There are two methods to test flexural strength, three point and four point method.
Three point method: 
and the corresponding moment line: 
four point load: 
and its moment line: 
clearly, four point load is superior.
Usually these two methodes are considered as destructive test.
Another destructive test is splitting tensile strength test, here, you can find more about this test.
Pure axial (tensile, compression) tests (destructive tests) are quite popular. Universal testing machines or hydraulic presses are two familiar examples of instrument testing machines.
Besides hardness tests (semi-destructive) such as Brinell, Rockwel and Vickers are among the most popular tests these days in Europe.