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Theory of Linear Elasticity
Published in Irving H. Shames, Clive L. Dym, Energy and Finite Element Methods in Structural Mechanics, 2018
Irving H. Shames, Clive L. Dym
Now, employing the Airy stress function, we have () Tx(v)=∂2Φ∂y2cos(v,x)−∂2Φ∂x∂ycos(v,y)Ty(v)=−∂2Φ∂x∂ycos(v,x)−∂2Φ∂x2cos(v,y)
Thermal Stresses In Circular Cylinders
Published in Naotake Noda, Richard B. Hetnarski, Yoshinobu Tanigawa, Thermal Stresses, 2018
Naotake Noda, Richard B. Hetnarski, Yoshinobu Tanigawa
The isothermal problem can be solved with the Airy stress function χ¯¯ which is the thermal stress function for the isothermal problem. Taking the term with respect to the temperature change in Eq. (6.34) as zero, the Airy stress function χ¯¯ is () χ¯¯=C¯¯1+C¯¯2lnr+C¯¯3r2
Vibration of Shells
Published in Henry R. Busby, George H. Staab, Structural Dynamics, 2017
Henry R. Busby, George H. Staab
Assuming no loading in the x and y directions results in px = py = 0. In addition, we assume that the inertial forces in the x and y directions are small and can be neglected. Therefore, we have that ρh(∂2u/∂t2) ≈ ρh(∂2v/∂t2) ≈ 0. With these assumptions, we can introduce a stress function F (analogous to the Airy stress function in elasticity) defined by Nx=1R2∂2F∂θ2,Nθ=∂2F∂x2,Nxθ=Nθx=−1R∂2F∂x∂θ
A closed-form hydro-mechanical solution for deep tunnels in elastic anisotropic rock
Published in European Journal of Environmental and Civil Engineering, 2018
Nam-Hung Tran, Duc-Phi Do, Dashnor Hoxha
The Airy stress function must satisfy the equation which can be obtained by substituting the Equations (2), (3), (5) and (9) in Equation (4). In the general context of a coupled hydro-mechanical problem, this equation can be deduced as follows (see Bobet, 2011):