Basic Equations: Differential Form - Mechanics of Structures: Variational and Computational Methods - page 37

the structure is such that the maximum principal stress at a point reaches the stress at yield
(or other “failure” level) in a tensile test for the material. Choose the principal directions
such that σ >σ >σ

3 . Then, yield occurs when

1 2

σ 1 = σ ys (1.92)

A similar expression is obtained if a yield stress for a compressive test is available. Also a
maximum strain theory is readily derived.

Maximum Shear Theory

According to maximum shear theory, failure occurs in a body in a complex state of stress
when the maximum shear stress at a point (Eq. 1.91), e.g.,

| σ 1 − σ 2 | / 2, reaches the value

of the shear yield stress of the material in a tensile test, σ ys / 2. Thus yield (failure) for the

complex state of stress occurs if

( | σ − σ | | σ − σ | | σ − σ | ) = σ

max , , (1.93)

1 2 2 3 3 1 ys

The term stress intensity is sometimes used to indicate the highest of these absolute values.
The failure theory of Eq. (1.93) is also called the Tresca theory . Frequently, the theory of
Eq. (1.93 ) is expressed as

σ − σ = σ

(1.94)

max min ys

where σ max and σ min are the maximum and minimum principal stresses, respectively. For

the case σ 1 >σ 2 >σ 3 , Eq. (1.94) would be σ 1 − σ 3 = σ ys .

von Mises 22 Criterion

Another failure criterion is based on the maximum distortion energy at a location in the
structure reaching the maximum distortion energy at yield in a tensile test. This leads to
the following expression as a criterion of failure by yielding:

(σ − σ )

2

+ (σ − σ )

2

+ (σ − σ )

2

1 2 2 3 1 3

= σ (1.95)

ys

2

For another failure mode, such as fatigue, ultimate stress, or fracture stress, simply replace
σ ys by the appropriate tensile stress level. The quantity on the left-hand side of Eq. (1.95) is
sometimes referred to as the equivalent stress and is often available as output of structural
analysis software. The underlying theory for Eq. (1.95) can also be based on the maximum
shear stress on an octachedral plane (a plane which intersects the principal axes at equal
angles) and hence this criterion is sometimes referred to as the octachedral shear stress theory .
Other names for this failure criterion are the von Mises theory or the Maxwell-Huber-Hencky-
von Mises theory .If

σ = 0, Eq. (1.95) reduces to

3

1 2

σ − σ 1 σ 2 + σ = σ ys (1.96)

22 Richard von Mises (1883-1953), an aerodynamicist and mathematician, was born and educated in Austria. He
did research in flight mechanics and was the founding editor of the journal Zeitschrift f ur angewandte Mathematik
und Mechanik . Early in his career he proposed two fundamental axioms in probability theory: the axioms of
convergence and randomness. Later he proposed the famous birthday problem which asks for a probability of at
least 50%, how many people must be in a room so that some have the same birthday.