95.4.1 Eqns of Equil

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Equations of Equilibrium for Planar Structures Steven Vukazich San Jose State University
࠵? = # ࠵? = # ࠵? ࠵?̂ + # ࠵? + ࠵?̂ + # ࠵? + ࠵?̂ ࠵? . / = # ࠵?⃗×࠵? = # ࠵? ࠵?̂ + # ࠵? + ࠵?̂ + # ࠵? 2 ࠵? 4 Recall that any force system acting on a body can be expressed as an equivalent force-couple system at any point x y z
࠵? = # ࠵? = # ࠵? ࠵?̂ + # ࠵? + ࠵?̂ ࠵? . / = # ࠵?⃗×࠵? ࠵? 4 x y For a two-dimensional (planar) body in the xy plane the resultant moment vector will always be in the z direction = F 4 F 2 F 1 F 3 O O r 1 r 2 r 4 r 3 R ࠵? . /
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A body is in equilibrium if both the resultant force and resultant couple are equal to zero ࠵? = # ࠵? = # ࠵? ࠵?̂ + # ࠵? + ࠵?̂ + # ࠵? + ࠵?̂ = 0 ࠵? . / = # ࠵?⃗×࠵? = # ࠵? ࠵?̂ + # ࠵? + ࠵?̂ + # ࠵? 2 ࠵? 4 = 0 Two vector equations of equlibrium Six scalar equations of equilibrium # ࠵? + = 0 # ࠵? = 0 # ࠵? 2 = 0 # ࠵? = 0 # ࠵? + = 0 # ࠵? 2 = 0
For a general two-dimensional body, the six scalar equations simplify to three Three scalar equations of equilibrium # ࠵? + = 0 # ࠵? = 0 # ࠵? 2 = 0 Vector equations ࠵? = # ࠵? = # ࠵? ࠵?̂ + # ࠵? + ࠵?̂ ࠵? . / = # ࠵?⃗×࠵? ࠵? 4 Can take moments about any point
࠵? = # ࠵? = # ࠵? ࠵?̂ + # ࠵? + ࠵?̂ = 0 x y For a body where all of the forces are concurrent, rotational equilibrium is satisfied = F 4 F 2 F 1 F 3 O O R Two scalar equations of equilibrium # ࠵? + = 0 # ࠵? = 0 Planar body with concurrent forces
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Choose the free body to isolate; Draw a Free Body Diagram (FBD) of the body; Isolate the body from all of its surroundings, Magnitudes and directions of all known and unknown forces acting on the body should be included and clearly indicated, Indicate dimensions on the FBD, Write the equations of equilibrium and solve the equations for the unknown quantities. General procedure for the Analysis of Bodies in Static Equilibrium

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