chapter 3 trusses - موقع المهندس حماده شعبان 3 final2 (2).pdfb + r > 2 j...

July 2011

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Chapter 3

Trusses

،زوشاخ ؽشغ ذاس٠ ؽح

ااد أدا عاتمح ؼذ٠ذ ارؽااخ

ازوس٠الؼ١ راؼح عاا ػ

ببب األببس رافببب اغضببة

.الد ض١ؼاخ أعأ

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Trusses

ذرم ف ػذد ـــ اــــماط (members)ػثاسج ػ عػح اؼـاصـــش اإلــؾائ١ح *

(joints) ٠ى ذطث١ك ل(forces) فمظ ػذ اـ(joints) ف و فررذ لــــ داخــــ١ح

(member) إا(tension) أ(compression).

إا (compression)أ (tension)ا ــػـمج ــــح اـل١ عادــإ٠ طبـــــــع١ى ا *

(members)أ ف تؼـــض (joint method)فغرخذ طش٠مــح (member)ف و

(section method)فغرخذ طش٠مح

٠ى اعرخذا إؼذ اطش٠مر١ تذال ػ األخش أ اذط ت١ا إل٠ــعاد ام اطتح *

.اؼثشج ف ره أ٠ا أعشع أع ف إ٠عاد اطب

أ (joint)فمظ ٠ؼثشػا تغ خاسض اـ (tension)إا (member) امج ف و *

(compression) اـ فمظ ٠ؼثش ػا تغ داخ ػ (joint).

أتذأ ا١ ؼر ال ذذ

.ػ فاخ األا غذا


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Analysis of Trusses

considering:

b = number of members.

r = number of reactions.

j = number of joints.

:tabilityS

:tructure will be unstable in the following casesS

1) b + r < 2 j

2) b + r ≥ 2 j (but all reactions are parallel)

3) b + r ≥ 2 j (but all reactions are concurrent)

: of stability) providedeterminacy (D

b + r = 2 j statically determinate

b + r > 2 j statically indeterminate

degree of indeterminacy = b + r – 2 j

ذظذ طش٠مح أفض ؼ ره،

(ذاط إد٠غ) اتؽصا ػا


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اػرثاسبا غ١بش ظـبـدج members (zero)٠فض ذؽذ٠بذ ابـ trussأ لث تذء ؼ *

(.خصصا إرا وا اطب إ٠عاد و ام)تاؾىــ

١غبا ػب اعبرماح اؼبذج (only 2 members) ١ظ ذ٠ا لبج بذ٠ا (joint)إرا *

(.A, D)ومطر ((٠zero memberى والا

اشا ا ػ اعرماح اؼذج ١ظ ذ٠ا (only 3 members) ذ٠ا (joint)إرا *

(. C, D)ومطر ((zero memberاصاس ((memberلج ٠ى اـ

zero members

٠ىبببب ببببؼه أبببب١ظ بببب اىبببب أ

تفظ اإلىااخ اراؼح اؽا أفض


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.((membersإرا وـا اـــطب إ٠ـــعاد امــــ ف وـــ اـــ غرخذ ز اطش٠مح ـ

.((support reactionsـ ٠فض إ٠عاد 1

(.٠ى أؼ١اا االعرغاء ػ ؼغاتا ى زا ١ظ ؽشطا إر)

تـبـاعرخذا ـؼـبـادر (only 2 members) بذ٠ا (joint)ـ ثذأ تئ٠عباد اعا١ب بـ 2

االذضا

.ار ٠ى إ٠ـعاد أؼـــذ اـعــ١ ـا ـثاؽــــــــشج االذضاـ ٠فض اثذء تــؼادح 3

ذ٠ــــبـا أوصبش ب (joints)، ىب ٠ىب بز ابـشج بـ 3 ,2ـب بـم ترىبشاس اخــطذ١بـ 4

2 - members)) تـؾشط أال ٠ــض٠ذ ػــــذد اعا١ ػذــــــــا ػ عــــــ١.

.((membersؼر ٠ر إ٠ـــعاد امــــ ف ظ١غ ( 4)ـ غرش ف ذطث١ك اخطج 5

أ (tension)ــــببـ بب ((memberببع امــببـج فبب وـــببـ ـبب بب ابب ظببذا ذؽذ٠ببذ 6

(compression) .

.ـ ال ذغ وراتح اؼذاخ7

joint method

د .إرا ظد أه ضد فمذ ض

Σ

Σ


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فمبظ ((membersغرخذ ــــبـز اطبـش٠مح إرا وبـا اطبب إ٠عباد امبـ فـــبـ تـبـؼض

.ــ١ظ ػــذ ظـاــثـ( truss)ـــ ف اـــرصــشب ـــل دــاــــو ا إراـــــصــصـخ

.((support reactionsـ ٠فض إ٠عاد 1

( أداوا ف اؾى اء ػ ؼغاتا٠ى أؼ١اا االعرغ)

ـ ؽاي إ٠عاد مطغ ٠ـش تاـمـ اـطب ؼغاتــا ٠ـفض أال ٠ـمطغ أوــــصش شـــــالز2

members) .)

ـطثك ــؼادالخ اإلذـــضا ( األعخراس )ـ م تفص أؼذ ظاث امطغ ٠غاس أ ٠ـ١ 3

أوصش شج ػذ ارماء و عب١ إل٠عباد اـبـعي ـ ٠فض ذطث١ك ؼادح4

.اصاس ؼادح اؼذج

اؼببـد االذعببافب ٠ىب اعرخــببـذا ((members - 2ـب فبب ؼباي ذبباص5

.ػ١ا إل٠عاد اعي اصاس

((membersـ ٠ى ذىشاس امطغ دساعح أوصش مطغ ف و بشج إل٠عباد امب فب 6

. ٠ؾا امطغ

إل٠عبباد ( section method)بغ ( joint method)ـب ٠ىب دببط اعبرخذا طش٠مببح 7

. ٠ؾا امطغ ((membersل ف

ethodmection s

افأد أؼذ اخر١اساذ ،ال ذطؼ ف رق صظره

Σ , Σ , Σ


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طش٠ك اعاغ ٠ش فق تؼض اؼمثاخ

a)

statically indeterminate to the 4th degree

b)

statically determinate.

c)

statically indeterminate to the first degree

d)

the left panel is a collapsible component

(6)

parallel are reactions all satisfied

d

be not can F

< r b

j

r

b

r b

j

r

b

r b

j

r

b

r b

j

r

b

x 0

11

6

2

9

9 2 19

9

4

15

8 2 16

8

3

13

10 2 24

10

4

20

unstable

3-2 classify each of the following trusses as statically determinate, statically indeterminate, or unstable.


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a)

stable, statically determinate.

b)

stable, statically indeterminate to 1°.

c)


d)

(r + b) – 2 j = 1


6

10

3

12)(

6

9

4

12)(

6

9

4

2)(

8

13

3

j

b

r

jbr

j

b

r

jbr

j

b

r

jbr

j

b

r

3-3 classify each of the following trusses as statically determinate, statically

indeterminate, or unstable. If indeterminate, state its degree.

دا وغح خاب ا


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Gy

Gx FGF = 20 kN

FAG = 15 kN Ax

FAB = 10 kN

20 kN

18 kN

7.5kN

FEF

FBC = FCD

15 kN

FCD

FED = 8.33 kN

5 kN

FCD = 6.67 kN

5 kN

10 kN 6.67 kN

FBF

G

A C

F

B

D

FBE = 4.17 kN

FAf = 18 kN 15 kN FAF = 18 kN

;

عم تذساعح و مطح ػ ؼذج تؾبشط أ ٠ىب بذ٠ا عبال ػب membersإل٠عاد ام ف *

( C)أ ( T) ؼادح اؼذج عب١ر فبشا االذعبا ا اؼذ عال األوصش عثذأ تاؼادح ار ذؼط

. األغة االذعاؼغة

.تا٢ح وراتح ارائط ثاؽشج ػ اشع( , )ع١ر ػ ؼغاتاخ *

joint F:

3-8 Determine the force in each member of the roof truss. State if the members

are in tension or compression. Assume all members are pin connected.

تخ االلرصاداإلفشاط ف

56.3o

56.3o


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FAH

FAB

15

A

(From symmetry)

joint A:

joint B:

joint H:

3-9 Determine the force in each member of the roof truss. State if the members are in tension or compression. Assume all members are pin connected.

FBH

B FAB

10 kN

.ال ؽء ٠أذ تاعا

FBC


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from and:

joint G:

3.9ذاتغ إظاتح عؤاي

16.67 kN 16.67 kN

FGC

G

ألتائه و١ف ذى لذج غ١شن أ

بببز اماػببباخ أبببد ذؽببب وببب

!!اغبببببببببببباثح ػبببببببببببب فغببببببببببببه


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(From symmetry)

(FCG) is a zero-force member

( – –

–

خ١ببش ابباط بب طبباي

ػ ػش ؼغ

+

E)ػذ مطح FGF ترؽ١ (

3-10 The Howe truss is subjected to the loading shown. Determine the force in members GF, CD, and GC, and state if the members are in tension or

compression.

5 kN

2 kN

9.5 kN

D

F

FGF

FDG

FCD

3 m

2 m 2 m

G

تأخز مطغ ٠ش تاؼاصش اطتح دساعح

:اعضء األ٠ ذطث١ك ؼادالخ االذضا

وا أى ػذ M∑٠فض ذطث١ك ؼادح )

.(ارماء و ع١


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(From symmetry)

وه را عؼ١ذا تا أد ف١ ٠ؼذ

.ع١ح أخش ثاء اصمح تافظ

3-11 The Howe truss is subjected to the loading shown. Determine the force in

members GH, BC, and BG, of the truss and state if the members are in tension or

compression. (Fig. 3-11)

B

9.5 kN

H

G

FBC

FGH

FBG

5 kN

2 m 2 m

3 m 2 kN

36.87o

56.31o


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.أع أ ذصك تفغه تؼذ فاخ ا٢اا

Solution:

(From symmetry)

3-14 Determine the force in members IH, BC, and BH of the bridge truss. Solve

for each unknown using a single equation of equilibrium. State if the membrs are in tension or compression. Assume all members are pin connected.

5 kN 3 kN

O

H FIH

FBC

FBH

2 m

4 m

4 m

B

I

63.43o

26.56o

A

2 m

Ey

Ay


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ذ ف اذ١ا ٠ ذ اص ؽثه هللا اص

.ؽثه ااطف١ا ػذ ااط ٠

تأخز مطغ ٠ش تاؼاصش اصالشح اطتح

.دساعح اعضء األ٠غش وا ث١ تاؾى

–

(reactions ال داػ ؽغاب )

3-17 Determine the force in members GF, FC, and CD of the cantilever truss. State if the membrs are in tension or compression. Assume all members are pin

connected.


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Intire truss:

;

ا تاغد ف ذم١١

.لذساذه فغاثا ذخطأ

3-18 Determine the forces in members KJ, CD, and CJ of the truss. state if the

members are in tension or compression. (Fig. 3-18).

5 kN 15 kN 15 kN

3 m

FKJ

FCJ

FCD

Ay

A

C

33.69o

18.43o

J

3 m 3 m

3 m


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تأخببببببز امطببببببغ اثبببببب١ اعببببببرخذا ؼادببببببح

moment ػبببذ ارمببباء وببب(2 members )

رغ١ ؼغاب اصاس ؼادب اؼبذج ذم١ب

األخطاء اؽرح

joint D:

(Gػذ مطح FDEترؽ١ )

3-19 Determine the forces in members JI, JD, and DE of the truss. state if the

members are in tension or compression.

أ ٠ىه فىشخ و١ف

ذؼ١ؼ اؽ١اج ار ذراا؟

FDJ

97.5 kN 97.5 kN

D

J H G

FJI

5 10 20

49.17

43.18FDE

E

J


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from symmetry

joint A:

joint D:

3-20 Determine the forces in each member of the truss in terms of the load P and

state if the members are in tension or compression.

FDB

P

0.687 P 0.687 P

P/2

FAB

FAD

A

D

ؼف امبببببج غ اؾبببببشل١١ ببببب١ ضببببب٠ ببببب

!! ١شجص ؼف اث ؼ وصش ا ١ أ


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For maximum loads, choose the minimum, Pmax = 4.25 kN

From symmetry

Ay = Cy = 0.5 P

joint A:

joint B:

FAD cos θ – FAB cos 45 = 0

FAD cos 14 = FAB cos 45

FAD = 0.729 FAB

FAD sin θ + .5 P – FAB sin 45 = 0

(0.729 FAB * sin 14) + 0.5 P - FAB sin 45 = 0

FAB = 0.942 P ( C )

FAD = 0.687 P ( T )

- FBD + 2 FAB sin 45 = 0

FBD = 1.33 P ( T )

FAB = 0.942 P = 4 kN, Pmax = 4.25 kN

FBD = 1.33 P = 10 kN, Pmax = 7.52 kN

3-22 Members AB and BC can each support a maximum compressive force of 4 kN, and members AD, DC, and BD can support a maximum tensile force of 10

kN. If a = 1.8m determine the greatest load P the truss can support.

٠مببباط ػبببش ابببشء

ف بببب١ظ ببببتببببا خ

.تمذس ا ػاػ

Ay = 0.5 P

FAD

FAB

FBC

FBD

FAB

B

θ = 14°


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from symmetry

Due To Symmetry

از٠ ٠ؽص ػ اعائض

.٠ذفؼ شا مذا ا

3-24 Determine the forces in members GF, FB, and BC, of the fink truss and

state if the members are in tension or compression.

FBC

FGF

FFB

4 kN

30o 60

o

B

3 cos2 30

o

4.5 tan 30

o


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.تااظؽ١ ذعػ فاشء ػ د٠ خ١اصك

3-25 Determine the force in members GF, CF, and CD, of the roof truss and

indicate if the members are in tension or compression.

FGF

Ey

FCD

FCF

1 m 1 m

F

1.5 m

C

E


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This is a compound truss

Where ACB and FED are connected by

Three bars FA, EC, and DB.

Since

b + r = 2 j

9 + 3 = 2(6) = 12

Statically determinate

Truss is internally unstable.

افاؽ ٠ؼرمذ أ

.اعاغ عشد ػ١ح ؼع

3-26 Classify the truss and determine if it is stable.


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From symmetry:

٠ى ؼغاب صف اؾى فمظ

joint A:

joint B:

joint F:

6 kN

A FAB

FAF

8 kN

FBE

FBF

8 kN

B

3-30 Specify the type of compound truss and determine the force in each member.

State if the members are in tension or compression. Assume all members are pin connected.

اؽم١مببببح بببب ببببا ؽبببب

بببا ػ١ببب، اغبببؼح ببب

.٠ظ ا٢خش أا ػ١ب

2 kN

2 kN 10 kN

F FEF

FCF


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٠ش طش٠ك اعاغ

فق تؼض اؼمثاخ

3-31 Determine the forces in members IH , AH, and BC, of truss. State if the

members are in tension or compression.

5 m 5 m

FIH

FAH

FBC

10 kN 14 kN

5 m

B

J I

A

26.56o


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Supports Reaction:

Method of joints: By inspection,

members BN, NC, DO, OC, HJ, JE, and LG are zero Force members.

غاثا ال ٠أذ اعاغ

.حـ ػ سغثـت. . . ذفح ــص

3-39 Determine the force in members EF, EB, and LK, of the Baltimore bridge truss and state if the members are in tension or compression. Also, indicate all

zero-force members. (Fig.3-39).

FIK

FEP

FEF

Iy 2 kN 3 kN

K

P

F

4 m 2m

4 m


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مطببببغ ٠ببببش تببببام اببببصالز اطتببببح تأخببببز

دساعح اعضء األ٠غش

(

BCF

BGF

BA

kN9 kN6

HGF

G

X X X m3 m3

X

X

m3

m5.1

m3X X

X

X

m5.1

(

6.26

3

5.1tan 1

5.4

33.56

3

5.4tan 1

3-41 Determine the force in members BG, HG, and BC, of the truss and state if the members are in tension or compression. Also, indicate all zero-force

members.

أ ذغببأي ببشذ١ ببهخ١ببش

.خطأ شج اؼذج أ ذ


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