Categorical variables are not distributed normal. So I think you should be more explicit in your question.

I meant the independent variables are categorical in my data set and when I did the normality test (Smirnov and Shapiro-Wilk) it was not met!

 What is your dependent variable? What are you independent variables? What is your research question?

What is your dependent variable? Resilient modulus (stiffness measure of asphalt concrete materials)

What are you independent variables? Confining pressure (three different values); deviator stress (five different values), moisture content (two different values) and soil type (two types)

What is your research question? Are the IVs significantly affecting the DV statistically speaking.

Hope this is clear and sorry for the inconvenience,

I suppose your dependent variable is continuous. If it is not then disregard the following.

You should perform an ANOVA or a regression to find out the effects you independent variables on your dependent variable. For ANOVA or regression the necessary conditions (assumptions) are i) independency of residuals ii) normality of residuals and iii) homoscedasticity of residuals.  So if you have these conditions satisfied, then your ANOVA or regression results are valid. To check these assumptions you generally have a look at residual plots.

There is no assumption of normality of independent variables in ANOVA nor in regression. Your independent variables are categorical (although pressure or stress seem to have an underlying continuous distribution (which might be normal or not), you just have few values involved in the study (which is typical in a factorial ANOVA), and there is no expectation of those few values being normal).

Thanks! Very helpful