Theory of Structures
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Transcript of Theory of Structures
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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
= :
-
)9002-8002( serutcurtS fo yroehT deegaM ludbA siaQ .rD
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
-
)9002-8002( serutcurtS fo yroehT deegaM ludbA siaQ .rD
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)
-
)9002-8002( serutcurtS fo yroehT deegaM ludbA siaQ .rD
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)
.
-
)9002-8002( serutcurtS fo yroehT deegaM ludbA siaQ .rD
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
:-
-
)9002-8002( serutcurtS fo yroehT deegaM ludbA siaQ .rD
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 :. (
-
)9002-8002( serutcurtS fo yroehT deegaM ludbA siaQ .rD
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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