Shells - The differential equation

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A shell element in equilibrium

Equilbrium

Above an infinite small shell elements is illustrated.

By the demanding equilibrium i the x-direction, the following equation is obtained:


Equilibrium in y-direction gives the equivalent:


Traction forces can be neglected, and the shear forces are all the same, which gives us:


Kinematic compatibility

The strains in the material are given as:


We then differentiate twice with respect to x and y:


The two first equations in (5) put in the last equation gives:


This equation expresses the kinematic compatibility requirement. By applying Hooke's law we get the relationship between stress and strain, which gives us:


This equation contains the kinematic compatibility demand expressed with the stresses.

The shells differential equation can be deduced from this compatibility equation and the equilibrium equations. This deduction is complicated, and the differential equation is in practice impossible to solve. We have to take another approach as we will see below.

Airy's stress function

Airy introduced the so called Airy's stress funtion. This is defined so that it satisfies the following conditions:


By introducing this function, a differential equation can easily be established. Again, we neglect traction forces in the following deducations. Then, this function will automaticly satisfy the equilibrium equations, (3), as we can see here:


We see that this is satisfied no matter what he stress function is.

We then insert the stress function in the compatibility equation, (7), which gives:


Some rearrangement gives:


We now introduce the differential operator:


We then get the equation:


This is the Shell's differential equation, which is a 4. order, homogeneous differential equation. The functions satisfying this equation are biharmonic funtions, also containing hyperbolic trigonometric functions. In addition, the solution has to satisfy the boundary conditions