# Stress reduced by hypotheses Hubert-M-H

A word of introduction to the guide. At the beginning, as usual, a bit of theory, I will explain the H – M – H hypothesis (plasticity condition), then I will give the formulas and, for example, I calculate the stresses reduced in the beam.

Hypothesis Hubert-Mises-Hencky (plasticity condition) – one of the performance hypotheses, in which the energy of the form deformation is compared, namely:
the material passes into a plastic state at a given point, when the energy density of the form strain (energy of deviators) reaches a certain limit value, characteristic for a given material.
Stress state as in the beam:
– elastic state (desired),
– the point at which the plasticity occurs,
– impossible state; before, plasticity (destruction) occurred.

Now I will quote the necessary formulas to calculate the reduced stresses. There are exactly three formulas: the first is for normal stresses, the next for tangential stresses, and the last for calculating the reduced stresses.

Pattern for normal stresses.

where:
M – this is the value of the bending moment in the place of the calculated reduced stress
z – distance of the point to the center of gravity
– moment of inertia relative to the horizontal axis

Pattern for tangential stresses.

where:
– value of cutting force at the place of calculated reduced stress
– static moment at point C. We calculate it as follows:
area over the calculated point, multiply the distance of the center of gravity of the surface area over the point to the center of gravity of the whole figure.
– moment of inertia relative to the horizontal axis
– the width of the field on which the point is located

The last formula for calculating the reduced stresses.

where:
– calculated normal stresses
– calculated tangential stresses

We will now go on to calculate the example to better understand how to use the above formulas. We have a fixed beam with an I-section. Description:
Dimensions are in m. A rectangular distributed force acts on the beam. Suppose we count the stresses reduced at the point located in the place of restraint.

Data:

Let’s remember about units to convert to the same when calculating.
We substitute our data for the patterns that appear higher.