Influence of Non-Structural Localized Inertia on Free Vibration Response of Thin-Walled Structures by Variable Kinematic Beam Formulations

Table 1

First 10 natural frequencies (Hz) for the cantilever C-shaped beam.

Mode	Classical and refined models based on TE					LE models		MSC Nastran^©
	EBBM	TMB	𝑁=4	𝑁=6	𝑁=8	6 L9	10 L9	FEM-2D	FEM-3D
1	1.75^b	1.75^b	1.76^b	1.76^b	1.76^b	1.76^b	1.76^b	1.74^b	1.75^b
2	2.99^b	2.98^b	2.93^b	2.76^b	2.65^b	2.56^b	2.56^b	2.49^b	2.54^b
3	10.97^b	10.90^b	10.90^b	7.66^t	6.73^t	6.29^t	6.27^t	6.09^t	6.23^t
4	18.67^b	18.35^b	12.25^t	10.89^c	10.89^c	10.87^c	10.87^b	10.71^b	10.83^b
5	30.63^b	30.20^b	17.46^t	13.83^t	12.37^t	11.46^t	11.41^t	10.99^t	11.34^t
6	51.81^b	49.83^b	29.94^b	24.87^t	22.54^t	21.18^t	21.10^t	20.29^c	20.97^t
7	59.75^b	58.25^b	34.66^t	29.93^b	29.87^t	29.25^t	29.19^t	28.82^t	29.06^t
8	65.87^a	65.87^a	45.76^t	32.49^b	30.21^b	29.89^b	29.86^b	29.55^b	29.75^b
9	98.20^b	93.73^b	56.94^c	41.26^c	37.12^c	34.93^c	34.76^c	33.44^c	34.53^c
10	100.22^b	94.43^b	59.22^c	57.05^c	53.02^c	50.23^c	49.88^c	48.06^c	49.51^c
DOFs	93	155	1395	2604	4185	3627	5859	38250	177000

aAxial; bbending; ccoupled; ttorsional