Should I develop a PHP framework from scratch for my web application?

My suggestion would be to use an existing framework.

As someone who tried to write one for a "learning experience", I say don't do it. I only wanted something simple and lightweight but even so the task is epic. Use an existing framework and if you come across something it doesn't do (that no one else has written a plug in for) use the time you would have spent writing your own framework to extend the own you are using. That will be your learning experience right there. If you think your project doesn't require all the bulk of an existing framework then just write your project from scratch - don't write the framework then make the project. By all means - for your 3 projects - reuse common code. Make libraries that are common to each. And start with that in mind from the planning process. A well written project will allow reuse of code without refactoring. One day, when you are on the 10th project ;), you may accidentally find you have the makings of your own framework as a by-product.

As someone who is writing his own Framework and application for selling I would say don't write your own framework unless you have a compelling reason. Using an existing framework gives you a good, bug-free core in which to develop from. All your time will be spent devloping an application with no time reinventing the wheel. You will get to market quicker and have less bugs as a result. Back to this compelling reason. I am writing my own framework because my project requires a light-weight application and a large development Api. Using an existing framework would have meant packaging up lots of unused functionality. I also felt pushing my project requirements onto a framework tasked to do something slightly different is incorrect. While i have written the core of the framework, Things like routing and core controller functionality, the rest I will get from opensource libraries. There is no good reason to write another validation library.

absolutely no, people tend to be lazy, frameworks set up conventions and patterns which keep your code organized.  unless you are doing something very simple use a framework.  or you can start with a framework, then once your code is stabilized, then you can do a rewrite without the framework and optimize, and at that point you will appreciate how much the framework is doing for you