As per $$IS$$ $$456:2000$$ for $$M20$$ grade concrete and plain barsin tension the design bond stress $${\tau _{bd}} = 1.2\,\,MPa.$$ Further, $$IS$$ $$456:2000$$ permits this design bond stress value to be increased by $$60\% $$ for $$HYSD$$ bars. The stress in the $$HYSD$$ reinforcing steel bars in tension $${\sigma _s} = 360\,MPa.$$ Find the required development length, $${L_d},$$ for $$HYSD$$ bars in terms of the bar diameter, $$\phi $$ ______________.

The creep strains are

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: