Beam balance theory

Figure 1 shows a diagram illustrating the theory of the balance. Following Glazebrook (19) we have the following equations for a beam type of balance. 

As mentioned above, when the transverse dimensions of the beam are of the same order of magnitude as the length, the simple beam theory must be corrected to introduce the effects of the shear stresses, deformations, and rotary inertia. The theory becomes inadequate for the high frequency modes and for highly anisotropic materials, where large errors can be produced by neglecting shear deformations. This problem was addressed by Timoshenko et al. (7) for the elastic case starting from the balance equations of the respective moments and transverse forces on a beam element. Here the main lines of Timoshenko et al. s approach are followed to solve the viscoelastic counterpart problem. 

O Equation 24.41 is a set of differential equations with coupled shear and peel stresses. In the above equations, kid and Ad reflect the extension, bending, and extension-bending coupling effects of the adherends Ad indicates the unbalanced condition of the substrates that couples the shear and peel stresses Ad denote the shear stiffness. The shear and peel stresses become decoupled when Ad = 0> which is seen in the case of balanced joints with identical substrates. If Ad = 0,0 Eq. 24.45 becomes the governing equations based on the Euler beam theory. When the shear and peel stresses are solved analytically fromO Eq. 24.45, forces and displacements can be derived fromO Eqs. 24.7 and 24.44, respectively. 

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