Question 15.12: A cantilever 800 mm long with a prop 500 mm from its built-i...
A cantilever 800 \mathrm{~mm} long with a prop 500 \mathrm{~mm} from its built-in end deflects in accordance with the following observations when a concentrated load of 40\, \mathrm{kN} is applied at its free end:
What will be the angular rotation of the beam at the prop due to a 30 \mathrm{kN} load applied 200 \mathrm{~mm} from the built-in end together with a 10 \,\mathrm{kN} load applied 350 \mathrm{~mm} from the built-in end?
| Distance from fixed end (mm) | 0 | 100 | 200 | 300 | 400 | 500 | 600 | 700 | 800 |
| Deflection (mm) | 0 | 0.3 | 1.4 | 2.5 | 1.9 | 0 | -2.3 | -4.8 | -10.6 |
The initial deflected shape of the cantilever is plotted to a suitable scale from the above observations and is shown in Fig. 15.24(a). Thus, from Fig. 15.24(a) we see that the deflection at \mathrm{D} due to a 40\, \mathrm{kN} load at \mathrm{C} is 1.4 \mathrm{~mm}. Hence the deflection at \mathrm{C} due to a 40\,\mathrm{kN} load at \mathrm{D} is, from the reciprocal theorem, 1.4 \mathrm{~mm}. It follows that the deflection at \mathrm{C} due to a 30 \,\mathrm{kN} load at \mathrm{D} is equal to (3 / 4) \times(1.4)=1.05 \mathrm{~mm}. Again, from Fig. 15.24(a), the deflection at \mathrm{E} due to a 40\, \mathrm{kN} load at \mathrm{C} is 2.4 \mathrm{~mm}. Thus the deflection at \mathrm{C} due to a 10\, \mathrm{kN} load at \mathrm{E} is equal to (1 / 4) \times(2.4)=0.6 \mathrm{~mm}. Therefore the total deflection at \mathrm{C} due to a 30\, \mathrm{kN} load at \mathrm{D} and a 10\,\mathrm{kN} load at \mathrm{E} is 1.05+0.6=1.65 \mathrm{~mm}. From Fig. 15.24(b) we see that the rotation of the beam at B is given by
\theta_{\mathrm{B}}=\tan ^{-1}\left(\frac{1.65}{300}\right)=\tan ^{-1}(0.0055)
or
\theta_{\mathrm{B}}=0^{\circ} 19^{\prime}
