Objective
To perform a modal analysis of a beam with a variable cross-section.
Problem statement
To determine the natural frequencies and mode shapes of a beam with a variable cross-section, rigidly fixed at both ends.
Design model
A beam with a variable cross-section and material density ρ, rigidly fixed at both ends.
Analytical solution
Société Française des Mécaniciens – Commission Validation de Progiciels de Calcul de Structures, Groupe de travail Dynamique, Paris, 1989.
Geometry
Length l = АВ = 0,6 m
Thickness h = 0,01 m
Cross-section width b0 = 0,03 m
Cross-section variation (for α = 1) b = b0e-2αx.
Material properties
Modulus of elasticity Е = 2 * 108 tf/m2
Poisson's ratio v = 0,25
Material density ρ = 7800 kg/m3
Boundary conditions
The beam is restrained in all degrees of freedom (DOF) of the 2D problem at points A and B.
Loads
The self-weight of the beam is included in the modal analysis as (b*h*ρ*g).
Note
The problem is solved in 2D formulation (model type 2 – XOZ-plane).
The model is generated with FE type 2 – FE of 2D frame.
The variable cross-section of the beam is modelled using a set of finite elements with different widths.
The mass weight is defined using the load «Weight of distributed dynamic mass», depending on the width of each element cross-section.
A modal analysis is performed.
The number of mode shapes considered is – 10.
Nodes: 21. Elements: 20.
Output data
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1st natural vibration mode shape
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2nd natural vibration mode shape
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3rd natural vibration mode shape
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4th natural vibration mode shape
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Comparison of calculation results
| Parameters | Mode shape | Analytical solution | LIRA-FEM | Error, % |
| Frequency, Hz | 1 | 143,303 | 145,882 | 1,7679 |
| 2 | 396,821 | 400,303 | 0,8698 | |
| 3 | 779,425 | 783,197 | 0,4816 | |
| 4 | 1289,577 | 1293,323 | 0,2896 |
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