Download - Strength of materialprint
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Norton University Academic Year: 2013-2014
Department: Civil Engineering Page 1
Subject: Strength of Material
Group: M8
Session: Morning
(7 ) 7 5 35AM kgf F d kgf m kgfm
B A5m
M(A)
10N10N
(10 ) 10 2 20AM N F d N m kNm
B
A
8m
M(A)
4kN
4kN
(4 ) 4 8 32
(4 ) 32
A
A
M kN F d kN m kNm
M kN kNm
I. Relocating a force Move force from A to B
a.
b.
c.
A B5m
M(A)
7kgf7kgf
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Norton University Academic Year: 2013-2014
Department: Civil Engineering Page 2
Subject: Strength of Material
Group: M8
Session: Morning
0
0
245 20 10 2
2
245 20 10 2
2
(20 ) (10 2 ) (10 2 )A A A
Fx Fx Cos ft
Fx Fx Sin ft
M ft M ft M ft
(20 ) (10 2 10 ) (10 2 10 )
(20 ) 0
A
A
M ft ft m ft m
M ft
A
B
60°
30°
30°
10m
40kN
20ft
(40 ) 40 9 360
(40 ) 360
A
A
M kN F d kN m kNm
M kN kNm
0tan 45 10 1 1010
dxdy m m
m
d.
slope:
+
e.
45°
ABFx
Fy
dx
dy
20ft
10m
20ftF
45°
45°
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Norton University Academic Year: 2013-2014
Department: Civil Engineering Page 3
Subject: Strength of Material
Group: M8
Session: Morning
A
BFy
Fx
F
12m
4
35
4
3
3tan 0.75
4
arctan(0.75) 36.869
4 3
Fx Fx
B A10m
M(A
)
6kN
6kN
(6 ) 6 10 60AM kN F d kN m kNm
A B4ft
M(A
)
7kips7kips
(7 ) 7 4 28 .AM kips F d kips ft kips ft
f. slope:
g.
h.
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Norton University Academic Year: 2013-2014
Department: Civil Engineering Page 4
Subject: Strength of Material
Group: M8
Session: Morning
A
B
7ft
3lb
3lb
(3 ) 3 7 21 .AM lb F d lb ft lb ft
50MN
A
B
6m50MN
60°
60°
45°
0
0
0
0
0
50 60 25
50 60 43.30
6 45 4.24
6 45 4.24
25 4.24 106
43.30 4.24 183.59
106 183.59 289.59
289.59
A
A
A
A
M
M Mx My
Fx Cos MN
Fy Sin MN
dx Cos m
dy Sin m
Mx Fx dy MNm
My Fy dx MNm
M MNm
M MNm
A B
30° 30°
20kips 20kips
30 tan 30 55
35 1.666 3
3
20A
dtan d ft
ft
d ft ft
M kips F d
20 1.666 3 33.333 3kips ft
20 33.333 3 .AM kips kips ft
i.
j.
k.
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Norton University Academic Year: 2013-2014
Department: Civil Engineering Page 5
Subject: Strength of Material
Group: M8
Session: Morning
A
54°
3MN
60°
B
54°
4m
0
0
0
0
3 54 1.76
3 54 2.42
4 60 2
4 60 3.46
1.76 3.46 6.08
2.42 2 4.84
6.08 4.84 1.24
1.24
A
A
A
M Mx My
Fx Cos MN
Fy Sin MN
dx Cos m
dy Sin m
Mx Fx dy MNm
My Fy dx MNm
M MNm
M MNm
l.
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Norton University Academic Year: 2013-2014
Department: Civil Engineering Page 6
Subject: Strength of Material
Group: M8
Session: Morning
II. Moment of Inertia
1. រក XI , YI , XYI
3
34
12
200 2013333333.33
12
X
h dI
cm
X
Y b
a
x250cm
300cm
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Norton University Academic Year: 2013-2014
Department: Civil Engineering Page 7
Subject: Strength of Material
Group: M8
Session: Morning
3
34
12
2000 20133333.33
12
Y
Y
hdI
I cm
0XYI ពីព្រោះ (X’X និង Y’Y ជាអកស័ឆ្លោះ) ដូចពនោះ 413333333.33XI cm , 4133333.33YI cm and 0XYI
2. រងវិលអកស ័ 11 ចូរក ; ;a b abI I I រមូមនតទូពៅ
2 2 2
2 2 2
2 2
cos ( ) sin ( ) sin (2 )
cos ( ) sin ( ) sin (2 )
sin( )cos( ) sin( )cos( ) (cos (2 ) sin (2 ))
m x y xy
n y x xy
mn x y xy
I I I I
I I I I
I I I I
ពោយ អកស័ថមីជាអកស័កាត់តាម ទី្រជលុំទុំងន់ររស់ចតលពកាណកកង ពេបាន
2 2 2
2 2 2
2 2
cos ( ) sin ( ) sin (2 )
cos ( ) sin ( ) sin (2 )
sin( )cos( ) sin( )cos( ) (cos (2 ) sin (2 ))
a X Y XY
b Y X XY
ab X Y XY
I I I I
I I I I
I I I I
ពោយ 413333333.33XI cm , 4133333.33YI cm and 0XYI 2 2 2
4
2 2 2
4
2
13333333.33cos (11 ) 133333.33sin (11 ) 0sin (22 )
12852746.74
133333.33cos (11 ) 13333333.33sin (11 ) 0sin (22 )
613919.88
13333333.33cos(11 )sin(11 ) 133333.33cos(11 )sin(11 ) 0(cos (2
a
b
ab
I
cm
I
cm
I
2
4
2 ) sin (22 ))
2472403.517 cm
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Norton University Academic Year: 2013-2014
Department: Civil Engineering Page 8
Subject: Strength of Material
Group: M8
Session: Morning
ដូចពនោះ 4
4
4
12852746.74
613919.88
2472403.517
a
b
ab
I cm
I cm
I cm
3. រកមង្កិលអកស័ពី C(a,b) មក (0,x,y)។ ចូររក ; ;x y xyI I I
ររូមនតរកមង្កិល
2
2
2
2
x X X
y Y Y
xy XY Y X
I I aQ a A
I I bQ b A
I I aQ bQ abA
ពោយ
23
23
4
4
2
20 200400000
2
20 20040000
2
250
300
13333333.33
133333.33
0
200 20 4000
X
X
X
Y
XY
Q cm
Q cm
b cm
a cm
I cm
I cm
I
A cm
ពេបាន
2
4
13333333.33 2 300 400000 300 4000
613333333.33
xI
cm
2
4
133333.33 2 250 40000 250 4000
270133333.33
yI
cm
4
0 300 40000 250 400000 300 250 4000
412000000
xyI
cm
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Norton University Academic Year: 2013-2014
Department: Civil Engineering Page 9
Subject: Strength of Material
Group: M8
Session: Morning
ដូចពនោះ 4
4
4
613333333.33
270133333.33
412000000
x
y
xy
I cm
I cm
I cm
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Norton University Academic Year: 2013-2014
Department: Civil Engineering Page 10
Subject: Strength of Material
Group: M8
Session: Morning
III. Truss Analysis by Method of Joints
1.កុំលុំង្រតិកមមររស់ទុំរ Ax; Ay; By
តាម(1)
2.សិកាលុំនឹងតុំន
តុំណ(A)
តាមSlope
0 ; Fx
0xA
0 ; Fy
6 4 4 2 0
20 (1)
0;
12 8 3 8 6 4 9 0
8 3 8 6 4 99
12
y y
y y
A
y
y
A B KN KN KN KN
A b KN
M
B m KN m KN m KN m
KN m KN m KN mB KN
m
20 9 11yA KN KN KN
2 2 3 22 3
3 213
yx
yx
yx
ACACAC
ACACAC
ACACAC
FFF
AC AH CH
FFF
FFF
2KN4KN
6KN
4KN4KN
A
C
D
E
B
FGH
Ax=0
Ay=11KN By=9KN
2M
2M
3M 3M 3M3M
A
11KN
A
11KN
A
11KN
A
11KN
A
11KN
A
11KN
A
11KN
A
11KN
A
11KN
A
11KN
A
11KN
A
11KN
A
11KN
A
11KN
A
11KN
A
11KN
A
11KN
A
11KN
A
11KN
A
11KN
A
11KN
A
11KN
?AHF
?yACF
?xACF
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Norton University Academic Year: 2013-2014
Department: Civil Engineering Page 11
Subject: Strength of Material
Group: M8
Session: Morning
តុំណ(H)
តុំណ(C)
0 ; Fy
11 0
11
13
2
13 = 11 19.83
2
311 16.5
2
y
y
y
x
AC
AC
AC AC
AC
F KN
F KN
F F
KN KN
F KN KN
0 ; Fx
0
16.5
xAH AC
AH
F F
F KN
0 ; Fy
4 0
4
HC
HC
F KN
F KN
0 ; Fx
16.5 0
16.5
HG
HG
F KN
F KN
H16.5KN
4KN
?HCF
?HGF
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Norton University Academic Year: 2013-2014
Department: Civil Engineering Page 12
Subject: Strength of Material
Group: M8
Session: Morning
តាម(1)&(3)ពយើងបាន
0 ; Fy
4 4 11 0
3 (1)
y y
y y
CD CG
CD CG
F F KN KN KN
F F KN
0 ; Fx 16.5 0
16.5 (2)
x x
x x
CD CG
CD CG
F F KN
F F KN
2 2
2 2 16.5
8.25 (3)
x x y y
y y
y y
CD CG CD CG
CD CG
CD CG
Slope F F F F
F F KN
F F KN
3 (1)
8.25 (3)
2 5.25
2.625
5.625
y y
y y
y
y
y
CD CG
CD CG
CG
CG
CD
F F KN
F F KN
F KN
F KN
F KN
22
2 2
35.625 8.438
2
3 2.625 3.938
2
(8.438) 5.625 10.14
3.938 2.625 4.732
x
x
CD
CG
CD
CG
Slope F KN KN
F KN KN
F KN
F KN
?yCDF
?yCGF
?xCDF
?xCGF
C
4KN
4KN
16.5KN
11KN
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Norton University Academic Year: 2013-2014
Department: Civil Engineering Page 13
Subject: Strength of Material
Group: M8
Session: Morning
តុំណ(D)
តុំណ(G)
0 ; Fx
8.438 0
8.438
x
x
DE
DE
F KN
F KN
28.438 5.625
3
13 8.438 10.14
3
yDE
DE
Slope F KN KN
F KN KN
0 ; Fy
6 5.625 5.625 0
5.25
DG
DG
F KN KN KN
F KN
0 ; Fy
5.25 2 5.25 0
2
y
y
GE
GE
F KN KN KN
F KN
3
2 32
13 F 2 3.605
2
xGE
GE
Slope F KN KN
KN KN
0 ; Fx
3.938 16.5 3 0
15.562
GF
GF
F KN Kn
F KN
D
6KN
5.625KN
8.438KN
G
2KN
16.5KN
3.938KN
5.25KN
5.25KN
?yGEF
?xGEF
?GFF
?DGF
?xDEF
?yDEF
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Norton University Academic Year: 2013-2014
Department: Civil Engineering Page 14
Subject: Strength of Material
Group: M8
Session: Morning
តុំណ(E)
តុំណ(F)
របារ កម្្ុំងកនលង របាររង AC 19.83KN Compression
AH 16.5KN Tension
HC 4KN Tension
HG 16.5KN Tension
CD 10.14KN Compression
CG 4.732KN Compression
DE 10.14KN Compression
DG 5.25KN Tension
GE 3.605KN Tension
0 ; Fx
8.438 3 0
11.438
x
x
EB
EB
F KN KN
F KN
211.438 7.625
3
13 11.438 13.747
3
yEB
EB
Slope F KN KN
F KN KN
0 ; Fy
4 5.625 7.625 2 0
0
EF
EF
F KN KN KN KN
F
0 ; Fx 15.562 0
15.562
FB
FB
F KN
F KN
E
4KN
3KN
2KN
3.938KN
5.625KN
?EFF
?yEBF
?xEBF
F15.562KN
0KN
?FBF
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Norton University Academic Year: 2013-2014
Department: Civil Engineering Page 15
Subject: Strength of Material
Group: M8
Session: Morning
GF 15.562KN Tension
EB 13.747KN Compression
EF 0KN No Tension or
Compression
FB 15.562KN Tension
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Norton University Academic Year: 2013-2014
Department: Civil Engineering Page 16
Subject: Strength of Material
Group: M8
Session: Morning
IV. Truss analysis by Method of Section
Reaction
10t10t
2t
2t
0.8t
0.8t
0.4t
N P
K M
H
F
D
BA
C
E
G
10t10t
2t
2t
0.8t
0.8t
0.4t
N P
K M
H
F
D
BA
C
E
G
1
2
2
1
3
4 4
5
5
3
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Norton University Academic Year: 2013-2014
Department: Civil Engineering Page 17
Subject: Strength of Material
Group: M8
Session: Morning
0
3 0.4 6 0.8 9 0.8 12 2 15 2 4 10 4 0
3 0.4 6 0.8 9 0.8 12 2 15 2 4 10
4
26.8
A
y
y
m t m t m t m t m t m t B
m t m t m t m t m t m tB
m
t
M
0 ( )
20 26.8 0
6.8
0 ( )
0.4 0.8 0.8 2 2 0
6
y
y
x
x
Y
A t t
A t
X
t t t t t A
A t
Section (1-1)
26.8t6.8t
6t BA
FAC
FCBXFDB
FCBY
3
4
5
section (1-1)
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Norton University Academic Year: 2013-2014
Department: Civil Engineering Page 18
Subject: Strength of Material
Group: M8
Session: Morning
0
4 ( 6.8 ) 0
6.8 ( )
0 ( )
6 0
36 , 6 4.5
4
5 306 7.5 ( )
4 4
X
X y
B
AC
AC
BC
BC BC
BC
m F t
F t Tension
X
F t
F t F t t
N t Compress
M
0 ( )
6.8 6.8 26.8 4.5 0
22.3 ( )
DB
DB
Y
F t t t t
F t Compress
Section (2-2)
26.8t
D
B
FDF
6t
4.5t
FBA
FDC
section (2-2)
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Norton University Academic Year: 2013-2014
Department: Civil Engineering Page 19
Subject: Strength of Material
Group: M8
Session: Morning
0 ( )
26.8 4.5 0
22.3 ( )
DF
DF
Y
F t t
F t Compress
0 ( )
6 ( )
D
BA
counterwise
F t Tension
M
0 ( )
6 6 0
0
DC
DC
X
F t t
F
Section (3-3)
0.8t
0.4t
26.8t6.8t
6t
F
D
BA
C
E
FGFX
FGFY
FFHFEG
3
4
5
section (3-3)
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Norton University Academic Year: 2013-2014
Department: Civil Engineering Page 20
Subject: Strength of Material
Group: M8
Session: Morning
0 ( )
(6 6 ) (3 0.4 ) (4 6.8 ) 4 0
36 1.2 27,2 4 0
1.9 ( )
F
EG
EG
EG
counterwise
m t m t m t m F
tm tm tm m F
F t Comprss
M
Section (4-4)
10t10t
2t N P
FN
MY
FN
MX
FNK FPM
3
4
5
section (4-4)
0 ( )
10 ( )
N
PM
counterwise
F t Compress
M
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Norton University Academic Year: 2013-2014
Department: Civil Engineering Page 21
Subject: Strength of Material
Group: M8
Session: Morning
0 ( )
2 0
32 ( 2 ) 1.5
4
5( 2 ) 2.5 ( )4
MNx
MNx MNy
MN
X
F t
F t F t t
F t t compression
0 ( )
10 20 1.5 0
5.5 ( )
NK
NK
Y
F t t t
F t Compression
Section (5-5)
10t10t
2t
2t
N P
K
1.5
t
2t
10t
FPM
FKM
section (5-5)
0 ( )
0
N
KM
counterwise
F
M
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Norton University Academic Year: 2013-2014
Department: Civil Engineering Page 22
Subject: Strength of Material
Group: M8
Session: Morning
របារ កុំលុំងកនលង (t) របាររង AB 6 (Tension)
AC 6.8 (Tension)
FC 0 (Not C and nor T)
DF 22.3 (Compression)
CB 7.5 (Compression)
EG 1.9 (Compression)
PM 10 (Compression)
NM 2.5 (Compression)
NK 8.5 (Compression)
KM 0 (Not C and nor T)
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Norton University Academic Year: 2013-2014
Department: Civil Engineering Page 23
Subject: Strength of Material
Group: M8
Session: Morning
V. Maxwell Cremona Diagram
Reaction
3000.0000 3000.0000 3000.0000 3000.0000
2000.0000
2000.0000
3kN
4kN
3kN
3kN
4kN
1kN
A B
C
D
E
H G F
7.5kN4.5kN
6kN 10 11
12 13
14 152
1 9 8
3
5
64
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Norton University Academic Year: 2013-2014
Department: Civil Engineering Page 24
Subject: Strength of Material
Group: M8
Session: Morning
0
(2 3 ) (3 4 ) (4 3 ) (6 4 ) (9 4 ) 12 0
907.5
12
0
12 7.5 4.5
0
3 3 6
A
y
y
y
y
x
x
M
m kN m kN m kN m kN m kN B
B kN
F
A kN
F
A kN kN kN
Scale: 1000mm/1kN
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Norton University Academic Year: 2013-2014
Department: Civil Engineering Page 25
Subject: Strength of Material
Group: M8
Session: Morning
From the Maxwell Cremona Diagram above
We obtain:
scale: 1000mm/1kN
9
12
10
8112.4904=8.112kN
8
12750.0000=12.750kN
11
3
4
12
5
6
13
14
3605.5513
15
7
13520.8173
1000.0000
6309.7147
5408.3269
5000.0000
3605.5513
11250.0000
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Norton University Academic Year: 2013-2014
Department: Civil Engineering Page 26
Subject: Strength of Material
Group: M8
Session: Morning
AC=2-10=8.112kN (C)
HA=10-9=12.750kN (T)
HG=11-8=12.750kN (T)
HC=10-11=1kN (T)
CD=4-12=6.310kN (C)
CG=12-11=5.408kN (C)
DE=6-13=9.915kN (Sh,C)
DG=13-12=5kN (T)
GE=13-14=3.606kN (C)
GF=14-8=11.250kN (T)
FE=14-15=0
FB=15-8=11.250kN
BE=15-7=13.521kN (C)
របារ កុំលុំងកនលង (kN) របាររង AC 8.112 (Compression)
AH 12.75 (Tension)
HG 12.75 (Tension)
HC 1 (Tension)
CD 6.31 (Compression)
CG 5.408 (Compression)
DE 9.915 (Compression)
DG 5 (Tension)
GE 3.606 (Compression)
GF 11.25 (Tension)
FE 0 (Not C and nor T)
FB 11.250
BE 13.521 (Compression)
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Norton University Academic Year: 2013-2014
Department: Civil Engineering Page 27
Subject: Strength of Material
Group: M8
Session: Morning