Fundamentals of the stress-strain state. Strength Theory
In analyzing the deformation of tension (compression), shearing, torsion and bending, normal and tangential stresses act simultaneously at most points of the stressed body, the magnitude of which depends on the orientation of the site passing through the given point.
In order to evaluate the strength of the structural material, it is necessary to know the stress state at the point of the material, which is characterized by a combination of normal and tangential stresses acting on all sites passing through the given point. Obviously, there are countless such sites. It can be shown that the stresses at any site passing through a given point can be expressed in terms of stress components acting on any three mutually perpendicular sites passing through a given point.
Let us select an elementary parallelepiped in the material of the construction in the neighborhood of the point O. Normal and tangential stresses act on its faces in the general case (Fig. 2.36). There are nine voltage components in total . According to the shear stress property . Thus, only six components are independent. When studying the stress state at a point, it is assumed that the stress components are known: we establish the relationship between the stress components acting on three mutually perpendicular sites and the stresses acting on the inclined site/HW (Figure 2.37).
Suppose that the normal to the area v is with the coordinate axes (Figure 2.38). On the inclined platform, the total voltage K (in general not perpendicular to the site) acts. Projections of the total voltage on the coordinate axis will be denoted by . Equations are equal to -
this for the tetrahedron considered, expressed in the stresses, can be written in the form(2.84)
where - direction cosines:
Since the total voltage on the slope
then, taking into account the expressions (2.84), the stresses on any on an inclined platform can be expressed through the stress components at a given point acting on three mutually perpendicular sites.
Main stresses. Among the innumerable sets of sites passing through the point under consideration, there are at least three mutually perpendicular sites on which there are no tangential stresses. These sites are called the main ones. Normal stresses acting on the main areas are called main stresses and are denoted by . And after determining the values of the main voltages, the indices are arranged so that the condition (algebraic values) is met.
If at the given stress point all three main voltages are equal to each other , then any site is main.
The practical feasibility of introducing the main stresses into consideration is due to the fact that one of them is the largest and the other is the smallest of the entire set of total stresses on the entire set of sites passing through the point under consideration.
Thus, and limit the range of values of the total voltages from above and below, allowing for these two values to estimate the set of all possible voltages at a given point.
The main stresses are determined from the system of equations(2.85)
Fig. 2. 38
Since , the direction cosines can not simultaneously be zero. Then, in order for the system of homogeneous equations (2.85) to have a nonzero solution, the determinant of the system must be zero:(2.86)
The values of principal stresses are calculated from condition (2.86) . The stress components must be known in advance.
Tangential stresses. To determine the position of the main areas, the direction cosines of the normals to the areas are found using Eqs. (2.85). In the case of a biaxial stress state, tangential stresses acting on an inclined platform are calculated by the formula
In planes inclined at an angle to the axes parallel to and ,
In the presence of three main sites (Figure 2.39), the values of the maximum tangential stresses acting at an angle of 45 ° to the corresponding planes are determined by the formulas
Fig. 2. 39
Since , the maximum tangential stress is calculated using the formula
Extreme tangential stresses act on areas inclined to the main at an angle of 45 °.
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