A concrete column of mass density p supports its own weight, as shown in the figure below. (Assume that the solid is subjected to a uniform gravitational body force of magnitude g per unit mass). a) Show that the stress distribution = pg(H y) and x = Txy = 0 satisfies the two-dimensional stress equilibrium equations. [10 marks] y H y g 00 b) Suppose that the concrete contains a large number of randomly oriented micro-cracks. A crack which lies at an angle & to the x-axis will propagate if (1) + to where and 17 are normal and shear stress acting on the plane defined by angle &, is the friction coefficient between faces of the crack and to is a critical shear stress that is related to the size of the micro-cracks and the fracture toughness of the concrete, and is therefore a material property (we will talk about fracture properties in Lecture 6). Assume that f = 0.5. Find the orientation of the micro-crack that is most likely to propagate. [20 marks] c) For the same friction coefficient as in b) (f = 0.5) find an expression for the maximum possible height of the column as a function of its material properties and free acceleration, g. Obtain the value of the maximum possible height of concrete columns (without reinforcement) if p = 3000 kg m, T0 = 50 MPa and g = 9.8 m s.