EGB275 Assessment1-Assignment 3 Sem 2 2023 (1)

1 QUT Faculty of Engineering School of Civil & Environmental Engineering EGB275 Structural Mechanics Assignment 3: Eigenvectors/Principal Stresses, Torsion, Unsymmetrical Bending, Combined Loading and Principle of Compatibility Due on Week 13 (Friday, 27/10/2023, 11.59 pm) (20% of the Overall Assessment) Question 1 a) A structural element is subjected to the stresses 𝜎𝜎 𝑋𝑋 = 𝑎𝑎 , 𝜎𝜎 𝑌𝑌 = 𝑏𝑏 , 𝜏𝜏 𝑋𝑋𝑌𝑌 = 𝑐𝑐 , where the values 𝑎𝑎 , 𝑏𝑏 and 𝑐𝑐 are assigned in the table below. i) Write down the stress tensor S corresponding to this stress regime. (1%) ii) Use matrix methods to find the principal stresses acting on this element, using the information in the stress tensor. (6%) iii) Use your results from part (ii) to determine the maximum in-plane shear stress acting on this element. (2%) iv) Use matrix methods to find the direction in which the major principal stress ( 𝜎𝜎 𝑚𝑚𝑚𝑚𝑚𝑚 ) is acting. Express your answer as the angle ( θ ) from the current x -axis to the direction of action of this stress. (7%) b) The matrix A takes the form 𝐴𝐴 = � 𝑆𝑆 1 𝑆𝑆 2 𝑆𝑆 3 𝑆𝑆 4 � , where the values of the elements 𝑆𝑆 1 , 𝑆𝑆 2 , 𝑆𝑆 3 , 𝑆𝑆 4 are assigned according to your student number, as per the table below. i) Find the eigenvalues of the matrix A , expressing your answer to at least four significant figures. (6%) ii) Write down the equations that must be solved to find the eigenvectors corresponding to these eigenvalues; that is, substitute your answers from part (i) into the relevant matrix formula. (Hint: You should have two matrix equations – one for each eigenvalue.) Note : You are NOT required to solve these equations. (3%) Parameter Calculation Table for Question 1 ( NO MARKS will be given if the following table with correct parameters calculated is not included in your submission). Please use the exact value for each parameter in your calculations. Student ID (Example: N10935108 – 1st Digit is 1, 2nd Digit is 0, 3rd Digit is 9, …) 𝑎𝑎 (MPa): ( − 1) 2 𝑛𝑛𝑛𝑛 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑑𝑑𝑖𝑖 𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦 𝐼𝐼𝐼𝐼 × (250 + 5 × 4 𝑑𝑑ℎ 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑑𝑑𝑖𝑖 𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦 𝐼𝐼𝐼𝐼 ) (265) 𝑏𝑏 (MPa): − sign( 𝑎𝑎 )(70 + 15 × 7 𝑑𝑑ℎ 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑑𝑑𝑖𝑖 𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦 𝐼𝐼𝐼𝐼 ) (-70) 𝑐𝑐 (MPa): ( − 1) 5 𝑡𝑡ℎ 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑑𝑑𝑖𝑖 𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦 𝐼𝐼𝐼𝐼 � 5 + 5 × 3 𝑦𝑦𝑑𝑑 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑑𝑑𝑖𝑖 𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦 𝐼𝐼𝐼𝐼� (-50) 𝑆𝑆 1 : 0.7 × 6 𝑑𝑑ℎ digit in your ID – 2 .6 (–1.9) 𝑆𝑆 2 : 5 .7 + 0.8 × 4 𝑑𝑑ℎ digit in your ID (8.1) S 3 : 4.5 + 0.4 × 3 𝑦𝑦𝑑𝑑 digit in your ID (8.1) 𝑆𝑆 4 : 1.5 - 0.2 × 7 𝑑𝑑ℎ digit in your ID (1.5)

2 Question 2 Two cantilever aluminium shafts of circular and square cross section are subjected to similar series of external torques as shown in Figure 1. The circular shaft is having 15 mm radius, r and the square shaft cross-section dimension, a is 30 mm x 30 mm, as shown in Figure 2. Given the shear modulus of both shafts = 27 GPa. Solve the followings: i) Plot the torque diagram. (7%) ii) Find the values of the maximum torsional shear stress developed in the two shafts. (10%) iii) Calculate the angle of twist at point A in the two shafts. (8%) Figure 1 Figure 2 A B C D 0.6 m 0.35 m 0.5m X 1 Nm X 2 Nm X 3 Nm

4 Question 4 A cross-section of circular concrete encased steel composite column with 3 m height is shown in Figure 4. The column diameter is X 7 mm and the steel profile is having a cross-sectional area of X 8 mm 2 . Given the elastic modulus of steel and concrete are 200 GPa and 32 GPa, respectively. Using the principle of compatibility, solve the following questions: i) If the column is fixed at the base and an axial centric force X 9 kN is applied to the column at the top through a rigid plate, find the axial forces and normal stresses in the steel profile and concrete. (12%) ii) Both ends of the column are fixed. If the column is subjected to 100 o C temperature rise and no axial load, determine the normal stresses induced in the steel and in the concrete. Given the coefficient of thermal expansion for steel and concrete are 12 × 10 −6 / ℃ and 8 × 10 −6 / ℃ , respectively. (13%) Figure 4 Steel profile Concrete

5 Parameter Calculation Table for Questions 2,3 and 4 ( NO MARKS will be given if the following table with correct parameters calculated is not included in your submission) Round your parameters to ONE decimal place only. Student ID (Example: N10935108 – 1st Digit is 1, 2nd Digit is 0, 3rd Digit is 9, …) 𝑋𝑋 1 (Nm): ( 100 + 3 𝑦𝑦𝑑𝑑 digit of your ID) × 2 (218) 𝑋𝑋 2 (Nm): 40 0 + 3 𝑦𝑦𝑑𝑑 + 4 𝑑𝑑ℎ digit of your ID (412) 𝑋𝑋 3 (Nm): (140 + 6 𝑑𝑑ℎ 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑦𝑦𝑜𝑜 𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦 𝐼𝐼𝐼𝐼 ) ÷ 2 (70.5) 𝑋𝑋 4 (m): 2. 5 + (1 𝑠𝑠𝑑𝑑 + 2 𝑖𝑖𝑑𝑑 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑦𝑦𝑜𝑜 𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦 𝐼𝐼𝐼𝐼 ) ÷ 10 (2.6) 𝑋𝑋 5 (kN): (120 + 1 𝑠𝑠𝑑𝑑 digit of your ID) × 4 (484) 𝑋𝑋 6 (kN): (200 +7 𝑑𝑑ℎ 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑦𝑦𝑜𝑜 𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦 𝐼𝐼𝐼𝐼 ) × 3 (600) 𝑋𝑋 7 (mm): 300 + 5 𝑑𝑑ℎ 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑦𝑦𝑜𝑜 𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦 𝐼𝐼𝐼𝐼 (305) 𝑋𝑋 8 (mm 2 ) : (2 𝑖𝑖𝑑𝑑 + 4 𝑑𝑑ℎ 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑦𝑦𝑜𝑜 𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦 𝐼𝐼𝐼𝐼 + 200) × 10 (2030) 𝑋𝑋 9 (kN): 750 − 7 𝑑𝑑ℎ + 5 𝑑𝑑ℎ digit of your ID (755)