University of New Mexico

Civil Engineering Department

Civil Engineering Materials Laboratory, CE 305L

NON-DESTRUCTIVE TEST OF A HOOKEAN ELEMENT (SPRING)

General

This experiment investigates the response of typical elastic material under axial compression forces by relating the deformation of a helical spring to the forces applied. The relations observed are analyzed to confirm some of the elastic properties of steel, e.g., a spring constant, work (strain energy), and hysteresis.

An ideal linear elastic spring has a load ( F ) deformation ( δ ) relationship as follows:

(1)

where k is the spring constant. It can be shown, e.g., in Levinson (Ref. 1), that the deformation and load are related through geometry and a material property of the spring as follows:

(2)

where R is the mean radius of the coil, d is the diameter of the solid circular wire, n is the number of coils, and G is the shear modulus of the material. Figure 1 shows the described geometric terms. Combining the above two equations yields a relation for the theoretical spring constant, viz.,

(3)

Levinson also shows that the maximum shear stress that occurs in the spring is equal to the following:

(4)

Note that the mean radius can be computed from the following:

(5)

where D_M is the inner diameter of the coil (i.e., the mandril diameter).

Definitions

•  Spring Constant ( k ): also called stiffness. Relationship between load and deformation, or slope (as defined above in equation 1)

•  Work ( W ): mathematically equivalent to strain energy. Graphically this is equivalent to the area under the load deformation curve. For P  =  f ( δ), then the following integral represents this area under the curve (or work):

(6)

Numerically this integral can be evaluated using a trapezoidal integration technique per the following equation:

(7)

•  Hysteresis ( H ): energy loss, or initial energy minus recovered energy. This may be determined graphically by observing the difference between the plotted curves of load and deformation during the loading and unloading phases. It can be computed, again, using trapezoidal integration as the difference in areas under the load curve and the unload curve, i.e.,

(8)

Objectives

Observe the load-deformation behavior of an elastic steel spring. Compute the experimental and theoretical values of the spring constant and ascertain the percent error between these values.

Equipment

1. Universal Testing Machine (UTM) with applicable loading head

2. Mechanical Dial Gage (1 inch range) with magnetic stand

3. Calipers

4. Tape Measure

Specimen

1. Steel helical spring (see photo)

Procedure

Download the prodecures for Non-Destructive Test of a Hookean Element here

1. Carefully measure the geometry of the spring (height ( H ), inside diameter ( D_M), wire diameter ( d )), in inches. Carefully count the number of coils on the spring. Calculate the mean radius ( R ). Record these values.

•  Calculate the theoretical spring constant k . Record this value.

3. Calculate the maximum shear stress and maximum deflection that will be generated in the spring for the maximum test load of 8000 lb. Record these values.

4. Place the spring in the UTM and lower the crosshead until it just makes intimate contact with the top of the spring. Place the dial gage between the lower platen and crosshead and adjust to a reading of 0.000 inch. Select the lowest load range on the UTM that will accommodate a maximum load of 8000 lb. Zero the UTM load output.

5. While “slowly” loading the spring in compression, record both the load and spring deflection simultaneously at increments of 1000 pounds to a maximum of 8000 pounds (i.e., 0, 1000, 2000, etc.). Remember that (0, 0) is a data point, too.

6. Record the unloading data in similar fashion from 8000 pounds to zero pounds in 1000 pound decrements (7000, 6000, etc.). Note that when you reach zero load, the dial gage may not read zero displacement (i.e., there maybe an offset at the conclusion of the test).

Required

•  Report all the spring geometry as recorded in the procedure above for the specimen tested.

•  Tabulate the deflection ( δ ) – load ( P ) data. Plot this load vs. deflection data (see Figure 2 for an example).

•  Determine the experimental spring constant as the ratio of the 8000 pound load to the deflection measured at the 8000 pound load.

•  Using trapezoidal integration, compute the work expended in compressing the spring to 8000 pounds ( W_load), and the energy recovered during unloading ( W_unload)

•  Calculate the hysteresis. This should be a positive value.

•  Report all the valudes ( k , δ_max , etc.) and compute percent errors were applicable.

References

1. Levinson, I.J. Machine Design , Reston, 1978.

Figure 1. Forces in Helical Spring (Levinson).

Figure 2. Typical Spring Force vs. Displacement Curve.