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.

Let’s start with normal stresses:

Contact stresses:

Reduced stresses:

We have a state impossible!

– impossible state; before, plasticity (destruction) occurred.

This is enough of the reduced stress. Once again, remember to run calculations in units of the same degree!

I invite you to the next guides, if everything is clear or to previous ones, if there are any ambiguities.