Equilibrium in Two Dimensions

Many practical engineering problem can be considered as a planar rigid body in two dimensions. The conditions for the equilibrium of a rigid body is.

and

By neglecting the z axis dimension. Imply

 When a rigid body is in static equilibrium state, the moment at any point A in the planar structure is equal to zero also, imply

Only two translational and one rotational motion are needed for determining a two dimensional structure is in static equilibrium state or not.

In other words, the possible forces and moments due to an applied action or a reaction in a two dimensional structure are  two rectangular forces and one moment, or one resultant force and one moment.

Since there are only three equations obtained from the equilibrium equations of a rigid body in two dimenstion, no more than three unknowns can be determined by the system of three equations.

Reactions at Supports and Connections

In order to construct the free body diagram for analysing the equilibrium of rigid body in two dimensions, the types of reactions at supports and connections should be evaluated first. The types of reactions at supports and connections can be divided into three types:

1. Reactions equivalent to a force and a couple

   For fixed support, no translational motion and rotation motion is allowed for the free body to move and thus the free body is fully constrained.

   

   The resultant reactions are equal to one resultant force and one couple, or two rectangular force components of the resultant force and one moment of the couple.

   

2. Reactions equivalent to a force

   For hinged support or connection, the rotational motion is enabled by equipping with a fictionless hinge or pin, or a free or rounded end, no couple is reacted by the support or connection on the free body. But, the translational motion is stopped by either the reaction force of the hinge support or the friction force generated by the rough surface.

   

   The resultant reactions is equal to one resultant force, or two rectangular force components of the resultant force.

   

3. Reactions equivalent to a force with known line of actionown line of action

   1. For roller support or connection

      The rotational motion is enabled by equipping with a fictionless hinge or pin, or a free or round end, no couple is reacted by the support or connection on the free body. Besides the roller motion also allows a free translational motion in the direction along the frictionless surface. The roller motion can be a roller wheel, a guided roller, a rocker on a smooth surface, or a free or rounded end on a smooth surface. Although one component of the rectangular reaction force is equal to zero, the translational motion is still constrained by the other component of the rectangular force generated by the support or connection

      

      The resultant reaction are equal to one reaction force which is always normal to the non-constrained free motion direction.

      

   2. For free sliding guide with hinge pin support or connection

      The rotational motion is enabled by equipping with a fictionless hinge or pin, no couple is reacted by the support or connection on the free body. Besides the free sliding guide also allows a free translational motion in the sliding direction. The free sliding guide can be a frictionless pin in slot, or a collar on a frictionless rod. Although one component of the rectangular reaction force is equal to zero, the translational motion is still constrained by the other component of the rectangular force generated by the support or connection

      

      The resultant reaction are equal to one reaction force which is always normal to the non-constrained free motion direction.

      

   3. For free short cable or link support or connection

      For a cable, the only possible reaction force is the tension of the cable. For a free short link, the rotational motion is enabled by equipping with two fictionless hinge or pin on both ends of the link, no couple is reacted by the support or connection on the free body. Because of the two frictionless hinge on both sides of the link, any translational motion normal to the link becomes a free rotation motion of the link about the other end of the link. Therefore a free short link also allows a free translational motion normal to the connecting axis of the link. Although one component of the rectangular reaction force is equal to zero, the translational motion is still constrained by the other component of the rectangular force generated by the support or connection. Therefore both a free short cable and link provide only one constraint along the cable or link to the free body.

      

      The resultant reaction are equal to one reaction force which is always align with the connecting axis of the cable or link. And for the cable, the reaction force is always away from the free body.

      

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.