Introduction to Elasticity/Polynomial solutions
Using the Airy Stress Function : Polynomial Solutions
Find the problem which fits this solution.
This is a homogeneous stress field. An infinite number of problems can satisfy these conditions.
Consider a cantilevered beam that is fixed at one end and has a vertical force F applied at the free end.
The boundary conditions on the beam are
We will use Maple to solve the problem.
First, assume a polynomial Airy stress function that has a high enough order. In this case a fourth order polynomial will suffice
Take the derivatives of the stress function to obtain the expressions for the stresses.
Next, use the command
We now find the tractions on as
On , we have
The stress function is order 4, so the stresses are order 2 in x and y. The tractions on or might therefore be polynomials in of order 2.
We calculate the coefficients of each power of x in these expressions as
The biharmonic equation is 4th order, so applying it to a 4th order polynomial generates a constant. And this constant must be equal to zero.
We also calculate the three force resultants on x=0 by integrating over y:
We now solve for the constants so as to satisfy (i) the strong boundary conditions, (ii) the biharmonic equation and (iii) the weak boundary conditions.
Notice that there are more equations than there are constants. Some of the equations are not linearly independent. However, Maple can handle this if there is a solution.
Substitute the solution into the original stress function and calculate the final stresses.
Displacement Boundary Condition
The displacement potential function must satisfy the relations and .
In this problem,
which means that
These can be integrated to find and in terms of , and constants. The constants can be determined from the displacement BCs applied so as to fix rigid body motion.
The displacements are given by
where , and = thickness of the beam.
General Approach For Beam Problems
- Find the highest order polynomial terms and for the normal and shear tractions on .
- Use a polynomial of order max() excluding constant and linear terms. For example, for a polynomial of order
- Substitute () into the biharmonic equation to get a set of constraint equations. Also compute the stresses.
- Apply boundary conditions to obtain the tractions at the boundary.
- For the strong BCs, find the coefficients of powers of x and y and equate with expressions for the tractions.
- For the weak BCs, find algebraic expressions.
- Solve the set of equations and back-substitute.