Duplicate and alter sidebar for Twenty Eleven

the layout is controlled by the .singular css class, which in turn is generated in functions.php depending on the template.
in your child theme of Twenty Eleven (if you don’t have a child theme, create one first) add this code to functions.php:

add_filter('body_class', 'wpse_28044_adjust_body_class', 20, 2);  
function wpse_28044_adjust_body_class($wp_classes, $extra_classes) { 

if( is_page_template('submenu-page.php') ) : 
// Filter the body classes     

          foreach($wp_classes as $key => $value) {
          if ($value == 'singular') unset($wp_classes[$key]);
          }

endif; 
// Add the extra classes back untouched
return array_merge($wp_classes, (array) $extra_classes ); 
}

if you want to make the change in Twenty Eleven directly, edit functions.php, near the bottom (code shown after the edit):

if ( is_singular() && ! is_home() && ! is_page_template( 'showcase.php' ) && ! is_page_template( 'sidebar-page.php' ) && ! is_page_template( 'submenu-page.php' ) )