# Paper Stress Concentrations

### Model Description

This is a simple demonstration of the basic principles and behavior underlying stress concentrations.  Stresses at points of load application can be much larger than the average stress in a member. The same is true when a member contains a discontinuity. Paper members with the same mid-span width are used to demonstrate the stress concentrations that can occur at load points as well as at discontinuities: holes and fillets.  This demonstration should take 8-10 minutes.

### Engineering Principle

The stress concentration factor (K) is defined for a particular cross-section as

$K = \frac{\sigma_{max}}{\sigma_{avg}}$

the maximum stress over the average stress. The maximum stress typically occurs near a discontinuity. The experimental results obtained for K are independent of the size of the member and the material used; they depend only upon the ratios of the geometric parameters involved (i.e., diameter of the circle, radius of fillet, or remaining width of cross-section).

### What You Need

Item Quantity Description/Clarification
Sheet of paper members 2 sheets per student Cut out the shapes. Two sheets/shapes per student. There are four shapes. Shape 3 has three (multiple)  different configurations.  Get some help if you have a big class.
Scissors 1 It important not to create stress concentrations when cutting out the members (i.e., do not create notches when cutting out the shapes), especially the third member with the square corners.
Single hole punch 1 Used to put the hole in Member 2.

### How It’s Done

Before Class: Cut out the members so that each student has at least one of each of the four paper member configurations (figure below). Note that the cross section at the middle of each member is the same (in this case 1 inch). Member 3 has three configurations – no fillet, a circular fillet and a larger/possible non-circular fillet. You decide if they have one of each – the reason for two sheets of members – per student. Also many students will place impact loads on the members, so they will need replacement members when this occurs.