Resistance of materials

See also: Resistance

The resistance of the materials is a branch of the Mécanique continuous mediums adapted to the structural deformations (Machine S - mechanical Engineering - or building S - Civil engineering).

This science makes it possible to bring back the law of total behavior of a structure (relation between requests - force S or couple - and displacements) to a local law of behavior of the Matériaux (relation between Contrainte S and Déformation S). The objective being the dimensioning of the structure according to a criterion of resistance or acceptable displacement.

According to the intensity of the constraint, there is initially elastic strain (when the request disappears, the material regains its shape and its initial position) then plastic deformation (when the request disappears, a certain deformation remains) and finally rupture when the intrinsic limits of material are exceeded.

History

First course of Resistance of the Materials given by August Wöhler to the University of Göttingen in 1842. (sources: to see discussion)

Assumptions of the RDM

the calculation of RDM is valid in a field limited by the following assumptions:

The matter is:

• elastic (not of plasticization),
• linear (not of non-linearity),
• homogeneous (not of variation of behavior in material),
• isotropic (not of variation of behavior following the direction).

The problem is:

• isostatic (kinematic part in équilbre),
• in small displacements (not of great displacement),
• quasi-static (not of dynamic effect),
• quasi-isotherm (not of change of temperature).

Concept of beam

The engineer uses the resistance of the materials before very designing the structural components and checking their resistance and their deformation. Some rapids calculations can be carried out easily if one limits to the beam in average plan , i.e. an object big length compared to its section and equipped with a plan of Symétrie (plane means).

See the complete article on the concept of beam in RDM

Requests

Simple

Base resolution

The Principe of Saint-Coming stipulates that a limiting condition (at the point M) can be replaced by a loading being equivalent without modifying the problem notably, if one places sufficiently " loin" of Mr.

• replacement of the limiting conditions by a loading,
• concept of error with " proximité" limiting conditions.

The Principe of superposition makes it possible to break up any request complexes simple requests all in all. This principle is directly related to the assumption of linearity.

The static balance gives the base of the resolution of the problem. It stipulates that:

• the sum of the Force S external with the system is equal to the null vector:

$\backslash \; sum\; \{\backslash \; underline\; \{F\_\; \{ext.\}\}\}\; =\; \{\backslash \; underline\; \{0\}\}$.

• the sum of the Moment S in a point, here at the point has, is equal to the null vector:

$\backslash \; sum\; \{\backslash \; underline\; \{M\_\; \{(A)\}\}\}\; =\; \{\backslash \; underline\; \{0\}\}$.

the Théorème of Castigliano defines displacement of the point, place of application of a force by the derivative of the elastic potential by report/ratio of this force.

According to the studied domains, there exist two types of size (outside and interior). they are differentiated compared to the studied part.

The efforts (or loading) gather the Forces and the moments. displacement engloblent the translations and rotations.

Mechanical constraints

• Law of Hooke

The normal constraint σ is proportional to relative lengthening ε unit by the constant of the Young modulus E:

$\backslash \; displaystyle\; \backslash \; underline\; \{\backslash \; underline\; \{\backslash \; sigma\}\}\; =\; E.\; \backslash \; underline\; \{\backslash \; underline\; \{\backslash \; epsilon\}\}$

with the relative Lengthening ε unit given by the relation lengths initial and final: $\backslash \; epsilon\; =\; \backslash \; frac\; \{l\_\; \{final\}\; -\; l\_\; \{initial\}\}\; \{l\_\; \{initial\}\}$

• Traction/Compression

This constraint is given normal to the force of traction. σ is equal to the force F divided by the Surface normal S:

$\backslash \; displaystyle\; \backslash \; sigma\_\; \{traction\}\; =\; \backslash \; frac\; \{F\}\; \{S\}$

• Inflection

the bending stress is described with moment M_3 the bending, the arrow x_2 and the moment of inertia I_3

$\backslash \; displaystyle\; \backslash \; sigma\_\; \{inflection\}\; =\; \backslash \; frac\; \{M\_3.\; x\_2\}\{I\_3\}$

with the Moment of inertia: $I\_3\; =\; \backslash \; int\_S\; \{x\_2^2\}\; dS$

• Shearing

$\backslash \; displaystyle\; \backslash \; tau\_\; \{moy\}\; =\; \backslash \; frac\; \{\backslash \; sigma\_\; \{shearing\}\}\; \{S\}\; =\; G.\; \backslash \; gamma$

with the moment of shearing: $G\; =\; \backslash \; frac\; \{E\}\; \{2\; (1+\; \backslash \; upsilon)\}$

Theoretical references

• the normal constraint σ: Constraint
• relative lengthening ε: Elongation at fracture
• the Young modulus E or the longitudinal modulus of elasticity: Modulus Young
• the modulus of rigidity G or the tangential modulus of elasticity: Modulus of Rigidity
• moment of inertia of inflection I: Moment of inertia

Composed

The beam is generally supposed to be made up of a Matériau Isotrope Homogène and is charged in its average plan (not of torsion thus). Under these conditions, the resultant of the external efforts is made up:

• of a longitudinal force of compression or traction;
• of a normal effort of shearing: the effort edge ;
• one moment bending .

One can still simplify while considering for example, a beam right, horizontal, of constant section, uniformly charged and resting on two simple supports. If one indicates by p the linear load and L the length of the beam, the solution of the problem holds in some simple formulas:

• the reaction of support is reduced to two vertical forces, equal each one to half of the load is pl/2
• the shearing action varies +pl/2 with - pl/2 with a zero value in medium of travée . One must check that the shear stress on support remains lower than the Shear strength maximum of the material
• the bending moment is null on support and maximum in medium of span where it is worth pl ² /8  One must check that the constraints in the median section exceed neither the Compressive strength, nor the Tensile strength maximum.

See too

• Mechanical
• Static Material
• of the solid
• Mechanical statics

External bond

• simple Calculations in line

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