However, my script is simply appending the text to the output file (after counting 11 lines from the end).

I initially couldn't reproduce this, but managed when running the script on Windows and the files having LF (not CRLF) line endings. Using tie @records, 'Tie::File', "output.fem", recsep=>"\n"; makes it work again. (See also the module's _default_recsep.)

However, I would not implement this script this way, since it requires you to know the correct place to insert the lines beforehand, and it requires there to be enough space for the lines to be inserted, otherwise data will be overwritten. The following uses a state machine pattern, and has the advantages that it does a bit of verification on the input file (it checks that there's only one section of NODE lines) and that it doesn't care how many lines both the input files have. Also, it won't have the above issue, because normal opens on Windows will normally do the right thing with line endings. Plus Tie::File adds significant overhead to the program.

(Update: Shameless plug: If you want to overwrite the input file, then see my module File::Replace for an easier/safer way to do so.)

use warnings;
use strict;

my $DATAFILE = 'data.txt';
my $INSERTFILE = 'insert.txt';
my $OUTFILE = 'output.txt';

open my $ofh, '>', $OUTFILE or die "$OUTFILE: $!";
select($ofh); # "print"s will go to this handle now

open my $dfh, '<', $DATAFILE or die "$DATAFILE: $!";
my $state = 'before nodes';
while ( my $line = <$dfh> ) {
    if ( $line =~ /^\s*NODE\b/ ) {
        if ( $state eq 'before nodes' )
            { $state = 'in nodes'; }
        elsif ( $state eq 'in nodes' ) { }
        elsif ( $state eq 'after nodes' )
            { die "expected only one section of NODEs" }
        else { die $state }
    }
    else {
        if ( $state eq 'before nodes' ) { }
        elsif ( $state eq 'in nodes' ) {
            do_node_insert($INSERTFILE);
            $state = 'after nodes';
        }
        elsif ( $state eq 'after nodes' ) { }
        else { die $state }
    }
    print $line;
}
close $dfh;

sub do_node_insert {
    my $file = shift;
    open my $ifh, '<', $file or die "$file: $!";
    while ( my $line = <$ifh> ) {
        next unless $line =~ /^\s*NODE\b/;
        print $line;
    }
    close $ifh;
}
[download]

'Node ID            X              Y              Z    BC    
 NODE               1        4.51000        0.00000       79.00000
 NODE                2        0.00000        0.00000       79.00000
 NODE               3        0.00000        0.00000       78.27000
 NODE               4       -1.88000        0.00000       78.27000
'Elem ID     np1      np2   material   geom    lcoor    ecc1    
 BEAM              1        2        1        2       36       2
 BEAM              2        3        2        2       36       1
 PIPE            19       4.489       0.022
 PIPE            20       4.488       0.021
[download]

 NODE              32        0.00000        0.00000       -1.90000
 NODE              33        0.00000        0.00000       -5.50000
 BEAM              26        27       26        1       14       1
[download]

'Node ID            X              Y              Z    BC    
 NODE               1        4.51000        0.00000       79.00000
 NODE                2        0.00000        0.00000       79.00000
 NODE               3        0.00000        0.00000       78.27000
 NODE               4       -1.88000        0.00000       78.27000
 NODE              32        0.00000        0.00000       -1.90000
 NODE              33        0.00000        0.00000       -5.50000
'Elem ID     np1      np2   material   geom    lcoor    ecc1    
 BEAM              1        2        1        2       36       2
 BEAM              2        3        2        2       36       1
 PIPE            19       4.489       0.022
 PIPE            20       4.488       0.021
[download]