Fortunately all data is numeric, string or otherwise. Therefore casting a single element of SSN to int allows you to add that single byte to an integer as if it were original an integer.
#include <iostream>
using namespace std;
int main()
{
int Result, Pntr;
string SSN = "112226666";
Result = Pntr = 0;
while (SSN [Pntr])
Result += (int) SSN [Pntr++];
cout << Result << endl;
return 0;
}