The slope at P will be
$ \begin{array}{l}\mathrm R\mathcal l-\frac{\mathrm M\mathcal l}{6\mathrm{EI}}\times\left(\frac{2\mathcal l}3+\frac{\mathcal l}9\right)-\frac{\mathrm M\mathcal l}{6\mathrm{EI}}\left(\frac{\mathcal l}3+\frac{\mathcal l}9\right)+\frac{\mathrm M\mathcal l}{6\mathrm{EI}}\left(\frac{\mathcal l}3+\frac{\mathcal l}9\right)\\=0\\\mathrm R\mathcal l=\frac{\mathrm M\mathcal l}{18\mathrm{EI}}\left[2\mathcal l+\frac{\mathcal l}3+\mathcal l+\frac{\mathcal l}3+\mathcal l+\frac{\mathcal l}3\right]\\\;\mathrm R=\frac{\mathrm M}{18\mathrm{EI}}\left[3\mathcal l\right]=\frac{\mathrm M\mathcal l}{6\mathrm{EI}}=\frac{\mathrm M\mathcal l}{6\mathrm{EI}}\end{array} $
A 3 m long simply supported beam of uniform cross section is subjected to a uniformly distributed load of w = 20 kN/m in the central 1 m as shown in the figure.
If the flexural rigidity (EI) of the beam is 30 x 10^{6} N-m^{2}, the maximum slope (expressed in radians) of the deformed beam is
Two beams PQ (fixed at P and with a roller support at Q, as shown in Figure I, which allows vertical movement) and XZ (with a hinge at Y) are shown in the Figures I and II respectively. The spans of PQ and XZ are L and 2L respectively. Both the beams are under the action of uniformly distributed load (W) and have the same flexural stiffness, EI (where, E and I respectively denote modulus of elasticity and moment of inertia about axis of bending). Let the maximum deflection and maximum rotation be ${\mathrm\delta}_{\max1}$ and ${\mathrm\theta}_{\max1}$ respectively, in the case of beam PQ and the corresponding quantities for the beam XZ be ${\mathrm\delta}_{\max2}$ and ${\mathrm\theta}_{\max2}$ respectively.
Which one of the following relationships is true?
For the cantilever beam of span 3 m (shown below), a concentrated load of 20 kN applied at the free end causes a vertical displacement of 2 mm at a section located at a distance of 1 m from the fixed end. If a concentrated vertically downward load of 10 kN is applied at the section located at a distance of 1 m from the fixed end (with no other load on the beam), the maximum vertical displacement in the same beam (in mm) is __________