statics unit 20

8/21/2019 Statics Unit 20

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Introduction to Statics

.PDF Edition Version 0.95

Unit 20

Equilibrium ofNon-Coplanar Force

Systems

Helen Margaret Lester PlantsLate Professor Emerita

Wallace Starr VenableEmeritus Associate Professor

West Virginia University, Morgantown, West Virginia

Copyrigt !"#" by Wallace Venable

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Unit 20Equilibrium of

Non-Coplanar Force Systems

Frame 20-1

Introduction

Except for nit !" #$ere #e on%& %oo'ed at partic%es" a%% t$e pro(%ems &ou $a)e so%)ed

so far $a)e dea%t #it$ (odies %oaded (& cop%anar force s&stems. *o#e)er" #e do in$a(it

a t$ree-dimensiona% uni)erse so &ou must %earn to cope #it$ pro(%ems in)o%)in+ t$e

e,ui%i(rium of (odies su(ect to %oads in t$ree dimensions.

Doin+ so in)o%)es no ne# t$eor& (ut t$e practice can (e a mite mess&. $is unit #i%%

t$erefore consist of se)era% some#$at mess& pro(%ems.

After ta'in+ a %oo' at supports in /-d" &ou #i%% re)ie# pro(%ems in)o%)in+ concurrent

non-cop%anar force s&stems. ext &ou #i%% consider s&stems of para%%e% non-cop%anar

forces. Last &ou #i%% #or' pro(%ems #it$ non-concurrent" non-para%%e%" non- cop%anar

forces. $ats #$en t$e part& +ets rou+$.3

*o#e)er &ou can #orr& a(out t$at #$en &ou +et t$ere.

4o to t$e next frame to (e+in.


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orrect response to precedin+ frame

o response

Frame 20-2

Supports and Connections

$ere are so man& sorts of supports and connections t$at can (e used in t$ree-

dimensiona% pro(%ems t$at it is impossi(%e to e)en attempt to discuss t$em a%%. 6e #i%%

examine a fe# common cases" (ut (e&ond t$at" t$e on%& ad)ice t$at can (e +i)en is"

7Fi+ure out $o# eac$ de)ice #or's in terms of #$at resistin+ forces and coup%es it can

pro)ide and dra# &our free (od& dia+rams accordin+%&.7

6$ic$ of t$e fo%%o#in+ %oads can t$e $in+es resist8

$ Force para%%e% to axis$ Force para%%e% to yaxis$

Force para%%e% to zaxis

$oment a(out axis

$ oment a(out yaxis$ oment a(out zaxis


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$e $in+es s$o#n can resist a%% %oads except a moment a(out t$e & axis.

Frame 20-/


$ec' t$e reactions #$ic$ t$e (a%%-and-soc'et oint can pro)ide.

$Force para%%e% to axis

$ Force para%%e% to yaxis$ Force para%%e% to zaxis$ oment a(out axis$ oment a(out yaxis$ oment a(out zaxis


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$e (a%%-and-soc'et s$o#n can resist forces in a%% t$ree directions" (ut no moments.

Frame 20-:


$ec' t$e reactions eac$ of t$e fo%%o#in+ can pro)ide.

F%an+ed #$ee% on rai%

$Force para%%e% to axis

$ Force para%%e% to yaxis$ Force para%%e% to zaxis$ oment a(out axis$ oment a(out yaxis$ oment a(out zaxis;$aft in a (earin+

$ Force para%%e% to axis$ Force para%%e% to yaxis$ Force para%%e% to zaxis$ oment a(out axis$ oment a(out yaxis$ oment a(out zaxis


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Frame 20-5

eview

*ere is a t#o-dimensiona% pro(%em in concurrent

forces a c&%inder restin+ in a +roo)e.

o so%)e it #$ic$ e,uation #ou%d &ou #rite8$ #F = 0$ #MG= 0


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#F = 0#MG= 0is true" of course" (ut #ou%d +i)e us no information" since a%% t$ree forces$a)e a moment arm of


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Frame 20-=

Concurrent .orces in Space

A (a%%oon is moored (&

t$ree ca(%es as s$o#n.

$e %ift on t$e (a%%oon is

!00 e#tons. Dra# t$e

F>D of t$e (a%%oon and

#rite a )ector expression

for t$e tension in eac$

cord.


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Frame 20-?


sin+ &our ans#ers from t$e precedin+ frame" #rite #F = 0and form t$e coefficiente,uations.


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Frame 20-9


sin+ &our coefficient e,uations from t$e precedin+ frame" so%)e for t$e ma+nitude of

t$e tension in eac$ ca(%e.


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Frame 20-10


List from memor& if possi(%e3 t$e steps ta'en in so%)in+ t$e precedin+ pro(%em.

1. @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@

2. @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@

/. @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@

:. @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@

5. @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@


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1. Dra# F>D

2. 6rite forces as )ectors

/. 6rite #F = 0:. 6rite coefficient e,uations

5. ;o%)e simu%taneous%&

Frame 20-11


Do Pro(%em 20-1 in &our note(oo'.


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Frame 20-12

%ransition

ou s$ou%d no# (e a(%e to #or' an& pro(%em dea%in+ #it$ forces in t$ree dimensions --

as %on+ as t$e& meet at a point.

$e next section #i%% dea% #it$ s&stems of para%%e% forces in space. Bnce more &ou #i%%

find t$at &our pro(%ems are )er& simi%ar to ones &ou #or'ed in t#o dimensions -- (ut

more tedious.

6it$ t$at c$eerin+ t$ou+$t in mind" %a& in #$ate)er supp%ies &ou need for a 15-

minute sei+e and turn to t$e next frame.


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o response

Frame 20-1/

eview//Parallel .orces in a Plane

Dra# t$e F>D and #rite t$e e,uations necessar& to so%)e for t$e reactions at Aand B .


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Frame 20-1:

Parallel .orces in Space

A $omo+eneous trian+u%ar p%ate #ei+$in+ 50 %( is suspended in a $oriD +i)en" #rite #F = 0and #MB= 0.


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Frame 20-15


Ceduce t$e e,uations from t$e precedin+ frame to coefficient e,uations and so%)e for

t$e tension in eac$ cord.


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$is resu%t mi+$t $a)e (een predicted from t$e s&mmetr& of t$e fi+ure (ut suc$

predictions s$ou%d (e made #it$ extreme caution.3

Frame 20-1!


A t$ree #$ee%ed do%%& #ei+$s 20 'i%o+rams. 6$en

carr&in+ a 1!00 e#ton %oad" t$e #$ee% pressure

at Ais /00 e#tons" as is t$e #$ee% pressure at

B. $e pro(%em is-- #$ere is t$e center of +ra)it&

of t$e %oad %ocated8 Dra# a F>D of t$e do%%&.


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Frame 20-1=


sin+ &our F>D from t$e precedin+ frame #rite t$e necessar& e,uations" reduce to

coefficient e,uations" and so%)e. Locate t$e %oad on a top )ie# of t$e p%atform.


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Frame 20-1?


Do Pro(%em 20-2 in &our note(oo'.


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Frame 20-19

%ransition

o# &ou can #or' t$e eas& ones in t$ree dimensions. $e rest of t$e unit #i%% s$o#

&ou $o# to do t$e $ard ones.

$e pro(%ems comin+ up #i%% dea% #it$ non-concurrent" non-para%%e%" non- cop%anar

s&stems. nfortunate%&" t$e& are not non-existent.3

$e met$od is t$e same dra# F>D" #rite #F = 0" #rite #MP= 0" form coefficiente,uations and so%)e. $e comp%ications arise from t$e fact t$at most sin+%e (od&

pro(%ems $a)e six un'no#ns Patience and attention to detai% #i%% $e%p (ut most%& it

#i%% ta'e time -- at %east an $our to finis$ t$e unit.

$e first pro(%em is a re%ati)e%& eas& one. ;$arpen &our penci%" find &our ca%cu%atorand (e+in.


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o response

Frame 20-20

.orces in Parallel Planes

$e %ar+er pu%%e& #ei+$s 100 pounds and is 1? inc$es in diameter. $e sma%%er pu%%e&

#ei+$s 50 pounds and is 12 inc$es in diameter. omp%ete t$e free (od& of t$e pu%%e&s

and s$aft.


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Frame 20-21


sin+ t$e free (od& from t$e precedin+ frame" #rite t$e e,uation #F = 0and

#MB= 0.ote ;ince t$ese pro(%ems are ,uite %a(orious &ou #i%% not %i'e%& #ant to #or' )er&

man&. onse,uent%& it mi+$t (e a +ood idea to so%)e t$em in a form suita(%e for

inc%usion in &our note(oo'.3


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Frame 20-22


1. 6or' out t$e cross products in t$e moment e,uation from t$e precedin+ frame.

2. *o# man& coefficient e,uations can &ou +et from t$e moment e,uation8 @@@@@

/. *o# man& coefficient e,uations can &ou +et from t$e force e,uation8 @@@@@

:. *o# man& un'no#ns $a)e &ou8 @@@@@

5. an &ou so%)e t$e pro(%em8

!. Do &ou #ant to8


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1.

2. //. 2

:. 5

5. es

!. certain%& dont #ant to.

Frame 20-2/

igid 0odies//1oncoplanar .orces

6rite prefera(%& from memor&3 t$e steps &ou too' in #or'in+ t$e precedin+ pro(%em

1. @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@

2. @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@

/. @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@

:. @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@

5. @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@


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1. Dra# free (od& dia+ram

2. 6rite a%% forces as )ectors

/. 6rite #F = 0and #M0= 0:. >rea' into coefficient e,uations

5. ;o%)e

Frame 20-2:

igid 0odies//1oncoplanar, 1onConcurrent, 1onParallel .orces

$e derric' s$o#n supports a

#ei+$t of 1000 pounds. $ere is a(a%% and soc'et at Aand a(%es run

from Dto Cand from Dto B .

$e free (od& is s$o#n.

6rite t$e tension in eac$ ca(%e in

)ector form.


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Frame 20-25

igid 0odies//1onCoplanar, 1onConcurrent, 1onParallel .orces

$e free (od& of t$e derric' from t$e

precedin+ frame is redra#n for &our

con)enience. 6rite #F = 0and #MA= 0.


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Frame 20-2!

igid 0odies//1onCoplanar .orces

>rea' t$e Force E,uation and t$e oment E,uation &ou ust #rote into component

e,uations.

*o# man& e,uations $a)e &ou8 @@@@@

*o# man& un'no#ns8 @@@@@


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Frame 20-2=

igid 0odies//1onCoplanar .orces

f &ou need practice in so%)in+ simu%taneous e,uations so%)e for t$e reactions at Aand

t$e tensions in t$e ca(%es. Bt$er#ise +o to t$e next frame.


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TBD= 867 lbTCD

= 600 lbRx= 667 lbRy= 2200 lbRz= 0 lb

Frame 20-2?

2eneral *3uilibriu'

A %ar+e si+n supp%& &our o#n messa+e3 of

uniform densit& #ei+$s /!0 pounds and is

supported (& t#o ca(%es and a (a%% and

soc'et at A. Find t$e tension in eac$ ca(%eand t$e components of t$e reaction at A.


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Frame 20-29

2eneral *3uilibriu'

o ma'e sure &ou $a)e at %east one of t$ese %itt%e o&s in &our notes" do Pro(%em 20-/

in &our note(oo'. $eer up" its not as (ad as t$e %ast one-- ,uite.3


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Frame 20-/0

Closure

$at does it *a%%e%ua$ >& t$is time &ou undou(ted%& a+ree #it$ t$e@ t$esis t$at

t$ese pro(%ems are tedious. t is to (e $oped t$at &ou a%so a+ree t$at +i)en enou+$

time and patience &ou can so%)e t$ree-dimensiona% e,ui%i(rium pro(%ems exact%& as

&ou did t$ose in t#o dimensions.

n fact a%% statica%%& determinate e,ui%i(rium pro(%ems #i%% &ie%d to t$e t#o e,uations

Ei G 0 and 0 G 0.

Possi(%& t$at is #$& t$e& are 'no#n as t$e e,ui%i(rium e,uations.

statics unit 20

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