2 ReCall Equilibrium Conditions Rigid Body in Static Equilibrium Characterized byBalanced external forces and momentsWill impart no Tendency toward Translational or Rotational motion to the bodyThe NECESSARY and SUFFICIENT condition for the static equilibrium of a body are that the RESULTANT Force and Couple from all external forces form a system equivalent to zero
3 Equilibrium cont. Rigid Body Equilibrium Mathematically Resolving into Rectangular Components the Resultant Forces & Moments Leads to an Equivalent Definition of Rigid Body Equilibrium
4 2D Planar System In 2D systems it is assumed that The System Geometry resides completely the XY PlaneThere is NO Tendency toTranslate in the Z-DirectionRotate about the X or Y AxesThese Conditions Simplify The Equilibrium Equations
5 2D Planar System: No Z-Translation → NO Z-Directed Force: No X or Y Rotation → NO X or Y Applied Moments
6 2D Planar System:If r Lies in the XY Plane, then rz = 0. With Fz = 0 the rxF Determinant:So in this case M due to rxF is confined to the Z-Direction:
7 Example Crane problem Solution PlanCreate a free-body diagram for the craneDetermine Rcns at B by solving the equation for the sum of the moments of all forces about Pin-ANote there will be no Moment contribution from the unknown reactions at A.Determine the reactions at A by solving the equations for the sum of all horizontal force components and all vertical force components.A fixed crane has a mass of kg and is used to lift a kg crate. It is held in place by a pin at A and a rocker at B. The center of gravity of the crane is located at G.Determine the components of the reactions at A and B.
8 Example Crane problem Reaction AnalysisPIN at AX & Y ReactionsROCKER at BNORMAL Reaction Only+X Rcn in This CaseSolution Plan cont.Check the values obtained for the reactions by verifying that the moments about B of all forces Sum to zero.
9 Example Crane problem Draw the Free Body Diagram (FBD)Determine the reactions at Pt-A by solving the eqns for the sum of all horizontal & vertical forcesDetermine Pt-B Rcn by solving the equation for the sum of the moments of all forces about Pt-A.
10 Example Coal Car Solution Plan Create a free-body diagram for the car with the coord system ALIGNED WITH the CABLEDetermine the reactions at the wheels by solving equations for the sum of moments about points above each axle, in the LoA of the Pull CableDetermine the cable tension by solving the equation for the sum of force components parallel to the cableA loaded coal car is at rest on an inclined track. The car has a gross weight of 5500 lb, for the car and its load as applied at G. The car is held in position by the cable.Determine the TENSION in the cable and the REACTION at each pair of wheels.
11 Example Coal Car Create Free Body Dia. Determine the reactions at the wheelsDetermine cable tension
12 Example Cable Braced Beam Solution PlanCreate a free-body diagram for the frame and cable.Solve 3 equilibrium equations for the reaction force components and Moment at E.Reaction AnalysisThe Support at E is a CANTILEVER; can ResistPlanar ForcesPlanar MomentAlsoThe frame supports part of the roof of a small building. The tension in the cable is 150 kN.Determine the reaction at the fixed end E.
14 Example Cable Braced Beam Create a free-body diagram for the frame and cablesincosForces are SLIDING VECTORS; they can be applied at ANY POINT on the Force LoASolve 3 equilibrium eqns for the reaction force components and couple
15 Rigid Body Equilibrium in 3D SIX scalar equations are required to express the conditions for the equilibrium of a rigid body in the general THREE dimensional caseThese equations can be solved for NO MORE than 6 unknownsThe Unknowns generally represent REACTIONS at Supports or Connections.
16 Rigid Body Equilibrium in 3D The scalar equations are often conveniently obtained by applying the vector forms of the conditions for equilibriumSolve the Above Eqns with the Usual Techniques; e.g., Determinant OperationsA Clever Choice for the Pivot Point, O, Can Eliminate from the Calculation as Many as 3 Unknowns (a PoC somewhere)
17 Example 3D Ball-n-Socket Solution PlanCreate a free-body diagram of the SignApply the conditions for static equilibrium to develop equations for the unknown reactions.Reaction AnalysisBall-n-Socket at A can Resist Translation in 3DCan NOT Resist TWIST (moment) in Any DirectionCables Pull on Their Geometric AxisA sign of Uniform Density weighs 270 lb and is supported by a BALL-AND-SOCKET joint at A and by two Cables.Determine the tension in each cable and the reaction at A.
18 3D Ball-n-Socket Prob cont. Create a free-body diagram of the SignThe x-Axis rotation Means that the Sign is Only Partially Constrained. MathematicallyBut If the Sign is NOT Swinging (i.e., it’s hanging STILL) The Following Condition Must ExistLOAs for Tec & Tbc intersect the X-axis thus giving them a ZERO lever-arm relative to this axis. They thus impart no twist about the x-axisWith The Absence of the Mx Constraint This Problem Generates only 5 UnknownsMX = 0 by X-Axis P.O.A. for W, TEC & TBCNote: The Sign can SWING about the x-Axis (Wind Accommodation?)
19 3D Ball-n-Socket Prob cont.2 State in Component-Form the Two Cable-Tension Vectors
20 3D Ball-n-Socket Prob cont.3 Apply the conditions for static equilibrium to develop equations for the unknown reactions.Solving 5 Eqns in 5 Unkwns
21 Solve 3D Problem by Mechanics & MATLAB >> F = [ ]F =>> rxF = cross(r,F)rxF =WhiteBoard WorkSolve 3D Problem by Mechanics & MATLABST * H 5-82 • see ENGR36_H13_Tutorial_3D_5DoF_HangingSign_H13e_5-82_1207.pptxDetermine the tensions in the cables and the components of reaction acting on the smooth collar at G necessary to hold the 2000 N sign in equilibrium. The sign weight may concentrated at the center of gravity.