Page - 787 - in Book of Full Papers - Symposium Hydro Engineering

validity of such supposition for gravity dams models will be further assessed. For a simply supported beam, the mode shape has the form [2]: 𝜙 (𝑥 ) = 𝑋 ∙ 𝑠 𝑖 𝑛 ( 𝑚 ∙𝜋 ∙𝑥 𝐿 ) [8] Where 𝐿 is the length of the beam; 𝑚 = 1,2,3,… is a natural number that indicates the vibration mode; and 𝑋 is a constant for generalization purposes. Substituting Eq. [8] in Eq. [6], one arrives in Eq. [9], after arithmetical manipulations: ( 𝑚 ∙𝜋 𝐿 ) 4 − 𝑎 4 − 𝑎 4 ∙ 𝑟 2 ∙ ( 𝑚 ∙𝜋 𝐿 ) 2 ∙ (1 + 𝐸 𝑘 ′𝐺 )+ 𝑎 4 ∙ 𝑟 2 ∙ (𝑎 4 ∙ 𝑟 2 ∙ 𝐸 𝑘 ′𝐺 ) = 0 [9] Following Pedroso [4], for the first vibration modes the last term of the sum in the left-hand side of Eq. [9] can be disregarded since it can be proven to be very small when compared to the third term. With this consideration, Eq. [9] can be simplified to Eq. [10]: 𝑎 4 = 𝛽 4 ∙ 1 1+𝑟 2∙𝛽 2∙(1+𝛾 ′) [10] Where the terms 𝛽 and 𝛾 ′ are defined as: 𝛽 = 𝑚 ∙𝜋 𝐿 [11] 𝛾 ′ = 𝐸 𝑘 ′∙𝐺 [12] From Eq. [10] it is possible to define 𝜏 as the correction factor for shear deformation and rotational inertia of the cross-section effects: 𝜏 = 1 1+𝑟 2∙𝛽 2∙(1+𝛾 ′) = 1 1+𝑟 2∙𝛽 2+𝑟 2∙𝛽 2∙𝛾 ′ [13] With Eq. [7], [10] and [13] one arrives at the Eq. [14]: 𝜔 𝑚 𝑓 +𝑠 +𝑟 = ( 𝑚 ∙𝜋 𝐿 ) 2 ∙ √ 𝐸 𝐼 ?̅? ∙𝐿 4 ∙ 𝜏 [13] With 𝑚 = 1,2,3,… defining the vibration mode considered. Eq. [13] defines the natural frequency of a deep beam considering the flexural, shear deformation, and rotational inertia of the cross-section effects. It is interesting to observe the resemblance of Eq. [13] to the classical natural frequencies equation of Euler’s beam: the only difference is the correction factor 𝜏 . It is important to note that Eq. [13] is valid for a beam with constant properties along its length and only for its first vibration modes. 787