Mohr's first moment-area theorem: change in slope between two points = area of the M/EI diagram between them.
Question 18Medium
A cantilever of span L carries a UDL w over its whole length and an upward concentrated load P at its free end. For the vertical displacement at the free end to be zero, P must be:
A simply supported beam of span L carries a UDL w over the whole span. If the UDL is replaced by a concentrated load W at the centre that produces the same central deflection (all other parameters unchanged), the ratio W/w is:
A simply supported uniform rectangular bar of breadth b, depth d and length L carries a load W at mid-span. The same bar extends by e under the same load applied as axial tension. The ratio of the maximum deflection to the elongation is:
A non-yielding (unyielding) support does not sink or move however large the reaction becomes. The book's key gives (d), which actually describes a fixed support.
Question 23Easy
'The change in slope between two points on a straight member under flexure is equal to the area of the M/EI diagram between those two points' is a theorem of which method?
Clapeyron's theorem of three moments (1857) relates the moments at three successive supports of a continuous beam.
Question 25Medium
A simply supported beam A carries a point load at its mid-span. An identical beam B carries the same total load uniformly distributed over the entire span. The ratio of the maximum deflections of beams A and B is:
Energy in an element dx = M²dx/(2EI); integrate over the length: U = ∫₀ᴸ M²dx/(2EI).
Question 27Hard
A uniform circular bar of diameter d and length l extends by an amount e under a tensile pull W. When it is simply supported and carries the same load W at mid-span, its deflection is:
δtip = Wa³/3EI + Wa²(L − a)/2EI. a = L/3: 4WL³/81EI. a = 2L/3: 14WL³/81EI. Ratio = 4/14 = 2/7.
Question 29Medium
A rolled steel joist, simply supported at its ends, carries a UDL that causes a maximum deflection of 10 mm and an end slope of 0.002 radian. The length of the joist is: