The following step-by-step procedure can be used to take advantage of
structural symmetry in the analysis of structures.
1. Check the given structure for symmetry, as discussed in Sec-
tion 10.1. If the structure is found to be symmetric, then proceed
to step 2. Otherwise, end the analysis at this stage.
2. Select a substructure (half the structure) on either side of the axis
of symmetry for analysis. The cross-sectional areas and moments
of inertia of the members of the substructure, which are located
along the axis of symmetry, should be reduced by half, whereas full
values of these properties should be used for all other members.
3. Decompose the given loading into symmetric and antisym-
metric components with respect to the axis of symmetry of the
structure by using the procedure described in Section 10.2.
4. Determine the response of the structure due to the symmetric
loading component as follows:
a. At each joint and end of the substructure, which is located
at the axis of symmetry, apply restraints to prevent rotation
and deflection perpendicular to the axis of symmetry. If there
is a hinge at such a joint or end, then only the deflection, but
not rotation, should be restrained at that joint or end.
b. Apply the symmetric component of loading on the sub-
structure with the magnitudes of the concentrated loads at
the axis of symmetry reduced by half.
c.
d.
Analyze the substructure to determine its response.
Obtain the symmetric response of the complete structure by
reflecting the response of the substructure to the other side
of the axis of symmetry.
5. Determine the response of the structure due to the antisym-
metric loading component as follows:
a. At each joint and end of the substructure located at the axis
of symmetry, apply a restraint to prevent deflection in the
direction of the axis of symmetry. In the case of trusses, the
axial forces in members located along the axis of symmetry
will be zero. Remove such members from the substructure.
b. Apply the antisymmetric component of loading on the sub-
structure with the magnitudes of the loads and couples, ap-
plied at the axis of symmetry, reduced by half.
Analyze the substructure to determine its response.
Obtain the antisymmetric response of the complete structure
by reflecting the negative of the response of the substructure
to the other side of the axis of symmetry.
c.
d.
6. Determine the total response of the structure due to the given
loading by superimposing the symmetric and antisymmetric re-
sponses obtained in steps 4 and 5, respectively.
The foregoing procedure can be applied to statically determinate as
well as indeterminate symmetric structures. It will become obvious in
subsequent chapters that the utilization of structural symmetry consid-
erably reduces the computational effort required in the analysis of stat-
ically indeterminate structures./n 7. Input
STAAD SPACE
START JOB INFORMATION
ENGINEER DATE 27-Mar-22
END JOB INFORMATION
INPUT WIDTH 79
UNIT FEET KIP
JOINT COORDINATES
1000; 2096 0; 3 30 96 0; 4 30 0 0; 5 60 0 0; 6 60 96 0; 70 72 0;
8 30 72 0; 9 60 72 0; 10 0 48 0; 11 30 48 0; 12 60 48 0; 13 0 24 0;
14 30 24 0; 15 60 24 0;
MEMBER INCIDENCES
11 13; 22 3; 3 3 8; 4 5 15; 5 6 3; 672; 78 11; 87 8; 996;
10 8 9; 11 10 7; 12 11 14; 13 10 11; 14 12 9; 15 11 12; 16 13 10;
17 14 4; 18 13 14; 19 15 12; 20 14 15;
START USER TABLE
TABLE 1
UNIT INCHES KIP
PRISMATIC
1
30 2000 2000 2000 30 30 0 0
END
UNIT INCHES KIP
DEFINE MATERIAL START
ISOTROPIC STEEL
E 29000
POISSON 0.3
DENSITY 0.000283
ALPHA 6.5e-006
DAMP 0.03
TYPE STEEL
STRENGTH FY 36 FU 58 RY 1.5 RT 1.2
END DEFINE MATERIAL
MEMBER PROPERTY
1 TO 20 UPTABLE 1 1
CONSTANTS
MATERIAL STEEL ALL
UNIT FEET KIP
SUPPORTS
145 FIXED
UNIT INCHES KIP
LOAD 1 LOADTYPE Dead TITLE LOAD CASE 1
JOINT LOAD
2 FX 25
7 FX 20/n Q;
Analyse the shown structure using
Symmetry and antisymmetry to det.
Supports reactions and draw axial,
Shear, moment diagram.
lok
20k
20k
J
D
A
2 12/1
31/1
3k/1
30
It
710
B
*
E = 29,000 kesi
A = 30 in ²
I = 2000 in 4
301
L
FI
AL
TIM
e
*
24
24
24 model 1: analyze the entire structure
Support reactions, axial, Shear, moment
diagram.
2- analyze the structure as symetry and
antisymetry at loading.
symetry
+
anti symetric
=
Full structurey
3. Compare results from ) and (2)
