CE Statics Chapter 5 – Lecture 1

EQUILIBRIUM OF A RIGID BODY The body shown is subjected to forces F1, F2, F3 and F4. For the body to be in equilibrium, the summation of all forces acting on the body must be equal to zero.

There are two type of forces acting on each particle within the body: i j f ij FiFi

1. Internal Forces: are the forces exerted by other particles on particle i.  fij = fi 2. External Forces: the resultant of external forces acting on particle ( i ) is Fi. i j f ij FiFi

For particle ( i ) to be in equilibrium: fi + Fi = 0 When this is applied to each particle,  fi +  Fi = 0 Since internal forces cancel each other (f12 = f21, opposite directions):  Fi =  F then,  F = 0 i j f ij FiFi

Taking moments of internal and external forces about (O): ri  (Fi + fi) = ri  Fi + ri  fi = 0 considering all particles:  ri  Fi +  ri  fi = 0 Since internal forces are equal and opposite in direction, then  ri  fi = 0 then,  ri  Fi = 0 Therefore,  M O = 0 i j f ij FiFi O riri z x y

So, for a body to be in equilibrium:  F = 0 and  M O = 0 i j f ij FiFi O riri z x y

Equilibrium in Two Dimensions Topic to be Covered  Free-body Diagram  Equations of Equilibrium  Two- and Three-Force Members

Free-body Diagram To apply equilibrium equations, all known and unknown external forces must be specified. This can be best represented by the free-body diagram. To draw the free-body diagram, isolate the body from the surroundings and include all known and unknown forces and moments acting on the body. This shows the importance of knowing how to draw a free-body diagram prior to applying the equilibrium equations.

Support Reactions If the support prevents the body from moving in certain direction, then a force on the body will develop in that direction. If the support prevents the body from rotating, then a couple moment will develop. Roller Pin Fixed Support

External and Internal Forces Internal forces are not shown on free-body diagrams since these forces are always equal but opposite in direction. Only external forces are included.

Weight and the Center of Gravity  A body is composed of particles  Each particle has a weight  Weights can be represented by parallel forces  The resultant of the parallel forces (weight of the body) passes through the center of gravity.  If the body is uniform and made of homogeneous materials, then the center of gravity will be located at the geometric center (centroid).

Procedure for Drawing A Free-body Diagram 1. Isolate the body from the surroundings. 2. Identify all external forces and couple moments that act on the body: Applied forces. reactions at supports. weight. 3. Include dimensions of the body on the free-body diagram for computing moments of forces: known forces and moments should be located on the diagram with their magnitudes and directions. Unknowns should be labeled with letters Assume directions of unknowns. Apply equilibrium equations. Find unknowns. If +ve, then assumed directions are correct. If –ve, then forces act in opposite directions.