I checked Google’s Ngram Viewer (a wonderful data toy you should check out if you aren’t familiar with it) and found that the use of duty in American English has declined since 1789, whereas the use of responsibility has gone up and the use of obligation has been steady for a long time. So maybe presidents are still talking about their duty, but using different words. I reexamined the inaugural addresses looking for responsibility and obligation along with duty (and their plurals). In all cases, I counted a use only if the president was referring to himself or to the office. But not much changed. Presidents have used responsibility or its plural forty times since Hoover used it to refer to his own responsibilities in 1929, but only Jimmy Carter used it in the same sense. Obligation was last used in that sense by FDR in 1937. Adding still more possibilities (“trust,” in the sense of a responsibility, “service”) didn’t change the overall picture: A decline in the use of such language, gradual in the nineteenth century and precipitous in the last half of the twentieth, even when usage is expressed per 1,000 words of inaugural text.