Bending of Thin Walled Beams Questions and Answers

Q: Match each design change in List I with its effect on bending performance in List II. List I A. Increasing the cap separation B. Widening a flange greatly C. Moving to a stiffer material D. Lengthening a cantilever List II 1. Raises stiffness in proportion to the modulus 2. Raises the second moment with the square of the separation 3. Increases tip deflection with the cube of the length 4. Brings diminishing returns because of shear lag Select the correct match.

Correct answer: A-2, B-4, C-1, D-3Four changes acting through four different powers. Cap separation enters the second moment as its square. Widening a flange runs into shear lag, so the return diminishes. Modulus enters \( EI \) linearly. And cantilever deflection goes as the cube of the length - which is why length is the most damaging variable of the four.

Q: All of the following raise the bending stiffness of a beam EXCEPT

Correct answer: Moving material closer to the neutral axis\( I \) depends on \( \int y^2\,dA \), so material near the axis contributes almost nothing and moving it inward reduces the stiffness. The other three all raise it: a higher modulus, material further out, and greater depth - the last two being the same measure described differently.

Q: Which statement about the product of inertia is INCORRECT?

Correct answer: It is always a positive quantityThe product of inertia is \( \int xy\,dA \) and can be positive, negative or zero according to how the material is distributed between the quadrants. Its sign determines which way the neutral axis tilts, so losing it would lose half the information. The other three statements are all correct.

Q: Which of these does NOT change when the reference axis is moved parallel to itself?

Correct answer: The area of the sectionArea is a property of the shape alone and does not depend on where the axes are drawn. The first moment, second moment and product of inertia all involve distances from the axes and therefore all change under a parallel shift - which is what the parallel axis theorem quantifies.

61.

Consider the following statements about an unsymmetric section under a vertical moment:

1. The neutral axis is generally inclined to the horizontal.

2. The beam deflects horizontally as well as vertically.

3. The simple bending formula gives the correct stress.

Which of the statements given above is or are correct?

A.1 and 2 only
B.1 and 3 only
C.2 and 3 only
D.1, 2 and 3

Answer: A

The first two are correct and describe the coupling the product of inertia introduces. The third is false: the simple formula assumes bending about a principal axis and gives the wrong answer here, so the general expression is required.

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62.

Match each section property in List I with what it governs in List II.

List I

A. Second moment of area

B. Product of inertia

C. Centroid

D. Shear centre

List II

1. Where the neutral axis passes

2. Bending stiffness about a given axis

3. Whether a transverse load also twists the section

4. Coupling between the two bending planes

Select the correct match.

A.A-4, B-2, C-1, D-3
B.A-2, B-4, C-1, D-3
C.A-1, B-4, C-2, D-3
D.A-2, B-1, C-4, D-3

Answer: B

The four properties answer different questions, and confusing the centroid with the shear centre is the commonest error. Bending is governed by the centroid and the second moments; twisting under a transverse load is governed by the shear centre.

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63.

Match each section in List I with its bending behaviour in List II.

List I

A. Rectangle

B. Z-section

C. Thin circular tube

D. I-section

List II

1. Deflects sideways under a vertical moment

2. Two axes of symmetry, product of inertia zero

3. Material concentrated in flanges, efficient in bending

4. Equal second moment about every diameter

Select the correct match.

A.A-2, B-4, C-1, D-3
B.A-3, B-1, C-4, D-2
C.A-2, B-1, C-4, D-3
D.A-1, B-2, C-4, D-3

Answer: C

The tube is the special case in which every axis is principal and the two principal second moments are equal, so it bends identically in every direction. The Z is the opposite extreme and is the standard unsymmetric example.

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64.

Match each formula in List I with the quantity it gives in List II.

List I

A. b h cubed over twelve

B. A d squared

C. pi r cubed t

D. M y over I

List II

1. Parallel axis transfer term

2. Second moment of a rectangle about its centroid

3. Direct stress in symmetric bending

4. Second moment of a thin tube about a diameter

Select the correct match.

A.A-1, B-2, C-4, D-3
B.A-2, B-4, C-1, D-3
C.A-3, B-1, C-4, D-2
D.A-2, B-1, C-4, D-3

Answer: D

These four expressions cover most of what a bending calculation on a thin-walled section requires. The tube result in particular is worth remembering, since the cube on the radius explains why tubes are such efficient bending members.

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65.

Match each effect in List I with the assumption of simple bending theory it violates in List II.

List I

A. Shear lag in a wide flange

B. Non-zero product of inertia

C. A buckled compression panel

D. Two bonded materials

List II

1. Bending occurs about a principal axis

2. Plane sections remain plane

3. A single homogeneous material throughout

4. The whole section remains fully effective

Select the correct match.

A.A-2, B-1, C-4, D-3
B.A-1, B-2, C-4, D-3
C.A-2, B-4, C-1, D-3
D.A-3, B-1, C-4, D-2

Answer: A

Each of the four complications that airframe work runs into corresponds to one of the assumptions behind the elementary formula. Recognising which assumption has broken tells you immediately which correction to reach for.

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66.

Match each design change in List I with its effect on bending performance in List II.

List I

A. Increasing the cap separation

B. Widening a flange greatly

C. Moving to a stiffer material

D. Lengthening a cantilever

List II

1. Raises stiffness in proportion to the modulus

2. Raises the second moment with the square of the separation

3. Increases tip deflection with the cube of the length

4. Brings diminishing returns because of shear lag

Select the correct match.

A.A-4, B-2, C-1, D-3
B.A-2, B-4, C-1, D-3
C.A-1, B-4, C-2, D-3
D.A-2, B-1, C-4, D-3

Answer: B

Correct answer: A-2, B-4, C-1, D-3

Four changes acting through four different powers. Cap separation enters the second moment as its square. Widening a flange runs into shear lag, so the return diminishes. Modulus enters \( EI \) linearly. And cantilever deflection goes as the cube of the length - which is why length is the most damaging variable of the four.

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67.

All of the following raise the bending stiffness of a beam EXCEPT

A.Choosing a material of higher Young's modulus
B.Moving material further from the neutral axis
C.Moving material closer to the neutral axis
D.Increasing the depth of the section

Answer: C

Correct answer: Moving material closer to the neutral axis

\( I \) depends on \( \int y^2\,dA \), so material near the axis contributes almost nothing and moving it inward reduces the stiffness. The other three all raise it: a higher modulus, material further out, and greater depth - the last two being the same measure described differently.

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68.

Which of the following is NOT an assumption of simple bending theory?

A.Plane sections remain plane
B.The material is linearly elastic
C.Bending occurs about a principal axis
D.The section has a non-zero product of inertia

Answer: D

Correct answer: The section has a non-zero product of inertia

Simple bending theory assumes bending about a principal axis, and about a principal axis the product of inertia is zero by definition. A non-zero product is precisely what takes the problem out of simple theory and into unsymmetrical bending. The other three are genuine assumptions.

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69.

Which statement about the product of inertia is INCORRECT?

A.It is always a positive quantity
B.It vanishes for a section with an axis of symmetry
C.It couples the bending equations in two planes
D.It determines the inclination of the neutral axis

Answer: A

Correct answer: It is always a positive quantity

The product of inertia is \( \int xy\,dA \) and can be positive, negative or zero according to how the material is distributed between the quadrants. Its sign determines which way the neutral axis tilts, so losing it would lose half the information. The other three statements are all correct.

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70.

Which of these does NOT change when the reference axis is moved parallel to itself?

A.The first moment of area
B.The area of the section
C.The second moment of area
D.The product of inertia

Answer: B

Correct answer: The area of the section

Area is a property of the shape alone and does not depend on where the axes are drawn. The first moment, second moment and product of inertia all involve distances from the axes and therefore all change under a parallel shift - which is what the parallel axis theorem quantifies.

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