Abstract
If lateral stiffness distribution of continuum structure is unreasonable, its displacement distribution will be non-uniform under lateral load. The non-uniform displacement distribution will weaken the safety of the structure. The continuum structure is equivalent to the cantilever. The optimal lateral stiffness distribution of the structure is obtained by adjusting the section size distribution. The wind load and earthquake action are simplified into three equivalent static loads: uniform load, inverted triangle load and inertia force related load. The shearing displacement and bending displacement distributions along the structural height are derived. The optimization objective is that the second derivative of the total displacement distribution is zero, that is, the displacement distribution is uniform. The section size of the cantilever is optimized and the optimal scheme is obtained. The optimization results are verified by numerical analysis and finite element analysis. The results show that the uniform displacement criterion can be realized by the cantilever with section size distribution of power function. The optimal parameters of the section under different loads are different. A practical design example is presented to prove the applicability.
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Abbreviations
- A(y):
-
Area of section at height y
- c :
-
Generalized parameter to be optimized
- D B(y):
-
Displacement caused by bending moment at height y
- D S(y):
-
Displacement caused by shear force at height y
- E :
-
Elastic modulus of the structure
- e :
-
Base of the natural exponential function
- f(c, y):
-
Total displacement of structure at height y
- G :
-
Shear modulus of the structure
- H :
-
Total height of the structure
- h :
-
Radius of structural bottom section
- \({\bar h}\) :
-
Radius of structural top section
- I(y):
-
Moment of inertia at any height y
- M(y):
-
Bending moment of the structural section
- \(\bar m(y)\) :
-
Mass distribution at height y
- n :
-
Shape control parameter in power function
- q :
-
Intensity of uniform load
- q max :
-
Maximum intensity of inverted triangle load
- Q(y):
-
Shear force of the structural section
- \(\bar q(y)\) :
-
Intensity of inertia force related load
- r :
-
Radius of structural section
- R C(y):
-
Optimal radius distribution under combined load
- R E(y):
-
Optimal radius distribution under inertia force related load
- R I(y):
-
Optimal radius distribution under inverted triangle load
- R U(y):
-
Optimal radius distribution under uniform load
- y :
-
Height of the structure which corresponds to the y axis of the coordinate system
- α :
-
Shape control parameter in natural exponential function
- β :
-
Bottom radius control parameter in power function
- λ :
-
Mass-dependent constant to control the load amplitude
- μ :
-
Coefficient to consider the non-uniform distribution of shear strain in the cross-section
- ρ :
-
Effective cross-section density
- φ U :
-
Weight coefficient of RU(y)
- φ I :
-
Weight coefficient of RI(y)
- φ E :
-
Weight coefficient of RE(y)
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Acknowledgments
This work is supported by the National Natural Science Foundation of China under the Project Number 51878017, which is gratefully acknowledged.
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He, H., Wu, S. & Liao, L. Section Size Optimization of Continuum Structure with Bending and Shearing Deformation for Uniform Displacement Criterion. KSCE J Civ Eng 26, 2234–2245 (2022). https://doi.org/10.1007/s12205-022-0573-8
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DOI: https://doi.org/10.1007/s12205-022-0573-8