PSI - Issue 62

Andrea De Flaviis et al. / Procedia Structural Integrity 62 (2024) 871–878 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

873

3

Fig. 1. Generic cross-section with basic notation.

The displacement field of the middle surface is written in Eq. (1) and shown in Fig. 1, where s,y and z are respectively the tangential, transversal and axial direction. The displacement d (s,y,z) of a generic point in the thickness is given, according to Kirchhoff hypothesis, by Eq. (2), where the components can be written as a linear combination of known deformation modes U k (s), V k (s), Ω k (s), W k (s) with unknown amplitude functions φ k (z) for conventional modes , Ψ j (z) for shear modes, as written in Eq. (3), where prime stands for derivative with respect to z. (,) (,) () (,) () (,) () sz usz s vsz s wsz s z y s e e e u    (1)     T z s T z y s syz d syz d syz d syz u yv v w yv , , ( , , ) ( , , ) ( , , ) ( , , )     d (2)

          

K

         V U T T     J K k k K k k k k k k z s V s z U s z 1 1 ( ) ' ( ) ( ) ( ) ( ) ( )  

( , )

u s z





( , )

(3)

v s z



T

T

Ψ ' W Ω

( , )

( ) ( )

w s z

W s

    z

 

j

j

1

1

k

j

From these equations, the infinitesimal strain vector can be written, considering membrane (superscript m) and flexural (superscript f) components, as reported in Eq. (4). Then, considering a linear elastic law, it is possible to write the active plane stress vector, (Eq. (4)). So, the equilibrium equations can be derived through the Principle of Virtual Work or through Hamilton’s Principle (in the most general dynamic case) in a variational way, obtaining the coupled differential equations and boundary conditions reported in Eq. (5). It is worth saying that the boundary conditions are of global/modal nature and not punctual, thus enabling to study problems of free end sections, fixed end sections or simply supported end sections (Basaglia et al. (2011)). This concept will be clearer in the following, where also punctual supports are considered. It is worth noting that, if shear modes are neglected, Ψ = 0 , the first field equation and the first two boundary conditions lead to the classical form of the GBT problem, with only conventional modes. Furthermore, if all terms with two dots are neglected, static equations are obtained. About the terms appearing inside Eq. (5), they are symmetric mass and stiffness matrices reported in Piccardo et al. (2013).