1 007 The McGrw-Hill Compnies, Inc. All rights reserved. Eighth E CHAPTER 5 Distriuted VECTOR MECHANICS FOR ENGINEERS: STATICS Ferdinnd P. Beer E. Rus...

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CHAPTER

5

VECTOR MECHANICS FOR ENGINEERS:

STATICS Ferdinand P. Beer E. Russell Johnston, Jr. Lecture Notes:

Distributed Forces: Centroids and Centers of Gravity

J. Walt Oler Texas Tech University

© 2007 The McGraw-Hill Companies, Inc. All rights reserved.

Eighth Edition

Vector Mechanics for Engineers: Statics Contents Introduction

Theorems of Pappus-Guldinus

Center of Gravity of a 2D Body

Sample Problem 5.7

Centroids and First Moments of Areas and Lines

Distributed Loads on Beams

Centroids of Common Shapes of Areas Centroids of Common Shapes of Lines

Center of Gravity of a 3D Body: Centroid of a Volume

Composite Plates and Areas

Centroids of Common 3D Shapes

Sample Problem 5.1

Composite 3D Bodies

Determination of Centroids by Integration

Sample Problem 5.12

Sample Problem 5.9

Sample Problem 5.4

© 2007 The McGraw-Hill Companies, Inc. All rights reserved.

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Eighth Edition

Vector Mechanics for Engineers: Statics Introduction • The earth exerts a gravitational force on each of the particles forming a body. These forces can be replace by a single equivalent force equal to the weight of the body and applied at the center of gravity for the body. • The centroid of an area is analogous to the center of gravity of a body. The concept of the first moment of an area is used to locate the centroid. • Determination of the area of a surface of revolution and the volume of a body of revolution are accomplished with the Theorems of Pappus-Guldinus.

© 2007 The McGraw-Hill Companies, Inc. All rights reserved.

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Eighth Edition

Vector Mechanics for Engineers: Statics Center of Gravity of a 2D Body • Center of gravity of a plate

∑My

• Center of gravity of a wire

x W = ∑ x∆W = ∫ x dW

∑My

yW = ∑ y ∆W = ∫ y dW

© 2007 The McGraw-Hill Companies, Inc. All rights reserved.

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Eighth Edition

Vector Mechanics for Engineers: Statics Centroids and First Moments of Areas and Lines • Centroid of an area

• Centroid of a line

x W = ∫ x dW

x W = ∫ x dW

x (γAt ) = ∫ x (γt )dA

x (γ La ) = ∫ x (γ a )dL

x A = ∫ x dA = Q y = first moment wit h respect to y

x L = ∫ x dL y L = ∫ y dL

y A = ∫ y dA = Q x = first moment wit h respect to x © 2007 The McGraw-Hill Companies, Inc. All rights reserved.

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Eighth Edition

Vector Mechanics for Engineers: Statics First Moments of Areas and Lines • An area is symmetric with respect to an axis BB’ if for every point P there exists a point P’ such that PP’ is perpendicular to BB’ and is divided into two equal parts by BB’. • The first moment of an area with respect to a line of symmetry is zero. • If an area possesses a line of symmetry, its centroid lies on that axis • If an area possesses two lines of symmetry, its centroid lies at their intersection. • An area is symmetric with respect to a center O if for every element dA at (x,y) there exists an area dA’ of equal area at (-x,-y). • The centroid of the area coincides with the center of symmetry. © 2007 The McGraw-Hill Companies, Inc. All rights reserved.

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Eighth Edition

Vector Mechanics for Engineers: Statics Centroids of Common Shapes of Areas

© 2007 The McGraw-Hill Companies, Inc. All rights reserved.

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Eighth Edition

Vector Mechanics for Engineers: Statics Centroids of Common Shapes of Lines

© 2007 The McGraw-Hill Companies, Inc. All rights reserved.

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Eighth Edition

Vector Mechanics for Engineers: Statics Composite Plates and Areas • Composite plates X ∑W = ∑ x W Y ∑W = ∑ yW

• Composite area X∑ A = ∑ xA Y ∑ A = ∑ yA

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Eighth Edition

Vector Mechanics for Engineers: Statics Sample Problem 5.1 SOLUTION: • Divide the area into a triangle, rectangle, and semicircle with a circular cutout. • Calculate the first moments of each area with respect to the axes.

For the plane area shown, determine the first moments with respect to the x and y axes and the location of the centroid.

• Find the total area and first moments of the triangle, rectangle, and semicircle. Subtract the area and first moment of the circular cutout. • Compute the coordinates of the area centroid by dividing the first moments by the total area.

© 2007 The McGraw-Hill Companies, Inc. All rights reserved.

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Eighth Edition

Vector Mechanics for Engineers: Statics Sample Problem 5.1

• Find the total area and first moments of the triangle, rectangle, and semicircle. Subtract the area and first moment of the circular cutout. © 2007 The McGraw-Hill Companies, Inc. All rights reserved.

Qx = +506.2 × 103 mm 3 Q y = +757.7 × 103 mm 3 5 - 11

Eighth Edition

Vector Mechanics for Engineers: Statics Sample Problem 5.1 • Compute the coordinates of the area centroid by dividing the first moments by the total area.

x A + 757.7 × 103 mm 3 ∑ X = = ∑ A 13.828 ×103 mm 2

X = 54.8 mm

y A + 506.2 × 103 mm 3 ∑ Y = = ∑ A 13.828 ×103 mm 2 Y = 36.6 mm © 2007 The McGraw-Hill Companies, Inc. All rights reserved.

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Eighth Edition

Vector Mechanics for Engineers: Statics Determination of Centroids by Integration x A = ∫ x dA =

∫∫ x dx dy = ∫ xel dA y A = ∫ y dA = ∫∫ y dx dy = ∫ y el dA

x A = ∫ xel dA = ∫ x ( ydx ) y A = ∫ y el dA y = ∫ ( ydx ) 2

• Double integration to find the first moment may be avoided by defining dA as a thin rectangle or strip.

x A = ∫ xel dA a+x [ (a − x )dx ] =∫ 2 y A = ∫ y el dA = ∫ y [(a − x )dx ]

© 2007 The McGraw-Hill Companies, Inc. All rights reserved.

x A = ∫ x el dA =

2r 1 2 cos θ r dθ ∫ 3 2

y A = ∫ y el dA =

2r 1 2 sin θ r dθ ∫ 3 2 5 - 13

Eighth Edition

Vector Mechanics for Engineers: Statics Sample Problem 5.4 SOLUTION: • Determine the constant k. • Evaluate the total area. • Using either vertical or horizontal strips, perform a single integration to find the first moments. Determine by direct integration the location of the centroid of a parabolic spandrel.

© 2007 The McGraw-Hill Companies, Inc. All rights reserved.

• Evaluate the centroid coordinates.

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Eighth Edition

Vector Mechanics for Engineers: Statics Sample Problem 5.4 SOLUTION: • Determine the constant k. y = k x2 b = k a2 ⇒ k = y=

b a2

x2

or

b a2 x=

a b1 2

y1 2

• Evaluate the total area. A = ∫ dA 3 a

b x = ∫ y dx = ∫ 2 x dx = 2 3 0 0a a ab = 3 a

© 2007 The McGraw-Hill Companies, Inc. All rights reserved.

b

2

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Eighth Edition

Vector Mechanics for Engineers: Statics Sample Problem 5.4 • Using vertical strips, perform a single integration to find the first moments. a

b Q y = ∫ xel dA = ∫ xydx = ∫ x 2 x 2 dx 0 a a

b x4 a 2b = 2 = 4 a 4 0 a

Qx = ∫ yel dA = ∫

2

y 1 b ydx = ∫ 2 x 2 dx 2 02a

a

b2 x5 ab 2 = 4 = 2a 5 0 10

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Eighth Edition

Vector Mechanics for Engineers: Statics Sample Problem 5.4 • Or, using horizontal strips, perform a single integration to find the first moments. b 2 a+x a − x2 (a − x )dy = ∫ Q y = ∫ xel dA = ∫ dy 2 2 0

1 b 2 a 2 = ∫ a − 2 0 b

2 a b y dy = 4

a Qx = ∫ yel dA = ∫ y (a − x )dy = ∫ y a − 1 2 y1 2 dy b a 3 2 ab 2 = ∫ ay − 1 2 y dy = 10 b 0 b

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Eighth Edition

Vector Mechanics for Engineers: Statics Sample Problem 5.4 • Evaluate the centroid coordinates. xA = Q y ab a 2 b x = 3 4

x=

3 a 4

yA = Q x ab ab 2 y = 3 10

© 2007 The McGraw-Hill Companies, Inc. All rights reserved.

3 y= b 10

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Eighth Edition

Vector Mechanics for Engineers: Statics Theorems of Pappus-Guldinus

• Surface of revolution is generated by rotating a plane curve about a fixed axis.

• Area of a surface of revolution is equal to the length of the generating curve times the distance traveled by the centroid through the rotation. A = 2π y L

© 2007 The McGraw-Hill Companies, Inc. All rights reserved.

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Eighth Edition

Vector Mechanics for Engineers: Statics Theorems of Pappus-Guldinus

• Body of revolution is generated by rotating a plane area about a fixed axis. • Volume of a body of revolution is equal to the generating area times the distance traveled by the centroid through the rotation. V = 2π y A

© 2007 The McGraw-Hill Companies, Inc. All rights reserved.

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Eighth Edition

Vector Mechanics for Engineers: Statics Sample Problem 5.7 SOLUTION: • Apply the theorem of Pappus-Guldinus to evaluate the volumes or revolution for the rectangular rim section and the inner cutout section. • Multiply by density and acceleration to get the mass and acceleration. The outside diameter of a pulley is 0.8 m, and the cross section of its rim is as shown. Knowing that the pulley is made of steel and that the density of steel is = 7mass .85 ×and 10 3 weight kg m 3of the determineρthe rim.

© 2007 The McGraw-Hill Companies, Inc. All rights reserved.

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Eighth Edition

Vector Mechanics for Engineers: Statics Sample Problem 5.7 SOLUTION: • Apply the theorem of Pappus-Guldinus to evaluate the volumes or revolution for the rectangular rim section and the inner cutout section. • Multiply by density and acceleration to get the mass and acceleration.

(

3

m = ρ V = 7.85 × 10 kg m

(

3

3 −9 3 7.65 × 10 mm 10 m mm

)(

W = mg = (60 .0 kg ) 9.81 m s 2

6

)

© 2007 The McGraw-Hill Companies, Inc. All rights reserved.

3

)

m = 60 .0 kg W = 589 N 5 - 22

Eighth Edition

Vector Mechanics for Engineers: Statics Distributed Loads on Beams

L

W = ∫ wdx = ∫ dA = A 0

(OP )W = ∫ x dW L

(OP ) A = ∫ x dA = x A 0

• A distributed load is represented by plotting the load per unit length, w (N/m) . The total load is equal to the area under the load curve.

• A distributed load can be replace by a concentrated load with a magnitude equal to the area under the load curve and a line of action passing through the area centroid.

© 2007 The McGraw-Hill Companies, Inc. All rights reserved.

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Eighth Edition

Vector Mechanics for Engineers: Statics Sample Problem 5.9 SOLUTION: • The magnitude of the concentrated load is equal to the total load or the area under the curve. • The line of action of the concentrated load passes through the centroid of the area under the curve. A beam supports a distributed load as shown. Determine the equivalent concentrated load and the reactions at the supports.

© 2007 The McGraw-Hill Companies, Inc. All rights reserved.

• Determine the support reactions by summing moments about the beam ends.

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Eighth Edition

Vector Mechanics for Engineers: Statics Sample Problem 5.9 SOLUTION: • The magnitude of the concentrated load is equal to the total load or the area under the curve. F = 18 .0 kN • The line of action of the concentrated load passes through the centroid of the area under the curve. X =

© 2007 The McGraw-Hill Companies, Inc. All rights reserved.

63 kN ⋅ m 18 kN

X = 3 .5 m

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Eighth Edition

Vector Mechanics for Engineers: Statics Sample Problem 5.9 • Determine the support reactions by summing moments about the beam ends.

∑ M A = 0 : B y (6 m ) − (18 kN )(3.5 m ) = 0 B y = 10.5 kN

∑ M B = 0 : − A y (6 m ) + (18 kN )(6 m − 3.5 m ) = 0 A y = 7.5 kN

© 2007 The McGraw-Hill Companies, Inc. All rights reserved.

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Eighth Edition

Vector Mechanics for Engineers: Statics Center of Gravity of a 3D Body: Centroid of a Volume

• Center of gravity G − W j = ∑ (− ∆ W j ) rG × (− W j ) = ∑ [r × (− ∆ W j )] rG W × (− j ) = (∑ r ∆ W ) × (− j ) W = ∫ dW

• Results are independent of body orientation, x W = ∫ xdW

y W = ∫ ydW

z W = ∫ zdW

• For homogeneous bodies,

rG W = ∫ r dW

© 2007 The McGraw-Hill Companies, Inc. All rights reserved.

W = γ V and dW = γ dV x V = ∫ xdV

y V = ∫ ydV

z V = ∫ zdV 5 - 27

Eighth Edition

Vector Mechanics for Engineers: Statics Centroids of Common 3D Shapes

© 2007 The McGraw-Hill Companies, Inc. All rights reserved.

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Eighth Edition

Vector Mechanics for Engineers: Statics Composite 3D Bodies • Moment of the total weight concentrated at the center of gravity G is equal to the sum of the moments of the weights of the component parts. X ∑ W = ∑ xW

Y ∑ W = ∑ yW

Z ∑ W = ∑ zW

• For homogeneous bodies, X ∑ V = ∑ xV

© 2007 The McGraw-Hill Companies, Inc. All rights reserved.

Y ∑ V = ∑ yV

Z ∑ V = ∑ zV

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Eighth Edition

Vector Mechanics for Engineers: Statics Sample Problem 5.12 SOLUTION: • Form the machine element from a rectangular parallelepiped and a quarter cylinder and then subtracting two 1-in. diameter cylinders.

Locate the center of gravity of the steel machine element. The diameter of each hole is 1 in.

© 2007 The McGraw-Hill Companies, Inc. All rights reserved.

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Eighth Edition

Vector Mechanics for Engineers: Statics Sample Problem 5.12

© 2007 The McGraw-Hill Companies, Inc. All rights reserved.

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Eighth Edition

Vector Mechanics for Engineers: Statics Sample Problem 5.12

X = ∑ xV

∑ V = (3.08 in 4 ) (5.286 in 3 ) X = 0.577 in.

Y = ∑ yV

∑ V = (− 5.047 in 4 ) (5.286 in 3 ) Y = 0.577 in.

Z = ∑ zV

∑ V = (1.618 in 4 ) (5.286 in 3 ) Z = 0.577 in.

© 2007 The McGraw-Hill Companies, Inc. All rights reserved.

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