Theory of Structures

Dr. Qais Abdul Mageed Theory of Structures (2008-2009)

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Text Book: Elementary Theory of Structures, 2nd Edition, by: YUAN-YU HSIEH References:

1. Elementary Structural Analysis, by: NORRIS, WILBAR UTKU. 2. Statically Indeterminate Structures, by: CHU-KIA WONG. 3. Indeterminate Structural Analysis, by: KINNEY

First Semester:

4. Stability and Determinacy of Structures: 4.2. Stability and Determinacy of Beams. 4.3. Stability and Determinacy of Trusses. 4.4. Stability and Determinacy of Frames. 4.5. Stability and Determinacy of Composite Structures.

5. Axial Force, shear Force and Bending Moment Diagrams: 5.2. Axial Force, shear Force and Bending Moment Diagrams for Frames. 5.3. Axial Force, shear Force and Bending Moment Diagrams for Arched

Frames. 5.4. Axial Force, shear Force and Bending Moment Diagrams for Composite

Structures.

6. Statically Determinate Trusses: 6.2. Types of Trusses. 6.3. Stability and Determinacy of Complex Trusses. 6.4. Examples on Solving and Analyzing Trusses.

7. Influence Lines for Statically Determinate Structures: 7.2. Influence Lines for Statically Determinate Beams. 7.3. Maximum Effect of a Function due to external loading: 4.2.1. Due to Concentrated loading. 4.2.2. Due to Distributed loading.

Distributed Dead Load. Distributed Live Load (occupying any length of the structure). Distributed Live Load (of a specific length).

4.3. Influence Lines for Girders with Floor Systems. 4.4. Influence Lines for Statically Determinate Frames. 4.5. Influence Lines for Girders in Trusses. 4.6. Influence Lines for Statically Determinate Composite Structures. 4.7. Maximum Effect of a Function due to Multiple External Moving Loads.

5. Absolute Maximum Moment for Simply Supported Beams.


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6. Elastic Deformation of Structures (Deflection & Rotation). 6.1. Conjugate Beam Method. 6.2. Deflection of Beams and Frames. 6.2.1. Unit-Load Method (Virtual Work Method). 6.3. Deflection and Rotation of Trusses. 6.4. Deflection and Rotation of Composite Structures.

Second Semester:

1. Approximate Analysis of Statically Indeterminate Structures: 1.1. Approximate Analysis of Statically Indeterminate Trusses.

Trusses with Double Diagonal System. Trusses with Multiple Systems.

1.2. Approximate Analysis of Statically Indeterminate Portals. 1.3. Approximate Analysis of Statically Indeterminate Frames.

Frames Subjected to Vertical Loads Only. Frames Subjected to Lateral Loads Only.

2. Symmetry and Anti-Symmetry of Structures. 3. Analysis of Statically Indeterminate Structures by the Method of

Consistent Deformations.

4. Fixed End Moments of some Important Beams with Constant EI.

5. Analysis of Statically Indeterminate Beams and Rigid Frames by the Slope-Deflection Method.

5.1. Analysis of Statically Indeterminate Beams by the Slope-Deflection Method.

5.2. Analysis of Statically Indeterminate Rigid Frames without joint translation by the Slope-Deflection Method.

5.3. Analysis of Statically Indeterminate Rigid Frames with One Degree of Freedom of joint translation by the Slope-Deflection Method.

6. Analysis of Statically Indeterminate Beams and Rigid Frames by the

Moment Distributed Method. 6.1. Fixed-End Moments. 6.2. Stiffness, Distribution Factor and distribution of External Moment Applied

to a Joint. 6.3. Distributed Moment and Carry-Over Moment 6.4. Analysis of Statically Indeterminate Rigid Frames with One Degree of

Freedom of joint translation by the Moment Distributed Method.


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R R R90o

R2R1

F.B.D Link 1

Link 2

Ry

Rx

R

R

M Rx

M

Ry

Review:

1) Roller: One unknown element.

(2 Degree of Freedom)

2) Link or strut: One unknown element.

(Two Degree of Freedom) 3) Hinge: Two unknown elements.

(One Degree of Freedom)

4) Fixed: Three unknown elements.

(Zero Degree of Freedom)


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1. Stability and Determinacy of Structures: 1.1. Stability and Determinacy of Beams.

(r) = no. of reactions (c) = The total no. of equations of conditions. (Where: c=1 for an internal hinge, c=2 for an internal roller and c=0 for beams without internal connection) (c + 3) = The total no. of the equilibrium equations. The beam is set to be:

3cr +=+=+>

==

Stable & Indeterminate to the 2nd degree

if ( )

3cr +=Determinate if Stable if ( )

3cr +>Indeterminate if Stable if ( )

The degree of indeterminacy (m) can be obtained by: ( )3crm +=


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1.2. Stability and Determinacy of Trusses.

(b) = no. of bar elements of truss (r) = no. of reactions (j) = no. of joints. The truss is set to be:

j2rb +Indeterminate if Stable if ( )

The degree of indeterminacy (m) can be obtained by: ( ) ( )j2rbm +=


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1.3. Stability and Determinacy of Frames.

(b) = no. of frame members (r) = no. of reactions (j) = no. of joints. (c) = The total no. of equations of conditions. (Where: c=1 for an internal hinge, c=2 for an internal roller and c=0 for beams without internal connection) (c = no. of members connected at joint 1) The frame is set to be:

cj3rb3 ++Indeterminate if Stable if ( )

The degree of indeterminacy (m) can be obtained by: ( ) ( )cj3rb3m ++=

Frame b r j c 3b+r 3j+c Classification

10 9 9 0 39 27

Indeterminate to the 12th

degree

10 9 9 6 39 33

Indeterminate to the 6th degree

Unstable


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1.4. Stability and Determinacy of Composite Structures.

(E) = no. of equilibrium equations (U) = no. of unknowns The structure is set to be:

EU

The degree of indeterminacy (m) can be obtained by: EUm =

Composite Structure U E Classification

10 10 Determinate

11 9 Indeterminate to the 2nd degree

2. Axial Force, shear Force and Bending Moment Diagrams: Sign convention:

N: Axial Force (tension +ve, compression ve) V: Shear Force (turning structure clockwise +ve, counter clockwise ve) M: Bending Moment (compression outside of structure and tension inside

+ve, otherwise ve)

2.1. Axial Force, shear Force and Bending Moment Diagrams for Frames.


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2.2. Axial Force, shear Force and Bending Moment Diagrams for Arched Frames.


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2.3. Axial Force, shear Force and Bending Moment Diagrams for Composite

Structures.


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2

3

4

5

Link

Link

Link

Hinge

Link

3. Statically Determinate Trusses: 3.1. Types of Trusses.

A truss may be defined as a plane structure composed of a number of

members joined together at their ends by smooth pins so as to form a rigid

framework. Each member in a truss is a two-force member and is subjected

only to direct axial forces (tension or compression).

A rigid plane truss can always be formed by beginning with three bars

pinned together at their ends in the form of a triangle.

Common trusses may be classified according to their formation as simple,

compound and complex.

Simple Truss: ( ) A simple truss is formed by a basic triangle; each new joint is connected to

the basic triangle by two new bars.

Compound Truss: ( ) A compound truss is formed from two or more simple trusses connected

together as one rigid framework either by three links neither parallel nor

concurrent, or by a link and a hinge.


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Complex Truss: ( ) The truss which is neither simple nor compound is called a complex truss.

h1

h2

g

3.2. Stability and Determinacy of Complex Trusses.

h1

h2

g

For the shown complex truss there are two cases:

1. If h1=h2=h, then the truss is unstable. 2. If h1h2, then the truss is stable.

3.3. Examples on Solving and Analyzing Trusses.


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4. Influence Lines for Statically Determinate Structures:


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4.1. Influence Lines for Statically Determinate Beams.


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4.2. Maximum Effect of a Function due to external loading:


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4.3. Influence Lines for Girders with Floor Systems.


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4.4. Influence Lines for Statically Determinate Frames.


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4.5. Influence Lines for Girders in Trusses.


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4.6. Influence Lines for Statically Determinate Composite Structures.


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4.7. Maximum Effect of a Function due to Multiple External Moving Loads.


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5. Absolute Maximum Moment for Simply Supported Beams.


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)9002-8002( serutcurtS fo yroehT deegaM ludbA siaQ .rD

64 egaP

(dohteM maeB etagujnoC)

:

(

.

. (

(

.

( )

8IE) ( ) 4 lw

( P

dohteM etagujnoC (.)

xxd

x

y

w

B A

2/lw=AR

2/2lw=AM

2Mlw

2

= A

.gaiD ecroF raehS 2/lw

.gaiD tnemoM gnidneB

8IE4 lw

= B

( )

) erutavruC : (

IEM

xddy2

2

= :


74 egaP

nat = xdyd

:

IEM

xddy

xddxdyd

2

2

==

=

:

)1( ------ xd MIE = xd) ( )

yd xd ydxdMIE == : (

IExd xd == ydxdM

:

)2( ------ xd yxdMIExd == ( ) ( xd)

:

xd VdxdwVdwxdVw === MdxdVMdVxdMVxdwxdxd ====

:

xd Vw = )( ------ MVxdwxd == )4( ------ xd


84 egaP

( maeb etagujnoC )

) ( )

htgnel tinu rep )w(

)a(

)maeB etagujnoC( )b(

2IE2 lw

6IElw

3l

2IEeRlusnattlw

23

=

=

BMB A

3/4l ) (

B l

B A

-( )

IEM

(

( b-)

( ) ( ) ( ) ( ) ( MV w)

IE)M

: (

)5( ------ VMIExd = )6( ------ MMIExdxd =

( ) ( ) ( ) ( )

:

( ) . =V ( maeB lautcA)

. ( maeB etagujnoC)


94 egaP

lautcA ) () . = yM ( maeB

. ( maeB etagujnoC)

lautcA ) ( maeB etagujnoC) .

. ( maeB

( maeB etagujnoC)

( ) (. maeB lautcA)

:

dnE eerF dnE dexiF

dnE elpmiS dnE elpmiS troppuS roiretnI noitcennoC roiretnI

( )

maeb etagujnoC maeB lautcA )daoL citsalE ot detcejbus( )daoL deilppa ot detcejbuS(

00

==

M0

V0

==

dnE dexiF dnE eerF

00

M0

V0

dnE eerF dnE dexiF

00

=

M0

V0

= )rellor ro egnih( )rellor ro egnih( dnE elpmiS dnE elpmiS

00

=

M0

V0

= )rellor ro egnih( )rellor ro egnih( troppuS roiretnI noitcennoC roiretnI

00

M0

V0

)rellor ro egnih( )rellor ro egnih( noitcennoC roiretnI troppuS roiretnI

): (noitnevnoC ngiS

:

( x)

.


05 egaP

(:dohteM maeB etagujnoC)

. ( DMB) (

. (

( ) ( DMB) (

.

(

.

(

.

:

maeb etagujnoC maeB lautcA

a b ab

c a b c a b

l l

l l

l l

l l

)daoL citsalE ot detcejbus( )daoL deilppa ot detcejbuS(

)a(

)b(

)c(

)d(

)e(

)f(

) (

) -b (.


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) ( )Actual BeamB (

)Conjugate Beam ( )B ( :

B

EI8wl

EI8wl

EI24wl3l

43

EI6wlM

4

B

443

B

=

===

)Conjugate Beam Method :(-

1) Using the (Conjugate Beam Method), find ( ) for the loaded beam shown

below:

B

(b) S.F.D

(c) B.M.D

(a) Actual Beam x

y

RA=P

MA=Pl P

l

A B

EI3Pl 3

B =

-Pl

P

)(

x

y

RA=P

MA=Pl

l

A B(d) Conjugate Beam

Pl/EI

l/3 2/3l

:-


25 egaP

. ( DMB) (

(

(.) ( IE)

:. (

) (

2IElP

IEllP

2eRlusnatt1

2

=

=2l/3] * [

l/3] [

.

( B) (

.

l ==32

2IEMlP

2

BB

3IE3 lP

)nwoD( = B

:

.

etagujnoC eht gnisu BA noitrop ni noitcelfed mumixam etulosba eht dniF )2

.dohteM maeB

)tnatsnoc IE(

-:

. ( DMB) (

(

. ( IE)

BA :. (


35 egaP

) (

IELP

IE2LLP

2eRlusnatt1

2

=

=

.B ] [ A L/32L/3] [ * ( B ( A ) .

M0 ) B =) ( ) (

3IELP

3IE2LP

2LR1

2LR2L031

IELP

32

A

A

2

==

=

.A x )

L3

x2

03IE

xLP2IE

xP2V1

2

=

==

B AC

L L2

P

P2P 3

21

LP

D.M.B

3IE2LP

IE 2LP

IELP

etagujnoC maeB

C B A

IELP


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))

EI39PL4

EI39PL6

EI39PL2

EI33PL2

EI39PL2M

L3

2EI3

PL3

L3

2

EI4

L3

2Px

EI3PL

3x

EI4PxM

33333

max

2

2

22

max

===

==

)EI39

PL4M3

max =

6. Elastic Deformation of Structures (Deflection & Rotation).


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Theory of Structures

Documents

indeterminate structures

indeterminate frames

indeterminate trusses

indeterminate beams

determinacy of structures

indeterminate rigid

determinacy of frames

deflection of beams