Date of Graduation


Document Type


Degree Type



Statler College of Engineering and Mineral Resources


Civil and Environmental Engineering

Committee Chair

Roger H.L. Chen

Committee Co-Chair

P.V. Vijay

Committee Member

P.V. Vijay

Committee Member

Karl E. Barth


In this study, the exact solution of the Timoshenko beam vibration frequency equation under free-free boundary conditions was determined with an accurate shear shape factor. The exact solution was compared with a 3-D finite element calculation using ABAQUS program, and the difference between the exact solution and the 3-D FEM was within 0.15% for both the transverse and torsional modes. Furthermore, relationships between the resonance frequencies and Poisson’s ratio were proposed that can directly determine the elastic constants, unlike the equations provided by ASTM C215. The frequency ratio between the 1st bending mode and the 1st torsional mode, or the frequency ratio between the 1st bending mode and the 2nd bending mode for any rectangular prism or rod can be directly estimated. Likewise, the bending and torsional modes can be used to determine the elastic constants of any rectangular prism or rod. The proposed equations were used to verify the elastic constants of a steel rod and prism with less than 0.36% error. The transverse and torsional frequencies of concrete, aluminum and steel rods were tested. Rectangular concrete beam specimens with three different mix designs were produced, and the transverse and torsional frequencies of these beams were tested. Results show that using the equations proposed in this study, the Young’s modulus and Poisson’s ratio of a Timoshenko prism and rod can be determined more directly than those obtained from ASTM C215 and with excellent accuracy.

Embargo Reason

Publication Pending