Question

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 ơy--pg(H-y) and ơx-Txy-0 satisfies the two-dimensional stress equilibrium equations [10 marks] b) Suppose that the concrete contains a large number of randomly oriented micro-cracks. A crack which lies at an angle φ to the x-axis will propagate if where σφ and τφ are normal and shear stress acting on the plane defined by angle φ, f is the friction coefficient between faces of the crack and τ0 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 a bit later in this course) Assume that f- 0.75. 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 -0.75) find an expression for the maximum possible height of the column as a function of its material properties and free acceleration, g.

Answer 1

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