6.2. The first strength hypothesis
The first strength hypothesis is also called the maximum normal stress hypothesis because it accepts the maximum normal stress as the criterion. It can be formulated like this:
the limit material state in the combined stress state occurs when the maximum normal stress reaches the value of the limit stress in the uniaxial stress, i.e.
(6.1)
where
is the limit maximum normal stress;
is the limit stress in the uniaxial stress.
The material strength in the combined stress state is ensured if the maximum normal stress is lesser than the normal allowable working stress accepted for the uniaxial stress.
We see this hypothesis takes into account the maximum principal stress influence alone ignoring the influence of two other principal stresses on the material strength.
6.3. The second and third strength hypotheses
In accordance with the second hypothesis of the strength also called the maximum linear deformation hypothesis, the maximum linear deformation is accepted as the strength criterion.
Since the experiments do not verify this theory, we shall not explain it in detail and pass on to describing the third hypothesis, widely used nowadays.
In accordance with the third strength hypothesis also called the maximum shearing stress hypothesis: the material strength in the combined stress state is ensured if the maximum shearing stress is lesser than the shearing allowable working stress accepted for the uniaxial stress, i.e.
(6.2)
The maximum
shearing stress in the biaxial stress state occurs on the second at
the angle
to the principal stress direction and is equal to half the difference
of these stresses.
For the general state of stress the maximum shearing stress occurs on the ABCD plane (see Fig. 6.1):
(6.3)
The shearing allowable working stress in the uniaxial stress is connected with the normal allowable working stress by the relation followed from the previous formula, if is equal to zero in it.
Using the formula (6.3) we get
(6.4)
The expression implies some stress called the reduced or equivalent (rated) stress.
It must be understood as the stress which is to be established in the specimen in tension (or compression) so that its tensed strength was equal to the specimen strength being in the conditions of the combined stress state.
6.4. The energy hypotheses of strength
In accordance with the first of the energy hypothesis the material strength in the combined stress state is ensured if the specific potential energy of deformation is lesser than the allowable specific potential energy accepted for the uniaxial stress:
(6.5)
The specific potential energy of deformations in the general state of stress is equal to
(6.6)
This value is always positive. Therefore the energy hypothesis like the third one does not consider the difference between tension and compression, in other words, using this hypothesis one has to accept
(6.7)
The
allowable specific potential energy in the uniaxial stress is
determined by the formula under
(6.8)
Substituting the values u and uadm into (6.5) we get
or
(6.9)
The experiments show that better results are obtained if to accept not the entire deformation energy but only that part of it which is connected with the change of the body shape as criterion.
Hence, the strength condition according to the energy hypothesis due to distortion (it is also called the forth hypothesis too) has the form:
(6.10)
In a
particular case for the plane stress state
we get
(6.11)