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Engineering Mechanics: Statics

Chapter 3: Equilibrium

Part A: Equilibrium in Two Dimensions

Equilibrium

� In equilibrium,

� Before applying the equation, we must define the mechanical system to be analyzed and represent allforces acting on the body

� To do that, the body has to be isolated from all surrounding bodies

� A diagramatic representation of the isolated system treated as a single body = free-body diagram (FBD)

0 0R F M M= ∑ = = ∑ =v vv v v v

FBD is the most important step in the solution of problems in mechanics!

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Free-Body Diagram

Free-Body Diagram

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Free-Body Diagram

Free-Body Diagram

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Free-Body Diagram

Free-Body Diagram

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Equilibrium Conditions

� In two dimensions, equations of equilibrium may be written as

0 0 0x y OF F M∑ = ∑ = ∑ =

Two- and Three-Force Members

� A body under the action of two forces only = two-force member

� For a two-force member to be in equilibrium, the forces must be equal, opposite and collinear

� For a three-force member, equilibrium requires the lines of action of the three forces to be concurrent

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Problem 3/6

The 450-kg uniform I-beam supports the load shown. Determine the reactions at the supports.

Problem 3/24

A block placed under the head of the claw hammer as shown greatly facilitates the extraction of the nail. If a 200-

N pull on the handle is required to pull the nail, calculate the tension T in the nail.

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Problem 3/55

It is desired that a person be ableto begin closing the van hatchfrom the open position shownwith a 40-N vertical force P. As adesign exercise, determine thenecessary force in each of thetwo hydraulic struts AB. The masscenter of the 40-kg door is 37.5mm directly below point A. Treatthe problemas two-dimensional.

(Ans. F = 803 N)

Equilibrium

Part B: Equilibrium in Three Dimensions

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Equilibrium Conditions

� In three dimensions, equations of equilibrium may be written as

� Statical determinacy

� The supporting constraints are not more than the number required to establish equilibrium condition

� If the supports are redundant, the body is statically indeterminate

0; 0, 0, 0

0; 0, 0, 0

x y z

x y z

F F F F

M M M M

∑ = ∑ = ∑ = ∑ =

∑ = ∑ = ∑ = ∑ =

vv

vv

Free-Body Diagram

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Free-Body Diagram

Sample Problem 3/5

The uniform 7-m steel shaft has a mass of 200 kg and is supported by a ball-and-socket joint at A in the horizontal floor. The ball end Brests against the smooth vertical walls as shown. Compute the forces exerted by the walls and the floor on the ends of the shaft.

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Problem 3/67

The light right-angle boom which supports the 400-kg cylinder is supported by three cables and a ball-and-socket joint at O attached to the vertical x-y surface. Determine the reactions at O and the cable tensions.