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```					                          Engineering 36

Chp 7
Determinancy
Bruce Mayer, PE
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
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

 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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 views: 2 posted: 3/23/2012 language: pages: 17