loading on a symmetric
structure, the loading can be
decomposed into symmetric and
antisymmetric components.
Displacement and force boundary
conditions for symmetric and
antisymmetric loadings along the
axis of structural symmetry apply.
To obtain the total response, use
superposition of the symmetric and
antisymmetric results.
22
Example
Problems
23
6