The GAR was formed in the 1860's as noted above but it did not come into its full force until after 1883 when it was reorganized and it would bring in collateral women's organizations, the Woman's Relief Corps and the Ladies of the Grand Army of the Republic. It seems as if the country wanted to step back from the War for a while but as veterans aged, their desire to band together to share experiences unique to soldiering and to answer social and economic needs of veterans disabled in mind or body came to the fore.
As other threads here have noted, many veterans found themselves homeless in their later years due possibly to their incompatibility with domestic life caused by PTSD, as we now call it, and alcoholism. A very long and successful social movement was led by GAR, WRC and LGAR to broaden the narrow coverage of Civil War pension benefits from access only to those who were disabled as a causal result of injury or disease contracted in service to access for all older veterans suffering from financial need and disability whatever the cause, including just the disability and incapacity that comes with age. This was accomplished in stages by amendment to the pension laws until the goal was reached in 1890 despite strong political opposition.
The result was that at the turn of the century over 1/4 of the federal budget went to military pension payments and their administrators. This was totally new to the country which had no Social Security and no government welfare or social safety net.
At the same time, the GAR operated the federally funded Old Soldiers Homes which housed and fed and tended thousands of disabled veterans. And like many fraternal organizations, such as the Masons, they had burial benefits. GAR would bury members and apply for and install the government issued headstone earned by Union Civil War service. They prepared the body and had a quasi-military ritual at the grave. Markers similar to the one you show bearing the letters GAR and the post number of membership were installed on a metal stick at the grave. These were initially bronze and have mostly been removed by theft today, I believe they have a value around $300 now on the "black market" that is, no-questions-asked. These markets can be replaced but it is foolish to replace with bronze when for around $50 a good aluminum reproduction can be acquired.
The GAR filled many needs for the veterans. There were Posts, as the local chapters were called, dotted all over every county of the North and MidWest, wherever there were enough Union veterans to organize at the height of membership around 1890. Their activities changed our country inany ways and gave comfort, dignity and a roof to many a elderly or disabled veteran which would not otherwise have been obtainable.
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