Principles of Dairy Machine Design

Module 3. Stress

Lesson 9
STRESS ANALYSIS

9.1 Introduction

Important factor in determining the size of component is the load carried. Stress on member should not be so high that the member may fail in service.

9.2 Stress

Whenever any external force is applied on a body, it causes deformation. Any deformation of a body resulting from the action of external forces causes internal forces within the material. These internal forces per unit area are called stress. Depending upon the plane in which it acts, the stress may be Normal, Tangential or Oblique

9.2.1 Tensile stress

When a section is subjected to axial pull P acting normally across the section, the resistance set up is called Tensile and the corresponding strain is called Tensile Strain. (Fig 9.1)

Expressed as kg/cm² or N/m²

P = Load or Force acting on the body, kg or N

A = Cross sectional area of the body cm² or m²

Fig. 9.2 Stress

9.2.2 Compressive stress

When the section is subjected to axial pushes P acting normally across the section, the resistance set up is compressive stress and corresponding strain is compressive strain. Hence the original length will shorten.

Fig. 9.3 Compressive stress

9.3 Strain

When a straight bar is subjected to a tensile load, the length of the bar increases. The extent of elongation of the bar is called Strain. The elongation per unit length of the bar is called unit strain. Thus, any element in a material subjected to stress is said to be strained. The strain (e) is the deformation produced by stress. The various types of strains are explained below

Fig. 9.4 Strain

9.3.1 Tensile strain

A piece of material, with uniform cross-section, subjected to a uniform axial tensile stress, will increase its length from and the increment of length is the actual deformation of the material . The fractional deformation or the tensile strain is given by

Fig. 9.5 Tensile strain

9.3.2 Compressive strain

Under compressive forces, a similar piece of material would be reduced in length from

1to (1– δl)

Fig. 9.6 Compressive strain

The fractional deformation again gives the strain e_c et

Where

9.3.3 Shear strain

In case of a shearing load, a shear strain will be produced this is measured by the angle through which the body distorts.

Fig. 9.7 Shear strain

A rectangular block LMNP fixed at one face and subjected to force F. After application of force, it distorts through an angle Φ and occupies new position LM’N’P. The shear strain (es) is given by

= Φ (radians) …… since Φ is very small.

The above result has been obtained by assuming NN’ equal to arc (as NN’ is small) drawn with centre P and radius PN.

9.3.4 Volumetric strain

It is defined as the ratio between change is volume and original volume of the body, and is denoted by e_v

The strains which disappear with the removal of load are termed as elastic strains and the body which regains its original position on the removal of force is called an elastic body. The body is said to be plastic if the strains exist even after the removal of external force. There is always a limiting due to load up to which the strain totally disappears on the removal of load-the stress corresponding to this load is called elastic limit.

9.4 Elasticity

It is the property of a material, which enables it to regain its original shape and dimensions when the load is removed.

A material is said to be perfectly elastic if the deformation due to external loading entirely disappears on removal of the load. For every material, a limiting value of the load for a given resisting section is found, up to and within which, the resulting strain entirely disappears on removal of the load. The value of the intensity of stress corresponding to this limiting load is known as the Elastic limit of the material. Within this limit, the material behaves like a perfect spring, regaining its original dimensions on removal of the load. Beyond the elastic limit, the material gets into the plastic stage.

Robert Hooke discovered experimentally that within elastic limit, stress varies directly as strain.

This constant is termed as Modulus of Elasticity.

(i) Young’s modulus: It is the ratio between tensile stress and tensile strain or compressive stress and compressive strain. It is denoted by E. It is the same as modulus of elasticity.

(ii) Modulus of rigidity: It is defined as the ratio of shear stress τ to shear strain, e_s and is denoted by C,N or G. It is also called shear modulus of elasticity.

(iii) Bulk or volume modulus of elasticity: It may be defined as the ratio of normal stress (on each face of a solid cube) to volumetric strain and is donated by the letter K.