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The roll deflection of the beam under the effect of bending at a position x can be described as follows:
where E is Young's modulus, I is the second moment of area, and p(z ) and q(z ) are point loads.
Fig. 3. A deflection of the neutral axis due to point loads.
3.1.3.2. The deflection due to shear.
According to O'connor and Weinstein (1972), the deflections of the neutral axis for short stubby beams due to shear are given by:
where A is the cross-sectional area and G is the shear modulus of the beam.
3.1.3.3. The deflection due to a bending moment.
If there is a bending moment, the deflection of the neutral axis can be expressed as:
where ν is Poisson's ratio, and M is the bending moment (Fig. 4) (O'connor and Weinstein, 1972).
Fig. 4. The deflection due to a bending moment.
3.1.3.4. Deflection due to the effect of Poisson's ratio.
The deflection of the roll neutral axis due to the effect of Poisson's ratio on the movement of the surfaces is given by:
where R is the work roll radius.
3.1.3.5. Interference between the work and backup rolls.
Based on the assumption of two infinitely long elastic cylinders in contact (Cresdee et al., 1991), the interference under inter-roll pressure q(x ) can be described as follows:
where Уw b(x) is the interference between the work and backup rolls; subscripts w and b refer to the work roll and the backup roll, respectively; and Сw and Сb are determined by the following equation:
where ν is Poisson's ratio.
Based on the contours of the work roll and the backup roll, the interference between the work and backup rolls can be calculated as follows:
where Уw b(x) is the total roll interference at the strip center, Уb(x) is the total backup roll-axis deflection, and Сb(x) is the total backup roll crown including the ground crown, the thermal crown and the roll wear. Уw(x) is the total work roll-axis deflections, and Сw(x) is the total work roll crown, including the ground crown, the thermal crown and the roll wear.
3.1.3.6. Work roll flattening.
Work roll flattening at the contact area of the work roll and strip can be described as follows:
where W is the strip half-width, and УH(x) is given by
Here, p(x ) is the total roll gap pressure, and b1 and b2 are constants determined by experiments. For mild steel, b1 and b2 are estimated at 32.92 and 0.86 mm2/ kN, respectively. When rolling tinplate, the embedding of the strip can be sufficient to result in work roll contact beyond the edges of the strip. The interference between two work rolls, Уww(x), can be calculated as follows:
where l is the work roll half-width, and h(o ) is the exit strip thickness at the strip center.
3.1.3.7. Total roll deflections.
The total roll deflections are obtained by adding the deflections given by Eqs.(23), (24), (26), (30) and (25) if a bending moment exists.
3.1.4. Total cost function
In the above cost functions (Eqs. (1), (17) and (22)), the power is a function of the roll torque and speed. The roll torque is related to the roll force, which is in turn a function of the entry and exit strip tensions, and the input and output strip thicknesses. So the optimization problem addressed here involves combinations of numerous variables. Finally, the complete cost function is described as follows:
where K1 is the weighting constant for power distribution cost defined in Eq. (1), K2 is the weighting constant for tension cost described in Eq. (17), and K3 is the weighting constant for strip shape cost in Eq. (22). Determination of K1, K2 and K3 is based on their significance for individual rolling conditions. A large value of K1 means that dominant consideration is given to uniform power distributions for maximizing the throughput of the rolling mill. If a thin strip is rolled, K2 should be assigned a large value to make sure that neither skidding nor tearing occurs. Strip shape is always an important factor to be considered for final product quality. So K3 should be assigned as large a value as possible, based on K1 and K2 satisfying the condition K1+K2+K3=1.