Now let’s see how many and what types of mechanical stresses affect our complex system and what are the methods of response.
In mechanics, compression is one of the basic efforts that a body can be subjected to, along with traction, bending, cutting and torsion.
A body is subject to compression when a system of converging forces acts on it.
In a generic section of a beam subject to compression, the unitary voltage is calculated with the relation, in which:
• σ is the unit stress (N / mm2);
• N is the compression effort (N)
• A is the area of the cross section of the beam (mm2).
The formulas obtained for the calculation of the tensile stress are applied identically to compressed bodies, provided that these are not long compared to the dimensions of the cross section.
If a very long straight cylindrical solid is subjected to a gradually increasing compression force, having reached a certain load value, called the Eulerian critical load, it infects indefinitely in the plane of least stiffness.
Bending is one of the basic efforts that a body can be subjected to, along with compression, traction, cutting and torsion. The stress that causes it is called the bending moment.
For simplicity, it can be said that a body is subjected to a bending stress when, due to the constraints to which it is subjected, it reacts, opposing itself, to a system of forces applied to it which would tend to make it rotate around its own point.
In practice, a beam is bent when it is subjected to a load system that has a component perpendicular to the longitudinal axis, generating a bending moment that causes the beam to bend.
In the beam subjected to bending, tensile and compressive stresses arise, ideally separated by a layer of fibers called “neutral axis” (x) which does not undergo any lengthening or shortening.
In a generic section of a beam subject to bending the unitary voltage is calculated with the relation:
• σ is the unit stress (N / mm2);
• M is the bending moment (Nmm)
• y is the distance of an elementary area from the neutral axis (mm);
• J is the moment of inertia with respect to the neutral axis (mm4)
This is the typical behavior of prosthetic bridges
In physics, cutting stress is a state of tension in which the shape of a material tends to change (usually due to “sliding” forces – twisting from transverse forces) without particular changes in volume. It is one of the elementary efforts that a body can be subjected to, along with compression, traction, flexion and torsion.
The change in shape is quantified by measuring the relative variation of the angle between the initially perpendicular sides of a differential element of the material (shear deformation). A simple definition of shear stress represents this as components of the tension at a point that acts parallel to the plane on which they lie.
Torsion is one of the elementary efforts that a body can be subjected to, along with compression, traction, flexion and cutting. The stress that causes it is called torque.
The solution to the torsion problem is exact for circular section beams (full or hollow), while approximations are required for thin hollow wall sections, rectangular and consequently those composed of thin rectangles (such as the classic steel sections). For this reason our bar has a circular section.
For the bars with circular cross-section, an exact solution to the problem of the expression of the tangential stress with respect to the applied stress can be determined. Their axial-symmetry and the condition of continuity of the solid (neither breaking, nor interpenetration of material) guarantee the impossibility of bowings or distortions of the section; therefore there are only simple rotations around the axis of the beam of the infinite disks. The circular section therefore represents the ideal structure for the solidarity of our plants.
When the twist is applied the section will rotate by an angle φ and at the same time the beam will distort so that the lines parallel to the axis will form an angle γ. These two corners share the same arc of a circle; let L be the length of the beam and ρ the radius of the section, the relation γL = φρ is valid (for. 3-4)
It is interesting to note the analogy with the simple flexion in which the longitudinal deformation is proportional to the distance from the center of gravity to less than the curvature (here instead expressed as a gradient of the rotation angle).
Tangential stress distribution
The relation shows that the distortion of the beam is the same for all the points equidistant from the axis and grows linearly with it.
Buckling is a stress generated by a force associated with a bending.
Simple and deflected buckling
Simple buckling is a stress generated by a normal effort associated with a bending moment. The deflected buckling is a stress generated by a normal effort, associated with a deviated deflection, that is deriving from the sum of two components (bending moments Mx, My).
The buckling can also be generated by a normal eccentric effort: typical is the case of a pillar with a non-axial load.
Hence the need to place single systems in parallel.
Solidarized systems, on the other hand, are protected against the distribution of loads given by the bar and can therefore be inserted without respecting the parallelism.
An important and determining factor is obviously the elastic modulus of the material constituting the element. The more flexible the plant and the material, the more subject to the phenomenon.