MEC3455 Solid Mechanics

MEC3455Solid MechanicsTopic 2 Airy Stress FunctionsDr Bernard ChenClayton Campus Building 31/121

Prof. Soh Ai Kah Monash University Sunway Campus

ObjectivesDerivation of the biharmonic P.D.E and the Airy stress function as a solution (biharmonic function)Equations of equilibriumCompatibility equationsBoundary conditionsExample questionsIntroduction

Mapping of this topicStructure under loadingCompatibility equationsi.e. strain-displacement relationshipsMapping of this topicThis is the equation that we need to satisfy, for an Airy stress function to exist!The unique Airy stress function (solution) depends on boundary conditions (loadings on the structure)Consider a small element of dimensions dx, dy and with unity thickness (thickness of 1)

ydyxdx

Equations of equilibriumForce balance equations for the element, with body force included.

Equations of equilibriumEqn (1) Eqn (1) reduces to;

Eqn (2)Eqn (2) are the Equations of Equilibrium for a 2-dimensional problem (plane stress)Equations of equilibrium

Normal movement due to normal stressRotational movement due to shear stress distortionCompatibility equations9Using the engineering definitions of strain;

Normal strainsShear strainEqn (3a)Eqn (3b)Eqn (3c)Compatibility equations

Eqn (4)Compatibility equationsHow do we relate the Equation of Equilibrium to the Equation of Compatibility?i.e. stress to strain?Hookes Law!Since this is a 2D problem (plane), we will use the 2D Hookes Law. Also, with Hookes Law, we are assuming that the material is in its elastic region (no plastic deformation).

Eqn (5)Substituting Hookes law (Eqn (5)) into the Equation of Compatibility, Eqn (4) gives

Eqn (6)How do we relate the Equation of Equilibrium to the Equation of Compatibility?i.e. stress to strain?

Eqn (7)By substituting Eqn (7) into Eqn (6) (both equations with stress terms only), this gives us

Eqn (8)How do we relate the Equation of Equilibrium to the Equation of Compatibility?i.e. stress to strain?The stresses can be written in terms of the Airy stress function as:

How do we relate the Equation of Equilibrium to the Equation of Compatibility?Eqn (9)

Eqn (10)How do we relate the Equation of Equilibrium to the Equation of Compatibility?

Eqn (10)How do we relate the Equation of Equilibrium to the Equation of Compatibility?

Example 1MEC3455-Airy Stress Function19Example 2For the uniformly loaded cantilever beam shown below, the compatible stress field was found to be

Verify that this stress field satisfies equilibrium.

MEC3455-Airy Stress Function20Example 3Consider a thin cantilever loaded as shown in the below figure. Assume that the bending stress is expressed by

and z = xz = yz = 0. Determine the stress components y and xy as functions of x and y.