DETERMINATION OF CRITICAL FORCE BY EULER FORM

Lecture

For pivotally fixed, centrally compressed rod of constant cross section (Fig. 8.2). I Euler's formula is:

DETERMINATION OF CRITICAL FORCE BY EULER FORM

where E is the modulus of the longitudinal elasticity of the material of the rod;

J _min - the minimum moment of inertia of the cross section of the rod.

For rods with other types of binding, Euler's formula is written in the form:

Where

- given the length of the rod;

- coefficient of bringing the length.

The expression "reduced length" means that in the Euler formula using the coefficient DETERMINATION OF CRITICAL FORCE BY EULER FORM

all cases of fastening of the ends of the rod can be brought to the main, hinged fastening.

Length reduction factor DETERMINATION OF CRITICAL FORCE BY EULER FORM

sometimes it can be estimated by the number of half-waves n for which the rod will bulge, losing stability, namely, we can take

In fig. 8.2 shows the cases of fixing the ends of the rod that are most frequently encountered in practice and the corresponding values of the coefficient DETERMINATION OF CRITICAL FORCE BY EULER FORM

Fig. 8.2

Euler's formula is applicable only on the limits of Hooke’s law, when the critical stress DETERMINATION OF CRITICAL FORCE BY EULER FORM

does not exceed the limit of proportionality of the material of the rod, since this formula was introduced using the dependence

in due time received on the basis of Hooke's law.

The applicability of the Euler formula can be determined by evaluating the flexibility of the rod and comparing this flexibility with its limit value. Rod flexibility is equal to

Where

- minimum radius of inertia (geometric characteristic of the section);

- the minimum moment of inertia of the cross-sectional area of the rod.

Marginal flexibility value DETERMINATION OF CRITICAL FORCE BY EULER FORM

is obtained from the condition

Ultimate flexibility equal to

So, for mild steel, if we take E = 2x10 ⁵ MPa,

To increase the bearing capacity of structures, they strive to use rods of as little flexibility as possible. So the calculation of real structures with flexibility DETERMINATION OF CRITICAL FORCE BY EULER FORM