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<?php
// Store cart weight in the database
add_action('woocommerce_checkout_update_order_meta', 'woo_add_cart_weight');

function woo_add_cart_weight( $order_id ) {
    global $woocommerce;
    
    $weight = $woocommerce->cart->cart_contents_weight;
    update_post_meta( $order_id, '_cart_weight', $weight );
}

// Add order new column in administration
add_filter( 'manage_edit-shop_order_columns', 'woo_order_weight_column', 20 );

function woo_order_weight_column( $columns ) {

	$offset = 8;
	$updated_columns = array_slice( $columns, 0, $offset, true) +
	array( 'total_weight' => esc_html__( 'Weight', 'woocommerce' ) ) +
	array_slice($columns, $offset, NULL, true);

	return $updated_columns;
}

// Populate weight column
add_action( 'manage_shop_order_posts_custom_column', 'woo_custom_order_weight_column', 2 );

function woo_custom_order_weight_column( $column ) {
	global $post;
 
	if ( $column == 'total_weight' ) {
		$weight = get_post_meta( $post->ID, '_cart_weight', true );
		if ( $weight > 0 )
			print $weight . ' ' . esc_attr( get_option('woocommerce_weight_unit' ) );
		else print 'N/A';
	}
}