Chapter 4: Equilibrium Equilibrium means balance of forces to prevent body from translating, and balance of moments to prevent body from rotating. Vector analysis in 3-D is the preferred method of solution.

4.1 Conditions for Equilibrium Equilibrium means that the object is at rest (if originally at rest), or in constant velocity (if originally moving).

4.1 Conditions of Equilibrium Con’t Moment about any point O,

4.2 Free Body Diagrams Need to know how to represent support and contact conditions. If a support prevents translation in any direction, we have a reaction force in that direction. If a support prevents rotation in any orientation, then we have a couple moment exerted.

4.2 Free Body Diagrams Con’t Weight always acts at the center of gravity. W = m g Consider the case of springs.

4.2 Free Body Diagrams Con’t Consider the cantilever beam supported by a fixed support at A. Free body diagram

Free Body Diagrams of a Platform Consider the platform Exclude all other effects except the platform now!

4.3 Equations of Equilibrium

4.3 Supports and Reactions Supports are idealized first. Reaction forces (magnitude and direction) and moments then depends on the type of support. Roller allows motion along the plane. Reaction force is perpendicular to the surface. Rocker allows rotation at that point. Reaction force is perpendicular to the surface.

4.3 Supports and Reactions-2 Pin connected to a collar. Reaction force is perpendicular to the rod. Hinge allows motion both in x and y direction but no rotation. Reaction force in x and y only. F not along member Fixed allows no rotation and no translation. Reaction force vector and moment will result.

4.3 Analysis Precedure First draw the free body diagram for the loading shown to the right. Apply equations of equilibrium through force and moment balance.

Problem 4-3 (page 138, Section 4.1-4.3) 4.3 Draw the free-body diagram of the automobile, which has a mass of 5 Mg and center of mass at G. The tires are free to roll, so rolling resistance can be neglected. Explain the significance of each force on the diagram. Solution:

Problem 4-7 (page 139, Section 4.1-4.3) 4.7 Draw the free body diagram of the beam. The incline at B is smooth. Solution:

Problem 4-17 (page 154, Section 4.4-4.5) 4.17 Determine the stretch of each spring for equilibrium of 20-Kg block. The springs are shown in their equilibrium position. Solution:

4.17 Determine the stretch of each spring for equilibrium of 20-Kg block. The springs are shown in their equilibrium position. Solution-Con’t (slide 2)

4.4 Two-Force Members Two force members are trusses that have forces (tension or compression) but not couple moments.

4.4 Two and Three-Force Members Hydraulic cylinder is a two-force member If a member is subjected to three coplanar forces, then the forces should either be concurrent or coplanar for equilibrium.

Problem 4-26 (page 155, Section 4.4-4.5) 4.26 Determine the horizontal and vertical components of reaction at the pin A and the force in the short link BD. Solution:

4.5 Equilibrium in 3-D The concept of equilibrium in 3-D is similar. Here we need to solve all the known and unknown in 3-D. Once again we need to know the reaction forces and moments for each type of support, see Table 4-2 Ball- Only 3 forces Pin- All except 1 moment Bearing- 2 forces+2 moments Fixed- All 6 forces & moments

4.6 Equations of Equilibrium Procedure for Analysis: Draw Free body diagram for the body under analysis Mark all the reaction and external forces/moments. Use the above equations to solve.

Problem 4-68 (page 174, Section 4.6-4.7) 4.68 Member AB is supported by a cable BC and at A by a smooth fixed square rod which fits loosely through the square hole of the collar. If the force lb, determine the tension in cable BC and the x, y, z components of reaction at A. Solution:

4.68 Member AB is supported by a cable BC and at A by a smooth fixed square rod which fits loosely through the square hole of the collar. If the force lb, determine the tension in cable BC and the x, y, z components of reaction at A. Solution-Con’t (slide 2)

4.68 Member AB is supported by a cable BC and at A by a smooth fixed square rod which fits loosely through the square hole of the collar. If the force lb, determine the tension in cable BC and the x, y, z components of reaction at A. Solution-Con’t (slide 3)

4.7 Friction Friction is the force of resistance offered by a body that prevents or retards a body from motion relative to the first. Friction always acts tangent to the surface and opposing any possible motion. Friction is caused by small asperities as shown here. We should consider all the minor surface asperities to get a distributed load .

4.7 Friction-2 Frictional coefficient changes from static to kinetic when the value reduces.. Consider the motion of the following structure.

4.7 Friction-3

4.7 Tipping or impending motion Tipping during motion or sliding depends if the clockwise moment at the bottom corner is CW or CCW. Evaluate the location of N with respect to W.

Example of pipes stacked The concrete pipes are stacked. Determine the minimum coefficient of static friction so that the pile does not collapse. Solution: Draw the Free body diagrams first. Top pipe

Example of pipes stacked Solution: Bottom pipe

Example of man on a plank Solution:

Example of man on a plank-2 Solution continued If the plank is on the verge of moving, slipping would occur at A.

Problem 4-100 (page 192, Section 4.6-4.7) 4.100 Two boys, each weighing 60 lb, sit at the ends of a uniform board, which has a weight of 30 lb. If the board rests at its center on a post having a coefficient of static friction of with the board, determine the greatest angle of tilt before slipping occurs. Neglect the size of post and the thickness of the board in the calculations. Solution:

Problem 4-112 (page 195, Review Problems) 4.112 The horizontal beam is supported by springs at its ends. If the stiffness of the spring at A is , determine the required stiffness of the spring at B so that if the beam is loaded with the 800-N force it remains in the horizontal position both before and after loading. Solution:

4. 112 The horizontal beam is supported by springs at its ends 4.112 The horizontal beam is supported by springs at its ends. If the stiffness of the spring at A is , determine the required stiffness of the spring at B so that if the beam is loaded with the 800-N force it remains in the horizontal position both before and after loading. Solution-Con’t (slide 2)

Chapter 4: Equilibrium.. concludes