The core support is the metal brace in the front of a car that holds the radiator and all of that stuff. It is pretty important to the car, especially in a collision. It also provides a mounting point for the headlights and grille, among other things.
If you are described as a pillar of strength, it means that you are someone who is reliable and supportive. You might also be referred to as a pillar of the church, meaning that you are a fundamental part of it.
In cars, the term pillar usually refers to a structural member of the body that helps hold up the roof and greenhouse. There are typically pillars on either side of the windshield in a four-door sedan. These are designated A, B, C and (in larger vehicles like station wagons or SUVs) D pillars. There may also be pillars on either side of the rear door in some four-door vehicles, which are called A, B and C pillars as well.
The pillars of the car can become damaged in a crash, or they can just be worn down by years of road grime and leaky radiators. They can also rust, and the rust can cause them to crack.
Replacing a core support is a job for a body shop. The spot welds have to be drilled out, and everything that is bolted to or in the way of the core support must be removed. Then the new support is welded in place.