VECTOR MECHANICS FOR ENGINEERS: STATICS
VECTOR MECHANICS FOR ENGINEERS: STATICS
12th Edition
ISBN: 9781260912814
Author: BEER
Publisher: MCG
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Chapter 5.4, Problem 5.134P

Locate the centroid of the section shown, which was cut f an elliptical cylinder by an oblique plane.

Chapter 5.4, Problem 5.134P, Locate the centroid of the section shown, which was cut f an elliptical cylinder by an oblique

Fig. P5.134

Expert Solution & Answer
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To determine

The centroid of the section.

Answer to Problem 5.134P

The centroid of the section (x¯,y¯,z¯) is (0,516h,14h).

Explanation of Solution

Refer Figure 1 and Figure 2.

VECTOR MECHANICS FOR ENGINEERS: STATICS, Chapter 5.4, Problem 5.134P , additional homework tip  1VECTOR MECHANICS FOR ENGINEERS: STATICS, Chapter 5.4, Problem 5.134P , additional homework tip  2

Consider an elemental section of the given section.

Write an expression to calculate the volume of the element.

dV=2xydz (I)

Here, dz is the thickness of the element, dV is the volume of the element, y is the height of the element and x is the width of the element.

From the symmetry, write an expression to calculate the distance of centroid of the section from x-axis.

x¯=0 (II)

Here, x¯ is the distance of centroid of section from x axis.

From the symmetry, write an expression to calculate the distance of centroid of the section from x-axis.

x¯=0 (II)

Here, x¯ is the distance of centroid of section from x axis.

Write an expression to calculate the distance of centroid of element from z-axis.

z¯EL=z (IV)

Here, z¯EL is the distance of centroid of element from z axis.

Write an expression to calculate width of the element.

x=abb2z2 (V)

Here, a is the semi major axis of the elliptical plane and b is the semi minor axis of the elliptical plane.

Write an expression to calculate the height of the element.

y=(h/2)bz+h2=(h2b)(bz) (VI)

Here, h is the height of the element.

Write an expression to calculate the volume of the section.

V=2xydz (VII)

Write an expression to calculate the thickness of the section.

z=bsinθ (VIII)

Differentiate the equation to calculate the thickness of the element.

dz=bcosθdθ (IX)

Write an expression to find the distance of the centroid of the section from x axis.

x¯=x¯ELdVV (X)

Here, x¯ is the distance of centroid of section from x axis.

Write an expression to find the distance of the centroid of the section from x axis.

y¯=y¯ELdVV (XI)

Here, y¯ is the distance of centroid of section from y axis.

Write an expression to find the distance of the centroid of the section from z axis.

z¯=z¯ELdVV (XII)

Here, z¯ is the distance of centroid of section from z axis.

Conclusion:

Substitute (V), (VI) and (IX) in equation (VII) to find V.

V=bb(2abb2z2)((h2b)(bz))dz=ahb2π/2π/2(bcosθ)(b(1sinθ))bcosθdθ=abhπ/2π/2(cos2θsinθcos2θ)dθ=abh(θ2+sin2θ4+13cos3θ)|π/2π/2=12πabh (XIII)

write an expression to calculate the distance of centroid of the element from x-axis.

x¯=0

Write an expression to calculate y¯ELdV.

y¯ELdV=bb(12(h2b)(bz))((2abb2z2)((h2b)(bz))dz)=14ah2b3bb(bz)2b2z2dz=14ah2b3π/2π/2(bbsinθ)2b2(bsinθ)2(bcosθdθ)=14ah2b3π/2π/2(b(1sinθ))2(bcosθ)(bcosθdθ)=14abh2π/2π/2cos2θ2sinθcos2θ+sin2θcos2θdθ=14abh2π/2π/2cos2θ2sinθcos2θ+(12(1cos2θ))(12(1+cos2θ))dθ=14abh2π/2π/2cos2θ2sinθcos2θ+14(1cos22θ)dθ=14abh2((θ2+sin2θ4)+13cos3θ+14θ14(θ2+sin4θ8))|π/2π/2=532πabh2 (XIV)

Substitute equation (XIII) and (XIV) in equation (XI) to find y¯.

y¯=532πabh212πabh=516h

Write an expression to calculate z¯ELdV.

z¯ELdV=bbz(2aba2z2(h2b)(bz))dz=ahb2bbz(bz)b2z2dz=ahb2π/2π/2(bsinθ)(b(1sinθ))(bcosθ)(bcosθdθ)=ab2hπ/2π/2(sinθcos2θsin2θcos2θ)dθ=ab2hπ/2π/2(sinθcos2θ14(1cos22θ))dθ=ab2h(13cos3θ14θ+14(θ2sin4θ8))|π/2π/2=18πabh2 (XV)

Substitute equation (XIII) and (XV) in equation (XII) to find z¯.

z¯=18πabh212πabh=14h

Thus, the centroid of the section (x¯,y¯,z¯) is (0,516h,14h).

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Chapter 5 Solutions

VECTOR MECHANICS FOR ENGINEERS: STATICS

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