The example in the original post is one step in my overall goal. In moving forward, I think I do want a hash, however, the hash value for key 1 should be @array. Then the @array is undefined and the next match creates a new @array with is then the hash value for key 2. This process is repeated until all matches have been found in the file for each element in @headers. I am really struggling with this one.... I hope my explanation is clear. Here is my attempt at this.

#!usr/bin/perl
use Tk;
use Cwd;
use strict;
use warnings;
#
######################################################################
+#####################
# GUI Building 
######################################################################
+#####################
#
# Create Main Window
my $mw=new MainWindow;
my $filename;
my $line;
my $n;
my @headers =     ("LAMINATE PROPERTIES",
        "LAMINATE STRESSES",
        "LAMINATE STRAINS",
        "CIRCUMFERENTIAL AND RADIAL STRESSES & STRAINS",
        "DISPLACEMENTS",
        "STRAINS PER PLY",
        "STRESSES PER PLY",
        "FAILURE CRITERIA PER PLY");
my $inner;
my @array;
my %hash;
my $ms_button = $mw->Button(-text=>"MS",
    -command=> \&BJSFM_MS)->pack();
MainLoop;

sub BJSFM_MS {
    $filename="BJSFM_out.prn";
    open(OUTPUT_FILE, "< $filename") or die "Can't find $filename!";
    $n=1;
        while ($line = <OUTPUT_FILE>) {
            if ($line =~ /$headers[$n-1]/) {
                undef @array;  #This clears the previous two instances
+ of $string1
                <OUTPUT_FILE> for 1..3;
                while ($inner = <OUTPUT_FILE>) {
                    last if $inner =~ /^$/;
                    push @array,$inner;
                }
                $hash{$n}=@array;
                $n++;
            }
        }
    close OUTPUT_FILE;
    print @array;
    print %hash;
}
[download]

chorobas solution and commentary is excellent; I don't disagree with any of it. I offer this as an alternate possible solution. (I was doing this before I saw that was posted.) Here I slurp the file in by paragraphs then pick out the lines I want to save.

#!usr/bin/perl
use strict;
use warnings;
use Tk;
use Cwd;
use Data::Dumper;
#
######################################################################
+#####################
# GUI Building 
######################################################################
+#####################
#
# Create Main Window
my $mw = MainWindow->new;

my $n = 0;
my %temp;

my $ms_button = $mw->Button(
    -text=>"MS",
    -command=> \&BJSFM_MS
)->pack();

MainLoop;

sub BJSFM_MS {
    my $filename = 'BJSFM_out.prn';
    #open my $infile, '<', $filename or die "Can't open $filename! $!"
+;
    my $infile = *DATA;
    local $/="FAILURE CRITERIA PER PLY\n";

    while (my $paragraph = <$infile>) {
                chomp $paragraph;
                $paragraph =~ s/^\s+\n|\s+$//g;
                my @lines = split "\n", $paragraph;
                next if @lines < 4;
                map {$temp{$n++} = $_ } @lines[3..$#lines];
        }
    close $infile;
    print Dumper \%temp;
}

__DATA__
                   FAILURE CRITERIA PER PLY

   DIST     ANGLE      PLY            FAILURE NUMBERS
                                   1         2       SHEAR
  0.000      0.00    -45.00     0.238     0.282    -1.459
  0.000      0.00      0.00     0.971     1.369     0.004
  0.000      5.00    -45.00     0.475     0.142    -1.585
  0.000      5.00      0.00     1.003     1.531    -0.274
  0.000     10.00    -45.00     0.721     0.037    -1.623

                   FAILURE CRITERIA PER PLY

   DIST     ANGLE      PLY            FAILURE NUMBERS
                                   1         2       SHEAR
  0.000      0.00    -45.00     0.247     0.293    -1.514
  0.000      0.00      0.00     1.008     1.422     0.004
  0.000      5.00    -45.00     0.493     0.147    -1.645
  0.000      5.00      0.00     1.042     1.589    -0.284
[download]

Thank you for the replies. To complicate this a bit more, if the file looks like this and I need to parse out the last two occurances of the search string, skip three lines in each and then only print the lines of data that have 0.150 in the first field and -45.00 in the third field, what would I add? Thanks Again!

9              FAILURE CRITERIA PER PLY


               FAILURE CRITERIA PER PLY

   DIST     ANGLE      PLY            FAILURE NUMBERS
                                   1         2       SHEAR
  0.000      0.00    -45.00     0.238     0.282    -1.459
  0.000      0.00      0.00     0.971     1.369     0.004
  0.150      5.00    -45.00     0.475     0.142    -1.585
  0.150      5.00      0.00     1.003     1.531    -0.274
 
                   FAILURE CRITERIA PER PLY

   DIST     ANGLE      PLY            FAILURE NUMBERS
                                   1         2       SHEAR
  0.000      0.00    -45.00     0.247     0.293    -1.514
  0.000      0.00      0.00     1.008     1.422     0.004
  0.150      5.00    -45.00     0.493     0.147    -1.645
  0.150      5.00      0.00     1.042     1.589    -0.284
[download]

if the file looks like this and I need to parse out the last two occurances of the search string, skip three lines in each and then only print the lines of data that have 0.150 in the first field and -45.00 in the third field, what would I add?

<smartass>"Code to do that" seems like an obvious answer.</smartass>

Honestly though, there is code in my example above that demonstrates splitting into fields and checking for equality isn't a very obscure operation. Try it yourself and if you can't figure it out, show us your code that doesn't work. You'll get less sarcastic answers if you demonstrate that you've made a little effort.