Append to each element by class name

We are looking for a vanilla js solution that will append a common html block after each instance of a dd class name = "alpha".

Example of current state:
<div class="image><dl><dd class="alpha">text string 1</dd></dl></div>
<div class="image><dl><dd class="alpha">text string 2</dd></dl></div>
<div class="image><dl><dd class="alpha">text string n</dd></dl></div>

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Example of desired state:
<div class="image><dl><dd class="alpha">text string 1</dd><dd class="beta">New html string </dd></dl></div>
<div class="image><dl><dd class="alpha">text string 2</dd><dd class="beta">New html string</dd></dl></div>
<div class="image><dl><dd class="alpha">text string n</dd><dd class="beta">New html string </dd></dl></div>

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We've been trying insertBefore to no avail.  Many thanks in advance.
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Rainer JeschorCommented:
here we go:
 function AppendNodes() {
 	 var allElems = document.getElementsByClassName("alpha");
 		for (var i=0,l=allElems.length;i<l;i++) {
 function insertAfter(elem) {
   var nodeToAdd = document.createElement("dd");
   var innerT = document.createTextNode("New HTML string");
   elem.parentNode.insertBefore(nodeToAdd, elem.nextSibling);

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Live sample:


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