Material removal by traditional machining operations has always been the workhorse of manufacturing industry. In this video lecture we will learn how to analyse a single point orthogonal cutting operation.
Detailed description of above video lecture is given below.
Cutting Force and Thrust Force During Cutting
During an orthogonal cutting operation material is removed from the work piece with help of cutting force. Along with that, a thrust force, which is normal to cutting direction is also induced. The thrust force could be either upward, or downward.
Fig.1 Cutting force,Fc and thrust force,Ft induced in an orthogonal cutting operation 
Cutting Force Analysis to learn how to determine this forces
Material Removal by Shearing – Shear Plane Theory
If you take a close look at cutting operation it will be evident that, material is getting removed from work piece by shearing action. Where different layers of atoms slip one another to form chip. This is why there is a change in chip thickness from uncut to cut chip. In actual case this shearing mechanism happens in a region called shear zone.
Fig.2 Material removal happens because of shearing action happens in shear zone 
But analysis of such a case will be too complex for this lecture. So we will assume instead of zone the shearing happens in a single plane. This is known as shear plane theory. According to this theory, material from uncut region undergoes, a sudden shear transformation across this plane. As you can see in Fig.3 thin layers of material slip across this plane.
Fig.3 According to shear plane theory shearing action occurs all of a sudden at shear plane 
If you know orientation of this plane or shear angle, you can easily predict chip thickness using following equation.
Which is derived from trigonometric analysis by equating length of common edge from cut and uncut side.
Determination of Shear Angle – Merchant Analysis
But what is the orientation of shear plane ?. That is the big question in machining analysis.
It could be at any angle during a particular machining operation. As explained in this figures. As shear angle decreases thickness of the chip increases.

Fig.4 Increase of chip thickness with decrease in sear angle for same cutting operation 
One way to predict this shear angle is assume shear plane adjust itself to reduce energy required for cutting operation. Or shearing will take place in a plane where there is maximum shear stress. So that force and work required for cutting is minimum. Shear stress at an angle phi is given by following equation.
By differentiating this equation with respect to phi and setting it to zero, one can obtain the plane on which shear stress is maximum.It is given as follows
Where, beta is the friction angle between tool and work piece interface. This is known as, Merchant analysis. Even though result given by Merchant analysis, does not always match with experimental results, this is a fairly good assumption. There are various other models also available to do this.
Since material flow before, and after cutting operation is same we can write.
This is assuming, width of the material does not increase much during machining.