Engineering Mechanics: Statics in SI Units, 12e
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1 Engineering Mechanics: Statics in S Units, 1e 10 Moments of nertia 1 Chapter(Objectives Method(for(determining(the(moment(of(inertia(for(an(area( Definition(of(Radius(of(Gration Chapter(Outline 1. Definitions(of(Moments(of(nertia(for(reas(. Parallelis(Theorem(for(an(rea(. Radius(of(Gration(of(an(rea( 4. Moments(of(nertia(for(Composite(reas
2 10.1(Definition(of(Moments(of(nertia(for(reas The(centroid(of(area(is(first(order(of(moment(of(an( area,( d d Moment(of(inertial(is(the(second(order(of(moment( of(an(area( d d Moment(of(inertia(is(used(to(calculate,(for(eample,( when(calculating(a(moment(about(an(ais(for( distributed(load,(where(the(force(is(proportion(to( distance( Mainl(applied(in(Mechanics(of(Materials,(Structural( Mechanics,(Fluid(Mechanics,(Mechanical(Design 10.1(Definition(of(Moments(of(nertia(for(reas For(eample(the(pressure(of(fluid(p(is(proportional( to(depth(, p γ! where γ is Specific(Weight of fluid! The(force(on(a(small(area(is(dF pd((( Moment(of(a(force(about(the((ais(is(( dm df γ d, M γ d The(value( (((((((((((((is(called(( d the(moment(of(the(area(( about(ais γ d 4
3 10.1(Definition(of(Moments(of(nertia(for(reas Moment(of(nertia(( Consider(the(area((on(the(pla(#( The(moment(of(inertial(of(the(small(are( about( and( is(( d d d d The(moment(of(inertia(about((and( ais is( d d (Definition(of(Moments(of(nertia(for(reas Polar(Moment(of(nertia(( The(second(order(of(moment(of(the(area(d(about(O(or(about( the(z(ais(is((this(ais(is(called(polar(ais)( dj O r d (r(is(the(perpendicular(distance(from(pole((z(ais)(to(the(area(d Polar(moment(of(inertia( J O r d + r + 6
4 10.(Parallel(is(Theorem(for(an(rea The(theorem(is(used(to(find(the(moment(of(inertia( of(an(area(about(an(arbitrar(ais(parallel(to(the( ais(passing(the(centroid(of(the(area((the(moment( of(inertia(about(the(centroid(must(be(known)( Consider(the(area,(in(which( ((is(passing(through( the(centroid( The(area(d(has(the(distance(from((to( (equal(to( (( The(moment(of(inertial(of(the(area(d(( about(!with(the(distance(from(the(centroid( (ais( ( equal(to(d ( ) d ' + d ( ' + d ) ' d + d d d ' d + d d 7 10.(Parallel(is(Theorem(for(an(rea ' d + d ' d + The(first(term(is(the(moment(of(inertial(about(the(ais(passing(the( centroid( The(second(term(is(zeros(since( (passes(the(centroid( ' d d 0; 0 The(third(term(is(( d Hence( ( and( + d + d nd( JO JC + d ( where( ( and d d + d J C! +! d d 8
5 10.(Radius(of(Gration(of(an(rea Radius(of(gration(is(used(in(column(design(is( k k ko JO This(is(similar(to(an(equivalent(radius(when(finding( the(moment(of(inertia( The(equation(is(similar(to(the(moment(of(inertia k, where d d 9 Procedure(for(nalsis Consider(ntegration(of(d! Eample(Find( ( ( Case(1( Consider(d(parallel(the((ais( ( Case( d d and d Case(1 Consider(d(perpendicular(to(the((ais( d d but d Find find of small area from parallel ais theorem Case( 10
6 11 Eample(10.1 Determine(the(moment(of(inertia(for(the(rectangular(area(with( respect(to(( (a) the(centroidal( (ais,(( (b)(the(ais( b (passing(through(the(base(of(the(rectangle,(( (c)(the(pole(or(z (ais(perpendicular(( to(the( # (plane(and(passing(( through(the(centroid(c. 1 Solution Part((a)( Differential(element(chosen,(distance( (from( (ais.( Since(d!!b!d,( Part((b)( B(appling(parallel(ais(theorem, / / / / 1 1 ' ') ( ' ' bh d b bd d h h h h!! bh h bh bh d b! " # $ % & + + '
7 Solution Part((c)( For(polar(moment(of(inertia(about(point(C,( First(determine,( Then: J ' C 1 1! hb + ' 1 1 bh( h + b ) 1 Eample(10. Determine(the(moment(of(inertia(for(the(area(about((ais. Case(1 Case( 14
8 Solution Solution(((case(1)( Differential(element(area;(!! d ( 100 ) d Moment(of(nertia 0 00mm d 107(10 6 ) 0 00mm & # $ ! d % " mm 4 (100 ) d 15 Solution Solution(((case()( To(determine(the(moment(of(inertia(of(the(differential(element( area(d,(b(parallelais(theorem.(( For(a(small(rectangle(element( Then;(( 1 d! d!!! 1 b(parallelais(theorem b d and h 16
9 Eample(10. Determine(the(moment(of(inertia(for(the(area(about((ais. Case(1 Case( 17 Solution Solution(((case(1)( Differential(element(area;(!! d d Moment(of(nertia 18
10 Solution Solution(((case()( To(determine(the(moment(of(inertia(of(the(differential(element( area(d,(( For(a(small(rectangle(element( Then;((!!! ntegrating(with(respect(to( b d and h (Moments(of(nertia(for(Composite(reas The(composite(area(is(composed(of(areas(with(man(simple(geometries( The(moment(of(inertia(of(composite(areas(can(be(found(b(the(sum(of(individual( moment(of(inertia(with(sign((holes(are(negative)( Procedure(for(nalsis( Composite(areas( Divide(a(composite(area(into(simple(geometrical(areas(and(locate(the(centroids( of(each(of(the(areas(and(find(the(moment(of(inertias(about(each(of(their(own( centroids( Parallel(is(Theorem( ppl(the(parallel(ais(theorem(to(calculate(the(moment(of(inertia(about(the( required(ais( Summation( Sum(all(of(the(moment(of(inertia,(beware(of(signs 0
11 Eample(10.4 Compute(the(moment(of(inertia(of(the(composite(area(about( the((ais. 1 Solution Composite(Parts( Composite(area(obtained(b(subtracting(the(circle(form(the( rectangle.( Centroid(of(each(area(is(located(in(the(figure(below.
12 Solution Parallel(is(Theorem( Circle( 1 4 ' + d π 5 + π 5 4 Rectangle( 1 ' + d Summation( 6 4 ( ) ( ) ( 75) 11.4( 10 ) mm 6 4 ( )( ) ( )( 150)( 75) 11.5( 10 ) mm For(moment(of(inertia(for(the(composite(area, ( 10 ) ( 10 ) 6 4 ( ) mm
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