04 rc beam.pdf

Bending Moment in RC Beam


N.A = Nutral Axis. D

kd

d jd

d-kd

As

Ac

C

T

kd/3

b fc

fs

M

C = T

M = C jd = T jd



kd

d jd

d-kd

As

Ac

C

T

kd/3

b fc

fs

M

k = 11+

fs

nfc

; n = EsEc

jd = d kd3 j=1k3



kd

d jd

d-kd

As

Ac

C

T

kd/3

b fc

fs

M

C = T, T = fs As and C = 12 fc b kd

M = T jd = fs As jd

As = Mfsjd

Concrete

W/C fC 28-day (ksc)

0.35 420 335

0.40 350 280

0.50 280 225

0.60 225 180

0.70 175 140

0.80 140 115

28 Ec ...

Ec = w1.5 4270 fC Where;

w = concrete density (t/m3)

Steel bar

Steel bar

(fc) ksc

(fc) ksc

Round bar (RB)

6-9 mm 2500 4100

> 12 mm 2400 4100

Deformed bar (DB)

SD24 2400 4900

SD30 3000 4900

SD35 3500 5700

SD40 4000 5700

28 Es ... Es = 2.04 106 kg/cm2

6-9mm 6-9mm [email protected] [email protected] 2 [email protected] [email protected] 10-DB24 # [email protected]

General Condition of RC beam

1. (d)

2. (L/360) L/20 L/23 L/26 L/10

3. 2.5 1.23

4. 0.0025 0.0015

5.

Steel spacing and covering

Z

Z

2.5 cm

X

X X 1.34 X 2.5 Z 2 Z 3 15 Z 4 15

N.A = Nutral Axis.

kd

d jd

d-kd

Ac

C

T

kd/3

b fc

fs

k = 11+

fs

nfc

; n = EsEc

As = Mfsjd

As

Diameter Type

weight (kg/m)

o (cm)

Number of reinforced steel (bar)

1 2 3 4 5 6 7 8 9

mm In As (cm2)

6 1/4

RB

0.222 1.885 0.283 0.565 0.848 1.131 1.414 1.696 1.978 2.262 2.544

9 3/8 0.499 2.828 0.636 1.272 1.909 2.545 3.182 3.820 4.455 5.090 5.726

12 1/2 0.888 3.770 1.131 2.262 3.393 4.526 5.650 6.785 7.915 9.045 10.180

15 5/8 1.390 4.710 1.767 3.534 5.295 7.065 8.830 10.600 12.360 14.140 15.900

19 3/4 2.230 5.965 2.835 5.670 8.500 11.340 14.170 17.020 19.850 22.680 25.500

22 7/8 2.980 6.905 3.801 7.602 11.400 15.200 19.000 22.800 26.600 30.400 34.200

25 1 3.850 7.850 4.909 9.818 14.730 19.640 24.530 29.450 34.380 39.280 44.200

28 1 1/8 4.830 8.790 6.158 12.320 18.480 24.630 30.800 36.960 43.120 49.280 55.420

12 1/2

DB

0.888 3.771 1.130 2.260 3.390 4.520 5.650 6.780 7.910 9.040 10.170

16 5/8 1.580 5.029 2.010 4.020 6.030 8.040 10.050 12.060 14.070 16.080 18.090

19 3/4 2.230 5.971 2.840 5.680 8.520 11.360 14.200 17.040 19.880 22.720 23.560

20 3/4 2.470 6.290 3.140 6.280 9.420 12.560 15.700 18.840 21.980 25.120 28.260

22 7/8 2.980 6.910 3.800 7.600 11.400 15.200 19.000 22.800 26.600 30.400 34.200

25 1 3.850 7.857 4.910 9.820 14.700 19.600 24.500 29.400 34.400 39.300 44.200

28 1 1/8 4.830 8.800 6.160 12.320 18.480 24.640 30.800 36.960 43.120 49.280 55.440

32 1 1/4 6.310 10.060 8.040 16.080 24.120 32.160 40.200 48.240 56.280 64.320 72.360

Example 1 Bending RC Beam

. 0.20x0.40 3000 kg.m n = 11, fc = 210 ksc, fy = 2400 ksc

. fc = 0.45f c = 0.45*210 = 94.5 ksc fs = 0.50fy = 0.50*2400 = 1200 ksc k jd k = 1

1+fs

nfc

= 11+

12001194.5

= 0.4642

j = 1 k

3 = 1 0.4642

3 = 0.8453

2.5 cm => d = 40-5 = 35 cm jd = 0.8453*0.35 = 0.2958 m As = M

fsjd = 3000

12000.2958 = 8.45 cm2

Design 3-RB19 As = 8.50 cm2


. 0.20x0.40 3000 kg.m n = 11, fc = 210 ksc, fy = 2400 ksc

Design 3-RB19 As = 8.52 cm2

d = 35 cm

covering = 2.5 cm

covering = 2.5 cm

20- (2*2.5) = 15 cm

x

x = 15 (31.9) 2

= 4.65 cm > 2.5 cm O.K.

diameter 2.5 cm

C + Cs = T1 + T2

12 fc bkd + A

s fs = As1 fs + As2 fs

N.A = Nutral Axis. d-d

jd

Ac

C

T1

kd/3

b fc

As

Ast=As1+As2

Cs

T2

+ As1 As2

T1 C

M1 = Rbd2

R = 12 fc k j

As1=M1fsjd


jd

Ac

C

T1

kd/3

b fc

As

Ast=As1+As2

Cs

T2

+ As1 As2

T2 Cs

M2 = MM1 = As f s d d = As2 fs (d d

)

As2 =M2

fs(dd)


jd

Ac

C

T1

kd/3

b fc

As

Ast=As1+As2

Cs

T2

+ As1 As2

Ast = As1 + As2 = M1

fs jd+

M2fs(d d)


jd

Ac

C

T1

kd/3

b fc

As

Ast=As1+As2

Cs

T2

+ As1 As2

Cs=T2, Asf s = As1fs

f s = fsk(d d )1k

. f s f s = 2fsk(d d )

1k

As = 12As2

1k

kd

d

As = M2fs(dd)


jd

Ac

C

T1

kd/3

b fc

As

Ast=As1+As2

Cs

T2

+ As1 As2

1.

2. k, j, R, As

3. M 1 =Rbd2 M

M1


. 0.25x0.60 16,500 kg.m n = 8, fc = 250 ksc, fy = 3,000 ksc

. fc = 0.45f c = 0.45*250 = 112.5 ksc fs = 0.50fy = 0.50*3000 = 1500 ksc k jd k = 1

1+fs

nfc

= 11+

15008112.5

= 0.375

j = 1 k

3 = 1 0.375

3 = 0.875

R = 12 fc kj =1

2 112.5 0.375 0.875 = 18.45 ksc

d = 60-5 = 55 cm d= 5 cm f s = 2fskd d

1k = 2 1500 0.3755 55

10.375 = 1364 ksc

M1 = Rbd2 =18.45 0.25 552 = 13,953 kg.m

As1 =M1fsjd

= 13,95315000.8750.55

= 19.33 cm2


. 0.25x0.60 16,500 kg.m n = 8, fc = 250 ksc, fy = 3,000 ksc

M1=13,953 kg.m < M=16,500 kg.m 2 = M 1 = 16,500 13,953 = 2,547 kg.m

As2 =M2

fs(dd) =

2,547

1,500(0.550.05) = 3.40 cm2

As = M2fs(dd)

=2,547

1,364(0.550.05) = 3.73 cm2 ==>

Ast = As1 + As2 = 19.33 + 3.40 = 22.73 cm

2 ==>

Design Design 2-RB16 As = 4.020 cm2

Design 3-DB32 As = 24.120 cm2 (x) = (25-(2*3)-(3*3.2))/2 = 4.7 cm > 2.5 cm O.K.

N.A.

H

H

N.A. V

V

H

V

T

H V

T

T2 = V2 + H2

Shearing Stress ()

= Vbd V = bd

. 1.32 f c ksc

=0.29 f c ksc Vc = 0.29 f c bd kg Vc ==> (Stirrup)

V = V Vc V = AVfV sin

= 90 Av = 2As of stirup fv = fs

V = Avfvds

V = 2Asfsds

(s ) ()

1. d

2 0.30 m

2. (V) 1.32 f c ksc

3. Av = 2As > 0.0015bs

4. V > 0.795 f c ksc d

4

Example 3 Shearing RC Beam

. 0.15x0.40 4.00 m 2,000 kg/m n = 11, fc = 250 ksc, fy = 2400 ksc

. fc = 0.45f c = 0.45*250 = 112.5 ksc fs = 0.50fy = 0.50*2400 = 1200 ksc

= 0.29 f c = 0.29 250 = 4.58 ksc Loading Analysis = = 4.58*15*30 = 2,061 kg V = V Vc = 4,000-2,061 = 1,939 kg Design [email protected] resist shear 1,970 kg

2000 kg/m

4000 kg 4000 kg

V=4000 kg

V=-4000 kg M=+4000 kg.m

Example 4 RC Beam Design

. A-A 0.20x0.40 n = 10, fc = 250 ksc, fy = 2400 ksc

. fc = 0.45f c = 0.45*250 = 112.5 ksc fs = 0.50fy = 0.50*2400 = 1200 ksc

= 0.29 f c = 0.29 250 = 4.58 ksc

820 kg/m

1640 kg 3954 kg

917 kg/m 480 kg

4.00 m 2.00 m


. A-A 0.20x0.40 n = 10, fc = 250 ksc, fy = 2400 ksc

V=2314 kg V=1640 kg

M=-1093 kg M=-2794 kg

M=+547 kg


. A-A 0.20x0.40 n = 10, fc = 250 ksc, fy = 2400 ksc

k jd k = 1

1+fs

nfc

= 11+

120010112.5

= 0.4839 j = 1 k3 = 1 0.4839

3 = 0.8387

d = 40-5 = 35 cm d= 5 cm jd = 0.8387*0.35 = 0.2935 m R = 1

2 fc kj =1

2 112.5 0.4839 0.8387 = 22.83 ksc

f s = 2fskd d 1k

= 2 1200 0.48395 35 10.4839

= 1,586 ksc

M1 = Rbd2 =22.83 0.20 352 = 5,593 kg.m

M=-1,093, +547, -2,794 kg.m M1 = 5,593 kg.m M < M1 O.K. Design 2-DB12 (As=2.260 cm2)


. A-A 0.20x0.40 n = 10, fc = 250 ksc, fy = 2400 ksc

1) M=-1,093 kg.m As = M

fsjd = 1093

12000.2935 = 3.10 cm2

Design 3-DB12 (As = 3.390 cm2) 2) M= +547 kg.m As = M

fsjd = 547

12000.2935 = 1.55 cm2

Design 2-DB12 (As = 2.260 cm2) 3) M= -2,794 kg.m As = M

fsjd = 2794

12000.2935 = 7.93 cm2

Design 2-DB12 + 2-DB19 (As = 7.940 cm2)


. A-A 0.20x0.40 n = 10, fc = 250 ksc, fy = 2400 ksc

V= 1,640 2,314 kg = = 4.58*20*40 = 3,664 kg 1) V= 1,640 kg V < V Design [email protected] 2) V= 2,314 kg V < V Design [email protected]


. A-A 0.20x0.40 n = 10, fc = 250 ksc, fy = 2400 ksc

3-DB12 2-DB12 + 2-DB19

2-DB12

[email protected]

3-DB12

2-RB6

2-DB12

2-DB12

2-RB6

2-DB12

2-DB19

2-DB12

04 rc beam.pdf

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