Page 1 of 3 Assignment 3 Due date: 30 May 2016 Weighting: 20% Marks: 200 Question 1: Symmetric bending (20 marks) A cantilever as illustrated in the figure to the right is loaded at its end with a mass of 0.5 kg. The cantilever is a steel ruler with E = 200 GPa. Take gravitational acceleration to be 9.81 m/s2. (a) Calculate the deflection at mid-length and the free end of the beam using the elastic formula [10 marks]. (b) Calculate the maximum normal stress in the beam [10 marks]. Question 2: Bending about two axes (20 marks) The moment M = 2 kN.m is applied to the beam as illustrated at the right. (a) Determine the stresses at each corner of the beam section [15 marks]. (b) Specify the orientation of the neutral axis relative to the principal centroidal axes of the section using a sketch [5 marks]. Page 2 of 3 Question 3: Principal centroidal axes (50 marks) Consider the unequal angle section illustrated in the figure to the right. (a) Calculate the location of the centroid as defined by zc and yc. [10 marks] (b) Determine the values of Iz, Iy, and Izy for the axes illustrated [10 marks]. (c) Calculate the values of Imax and Imin [10 marks]. (d) Show your calculated orientation of the principal centroidal axes (PCAs) on a sketch of the section, noting the axes associated with Imax and Imin values [10 marks]. (e) Discuss the agreement of your calculated values for Iz, Iy, and the orientation of the PCAs with available data for the section L 150 x 100 x 12 listed in the tables: European unequal angles.jpg, and discuss the differences [10 marks]. Question 4: Unsymmetric bending (50 marks) This question requires you to use the tabulated properties for the section L 130 x 65 x 10 listed in the file: European unequal angles.jpg. A 2 m long cantilever beam of this section is loaded with P = 1.3 kN at its free end and passing through its shear centre as illustrated in the figure to the right. Take E = 200 GPa. (a) From the tabulated data and the specified loading, determine the value of Imax and the orientation of the neutral axis and show both of these features on a sketch of the section [10 marks]. (b) Determine the magnitude of the bending moment that is acting on the section at the supported end of the cantilever [5 marks]. (c) Identify which of the 3 points: i, ii, or iii (as illustrated on the sketch) will have the highest magnitude of stress and which will have the lowest magnitude of stress by considering distance of each of these points from the neutral axis. (You may find that preparing a carefully scaled sketch of the section and neutral axis is the easiest approach for this part.) [15 marks]. (d) Calculate the value of the stress at the point ii and identify if this is a compressive or a tensile stress [20 marks]. Page 3 of 3 Question 5: Shear stress in beams (60 marks) Consider the beam section illustrated in the figure to the right. The beam has a shearing force of V = 5 kN acting vertically through the shear centre as illustrated. (a) Determine the shear stress at the three points in the beam section, labelled (i), (ii), and (iii) [20 marks]. (b) Produce a plot showing the direction and distribution of shear stress across the beam section, noting the shape of the distributions and key values [10 marks]. (c) Determine the distance e (as shown on the figure) which defines the location of the shear centre of this section [30 marks].