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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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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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orrect response to precedin+ frame
$e $in+es s$o#n can resist a%% %oads except a moment a(out t$e & axis.
Frame 20-/
Supports and Connections
$ec' t$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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orrect response to precedin+ frame
$e (a%%-and-soc'et s$o#n can resist forces in a%% t$ree directions" (ut no moments.
Frame 20-:
Supports and Connections
$ec' t$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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orrect response to precedin+ frame
#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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orrect response to precedin+ frame
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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orrect response to precedin+ frame
Frame 20-?
Concurrent .orces in Space
sin+ &our ans#ers from t$e precedin+ frame" #rite #F = 0and form t$e coefficiente,uations.
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orrect response to precedin+ frame
Frame 20-9
Concurrent .orces in Space
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
Concurrent .orces in Space
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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orrect response to precedin+ frame
1. Dra# F>D
2. 6rite forces as )ectors
/. 6rite #F = 0:. 6rite coefficient e,uations
5. ;o%)e simu%taneous%&
Frame 20-11
Concurrent .orces in Space
Do Pro(%em 20-1 in &our note(oo'.
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orrect response to precedin+ frame
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 $oriD +i)en" #rite #F = 0and #MB= 0.
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orrect response to precedin+ frame
Frame 20-15
Parallel .orces in Space
Ceduce t$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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orrect response to precedin+ frame
$is resu%t mi+$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!
Parallel .orces in Space
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$e do%%&.
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orrect response to precedin+ frame
Frame 20-1=
Parallel .orces in Space
sin+ &our F>D from t$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$e p%atform.
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orrect response to precedin+ frame
Frame 20-1?
Parallel .orces in Space
Do Pro(%em 20-2 in &our note(oo'.
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orrect response to precedin+ frame
Frame 20-19
%ransition
o# &ou can #or' t$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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orrect response to precedin+ frame
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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orrect response to precedin+ frame
Frame 20-21
.orces in Parallel Planes
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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orrect response to precedin+ frame
Frame 20-22
.orces in Parallel Planes
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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orrect response to precedin+ frame
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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orrect response to precedin+ frame
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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orrect response to precedin+ frame
Frame 20-25
igid 0odies//1onCoplanar, 1onConcurrent, 1onParallel .orces
$e free (od& of t$e derric' from t$e
precedin+ frame is redra#n for &our
con)enience. 6rite #F = 0and #MA= 0.
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orrect response to precedin+ frame
Frame 20-2!
igid 0odies//1onCoplanar .orces
>rea' t$e Force E,uation and t$e oment E,uation &ou ust #rote into component
e,uations.
*o# man& e,uations $a)e &ou8 @@@@@
*o# man& un'no#ns8 @@@@@
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orrect response to precedin+ frame
Frame 20-2=
igid 0odies//1onCoplanar .orces
f &ou need practice in so%)in+ simu%taneous e,uations so%)e for t$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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orrect response to precedin+ frame
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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orrect response to precedin+ frame
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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orrect response to precedin+ frame
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.