[{"data":1,"prerenderedAt":36},["ShallowReactive",2],{"topic-en-theoretical-mechanics\u002Fstatics\u002Ffundamentals-of-forces-and-moments\u002Freduction-force-system-given-point":3},{"topic":4,"trail":17,"children":31,"tasks":32,"alternates":33},{"id":5,"name":6,"locale":7,"path":8,"seo_title":6,"seo_description":9,"seo_text":10,"content_html":11,"content_chunks":12},185,"Reduction of a Force System to a Given Point","en","theoretical-mechanics\u002Fstatics\u002Ffundamentals-of-forces-and-moments\u002Freduction-force-system-given-point","Reduce a general force system to a given point using the resultant force vector and resultant moment for statics and equilibrium analysis.","This topic explains how a general force system is reduced to a specified point. It introduces the resultant force vector and resultant moment used to analyze equivalent force systems and equilibrium.","\u003Cp>\u003Cstrong>Reduction of a force system to a specified point\u003C\u002Fstrong> replaces a general set of forces and couples by a simpler equivalent system: one force applied at the selected point and one couple moment. This representation is fundamental for subsequent equilibrium analysis.\u003C\u002Fp>\u003Ch2>Moving a force to a specified point\u003C\u002Fh2>\u003Cp>A force $\\vec F$ applied at point $A$ can be moved parallel to itself to point $O$ if a couple with moment $\\vec M_O=\\vec r\\times\\vec F$ is added at the same time, where $\\vec r=\\overrightarrow{OA}$. The resulting force-and-couple system is equivalent to the original force.\u003C\u002Fp>\u003Ch2>Resultant force vector and resultant moment\u003C\u002Fh2>\u003Cp>A general force system acting on a rigid body can be reduced to a specified point $O$ as a \u003Cstrong>resultant force vector\u003C\u002Fstrong> and a \u003Cstrong>resultant moment\u003C\u002Fstrong>:\u003C\u002Fp>\u003Cp>$$\\vec R=\\sum_i\\vec F_i,$$\u003C\u002Fp>\u003Cp>$$\\vec M_O=\\sum_i(\\vec r_i\\times\\vec F_i)+\\sum_j\\vec M_j,$$\u003C\u002Fp>\u003Cp>where $\\vec r_i$ is the position vector from $O$ to the point of application of $\\vec F_i$, and $\\vec M_j$ are applied couple moments. The resultant force vector is independent of the reduction point, whereas the resultant moment generally depends on it.\u003C\u002Fp>\u003Cp>For a planar problem, the resultant force vector is described by $R_x=\\sum F_{ix}$ and $R_y=\\sum F_{iy}$, while the resultant moment about $O$ is the algebraic sum of the moments of all forces and applied couples.\u003C\u002Fp>\u003Ch2>Changing the reduction point\u003C\u002Fh2>\u003Cp>If a system has been reduced to point $O$, moving the reduction point to $A$ does not change the resultant force vector. The resultant moment transforms according to\u003C\u002Fp>\u003Cp>$$\\vec M_A=\\vec M_O+\\overrightarrow{AO}\\times\\vec R.$$\u003C\u002Fp>\u003Cp>Thus, the same force system has the same resultant force vector at every reduction point, but generally different resultant moments when $\\vec R\\ne0$.\u003C\u002Fp>\u003Ch2>Main cases after reduction\u003C\u002Fh2>\u003Cul>\u003Cli>if $\\vec R=0$ and $\\vec M_O=0$, the system is balanced;\u003C\u002Fli>\u003Cli>if $\\vec R=0$ but $\\vec M_O\\ne0$, the system is equivalent to a force couple;\u003C\u002Fli>\u003Cli>if $\\vec R\\ne0$, whether the system can be reduced further to a single resultant force depends on the relation between the resultant force vector and resultant moment.\u003C\u002Fli>\u003C\u002Ful>\u003Ch2>Planar example\u003C\u002Fh2>\u003Cp>Suppose a 10 kN force acts in the $+y$ direction at a point 2 m to the right of $O$. When the force is moved to $O$, an additional moment $M_O=10\\cdot2=20$ kN·m must be introduced. With the usual planar sign convention, this moment is positive because the original force tends to rotate the body counterclockwise about $O$.\u003C\u002Fp>\u003Ch2>Reduction procedure\u003C\u002Fh2>\u003Col>\u003Cli>choose the reduction point $O$;\u003C\u002Fli>\u003Cli>calculate the components of all forces and form the resultant vector $\\vec R$;\u003C\u002Fli>\u003Cli>calculate the moment of every force about $O$;\u003C\u002Fli>\u003Cli>add all applied couple moments;\u003C\u002Fli>\u003Cli>write the equivalent system $\\vec R$ and $\\vec M_O$;\u003C\u002Fli>\u003Cli>if required, determine whether the system can be simplified further.\u003C\u002Fli>\u003C\u002Fol>\u003Cp>The conditions $\\vec R=0$ and $\\vec M_O=0$ lead directly to the general equilibrium equations for a rigid body.\u003C\u002Fp>",[13],{"id":14,"code":15,"type":16,"locale":7},145,"statics-force-system-reduction","definition",[18,22,26,30],{"id":19,"name":20,"path":21},81,"Theoretical Mechanics","theoretical-mechanics",{"id":23,"name":24,"path":25},136,"Statics","theoretical-mechanics\u002Fstatics",{"id":27,"name":28,"path":29},274,"Fundamentals of Forces and Moments","theoretical-mechanics\u002Fstatics\u002Ffundamentals-of-forces-and-moments",{"id":5,"name":6,"path":8},[],[],{"en":34,"uk":35},"https:\u002F\u002Fmechclassroom.com\u002Fen\u002Ftopics\u002Ftheoretical-mechanics\u002Fstatics\u002Ffundamentals-of-forces-and-moments\u002Freduction-force-system-given-point","https:\u002F\u002Fmechclassroom.com\u002Ftopics\u002Fteoretychna-mekhanika\u002Fstatyka\u002Fosnovy-syl-i-momentiv\u002Fzvedennia-systemy-syl-do-zadanoho-tsentra",1787712535209]