MECH 396/398: Combined Loading of Straight and Curved Beams
Department of Mechanical and Materials Engineering
1
Queen’s University
Faculty of Applied Science
Department of Mechanical and Materials Engineering
COMBINED LOADING OF STRAIGHT AND CURVED BEAMS
Location Jackson Hall Rm 213
Objectives
- To calculate analytically the stress/strain distribution for combined loading conditions of two curved beam specimens.
2.To experimentally observe the stress/strain distributions in the two loaded beams by using:
- strain gauges
- photo-elastic analysis
Safety Considerations
This laboratory is conducted without the use of obviously dangerous equipment, but it does require the manipulation and suspension of relatively heavy metal weights at a significant height. Care must be taken to handle the weights in such a way that they do not fall and cause injury.
Preparation Notes
In this lab you will need to sketch the coloured photoelastic patterns seen in loaded beams. You should bring plain paper and a few coloured pencils, and/or a digital camera.
Background
Strain Gauges
The most common method of measuring mechanical strain in an object is with strain gauges. The basic principle underlying the operation of strain gauges is that the resistivity of wire changes when it undergoes mechanical strain. If the resistance element is attached directly to the object being strained it will undergo an equal strain and in this way, the measured change in resistance can be correlated to the strain in the object.
Stress Distribution in a Straight Beam
When a straight beam is loaded in tension or in compression, the stress in any cross-section of the beam is
where P = load
A = cross-sectional area
When a straight beam is loaded in bending, the stress at any point in the beam is determined by
where y = distance across the cross-section
M = bending moment at that location along the beam
Izz = moment of inertia perpendicular to the cross-sectional area
According to the principle of superposition, the total stress acting in a certain direction on a beam is the sum of applied stresses acting in that direction [1].
Stress Distribution in a Curved Beam
For the case where the beam is loaded only in tension or compression the stress components are:
sin
where
t = thickness
a, b, r = shown in Figure 1.
Figure 1: Dimensions of general curved beam and geometry of loading
In the case where the beam is loaded in a pure bending moment the stresses are:
where
Photo-elastic Analysis
Experimental stress analysis is used not only to find the magnitude of stresses. Often it is important to be able to interpret the entire stress distribution field. Photo-elastic analysis is one method that allows us to observe the stresses over an entire component. Photo-elastic analysis can also be an important design tool since it can yield valuable information on how to optimize the design and reduce stress concentrations.
The basic principles underlying photo-elastic analysis are relatively simple. Certain materials, notably plastics, behave isotropically when unstressed but become optically anisotropic when stressed. The index of refraction is dependent on the stress applied. When a polarized beam of light passes through a photo-elastic coating on a part subjected to stress it is split into waves traveling at different speeds. Once the waves have excited the coating, they will be put out of phase. This phase difference is called the retardation. Observing the stressed component with a polariscope, an interference pattern can be seen. This pattern indicates the stress distribution over the component.
If the strains along x and y are and and the speed of light in these directions is Vx and Vy then the retardation between the beams is: [2]
Brewster’s Law states that: "The relative change in index of refraction is proportional to the difference of principal strains" and is interpreted numerically by:
where K is called the strain-optical coefficient and is a physical property of the material.
Combining these expressions:
with the first expression being for transmission and the second for reflection. The basic relation for strain measurement with the reflection photo-elastic technique is therefore:
At every point the retardation between the light beams is:
: retardation
: wave length
N: fringe order.
Simplifying:
t: thickness of the coating
K: strain optical coefficient of the coating.
The difference between the principal stresses in the component is:
In order to determine the stress, the fringe order must be determined. When examined under a polariscope, the retardation between the two beams of light increases proportionally with the stress. Every time the retardation is equal to:
a particular colour disappears and is replaced by its complementary colour. The colour sequence is summarized in Table 1. The colours progressively change from the no load condition to the maximum load. The colour sequence is black, yellow, red, blue, yellow, red, green, yellow, red, green... From this the fringe order, N, can be determined. Note that only the magnitude, x-y can be derived, away from boundaries. However, at boundaries one stress will be zero, and therefore the other can be found.
Fringes appear as continuous bands that end at boundaries or occur as continuous loops that do not intersect. An area covered by a single uniform colour indicates that the strain is uniform over that entire area.
Table 1: Photo-elastic pattern colour sequence appearance.
No colour means “no stress”
Blackis a zero fringe
Yellow
Red
1st Fringeis black and labelled as 1
Blue-Green
Yellow
Red
2nd Fringeis black and labelled as 2
Green
Yellow
Red
3rd Fringeis black and labelled as 3
etc.
More Information
More information on strain gauges and photo-elastic technology may be found in reference books [2] and websites [3, 4].
Apparatus
The apparatus used in this lab is as follows:
1.two beam specimens
2.support frame
3.loading apparatus
4.weights
5.strain gauges fixed to the specimens
6.digital strain indicator
7.Model 031 reflection polariscope
There are two specimens; each of them is roughly a C-shape. The smaller specimen has a relatively large-radius continuous curve, Figure 2. The larger one has two small-radius curves and a straight section in the middle, Figure 3. For purposes of this lab, the larger one is referred to as the straight beam. Stress distributions within the straight section may be calculated using straight-beam equations, for effects of a force P acting with a moment arm d from any point in the cross-section of the straight section.
Figure 2: Schematic of curved beam specimen. Dimensions in inches.
Material properties are given in Tables 2 and 3.
Table 2: Specifications for specimen material
Photo-elastic Properties
t 0.116 ± 0.002 in
K 0.15
f 655 x 10-6 [estimated – may be verified by experiment]
22.7 x 10-6 in
Mechanical Properties for 6061 Aluminum
Tensile Strength32,000 psi
Yield Strength 28,000 psi
Shear Strength 22,000 psi
Young’s Modulus 10,000,000 psi
Shear Modulus 3,790,000 psi
Poisson’s Ratio 0.33
Figure 3: Schematic of beam with straight section. Dimensions in cm.
Procedure
Part A: Strain Gauge Analysis
1.Make a sketch of the apparatus. Record all appropriate dimensions and identify the locations of the strain gauges as precisely as possible.
2.Ensure that the strain gauges are properly connected and zeroed under no load conditions.
3.For four loads to be specified by the TA, record the strain readings for each specimen.
Part B: Photo-elastic analysis
For one of the loads from Part A and both specimens:
1.Make a sketch of the apparatus.
2.Sketch the photo-elastic patterns.
3.Identify the fringe orders (N) at the strain gauge locations.
Results
Part A: Strain Gauge Analysis
- From the experimental results determine the location of the neutral axis for both beams.
- Plot the strain distribution across both specimens. Is this linear? why or why not?
- From the axial strain readings, determine the axial stress values and plot these with respect to their location on the components.
Part B: Photo-elastic Analysis
1.On your sketch of the photo-elastic pattern indicate the whole value fringe orders (changes from red to green), and the fringe orders at each strain gauge location.
2.Calculate the maximum shear stress at each gauge location.
Before leaving the laboratory, your results must be checked and verified (signed) by the TA. These must be included as an appendix in the final report.
Report
An individual report is due one week after completion of this laboratory. For this particular laboratory, students should ensure that answers to the following Supplemental Questions are included in appropriate sections of their reports.
Supplemental Questions
- Do your experimental results for the strain distribution agree with the theoretical values? Why or why not?
2.Perform an uncertainty analysis on your theory and results. Note that for the theoretical component it is only necessary to perform the analysis on the straight beam as it is very tedious for the non-linear case.
3.Determine the loading that would produce failure of the components.
- Explain the relative advantages and disadvantages of reflection and transmission photo-elasticity.
- Briefly describe how you would select a photo-elastic coating.
- From your sketches of the photo-elastic patterns, where are the areas of stress concentrations?
References
- R.C. Hibbeler, Mechanics of Materials, 3rd edition, Prentice Hall, 1997.
- J.W. Dally, W.F. Riley, Experimental Stress Analysis, 3rd Edition, McGraw Hill, 1991.
- “Strain Gages and Accessories”, Vishay Measurements Group,
- “Optical Measurement and Analysis of Stresses/Strains in Test Parts and Structures”, Vishay Measurements Group,
- “Introduction to Photoelasticity”, University of Cambridge, Department of Materials Science and Metallurgy,
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