A rectangular concrete beam $$250$$ $$mm$$ wide and $$600$$ $$mm$$ deep is pre-stressed by means of $$16$$ high tensile wires, each of $$7$$ $$mm$$ diameter, located at $$200$$ $$mm$$ from the bottom face of the beam at a given section. If the effective pre-stress in the wires is $$700$$ $$MPa,$$ what is the maximum sagging bending moment (in $$kN$$-$$m$$) (correct to $$1$$-decimal place) due to live load that this section of the beam can with stand without causing tensile stress at the bottom face of the beam? Neglect the effect of dead load of beam.

Two steel columns $$P$$ (Length $$L$$ and yield strength $${f_y} = 250\,\,MPa$$) and $$Q$$ (length $$2L$$ and yield strength $${f_y} = 500\,\,MPa$$) have the same cross-sections and end-conditions. The ratio of buckling load of column $$P$$ to that of column $$Q$$ is:

A symmetric $${\rm I}$$-section (with width of each flange $$=50$$ $$mm$$, thickness of each flange $$=10$$ $$mm,$$ depth of web $$= 100$$ $$mm,$$ and thickness of web $$=10$$ $$mm$$ ݉݉) of steel is subjected to a shear force of $$100$$ $$kN.$$ Find the magnitude of the shear stress (in ܰ$$N/m{m^2}$$) in the web at its junction with the top flange. ___________

The state of $$2D$$-stress at a point is given by the following matrix of stresses: $$$\left[ {\matrix{ {{\sigma _{xx}}} & {{\sigma _{xy}}} \cr {{\sigma _{xy}}} & {{\sigma _{yy}}} \cr } } \right] = \left[ {\matrix{ {100} & {30} \cr {30} & {20} \cr } } \right]MPa$$$

What is the magnitude of maximum shear stress in $$MPa$$ ?