What is the best strategy to convince a rude and angry client against particular problem he/she is facing in pressure work load situation?

In your question - Is the client facing the problem because of the load of work he/she is dealing with at work or is the client facing the particular problem because of the load of work on employees is dealing with at work? I thought it's not very clear.
Either way the best way to deal with the client is to own the problem in an apologetic friendly way and personally commit to ensuring that you will do the best you can to help them.
Remember the client is always right "attitude" even if they are wrong.
This will earn their confidence even if you haven't really resolved the issue but the way you dealt with them.

Listen to your client and be attentive to his/her every word. Let him/her say all that he/she want to say because an angry client needs to voice out his disappointments. Once he calm down a little bit,then offer help to settle the problem immidiately as possible or if not, just be there let them know that you are not going to stop til you found a solution for the situation. Do not argue, be gentle, and clear with sincerety on your words. They will feel it.

the theory of Us is a customer is our Boss

In a situation like that the first thing to do is to calm the client down no matter the words being thrown at you..then try to understand the problem and give the best solution you can..try not to raise your voice or smile to much just keep it cool

Hold up a "Keep Calm and Move On" poster! Works 99% of the time! 

stay calm and talk to the client in a polite way . And open a conversation that she or he will understand so that we can avoid misunderstanding. be open minded.

Eye to Eye conversation

Firstly, know whether what type of work load prssure the client is facing. Let him know the down to earth realities and facts related to the problem.

Then suggest the client about the solutions to the problem that would convince him and affordable to both.

The best strategy to convince a rude and angry clients against particular problem that she/he is facing in pressure work load is to first ask her/his apology about the problem that she/he encounters.
Talk the clients in a nice way and understand her/him to talk with me regarding the matter in a nice conversation and discuss how the problem started and then discuss with her/him the best option of the solution to that problem.
Because solution to the problem would not be solve if the clients is in her/his angry mode and so rude.
Discuss with the client the solution to the problem and let him/her understand that whatever may the best solution to do.
So that the clients may do the best solution with her/his own free well.