Bending, Basic concepts, Differential and integral dependences...


Basic concepts

The bend is the deformation , associated with the curvature of the axis of the beam (or a change in its curvature). A straight beam that receives the bending load, a beam. In general, when there is a bend in the cross sections of the beam, there are two internal force factors: the shear force Q and the bending moment . If there is only one force factor in the beam cross-sections, , a , then the bend is called clean. If a bending moment and a transverse force act in the cross section of the beam, the bend is called transverse.

The bending moment and the lateral force Q are determined by the section method. In an arbitrary cross-section of the beam, the quantity Q is numerically equal to the algebraic sum of the projections on the vertical axis of all external (active and reactive) forces applied to the cut off part; the bending moment in an arbitrary cross-section of the bar is numerically equal to the algebraic sum of the moments of all external forces and pairs of forces located on one side of the section.

For the coordinate system, it is shown) in Fig. 2.25, the bending moment from the loads located in the plane xOy, acts on the axis r, and the shearing force is in the direction of the axis y. force , bending moment

If the lateral load acts so that its plane coincides with a plane containing one of the main central axes of inertia of the sections, then the bend is called straight.

Two types of displacements are typical for bending:

• curvature of the longitudinal axis of the beam Ox, corresponding to the displacement of points of the axis of the beam in the direction Oy,

• rotation in space of one cross section with respect to the other, i.e. rotation of the section relative to the axis r in the XOy


Fig. 2.25

Differential and integral dependencies for bending

Let the continuous distributed load q (x) act on the beam (Fig. 2.26, a). The two cross sections m-t and n-n select the section of the beam with the length dx. We assume that on this section d (x) = const, because the length of the section is small.

Internal force factors and , acting in section n-n, receive some increment and will be equal to . Consider the equilibrium of the element (Figure 2.26, b):

a) , from here



Fig. 2.26

The member can be omitted, since it has a second order of smallness compared to the others. Then


Substituting the equality (2.69) into the expression (2.68), we obtain


Expressions (2.68) - (2.70) are called differential dependences for beam bending. They are valid only for beams with an initially rectilinear longitudinal axis.

The character rule for and is conditional:

are considered positive if they tend to rotate the beam element clockwise. In Fig. 2.27 , a, b show positive and negative directions

• The bending moment is considered positive if the beam element is bent by the convexity downward, i.e. its compressed fibers are located in the upper part. In Fig. 2.27, в, г are the directions , taken as positive and negative.

Graphically, and are displayed in the form of diagrams. Positive values ​​are plotted upward from the axis of the bar, negative - down.

Fig. 2.27

Normal stresses with pure beam bending

Consider a model of pure bending (Figure 2.28, a, b). After the loading process ends, the longitudinal axis of the beam X is curved, and its cross sections will turn relative to its original position on the angle/0. In order to clarify the law of distribution of normal stresses along the cross-section of the beam, we make the following assumptions:

• With pure straight curvature of the sire, the hypothesis of flat sections is suggested: the cross sections of the bar, flat and normal to the axis before deformation, remain flat and normal to its axis during and after deformation;

• the fibers of the beam do not press against each other when deformed;

• the material operates within the elastic range.

As a result of the deformation of the bend, the axis x curves and the section turns around the conditionally clamped section . Define the longitudinal deformation of an arbitrary fiber AB, located at a distance y from the longitudinal axis (see Figure 2.28, a).

Let be the radius of curvature of the axis of the beam (see Fig. 2.28, b). The absolute elongation of the fiber AB is . The relative elongation of this fiber

Since, according to the assumption, the fibers are not pressed against each other, they are in a state of uniaxial tension or compression. Using Hooke's law, we obtain the dependence of the change in stresses over the cross section of the bucket:


The value of is constant for a given section, so changes in the height of the section depending on the coordinate -

Fig. 2.28

Fig. 2.29

you y. When bending, some of the fibers of the beam stretch, part - is compressed. The boundary between the areas of tension and compression is a layer of fibers that only curves, without changing its length. This layer is called neutral.

The stress σ * ​​in the neutral layer must be zero, respectively This result follows from the expression (2.71) with . Consider the expressions for Since, for pure bending, the longitudinal force is zero, we write: (Figure 2.29), and since ', then , that is, . It follows that the axis Οζ is central. This axis in the cross section is called the neutral line. For pure direct bending, Then

Since , then

It follows that the axes Οζ and Oy sections are not only central but also the main axes of inertia. This assumption was made earlier in the definition of the concept of "direct bend". Substituting in the expression for the bending moment the value of from the expression (2.71), we get

or , (2.72)

where is the moment of inertia about the main central axis of the section Οζ.

Substituting the equality (2.72) into the expression (2.71), we obtain


Expression (2.73) defines the law of voltage variation over the section. It can be seen that does not vary with respect to coordinate 2 (that is, the normal stresses are constant across the section width), but with respect to the height of the section, depending on the coordinate y

Fig. 2. 30

(Figure 2.30). The values ​​of appear in the fibers farthest from the neutral line, i. E. at . Then . Having designated , we get


where - the moment of section resistance to the bend.

Using the formulas for the principal central moments of inertia of the basic geometric shapes of sections, we obtain the following expressions for :

• a rectangular section: , where is the side parallel to the r axis; h - the height of the rectangle. Since the z axis passes through the middle of the height of the rectangle,

Then the moment of resistance of the rectangle

• circle: . Since , then

• ring: , where - respectively outer and inner diameters of the ring; . Then .

thematic pictures

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