工程力学英文版课件04 Equilibrium of a Rigid Body
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[Example 1] A crane as shown in figure, P=700kN, W=200kN (The maximum lifting weight). Determine (a) The range of the weight of the balance member Q. (b) when Q=180kN, the reactive forces at wheel A and B.
Identify each loading and give dimensions. The forces and couple moments that are known should be labelled with their proper magnitudes and directions. Letters are used to represent the magnitudes and direction angles of unknown forces and couple moments. Establish an x, y coordinate system so that these unknowns can be identified.
4
Procedure for drawing a free-body diagram To construct a free-body diagram for a rigid body or group of
bodies considered as a single system, the following steps should be performed: Draw outlined shape. Imagine the body to be isolated or cut ‘free’ from its constraints and connections and draw its outlined shape. Show all forces and couple moments. Identify all the external forces and couple moments that act on the body. Those generally encountered are due to
Fx Fy
0 0
M A 0
(1)
9
Alternative sets of equilibrium equations Although equations (1) are most often used for solving coplanar
equilibrium problems, two alternative sets of three independent equilibrium equations may also be used. One such set is:
11
§5-3 Equilibrium of a Rigid Body
Equilibrium of a two-force body When a body is subjected to no couple moments and forces
are applied at only two points on a body, the body is called a twoforce body. For equilibrium of a two-force body the force acting at one point must be equal in magnitude, opposite in direction, and have the same line of action as the force acting at the other point.
1
Equilibrium of a Rigid Body
§5–1 Introduction §5–2 Equations of Equilibrium §5–3 Equilibrium of a Rigid Body §5–4 Statical Determinacy and Constraint
14
Solution: (a) When the crane is lifting the maximum weight, i.e. W=200 kN, to avoid the crane turn to right side, the following equation must be satisfied:
5
(1) Applied loadings, (2) reactions occurring at the supports or at points of contact with other bodies (see Table 4-1, 4-2), and (3) the weight of the body.
3
2. Construction of free-body diagrams Successful application of the equation of equilibrium requires a
complete specification of all the known and unknown external forces that act on the body. It is necessary to show all the forces and couple moments that the surroundings exert on the body so that these effects can be accounted for when the equations of equilibrium are applied. For this reason, a thorough understanding of how to draw a free-body diagram is of primary importance for solving problems in mechanics.
M M
A B
0 0
(3)
M C 0
Here it is necessary that points A, B and C do not lie on the same
line.
Any one of the three sets of equations may be used to solve an equilibrium problem. The problem can be simplified if we select equations that result in only one unknown in each equation. Only three of the equations can be independent.
MFaA
0 0
(2)
M B 0
When using these equations, it is required that a line passing
through points A and B is not perpendicular to the a axis.
10
A second alternative set of equations is:
6
IMPORTANT POINTS • No equilibrium problem should be solved without first drawing
the free-body diagram, so as to account for all the forces and couple moments that act on the body.
7
• Study Table 4-1
• Internal forces are never shown on the free-body diagram, since they occur in equal but opposite collinear pairs and therefore cancel out.
Many mechanical elements act as two- or three-force body, and the ability to recognize them in a problem will considerably simplify an equilibrium analysis.
12
Equilibrium of a three-force body A body acted on by three forces is called a three-force body.
If a body is subjected to only three forces, then it is necessary that the forces be either concurrent or parallel for the body to be in equilibrium.
2
§5-1 Introduction
1. Support conditions for bodies in a plane The supports develop reactions in response to the weight of
the body and to loads (external forces or moments) that are applied to the body. They prevent the body from moving. That is, the body is in equilibrium under the action of the loads and reactions. There are several types of supports for bodies loaded by forces acting in a plane. See page 142 in textbook.
• The weight of a body is an external force, and its effect is shown as a single resultant force acting through the body’s centre of gravity G.
• Couple moment can be placed anywhere on the free-body diagram, since they are free vectors. Forces can act at any point along their lines of action, since they are sliding vectors.
• If a support prevents translation of a body in a particular direction, then the support exerts a force on the body in that direction.
• If rotation is prevented, then the support exerts a couple moment on the body.
8
§5-2 Equations of Equilibrium
Equations for equilibrium of a rigid body When a body is subjected to a system of forces, which all lie in
the x-y plane, then the forces can be resolved into their x and y components. Consequently, the conditions for equilibrium in two dimensions are:
[Example 1] A crane as shown in figure, P=700kN, W=200kN (The maximum lifting weight). Determine (a) The range of the weight of the balance member Q. (b) when Q=180kN, the reactive forces at wheel A and B.
Identify each loading and give dimensions. The forces and couple moments that are known should be labelled with their proper magnitudes and directions. Letters are used to represent the magnitudes and direction angles of unknown forces and couple moments. Establish an x, y coordinate system so that these unknowns can be identified.
4
Procedure for drawing a free-body diagram To construct a free-body diagram for a rigid body or group of
bodies considered as a single system, the following steps should be performed: Draw outlined shape. Imagine the body to be isolated or cut ‘free’ from its constraints and connections and draw its outlined shape. Show all forces and couple moments. Identify all the external forces and couple moments that act on the body. Those generally encountered are due to
Fx Fy
0 0
M A 0
(1)
9
Alternative sets of equilibrium equations Although equations (1) are most often used for solving coplanar
equilibrium problems, two alternative sets of three independent equilibrium equations may also be used. One such set is:
11
§5-3 Equilibrium of a Rigid Body
Equilibrium of a two-force body When a body is subjected to no couple moments and forces
are applied at only two points on a body, the body is called a twoforce body. For equilibrium of a two-force body the force acting at one point must be equal in magnitude, opposite in direction, and have the same line of action as the force acting at the other point.
1
Equilibrium of a Rigid Body
§5–1 Introduction §5–2 Equations of Equilibrium §5–3 Equilibrium of a Rigid Body §5–4 Statical Determinacy and Constraint
14
Solution: (a) When the crane is lifting the maximum weight, i.e. W=200 kN, to avoid the crane turn to right side, the following equation must be satisfied:
5
(1) Applied loadings, (2) reactions occurring at the supports or at points of contact with other bodies (see Table 4-1, 4-2), and (3) the weight of the body.
3
2. Construction of free-body diagrams Successful application of the equation of equilibrium requires a
complete specification of all the known and unknown external forces that act on the body. It is necessary to show all the forces and couple moments that the surroundings exert on the body so that these effects can be accounted for when the equations of equilibrium are applied. For this reason, a thorough understanding of how to draw a free-body diagram is of primary importance for solving problems in mechanics.
M M
A B
0 0
(3)
M C 0
Here it is necessary that points A, B and C do not lie on the same
line.
Any one of the three sets of equations may be used to solve an equilibrium problem. The problem can be simplified if we select equations that result in only one unknown in each equation. Only three of the equations can be independent.
MFaA
0 0
(2)
M B 0
When using these equations, it is required that a line passing
through points A and B is not perpendicular to the a axis.
10
A second alternative set of equations is:
6
IMPORTANT POINTS • No equilibrium problem should be solved without first drawing
the free-body diagram, so as to account for all the forces and couple moments that act on the body.
7
• Study Table 4-1
• Internal forces are never shown on the free-body diagram, since they occur in equal but opposite collinear pairs and therefore cancel out.
Many mechanical elements act as two- or three-force body, and the ability to recognize them in a problem will considerably simplify an equilibrium analysis.
12
Equilibrium of a three-force body A body acted on by three forces is called a three-force body.
If a body is subjected to only three forces, then it is necessary that the forces be either concurrent or parallel for the body to be in equilibrium.
2
§5-1 Introduction
1. Support conditions for bodies in a plane The supports develop reactions in response to the weight of
the body and to loads (external forces or moments) that are applied to the body. They prevent the body from moving. That is, the body is in equilibrium under the action of the loads and reactions. There are several types of supports for bodies loaded by forces acting in a plane. See page 142 in textbook.
• The weight of a body is an external force, and its effect is shown as a single resultant force acting through the body’s centre of gravity G.
• Couple moment can be placed anywhere on the free-body diagram, since they are free vectors. Forces can act at any point along their lines of action, since they are sliding vectors.
• If a support prevents translation of a body in a particular direction, then the support exerts a force on the body in that direction.
• If rotation is prevented, then the support exerts a couple moment on the body.
8
§5-2 Equations of Equilibrium
Equations for equilibrium of a rigid body When a body is subjected to a system of forces, which all lie in
the x-y plane, then the forces can be resolved into their x and y components. Consequently, the conditions for equilibrium in two dimensions are: