Tension and compression

Напряженность и сжатие можно охарактеризовать следующими параметрами: деформации, напряжение и стресс. Стресс Стресс - интенсивность внутренних сил. Стресс выражается в силе на единицу площади (мегапаскалях, фунт сила на квадратный дюйм) и негативными для сжатия. Внешние силы причиной деформации членов. Деформации деформации - увеличение в оригинальном размере члена. Он измеряется в единицах длины.

Elongation

Elongation - the increase in gauge length of a body subjected to a tensile force, referenced to a gauge length on the body. The parameter is considered as strain.

Strain

Strain - deformation of member divided by the original length of member.
According to Hooke's law for elastic deformation, stress is proportional to strain. The ratio of elastic stress/strain is a constant of the material. This constant is known as Young's modulus or modulus of elasticity. This parameter has units of stress (MPa, GPa).

Polyethylene : E = 1 GPa = 1000 MPa
Glass : E = 60 GPa = 60 000 MPa
Aluminum : E = 73 GPa = 73 000 MPa
Steel : E = 207 GPa = 207 000 MPa

For the pressure vessel, the summed force in the bolts is equal to the sum of the force from inner pressures. This condition helps to find the necessary number of bolts n.

Long cables are very flexible, the primary source of deformation in the cables is axial.

There is a compressive stress in the base due to the weight of a heavy block and it's own weight. The figure shows the optimal shape of the base. For such a design, stresses in the bottom and at the top of the base are equal.

SHEAR AND TORSION

Shear stress

Shear stress - the stress component tangential to the plane on which the forces act. Shear stress is expressed in shear force per unit of area (megapascals, pound-force per square inch).

Shear stress is highest in the most remote rivet from the center of sum area. There is practically no shear stress in the central rivet.

Torque transforms a square at the cylinder surface into a rhomb. Absolute values of shear stress at all side surfaces of the rhomb are equal. The stress state of pure shear is equivalent to bi-axial tension-compression state. Tensile stress can result in the brittle fracture of shafts under torsion. The figure shows typical failure of a cylindrical shaft made from brittle material.

There are two angles that help to describe torsion: shear angle g and angle of twist j. The shear angle does not depend on the length of a shaft with constant torque. The longer the shaft, the bigger the angle of twist.

A cross section without a sharp corner corresponds to uniform shear stress and effective use of the material. Rigidity depends on polar moment of inertia J which is proportional to r⁴. The bigger the radius r, the bigger the rigidity.

Rigidity of an open thin-walled shell is sufficiently smaller than rigidity of a closed section.

Torsional stress

Torsional stress - the shear stress on a transverse cross section resulting from a twisting action. Torsional stress is at a maximum at the surface. It is equal to zero in the center. The stress is proportional to the applied torque. For a rectangular cross section the maximum shear stress acts along the longer side, closest to the center point.

Shear stress is at a maximum for the shaft with the highest torque. The moment is the highest for the shaft with lowest rotation speed - the last shaft in the kinematic chain.

STRESS-STRAIN STATE

Normal stress in simple tension is given by s=Force/Area. If we cut the section at an angle j, there are two stress components perpendicular (s_n) and parallel (t) to the incline plane.

The maximum shear stress occuring at j=45^o is equal to half of maximum axial stress s.

For a bi-axial state of stress, normal stress s_n and shear stress t on an inclined plane depend on the two shown stress components.

For common plane stress state there are two perpendicular planes (principle planes) where there is no shear stress and normal stresses are a minimum and maximum. The two components are known as the principle stresses.

Maximum shear stress acts at the planes inclined 45^o to principle planes.

Hooke's law generally includes two constants of the material: Young's modulus E and Poisson's ratio m.

In the general case, maximum shear stress depends on two principle stresses only - maximum and minimum.

According to the first theory of strength fracture occurs if the maximum principle stress exceeds its critical value.

According to the second theory of strength fracture occurs if the maximum tensile strain exceeds its critical value. This can be transformed into an equation with the equivalent stress depending on all three stress components and Poisson's ratio.