Abstract
The paper is concerned with optimization of a damped column subjected to a follower load. The aim is to determine the colum of least volume which has the same critical load as a uniform reference column. The stability analysis is based on the finite element method. The optimization problem is solved by sequential linear programming. By only including a constraint on the flutter load in the volume minimization, a very large volume reduction is possible but the static buckling load (by a pure conservative loading) becomes very small.In applications, it may be important that the optimal column also is capable of supporting a conservative load. Consequently, the volume is minimized with constraints on both the flutter load and the static buckling load. The constraint on the buckling loadp b has the formp opt b ≥cp 0b , 0≤c≤1, where the upper index “opt” refers to the optimal design while the upper index “0” refers to the uniform initial design. It is found that, as the constantc approaches 1, the optimal column approaches the optimal Euler column of Tadjbakhsh and Keller (1962).
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Abbreviations
- c :
-
slack parameter on the constraint on the static buckling load; defined by (9)
- c int,c ext :
-
dimensionless internal and external damping parameters defined by (3)
- d j :
-
eigenvalue margin defined by (9)
- d :
-
vector of time-independent nodal displacements and rotations
- ℓ e :
-
length of thee-th finite element
- L :
-
total length of the column
- ℓ:
-
vector of element lengths defined by (11)
- m, m(x) :
-
mass distribution function
- m i :
-
design variables; the mass distribution function evaluated at the nodal points
- \(\bar m_i ,\underline m _i \) :
-
upper and lower bounds on the design parameters
- m :
-
design vector with elementsm i
- M :
-
mass matrix
- N e :
-
the number of finite elements used
- p :
-
load parameter
- Q :
-
load matrix
- S :
-
stiffness matrix
- t :
-
time
- x :
-
distance along the column, measured from the clamped end
- y :
-
lateral deflection of the column
- y :
-
vector of nodal displacements and rotations
- λ:
-
complex eigenvalue
- b :
-
refers to buckling (static instability by conservative loading)
- d :
-
refers to divergence (static instability by nonconservative loading)
- f :
-
refers to flutter (dynamic instability by nonconservative loading)
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Langthjem, M.A., Sugiyama, Y. Optimum design of Beck's column with a constraint on the static buckling load. Structural Optimization 18, 228–235 (1999). https://doi.org/10.1007/BF01223304
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DOI: https://doi.org/10.1007/BF01223304