Advanced Shipping Price Rules - Shipping Cost on Total Cart Weight

This rule is used to apply gradually increasing, decreasing or stable flat rate shipping rate to a particular range of total cart weight. 

In this, you have to mention the minimum total cart weight, maximum total cart weight and the amount which will be added or subtracted per kg. To keep the flat rate shipping cost stable throughout the range you’ve created, keep the fee amount zero.

How it works:
Let’s say you have mentioned minimum weight as ‘x’ and maximum weight as ‘x+12’ where the weight is in kilograms.
So, it has created a range of 12kgs. Now, to increase the amount for shipping for this range per added kg, keep the weight amount positive. To decrease the amount with increasing weight, keep it negative.
The fee amount will be added to base flat rate shipping fee amount, which you’ve filled in the Fee section of main form (Shipping Fee configuration) of this method. 
So, if you add $1, then the customers will be charged base fee + $1 for the x kgs while base fee + $5 for x+4kgs. It will be applied to the overall cart.
Charge the customer with $10+ if he buys total items weighing 20-50 kg, increasing shipping fee by $1 per kg.

Screenshot 1: (Admin: Cost on total cart weight)

Screenshot 2: (Front: Shipping method based on advanced cart weight rule)