shear moment diagrams
TRANSCRIPT
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7/25/2019 Shear Moment Diagrams
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April 2015
()
.
Design of Beams for Bending
( )
. (51)
.
Strength of Materials
Week 5
(03)
. .
(eng-hs.net) .
( )
(eng-hs.net)
.
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(2)
Design of Beams for Bending
4normal stress()beam
: (bending moment)M
I
yM
M
all
((W-flange shapes)(S shapes
. S
(W-flange shapes)(S shapes):
5-
S
2-
.
:
yI
M
/
I / ysection modulus(S):
MS
S
M
MS
: .
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(3)
Drawing Shear & Bending-Moment Diagrams
M:
.reactions-5
2-V:
concentrated
distributed
.
0-M:
V-Diagram
M-Diagram M
:
.
AMM previousnew
Beam bending moment
V-Diagram
M-Diagram
moment beam
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(4)
kNA
kNA
kNC
C
M
kNCA
CA
F
y
y
y
y
A
yy
yy
y
5.19
5.19
5.220
07.2*1209.0*8.1*458.1*
0
201
01208.1*45
,0
reactions
V:
concentrated
distributed
.
M:
V-diagram
M-diagram M
:
AMM previousnew
.
kNV 120max
.
+
mkNM .108max
120 120
+
-
--
100.5
19.5
V (kN)
M (kN.m)
5-10
108
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(5)
5.13
mkNw
w
Fy
/2
05.12)()5.1(
0
V:
concentrated
distributed
.
M:
V-diagram
M-diagram M
:
AMM previousnew
.
mNkNM
NkNV
.5.1121125.0
9009.0
max
max
.
1.5 kN1.5 kN
w
0.9 kN
0.6 kN
0.6 kN
- 0.9 kN
-
-
+ +
V (kN)
M (kN.m)+ +
0.1125 kN.m
0.09 kN.m 0.09 kN.m
-
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kNA
kNC
C
M
kNCA
CA
F
y
y
y
A
yy
yy
y
58.2
02.14
01.4*2.2*35.1*103*
0
6.16
02.2*310
0
.
V:
concentrated
distributed
.
M:
V-diagram
M-diagram M
:
AMM previousnew
.
+
MPa
I
cM
89.10
)10*200(*)10*100(*12
1
)10*100(*)10*26.7(
333
33
6.6
+V (kN)
+
-
2.582.58
7.427.42
3.87
7.26
- -
+ M (kN.m)
reactions
5-15
Fig. P5.15
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kNA
kNB
B
M
kNBA
BA
F
y
y
y
A
yy
yy
y
19.192
56.176
0375.3*25.2*75
5.1*10075.0*1005.4*
0
75.368
025.2*75100100
0
MPa
I
cM
52.6510*462
)10*2
420(*)10*14.144(
6
33
:
.
M:
V-diagram
M-diagram M
:
AMM previousnew
.
V:
concentrated
distributed
.
+
+
+
--
+ +
192.19
92.19
7.81
176.56
V (kN)
M (kN.m)
207.43213.29
144.14
reactions
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ma
MM
a
aaaM
aAa
M
M
a
CA
a
aC
aaC
M
DA
A
yA
leftB
yy
y
y
rightB
926.0
54
10)1054(
0)(
2
20
0)(
1054
)4.520(
1054
)4.5(10
02
)4.5(20)4.5(
0)(
2
2
2
5.4 - aa
MA
20 kN/m20 kN/m
AB D Cy
20 kN/m
Ay
Mmax
MPa
I
cM
6.29
)1010302(
102
357)926.01054(
126
33
4610302 mmI
357 mm
2
2
2
22
5.2279.72
)8.102.29(5.2
)4.5(5.2
8
)4.5(20
8
aa
aa
a
alwMD
(moment) :
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mm
md
dd
gvw
dw
d
dw
I
cM
w
wM
8.24
108.24
)81.9(14
)860,7(1044.2
1044.2
24
1
2)125.1(
1028
125.1
)5.1()5.1(2
1
3
236
36
4
6
max
max
N/mw
* :
+
-
V (N)
M (N.m)
-1.5 w
1.5 w
1.125 w
1
2
.
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(1)
kNA
BA
F
kNB
BM
y
yy
y
y
y
A
4.29
07.0140
0
6.68
05.035.07.01400
V:
concentrated
distributed
.
0.5 m 0.2 m
ByAy
140 kN/m
0.29 m28
29.4
++
-0.21 m
40.6
M:
V-diagram
M-diagram M
:AMM previousnew
.
3.087
2.8
yI .
I
cMmax
5.57
+- -
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()
5.57
MPa
I
CM
mmI
mm
I
mm
II
IIII
logicmmmmy
total
total
95.86
101047.1
10)72.3375()10087.3(
1047.1
1009.1
)5.3772.33(250,2)75()30(12
1104.189
)2572.33(5.487)75()13(36
1
)(5.3772.33225,3
750,108
126
33
maxmaxmax
46
46
233
43
2321
321
(mm )Area .(mm)Area (mm )Seg
12,187.5255.487)75()13(21
12,187.5255.487
84,3755.37250,27530
0.750,108 0.225,3
5
2
0
5
0
2
datum
3013 13
75
y
.
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(2)
PxQPPx
xxPxQxPxPM
xAx
xxPM y
45.0)1875.04.88125.0(5.3
4.88125.01875.05.345.0
0)(2
)(7)45.0(
2
2
4.88125.01875.0
0)4.2()2.1()4.2(7)95.1()45.0(
0
.72.32)105.45(
102
203)(
1073
.83.44)105.45(
102
203)(
10100
6
3
3
6
3
6
PQA
APQ
M
mkNM
M
mkNM
M
I
cM
y
y
B
E
E
D
D
* (M) (D)(E)
() :
(:(P)(Q( (A *
* (M) (D)(E)(P)(Q:)
.
0.45
x 1.95
*
(MD)
(ME)
.((P)(Q
7 kN/m
P V
M
x
0.45 m
4.88125.01875.0 PQyA
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(3)
5.67
kNEB yy 12
mm
mb
b
I
cM
mkNM
all
146
146.0
)10240(12
1
)10120()108.16(1012
.8.16
33
336
max
V:
concentrated
distributed
.
M:
V-diagram
M-diagram
M :
AMM
previousnew
.
:
From symmetry
20 kN20 kN
8 kN
8 kN
20 kN20 kN
+
+
-
-
12 kN.m12 kN.m
16.8 kN.m
.
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(4)
kNA
kNB
B
M
kNBA
BA
F
y
y
y
A
yy
yy
y
70.2
10.8
04.2*8.1*6.3*3
0
8.10
06.3*3
0
.
M:
V-diagram
M-diagram M
:
AMM previousnew
.
V:
concentrated
distributed
.
+
mmb
b
I
cM
48
)10*150(**12
1
)10*75(*)10*16.2(10*12
33
336
++
-
+
-
1.215
0.9 m
2.16
3.62.7
reactions
5-70
V (kN)
M (kN.m)
Fig. P5.70
1.5 m - 4.5
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(5)
101610
10*249510*165
10*75.411
5.206
5.181
02.7*
6.3*2.7*407.2*100
0
388
02.7*40100
0
33
6
3
W
mmM
S
kNA
kNC
C
M
kNCA
CA
F
y
y
y
A
yy
yy
y
:
.
M:
V-diagram
M-diagram M
:
AMM previousnew
.
V:
concentrated
distributed
.
+
+
-
206.5
98.5
1.5
181.5
V (kN)
+
411.75
M (kN.m)
reactions
P 5.72
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(6)
kNA
kND
D
M
kNDA
DA
F
y
y
y
A
yy
yy
y
5.73
5.60
06.3*
7.2*8.1*309.0*80
0
134
08.1*3080
0
:
.
M:
V-diagram
M-diagram M
:
AMM previousnew
.
V:
concentrated
distributed
.
3.47310
10*41310*160
10*15.66 336
3max
S
mmM
S
+
+
-
73.5
6.50
60.5
V (kN)
M (kN.m)
66.1560.3
reactions
5-78
Fig. P5.78
M++ +
-
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April 2015
(7)
kNA
EA
F
kNE
E
M
y
yy
y
y
y
A
3.130
012812832
0
7.157
0)7.9(128
)5.5(128)3.1(32)6.12(
0
V:
concentrated
distributed
.
:Reactions*
M:
V-diagram
M-diagram M
:
AMM previousnew
.
mmb
b
I
cM
mkNM
all
7.263
10)128.37.169(
)105.262()1025.582(
10154
.25.582
6
336
max
V (kN)+ +
-
-
130.3 kN
98.3 kN
-29.7 kN
-157.7 kN
M (kN.m)
+
457.5 kN.m
582.25 kN.m
169.4 kN.m
46
46
232
46
31
21
10)128.37.169(
10564.1
)250()25()25(121
107.169
)475()19(12
1
2
mbI
mmb
bbI
mm
I
III
total
total
.
* :
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(8)
wCB yy 45.0
43
23
2
43
231
23
21
10520
)5070(200,1)20()60(12
1
10840
)5030(200,1)60()20(12
112
1
50400,2
000,120
mm
I
mm
I
dAhbI
III
mmy
total
(mm3)Area .(mm)Area (mm2)Seg
36,0003020 * 60 = 1,2001
84,0007060 * 20 = 1,2002
120,000 2,400
:roidCent*
V:
concentrated
distributed
.
M:
V-diagram M-
diagram M :
AMM previousnew
.
0.2 w0.25 w
0.25 w
-
0.2 w
++
-V (kN)
M (N.m)
0.01125
+
--
20 mm
60 mm
60 mm
20 mm
y
datum
.
0.2 w0.2 w
:rom symmetryF
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April 2015
(9)
5.85
mkNw
w
mkNw
w
I
CM
mkNw
w
I
CM
mmI
cc
tt
total
/8.176)(
)1036.1(
)1050()02.0(10130
/524)(
1036.1
)1030()01125.0(10130
/4.193)(
1036.1
)1050()01125.0(1080
1036.1
max
6
36
2max
6
36
1max
6
36
46
Choose minimum w mkNw /8.176max
:(Ct)max ((M)(T*
:(Cc) (M)(C)*
.
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(21)
33
6
3
126
33
6
max
max
10754
10165
104.124
07.22
46.1
1010165
102
310)10120(
10165
120
120
mm
MS
maLl
ma
a
I
cM
aM
kNBA yy
)(1.46410 CheapestWchoose
.
a)
b)
120120 N
ByAy
a
+
-
120
120
V (kN)
120 a
240
M (kN.m)
120120
120
120
V (kN)
M (kN.m)
124.4
+ + +
+
-
+
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April 2015
(2)
kN
BA
kNP
mkNmN
www
yy
LDtotal
84.15
2
8.421638.1
8.46.13
/38.1/380,1
)600()6.1()350()2.1(
6.12.1
mmmh
h
h
I
cMall
168168.0
)1075(12
1
2)1084.15(
)10509.0(33.
3
6
V:
concentrated
distributed
.
M:
V-diagram
M-diagram
M :
AMM previousnew
.
:rom symmetryF
1.38 N/m
ByAy 4.8 kN 4.8 kN
4 m 8 m 4 m
.
14.04 kN8.52 kN
5.52 kN
-5.52 kN
-8.52 kN-14.04 kN
V (kN)
+
+-
-
56.16 kN.m
45.12 kN.m
M (kN.m)
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(22)
P
PA
PCA
F
PC
CP
M
y
yy
y
y
y
A
458.0
458.0
0
0
458.1
0)8.4()7(
0
V:
concentrated
distributed
.
M:
V-diagram
M-diagram
M :
AMM previousnew
.
kPP
P
mm
I
kNP
P
I
cM
208
10691
10182
417)2.2(
10150
10690
218
2417)18220()18()220(
121210315
7.193
)10315(
102
417)17.1(
10150
6
3
6
46
236
6
3
6
Choose minimum load P = 193.7 kN
*
(P)
:
*
(P)
: 2
+
P
-
V N
M N.m
0.952.55
0.95 P1.17 P
2.2 P
-0.458 P
2
.
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April 2015
(23)
49
21
43232
43
231
10*470
10129)4454(*152,1)12(*)96(*12
1
10*341
)2444(*576)48(*)12(*12
1
44728,1
032,76~
mIII
mmI
mm
I
mmArea
yAreay
total
Seg Area (mm2)
(mm)Area . (mm3)
1 24 13,824
2 54 62,208
1,728 76,032
PA
PPCA
F
PC
CPP
M
y
yy
y
y
y
A
467.0
0
0
533.1
075.0*9.0*25.0
0
Reactions, V & M:
Centroid and I: datum
.
57648*12
152,112*96
++
-
+
- -
0.533 P0.117 P
0.467 P
0.15 P
P
V (N)
M (N.m)
16
44
96
12
datum
0.117 P
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April 2015
kNP
P
kNP
P
kNP
P
kNP
P
c
IM
I
cM
67.15
)10*16()10*470(*)10*80(15.0
97.9
)10*44(
)10*470(*)10*140(15.0
31.7
)10*44(
)10*470(*)10*80(117.0
1.35
)10*16(
)10*470(*)10*140(117.0
max
3
96
max
max
3
96
max
max
3
96
max
max
3
96
max
5.159
Checking for M = 0.15 P:
Checking for M = 0.117 P:
Always choose the minimum load,Pmax = 7.31 kN
permissible
tension
16
44
96
12
datum
0.15 P