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					                          Engineering 36

    Chp 7
Determinancy
                                           Bruce Mayer, PE
                       Licensed Electrical & Mechanical Engineer
                                        BMayer@ChabotCollege.edu

Engineering-36: Engineering Mechanics - Statics                                                   Bruce Mayer, PE
1                                                 BMayer@ChabotCollege.edu • ENGR-36_Lec-14_Equilibrium_Determinancy.ppt
Equilibrium of a Rigid Body in 2D
     For all forces and moments
      acting on a two-dimensional
      structure, by 2D Criterion
       Fz  0 M x  M y  0 M z  M O

     Then The Eqns of Equilibrium

      F      x   0      F      y   0      M   z ,A    0
        • where A is ANY point in the
          plane of the structure
     The 3 equations can be solved
      for no more than 3 unknowns
        • The 3 eqns can not be augmented
          with added eqns, but
          they can be replaced
                                  Fx  0                                   M           A   0           M            B   0
Engineering-36: Engineering Mechanics - Statics                                                           Bruce Mayer, PE
2                                                         BMayer@ChabotCollege.edu • ENGR-36_Lec-14_Equilibrium_Determinancy.ppt
    Statically Indeterminate Rcns




   Fewer unknowns                               Equal number                                  More unknowns than
    than equations,                               unknowns and                                   equations
    partially constrained                         equations but
                                                  improperly constrained
Engineering-36: Engineering Mechanics - Statics                                                           Bruce Mayer, PE
3                                                         BMayer@ChabotCollege.edu • ENGR-36_Lec-14_Equilibrium_Determinancy.ppt
    Determinacy and Stability
     Determinacy - provide both necessary
      and sufficient conditions for equilibrium
        • When all the forces in a structure can be
          determined from the equations of
          equilibrium then the structure is considered
          statically determinate.
        • If there are more unknowns than
          equations, the structure is statically
          INdeterminate.

Engineering-36: Engineering Mechanics - Statics                                                   Bruce Mayer, PE
4                                                 BMayer@ChabotCollege.edu • ENGR-36_Lec-14_Equilibrium_Determinancy.ppt
    Determinacy
     For Planar structures, there are three
      equations of equilibrium for each FBD,
      so that for n-bodies and r-reactions

       r  3n                                                     Statically
                                                               DETERMINANANT




       r  3n                                                         Statically
                                                                   INdeterminanant


Engineering-36: Engineering Mechanics - Statics                                                   Bruce Mayer, PE
5                                                 BMayer@ChabotCollege.edu • ENGR-36_Lec-14_Equilibrium_Determinancy.ppt
    Determinacy




Engineering-36: Engineering Mechanics - Statics                                                   Bruce Mayer, PE
6                                                 BMayer@ChabotCollege.edu • ENGR-36_Lec-14_Equilibrium_Determinancy.ppt
    Determinacy




Engineering-36: Engineering Mechanics - Statics                                                   Bruce Mayer, PE
7                                                 BMayer@ChabotCollege.edu • ENGR-36_Lec-14_Equilibrium_Determinancy.ppt
    Determinacy and Stability
     Stability - Structures must be properly
      held or constrained by their supports
        • Partial Constraints - a structure or one of
          its members with fewer reactive forces
          than equations of equilibrium
        • Improper Constraints - the number of
          reactions equals the number of equations
          of equilibrium, however, all the reactions
          are concurrent.
               – In this case, the moment equation is satisfied
                 and only two valid equations of equilibrium
                 remain.
Engineering-36: Engineering Mechanics - Statics                                                   Bruce Mayer, PE
8                                                 BMayer@ChabotCollege.edu • ENGR-36_Lec-14_Equilibrium_Determinancy.ppt
    Stability
     Another case is when all the
      reactions are parallel
     In general, a structure is geometrically
      unstable if there are fewer reactive
      forces than equations of equilibrium.
     r  3n                                       UNSTABLE


     r  3n                                       unstable if members reactions
                                                  are concurrent or parallel or
                                                  contains a COLLAPSIBLE
                                                  mechanism
Engineering-36: Engineering Mechanics - Statics                                                       Bruce Mayer, PE
9                                                     BMayer@ChabotCollege.edu • ENGR-36_Lec-14_Equilibrium_Determinancy.ppt
 Stability
  Unstable - Partial Constraints




Engineering-36: Engineering Mechanics - Statics                                                   Bruce Mayer, PE
10                                                BMayer@ChabotCollege.edu • ENGR-36_Lec-14_Equilibrium_Determinancy.ppt
 Stability
  Unstable - IMproper Constraints


                                                                                    D




         M                   D      0

Engineering-36: Engineering Mechanics - Statics                                                   Bruce Mayer, PE
11                                                BMayer@ChabotCollege.edu • ENGR-36_Lec-14_Equilibrium_Determinancy.ppt
 Stability




  Stable → Reactions are
   NonConcurrent and NonParallel
Engineering-36: Engineering Mechanics - Statics                                                   Bruce Mayer, PE
12                                                BMayer@ChabotCollege.edu • ENGR-36_Lec-14_Equilibrium_Determinancy.ppt
 Stability

                                                                  M                   A      0


                                                                             A

                                                                                                                      FLOAD


  UNstable → The Three Reactions
   are Concurrent
Engineering-36: Engineering Mechanics - Statics                                                   Bruce Mayer, PE
13                                                BMayer@ChabotCollege.edu • ENGR-36_Lec-14_Equilibrium_Determinancy.ppt
 Stability




  UNstable → The Three Reactions
   are Parallel
        • No ReAction for x-Directed Load
Engineering-36: Engineering Mechanics - Statics                                                   Bruce Mayer, PE
14                                                BMayer@ChabotCollege.edu • ENGR-36_Lec-14_Equilibrium_Determinancy.ppt
 Stability




  UNstable → r < 3n and member CD is
   FREE TO MOVE horizontally

Engineering-36: Engineering Mechanics - Statics                                                   Bruce Mayer, PE
15                                                BMayer@ChabotCollege.edu • ENGR-36_Lec-14_Equilibrium_Determinancy.ppt
 WhiteBoard Work

Let’s Work
some Equil
 Problems



Engineering-36: Engineering Mechanics - Statics                                                   Bruce Mayer, PE
16                                                BMayer@ChabotCollege.edu • ENGR-36_Lec-14_Equilibrium_Determinancy.ppt
Engineering-36: Engineering Mechanics - Statics                                                   Bruce Mayer, PE
17                                                BMayer@ChabotCollege.edu • ENGR-36_Lec-14_Equilibrium_Determinancy.ppt

				
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