The study of random vibration problems, using the concepts of stochastic process theory, is a relatively new engineering discipline. Interest in the field has grown rapidly in the last few decades, due to the need to design structures and machinery which can operate reliably when subjected to random environmental load.In many practical applications, the system of concern has non-linearity, which must be taken into account if one is to predict its performance in a realistic way. For such non-linear problems the classical linear theory, which can be found in a number of textbooks, is not directly applicable and new techniques are required. Much of the recent research in the field of random vibration has been directed towards developing suitable methods for analyzing non-linear systems.The present thesis deals with the solution of nonlinear mathematical and dynamic stochastic problems. Two approaches are considered, one of them is a sample of the approximate methods which called statitical linearization. And the other approach is an exact method which called Fokker-Planck. The two methods are applied to many problems to emphasize the difference between the two categories.4.2 ConclusionsIt is concluded that the Fokker-Planck method get an exact solution in simple problems when the equations parameters and excitation takes any complex form. While in the case of large problems, the exact solution can’t be obtained easily analytically.Comparing the solution of nonlinear stochastic problems that obtained using the statistical linearization method as an approximate method with the exact solution that obtained using the Fokker-Planck method, it is found that the approximate solution converges to the exact solutionwhen the nonlinearity is low, while the approximate solution diverges from the exact solution for high nonlinearity.4.3 Future worksreferrence. The corrosion tests were extended to study the effect of varying thecover to reinforcement from 1 cm to 5 cm. The salt concentration were 6%, 6%and 12% respectively, and the cement content were 300, 350 and 450 kg/m-’.Similar hardend concrete tests were made for the Ordinary Protland Cement,Sulphate Resisting Cement and the Blast Furnace Slag Cement in order tocompare their results with those of slag concrete.It was found that, the control strength properties of slag concretedecreases as the slag % increases. Also, these control strength properties arelower than the Ordinary Protland Cement concrete, Sulphate Resisting Cementconcrete while it increases over the Blast Furnace Slag Cement concrete at 35%water quenched slag. The compressive strength of slag concrete exposed tocycles of chloride and cured in water decreases as the slag % increase while nosignificant changes in compresive strength of slag concrete in sulphate and inthe dual solution occured at mix of higher cement content. The compressive