This paper describes a laboratory experiment for illustrating basic Strength of Materials concepts. This experiment was introduced in the Mechanical Engineering Technology Applied Strength of Materials course at Purdue University. The experiment illu strates combined normal stress due to bending and axial loading. This is an excellent tool for communicating basic Mechanics of Materials concepts while also stimulating interest and visual understanding. The experiment was developed for undergraduate e ngineering technology students but it is equally valuable for engineering students.

The purpose of this experiment is to illustrate the basic strength of materials concepts, specifically combined (i.e. superimposed) axial and flexural normal stresses. This combined normal stress experiment uses a custom built C-clamp to illustrate co mbined axial normal stress and bending normal stress in a T-shaped cross section. The first step in the experiment is to gather strain data at the tension and compression sides of the clamp-body as the clamping force is increased. The next step is to me asure cross-sectional dimensions and calculate the centroid location and moment of inertia. Then using Hooke’s Law, the flexure formula, and the axial stress equations, plus the experimental strain data and area properties, the students are systematicall y led through a process to find the clamping force. All the information necessary for replicating this experiment is provided in this paper--including descriptive diagrams and step-by-step instructions.

The following equipment and supplies are needed to perform this experiment:

• Custom made steel C-clamp with strain gages mounted on the back member. (See Figures 1 and 2.)
• Two strain indicators for displaying the strain from the strain gages.
• 1 in. micrometer, 0.001 in. graduations.
• Dial calipers, 0.001 in. graduations.
• 36 in. steel rule.

Figure 1. The C-clamp.

Figure 2. The C-clamp cross-section at line AA.

The following section is an outline of the experimental procedure.

1. Calibrate the strain indicators. One strain indicator will be attached to the strain gage on the tension side of the clamp-body. The other will be connected to the compression side.
2. Turn the C-clamp hand-wheel in ¼-turn increments recording the tensile and compressive strain values on the data sheet. Do not turn the hand-wheel more than 2 full turns.
   BE CAREFUL NOT TO PHYSICALLY DAMAGE THE STRAIN GAGES!
3. Measure the cross-section of the C-clamp at the strain gages. Record this information on the data sheet provided. (See Attachment I.)

The following outline provides a step-by-step procedure for analyzing the data and ultimately determining the incremental clamping forces. A review of the basic strength of materials principles is included.

1. Before calculating the clamping force, some cross-sectional area properties must be determined. Figure 3 shows the C-clamp cross section at line AA.


Figure 3. Cross-sectional dimensions.

2. This cross-section is a composite area and must be broken into two parts, area 1 and area 2. The centroid location, , can be found from the following equation [1,2]:
   
   

   (1)

   where:
   
   = location of centroid for the total composite area
   = location of centroid for area 1
   = location of centroid for area 2
   A₁ = area 1
   A₂ = area 2
   
   
   Calculate the centroid location, , using equation (1) and record this value on the data sheet in Attachment I.
3. The moment of inertia, I, can be found from the following equation [1,2]:
   
   

   (2)

   where:

   I = moment of inertia for the total composite area
   = moment of inertia for area 1 with respect to the centroid of area 1
   = moment of inertia for area 2 with respect to the centroid of area 2
   A₁ = area 1
   A₂ = area 2
   d₁ = distance from centroid of area 1 to centroid of the total composite area
   d₂ = distance from centroid of area 2 to centroid of the total composite area
   
   
   Calculate the moment of inertia, I, using equation (2) and record this value on the data sheet.
   
4. Next the internal forces in the vertical member of the clamp-body must be found. The vertical member experiences both axial tension and flexural bending. Figure 4a is a Free Body Diagram (FBD) of the clamp-body. Figure 4b is a FBD of the clamp-body upper half cut at line AA.


Figure 4. Free Body Diagram of the C-clamp.

 

The variables are defined as follows:

F = external clamping force
P = internal axial load
M = internal bending moment
d = distance from the clamping force to the centroid of the cross-section


Performing a static force analysis on the FBD gives the following:

5. The internal force P causes axial normal stress equal to the following [1,2]:

(3)

where:

s_P = axial normal stress
P = internal axial load
F = external clamping force
A = total cross-sectional area at the location of the stress

6. The internal bending moment, M, causes flexural normal stress equal to the following [1,2]:

(4)

where:

s_M = flexural normal stress
M = internal bending moment
F = external clamping force
c = distance from the centroid to the extreme fiber
d = distance from the clamping force to the centroid of the cross-section
I = moment of inertia of the cross-sectional area

7. These stresses in equations (3) and (4) are additive since they are of the same type, act in same direction, and act over the same cross-sectional area. The stress on the tension side of the clamp is:

(5)

where:

s_T = total normal stress on the tension side of vertical member


The stress on the compression side of the clamp is:

(6)

where:

s_C = total normal stress on the compression side of vertical member


8. Substituting equations (3) and (4) into equations (5) and (6) respectively gives the following equations.

and

(7)

where:

c_T = distance from the centroid to the tension side surface
c_C = distance from the centroid to the compression side surface


Both c_T and c_C are shown in Attachment I.

9. Knowing that Hooke’s Law is s =Ee and substituting it into equations (7) gives:

and

(8)

where:

E = the modulus of elasticity, which is 30x10⁶ psi for steel
e_T = strain on the tension side surface
e_C = strain on the compression side surface

10. Rearranging these equations (8) and solving for the external clamping force gives:

and

(9)

where:

F_T = calculated external clamping force using the tensile strain
F_C = calculated external clamping force using the compressive strain

From equations (9) determine the clamping forces using the tensile strain values and record these on the data sheet. Then calculate the clamping forces again using the compressive strain values. The clamping force values should be approximately equal using both methods.

The calculated clamping forces are very accurate using the procedure described in this paper. The experimental tensile and compressive strains produce calculated clamping forces that differ by only ± 2% from 0-2 turns. By using a load cell, it was determined that the clamping force at 2 turns is approximately 125 lbs. It was also determined that the experimentally calculated forces differ from the actual forces by only ± 2% from 0-2 turns.