ABSTRACT
As the inclined plane is an ellipse with area Alcos 8, the direct stress UnO normal to the plane and shear stress To along the plane, in the direction of maximum inclination, are given by:
A 2A
It is obvious by inspection that the maximum normal stress, equal to FIA, acts on radial planes. The magnitude and direction of the maximum value of To can be found by differentiating equation 1.3b:
The maximum value of TO is found by putting dTold8=0, thus:
TOmax= 2A (1.4)
-1.0
Solution
1.3.1 Simple biaxial stress system
N1 =a11 cos () TI =a j lsin () F2 =a21 tan ()
(Ub)
(Uc)
Substituting equations 1.5b, 1.5e into equation 1.6:
Resolving forces in the direction of 'e:
'e1sec ()= Tl + T2 Substituting equations 1.5c, 1.5f into equation 1.8:
(1.10) This is not the maximum value of shear stress in the plate. As the third principal stress is zero, the maximum value of , in the plate acts on a plane at 45° to both (J 1 and (J 2 and has the value
EXAMPLE 1.2 BIAXIAL PRINCIPAL STRESSES
A flat piece of slate with uniform thickness 20 mm is cut into the shape of a square with 100 mm long squared edges. A test is devised which allows uniform compressive stress alto be applied along two opposite edges and uniform tensile stress a 2 along the other two opposite edges, as shown in Figure 1.4(a). The stresses (J 1 and (J 2 act normally to the edges of the test specimen. The test is performed by increasing the magnitudes of a 1 and (J 2 simultaneously, but keeping the magnitude of a 1 always four times the magnitude of (J 2' If failure of the slate occurs when the shear stress on any plane exceeds 1 MPa, what would be the values of (J 1 and (J 2 at the moment of failure?
