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Allowable (permissible) voltage– this is the voltage value that is considered extremely acceptable when calculating dimensions cross section element designed for a given load. We can talk about permissible tensile, compressive and shear stresses. The permissible stresses are either prescribed by the competent authority (say, the control bridge department railway), or are selected by a designer who knows well the properties of the material and the conditions of its use. The permissible voltage is limited to the maximum operating voltage designs.

When designing structures, the goal is to create a structure that, while being reliable, at the same time would be extremely light and economical. Reliability is ensured by the fact that each element is given such dimensions that the maximum operating stress in it will be to a certain extent less than the stress that causes the loss of strength of this element. Loss of strength does not necessarily mean destruction. Machine or building construction is considered to have failed when it cannot perform its function satisfactorily. A part made of a plastic material, as a rule, loses strength when the stress in it reaches the yield point, since due to too much deformation of the part, the machine or structure ceases to meet its intended purpose. If the part is made of brittle material, then it is almost not deformed, and its loss of strength coincides with its destruction.

Margin of safety. The difference between the stress at which the material loses strength and the permissible stress is the “margin of safety” that must be provided for, taking into account the possibility of accidental overload, calculation inaccuracies associated with simplifying assumptions and uncertain conditions, the presence of undetected (or undetectable) defects in the material and subsequent reduction in strength due to metal corrosion, wood rotting, etc.

Safety factor. The safety factor of any structural element is equal to the ratio ultimate load, causing loss of strength of the element, to a load that creates permissible stress. In this case, the loss of strength means not only the destruction of the element, but also the appearance of residual deformations in it. Therefore, for a structural element made of plastic material, the ultimate stress is the yield strength. In most cases, operating stresses in structural elements are proportional to the loads, and therefore the safety factor is defined as the ratio of the ultimate strength to the permissible stress (safety factor for ultimate strength). So, if the tensile strength of structural steel is 540 MPa, and the permissible stress is 180 MPa, then the safety factor is 3.

Ultimate voltage calculate the stress at which a material develops dangerous condition(fracture or dangerous deformation).

For plastic materials the ultimate stress is considered yield strength, because the resulting plastic deformations do not disappear after removing the load:

For fragile materials where there are no plastic deformations, and fracture occurs of the brittle type (no necking is formed), the ultimate stress is taken tensile strength:

For ductile-brittle materials, the ultimate stress is considered to be the stress corresponding to a maximum deformation of 0.2% (one hundred.2):

Allowable voltage- the maximum voltage at which the material should work normally.

The permissible stresses are obtained according to the limit values, taking into account the safety factor:

where [σ] is the permissible stress; s- safety factor; [s] - permissible safety factor.

Note. It is customary to indicate the permissible value of a quantity in square brackets.

Allowable safety factor depends on the quality of the material, operating conditions of the part, purpose of the part, accuracy of processing and calculation, etc.

It can range from 1.25 to simple details up to 12.5 for complex parts operating under variable loads under conditions of shock and vibration.

Features of the behavior of materials during compression tests:

1. Plastic materials work almost equally under tension and compression. The mechanical characteristics in tension and compression are the same.

2. Brittle materials usually have greater compressive strength than tensile strength: σ vr< σ вс.

If the permissible stress in tension and compression is different, they are designated [σ р ] (tension), [σ с ] (compression).

Tensile and compressive strength calculations

Strength calculations are carried out according to strength conditions - inequalities, the fulfillment of which guarantees the strength of the part under given conditions.

To ensure strength, the design stress should not exceed the permissible stress:

Design voltage A depends on load and size cross-section, permitted only from the material of the part and working conditions.

There are three types of strength calculations.

1. Design calculation - the design scheme and loads are specified; material or dimensions of the part are selected:

Determination of cross-section dimensions:

Material selection

Based on the value of σ, it is possible to select the grade of material.

2. Check calculation - the loads, material, dimensions of the part are known; necessary check whether the strength is ensured.

Inequality is checked

3. Determination of load capacity(maximum load):

Examples of problem solving

The straight beam is stretched with a force of 150 kN (Fig. 22.6), the material is steel σ t = 570 MPa, σ b = 720 MPa, safety factor [s] = 1.5. Determine the cross-sectional dimensions of the beam.

Solution

1. Strength condition:

2. The required cross-sectional area is determined by the relation

3. The permissible stress for the material is calculated from the specified mechanical characteristics. The presence of a yield point means that the material is plastic.

4. We determine the required cross-sectional area of ​​the beam and select dimensions for two cases.

The cross section is a circle, we determine the diameter.

The resulting value is rounded up d = 25 mm, A = 4.91 cm 2.

Section - equal angle angle No. 5 according to GOST 8509-86.

The closest cross-sectional area of ​​the corner is A = 4.29 cm 2 (d = 5 mm). 4.91 > 4.29 (Appendix 1).

Test questions and assignments

1. What phenomenon is called fluidity?

2. What is a “neck”, at what point on the stretch diagram does it form?

3. Why are the mechanical characteristics obtained during testing conditional?

4. List the strength characteristics.

5. List the characteristics of plasticity.

6. What is the difference between an automatically drawn stretch diagram and a given stretch diagram?

7. Which mechanical characteristics are selected as ultimate voltage for ductile and brittle materials?

8. What is the difference between ultimate and permissible stress?

9. Write down the condition for tensile and compressive strength. Are the strength conditions different for tensile and compressive calculations?

Answer the test questions.

Allows you to determine ultimate stress(), in which the sample material is directly destroyed or large plastic deformations occur in it.

Ultimate stress in strength calculations

As ultimate voltage in strength calculations the following is accepted:

yield strength for a plastic material (it is believed that the destruction of a plastic material begins when noticeable plastic deformations appear in it)

,

tensile strength for brittle material, the value of which is different:

To provide a real part, it is necessary to choose its dimensions and material so that the maximum that occurs at some point during operation is less than the limit:

However, even if the highest calculated stress in a part is close to the ultimate stress, its strength cannot yet be guaranteed.

Acting on the part cannot be installed accurately enough,

the design stresses in a part can sometimes be calculated only approximately,

Deviations between actual and calculated characteristics are possible.

The part must be designed with some design safety factor:

.

It is clear that the larger n, the stronger the part. However very big safety factor leads to waste of material, and this makes the part heavy and uneconomical.

Depending on the purpose of the structure, the required safety factor is established.

Strength condition: the strength of the part is considered ensured if . Using the expression , let's rewrite strength condition as:

From here you can get another form of recording strength conditions:

The relation on the right side of the last inequality is called permissible voltage:

If the limiting and, therefore, permissible stresses during tension and compression are different, they are denoted by and. Using the concept permissible voltage, Can strength condition formulate as follows: the strength of a part is ensured if what occurs in it highest voltage does not exceed permissible voltage.

Permissible stresses. Condition of strength.

The tensile strength and yield strength determined experimentally are average statistical values, i.e. have deviations upward or downward, therefore, the maximum stresses in strength calculations are compared not with the yield strength and strength, but with slightly lower stresses, which are called permissible stresses.
Plastic materials work equally well in tension and compression. The dangerous stress for them is the yield point.
The permissible stress is indicated by [σ]:

where n is the safety factor; n>1. Brittle metals work worse in tension, but better in compression. That's why dangerous voltage for them, the ultimate strength is σtemp. Allowable stresses for brittle materials are determined by the formulas: where n is the safety factor; n>1. Brittle metals work worse in tension, but better in compression. Therefore, the dangerous stress for them is the tensile strength σtemp. Allowable stresses for brittle materials are determined by the formulas:

where n is the safety factor; n>1.

Brittle metals work worse in tension, but better in compression. Therefore, the dangerous stress for them is the tensile strength σv.
Allowable stresses for brittle materials are determined by the formulas:

σtr - tensile strength;

σs - compressive strength;

nр, nс - safety factors for ultimate strength.

Strength condition for axial tension (compression) for plastic materials:

Strength conditions for axial tension (compression) for brittle materials:

Nmax is the maximum longitudinal force, determined from the diagram; A is the cross-sectional area of ​​the beam.

There are three types of strength calculation problems:
Type I tasks - verification calculation or stress check. It is produced when the dimensions of the structure are already known and assigned and only a strength test needs to be carried out. In this case, use equations (4.11) or (4.12).
Type II problems - design calculations. Produced when the structure is at the design stage and some characteristic dimensions must be assigned directly from the strength condition.

For plastic materials:

For fragile materials:

Where A is the cross-sectional area of ​​the beam. Of the two obtained area values, select the largest.
III type tasks - determination of permissible load [N]:

for plastic materials:

for brittle materials:

Of the two permissible load values, select the minimum.

Strength and stiffness calculations are carried out using two methods: permissible stresses, deformations And permissible load method.

Voltages, for which a sample from of this material collapses or in which significant plastic deformations develop are called extreme. These stresses depend on the properties of the material and the type of deformation.

Voltage, the value of which is regulated technical specifications, called permissible.

Allowable voltage– this is the highest stress at which the required strength, rigidity and durability of a structural element is ensured under the given operating conditions.

The permissible stress is a certain fraction of the maximum stress:

where is normative safety factor, a number showing how many times the permissible voltage is less than the maximum.

For plastic materials the permissible stress is chosen so that in case of any calculation inaccuracies or unforeseen operating conditions, residual deformations do not occur in the material, i.e. (yield strength):

Where - safety factor in relation to .

For brittle materials, permissible stresses are assigned based on the condition that the material does not collapse, i.e. (tensile strength):

Where - safety factor in relation to .

In mechanical engineering (under static loading), safety factors are taken: for plastic materials =1,4 – 1,8 ; for fragile ones - =2,5 – 3,0 .

Strength calculation based on permissible stresses is based on the fact that the maximum design stress in the dangerous section of the rod structure does not exceed the permissible value (less than - no more than 10%, more - no more than 5%):

Stiffness rating the rod structure is carried out on the basis of checking the conditions of tensile rigidity:

The amount of permissible absolute deformation [∆l] assigned separately for each design.

Permissible load method is that the internal forces arising in the most dangerous section of the structure during operation should not exceed the permissible load values:

, (2.23)

where is the breaking load obtained as a result of calculations or experiments taking into account manufacturing and operating experience;

– safety factor.

In the future we will use the method of permissible stresses and deformations.

2.6. Checking and design calculations

for strength and rigidity

The strength condition (2.21) makes it possible to carry out three types of calculations:

– check– according to the known dimensions and material of the rod element (the cross-sectional area is specified A And [σ] ) check whether it is able to withstand the given load ( N):

; (2.24)

– design– according to known loads ( N– given) and the material of the element, i.e., according to the known [σ], pick up required dimensions cross section providing it safe work:

– determination of permissible external load– according to known sizes ( A– given) and the material of the structural element, i.e., according to the known [σ], find the permissible value of the external load:

Stiffness rating rod structure is carried out on the basis of checking the stiffness condition (2.22) and formula (2.10) under tension:

. (2.27)

The amount of permissible absolute deformation [∆ l] is assigned separately for each structure.

Similar to calculations for the strength condition, the stiffness condition also involves three types of calculations:

– hardness check of a given structural element, i.e. checking that condition (2.22) is met;

– calculation of the designed rod, i.e. selection of its cross section:

– performance setting of a given rod, i.e. definition permissible load:

. (2.29)

Strength analysis any design contains the following main steps:

1. Determination of all external forces and support reaction forces.

2. Construction of graphs (diagrams) of force factors acting in cross sections along the length of the rod.

3. Constructing graphs (diagrams) of stresses along the axis of the structure, finding the maximum stress. Checking the strength conditions in places of maximum stress values.

4. Constructing a graph (diagram) of the deformation of the rod structure, finding the maximum deformation. Checking stiffness conditions in sections.

Example 2.1. For the steel rod shown in rice. 9a, determine the longitudinal force in all cross sections N and voltage σ . Also determine vertical displacements δ for all cross sections of the rod. Display the results graphically by constructing diagrams N, σ And δ . Known: F 1 = 10 kN; F 2 = 40 kN; A 1 = 1 cm 2; A 2 = 2 cm 2; l 1 = 2 m; l 2 = 1 m.

Solution. For determining N, using the ROZU method, mentally cut the rod into sections I−I And II−II. From the condition of equilibrium of the part of the rod below the section I−I (Fig. 9.b) we get (stretching). From the condition of equilibrium of the rod below the section II−II (Fig. 9c) we get

from where (compression). Having chosen the scale, we build a diagram of longitudinal forces ( rice. 9g). In this case, we consider the tensile force to be positive and the compressive force to be negative.

The stresses are equal: in the sections of the lower part of the rod ( rice. 9b)

(stretch);

in sections of the upper part of the rod

(compression).

On the selected scale we construct a stress diagram ( rice. 9d).

To plot a diagram δ determine the displacements of characteristic sections B−B And S−S(section movement A−A equals zero).

Section B−B will move up because top part shrinks:

The displacement of the section caused by tension is considered positive, and that caused by compression - negative.

Moving a section S−S is the algebraic sum of displacements B−B (δ V) and lengthening part of the rod with a length l 1:

On a certain scale, we plot the values ​​of and , connect the resulting points with straight lines, since under the action of concentrated external forces the displacements linearly depend on the abscissa of the sections of the rod, and we obtain a graph (diagram) of displacements ( rice. 9e). From the diagram it is clear that some section D–D doesn't move. Sections located above the section D–D, move upward (the rod is compressed); the sections located below move downwards (the rod is stretched).

Questions for self-control

1. How are the values ​​of axial force in the cross sections of a rod calculated?

2. What is a diagram of longitudinal forces and how is it constructed?

3. How are normal stresses distributed in the cross sections of a centrally stretched (compressed) rod and what are they equal to?

4. How is the diagram of normal stresses under tension (compression) constructed?

5. What is called absolute and relative longitudinal deformation? Their dimensions?

6. What is the cross-sectional stiffness under tension (compression)?

8. How is Hooke's law formulated?

9. Absolute and relative transverse deformations of the rod. Poisson's ratio.

10. What is the permissible stress? How is it selected for ductile and brittle materials?

11. What is called the safety factor and what main factors does its value depend on?

12. Name the mechanical characteristics of strength and ductility of structural materials.