| Introduction | ||
 Strain Gage Rosette on Bracket | In order to determine the load P and its location s, a strain gage rosette is attached to the circular bar near the wall. Each gage is oriented 120o apart as shown. What is known:
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| Question | ||
| What is the load P and distance s? | ||
| Approach | ||
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| Strain Gage Basics | ||
 Basic Linear Strain Gage | It is not possible (currently) to measure stress directly in a structure. However, it is possible to measure strain since it is based on displacement. There are a number of techniques to measure strain but the two more common are extensometers (monitors the distance between two points) and strain gages. Strain gages are constructed from a single wire that is wound back and forth. The gage is attached to the surface of an object with wires in the direction where the strain is to be measured. The electrical resistance in the wires change when they are elongated. Thus, the voltage change in the wires can be collaborated to the change in strain. Most strain gage measurement devices automatically collaborate the voltage change to the strain, so the device output is the actual strain. | |
| Strain Rosette | ||
 Strain Gage Rosette at Arbitrary Angles | Since a single gage can only measure the strain in only a single direction, two gages are needed to determine strain in the εx and εy. However, there is no gage that is capable of measuring shear strain. There is a clever solution to finding shear strain. Three gages are attached to the object in any three different angles. Recall, any rotated normal strain is a function of the coordinate strains, εx, εy and γxy, which are unknown in this case. Thus, if three different gages are all rotated, that will give three equations, with three unknowns, εx, εy and γxy. These equations are,  Any three gages used together at one location on a stressed object is called a strain rosette. | |
| Strain Rosette - 45o | ||
 Strain Gage Rosette at 45o | To increase the accuracy of a strain rosette, large angles are used. A common rosette of three gages is where the gages are separated by 45o, or θa = 0o, or θb = 45o, or θc = 90o. The three equations can then be simplify to  Solving for εx, εy and γxy gives,  | |
| Strain Rosette - 60o | ||
 Strain Gage Rosette at 60o | Similarly, if the angles between the gages are 60o, or θa = 0o, or θb = 60o, or θc = 120o., the unknown strains, for εx, εy and γxy will be,  | |
 Strain Gage Rosette on Bracket | In order to determine the location and weight of a particular hanging load P, a strain rosette is attached to the top of the bracket near the wall (assume strain gages are at the wall). The rosette has three gages, each are 120o apart as shown in the diagram. The bracket material is steel, E = 29,000 ksi and ν = 0.29. When the weight P is placed at a distance s on the rectangular bracket arm, the three gages measure the following strains, εa = 224.8 μ εb = -118.3 μ εc = 132.9 μ From these three strains, the load P and the distance s need to be determined. | |
| Transform Gage Strains to Strain Element | ||
| The first step is to transform the strain gage strains to strains that can be represented in a normal x-y strain element. This can be done by rotating the three unknown strains, εx, εy and γxy. into the three known stains, εa, εb and εc. using the basic strain rotation equation, εx' = (εx + εy)/2 + (εx + εy)/2 cos2θ + γxy/2 sin2θ Applying this equation three times, once for each gage where θa = 0o, or θb = 120o, or θc = 240o, gives,  These simplify to  Substituting the first equation into the second two and solving for εx and εy gives  | ||
 Gage Direction Strains Transformed to x-y Strain Element | Substituting the actual gage values gives the final strains in the x-y coordinate system, εx = 224.8 μ εy = [2(-118.3 μ) + 2(132.9 μ) - 224.8 μ]/3 = -65.2 μ γxy = [-2(-118.3 μ) + 2(132.9 μ)]/1.7321 = 290.0 μ Notice, the y-direction normal strain is not zero due to Poisson's effect even though the stress will be zero. | |
| Stresses at Gage Rosette | ||
 Actual x-y Stresses at Gage Location | Hooke's law can be used to calculate the x-direction stress and the shear stress.  | |
| Bending and Twisting Moment | ||
 Bending Moment (M) and Twisting Moment (T) and | The load P will cause both a bending moment and a twisting moment at the wall where the strain gages are located. The bending moment is Mbending = P (6 in) This moment will cause a bending stress at the top of the circular bar at the gage location.  P = 45.0 lb | |
| The twisting moment is Ttwisting = Ps This torque will cause a shear stress in the circular bar.  s = 6.0 in | ||
 Strain Gages on Hydraulic Chair Piston | Example | |
| A hydraulic piston is used to move a dental chair up and down. To help assist in designing the piston, three strain gages have been attached directly to the piston as shown in the diagram on the left. When the chair is raised, the strain gages give strains as εa = 80×10-6 m/m εb = 60×10-6 m/m εc = 20×10-6 m/m (1) Determine the principal strains and the principal strain directions for the given set of strains. (2) Compute the strain in a direction -60° (clockwise) with the x axis. | ||
| Solution | ||
The first step is to transform the strains to the standard x-y coordinate directions. This can be done by rotating the three unknown strains, εx, εy and γxy, into the three known stains εa, εb and εc. Using the equations for a 45° strain rosette, these strains are, Solving for εx, εy and γxy gives, εx = εa εy = εc γxy = 2εb - (εa + εc) Substituting the actual strains into the equations gives, εx = 80×10-6 m/m εy = 20×10-6 m/m γxy = 20×10-6 rad | ||
 Mohr's Circle for Principal Strains | To determine the principal strains, the strains are plotted on a Mohr's circle as shown in the diagram on the left. The coordinates of point A are εx = 80 and γxy/2 = 10 and the coordinates of B are εy = 20 and -γxy/2 = -10. The x axis is represented by the radius CA, and the y axis by the radius CB. Radius of the circle is R = ((80 - 50)2 + (10)2)1/2 = 31.62 The principal strains are, ε1 = (50 + 31.62)×10-6 = 81.62 ×10-6 m/m ε2 = (50 - 31.62)×10-6 = 18.38 ×10-6 m/m The angle between the maximum strain axis and the x axis is, θ = (1/2) (∠ACD) = (1/2) tan-1(10/30) = 9.217 ° | |
 Mohr's Circle for 60° Rotation | To compute the strains in a new direction, Mohr's circle can again be used. For strains in a direction 60° clockwise with the x axis, rotate the x axis two times 60° (i.e., 120°) as shown with line FG. The strain at that direction are, εx* = {50 - 31.62 cos(180 - 120 - 18.43)}×10-6 = (50 - 23.66) ×10-6 m/m = 26.34 ×10-6 m/m To compute the strain in a direction 90° with this line, it can be rotated two times 90° or 180°, which is diametrically opposite to it. εy* = {50 + 31.62 cos(180 - 120 - 18.43)}×10-6 = (50 + 23.66) ×10-6 m/m = 73.66 ×10-6 m/m Corresponding shear strain is, γxy* = 2 {31.62 sin(180 - 120 - 18.43)}×10-6 = 41.96×10-6 rad The new strains are represented as point F and G on the diagram at the left. | |
