Page 49 - Aircraft Stuctures for Engineering Student

34 Basic elasticity

stresses to a common set of axes (x, y) determine the principal stresses at the point and
their directions.
Ans. aI = aII = 15N/mm2. All directions are principal directions
P.1.5 A shear stress T,~ acts in a two-dimensional field in which the maximum
allowable shear stress is denoted by T~~ and the major principal stress by aI.
Derive, using the geometry of Mohr's circle of stress, expressions for the maximum
values of direct stress which may be applied to the x and y planes in terms of the three
parameters given above.

cy = 01 - rmax - ddax - <y
P.1.6 A solid shaft of circular cross-section supports a torque of 50 kNm and a
bending moment of 25 kNm. If the diameter of the shaft is 150 mm calculate the
values of the principal stresses and their directions at a point on the surface of the
shaft.

A~S. aI = 121.4~/mm~, e = 31043'
aII = -46.4~/m~, e = 121043'
P.1.7 An element of an elastic body is subjected to a three-dimensional stress
system a,, ay and a,. Show that if the direct strains in the directions x, y and z are
E,, and .zZ then
a, = Xe + ~GE,, ay = Xe + ~GE,, a, = Xe + ~GE,

where
uE
A= and e = E, + + E,
(1 +Y)(l -24
the volumetric strain.
P.1.8 Show that the compatibility equation for the case of plane strain, viz.

may be expressed in terms of direct stresses a, and cry in the form

P.1.9 In Fig. P.1.9 the direct strains in the directions a, by c are -0.002, -0.002
and +0.002 respectively. If I and I1 denote principal directions find E~., qI and 0.
Ans. = +0.00283, = -0.00283, 8 = -22.5" or +67.5"
P.1.10 The simply supported rectangular beam shown in Fig. P. 1.10 is subjected
to two symmetrically placed transverse loads each of magnitude Q. A rectangular
strain gauge rosette located at a point P on the centroidal axis on one vertical face