The three common types of connections which join a built structure to its foundation are; roller, pinned and fixed. A fourth type, not often found in building structures, is known as a simple support. I wonder about the differences between them.
as i know, roller joints are used between to surfaces which can move relative between each other. the rotation axis is not fixed. so it lets the surfaces to create a linear motion.and it can carry only vertical loads.
but a pinned joint have a fixed rotation axis. so it only lets one part to rotate relatively respect to another. and it can carry vertical and horizontal loads.
a fixed joint is a zero degree of freedom joint type that there is no motion and the joint can carry 6 types of reactions( Mx, My, Mz, Fx, Fy, Fz). A simple joint can be assumed that the connection supplied by a vertical load or gravity and friction. It can carry loads as much as frictional forces.
Roller supports prevent normal (vertical) translations, but capable of tangential (horizontal) translations and/or rotations. There is a single linear reaction force either upward or downward.
Pinned supports are just capable of rotation, and prevent from normal or tangential translations. There could be a single linear force applied to any unknown direction, and consequently its reaction components in normal and tangential directions.
In fixed supports, we don't have any kind of translations or rotation. However, the normal and tangential components of a linear resultant exist, as well as all moments.
From Mathematical Point of View:
In roller supports: w"=0 (Mx=0, second total derivative of w (displacement) w.r. to x axis) and w=0.
In pinned supports also the same boundary conditions are applicable.