Design of truss and frame structures with interval
and fuzzy parameters
M.V. Rama Rao Andrzej Pownuk
Department of Civil Engineering Department of Mathematical Sciences
Vasavi College of Engineering The University of Texas at El Paso
Hyderabad, Andhra Pradesh-500 031, India
500 West University Avenue, El Paso,
Texas 79968-0514, USA
dr.mvrr@gmail.com ampownuk@utep.edu
Abstract – In this paper a method of designing a structure
with interval parameters and fuzzy set parameters is presented.
This paper also outlines a procedure for designing a structure
with random set parameters. All procedures use solutions of the
interval equations which are based on the earlier works of both
authors. Safety of the structures is determined by using interval
limit state equations.
I. DESIGN OF STRUCTURS WITH INTERVAL PARAMETERS
According to the civil engineering codes (e.g. Eurocode
[1], Load and Resistance Factor Design [2] or [11]) limit state
design requires the structure to satisfy two principal criteria:
the ultimate limit state (ULS) and the serviceability limit state
(SLS). A limit state is a set of performance criteria (e.g.
vibration levels, deflection, strength), stability (buckling,
twisting, collapse) that must be met when the structure is
subjected to loads.
The equations of the limit state may have the following form
( )
L Q T
R D L Q T ϕ α ψγ α α α > + + + . (1)
where φ is resistance factor, ψ is load Combination factor, γ is
importance factor, α is dead load factor,
L
α is live load
factor,
Q
α is earthquake load factor,
T
α is thermal effect
(temperature) load factor, D is dead load, L is live load, Q
earthquake load, T is the thermal load, R is the resistance. In
structural mechanics several other safety criteria such as von
Mises, Drucker-Prager, Tresca or William Warke criteria can
be applied [3]. In general the structure is safe if
( ) 0 g p ≥ . (2)
where ( ) g p is some function, which describes the limit state
and p is a vector of parameters
1
( ,..., )
m
p p p = .
For example in the case of 3D elasticity problem the equation
(2) has the following form (von Mises criteria Error!
Reference source not found.)
2 2 2 2
1 2 2 3 1 3
6 ( ) ( ) ( ) 0 k p p p p p p
- - + - + - ≥
. (3)
where k is the magnitude of the shear stress at yielding in
pure shear,
i i
p σ = are principal stresses of the stress tensor.
II. TENSION-COMPRESSION OF STRUCTURS WITH INTERVAL
PARAMETERS
In the case of bar under tension the structure is safe if the
stress σ in the bar is smaller than the allowable stress in
tension
t
σ
t
σ σ ≤ ⇔
t
N
A
σ ≤ ⇔ 0
t
N
A
σ - ≥ (4)
where N is the axial force, A is the area of cross-section. If
the structure contains interval parameters (e.g. area of cross-
section, Young modulus, forces, geometrical dimensions)
1
( ,..., )
m
p p p = then the structure is safe if
( ) , ()
t
x p x σ σ ≤ (5)
for all
1 1 2 2
, , ... ,
m m
p p p p p p p ∈ × × × =
p and x ∈Ω
is the spatial variable. The condition (5) is satisfied if
() ()
t
x x σ σ ≤ (6)
where
{ } () inf (, ): x xp p σ σ = ∈ p , (7)
{ } () sup (, ): x xp p σ σ = ∈ p . (8)
If tension and compression are taken into account then the
safety condition has the form
() ()
t
x x σ σ ≤ . (9)
in tension and
() ()
c
x x σ σ ≤ . (10)
in compression. Here
c
σ is the allowable stress in
compression. Then in order to check the safety of the structure
with the interval parameters we need to know the value of the
interval stress [ ] () ( ), () x x x σ σ σ ∈ . In the dynamics problems
of computational mechanics the stress field also depends on
time i.e. [ ] (,) ( , ), (,) x t xt xt σ σ σ ∈ . Interval reliability of
structures is also discussed in the papers [12], [13].
978-1-4244-2352-1/08/$25.00 ©2008 IEEE