Application of Moment - Area Method for Cantilever Beams

Generally, the tangential deviation t is not equal to the beam deflection. In cantilever beams, however, the tangent drawn to the elastic curve at the wall is horizontal and coincidence therefore with the neutral axis of the beam. The tangential deviation or vertical intercept in this case is equal to the deflection of the beam as shown below.

General representation of deflection of cantilever beams

From the figure above, the deflection at B denoted as δ_B is equal to the deviation of B from a tangent line through A denoted as t_B/A. This is because the tangent line through A lies with the neutral axis of the beam.

Similarly the angle between the tangents will equal to slope of the point.

Problem : 1

The cantilever beam shown in Fig.1 has a rectangular cross-section of 50 mm wide and h mm high. Find the height h if the maximum deflection is not to exceed 10 mm.

FIG.1

Use E = 10 GPa.

Solution:

BENDING MOMENT DIAGRAM

The max. deflection ya = t A/B

$t_{A/B} = \dfrac{1}{EI}(Area_{AB}) \, \bar{X}_A$

$-10 = \dfrac{1}{10\,000\left( \dfrac{50h^3}{12} \right)}\Bigg[ -\frac{1}{2}(2)(4)(\frac{10}{3}) - \frac{1}{2}(4)(16)(\frac{8}{3}) \Bigg] \, (1000^4)$

$-10 = \dfrac{3}{125\,000h^3}\left[ \, -\dfrac{296}{3} \, \right](1000^4)$

$h^3 = \dfrac{-296(1000^4)}{125\,000(-10)}$

$h = 618.67 \, \text{mm} \,\,$ answer

Problem : 2

For the cantilever beam shown in Fig.2, determine the value of EIδ at the left end. Is this deflection upward or downward?

Fig.2

Solution:

$EI \, t_{A/B} = (Area_{AB}) \, \bar{X}_A$

$EI \, t_{A/B} = 2(2)(3) - \frac{1}{2}(4)(1)(\frac{8}{3})$

$EI \, t_{A/B} = \frac{20}{3} = 6.67 \, \text{ kN}\cdot\text{m}^3$

∴ EIδ = 6.67 kN·m³ upward answer

Comments

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