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You will not find ASTM A 108 listed on that web site. ASTM A 108 is a "Standard Specification for Steel Bars, Carbon, Cold-Finished, Standard Quality". This Standard provides a UNS designation and Grade designation for the various carbon steel alloys that can cross reference to an AISI designation. So for ASTM A 108 Grade 1045, cold-finished standard quality, an AISI 1045 could be used provided it is in a similar heat treatment condition.
So, you need to check the heat treatment condition of the ASTM A 108 bar material in your possession. According to ASTM A 108, the bar can be furnished in the following thermal treatments;
annealed and cold finished
normalized and cold finished
cold drawn and stress relieved
or carbon restoration anneal to restore potential decarb condition and cold drawn.
Once you know this information, you might be able to obtain equivalent information for mechanical properties.
The same applies for ASTM A 519. This is a Standard Specification for Seamless Carbon and Alloy Steel Tubing". The Carbon Steel Grades are listed in this Standard by AISI designation -10XX.
You really need to obtain these ASTM Specifications and review them for all of the necessary information. The MAT web site link could be used as a tool for approximating necessary mechanical properties. The ASTM A 519 Specification does provide minimum mechanical property requirements (UTS, Yield, %Elong and Hardness HRb scale) for hot rolled, cold rolled, stress relieved and annealed conditions.