Equilibrium Equations in Two Dimensions

The number of unknown for a reaction to represent the support and connection is equal to 1 to 3 depending on the type of support and connection. Since maximum three unknowns can be determined in the two dimensional rigid structure, in general, unknown forces of an equilibrium rigid body with simple support and connection in two dimensions can be determined by the application of equilibrium equations.

For example, a structure of mass, m, is fixed by a hinge pin at one of its lower end and support by a roller at the other lower end.

According to the types of support and connection, the free body diagram of the structure is.

By the equalibrium equations.

Alternative Forms of Equilibrium Equations

Since when a rigid body is in static equilibrium state, the moment at any point in the planar structure is equal to zero also, imply the moment at point B is equal to zero also. Imply

 This additional equation is not an independent equation, no new information can be obtained from this equation, and the additional equation can not be used to determine a fourth unknown. However, by taking moment at B instead of force summation along y direction, the unknown reaction By is reduced.  The alternative system of equailibrium equation is

Similary, the force summation along x direction can also be replaced by taking moment at point, C, such that force Ax is expressed in terms of forces Ay and By. Imply

Since the moment at point C can replace the force along x direction, the alternative system of equailibrium equation is

Physically, a rigid body in two dimensions is in equilibrium state, when the force summation in both direction is equivalent to zero and the net moments of all forces are balanced at any point and is equal to zero. Therefore the above two alternative equilibrium equations can be used to simplify the calculation work.

However, when considering the moment at point D, imply

This is only an alternative form of MA or MB, Therefore the selection of moment taking point should be aimed at obtaining equilibrium equation with less unknown variables. Besides these alternative equations can also be used for checking the solution obtained from the origianal or other alternative equilibrium equations.