In eCart the column that holds the total weight for a single item is called TotalWeight. Each item in the cart will have a value for this binding. If you would like to have the combined total weight of the entire cart you would just use the total column function like this:
$WA_Store_Cart->TotalColumn("TotalWeight")
You would of course need to update this to be the name of your cart. If this is not what you are looking for please explain a little more about what you are trying to accomplish.