1.Concepts about the science of "Strength of Materials" (tasks, consistency, uniformity, priority, brief history The science of "Strength of Materials" is considered a science of general engineering and teaches the methods of calculation of constructions (structures),buildings, parts of machines and mechanisms for durability (strength), stiffness and sustainability. Design works of construction elements and machine details are carried out on the basis of engineering calculations aimed at ensuring their safe operation without damage and corrosion during using, as well as economic efficiency.
All types of constructions are considered for strength during the construction period, and the dimensions of construction elements are determined based on the calculation results. Incorrect calculation of the strength of a structural element that seems insignificant at first glance can lead to unpleasant consequences during use, even to the destruction of the structure.
Durability(strength)is the ability of the above to resist decay under the influence of external forces. Stiffness is the ability of parts of engineering structures to resist deformation under the influence of external forces.
In the design of structural elements and machine details, it is not enough to consider calculations for durability and stiffness in order to form a complete picture of their operation. Some of their elements, which work on compression deformation are definitely calculated for sustainibility. Deformations that usually occur in structures change according to Hooke's law when the amount of external force is less than the critical value. If the amount of external force increases by a small amount from the critical value, it will cause a sharp increase in deformation.
2-10-42.Bending. Determination of normal stresses. Nav'e formula (straight and curved, pure and transverse, neutral floor, axis, curvature, distribution, moment of resistance). In the transverse sections of the beam in pure bending, only the bending moments acting in the plane passing through one of its major central axes are affected.
The bending moment is the equally acting moment of the internal forces distributed in the section.
Equations of statics will not be enough to determine the magnitude and distribution of internal forces in cross-sections of beams. That is, this problem is statically indeterminate. Therefore, we will consider the conditions of deformation of the beam.
D AB=A′B′-AB =( r +y)dTH- r dTH=ydz A′B′=( r +y)dTH D AB=ydTH. So, the relative deformation of this fiber layer is equal to
Here is the distance from the neutral axis to the visible fiber; - the radius of curvature of the neutral floor of the beam.
If we consider the linear elongation and compression of each fiber in bending according to Hooke's law ,
(8.2)
will be It can be seen that the normal stresses generated in pure bending change proportionally to the distance from the neutral floor according to the height of the cross-section of the beam, that is, according to the linear law (Fig. 16.2). According to the formula u=0 at =0 and u=u max at = max . Therefore, the normal stresses reach their maximum values at the extreme points of the cross-section, which are farthest from the neutral axis.
Formula (8.2) is not suitable for practical calculations. Therefore, let's consider a part of the balance of the beam (Fig. 8.3).
equilibrium equations of statics for space .
1) X=0 (instantaneous)
2) U=0 (instantaneous)
3) Z=0.
We use the formula (8.2):
Here , then
This integral forms the static moment of the cross section of the beam relative to the neutral axis. Its equal to zero indicates that the neutral axis in bending passes through the center of gravity of the section.
M Z =0 (instantaneous)
Mu=0
( 8.2 ) into it,
.
, if we assume that this integral is the centrifugal moment of inertia and its equal to zero indicates that the neutral x- axis of the section and the u- axes perpendicular to it are the principal axes. So, the line of force is gi and the neutral axis are mutually perpendicular.
6. M x =0.
From this
. ( 8.3 ) Here - the curvature of the neutral floor of the beam.
As mentioned above, the neutral axis of the cross section passes through its center of gravity. Therefore, the longitudinal axis of the beam, which forms the geometric position of the centers of gravity along its length, is located in the neutral plane. It follows that the relation (8 . 3) represents the curvature of the bent axis of the beam. In general, this equation is the most fundamental connection in bending theory.
Thus, the curvature of the axis of the beam in bending is directly proportional to the bending moment, and the so-called "uniformity of section in bending" is inversely proportional to the magnitude of EJ x . the curvature of the neutral layer, that is, the connection (8.3) into ( 8.2 )
( 8.4 ) _
the formula is formed. This formula is called Nav'e formula. With its help, it is possible to find the normal stress at an arbitrary point on an arbitrary section of a beam in bending.
In general, the formula (8.4 ) can be used not only in pure bending, but also in transverse bending.
M in this formula is the bending moment related to the section where the tension is being found, and its value is obtained from the curve of bending moments.
