The curvature of a beam in pure bending is
Answer: A
Curvature is the moment over EI, which integrates twice to give the deflected shape. The relationship is the starting point for every beam deflection method, from direct integration to Macaulay's notation and the moment-area theorems.
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Macaulay's method is used to
Answer: B
The bracket notation lets terms switch on at the station where a load is applied, so a single expression covers the whole beam and can be integrated once for the whole length. It avoids writing a separate equation for every span between loads.
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The moment-area theorems relate the deflected shape of a beam to
Answer: C
The first theorem gives the change in slope as the area under the moment-over-stiffness diagram and the second gives the deviation as its first moment. They are particularly quick for cantilevers and for beams of varying section.
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A section has \( I = 5.4 \times 10^7 \) mm\( ^4 \) with its extreme fibre \( 150 \) mm from the neutral axis. Its section modulus is
Answer: D
Correct answer: 3.6e5 mm^3
\[ Z = \frac{I}{y} = \frac{5.4\times10^7}{150} = 3.6\times10^5\ \mathrm{mm^3}. \] The section modulus packages the two quantities that determine the maximum stress into one, so sections can be compared directly without carrying \( I \) and \( y \) separately.
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A beam of section modulus \( 3.6 \times 10^5 \) mm\( ^3 \) is limited to \( 200 \) MPa. The largest bending moment it may carry is
Answer: A
Correct answer: 72 kN m
\[ M = \sigma Z = 200 \times 3.6\times10^5 = 7.2\times10^7\ \mathrm{N\,mm} = 72\ \mathrm{kN\,m}. \] The conversion is where the arithmetic goes wrong: newton millimetres to kilonewton metres is a division by \( 10^6 \), not by \( 10^3 \).
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A simply supported beam of span \( 6 \) m carries a central point load of \( 12 \) kN. The maximum bending moment is
Answer: B
For a central point load the maximum moment is \( PL/4 = 12 \times 6/4 = 18 \) kN m, occurring under the load. The corresponding cantilever result would be \( PL \), four times as large, which is why simple supports are preferred where they can be arranged.
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Assertion (A): The section modulus is a convenient figure for comparing candidate sections.
Reason (R): It combines the second moment of area and the extreme fibre distance into one number.
Select the correct answer from the options below.
Answer: C
Both are true and the second explains the first. Since the maximum stress is simply the moment divided by the modulus, two sections can be ranked directly without carrying their second moments and fibre distances separately.
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Assertion (A): For an unsymmetric section the maximum stress need not occur at the topmost fibre.
Reason (R): The neutral axis of such a section is inclined to the horizontal.
Select the correct answer from the options below.
Answer: D
Both statements are true and the second is the reason for the first. The critical point is whichever lies furthest from the inclined neutral axis, and that may well be a side corner rather than the top or bottom of the section.
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Assertion (A): Macaulay's method avoids writing a separate equation for each span of a loaded beam.
Reason (R): Its bracket terms vanish upstream of the station where each load is applied.
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. Each Macaulay bracket is defined to be zero while its argument is negative, so a term for a load at 3 m contributes nothing to the expression before that station and switches on afterwards. One expression therefore covers every span, and the constants of integration are found once.
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Consider the following statements about beam deflection methods:
1. Curvature is the bending moment divided by the bending stiffness.
2. Macaulay's method handles discontinuous loading in a single expression.
3. The moment-area theorems use the area under the shear force diagram.
Which of the statements given above is or are correct?
Answer: B
Correct answer: 1 and 2 only
Statements 1 and 2 are correct: curvature is \( M/EI \), and Macaulay’s brackets handle discontinuous loading in a single expression. Statement 3 names the wrong diagram - the moment-area theorems use the area under the \( M/EI \) diagram, not the shear force diagram.
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