Beam with a uniform distributed load ”q (N/m)”

Consider a beam with a uniform distributed load ”q (N/m)” being as shown in Figure Q1. The beam has flexural stiffness equal to EI, with E and I the Young’s modulus and second moment of area of the cross-section, respectively.

Q1

(a) Calculate the reactions by using the double integration of the bending differential equa-
tiond2w/dx2 =M/EI. [7 marks]
(b) Calculate the reactions by using the stationarity of the complementary energy and by
neglecting the transverse shear strain energy. [8 marks]
(c) Calculate the rotation of point C by using the beam integration of the bending differential
equationd2w/dx2 =M/EI. [4 marks]
(d) Calculate the rotation at point C by using the stationarity of the complementary energy
and by neglecting the transverse shear strain energy. [6 marks]
Total for Question 1: 25 marks
Part of an aircraft structure is to be analysed as a pin-jointed frame carrying 10kN, 15kN, 20kN and 30kN loads as shown in figure Q2. The solid bars have circular cross-section with 40 mm diameter. Take the Young modulus as E =70 GPa.
(a) Using complementary virtual work, calculate the vertical displacement at point C. [25 marks]

Q2

Total for Question 2: 25 marks
For the beam section shown in figure Q3 the thickness of all of the sections is 4 mm. The beam is loaded with a negative transverse shear force of 40 kN passing through the shear centre (as shown in figure Q3). The x-axis is an axis of symmetry.
(a) Calculate the second moment of area Ixxabout the horizontal x-axis. [5 marks]
(b) Calculate the distribution of the shear flows in the cross-section. Also, locate the point in the cross-section with the maximum shear flow and calculate its value. [12 marks]
(c) Calculate the location of the shear centre (ec). [8 marks]

Q3

Total for Question 3: 25 marks
For the beam section shown in figure Q4 the x and y axes pass through the section centroid. The thickness of all of the sections is 10 mm. The beam is loaded with a positive couple of 6 kNm acting around the x-axis.
(a) Calculate the location of the centroid xCand yC. [2 marks]
(b) Calculate Ixx, Iyyand Ixy. [6 marks]
(c) Calculate the angle of the neutral axis from the x-axis. [7 marks]
(d) Calculate the maximum stress in the beam section. [10 marks]

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