June 2011 Chapter 2 Analysis of structures - eng-hs.net 2011(2).pdf · 2-5 Classify each of the...

June 2011

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Chapter 2

Analysis of

structures

،زوشاخ ششح ذاس٠ ذح

ااد أدا صاتمح عذ٠ذ ارذااخ

الع١ ازوس٠ عرادح جاا

.كال ذشض تأل ار١ز اط

June 2011






Analysis of structures

Considering:

n = number of parts in the structure.

r = number of reactions.

Stability:

Structure will be unstable in the following cases:

1) if r < 3 n

2) if r ≥ 3 n but all reactions are parallel.

3) if r ≥ 3 n but all reactions are concurrent.

Determinacy (provided of stability):

r = 3 n statically determinate.

r > 3 n statically indeterminate.

degree of indeterminacy = r - 3 n ممم ذعممم أ٠ممم ذمممزةإرا،

.فى اطشق ذف تاغشض

June 2011






examples of unstability

partial constraints

concurrent reactions

parallel reactions

ا أ ٠ى ذ٠ه

ا ذضر١مظ أج

June 2011






examples of determinacy

ــذ أــــح تـــال ر١ج

No gains without

pains

June 2011






Solution:

Compound beams, which are composed of pin-connected members are

must be disassembled.

Applying r =3n or r > 3n, the resulting classifications are indicated.

أبقىظهرا والقطعأرضا الالمنبتإن

2-3 Classify each of the beams in Fig. 2-18a through 2-18d as statically

determinate or statically indeterminate. If statically indeterminate,

report the number of degrees of indeterminacy. The beams are

subjected to external loadings that are assumed to be known and can act

anywhere on the beams.

June 2011






Solution:

Applying r = 3n or r > 3n, the resulting classifications are indicated.

الطيورعمياشكالهاتقع

2-4 Classify each of the pin-connected structures shown in Fig. 2-19a

through 2-19d as statically determinate or statically indeterminate. If

statically indeterminate, report the number of degrees of indeterminacy.

The structures are subjected to arbitrary external loadings that are

assumed to be known and can act anywhere on the structures.

June 2011






Solution:

Frame structures consist of members that are connected together by

rigid joints. Sometimes the members form closed loop, In order to

classify these structures, it’s necessary to use the method of sections

and “cut” the loop apart.

2-5 Classify each of the following frames as statically determinate or

statically indeterminate. If statically indeterminate, report the number

of degrees of indeterminacy. The frames are subjected to external

loadings that are assumed to be known and can act anywhere on the

frames.

عودلسانكعميلينالكالموبذل(.االمامعمي)السالميكثرمحبيك

June 2011






Solution

The structures are classified as indicated.

The member is stable since the reactions are non-concurrent and non-

parallel. It’s also statically determinate.

The compound beam is stable. It’s also indeterminate to the second

degree.

The member is unstable since the three reactions are concurrent at B.

(b)

بالكتمانحوائجكمقضاءعمىاستعينوا

2-6 Classify each of the following structures as stable or unstable. The

structures are subjected to arbitrary external loads that are assumed to

be known.

June 2011






The beam is unstable since the three reactions are all parallel.

The structure is unstable since r = 7, n = 3, so that r < 3n, 7 < 9.

Also, this can be seen by inspection, since AB can move horizontally

without restraint.

6-2ذاتع د اثاي سل

(d)

تمرداالمئيمأكرمتأنتنوا *** ممكتوالكريمأكرمتأنتإذا

June 2011






a) (partial constraints)

b)

(Statically determinate)

c)

Statically indeterminate to the first degree

d)

Statically indeterminate to the first degree

:Solution

2-9 Classify each of the structures as statically determinate, statically

indeterminate, stable, or unstable. If indeterminate, specify the degree of

indeterminate.

االمانيرءوسمالالمفمسين

June 2011






Solution:

( a)

Statically determinate

(b)

Indeterminate to 3˚

(c)

Statically determinate

(d) parallel reactions on lower beam unstable.

. صاس ع اذسب ص

2-10 classify each of the structures as statically determinate, statically

determinate, or unstable. If indeterminate, specify the degree of

indeterminacy. (Fig. a, b ,c ,d)

June 2011






Solution:

n = number of parts

r = number of reactions

a)

n = 1

r = 4

r > 3n

statically indeterminate to 1o.

b)

unstable since all reactions are parallel.

c)

n = 1

r = 6

r > 3n


d)

unstable since all reactions are

parallel.

2-11. Classify each of the structures as statically determinate, statically

indeterminate, or unstable. If indeterminate, specify the degree of

indeterminacy.

ك ث ع تاهلل أغا ذو .اف و ١

June 2011






Solution:

(a)

Statically indeterminate to 3˚

(b)


(c)


(d) Parallel reactions unstable

.اشى صالح اضعفاء



indeterminacy. The supports or connections are to be assumed as stated.

(Fig. a, b ,c ,d)

June 2011






a) The frame has no closed loops

r = 15

Statically indeterminate to 6o

b) Cutting the loops as shown

n = 4

r = 4 * 2 (pins) + 6 * 3 (reactions)

= 8 + 18 = 26


c) Cutting the loops as shown

n = 4

r = 4 * 3 (supports) + 9 *3 (sections)

r > 3 (n)


:Solution

2-13 classify each of the frames as statically determinate, statically


indeterminacy. All internal joints are fixed connected.

ااض ال ٠ضا اضرمث

June 2011






a)

n = 6

r = 26

r > 3 n

statically indeterminate to (r – 3 n) = 8

b)

n = 1

r = 2 + 1 = 3

r = 3 n

statically determinate

c)

n = 7

r = 27


6 6 2 3

3 3 3

1

T

3

3

3

3

6

3

3

3

2

دضة اطش٠مح تاضؤاي اضاتك أ٠ى اذ تز اطش٠مح

:Solution

2-14 classify each of the frames as statically determinate, statically


indeterminacy. All internal joints are fixed connected.

. ى تضثة زا٠ان ، ااس ال ٠ىشه ع١ب ف١ه

June 2011






d)

n = 9

r = 33


e)

n = 5

r = 21


f)

n = 9

r = 35


6 3 3

3

6 6

3 3

2 6 6 6

6 3

3 3

اجاح از ذشغث، ذخ١ أه ذضمرذم، ذخ١

ذخ١ أه لادس ع ذذم١م، ارا تم؟

14-2ذاتع د اثاي سل

June 2011






Solution:

a)

n = 2

r = 3+1+2 = 6

r = 3 n

statically determinate

b)

Improper constraints.

n = 1

r = 3

r = 3 n

but all reactions are parallel (unstable)

c)

n = 1

r = 2 + 1 + 1 + 2 = 6

r > 3 n




indeterminacy. The supports or connections are to be assumed as stated.

ذخص االعرمماد افاصمذ

.تأ إىا١اذه ذذدج

June 2011






Fig (2-17)

Ay

Ax

45 kN

Solution:

اج أ ذع فش األعاي تمفش

(١رأش) .اطش٠مح ذرلع رائج خرفح

2-17 Determine the reactions on the beam, the support at B can be

assumed to be roller, Neglect the thickness of the beam. (Fig. 2-17).

By

7.5(3) = 22.5 kN

4.5 m 1.5 m

June 2011






By Ay

7.5 kN/m

Fig (2-18)

Solution:

٠عمم١ا اسضمما ا١مم فمم تمم لممشاساخ صمماتمح

اذخزا فأ٠ ذذة أ ذى تعذ عشش صاخ؟

2-18 Determine the reactions at the supports A and B. Assume A is a

roller, and B is a pin. (Fig. 2-18).

36 kN 36 kN

2.4 m 2.1 m 2.1 m 2.1 m

June 2011






Ay

F1

By

F2 F3

Solution:

-

– – –

– – –

2-19. Determine the reactions on the beam.

ح... اعثذ ٠مشع تاعصا ال اذش ذىف١ ا

0.9 0.45 1.35 0.9 0.9

June 2011






Ax

Ay FB

(30)(7.2) + (0.5)(15)(7.2) =272.25 kN

3.6 m 3.6 m 810 kN.m

Solution:

ص١ح ام اخشائظ غ١ش اح

.إرا ذى ذعشف جره

2-20 Detremine the reactions on the beam. (Fig. 2-20)

Fig (2-20)

June 2011






Solution:

y

2-21 Detremine the reactions on the beam. (Fig. 2-21)

Fig (2-21)

X

X F2

F1

1.3 3.9

2.6

A y

FB A x

7.8m

ا١ممم درممم ال ذمممذ اتمممذأ

.غذا ع فاخ األا

June 2011






By Ay

45 kN 54 kN

1.8 0.6 2.1 0.3

Solution:

ا جذا ف صفشا

.سوب امطاس اصذ١خ

2-22 Determine the reactions at the supports A and B. the floor deck CD,

DE, EF and FG transmit their loads to the girder on smooth support,

assume A is a roller and B is a pin. (Fig. 2-22)

Fig (2-22)

June 2011






By

18 kN Cy

Cx

Cx

24 kN

Ay

Bx

Mo

Cy

Solution:

Section AC

Section CB

ذد أ ذع١ا ع اا اذ١اج وا ٠ع١ا أغة

.ااس أ أ ذ٠ه إصااخ ألره عا أجع

2-23 Determine the reactions at the supports A and B of the compound

beam. There is a pin at C. (Fig. 2-23)

Fig (2-23)

June 2011






Ay

Ax

FC

2.16 kN

0.9 m 0.9 m

3 m 30o

1.5 m

2.6 m

Solution:

فضم١ر تمه ،ذرجم إرا ال ذعشف إ أ٠

.ال ذش٠ذاطاف ع األسجخ ف ىا

2-26 Determine the reactions at the smooth supports C and pinned support

A. Assume the connection at B is fixed connected. (Fig. 2-26).

Fig (2-26)

June 2011






Bx

By

FA

6 kN

1.5 m 1.5 m

1.5 m

Solution:

If you fail to plan,

you plan to fail.

2-27 Determine the reactions at the smooth support A and pin support B.

The connection at C is fixed. (Fig. 2-27).

Fig (2-27)

June 2011






Bx

By

Ay

25 kN 35 kN 50 kN 10 kN

2.5 kN

4 m 3 m 3 m

4 m

3 m

Solution:

طعاه واله ٠ىه ذف١ش صاعح ١٠ا لد

.صرصع تا اعجزاخ. . . أش١اء أخش

2-31 Detremine the reactions at supports A and B of frame. Assume that

support at A is a roller. (Fig. 2-31).

Fig (2-31)

June 2011






Bx By

Ay

20 kN 30 kN 40 kN

84 kN

8 m 6 m

Solution:

Bx – (7 * 12)

Bx 84 kN

(7 * 12 * 3.5) + (30 * 8) + (20 * 14) – (Ay * 8) 0

Ay 101.75 kN

Ay + By – 40 – 30 – 20 0

By 11.75 kN

2-33. Determine the horizontal and vertical components of reaction at the

supports A and B. the joints at C and D are fixed connections.

٠مممم ممممشء مممم ا مممم ذ إرا ا مممم ش عشض فى سداء ٠شذذ٠ ج١

June 2011






Ey

Ex

Ay By 25 kN

180 kN

C

D

Solution:

2-34. Determine the reactions at the supports A, B, and E. Assume the

bearing support at B is a roller.

.خاطش اصرغ تشأ٠

(ع ت أت طاة)

June 2011






Ay By Cy Dy

180 kN 120 kN 40 kN

Solution:

– –

– –

:تذ اعادر١ ٠رج أ

– – –

– – –

2-35 Determine the reactions at the supports A, B, C, and E.

... دلاخ لة اشء لائح

اذ١اج دلائك ثا إ

June 2011






Cy Ay

Ax

Dx

67.5 k (30)(3)=150 kN

1.8 1.5 1.5 1.5

Solution:

2-38 Determine the reactions at the supports A, C, and D. B is pinned.

مما ذخمماف مم لممذ ٠ذممذز ممه إرا

.ا داد ع ارفى١ش ف١

June 2011






Cy

25 kN

Ay

80 k

2 m 2 m 2 .5 m

2 .5 m

Solution:

ح،ـماطعاخ ااذف١( ال) ٠عطه، ( ال: )ذع لي

.اــــــ ــــداع الـــرـطاسئ ا( ال)

2-39 Detremine the reactions at supports A and C. Assume that support

at A is a roller, B is a fixed- connected joint, and C is a pin. (Fig. 2-39)

Fig (2-39)

Cx

June 2011






Ay

Ax

50 kN By

350 kN

2 .5 m 2 .5 m 2 m

2 m

75 kN

Solution:

صأي اضرذ١ أ٠ ذم١؟ : اى

.ف أدال اعاجز: فأجات

2-40 Detremine the reactions at supports A and B. Assume that support

at B is a roller, C is afixed- connected joint. (Fig. 2-40)

Fig (2-20)

June 2011






By

Bx

Ay

Ax

MA

45 kN 27 kN

67.5 kN

Solution

أم ـكتمدهلمأخرب

2-41. Determine the reactions at the supports A and B.

June 2011






FB

Ay Ax

30 kN

4 m

3 m

Solution:

ذخ١ اجاح از ذشغث، ذخ١ أه ذضرذم،

ذخ١ أه لادس ع ذذم١م، ارا تم؟

2-44 Detremine the reactions at supports A and B. The joints C and D

are fixed connected. (Fig. 2-44).

Fig (2-44)

June 2011






Solution:

2-45 Detremine the horizontal and vertical coomponents of reaction at

supports A and B.

X X

100 kN

2.5 m 2.5 m

3 m

3 m

X X

X X

4 m

B y

B x

A y

A x

48

X

X

.أد أوثش أفض ا ذعرمذ ع فضه

June 2011






Cy

Cx

30 kN 8 kN 6 kN

8 kN

Ay

Ax

Solution:

–

– – –

–

– –

2-47 Determine the reactions at the supports A and C. The frame is pin

connected at A, B, and C and the two joints are fixed connected.

... آفمممح ممم جضممم شمممءىممم

صطا ع١ اثشد در اذذ٠ذ

June 2011






1.8 m 1.2 m 0.9 m

1.8 m

1.8 m

21.6 kN

18 kN 13.5 kN

Ay

Ax

Bx

By

Bx

By

Cx

Cy

Solution: Member AB:

Member BC:

رى أد أ تذ ف لائح ارااذه

2-48 Detremine the horizontal and vertical components of force at the

connections A, B and C. Assume each of these connections is a pin. (Fig. 2-48).

Fig (2-48)

June 2011






Ax

Ay Cy

Cx

By

Solution:

E نقطة عند BEبفصل

y

– – اخرالف اشأ ال

د ل ٠ ض١ حفضذ

2-50. Determine the horizontal and vertical components of reaction at the

supports A, B, and C. Assume that the frame is pin connected at A, B, D,

E, and F, and there is a fixed connected joint at C.

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June 2011 Chapter 2 Analysis of structures - eng-hs.net 2011(2).pdf · 2-5 Classify each of the...

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