Lab 2 - Lab 2 on cross-sectional properties PDF

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Lab 2 on cross-sectional properties...

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Department of Civil Engineering Auburn University

Laboratory Report On Cross-Sectional Properties

Submitted to Hayden Yendle

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Group Members

September 18, 2019

Introduction The purpose of this lab was to strengthen students’ knowledge of centroids, moment of inertia, and radius of gyration or the cross-sectional properties of different shapes. These cross-sectional properties help engineers understand what shapes will be stronger under different kinds of forces. In Exercise 5, students measured a W-shaped cross-section (Figure 2) and a single angle cross-section (Figure 3). Students gained practice using a micrometer to measure the thicknesses of these two shapes. Calculations were then performed to determine centroids, moments of inertia, and radii of gyration for the shapes. These calculations were then compared to values given in the AISC Manual which allowed for a percent error to be calculated as well. Exercise 6 had students create different shapes from folding thick paper. Loads were then put on the different shapes and the results were observed. This allowed students to see what happens to different shapes when different loads were applied. This lab was important for students to both calculate cross-sectional properties to see the math behind the strengths of certain shapes, and also to allow students to observe what happens to a shape under different loads.

Results and Discussion of Exercise 5 Exercise 5 has three parts to it where three different shapes were measured, and various calculations were done for analysis. The first shape was given in the lab procedure and can be seen below in Figure 1.

Calculation Ix about centroidal axis Iy about centroidal axis rx ry Figure 1: Part 1 Shape

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Value

AISC Value

Percent Error

94.5 in4

88.6 in4

6.69%

2.40 in4 4.51 in 0.718 in

2.36 in4 4.62 in 0.753 in

1.53% 2.38% 4.65%

The horizontal axis was defined as the x-axis and the dotted line running top to bottom was defined as the y-axis. The origin was defined to be the bottom left corner of the shape. The centroid and the moments of inertia about different axis were calculated. As shown in Table 1 above, the centroid was found to be at (1.50 in., 1.61 in.), Ix was found to be 5.84 in4, Iy was 7.47 in4, and I about a horizontal axis passing through the bottom of the shape was calculated to be 24.9 in4. Examples of these calculations can be found in Appendix B. Since the moment of inertia about the y-axis is greater than the x-axis, it can be concluded that the y-axis is the stronger axis when a bending moment is applied.

Calculation Centroid

Value (1.50 in., 1.61 in.)

Ix about centroidal axis

5.84 in 4

Iy about centroidal axis

7.47 in 4

I about base

24.9 in

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Table 2: Calculations for Part 2 of Exercise 5 Figure 2: W-Shape for Part 2 of Exercise 5

The second part of the exercise was to measure and perform calculations on the crosssection of a W-flange (shape can be seen in figure 2 above along with the measurements that were recorded). The measurements were taken using a micrometer for thickness measurements (accurate to 4 decimals) and a ruler for length measurements (accurate to 1/16 of an in.). When locating the centroid of the shape, the bottom left corner was used as the origin. As seen in Table 2 above, Ix was found to be 94.5 in4, Iy was 2.40 in4, rx was 4.51 in, and ry was found to be 0.718 in. Example calculations for these values can be found in Appendix B. Since the moment of inertia is significantly higher for the x-axis it can be concluded that the x-axis is the stronger axis and significant loads applied about the y-axis could cause the structure to fail. The calculated values were then compared against values found in the AISC manual and percent error was calculated. Shown in Table 2, the percent error was under 7% for all of the calculations which is fairly accurate. There are several variables that could cause this error. The first option is human error, this could be a result of misreading the micrometer or ruler. Another option is the fact that the exact measurements 2

taken do not match what was compared against in the AISC manual. For example, the dimensions measured were rounded up to closest shape in the AISC manual.

Calculation Ix Iy I xy Tan(angle) rx ry rmin

Value

AISC Value

1.307 in

4

0.452 in

4 4

0.440 in 0.423 0.925 in 0.544 in 0.411 in

Percent Error 4

15.1%

4

16.1%

N/A 0.426 0.937 in 0.555 in N/A

N/A 0.8% 1.3% 2.0% N/A

1.54 in

0.539 in

Table 3: Calculations for Part 3 of Exercise 5 Figure 3: Single Angle Shape for Part 3 of Exercise 5

The final part of this exercise was to measure and calculate the cross-sectional properties for a single angle (L shape). Again, the bottom left corner of the shape was taken as the origin for computing the centroid and example calculations for this exercise can be found in Appendix B. Ix was calculated to be 1.307 in4, Iy was 0.452 in4, and Ixy was found to be 0.440 in4. The radius of gyration was also calculated, rx was found to be 0.925 in, ry was 0.544 in, and rmin was 0.411 in. Also calculated was the angle of principle axis, which was 22.92 degrees, yielding a tan(angle) of 0.423. All of the calculations for this portion of the exercise can be found above in Table 3. These calculations were then compared against the known values of general shapes found in the AISC Manual. With these known values, percent error was then calculated. The error for Ix and Iy was found to be a little high at 15 and 16 percent but the rest of the calculations were below a two percent error. Again, the reasons for the error could be from human error, misreading of measuring devices, or from the fact that the measured values were not exact to the values compared to in the AISC Manual (measurements were rounded up).

Results and Discussion for Exercise 6 In exercise 6, students formed various different cross-sections with thick paper and noted how each responded to different stresses. The first step of the exercise required that an

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Figure 4: 11"x1" Piece of Paper

11”x1” piece of paper be placed between two objects and a force applied at the center as shown below in Figure 4.

Next, the 11”x1” strip of paper was rotated, so that the member was now 1” deep and a force applied at the midpoint again. The member was stronger when it was rotated to where it was 1” deep, however in both cases the member was not very strong. In the next step, six sheets of 11”x1” sheets of paper were joined into a single member by a rubber band on either end. This was identical to the test performed in step 1 except the thickness was now six times as much, meaning that the member was stronger than in step 1. Steps 4, 5, and 6 use a member that is 11”x2” and are loaded in different ways. In step 4, the member is oriented and loaded the same as pictured in Figure 4 above and seemed to have about the same amount of strength as the member in step 1. The member was then turned vertically and loaded as a column as shown below in Figure 5. This was not very strong as it bent a large amount with minimal force.

Figure 5: 11"x2" Strip of Paper turned vertically and axially loaded

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In step 6, the same strip was taken and folded along its length to form a 90-degree angle and loaded like a beam (step 4) and as a column (step 5). This shape seemed more stable than when it was un-folded and flat. Steps 7, 8, and 9 used the same 11”x6” strips of paper. In step 7, the strip was held at either end and a torsional force was applied to the strip, which was not very strong. In step 8, the strip was rolled into a 11” long cylinder with an approximate diameter of 2” with the ends being secured with rubber bands. Torque was again applied to this shape which was much stronger than the shape in step 7. The same shape was used in step 8 except the seam was taped on the inside and outside. This extra bondage provided by the tape made the member stronger. For steps 11, 12, 13, and 14, 11” paper strips were used to create one of the shapes pictured below in Figures 6,7, and 8.

In

Figure 6: C-Shaped Beam

Figure 7: I-Shaped Beam

Figure 8: Box Shaped Beam

step 11, each shape shown above (Figures 6,7, and 8) was loaded as a 1” deep beam (majoraxis bending) and also as a 1” wide beam (minor-axis bending). In both loading scenarios, it was concluded that the C-shaped beam performed the worst and that the I-beam and the box resisted the forces very well, but the box was determined as the strongest shape. In step 12, a torsional force was applied to each of the shapes. Again, the C-shaped beam failed first with the I-beam and the box having similar results, both resisting a large amount of torsion. The box had a slightly higher resistance to torsion where it failed somewhat uniformly, while the others failed in the center. Step 13 required that the tape in the corner of the box be cut and then the torsion be reapplied. This made a big difference in the amount of torsion the box could withstand and proves that the seams of the box must be intact in order to provide the most support against torsional forces. In the final step, the box was retaped and each of the shapes were turned vertically and loaded as a column. As with the previous steps, the C5

shaped beam performed the worse. However, the I-beam outperformed the box beam in this test. The I-beam and box beam also easily outperformed the shapes in steps 5 and 6 as well. In conclusion, different shapes perform better under different types of stresses. For example, it was clear to see that the I-beam was the best at resisting deflection when the flange is in the vertical direction. The I-beam also supports the compressive force, the forces in a column, best along with the cylindrical shape. If a torsional force is the issue, then a cylindrical shape is the best fit. It is also evident that the bending moment is greater the further it is away from the centroid due to the fact that moment is equal to force times distance.

References 6

Kajevic, Ben. All photos were taken by Ben. Auburn University, Auburn, AL. Marshall, Assignment Sheet posted to Canvas, Cross-Sectional Properties Assignment. Auburn University, Auburn, AL. Ramey, G. Ed. (1993). Mechanics of Solids and Structures Laboratory Manual. Auburn University, Auburn, AL. Steel Construction Manual. American Institute of Steel Construction. 15th Edition. United States of America. May 2017. Yendel, Hayden (September 11th, 2019). Blackboard Instructions for Lab. Auburn University, Auburn, AL.

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Appendix A: Notation Symbol A In. Imin Ix Iy Ixy rx ry rmin ´x ´y xi yi  p

Description Area Inches Minimum Moment of Inertia Moment of Inertia in x-Axis Direction Moment of Inertia in y-Axis Direction The product of x and y Moments of Inertia Radius of Gyration in x Direction Radius of Gyration in y Direction Minimum Radius of Gyration Centroid in x-Axis Centroid in y-Axis Center of each individual member in x-Axis Center of each individual member in y-Axis Summation Angle of Incline of the Principle Moment of

Page Number B-1 1 B-1 1 1 3 1 1 3 B-1 B-1 B-1 B-1 B-1 B-2

“

Inertia Inches

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Table A-1: Notation Use

A-1

A-1

A-1...