One of the basic assumptions in Greenwood-Williamson model is that asperity summits have a constant radius. Consider that asperity summits have variable radius and the radius (R) depends on the height of asperity (z) as given by: R(z)-Ro exp(-kz). With this, derive the expressions for load and contact area for elastic-plastic contact of two Gaussian rough surfaces.

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Question 1 The wedge shaped bar shown in Figure Q1 is to be analysed using a single one dimensional quadratic element as shown. The bar has a rectangular cross section with a constant width and a linearly varying depth. 500 k = 10E 2 100 3 Figure Q1. +1] a) Calculate the strain shape function matrix, [B], for the element. b) Using the strain shape function matrix derived in (a), show that the stiffness matrix for the element is given by the equation: I 255+ ALL DIMENSIONS IN mm (3-5) d h 100- with units of N/mm Where: E = Young's modulus of the material 10 c) Show how the strain shape function matrix may be used to determine the stress from the nodal displacement for this type of element. Do not attempt to calculate the stress. LEGION Question 2 Figure Q2 shows a two-dimensional quadratic element that has been distorted such that one Q Search

The diameter (d) of a solid rod (i.e., with a circular cross-section) is 16 mm and is made from a homogeneous material. The length (L) of the rod is 1.75 m and Young’s Modulus for the material (E) is 250 GPa. When the rod is placed under tension it experiences deformation and a stress (?) of 650 MPa. Calculate the following:(1) Strain energy (U) to 2 decimal places in Joules.(2) Strain energy per unit volume (U/V) in J/m3(3) The change in length ∆L to 2 decimal places in mm.(4) The strain (?) due to deformation to 2 significant figures. (show all work)