University of Mississippi  Department of Civil Engineering 
CE407: Civil Engineering Laboratory II 

Experiment S3: Stress Analysis of a Circular Ring-Numerical Experiments
Dr. Chris L. Mullen
Last Updated: 10-31-01

1.2 The objective is to use the finite element and photoelastic methods to display the principal stress distribution throughout the ring. Specifically, contours of maximum and minimum principal stresses are to be plotted using Patran's Graphical User Interface (GUI), and an X-Y plot is to be made of the normal stresses on two important cross-sections of the ring.. These plots are to be used to compare with hand sketches of the photoelastic images of a frozen-stress specimen displayed using the Teaching Polariscope (see Experiment 9).
2.0 THEORY
2.1 Definitions.

Plane Stress -special case of a planar continuum which is subject to zero out-of-plane stress. Thus, there are only three nonzero stress components: the two in-plane normal stresses, SX and SY, and the one in-plane shear stress, TXY.
Hooke's Law - strains are related to stresses through this law for an assumed linear elastic homogeneous isotropic continuum. Applied to plane stress conditions, Hooke's law requires that only strains EX , EY, and EXY, remain nonzero. Two constants of proportionality are required by Hooke's law: Youngs modulus, E, and Poissons ratio, NU.
Plane Stress Element-finite element used to model a 2-D continuum subject to plane stress conditions. In this case, quadrilateral elements will be used exclusively which have four nodes at the corners each of which has two degrees-of -freedom. Definition of the material behavior of the element only requires assignment of the two constants, E and NU.
Contour Plot- Patran will compute the stresses on the interior of the element from the solution of the unconstrained degrees-of-freedom for the entire model subject to applied loading. Contour plots will then be made using interpolation between values of interior stresses in adjoining elements.
Frozen-Stress Specimen- photoelastic specimen which has been preloaded and the fringe patterns made permanent for future viewing with a polariscope. The magnitude of the load can only be inferred from the fringe-pattern and knowledge of the stress-optic coefficient.

Estimate the complete solution of the internal forces and the deformation under the load using the curved beam analysis. Estimate the normal and shear stresses on the cross-section beneath the load and the cross-section 90 deg from the load point.

Compute the solution to the stress analysis problem using the finite element software described below. Take advantage of the double symmetry of the geometry.
3.0 SOFTWARE
The SAP2000 on the computers in the CECGL or PATRAN version 6.0 finite element program available on on the Silicon Graphics Inc. (SGI) machine, Sweetgum, is to be used to construct a model of the first quadrant of the ring. The existing database file, ring0.db, ( 2 x 3 elements) or the ring.db (4 x 9 elements), created as a new file in an interactive session during lab, may be used as a starting point or as a reference.
4.0 NUMERICAL MODELING PROCEDURE
For the PATRAN quarter ring model, use the measured dimensions (in meters, m) of the frozen-stress ring specimen to define the geometry and a unit (P= 1 N) preload force. Assume values for E= 7.24 e10 Pa and NU=0.3 (i.e,. typical glass fiber). The principal model generation steps are:

Generate other nodes on the section using translational transformations under the FE/Trannform/Node/Translation menu.
Define a four-node Quad/Quad4/Standard solid element using the FE/Create/Element menu using four nodes defined in Step 1. Create additional elements along the 90 deg section in the same manner. Generate the rest of the elements on the quadrant by using the FE/Transform/Element/Rotation menu.
Define a linear elastic isotropic material set using the Material/Create menu.
Define a 2D Solid/Plane Stress property set using the Property/Create menu.
Define the symmetry boundary conditions using the Load/BC/Create/Displacement/Nodal menu. Two named sets are required: one for each cross-section at a symmetry plane. Restrain the x-movement under the load and the y-movement at the 90 deg section.
Define the applied load using the Load/BC/Create/Force/Nodal menu. Define a load case using the Load Case/Create menu.
Perform a static analysis using the Analysis/Entire Model/Full Run menu.
After some time, check to see if a results file has been generated and load in results using the Analysis/Read Results menu.
Select results using the Results/Basic menu and choose Select Results Cases/Default menu bar.
Display deformed shape using the Select Deformation Results menu bar.
Display stress contours using the Select Fringe Results/Stress Components menu bar. Choose appropriate Results Quantity (XX implies SXX, XY implies TXY, Major implies S1, Max Shear implies TMAX).

Plot the distribution of M and N along the neutral axis of the ring based on the curved beam analysis.
Compute the deflection of the ring under the load point using the curved beam analysis.
In particular, before you interpret stresses and make any comparisons with your photoelastic ring, check that your results make sense by computing the net reactions and bending moments at each cross-section as well as by checking the deflection under the load against the curved beam results.
After verifying that results are sensible, plot the contours of S1 and S2.
Plot the contour of TMAX and compare with your sketch for the photoelastic specimen.