Q2) Figure Q2a shows a beam of length 2L with a rectangular cross-section (bxh). The beam is subject to a pure bending moment M. The boundary condition of one half of the beam is shown in Fig. Q2b. Assume the beam's thickness (b) is small compared to its other dimensions.
a) Show that the stress distribution, as calculated by the theories of bending, can be represented by an Airy stress function.
b) Using the stress function obtained in part(a), determine the displacement field u(x,y) and v(x,y).
c) Is the compatibility equation satisfied?
d) Compare the displacement field obtained using the Airy's stress function with the corresponding results obtained based on the theories of bending.
[N.B. For the values of M, 2L, b, h, E and v, please refer to Table Q2.]
[15 marks]
Table Q2
Workspace number | M (kNm) | Length (2L) (m) | b (mm) | h (mm) | E (GPa) | v
1-30 | 1 | 1.0 | 5 | 100 | 90 | 0.25
31-60 | 1.2 | 1.1 | 6 | 105 | 95 | 0.26
61-90 | 1.4 | 1.2 | 7 | 110 | 105 | 0.27
91-120 | 1.6 | 1.3 | 8 | 115 | 110 | 0.28
121-150 | 1.8 | 1.4 | 9 | 120 | 115 | 0.29
151-176 | 2.0 | 1.5 | 10 | 125 | 120 | 0.3