Finite Element Formulation for Stability and Free Vibration Analysis of Timoshenko Beam

Table 2

Dimensionless frequency parameter 𝜆_𝑖 for the simply supported Timoshenko beam.

Mode	Euler theory	ℎ/𝑙=0.01
		Ferreira [10]	Lee and Schultz [12]	Proposed element
1	3.14159	3.1425	3.14133	3.14133
2	6.28319	6.2908	6.28106	6.28106
3	9.42478	9.4503	9.41761	9.41764
4	12.5664	12.6271	12.5494	12.5496
5	15.7080	15.8267	15.6749	15.6754
6	18.8496	19.0552	18.7926	18.7937
7	21.9911	22.3186	21.9011	21.9034
8	25.1327	25.6231	24.9988	25.0034
9	28.2743	28.9749	28.0845	28.0927
10	31.4159	32.3806	31.1568	31.1705