shear moment diagrams

7/25/2019 Shear Moment Diagrams

1/24

April 2015

()

.

Design of Beams for Bending

( )

. (51)

.

Strength of Materials

Week 5

(03)

. .

(eng-hs.net) .

( )

(eng-hs.net)

.


2/24

April 2015

(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

: .


3/24

April 2015

(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


4/24

April 2015

(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


5/24

April 2015

(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

-


6/24

April 2015

(6)

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


7/24

April 2015

(7)

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


8/24

April 2015

(8)

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) :


9/24

April 2015

(9)

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

.


10/24

April 2015

(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

+- -


11/24

April 2015

()

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

.


12/24

April 2015

(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


13/24

April 2015

(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

.


14/24

April 2015

(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


15/24

April 2015

(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


16/24

April 2015

(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++ +

-


17/24

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

.

* :


18/24

April 2015

(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


19/24

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)*

.


20/24

April 2015

(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

+ + +

+

-

+


21/24

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)


22/24

April 2015

(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

.


23/24

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


24/24

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

shear moment diagrams

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