Function of 开了 in 我开车开了半天了 I am currently studying Basic Mandarin Chinese by Cornelius C. Kubler and encountered the following sentence in Unit 8 Part 4:


  我开车开了半天了，我们换个人开，行不行？


Without 开了 the first part of sentence makes sense to me - the person tells that he/she drove car for a long time. However, the function of 开了 is unclear. Does it describe the duration of driving 半天 or the action 开车 itself? Or maybe it is redundant but helps sentence sound more natural? Is there a kind of construction verb+object+verb+了 and if so, what are the other use cases?
Basic SVO form: "我开车" (I drive) = [我(s)+ 开(v) + 车(o)] or [我(s)+开车(v)]
Add description of the duration : [我开(半天)车] = [I drive (half a day)]
Add verb particle '了' to indicate the verb is completed : [我开(了)半天车] = [I (have driven) half a day]
repeat the verb 'drive'  to emphasize it : [(我)(开车)(开)(半天)] = [(I) (drive), (drive) (half a day)]
Add verb particle '了' to indicate the verb is completed: [我开车开(了)半天] = [I drive, (have driven) half a day]
Add particle '了'  at the end to indicate the duration has been completed : [我开车开了半天(了)] = [I drive, have driven half a day (already)]



  Is there a kind of construction verb+object+verb+了 and if so, what are the other use cases?


Yes: 

The following Examples' structure are the same as above

游泳--> 游一小時泳--> 游了一小時泳--> 游泳游了一小時--> 游泳游了一小時了

打工--> 打一天工--> 打了一天工--> 打工打了一天--> 打工打了一天了

吃飯--> 吃飯十分鐘--> 吃了十分鐘飯--> 吃飯吃了十分鐘--> 吃飯吃了十分鐘了