A beam section has \( I = 5.4 \times 10^7 \) mm\( ^4 \) and carries \( 40 \) kN m. The stress at a point \( 150 \) mm from the neutral axis is nearest to
Answer: A
Substituting into \( My/I \) in newtons and millimetres, \( (40 \times 10^6 \times 150)/(5.4 \times 10^7) = 6 \times 10^9/5.4 \times 10^7 = 111 \) MPa. For a common aluminium alloy this leaves a reasonable margin against yield.
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Assertion (A): A Z-section beam loaded by a purely vertical moment deflects sideways as well as downward.
Reason (R): The Z-section has no axis of symmetry, so its product of inertia is non-zero.
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Answer: B
Both are true and the second explains the first. A non-zero product of inertia couples the two curvature equations, so the neutral axis inclines and the beam bends about an axis that is not the one the moment was applied about. It is the standard demonstration of unsymmetric bending.
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Assertion (A): Material placed far from the neutral axis is disproportionately effective in bending.
Reason (R): Each element contributes its area times the square of its distance from the axis.
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Answer: C
Both statements are true and the second is the reason for the first. Doubling the distance quadruples the contribution, which is why a spar consists of heavy caps joined by a light web and why a deep wing box is worth so much.
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Assertion (A): The boom idealisation neglects the booms' own centroidal second moments of area.
Reason (R): The transfer term dominates because the booms sit far from the neutral axis.
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Answer: D
Both are true and the second explains the first. A boom of 500 square millimetres at 100 millimetres contributes five million in transfer against a few thousand of its own, so dropping the latter changes nothing of consequence.
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Assertion (A): Simple bending theory may be applied directly to an unsymmetric section.
Reason (R): The neutral axis of any section in bending passes through the centroid.
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Answer: A
The assertion is false while the reason is true. The neutral axis does pass through the centroid whatever the symmetry, but for an unsymmetric section it is not perpendicular to the applied moment, so the simple formula gives the wrong stress and the general expression must be used.
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Assertion (A): Shear lag makes the measured stress in a wide flange lower than simple theory predicts.
Reason (R): Load reaches the outer flange material by shear, and that transfer takes distance.
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Answer: B
Both statements are true and the second is the reason for the first. Material far from a web is never fully stressed, so counting the whole flange as working overstates the section's capability. Designers use an effective width instead.
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Assertion (A): A buckled compression panel is treated as only partly effective in bending.
Reason (R): After buckling, the middle of the panel sheds load while material near the stiffeners keeps carrying it.
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Answer: C
Both are true and the second explains the first. Counting an effective width attached to each stiffener reproduces the load the panel really carries. Treating the panel as fully effective would overstate the section's bending capacity.
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Assertion (A): The product of inertia vanishes for a section with one axis of symmetry.
Reason (R): Contributions from either side of the axis cancel in pairs.
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Answer: D
Both statements are true and the second explains the first. For every element at a positive coordinate there is a mirror element at the negative one, so the integral of the product cancels. Such reference axes are therefore already principal axes.
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Assertion (A): The principal second moments of area are the largest and smallest over all axis orientations.
Reason (R): The product of inertia is zero about the principal axes.
Select the correct answer from the options below.
Answer: A
Correct answer: Both A and R are true, and R is the correct explanation of A
Both are true and the reason gives the mechanism. As the axes rotate, \( I_{xx} \) and \( I_{yy} \) vary and \( I_{xy} \) passes through zero; that orientation is where one second moment reaches its maximum and the other its minimum. Vanishing product of inertia and stationary second moments are the same condition.
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Assertion (A): Bending stiffness can be raised without changing the material.
Reason (R): Stiffness depends on the product of Young's modulus and the second moment of area.
Select the correct answer from the options below.
Answer: B
Both statements are true and the second explains the first. Redistributing the same material further from the neutral axis raises the second moment without adding a gram, which is nearly always cheaper in an airframe than changing to a stiffer and denser material.
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