Principles of Dairy Machine Design

Module 5. Properties of material, failures and factor of safety

Lesson 24
THEORIES OF FAILURE, STRESSES IN ELEMENTARY MACHINE PARTS

24.1 Introduction

It has already been discussed in the previous chapter that strength of machine members is based upon the mechanical properties of the materials used. Since these properties are usually determined from simple tension or compression tests, therefore, predicting failure in members subjected to uniaxial stress is both simple and straight-forward. But the problem of predicting the failure stresses for members subjected to bi-axial or tri-axial stresses is much more complicated. In fact, the problem is so complicated that a large number of different theories have been formulated. The principal theories of failure for a member subjected to bi-axial stress are as follows:

24.2 Principal Theories of Failures

• Maximum principal (or normal) stress theory (also known as Rankine’s theory).

• Maximum distortion energy theory (also known as Hencky and Von Mises theory).

• Maximum shear stress theory (also known as Guest’s or Tresca’s theory).

• Maximum strain energy theory (also known as Haigh’s theory).

• Maximum principal (or normal) strain theory (also known as Saint Venant theory).

Since ductile materials usually fail by yielding i.e. when permanent deformations occur in the material and brittle materials fail by fracture, therefore the limiting strength for these two classes of materials is normally measured by different mechanical properties. For ductile materials, the limiting strength is the stress at yield point as determined from simple tension test and it is, assumed to be equal in tension or compression. For brittle materials, the limiting strength is the ultimate stress in tension or compression.

24.2.1 Maximum principal or normal stress theory (Rankine’s Theory)

According to this theory, the failure or yielding occurs at a point in a member when the maximum principal or normal stress in a bi-axial stress system reaches the limiting strength of the material in a simple tension test. Since the limiting strength for ductile materials is yield point stress and for brittle materials the limiting strength is ultimate stress, therefore according to the above theory, taking factor of safety (F.S.) into consideration, the maximum principal or normal stress (σt1) in a bi-axial stress system is given by

σt1 = σyt/F.S for ductile material

= σu/F.S for brittle material

Where,

σyt = Yield point stress in tension as determined from simple tension test, and

σu = Ultimate stress.

24.2.2 Maximum distortion energy theory (Hencky and Von Mises Theory):

According to this theory, the failure or yielding occurs at a point in a member when the distortion strain energy (also called shear strain energy) per unit volume in a bi-axial stress system reaches the limiting distortion energy (i.e. distortion energy at yield point) per unit volume as determined from a simple tension test. Mathematically, the maximum distortion energy theory for yielding is expressed as

(σ_t1)² + (σ_t2)² – 2σ_t1 × σ_t2 = (σ_yt /F.O.S) ²

Where,

σ_yt is yield stress

F.O.S. = Factor of safety.

This theory is mostly used for ductile materials in place of maximum strain energy theory.

24.2.3 Maximum shear stress theory (Guest’s Or Teresa’s Theory)

According to this theory, the failure or yielding occurs at a point in a member when the maximum shear stress in a bi-axial stress system reaches a value equal to the shear stress at yield point in a simple tension test. Mathematically,

ז_max = ז_yt /F.O.S.

Where

ז_max = Maximum shear stress in a bi-axial stress system,

ז_yt = Shear stress at yield point as determined from simple tension test,

F.O.S. = Factor of safety.

24.2.4 Maximum strain energy theory (Haigh’s Theory)

According to this theory, the failure or yielding occurs at a point in a member when the strain energy per unit volume in a bi-axial stress system reaches the limiting strain energy per unit volume as determined from simple tension test. We know that strain energy per unit volume in a bi-axial stress system,

U₁=1/2E[σ_t1² + σ_t2² – ((2σ_t1* σ_t2)/M)]

U₂=1/2E[σ_yt/F.O.S]²

According to the above theory U₁=U₂

1/2E[σ_t1² + σ_t2² – ((2σ_t1* σ_t2)/M)] =1/2E[σ_yt/F.O.S]²

Or [σ_t1² + σ_t2² – ((2σ_t1* σ_t2)/M)]= [σ_yt/F.O.S]²

This theory may be used for ductile materials.

24.2.5 Maximum principal strain theory (Saint Venant’s Theory)

According to this theory, the failure or yielding occurs at a point in a member when the maximum principal (or normal) strain in a bi-axial stress system reaches the limiting value of strain as determined from a simple tensile test. The maximum principal (or normal) strain in a bi-axial stress system is given by

Є_max= (σ_t1/E)-( σ_t2/m*E)

According to the above theory,

Є_max= (σ_t1/E)-( σ_t2/m.E)= Є= (σ_yt/E*F.O.S)………………………….

Where,

σ_t1 and σ_t2 = Maximum and minimum principal stresses in a bi-axial stress system,

ε = Strain at yield point as determined from simple tension test,

1/m = Poisson’s ratio,

E = Young’s modulus, and

F.O.S. = Factor of safety.

From equation (i), we may write that

σ_t1-( σ_t2/m)= (σ_yt/F.O.S)

This theory is not used, in general, because it only gives reliable results in particular cases.