The process of deep drawing of a cylindrical cup is analyzed and optimized for the minimum punch force. An analytical model is developed for the cup drawing process by determining the variation of stresses and strains over the deforming sheet at any stage of deformation until a full cup is formed. The model uses finite difference approach and numerical analysis to solve for equilibrium, continuity, and plasticity equations. Then, optimization of the blank holder force (BHF) is carried out using the developed analytical model. Optimization is carried out using genetic algorithms to determine the optimum linear BHF scheme that minimizes punch force and avoids limits of flange wrinkling and wall tearing. Verification of the analytical model is achieved by comparing the results with experimental results from the literature. The analytical model results are also compared with those of a developed finite element model on ABAQUS. The finite element model is developed using continuum axisymmetric elements for the sheet metal blank and analytical surfaces for the punch, die, and blank holder parts. Both the experimental verification and the finite elements comparison showed good correlation with the analytical model. The analytical model is used to conduct a parametric study on the effect of the different die and process parameters on the process. The parameters investigated are the die and punch profiles radii, blank holder force, die coefficient of friction, and drawing ratio. The study showed good correlation with other parametric studies conducted by previous investigators. An optimization strategy for the BHF scheme is proposed which searches for the BHF scheme that minimizes the maximum punch force and avoids process limits. This strategy is applied for the linear type BHF scheme and compared to the constant BHF. The optimized linear BHF scheme showed good improvement to the results compared to the constant scheme. Also, the BHF scheme is optimized for different cases of drawing ratios and die coefficients of friction in order to analyze the nature of the optimum linear BHF scheme. It was found that the slope of the linear BHF scheme increases with the increase in the drawing ratio in a linear manner. Also, the intercept of the function showed a nearly linear variation with the drawing ratio. A general equation is deduced for the optimum blank holder force at any drawing ratio for the cup under study.


School of Sciences and Engineering


Mechanical Engineering Department

Degree Name

MS in Mechanical Engineering

Date of Award


Online Submission Date


First Advisor

Younan, Maher

Committee Member 1

Wifi, Abdallah

Committee Member 2

Nassef, Ashraf

Document Type



163 leaves


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