Lec.6 strength design method rectangular sections 2

Analysis & Design of Reinforced Concrete Structures (1) Lecture.6 Strength Design Method

49

Dr. Muthanna Adil Najm

Analysis Examples:

Ex.1)

Determine the moment capacity of a beam with 375mm width, 675mm depth and

reinforced with 4 Ø 28 mm bars, if MPafc 28 and MPaf y 420 .

Sol.)

mmd 60075675 and

22

246461644

284 mmAs

011.0600375

2464

bd

As

Check steel percentage for ACI requirements.

0033.0420

4.14.10031.0

420

2825.025.0min

yy

c

ff

f

0033.0min

0181.08

3

420

2885.085.0

005.0003.0

003.085.0 1max

y

c

f

f

0181.0

0033.0011.0

max

min

mmbf

fAa

c

ys116

3752885.0

4202464

85.0

mma

c 4.13685.0

116

1

Check net tensile strain:

Strength Design Method

Analysis & Design of Rectangular Section Beams.

375mm

600mm

84Ø2

tε

= 0.003 cε

c

d-c d 675mm


50


005.001.0003.06.136

4.136600003.0

c

cdt Tension controlled

section.

mkNa

dfAM ysn

9.560

2

1166004202464

2

mkNMn .8.5049.5609.0

Ex.2)

Determine the moment capacity of the beam section shown below,

MPafc 28 and MPaf y 420 .

Sol.)

2

2

3054101834

363 mmAs

027.0375300

3054

bd

As


0033.0420

4.14.1min

yf

0206.07

3

420

2885.085.0

004.0003.0

003.085.0 1max

y

c

f

f

0206.0027.0 max

Section is not ductile, Check tensile strain.

mmbf

fAa

c

ys6.179

3002885.0

4203054

85.0

mma

c 3.21185.0

6.179

1

004.00023.0003.03.211

3.211375003.0

c

cdt

Section is not tension controlled, and not in the permissible transition zone

between strain of 0.004 to strain of 0.005. The section is not ductile and may not be used per ACI section 10.3.5

Ex.3)

Determine the moment capacity of the beam section shown below,


Sol.)

300 mm

375mm

450 mm 3Ø36

250 mm

375mm

450 mm 3Ø28


51


2

2

184861634

283 mmAs

02.0375250

1848

bd

As


0206.07

3

420

2885.085.0

004.0003.0

003.085.0 1max

y

c

f

f

0206.002.0 max

mmbf

fAa

c

ys4.130

2502885.0

4201848

85.0

mma

c 4.15385.0

4.130

1

Check tensile strain:

004.0

005.00043.0003.0

4.153

4.153375003.0

c

cdt

Beam is in the permissible transition zone and

841.03

250002.00043.065.0

mkNa

dfAM ysn

5.240

2

4.1303754201848

2

mkNMn .3.2025.240841.0

Design Examples:

Ex.4)

Calculate the required amount of reinforcement to resist a moment of

mkNMu .150 for a beam with section of b = 250 mm and h = 500 mm, , if


Sol.)

Assume using 25 bars d = 500 – 40 – 12 -25/2 = 435 mm

65.172885.0

420

85.0

c

y

f

f

52.3

4352509.0

101502

6

2

bd

MR u

n


52


009.0420

65.1752.3211

65.17

1211

1

y

n

f

R


0033.0420

4.14.10031.0

420

2825.025.0min

yy

c

ff

f

0033.0min

0181.08

3

420

2885.085.0

005.0003.0

003.085.0 1max

y

c

f

f

0181.0

0033.0009.0

max

min

2979435250009.0 mmbdAs

Use 2 Ø 25 bars;

2

2

., 98249124

252 mmA provs

2.,

2., 979982 mmAmmA reqsprovs

Ex.5)

Design a simply supported rectangular beam with a span of 5 m and carrying a

total service dead load of 18 kN/m and service live load of 30 kN/m , if


Sol.)

mkNLDwu /70306.1182.16.12.1

mkN

lwM u

u .8.2188

570

8

22

The concrete dimensions will depend on the designer choice of reinforcement

ratio. To minimize the concrete section it is desirable to select the maximum

permissible steel ratio.

5 m

D = 18 kN/m & L =30 kN/m

250 mm

435

mm

500

mm

5Ø22


53


0181.08

3

420

2885.085.0

005.0003.0

003.085.0 1max

y

c

f

f

nu MM

c

yyu

f

fbdfM

59.012

28

4200181.059.014200181.09.0108.218 26 bd

32 38079682mmbd

b d

200 mm 436 mm d > 2b try greater ' b ' value

250 mm 390 mm 3/2 b < d < 2b OK.

300 mm 356 mm d < 3/2 b Not Good

Use section with b = 250 mm and d = 390 mm

217653902500181.0 mmbdAs

Use 3 Ø 28 mm

2

.,2

2

., 1765184861634

283 mmAmmA reqsprovs

h = 390 + 40 (cover) + 12 (Stirrups) + 28/2 (bar diameter /2) = 456 mm

Use h = 460 mm

d = 460 - 40 (cover) - 12 (Stirrups) - 28/2 (bar diameter /2) = 394 mm

Check Section:

mmbf

fAa

c

ys4.130

2502885.0

4201848

85.0

mkNMmkNa

dfAM uysn

8.2187.229

2

4.13039442018489.0

2

Section is OK.

Check deflection requirements:

From ACI Table 9.5.a, the minimum

permissible beam thickness for simply

supported beam is:

mml

h 5.31216

5000

16

mmhmmh requiredavailable 5.312460 250 mm

394

mm

460

mm

Ø283


54


Section is OK.

Ex.6)

Resolve example (4) using steel percentage of max5.0

Sol.)

mkNLDwu /70306.1182.16.12.1

mkN

lwM u

u .8.2188

570

8

22

0181.08

3

420

2885.085.0

005.0003.0

003.085.0 1max

y

c

f

f

009.02

0181.0

2

max

nu MM

c

yyu

f

fbdfM

59.012

28

420009.059.01420009.09.0108.218 26 bd

32 69881141mmbd

b d

200 mm 591 mm d > 2b try greater ' b ' value

300 mm 482 mm 3/2 b < d < 2b OK.

350 mm 447 mm d < 3/2 b Not Good

Use section with b = 300 mm and d = 490 mm

21323490300009.0 mmbdAs

Use 3 Ø 25 mm

2

.,2

2

., 1323147349134

253 mmAmmA reqsprovs

h = 490 + 40 (cover) + 10 (Stirrups) + 25/2 (bar diameter /2) = 553 mm Use h = 560 mm

d = 560 - 40 (cover) - 10 (Stirrups) - 25/2 (bar diameter /2) = 498 mm

Check Section:

mmbf

fAa

c

ys6.86

3002885.0

4201473

85.0


55


mkNMmkNa

dfAM uysn

8.2182.253

2

6.8649842014739.0

2

Section is OK.

Check deflection requirements:

From ACI Table 9.5.a, the minimum

permissible beam thickness for simply

supported beam is:

mml

h 5.31216

5000

16

mmhmmh requiredavailable 5.312560

Section is OK.

300 mm

498

mm

560

mm

5Ø23

Education

Lec.6 strength design method rectangular sections 2