Chapter 4 flat Force 1

Chapter 4 flat Force System

Teaching Objectives

1. Master the method of simplifying any force system on the plane to a point. The parsing method is used to obtain the primary vector and the primary moment. Familiar with the simplified results of any force system on the plane.

2. gain an in-depth understanding of the three equilibrium equations of any force system in a plane.

3. Understand the simplified assumptions of a simple truss, master the Node Method and intercept method for calculating the internal force of its member.

Highlights of this Chapter

1. Simplify any point of force in the plane and simplify the force system.

2. Analytical conditions for the equilibrium of any force system in the plane, and various forms of the equilibrium equation.

3. Solution to the balance problem between objects and thing systems.

Difficulties in this Chapter

The concept of the main vector and the main moment. The plane force is simplified to the combination of force, and the force line is determined. The balance of the thing system.

Teaching Process

Overview

Any force system of each force distribution in the same plane is called a plane force system. The Study of flat force is of great significance not only in theory but also in practical engineering applications. First, the flat force system is a common force system in Engineering. In addition, although the Force line of many engineering structures and components is not in the same plane, the force system usually has a pair of called planes, it can be simplified to the force system acting in the symmetric plane.

1. Simplified one point in the plane
   

It is located on the rigid body and acts on a flat force system. Simplified Center

Vector

Primary moment

Conclusion: The plane force system is simplified to any point in the action plane. Generally, a force and a force pair can be obtained. The force is equal to the main vector of the Force System in size and direction by simplifying the center, the size and direction of the primary vector are irrelevant to the simplified center. The Force Moment of the primary vector is equal to the main moment of the simplified center. The size and direction of the primary moment are related to the simplified center.

2. Fixed End constraints (insert end constraints)

Concept; a part of an object is a constraint inherent in another object.

Example: wire rod.

When the main power is a plane force system, the Force System of the object in the solid embedded part is also a plane force system, which is generally complicated, but can be simplified to a force and a force pair to the point, the force size and direction are unknown, expressed

1. Changes in the main moment of a simplified Center
   

Point simplification

Point simplification

To make it, there must be or

For the flat force system, the change of the main moment can also be expressed by the number of generations, that is

4. The final result of simplification of the flat Force System

1) simplified results

(1) plane force system balance

(2) The plane force system is simplified to a power couple. The size and steering of the Power moment are determined by the main moment, and are irrelevant to the simplified center.

(3) The flat force system is simplified into a force. The force is over simplified center, and the size and direction are determined by the main vector.

(4) The flat force system is simplified to a force. The interaction line of the force is on the point side, so that the moment is the same as the steering of the main moment. Figure

2) moment theorem of force

(4.7)

That is, the moment of the force of the plane force on any point in the surface is equal to the algebraic sum of the moment of the forces on the same point in the Force System. It is called the force moment theorem of the plane force system.

3) equations of the force acting line

The analytical expression of the point moment of the plane internal force shows that there is a point in the Force line.

For example, 4.1 is used to calculate the force of the distributed load on the beam and the position of the line.

1. The load set is
2. Loads are distributed in the first line. The load set on the left side is zero, and the load set on the right side is

Solution: 1) the force of "Uniform Load" can be treated as the gravity of homogeneous rod, so the force of force is in the center of the beam,

1. When the load is unevenly distributed, the points can be used to calculate the force size and the position of the line. Take the microelement at the far end of the beam. Due to the linear distribution of loads, the concentration at the beam is as follows:

The force size is

Use the moment theorem to calculate the position of the force acting line. Set the distance between the active line and the end

Example 2: It is known that the four vertices of the rectangular plate apply four forces and one force couple respectively. Here, moment, steering, the unit of length in the figure is. Test results: 1) force system directed point simplification result 2) force system directed point simplification result 3) force system simplification final result

Solution: 1. vector of the computing power system:

Analytical Expression

2 main moment simplified to the point

That is, the flat force system simplifies a force and a force even. The Force crossing point has the same size and direction as the main vector of the force system. The power moment of this power doll is equal to the main moment,

3. Master moment of point simplification

Calculation Based on the moment of two points

A force and a force pair are still obtained when the flat force system is simplified. When the force is over, its size and direction are still the same as the main vector of the Force System. The Force Moment of the force couple is equal to the main moment,

4. Force the final result of the simplification

Because of the main vector, the final result of the force system simplification is a force. Its size and direction are the same as those of the main vector. The line equation is: the coordinate of the intersection of the force for the axis is (-3, 0)

Ii. balanced equations of the plane force system and their applications.

1. Equilibrium Equation of plane Force System
   
2. Basic Form:
   

The flat force system is a combination of flat interaction force and flat force coupling. Therefore, it is necessary to balance the flat force system.

,

The analytical expression is:

That is to say, the necessity and sufficient resolution condition of the plane force system balance are: the algebraic and equal values of the Forces projected on the two Cartesian axes of the Force System in the working plane are zero, in the force system, the algebraic sum of the moment of any point in the plane is equal to zero.

2) Two-torque type

And the axis is not perpendicular to or connected to two points.

Proof: necessity. If the flat force is balanced, it is easy to know.

Adequacy, it is known that the force is perpendicular to the axis, which is in conflict with the known. Therefore, (4.11) is established and the force system is balanced.

3) triplicate moment (4.12)

And, and are not collocated

1. For example, the equilibrium equation of the plane parallel force system.
   

Basic Format: (4.13)

2-torque: (4.14)

Example 1: In the structure shown in the figure, all three are hinge constraints. The horizontal bar is subjected to a concentrated load, and the size of each part of the structure is shown in the figure. If sum is known, try to obtain the force and binding force of the pole.

Solution: study object: Rod

1. Force Analysis: Yi Zhi is a two-force rod.
   
2. Solving the column equilibrium equation:
   
3. Study object: Rod
   
4. Force Analysis: Yi Zhi is a two-force rod.
   
5. Solving the column equilibrium equation:
   
   1. Basic Equation:
      

Available

1. Triplicate:
   

Available

Example 2: the horizontal beam is subjected to a distributed load and a force couple. It is known that, regardless of the beam weight, the anti-force of the bearing is obtained.

Solution: 1. Study object: Beam

2. Force Analysis:

3. Solving the column equation:

Resolution

Example 3: the bearing capacity and each part of the plane steel frame are fixed end constraints. If the graph, and so on are known, the binding force on the trial end is used.

Solution: 1. research object, Steel Frame

2. Force Analysis:

3. Solving the column equation:

Resolution