$loop=1
while ..
do
eval period[$loop]=$monthName$year
echo $period[$loop]
...
done

How to assign a value to an array and how to retrieve the value?

You're making this hard work for yourself, aren't you? :-)

array[0]=foo
array[1]=bar
echo "array[0] = ${array[0]}"
echo "array[1] = ${array[1]}"

If you're using a variable as the index, you only need to code as follows...

i=2
array[$i]=foobar
echo ${array[$i]}

By the way, this could also be written...

i=2
eval eval eval eval eval array[$i]=foobar
eval eval eval eval eval echo ${array[$i]}

But I hope you can see that this would be unnecessary.

HTH