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HW13-aCentroidandMomentofInertia

You may (1) print this �ile and write directly on the hard copy; or (2) annotate the PDF digitally. When done, upload the PDF to Canvas. Keep the master copy in your course binder. All expectations from LE00 are in effect.

PartI.BASICS (1) Let’s say that you wanted to �ind the location of the

centroidal axes (x and y) for each area in the table. Each area represents the cross-sectional geometry of a beam. In the �irst column, visualize lines of symmetry (if any). The centroid of the cross-section (the center of area) must lie somewhere on each line of symmetry. Draw the centroid with a circle and label it. Draw the centroidal axes and dimension and : x y

= the horizontal distance between the extreme left x �iber and the centroid

= the vertical distance between the extreme y bottom �iber and the centroid

In the second column, think about how you would calculate . If you can take advantage of a vertical line x of symmetry, then is equal to b/2 (where b is equal x to the maximum width of the cross-section’s geometry). Circle “ by inspection.” If you can’t take advantage of a vertical line of symmetry, then you have to calculate x using the composite area method. Here, you would circle “ bycalculation.” In the third column, think about how you would calculate . If you identi�ied a horizontal axis, you can y determine this “ byinspection.” That’s h/2 (where h is equal to the maximum height of the cross-section’s geometry). Otherwise, you must �ind it “ by calculation.” The �irst row has been done for you.

(2) Cantilever beam “Beam A” (�ixed on the right and free

on the left) supports a line load. The beam bends about the centroidal x-axis because the load is transverse or perpendicular to that axis. Fillintheblanks. The longitudinal axis is the ______ axis. The centroidal axes are the ______ and ______ axes. = _________ inches y The moment of inertia abouttheaxisofbending is:

HW13-a Centroid and Moment of Inertia | copyright Prof. Susan Reynolds 2021 | sreynold@mines.edu | page 1 of 4

(3) This beam (“Beam B”) is identical to the previous one, except that it is oriented “�latwise.” Fillintheblanks. The longitudinal axis is the ______ axis. The centroidal axes are the ______ and ______ axes. = _________ inches y The moment of inertia abouttheaxisofbending is:

(4) Compare the moments of inertia for Beam A and Beam B. The axis with the largest moment of inertia is called the strong-axis. Which beam depicts bending about the strong-axis: A or B?

(5) Find an object similar to a popsicle stick, yardstick,

fettuccine pasta noodle, or ruler that is rectangular in cross-section. Bend it about the centroidal strong-axis. Then bend it about the centroidal weak-axis. (Use the Beam A and Beam B pictures as a guide.)

Brie�ly describe your �indings.

(6) The formula for the centroidal moment of inertia of a solid circle is:

I = 4πr4

Fillintheblanks. The longitudinal axis is the ______ axis. The centroidal axes are the ______ and ______ axes. = _________ inches y The moment of inertia abouttheaxisofbending is:

(7) What is the moment of inertia for this hollow cylinder

about its centroidal x-axis? (Note: I only drew the cross-section.) Ans.1.69E6mm4

(8) What is the moment of inertia for this hollow “box beam” about its centroidal x-axis? Ans.125E6mm4

HW13-a Centroid and Moment of Inertia | CEEN 311 | copyright Prof. Susan Reynolds 2021 | sreynold@mines.edu |page 2 of 4

PartII.PROBLEM-SOLVING

(9) Solve the previous problem again. If you used an additive method last time, use a subtractive method this time. If you used a subtractive method last time, use an additive method this time. Which method was faster? Why? 🌶

(10) Compute the moment of inertia about the centroidal x-axis of this cross-section. 🌶 Answer: 250 in 4

HW13-a Centroid and Moment of Inertia | CEEN 311 | copyright Prof. Susan Reynolds 2021 | sreynold@mines.edu |page 3 of 4

(11) This “I” shape cross-section is often used for beams because the moment of inertia about the strong-axis is quite high, relative to the material used. Compute the moment of inertia about the strong-axis. Should you use an additive method (all solids) or a subtractive method (solids and voids)? Which is fastest? 🌶🌶 Given : b = 3” ; c = 0.5” ; e = 0.3” ; f = 13.4”

Ans: 283 in 4

(12) Now, compute the moment of inertia about the weak-axis. 🌶🌶

Ans: 13.9 in 4

HW13-a Centroid and Moment of Inertia | CEEN 311 | copyright Prof. Susan Reynolds 2021 | sreynold@mines.edu |page 4 of 4