How to Analyze Rolling of Wide Strips? | Metals | Metallurgy

Here we make use of slab method for analysis of rolling of wide strips. In wide strip rolling the lateral spread is very small and hence it may be assumed that the widening of strip during rolling is negligible and hence it becomes a case of plane strain deformation. The process is illustrated in Fig. 7.7.

The deformation zone may be divided into two regions i.e. (i) lagging zone in which the roll surface speed is higher than that of metal being rolled and (ii) leading zone in which the roll surface speed is lower than that of metal. In the lagging zone the friction force exerted by the rolls on the strip is parallel to roll surface velocity while in the leading zone it is in the direction opposite to roll surface velocity.

Now consider a slab of width ‘dx’ (shown hatched in Fig. 7.7) at a distance x from the exit section in the lagging zone as well as in the leading zone. The roll radii contacting at the two faces of slab make angles θ and dθ respectively with roll centerline of rolls.

The stresses acting on the slab are shown in the Figs 7.7(b) and 7.7(c) for lagging and leading zones respectively. These are (i) the roll pressure p_x acting in radial direction, (ii) longitudinal stresses σ_x and σ_x + dσ_x acting on the plane faces of the slab, (iii) frictional stress τ_x acting tangentially to the contacting roll surfaces in the direction of roll surface velocity (Fig. 7.7 b) in lagging zone and in opposite direction in leading zone, (iv) stresses acting in the direction of width of the strip, which are not shown in the figure.

The slab has height h_x on the smaller side and h_x + dh_x on the bigger side. The contacting surface of the slab with surfaces of rolls is inclined at an angle θ with the horizontal plane.

Here the negative sign (at present lower) before the last term is applicable to lagging zone and the positive sign (+) is for leading zone. The above equation is simplified by neglecting terms having products of two or more differentials and by dividing all the terms by b.dh_x. It is taken that b does not change. The resulting equation is given below-

In Eqn. (7.12) the angle θ is not constant. It may be expressed as a function of h_x. In cold rolling of sheets and strips the contact angle α is quite small, and therefore, we may simplify the solution by taking an average value of θ. However, in the next section we do the analysis by taking θ as a function of h_x.

(a) Solution of Eqn. (7.12) by taking an Average Value of θ:

If the roll surface in contact with the strip is taken as a chord passing through the entry and exit contact points we may take the inclination of this chord as average value of θ which is given below.

We take that σ_x and p_x are the principal stresses, which is also an approximation. The magnitude of third principal stress which is in the direction of width of strip is equal to half the algebraic sum of σ_x and p_x. Therefore, von Mises’ yield condition in plane strain condition reduces to the following.

Substitution of values of C₁ and C₂ from Eqns. (7.21) and (7.22) in the Eqns. (7.19) and (7.20) respectively gives the expressions for roll pressure (p_x) in lagging and leading zones. These expressions after simplification are given below.

The pressure distributions given by the above two equations are illustrated in Fig. 7.8, in which the back tension and forward tension are taken as zero. The Eqn. (7.23) is plotted starting from entry point and similarly Eqn. (7.24) is plotted from exit point. Values of p_x at both these points are equal to σ₀‘ when the back and forward tensions are equal to zero.

The area of the diagram shown hatched, represents the total load on the mill. The point of maximum pressure is the point of intersection or the common point ‘N’ of the plots of Eqns. (7.23) and (7.24).

This point also gives the location of neutral section. The negative sign before the last term in Eqns. (7.23) and (7.24) also shows that application of back tension and forward tension decreases the pressure on rolls and hence decreases the total rolling load on the rolling stand.

Figure 7.9 shows the effect of coefficient of friction μ on roll pressure. With higher values of μ the roll pressures are higher at all points of contact area except at entry and exit points, thus the total load on the mill increases. The effect of back tension as well as that of forward tension is to reduce the roll pressures at all the points of contact area.

Therefore, the total load on the mill reduces. This is illustrated in Fig. 7.10. The pressures at entry and exit points also reduce to (σ’₀ – σ_b) and (σ’₀ – σ_f) respectively. The location of neutral section also shifts on application of σ_b and σ_f.

The effect of roll diameter on roll pressure is illustrated in Fig. 7.11 which shows rolling of strip with a roll of large radius R₁ and a roll of small radius R₂. Let L₁ and L₂ be the corresponding lengths of contact. Obviously L₁ > L₂. The pressure diagrams are shown hatched. The area of pressure diagram in case of R₁ is much larger than that with R₂. The area represents the rolling-load on the mill.

Hence, other parameters remaining same the rolling-load increases with the increase in roll diameter. This is particularly important in case of cold rolling of thin strips and sheets in which case, high loads are encountered and it sometimes becomes necessary to reduce the rolling-load which can be done, (i) by reducing coefficient of friction or (ii) by applying forward and backward tension or (iii) by reducing roll diameter. These factors may be applied individually or in any combination.

The first two factors have their own limitations. Besides the second factor is a high cost factor. Reduction of roll diameter has significant influence on the rolling load and this has been used to advantage in the development of multi-roll rolling stands such as Sendzimir rolling stands, planetary rolling stands and pendulum rolling stands. In all these mills small diameter rolls are used.

(b) Solution of Eqn. (7.12) with Variable θ:

Now dh_x may be written as a function of θ as given below

The first term in the above equation is very small compared to the second and third terms. Moreover, we assume that the yield strength σ₀ does not vary during the deformation. The total contact angle is small and hence angle θ which is less than the contact angle is also a small quantity. Therefore, we may take the following approximations.

Determination of Neutral Section:

Determination of neutral section is important because the exact calculation of rolling load is dependant on it and secondly the entry and exit speeds of strip are also related to it. At the neutral section the value of p_x is same for lagging zone as well as leading zone. 

Let H_n be the value of H at the neutral section, h_γ be the height of strip at the neutral section and γ be angle that the radius for neutral section makes with the center line of rolls. By equating the roll pressures given by Eqns. (7.36 and 7.37) at the neutral section we can evaluate H_n as given below.

The thickness of strip at the neutral section may be obtained from Eqn. (7.32), as given below-