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Generally, for a given loading condition, the bending moment in a beam varies throughout its length due to which the maximum normal/bending stress at every section is different. For a given allowable stress (fixed by the material and Factor of Safety), the maximum bending stress will be maximum where the bending moment is maximum, and will be equal to the allowable. Everywhere else the maximum bending stress will be below this allowable value.

To save material we can vary the section modulus such that the maximum bending stress at every section remains the same, equal to allowable stress. such a beam will have, at every cross section, the same value of max bending stress and hence will be fully stressed.

Such beams are also called beams of uniform strength. Even though I get why they are called fully stressed beams, I don't understand why they are called beams of uniform strength.

Strength to my understanding is the capability of a structure to bear loads. The larger the loads it can carry the higher is its strength. I don't understand how to use this definition with the concept above.

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I am not familiar with the descriptive term "beam of uniform strength", however I've come across them as isotasic (meaning equal stress through out the beam).

The reason they are called uniform strength though, is that the beam can fail at any point along its length. This is because the development of the maximum stresses are each cross-section is the same.

One thing, that it is important though, is that the development of stresses depends on both geometry of the structure (e.g. cross-sections and legnth of the beam) and also loading. This is the reason why they are not heavily reported in textbooks.

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For a simply supported beam if we were to shape the beam by changing its section modulus we end up having a beam looking similar to its moment diagram, getting deeper at the middle.

The concern is that having the same stress in extreme fibers along the beam is not practical in the majority of the situations. What about shear stress which increases on the supports and is zero at the middle. What about lateral stability and local buckling.

Also, the deflection and deformation of the beam and hence its fatigue stress under repetitive loading becomes critical. The two shallow ends take up the brunt of deformation because as we know the stiffness of a section decreases by cube power of its depth $I= \frac{BH^3}{12}$.

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The two types of beams are incompatible by simply observing the stress formula,

$M = f_bS_x$

The name "uniform strength" indicates wherever on the beam, the moment capacity (the strength) is the same/identical, which can be achieved only if we hold both the stress and the section modulus to be constant. However, this defeats the purpose to "save" the material by varying the section modulus and requiring the stress to satisfy the unique condition, $f_b \le f_a$, that inevitably will result in varying moment capacity along the span of the beam. So the label "uniform strength" is not applicable. Instead, I think the bam can be said to have "conforming strength" in view of the moment diagram/demand.

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