[{"data":1,"prerenderedAt":37},["ShallowReactive",2],{"topic-en-theoretical-mechanics\u002Fstatics\u002Fequilibrium-of-coplanar-systems\u002Fconstraints-constraint-reactions":3},{"topic":4,"trail":18,"children":32,"tasks":33,"alternates":34},{"id":5,"name":6,"locale":7,"path":8,"seo_title":9,"seo_description":10,"seo_text":11,"content_html":12,"content_chunks":13},187,"Constraints and Constraint Reactions","en","theoretical-mechanics\u002Fstatics\u002Fequilibrium-of-coplanar-systems\u002Fconstraints-constraint-reactions","Constraints and Constraint Reactions in Statics","Common constraints and reactions in statics: supports, hinges, cables and contacts. Learn to construct free-body diagrams for equilibrium.","This topic examines mechanical constraints and the reaction forces that replace them in a free-body diagram. It covers common supports, hinges, cables and contacts and the directions of their reactions.","\u003Cp>\u003Cstrong>Constraints\u003C\u002Fstrong> are bodies or devices that restrict the possible motion of the body being analyzed. In statics, each constraint is replaced by its \u003Cstrong>reaction\u003C\u002Fstrong>, after which the body is treated as free under the action of the applied forces and constraint reactions.\u003C\u002Fp>\u003Ch2>Principle of releasing constraints\u003C\u002Fh2>\u003Cp>To write equilibrium equations, conceptually remove the constraints from the body. Replace each removed constraint by a reaction force or a system of forces and moments that reproduces its mechanical action. The number and directions of unknown reaction components are determined by the motions that the constraint prevents.\u003C\u002Fp>\u003Ch2>Common constraints in two dimensions\u003C\u002Fh2>\u003Ctable>\u003Cthead>\u003Ctr>\u003Cth>Constraint\u003C\u002Fth>\u003Cth>Idealized reaction\u003C\u002Fth>\u003Cth>Key feature\u003C\u002Fth>\u003C\u002Ftr>\u003C\u002Fthead>\u003Ctbody>\u003Ctr>\u003Ctd>Smooth surface\u003C\u002Ftd>\u003Ctd>Normal force $N$ perpendicular to the surface\u003C\u002Ftd>\u003Ctd>With friction neglected, there is no tangential reaction component\u003C\u002Ftd>\u003C\u002Ftr>\u003Ctr>\u003Ctd>Flexible cable or rope\u003C\u002Ftd>\u003Ctd>Tension $T$ along the cable\u003C\u002Ftd>\u003Ctd>The cable pulls the body away from the attachment point\u003C\u002Ftd>\u003C\u002Ftr>\u003Ctr>\u003Ctd>Roller or movable pin support in 2D\u003C\u002Ftd>\u003Ctd>One reaction, usually normal to the supporting surface\u003C\u002Ftd>\u003Ctd>Allows motion along the permitted direction and allows rotation\u003C\u002Ftd>\u003C\u002Ftr>\u003Ctr>\u003Ctd>Pin support in 2D\u003C\u002Ftd>\u003Ctd>Two components $R_x$ and $R_y$\u003C\u002Ftd>\u003Ctd>Prevents two independent translations but allows rotation\u003C\u002Ftd>\u003C\u002Ftr>\u003Ctr>\u003Ctd>Fixed support in 2D\u003C\u002Ftd>\u003Ctd>$R_x$, $R_y$, and a reaction moment $M$\u003C\u002Ftd>\u003Ctd>Prevents translation and rotation\u003C\u002Ftd>\u003C\u002Ftr>\u003Ctr>\u003Ctd>Ideal pin-connected member loaded only at its ends\u003C\u002Ftd>\u003Ctd>Force along the member axis\u003C\u002Ftd>\u003Ctd>The member acts as a two-force member\u003C\u002Ftd>\u003C\u002Ftr>\u003C\u002Ftbody>\u003C\u002Ftable>\u003Cp>Constraint reactions follow from the kinematic restrictions of the adopted model. Before writing equilibrium equations, replace each constraint by the appropriate reaction on the free-body diagram.\u003C\u002Fp>\u003Ch2>Smooth contact\u003C\u002Fh2>\u003Cp>For an ideally smooth surface, friction is neglected. The contact reaction acts along the common normal to the surfaces at the contact point. For a flat surface, the reaction direction is known in advance and only its magnitude is unknown.\u003C\u002Fp>\u003Ch2>Cables and two-force members\u003C\u002Fh2>\u003Cp>An ideal flexible cable or rope carries tension only, so the tension force acts along the cable. A straight member acted on only by forces at two pin-connected ends is a two-force member: the end forces are collinear with the member axis, equal in magnitude, and opposite in direction.\u003C\u002Fp>\u003Ch2>Pin and roller supports\u003C\u002Fh2>\u003Cp>A roller or movable support in a planar model produces one reaction in the direction in which it prevents motion. A pin support prevents two independent translations, so its reaction is usually represented by two unknown components $R_x$ and $R_y$. An ideal pin does not transmit a reaction moment.\u003C\u002Fp>\u003Ch2>Fixed support\u003C\u002Fh2>\u003Cp>A fixed support in two dimensions prevents two translations and rotation. Its action is therefore represented by two reaction components $R_x$, $R_y$ and a reaction moment $M$.\u003C\u002Fp>\u003Ch2>Free-body diagram\u003C\u002Fh2>\u003Cp>After releasing the constraints, construct a free-body diagram. Show all applied forces, applied couple moments, the body weight when relevant, and every constraint reaction. The equilibrium equations are applied to this isolated diagram.\u003C\u002Fp>\u003Ch2>Example\u003C\u002Fh2>\u003Cp>A beam supported by a pin at $A$ and a roller at $B$ has three unknown reactions in a typical planar arrangement: $A_x$, $A_y$, and $B_y$ when the roller reaction is vertical. This matches the three independent equilibrium equations available for a planar rigid body.\u003C\u002Fp>\u003Ch2>Common mistakes\u003C\u002Fh2>\u003Cul>\u003Cli>leaving a support on the free-body diagram while also drawing its reactions;\u003C\u002Fli>\u003Cli>adding a reaction moment at an ideal pin;\u003C\u002Fli>\u003Cli>assigning an arbitrary direction to a smooth-contact reaction instead of the normal direction;\u003C\u002Fli>\u003Cli>assuming a cable can carry compression;\u003C\u002Fli>\u003Cli>omitting one of the reaction components of a fixed support.\u003C\u002Fli>\u003C\u002Ful>",[14],{"id":15,"code":16,"type":17,"locale":7},149,"statics-constraint-reactions","table",[19,23,27,31],{"id":20,"name":21,"path":22},81,"Theoretical Mechanics","theoretical-mechanics",{"id":24,"name":25,"path":26},136,"Statics","theoretical-mechanics\u002Fstatics",{"id":28,"name":29,"path":30},275,"Equilibrium of Coplanar Systems","theoretical-mechanics\u002Fstatics\u002Fequilibrium-of-coplanar-systems",{"id":5,"name":6,"path":8},[],[],{"en":35,"uk":36},"https:\u002F\u002Fmechclassroom.com\u002Fen\u002Ftopics\u002Ftheoretical-mechanics\u002Fstatics\u002Fequilibrium-of-coplanar-systems\u002Fconstraints-constraint-reactions","https:\u002F\u002Fmechclassroom.com\u002Ftopics\u002Fteoretychna-mekhanika\u002Fstatyka\u002Frivnovaha-ploskykh-system\u002Fviazi-ta-reaktsii-viazei",1787712535320]