Stress (mechanics)

The mechanical stress (symbol ) is a term from the strength theory, a branch of engineering mechanics. It is the force per unit area, which acts in an imaginary sectional surface of a body, a liquid or a gas.

In general, the voltage ( engl. stress, French contrainte ) the amount of force F ( engl. force) per surface area A ( engl. area):

The mechanical stress has the same physical dimensions as the pressure, ie force per unit area. The pressure represents a special case of the stress

Normal and bending stress

At a normal load, the voltage on the surface is uniformly distributed, at a bending load results in a voltage waveform, and the bending stress is the maximum value of:

The normal stress, ie the stress at normal force stress ( train / pressure) results from:

Wherein the force in the direction of the surface normal and the area is.

The bending stress, ie the stress at the moment loads (bending ) results from:

Where the bending moment, the area moment of inertia, the distance from the cross- section centroid to extreme fiber and the moment of resistance.

The following diagram illustrates this on a cantilever beam:

As a vector has three components, and it depends on the orientation of the cut surface. The sense of direction is only defined when you define which side of the cut surface is considered, because the stress is the force per unit area exerted by the cut-away in thought material to the remaining material. It is therefore directed oppositely in two opposed cutting banks.

Tension and stress tensor

Acting at a certain point voltages will be described in its entirety by the stresses in the three cutting surfaces that intersect at the point, ie by three vectors of three components; these together form the stress tensor.

The simplest representation is the stress tensor, if one chooses the three cut surfaces are perpendicular to a direction of a Cartesian coordinate system. The three forces in the three -sectional areas correspond to the rows of the following matrix:

The importance shows the following sketch of a cut out very small volume element:

Shear, compression and tensile stress

The diagonal elements in this case represent the normal stress, that is, the forces acting perpendicular to the surface. They are depending on the direction of tensile stress (positive sign ) or compressive stress (negative sign ) (and their scalar pressure) called. The non-diagonal elements are referred to as shear stress. They act tangent to the surface, so put a strain on shear dar.

In a double index describes the first index, the direction in which the outer unit normal vector of the cut surface shows ( ie worked on which is cut bank) and the second index, the direction in which the tension acts.

The shear stress distribution is used to illustrate the occurring shear stresses to the reference axis within a burdened with a transverse force profile. Accesses the transverse force outside the shear center at occurs twist.

Principal stress and principal stress direction

The tensor allows to first describe the stress state independent of a particular coordinate system and adapt after a derivation of the respective calculation process, the component equations of the geometric properties of the body, for example in cylindrical coordinates. In the tensor stress tensor is defined as those second-rank tensor, the scalar multiplied by the outer surface normal of a cut surface gives the force vector per unit area.

Each stress state can be converted by the principal axis transform in a coordinate system in which all shear stresses vanish. If the three normal stresses are summarized in this coordinate system to a vector, it can be broken down into these two components:

  • The component transversely to the body diagonal is a measure of how much other cutting directions, the shear stresses are a maximum, depending on the cutting direction. But this share is relevant for the calculation of steel structures. It corresponds to the equivalent stress according to change of shape hypothesis. If it exceeds the yield stress of each steel grade, the steel plastically deformed.
  • The component in the direction of space diagonal describes the pressure; this share is irrelevant in the calculation of steel structures, because it leads to no cutting direction to shear stresses, and thus also no plastic deformation.

The principal stresses can be calculated by solving the equation: where E is the 3 x 3 identity matrix. Multiplying out the determinant leads to a cubic equation whose solutions, and representing this principal stresses. They are the eigenvalues ​​of the matrix P. voltage

The respective principal stress direction is defined by the equation, which is used for the calculated principal stress. The solutions are eigenvectors of the matrix S voltage and indicate the direction of the voltages at. In normalized form, they form an orthonormal basis of the three-dimensional space, provided that none of the principal stresses is zero.

The curves of the main power lines is called trajectory. A geometric representation of the voltage components in different coordinate directions in the plane stress state is given by the Mohr stress circle.

The connection to the deformation produces Hooke's law for elastic deformation. Important material constants are Young's modulus and Poisson's ratio. The plastic deformation describe the flow condition, the flow rule and the hardening law. The relation to rate of deformation in viscous liquids produces Newton's toughness approach. Important material constant in the dynamic viscosity.

The deformation behavior of a body is defined by the relationship between stress and deformation tensor. This relationship is called rheological law and is a material property of the body. The rheology is concerned with the flow and deformation of bodies.

The stresses occurring in the body forces acting without outside are called residual stresses.

A hydrostatic stress state exists when the three principal stresses are equal.

The voltage in gases is described in Boyle's Law and Gay -Lussac 's law.