Let us now solve the problem of bringing an arbitrary system of forces to a given center, that is, of replacing a given system of forces with another, equivalent to it, but much simpler, namely, consisting, as we will see, of only one force and a pair.
Let an arbitrary system of forces act on a solid body (Fig. 40, a).
Let us choose some point O as the center of reduction and, using the theorem proven in § 11, transfer all the forces to the center O, adding the corresponding pairs (see Fig. 37, b). Then a system of forces will act on the body
applied at the center O, and a system of pairs whose moments, according to formula (18), are equal to:
The converging forces applied at point O are replaced by one force R applied at point O. In this case, or, according to equalities (19),
To add all the resulting pairs, you need to add the moment vectors of these pairs. As a result, the system of pairs will be replaced by one pair, the moment of which or, according to equalities (20),
As is known, the value of R, equal to geometric sum of all forces is called the main vector of the system; a value equal to the geometric sum of the moments of all forces relative to the center O is called the main moment of the system of forces relative to this center.
Thus, we have proven the following theorem about the reduction of a system of forces: any system of forces acting on an absolutely rigid body, when reduced to an arbitrarily chosen center O, is replaced by one force R, equal to the main vector of the force system and applied at the reduction center O, and one pair with a moment equal to the main moment of the system of forces relative to the center O (Fig. 40, b).
Note that the force R is not the resultant of this system of forces here, since it replaces the system of forces not alone, but together with a pair.
From the proven theorem it follows that two systems of forces that have the same main vectors and main moments relative to the same center are equivalent (conditions for the equivalence of force systems).
Let us also note that the value of R obviously does not depend on the choice of center O. The value, when the position of the center O changes, can generally change due to changes in the values of the moments of individual forces. Therefore, it is always necessary to indicate relative to which center it is determined main point.
The method of bringing one force to a given point can be applied to any number of forces. Let us assume that at some points of the body (Fig. 1.24) forces are applied F 1 F 2 , F 3 And F4. It is required to bring these forces to the point ABOUT plane. Let us first present the force applied at the point A. Let us apply (see § 16) at the point ABOUT two forces that are separately equal in value to a given force, parallel to it and directed in opposite directions. As a result of bringing the force, we get the force , applied at point O, and a couple of forces with the shoulder . By doing the same with force , applied at the point IN, we'll get the power , applied at the point ABOUT, and a couple of forces with a shoulder, etc. A flat system of forces applied at points A, B, C And D, we replaced with converging forces , applied at a point ABOUT, and pairs of forces with moments, equal moments given forces relative to a point ABOUT:
Fig.1.24
The forces converging at a point can be replaced by one force equal to the geometric sum of the components,
This force, equal to the geometric sum of given forces, is called the main vector of the force system and denote .
Based on the magnitude of the projections of the main vector on the coordinate axes, we find the module of the main vector:
Based on the rule for adding pairs of forces, they can be replaced by the resulting pair, the moment of which is equal to the algebraic sum of the moments of the given forces relative to the point ABOUT and is called main point relative to the reference point
Thus, an arbitrary plane system of forces is reduced to one force(the main vector of the force system) and one moment(the main moment of the system of forces).
It is necessary to understand that the one hundred principal vector is not the resultant of a given system of forces, since this system is not equivalent to one force. Since the main vector is equal to the geometric sum of the forces in a given system, neither its magnitude nor its direction depends on the choice of the center of reduction. The value and sign of the main moment depends on the position of the center of reduction, since the arms of the component pairs depend on the relative position of the forces and the point (center) relative to which the moments are taken.
Special cases of reduction of a system of forces:
1) ; the system is in equilibrium, i.e. For the equilibrium of a plane system of forces, it is necessary and sufficient that its main vector and main moment be simultaneously equal to zero.
Lecture 5
Summary: Bringing force to a given center. Bringing a system of forces to a given center. Equilibrium conditions spatial system parallel forces. Equilibrium conditions for a plane system of forces. Three-moment theorem. Statically definable and statically indeterminate problems. Equilibrium of the system of bodies.
BRINGING THE SYSTEM OF FORCES TO A SPECIFIED CENTER. CONDITIONS OF EQUILIBRIUM
Bringing force to a given center.
The resultant of a system of converging forces is directly found using the addition of forces according to the parallelogram rule. Obviously, a similar problem can be solved for an arbitrary system of forces if we find a method for them that allows us to transfer all the forces to one point.
Theorem on parallel force transfer . A force applied to an absolutely rigid body can, without changing the effect it exerts, be transferred from a given point to any other point of the body, adding a couple with a moment equal to the moment of the transferred force relative to the point where the force is transferred.
Let a force be applied at point A. The effect of this force does not change if two balanced forces are applied at point B. The resulting system of three forces is a force equal to, but applied at point B, and a pair with a moment. The process of replacing a force with a force and a pair of forces is called bringing the force to a given center B.
Bringing a system of forces to a given center.
Main theorem statics (Poinsot).
Any arbitrary system of forces acting on a rigid body can, in general, be reduced to a force and a pair of forces. This process of replacing a system of forces with one force and one pair of forces is called bringing the system of forces to a given center.
The main vector of the system strength is called a vector equal to vector sum these forces.
The main point of the system strength relative to point O of the body, a vector is called equal to the vector sum of the moments of all forces of the system relative to this point.
Formulas for calculating the main vector and main moment
Formulas for calculating the modulus and direction cosines
main vector and main moment
Conditions for the equilibrium of a system of forces.
Vector shape.
For the equilibrium of an arbitrary system of forces applied to a rigid body, it is necessary and sufficient that the main vector of the force system is equal to zero and the main moment of the force system relative to any center of reduction is also equal to zero.
Algebraic form.
For the equilibrium of an arbitrary system of forces applied to a rigid body, it is necessary and sufficient that the three sums of the projections of all forces on the axis Cartesian coordinates were equal to zero and the three sums of the moments of all forces relative to the three coordinate axes were also equal to zero.
Conditions for the equilibrium of a spatial system
parallel forces.
A system of parallel forces acts on the body. Let's place the Oz axis parallel to the forces.
Equations 
For the equilibrium of a spatial system of parallel forces acting on a solid body, it is necessary and sufficient that the sum of the projections of these forces be equal to zero and the sum of the moments of these forces relative to two coordinate axes perpendicular to the forces are also equal to zero.
- projection of force onto the Oz axis.
FLAT FORCE SYSTEM.
Equilibrium conditions for a plane system of forces.
A plane system of forces acts on the body. Let's place the Ox and Oy axes in the plane of action of the forces.
Equations 
For the equilibrium of a plane system of forces acting on a solid body, it is necessary and sufficient that the sums of the projections of these forces onto each of the two rectangular coordinate axes located in the plane of action of the forces are equal to zero and the sum of the moments of these forces relative to any point located in the plane of action the forces were also zero.
Three-moment theorem.
For the equilibrium of a plane system of forces acting on a rigid body, it is necessary and sufficient that the sums of the moments of these forces of the system relative to any three points located in the plane of action of the forces and not lying on the same straight line are equal to zero.
Statically definable and statically indeterminate problems.
For any plane system of forces acting on a rigid body, there are three independent equilibrium conditions. Consequently, for any plane system of forces, no more than three unknowns can be found from equilibrium conditions.
In the case of a spatial system of forces acting on a rigid body, there are six independent equilibrium conditions. Consequently, for any spatial system of forces, no more than six unknowns can be found from equilibrium conditions.
Problems in which the number of unknowns is not greater than the number of independent equilibrium conditions for a given system of forces applied to a rigid body are called statically definable.
Otherwise, the problems are statically indeterminate.
Equilibrium of the system of bodies.
Let us consider the equilibrium of forces applied to a system of interacting bodies. The bodies can be connected to each other using hinges or in another way.
The forces acting on the system of bodies under consideration can be divided into external and internal.
External are called the forces with which bodies of the system under consideration are acted upon by bodies that are not included in this system of forces.
Internal are called the forces of interaction between the bodies of the system under consideration.
When considering the equilibrium of forces applied to a system of bodies, one can mentally divide the system of bodies into individual solid bodies and apply the equilibrium conditions obtained for one body to the forces acting on these bodies. These equilibrium conditions will include both external and internal forces of the system of bodies. Inner forces Based on the axiom of the equality of action and reaction forces at each point of articulation of two bodies, they form an equilibrium system of forces.
Let us demonstrate this using the example of a system of two bodies and a plane system of forces.
If we create equilibrium conditions for each solid system of bodies, then for body I 
.
for body II 
In addition, from the axiom about the equality of action and reaction forces for two interacting bodies we have 
.
The presented equalities are the equilibrium conditions external forces, acting on the system.
Sealing reaction.
Let's consider a beam, one end of which AB is embedded in the wall. This type of fastening of the end of the beam AB is called sealing at the point B. Let a plane system of forces act on the beam. Let us determine the forces that must be applied to point B of the beam if part of the beam AB is discarded. Distributed reaction forces are applied to the beam section (B). If these forces are replaced by elementary concentrated forces and then brought to point B, then at point B we obtain a force (the main vector of reaction forces) and a pair of forces with a moment M (the main vector of reaction forces relative to point B). Moment M called the closing moment or directive moment. The reaction force can be replaced by two components and .
The seal, unlike a hinge, creates not only a reaction unknown in magnitude and direction, but also a pair of forces with an unknown moment M in the seal.
The described method of bringing one force to a given point can be applied to any number of forces. Let us assume that at points of the body A,B,C And D(Fig. 19) forces applied 1 , 2 , 3 And 4 . It is required to bring these forces to the point ABOUT plane. Let us first give the force 1 , applied at the point A. Let's apply at the point ABOUT two forces ’ 1 And ’’ 1 , separately equal in modulus to a given force 1 , parallel to it and directed towards opposite sides. As a result of bringing the force 1 we'll get the power ’ 1 , applied at the point ABOUT, and a couple of forces 1 ’’ 1 (forces forming a pair are marked with dashes) with a shoulder a 1. By doing the same with force 2 , applied at the point IN, we get strength 2 , applied at the point ABOUT, and a couple of forces 2 ’’ 2 with shoulder a 2 etc.
A plane system of forces applied at points A, IN, WITH And D, we replaced by converging forces ’ 1 , ’ 2 , ’ 3 And ’ 4 , applied at the point ABOUT, and pairs of forces with moments equal to the moments of the given forces relative to the point ABOUT:
M 1 = P 1 a 1 = M o ( 1); M 2 = P 2 a 2 = M o (2);
M 3 = – P 3 a 3 = M o ( 3); M 4 = – P 4 a 4 = M o (4).
Forces converging at a point can be replaced by a single force " , equal to the geometric sum of the components,
" = " 1 + " 2 + " 3 + " 4 = 1 + 2 + 3 + 4 = i .(16)
This force, equal to the geometric sum of given forces, is called the main vector of the system of forces.
Based on the rule for adding pairs of forces, from can be replaced by the resulting pair, the moment of which is equal to the algebraic sum of the moments of the given forces relative to the point ABOUT:
M o = M 1 + M 2 + M 3 + M 4 = i = o (i).(17)
By analogy with the main vector, the moment M 0 pairs equal to the algebraic sum of the moments of all forces relative to the center of reduction ABOUT, called the main moment of the system relative to the given center of reduction O. Hence, in the general case, a flat system of forces as a result of reduction to a given point O is replaced by an equivalent system consisting of one force - the main vector - and one pair, the moment of which is called the main moment of the given system of forces relative to the center of reduction.
It is necessary to understand that the main vector ’ is not the resultant of a given system of forces, since this system is not equivalent to one force ’. Only in the special case when the main moment vanishes, the main vector will be the resultant of a given system of forces. Since the main vector is equal to the geometric sum of the forces of a given system, neither its magnitude nor its direction depend on the choice of the center of reduction. Magnitude and sign of the main moment M 0 depend on the position of the center of reduction, since the arms of the component pairs depend on the relative position of the forces and the point (center) relative to which the moments are taken.
The following cases of bringing a system of forces may occur:
1. " ≠ 0; M o ≠ 0 - general case; the system is reduced to the main vector and the main moment.
2. " ≠ 0; M o = 0; the system is reduced to one resultant equal to the main vector of the system.
3. " = 0; M o ≠ 0; the system is reduced to a pair of forces whose moment is equal to the main moment.
4. " = 0; M o = 0; the system is in equilibrium.
It can be proven that in the general case, when " ≠ 0 and M o ≠ 0, There is always a point relative to which the main moment of the system of forces is equal to zero.
Let us consider a plane system of forces that is reduced to the point ABOUT, i.e. replaced by principal vector " ≠ 0 , applied at the point ABOUT, and the main point M o ≠ 0(Fig. 20).
For definiteness, we assume that the main moment is directed clockwise, i.e. M o< 0. Let's depict this main moment with a pair of forces "" , the module of which is chosen equal to the module of the main vector " , i.e. R =R'' = R'. One of the forces that make up the pair is the force "" – apply at the reduction center ABOUT, another force – at some point WITH, the position of which is determined from the condition: M o = OS*R. Hence,
OS =. (18)
Let's put a couple of forces "" so that the strength "" was directed in the direction opposite to the main vector " . At the point ABOUT(Fig. 20) we have two equal and mutually opposite forces " And "" , directed along one straight line; they can be discarded (according to the third axiom). Therefore, relative to the point WITH the main moment of the system of forces under consideration is equal to zero, and the system is reduced to the resultant .
§ 18. Theorem on the moment of the resultant (Varignon’s theorem)
In the general case (see § 17), an arbitrary plane system of forces is reduced to the main vector " and the main point M 0 relative to the selected center of reduction, and the main moment is equal to the algebraic sum of the moments of the given forces relative to the point ABOUT
M o = o (i).(A)
It was shown that it is possible to select the center of reduction (in Fig. 20, the point WITH), relative to which the main moment of the system will be equal to zero, and the system of forces will be reduced to one resultant equal in magnitude to the main vector ( R = R'). Let us determine the moment of the resultant relative to the point ABOUT. Considering that the shoulder OS forces equal , we get
M o () = R*OC = R = M o.(b)
Two quantities, separately equal to the third, are equal to each other, therefore from equations (a) and (b) we find
M o () = o ( i).(19)
The resulting equation expresses Varignon's theorem: the moment of the resultant plane system of forces relative to an arbitrary point is equal to the algebraic sum of the moments of the component forces relative to the same point.
From Varignon’s theorem it follows that the principal moment of a plane system of forces relative to any point lying on the line of action of its resultant is equal to zero.
