General Mechanical

General Mechanical

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Problem about Euler beam

    • Yohan Ahn

      Hello, I'm leaving a post because I have a question while solving a simple cantliever beam problem. The problem is that the natural frequency calculated at Ansys is less than the theoretical value.

      I learned Euler's beam theory, and I was able to get a characteristic equation of a beam that was fixed at one end and free at the other.

      < cos(beta*L)*cosh(beta*L)=-1 >

      Calculated using the Newton-Raphson method, the value of beta*L was obtained as follows.

      < beta1*L=1.8751 >, < beta2*L=4.6941 >, < beta3*L=7.8547575 > ....

      When the following properties are applied, the natural frequency value is calculated as follows.

      < Properties, Structural Steel: Young's modulus of 200 GPa and density of 7850 kg/m^3

      Cross-Section: Rectangular, Length: 0.3m, Wide: 0.03m and Thickness: 0.005m >

      Natural frequencies: <45.2987>, <283.8846>, <794.8821>, ....


      Ansys model results using beam4 (Number of elements: 40)

      Natural frequecies: <45.296>, <283.78>, <794.17>, ...


      I learned in class that the exact solution at the natural frequency is always the same or less than the approximation. Ansys calculates using approximate solutions, so it must be less than exact solution, so I wonder why this is the result.
      Are there any additional tools that are reflected on the Ansys? This difference is confirmed that the error increases as the thickness of the beam increases, and the above result is a small difference.
      cf) Additionally, I wonder if Ansys uses Timoshenko Beam theory for the basic analysis setting of beam modeling.


      Thank you

    • Nanda Veralla
      Ansys Employee


      Hello Yohan,

      Yes, you are correct. Both beam elements BEAM188 & BEAM189 elements are based on Timoshenko beam theory which includes shear-deformation effects. The elements can account for unrestrained warping and restrained warping of cross-sections.

      The amount of error generally depends on approximations we make in our model. Can you check with both beam elements and check what’s the difference in error. The amount of error you are having is acceptable and doesn’t need to be worried about.

      For more information on beam elements, kindly refer to below links:

      5.3. Beam and Link (Truss) Elements (

      BEAM189 (

      BEAM188 (



      Guidelines for Posting on Ansys Learning Forum

      How to access ANSYS help links



    • Yohan Ahn

      Thank you, Nanda, for your quick and kind reply.

      Currently, the link you attached with my academy license is not accessible, but I checked the information about BEAM4 and BEAM188 you wanted to talk about using Ansys Help. As you answered, very small errors may occur in numerical analysis, etc., and I think it is within an acceptable range.
      Lastly, can you check the command using beam 4 that I used in the above problem? The picture below is the command used for the above problem.


      Thank you! :)

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