## 3 Effect of the Cross-Sectional Types on Natural Frequencies
### Rectangular Section Beam  
#### 3.1.1 Hinged-Hinged  
Figures 4-6 show the natural frequencies of a hinged-hinged beam with the rectangular section subjected to different beam models. All first subgraphs of these 3 figures show the overall frequency curves, from which the same variation trend of the natural frequencies can be found for all five beam theories, i.e., Bernoulli-Euler beam, Rayleigh beam, shear beam, Timoshenko beam, and modified Timoshenko beam.  
Figure 4: Natural frequencies of the 8-m-long beam: (**a**) overall frequency curve and (**b**) frequency difference compared with a modified Timoshenko beam.  
Figure 5: Natural frequencies of the 4-m-long beam: (**a**) overall frequency curve and (**b**) frequency difference compared with a modified Timoshenko beam.  
Figure 3: Cross-sectional forms: (**a**) rectangular section, (**b**) rectangular hollow section, and (**c**) circular section.  
The Bernoulli-Euler model provides acceptable frequencies at lower orders while loses its rationality and feasibility with the order increasing. For Rayleigh beam theory, which takes the effect of rotary inertia due to flexural deformation into account, its natural frequencies are substantially reduced compared with the Bernoulli-Euler beam. However, the result is still much higher than those of other beams. Therefore, we could lay more emphasis on the shear beam, Timoshenko beam, and the modified Timoshenko beam without considering Bernoulli-Euler beam and Rayleigh beam.  
It can be noted that the results obtained from the shear beam, [PERSON] beam, and the modified Timoshenko beam are generally common; however, the difference between the shear model and Timoshenko beam is much less. As can be seen from Figure 4b, when the beam length is 8 m, the frequency difference between the shear model and the modified [PERSON] beam is 12.3% at the order of 20 and 8.8% between the Timoshenko beam and the modified Timoshenko beam. The major distinction between the shear model and Timoshenko model is whether rotary inertia due to flexural deformation has been concerned, and the modified Timoshenko beam further takes rotary inertia due to shear deformation into account based on the Timoshenko beam. Therefore, the consequence shows that rotary inertia caused by the shear deformation is nonnegligible.  
As can be seen from Figures 5 and 6, when the beam length reduces to 4 or 2 m, the difference between the shear beam and the Timoshenko beam has also been reduced. Actually, for a 2-m-long beam, the frequency difference between the shear beam and modified Timoshenko beam is 15.0% at the 20 th order, almost the same as that between the Timoshenko beam and modified Timoshenko beam, which is 14.5%. The result illustrates that the effect of the rotary inertia caused by the flexural deformation becomes small when the beam tends to be non-slender. On the contrary, the significance of the rotary inertia due to shear deformation for non-slender beam emerges. In addition, the fact that the frequency difference has grown enormously for both shear beam and Timoshenko beam illustrates that the rotary inertia caused by the shear deformation has an increasing influence on natural frequencies with the beam is shortened. It is intelligible for the reason that the effect of shear deformation as well as the rotary inertia that it causes becomes notable when the beam tends to be non-slender. In the meantime, the effect of bending moment and rotary inertia that it causes decreases. As a consequence, such modification is essential if the high-frequency band is considered.