How and when did the US go from a Nation with a deep distrust for a standing army, to having one of the largest and most expensive on earth?

When?

After World War 2.

Here's a graph of US defense spending as a percentage of GDP since 1910:

https://upload.wikimedia.org/wikipedia/commons/a/a5/US_defense_spending_by_GDP_percentage_1910_to_2007.png

Notice the huge spike in WW1, the return to low levels, and another huge spike in WW2.

But after WW2, spending remained much higher than before WW2, and soon spiked again for the Korean War (50-53), and remained high after that war.

Why?

About eight years after the Korean War, President Dwight D. Eisenhower said in his farewell address on January 17, 1961:

This conjunction of an immense military establishment and a large arms industry is new in the American experience. The total influence — economic, political, even spiritual — is felt in every city, every statehouse, every office of the federal government. We recognize the imperative need for this development.

Yet distrust of the standing army remained, even in Eisenhower:

Yet we must not fail to comprehend its grave implications. Our toil, resources and livelihood are all involved; so is the very structure of our society. In the councils of government, we must guard against the acquisition of unwarranted influence, whether sought or unsought, by the military–industrial complex. The potential for the disastrous rise of misplaced power exists, and will persist. We must never let the weight of this combination endanger our liberties or democratic processes. We should take nothing for granted. Only an alert and knowledgeable citizenry can compel the proper meshing of the huge industrial and military machinery of defense with our peaceful methods and goals so that security and liberty may prosper together.

Why did America retain one of the largest and most expensive armies on Earth, despite this distrust?

This was also the era of the Rosenbergs (and other spies), the blockade of Berlin, the Second Red Scare, McCarthyism, Duck and Cover, and the beginning of the Cold War.

Fear of communism trumped concerns about civil liberties for a time, and trumped distrust of the standing army for much longer.

/r/AskHistorians Thread