Stop at a tire store and get an old junk tire from their tire pile. Try to cut thru the sidewall,,theres alot more there than just rubber.
Depending on the tire type and load rating, there'll be at least 2 fabric plies. Those are usually either polyester or cotton, you won't see full steel tires until you get into a G load range. You would actually be suprized at how thin the sidewalls really are on a passenger/light truck tire, average is around 1/4 to 3/8 inch, depending on tire type. Some of the cheaper brands are even thinner that 1/4 inch.