solutions to question 3,5,6

8/11/2019 Solutions to Question 3,5,6

1/8

Example 3. For the RC beam section shown, calculate uncracked moment of inertia,

flexural rigidity, the moment to initiate concrete cracking and the corresponding

curvature.

150

300

2

2

2

/6.2

/000,210

/250,26

mmNf

mmNE

mmNE

t

s

c

250

2mm602tensionst

A

25

2

mm300ncompressiostA 8250,26

000,210

c

s

E

E

m

Transformed concrete section

2

mm000,45

150300

2

mm4214602)18()1(

tstAm

2mm2100300)18()1( cstAm

All dimensions are in mm

mmyy 1.1532504214150000,45252100314,51

2mm314,51214,4100,2000,45 A

Depth of NA from beam top:

NA

y


2/8

150

300250

25 Transformed concrete section

2mm4214

2mm2100

2

mm000,45NA

mm1.153y

462

223

mm10412)1.153250(214,4

)251.153(100,2)1501.153(30015012

300150

gI

2126 mmN1082.10250,2610412 gcIE

kNm997.61.153

104126.2 6

y

IfM

gt

cr

17

12

6

mm1047.61082.10

10997.6

gc

crcr

IE

M

2mm602cst

A

2mm300tst

A

22 N/mm6.2,N/mm250,26 tc fE

y


3/8

Flextural Rigidity of the Cracked Section

dn

N A

Transformed concrete sectionR.C. Section

Neutral axis (NA) will pass through the centroid of the transformed concrete section

as the analysis is based on elastic theory.

So the distance of NA from the centroid of transformed section is zero.

tst

A

cstA

tst

Am

cst

Am )1(

00

dAyA

dAy

dA

dAyy

ycd

b

....0)()()())(1(2)( nntstncst

n

n dddAmcdAm

d

db


4/8

Example 5. Determine the flexural rigidity of the uncracked section.What moment will cause the stress in the bottom fibre to reach 4MPa and

at what curvature will this occur. What is the stress in the reinforcing bars

and in the top fibre of the concrete.

150

500

1000

300

50

2

2

/200

/20

mmkNE

mmkNE

s

c

23000mmAst


5/8

Solution5

1501000

300 500

50

600


4MPa

2000,2793000 mm

1020/200 m

Position of centroid from top surface

mmd

d

n

n

4.267

500300150000,1000,27

)2/500150(5003002/150150000,1600000,27

nd


6/8

Moment of inertia

4622

32

3

10583,144.267600000,274.2672/500150500300

125003002/1504.267150000,1

12150000,1

mm

I

Flexural rigidity 2146 10917.21014583000,20 mmNEI

Moment kNmy

IM 5.152

4.267650

10583,144 6

Curvature 1714

6

1023.510917.2

105.152

mm

EI

M

Stress in top fibre on)(compressi/80.210583,14

4.267105.152 26

6

mmNI

Myc

Stress in rebars 26

6

/8.341010583,14

4.267600105.152mmNm

I

Mys

Solution5

Note - convert the stress in concrete to stress in steel by multiplying with m


7/8

Example 6: Determine flexural rigidity of cracked section and thestress in the reinforcement when a moment of 130kNm is applied. What

the the curvature when this moment is applied.

200

400

1000

300

50

2

2

kN/mm200

kN/mm20

s

c

E

E

2mm3000stA


8/8

Solution6

1000

300550


dn

2mm000,30103000

1020/200 m

Position of centroid from top surface

mmd

dd

ddd

n

nn

nnn

1545002

000,500,165004000,30000,30

0000,500,16000,30500

550000,302/000,1

2

2

As a starting position, assume NA

is in within the flange

4623

109.5921154550000,30

3

154000,1mmI

200

Assumption is correct

solutions to question 3,5,6

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