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Φ Weight mm Kg/m 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38
0.222 0.395 0.617 0.888 1.21 1.58 2.00 2.47 2.98 3.55 4.17 4.83 5.55 6.31 7.13 7.99 8.90
1 .283 .503 .785 1.13 1.54 2.01 2.54 3.14 3.80 4.52 5.31 6.16 7.07 8.04 9.08 10.2 11.3
2 .566 1.01 1.57 2.26 3.08 4.02 5.09 6.28 7.60 9.04 10.6 12.3 14.1 16.1 18.2 20.4 22.6
Area of Cross-Section 3 4 5 6 .848 1.13 1.41 1.70 1.51 2.01 2.51 3.02 2.36 3.14 3.93 4.71 3.39 4.52 5.65 6.79 4.62 6.16 7.70 9.24 6.03 8.04 10.1 12.1 7.63 10.2 12.7 15.3 9.42 12.6 15.7 19.8 11.4 15.2 19.0 22.8 13.6 18.1 22.5 27.1 15.9 21.2 25.5 31.9 18.5 24.6 30.8 37.5 21.2 28.3 35.3 42.4 24.1 32.2 40.2 48.3 27.2 36.3 45.4 54.5 30.6 40.9 50.7 61.2 33.9 45.2 56.5 67.8
in cm2 7 8 1.98 2.26 3.52 4.02 5.50 6.28 7.92 9.05 10.8 12.3 14.1 16.1 17.8 20.4 22.0 25.1 26.6 30.4 31.7 36.2 37.2 42.5 43.1 49.3 49.5 56.6 56.3 64.3 53.6 72.6 71.4 81.6 79.1 90.4
9 2.54 4.52 7.07 10.2 13.9 18.1 22.9 28.3 34.2 40.7 47.0 55.4 63.6 72.4 81.7 91.8 102
10 2.83 5.03 7.85 11.3 15.4 20.1 25.4 31.4 38.0 45.2 53.1 61.6 70.7 80.4 90.8 102 113
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E١
WASCIאFא
U = 1.4 D + 1.7 L
(2-1)
Dead Loads אאאZD
Live loadsאאZL
אאא٠{٧٥אאאE٢ Wאא
U = 1.5 (D +L)
(2-2)
אאאא אאE٣ Wאאאא
U = 1.4 D + 1.7 (E + L)
(2-3)
(Lateral Load)אאZE (2-1)אאU
- ٢٣ -
אאא ã¹]<gè…‚jÖ]æ<ËÖ]<Üé×Ãj×Ö<íÚ^ÃÖ]<퉉ö¹] אאאא
א
אא
אא
٣
אא
٢٠١
א
אא
א
Kאא אאאא אאאא אאאא א אאאא
א א א א א KKא Kאאאאאא
אא Kאאאאא E١
א א א א E٢ Kאאאא
א א א א א א E٣
אא Kאא Wאאאאא KאאאאE١
KאאאאE٢ KאאאE٣
KאאאE٤
KאאאאאאאאE٥ Kאאאא • Kאא אאאKאא
- ٢٤ -
אא
٢٠١
א
אא
א
אא
Structural Analysis
אאW١J ٣ Wאאאאאא אאJ ١
Member Analysisאא אא (Beams)אא (slabs)א،
WאאK(Footings)א(Columns)
SlabsאEא
BeamsאאE٢ GirdersאאE٣ ColumnsאE٤
FootingsאE٥
אאJ ٢ Section Analysisאא
אאאא אKKאאאא–אאKאאא J Wא
UL،T،،،
KאאאאE١J ٣Fא KאאאE٢J ٣Fא
- ٢٥ -
אא
٢٠١
א
אא
א
Over hanging beam
Simple slab or beam
continous slab or beam
(Column)
Simple frame
(Column)
אאאE١J ٣F
U
L
U - section
L - section
T circular sec
T - section square sec
אאאE٢J ٣F
- ٢٦ -
Rectangular sec
אא
٢٠١
א
אא
א
אאאW٢J ٣ א Statically Systemאאא א
א א Kא א א
Kאאא
אאא
J WW١J ٢J ٣ אEאאFאא
אאE٣J ٣F (Knife Edge Support)
א
K
d
L
Knife edge support
E٣J ٣F
אאW٢J ٢J ٣
אא אא (Linear Elastic Theory)אא
אאא אאאאא Kאאא W
Three Moment EquationאאE Virtual Work MethodאאאE - ٢٧ -
אא
٢٠١
א
אא
א
Moment DistributionאE אאEאאא Fאא
אא٠{٤،אא٠{٨(lo)אא(d)א i.e
d / l0 < 0.8 for simple beams. d / l0 < 0.4 for continues beams. (Deep Beam)אא אאא
Kאאא(Plastic Analysis)א
(Effective Span l0)אאאאW٣J ٣ אאאE
J Wאאאאאאא l01אאא
•
l03 = l0 + d
•
•
l02 = l0 × 1.05 אא١{٠٥
E٤J ٣FאאאאZd
d
L L
1
L 2 = 1.05 L
C L
C L
אאאE٤J ٣F
- ٢٨ -
אא
٢٠١
א
אא
א
KאאאE
אFאEאFאאאאE١ J WאאאאאE(Rigid Connections)
•
l01 = Centerline to Centerline of Supportsאא
•
l02 = 1.05 × l0אא١{٠٥
א א F א א א אא E٢ אאאאKEאא – Kאאאאא
Or
L03 = Distance from Centerline to Centerline L03 =l0 +d
אאאE
Wאאאאאאא
•
(l01)אא
•
J Wאא L02 = l0 + d2
E٥J ٣Fאא
C
L
d1 d2
L L 1
C
L
E٥J ٣F - ٢٩ -
אא
٢٠١
א
אא
א
(Straining Actions)אאW٤J ٣ (Normal Forces) אא(Straining Actions)אא
אא(Bending Moments)א(Shear Forces)א אKאאאאא
Kאאאאאאאאא
(N)אא אאאאאא אאאאאא (M)א (Q)א
Kאאא
Simple Slab or Simple BeamEאאFE١ Wאא(D.L + L.L )אא
•
Z(N)אא
•
Wאא(Q)א
Q = K q × wt × L
(3-1)
EJ ٦J ٣Fאא٠{٥Z (shear factor) אZK q
Uniformly Distributed Load (D.L +L.L) אאZwt Span of Slab or beam אאZL
Wאא(M)אאJ
M = w × L2 / Km
EJ ٦J ٣Fאא
(3-2)
٨Z(moment factor)אZKm
- ٣٠ -
אא
٢٠١
א
אא
א
w t/m Loads L
wL 2 K q = 0.5
Q=
S.F.D
Q B.M.D
K m= 8 2
wL M= 8
E٦J ٣F Over Hanging Slab or BeamאאE٢
א א (D.L + L.L ) א א Z(N)אא W W(Q)א
•
(D.L+L.L) אא א א א א KKKK (D.L only) אא Eא אאF אא KEJ ٧J ٣Fאא(D.L + L.L)אאאאא i.e.
Q = K q × wt × L
(3-3)
،EאF٠{٥Z (shear factor) אZK q W Uniformly Distributed Load
(D.L +L.L) אאZW t
Span of Slab or beam
אאZL
- ٣١ -
אא
٢٠١
א
אא
א
•
W(M)אא
(D.L+L.L)אאאאE K EJ ٧J ٣F א א (D.L) אא EאאF אא
W
2
2
Max. M +ve. = wt×L /8 – w × L1 /2
(3-4)
(D.L+L.L)אאאאאאZwtW (D.L. only)אאאאZw
אאZL1 אאZL
(D.L. אאאא אE
J ٧J ٣Fאא(D.L+L.L.)אEאאFאאOnly) 2
Max. M-ve = w × L1 /2
(3-5)
- ٣٢ -
WE
אא
٢٠١
א
אא
א
w1 = D.L + L.L (t/m)
w
D.L
w
Loads L
1
L
L
w1L 2
1
wL1 S.F.D
Kq = 0.5 wL1 w1L 2
M -v e
B.M.D
Case of max +ve M w1L2 8 w t/m (D.L)
w1 t/m
w1(D.L + L.L) Loads
M -ve =
w1L21 2 B.M.D
Case of max -ve M 2
wL
8
E٧J ٣F Continues Slab or Beam
E٢
א א א א א אא א Kאאאא٪٢٠א
WאאאאאW
M = w×L2 / Km
(3-7)W(M)אא
E٨J ٣Fאא(moment factor)אZKm
EffectiveSpan of Slab or beam אאאZL - ٣٣ -
אא
٢٠١
א
אא
א
Q = K q × wt × L
(3-6
(Q)א
E٩J ٣Fאא (shear factor) אZK q
(D.L+L.L)אאאאאאZwt
Effective Span of Slab or beam אאאZL
L،٪٢٠ א،אא W
Where,
L = (L1 + L2) / 2 L1 < 1.2 L2
and, wt = (wt1 + wt2) / 2 or, L2 < 1.2 L1
w t/m or w1
WL2L1א
w t/m or w2 Loads
L or L
-24
L or L
2
1
-24
-9 +11 w L2 24
( Moment factor)
+11 w L2 9
Km
w L2 24 B.M.D
w L2 11
w L2 11
אE٨J ٣F
- ٣٤ -
אא
٢٠١
א
אא
א
0.4
0.6
0.6
0.4 K q ( Shear factor)
0.6 wL 0.4 wL S.F.D 0.4 wL 0.6 wL
אE٩J ٣F .אאאאW (Q)א Q = K q × wt × L
(3-8)
•
E١٠J ٣Fאא٠ (shear factor) אZK q
(D.L+L.L)אאאאאאZw t Effective Span of Slab or beam אאאZL W(M)אא • 2
M = w×L / Km
(3-9)
E١٢J ٣FE١١J ٣Fאא(moment factor)אZKm EffectiveSpan of Slab or beam אאאZL - ٣٥ -
אא
٢٠١
א
אא
א
L ،٪٢٠ א،א א W
Where,
WL2L1א
L = (L1 + L2) / 2 L1 < 1.2 L2
0.45
and, wt = (wt1 + wt2) / 2 or, L2 < 1.2 L1
0.6
0.5
0.5
0.5
0.5
0.5
Kq
אאאאאE١٠J ٣F
-24
-10
-12 +12
+10
-12 +12
Km ( Moment factor)
אאE١١J ٣F
-24
-10 +12
-12 +16
-12 +16
Km ( Moment factor)
אאאאE١٢J ٣F - ٣٦ -
אא
٢٠١
א
אא
א
אאאW٥J ٣ אאאאE١J ٣Fאאאא
אאאKאאK
WאEאFEאאF E
EE٦J ٣FאאFאאאא •
Kאאא • Kאא E
F א א א א א • E٧J ٣אאFEא
אאא • EאFE
E١٠J ٣FאאFאאאא •
א FאאKאאאא • E١٢J ٣FE١١J ٣Fא
- ٣٧ -
אאא ã¹]<gè…‚jÖ]æ<ËÖ]<Üé×Ãj×Ö<íÚ^ÃÖ]<퉉ö¹] אאאא
א
אאא
אאא
٤
אאא אאא
٢٠١
א
א
אא א אאאEאFאאא אא
א א ،א א אא א א אא א א א א Kא א א א Kאאאאא
אא Kאאאאא E١ KEאאאFאאאאאE٢
א א א א אא א E٢ Kאא
Kאאאאאאא E٣ אאאאאאא א E٤ Kא
Kאא אאאאאאאאאאא ٪١٠٠
Kאאאא • Kאא Kאא
אאאאאא
- ٣٨ -
אאא אאא
٢٠١
א
א
W١J ٤ Wאאא،א،א Solid SlabsאאE١
Hollow Block SlabsאאE٢
Flat SlabsאאE٣
Waffle SlabsאאאאE٤
Lift SlabsאאE٥
Pre-SlabsאאE٦
אא (Solid Slabs)אאא
Kאאאאא(Working Stress Design Method)
(Solid Slabs)אא٢J ٤ א F אא א א א א א א א
(Beams)אאאאאKEאKKKKKאאא
Kא(girders)א Wאאא
One way Solid Slabs
Two Way Solid Slabs
אאא
E
א
E
J W
One way Solid Slabs אאאאאאאW١J ٢J ٤ W
אאאאאא
i.e. r = L / B ≥ 2
E١J ٤אאFאאאא
the slab is one way slab אאB،אאL،אrW
- ٣٩ -
אאא אאא
٢٠١
א
א
m 1
d
2
b d
2
L
E١J ٤F
אFאא א אאE١J ٤Fא
F אאKEאאא א
KאאאאאאאאE
א א א אא א א א א Kאאאא אאאאאא٣J ٤
Section Analysis by Working Stress Method אא(Hook’s Law) (Bernoulli theorem)
א א א א K א א
KWאאאא
(Pure Bending Moment or Pure אא א W١J ٣J ٤
Flexure) א •
(Bond slip )EFאא •
א א א א א א א א• א א - ٤٠ -
אאא אאא
٢٠١
א
א
א (Plane cross sections) א • א KE٢J ٤FאאKא fc
c
x
3 C
x
Z=d -x 3
d
T fs n
s Strain diagram
R.C sec
Stress diagram
E٢J ٤F
Wאאאאאא
d
=
k
1
M b
(4-1)
J Wאאא
EאFאאZb אאZd
(f c) א א Z k1 KE١J ٤FאאK(f s)אא
KאאאZM
- ٤١ -
אאא אאא
٢٠١
א
א
J Wאא
A
s
=
M k 2 × d
(4-2)
(M)אאאZAs אאZd
א(f c)אאZK2 KE١J ٤FאאK(f s)א
KאאאZM
Diagonal TensionאאW٢J ٣J ٤ Horizontal אאאVertical Shearאאאאא KShear
Wqאא
q =
Q 0 . 87 × bd
(4-3)
אאZq Kאאאא ZQ BondאW٣J ٣J ٤
א א א א א (Bond Stress) א
Flexure א א א K א א KאאKShear Strengthא Flexure Bondא
E
Anchorage Bondא E
- ٤٢ -
אאא אאא
٢٠١
א
א
Flexural Rigidity אW٤J ٣J ٤ ،אאא (EI) א
آﻤﺎE с א Esא DeflectionEאF -: ﻳﻠﻲ n =E s / E c = 15 אאאE אאאElastic Deformation אא (ب
אאאאאא n = 10 . אאאאאא
אE١J ٤F
Design coefficients for bending (working Stress Method)
Fs=2000
Fs=1600
Fs=1400
Fs=1200
Fs=1000
fs
Fs=2200
fc α β k1 k2 α β k1 k2 α β k1 k2 α β k1 k2 α β k1 k2 α β k1 k2
45 50 .403 .428 .866 .857 .357 .330 866 857 .360 .385 .880 .872 .374 .345 1058 1048 .325 .349 .892 .884 .391 .360 1248 1237 .297 .319 .901 .894 .408 .375 1441 1430 .273 .909 .402 1818 .254 .915 .415 2013
55 .452 .849 .308 849 .407 .864 .322 1037 .372 .876 .335 1227 .340 .887 .347 1419 .292 .803 .371 1898 .273 .909 .383 2000
60 .474 .842 .289 842 .428 .857 .301 1029 .391 .870 .313 1217 .360 .880 .324 1408 .310 .897 .346 1793 .290 .903 .357 1987
65 .494 .835 .273 835 .448 .851 .284 1021 .411 .863 .295 1208 .379 .874 .305 1398 .328 .891 .328 1782 .307 .898 .334 1976
70 .512 .829 .259 829 .467 .844 .269 1013 .429 .857 .279 1200 .396 .868 .288 1389 .344 .885 .306 1771 .323 .892 .315 1962
- ٤٣ -
75 .529 .824 .247 524 .484 .839 .256 1007 .446 .851 .265 1192 .413 .862 .274 1380 .360 .880 .290 1769 .338 .887 .298 1952
80 .545 .818 .237 818 .500 .833 .245 1000 .462 .846 .253 1185 .429 .857 .261 1371 .375 .875 .276 1750 .353 .882 .283 1940
90 .574 .809 .219 809 .529 .823 .226 988 .491 .836 .233 1171 .458 .847 .240 1367 .403 .866 .252 1731 .380 .873 .269 1921
95 .588 .804 .211 804 .543 .819 .218 983 .504 .832 .224 1165 .471 .843 .230 1349 .416 .861 .242 1723 .393 .869 .248 1912
100 .600 .800 .204 800 .555 .815 .210 978 .517 .826 .216 1162 .484 .839 .222 1342 .429 .857 .233 1715 .406 .865 .239 1903
105 .612 .796 .193 796 .567 .811 .204 973 .529 .824 .209 1154 .496 .835 .214 1335 .440 .853 .225 1706 .417 .861 .230 1894
אאא אאא
٢٠١
א
א
٥J ٣J ٤ א אKE٣J ٤Fאאאאאא WE١F K
א
Dead loadsWאאא ١٠Z(t)אא
A
One - way slab
m 2.25
L=6
m
A
2
w = 0.6 t/m = 6 KN/m Loads
i.e.
M
m
B.M.d
2.25
E٣J ٤F
Own weight of slab =1×1× t ×γ c = t × γ c = 0.1× 2.5 = 0.25 t/m2 = 2.5 KN / m2 Flooring (א) = 0.15 t / m2 = 1.5 KN / m2 Live Loads (L.L.) (א = )א0.20 t/m2 = 2.0 KN / m2 = 0.60 t / m2 = 6.0 KN / m2
Total Loads (wt) (א)א Bending Moments (א) Section A-A:
M = wt ×L2/8 = 0.6 × (2.25)2 / 8 = 0.380 m.t /m - ٤٤ -
אאא אאא
٢٠١
א
א
Assume
f c =60 kg / cm2
(f cu = 250 kg / cm2)
f s = 1400 kg/cm2
And
(mild steel 37)
i.e.
From table (4-1)
k 1= 0.313
and k 2 = 1217
Eb=100cm.F١{٠אEdFאא = 6.10 cm.
0 . 313
0 . 380 × 100000 100
d = k1
M b
=
א א K ٨ Z EtF א א א
K
I.e. take t min = 8.0 cm.
d act. = t – cover = t – (1.5: 2.0 cm) = 8.0 – 1.5 = 6.5 cm.
Wא = 4.8 cm2/m
0.380 × 10 5 1217 × 6.5
= As main = M × 10 ×d
k
5
2
Choose 7Ø10 mm/m (5.5cm2/m) Check: As min = 0.25% Ac = 0.25 / 100 ×10×100 = 2.5 cm2 i.e.
As main chosen is okay.
As secondary = 0.20 A s main = 0.20 × 4.8 = 0.96 cm2 /m Choose As secondary = 5 Ø 8 mm/m (2.51 cm2)
WאאW
א K ٨Z (t) א E١ K٨א
אK ١٠ ٧אא אאE٢ ٨אאE٥Fא
Kאא٪٠{٢٠אאE٣
٪٠{٢٥EאאFאאE٤ Kאא
- ٤٥ -
אאא אאא
٢٠١
א
א
J WE٢F
אאאאאאאא א KE٤J ٤Fאאאא ٣{٠٠ KE٢L٢٠٠ZאאFא א (One way slab) r = Loads on Slab:
L 6 = =2 b 3
assume t s=10 cm.
O.w. of slab = 0.1 ×2.5 = 0.25 t/m2 = 2.5 KN/m2 Flooring
= 0.15 t/m2 = 1.5 KN/m2
L.L.
= 0.20 t/m2 = 2.0 KN/m2
Total load wt =
= 0.60 t/m2 = 6.0 KN/m2 3.0 m
3.0 m
W d.L + L.L Loads m 3.0
m 3.0
S
S wL2 24
--v M e
+ve M
wL2 24
m 6.0 B.M.d
+ve M
loads & B.M at section s-s
m L = 3.0
E٤J ٤F
Bending Moments: 2 × l 1 0 .6 × 9 w t M +ve = 11 = 11 = 0.49 mt / m 2 × l 1 0 .6 × 9 w t M −ve = 9 = 9 = 0.6 mt / m - ٤٦ -
m L = 3.0
אאא אאא
٢٠١
א
א
(Cases of א٢L٤٠٠אא W
K(case of total loads)אאloading) Assume
And
f c =60 kg / cm2
(f cu = 250 kg / cm2)
f s = 1400 kg/cm2
(mild steel 37—24/35)
i.e.
From table (4-1)
k 1= 0.313
and
k 2 = 1217
Eb=100cm.F١{٠אEdFאא
= 7.74 cm. 0 . 313
0 . 6 × 100000 100
d = k1
M b
=
Take t = 10.0 cm d act. = t – cover = t – (1.5: 2.0 cm) = 10.0 – 1.5 = 8.5 cm.
Wאאא
0 . 49 × 10 5 M +ve ×105 2 main = = 4.74 cm /m 1217 * 8 . 5 = As1 k2 × d Choose 7 Ø 10 mm/m (5.5cm2/m) Check: As min = 0.25% Ac = 0.25/100 ×10×100 = 2.5 cm2 i.e.
As1 main chosen is okay.
As1 secondary = 0.20 As1 main = 0.20× 4.74 = 0.948 cm2 /m Choose As1 secondary = 5 Ø 8 mm/m (2.51 cm2)
Wאאא = 5.8 cm2 / m
0.6 ×105 1217× 8.5
=
As2 main =
M −ve ×105 k2 × d
Choose 8 Ø 10 mm/m (6.28cm2/m) Check: As min = 0.25% Ac = 0.25/100 ×10×100 = 2.5 cm2 i.e.
As2 main chosen is okay.
As2 secondary = 0.20 As2 main = 0.20× 5.8 = 1.16 cm2 /m Choose As2 secondary = 5 Ø 8 mm/m (2.51 cm2) - ٤٧ -
אאא אאא
٢٠١
א
א
WאאאW٦J ٣J ٤ אW Eאא Fאאא (١ Kאאא٪٠{٢٥
EאאFאא אא (٢
אאKאאE 1 F 5
KE٥F
E٪ ٢٠ F א א א א א (٣ אא ٥L١אאאא
אאא ٤L ١ אאא Kאא
אאאא (٤ אאE٥F ٢٠ K١٠א
K٢{٠١{٥א (٥ E ٣L ١F א א א א (٦ Kאאאא
K٨ (٧
E٢F א א א א (٨ KE٥J ٤Fא
- ٤٨ -
אאא אאא
٢٠١
א
א
E٥J ٤F אW
אאאא٢L٤٠٠אאא Kאאאא
E١
Kא
E٣
Kא
W
E٢
אW
K ١٠א א
K١٥א אאWא
(Deflection)Fא אאאאא t min.. =L/30
WEא
א E١
t min.. =L/35 אא E٢
t min.. =L/40
א E٣
KאאאאאאL - ٤٩ -
אאא אאא
٢٠١
א
א
אW ١{٠٥ א א א א א KאאKאא
Tow Way SlabsאאאאאW٤J ٤ W١J ٤J ٤
אאאא،אאאאא K?٢?אאאאא،א i.e. L / b ≥ 1.0 ≤ 2.0 אZbאZL
J ٤FאאKאאאא
KאאאאאKE٦
אאאאאא אאאKא
אא אאאא
Kאאאאא אאא Wאאא
אאאE٦J ٤F
KL/b אE KאאאאאאאE
αEאאFאאא
βEאאFאאא - ٥٠ -
אאא אאא
٢٠١
א
א
אאאאW٢J ٤J ٤ Wאאאא t min.. =b/35 א E
t min. =b/45 אא E
Kb
אאאאאאW٣J ٤J ٤ אאאאאאאאא
א Kא א א אא א א א
KאאK(b)אא(L)א EאFאאאZb W
EאFאאאZL
א א א א א Z m b
Kbא(b)אא א א א א א Z m L KLא(L)אא
Kאm L m b
Wm L m bאאא m b = m L = 1.0Wא
m b =m L = 0.87Wא
m b =m L = 0.76W א א (r) א א א א (4-4)
r=
m ×l m ×b l
Wאא
b
א א (β) (α)אE٢J ٤Fא
Kאrאאא(L)א(b)אא - ٥١ -
אאא אאא
٢٠١
א
א
rא(β)(α)אE٢J ٤F r α β
אאEאFאא
1.0 0.35 0.35
1.1 0.40 0.29
1.2 0.45 0.25
1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.21 0.18 0.16 0.14 0.12 0.11 0.09 0.08 β = 0.35/ r2W
Wאאא
W α =Wt × α(b)אאא W β =Wt × β(L)אאא
EWL.LאאWD.LאאאFאאwt
Wאא
Wאא(Bending Moments)א Wאאאאא E١
×b2
M b = ± wα (4-5)אאא 10
×L
2
M L = ± wβ (4-6) 10
אאא
WאאאאאE٢
×b2
M b = ± wα 12 M L = ± w β
×L
12
(4-7)
אאא
2
אאא
(4-8)
- ٥٢ -
אאא אאא
٢٠١
א
א
אאאאW٤J ٤J ٤ אאא אאא אאKE٣J ٤F
EאאאאFאאא
אFא אא א
א א א א א E א אאE٢J ٤Fאאאא אא
K Rigid Beams א א א אא א K٢٠א١٥א
EFrא(β)(α)אE٣J ٤F r α β
1.0 .396 .396
1.1 .473 .323
אאא 1.2 .542 .262
1.3 .606 .212
1.4 .660 .172
1.5 .706 .140
1.6 .746 .113
1.7 .778 .093
1.8 .806 .077
1.9 2.0 .830 .849 .063 .053
אאא٥J ٤J ٤ J WE١F
אאאאאאאא א א K E٧J ٤F ٤{٥
KE٢L٢٠٠ZאאF א (Two way slab) r =
٤{٥Zאא
m × l = 0.87 × 4.5 = 1.0 m × b 0.87 × 4.5 l
b
From table (4-1)
α = β =0.35 Slab is continues
- ٥٣ -
i.e. t = 450/ 45 =10 cm
אאא אאא
٢٠١
א
א
4.50 m
4.50 m
0.40 m
4.50
m
1
1 0.20 m
0.40 m
4.50
m
0.20 m 0.40 m
E٧J ٤F Loads on Slab:
O.w. of slab = 0.1 ×2.5 = 0.25 t/m2 = 2.5 KN/m2 Flooring
= 0.15 t/m2 = 1.5 KN/m2
L.L.
= 0.20 t/m2 = 2.0 KN/m2
Total load wt =
= 0.60 t/m2 = 6.0 KN/m2
W α =Wt × α= Load in short direction = 0.6 × 0.35 = 0.21t/m2 Wt × β = Load in long direction = 0.6 × 0.35 = 0.21 t/m2=W β
- ٥٤ -
אאא אאא
٢٠١
א
א
Bending Moments:
WKאא
Mα Mβ
+ ve
=
+ ve
=
Mα
− ve
Mβ
− ve
× = wα l =
2 1
10 2 wβ × l 2
0.21 × 4.5 2 = = 0.425 mt / m 10 =
10
0 .21 × 4 .5 2 = 0 .425 mt / m 10
(Cases of א ٢L ٤٠٠ א א W
K(Case of total loads)אאloading) Assume f c =60 kg / cm2 f s = 1400 kg/cm2
And
(f cu = 250 kg/cm2)
(mild steel 37—24/35) i.e. From table (4-1)
k1= 0.313
and
k 2 = 1217
Eb=100cm.F١{٠אEdFאא = 6.45 cm. 0 . 313
0 . 425 × 100000 100
M b
d = k1
=
Take t = 10.0 cm d act. = t – cover = t – (1.5: 2.0 cm) = 10.0 – 1.5 = 8.5 cm.
Wאאא 0.425 × 10 5 = 4.11 cm /m 1217 × 8.5 2
= A main = M α k
+ ve
sL
2
× 10 5
×d
Choose 6 Ø 10 mm/m (4.71cm2/m) 0.425 × 10 5 = 4.65 cm /m = 1217 × 7.5 2
Mβ A Secondary = k
+ ve
sl
2
× 10 5
×d
Choose 6 Ø 10 mm/m (4.71cm2/m)
Wאאא = 4.11 cm2/m
0.425 × 10 5 = 1217 × 8.5
Asu main =
Mβ
− ve
× 10 5
k2 × d
Choose 6 Ø 10 mm/m (4.71cm2/m) - ٥٥ -
אאא אאא
٢٠١
א
א
0.425 × 10 = 4.65 cm /m = 1217 × 7.5 2
A
su
Mβ sec ondary =
− ve
× 10 5
k2 × d
Choose 6 Ø 10 mm/m (4.71cm2/m)
3
10 m m m
3
10 m m m
10 mm m
10 mm m
3 3
4.50
10 m m m
m 1
10 m m m
4.50
m
3
3
1
3
3
10 mm m
10 mm m
E٨J ٤Fאא
4.50 m
4.50 m
6
3 +3
3 +3
10 m m m 10 m m m
10 m m m
10 m m m 10 m m m
S ec ( 1-1 )
אE٨J ٤F
J WE٢F
א א ٣{٥Zאאאאאא
KE٩J ٤Fא
א
٣{٥Zאא
(Two way slab) r =
m × l = 1.0 × 3.5 = 1 .0 m × b 1.0 × 3.5 l
b
- ٥٦ -
אאא אאא
٢٠١
א
א
א٢٠אאא
E٣J ٤Fא From table (4-3)
α = β =0.396
Slab is Simple
i.e. t = 350/ 35 =10 cm Loads on Slab:
O.w. of slab = 0.1 × 2.5 = 0.25 t/m2 = 2.5 KN/m2 = 0.15 t/m2 = 1.5 KN/m2
Flooring L.L.
= 0.20 t/m2 = 2.0 KN/m2
Total load wt =
= 0.60 t/m2 = 6.0 KN/m2
W α =W β= 0.6 × 0.396 = 0.21t/m2
Bending Moments:
WKEFאאא
Mα
+ ve
=
Mβ
+ ve
× = wα l 8
2 1
0.2376 × 3.5 2 = = 0.364 mt / m 8 (f cu = 250 kg/cm2) Assume f c =60 kg / cm2 2 And f s = 1400 kg/cm (mild steel 37—24/35) i.e. From table (4-1) k1= 0.313 and k2 = 1217
Eb=100cm.F١{٠אEdFאא = 6.0 cm. 0 . 313
0 . 367 × 10 100
5
M b
d = k1
=
Take t = 10.0 cm d act. = t – cover = t – (1.5: 2.0 cm) = 10.0 – 1.5 = 8.5 cm.
= 3.52 cm2/m
Wא 0.364 × 10 5 1217 × 8.5
× 10 5
= AsL main = M α +ve *d
k
2
Choose 7 Ø 8 mm/m (3.52cm2/m) 0.364 × 10 5 = 3.98 cm /m = 1217 × 7.5 2
Mβ A Secondary = k
+ ve
sl
2
× 10 5
×d
Choose 8 Ø 8 mm/m (4.02cm2/m) - ٥٧ -
אאא אאא
٢٠١
א
א
א KE β Fא א א E α Fא א א א Kאאאאא
KאאאאאאW i.e. A s α. ≥ A s β. 2
10 mm m
8 m
0.2
5
m
8 mm m
4
8 mm / m
Concrete seat
4
8 m
2
m 3.50
10 mm m 8
8 mm m
m 3.50
Sec ( i - i )
i
8
i
E٩J ٤F
אאאא: ٥J ٤ אאאאE١ KאאאאK٢٠
K(Ø=8mm)٨אE٢ E٤L١FאאאאאE٣ אE٥FאKאא
K٢{٠W١{٥אאE٤K Kאאאא
KאאאאאאE٥
- ٥٨ -
אאא אאא
٢٠١
א
א
אאאאאאאאאאW٦J ٤ EאאאאFאאאאא אא (Flexure)אא
Kאאאאא
K(No need to check shear stress)אא EDeflection in SlabsFאאW٧J ٤ KאאאאאW אאא אאאאא אא Kאא א א א א א א
אא١٩٨٥ BS8110 CP110אאא ACIא
אאאKאאא ١٩٩٥א
א E٤J ٤F א א (ACI-318-71) ١٩٧١ Kאאאאאאא
(L/d)אאאE٤J ٤F ACIאאאאאאא
א
א
א
א
א (Fy kg/cm2) ٢٨٠٠ אאא
١٢{٥
٣٥
٣٠
٢٥
١٠
٢٦
٢٣
٢٠
٢٨٠٠
אא
١١
٣١
٢٧
٢٢
٣٥٠٠
אאא
٩
٢٣{٥
٢٠{٥
١٨
٣٥٠٠
אא
١٠
٢٨
٢٤
٢٠
٤٢٠٠
אאא
٨
٢١
١٨{٥
١٦
٤٢٠٠
אא
אא
אא
אא
١٠{٠٠אא (L/d)א אאE٥J ٤Fא Kאאאא - ٥٩ -
אאא אאא
٢٠١
א
א
(L/d)אאאE٥J ٤F E.C.O.Pאאאאאאא א
א
א
א (Fy kg/cm2)
٣٥
٣٠
٢٥
אא
٢٦
٢٣
١٨
אאא
٢٨
٢٤
٢٠
٢١
١٧
١٤
٢٨٠٠J ٢٤٠٠
א
٢٨٠٠J ٢٤٠٠
אא
אא
א
٤٢٠٠J ٣٥٠٠
אאא אא
אאאW אאא(ACI-318-83)K١٩٨٣אאא Kאאאא
אאאאא א א א K ٣٥J ٣٣ –א K ١٩٨٣ ACI-318-83 Kאאא
E١٢F١٢٠(Drop panels)אאאא E٢J ٣J ٥J ٩Fאא E١٠F١٠٠אא
٢{٠٠אאא אאאא E٩F٩٠
- ٦٠ -
E١
E٢ E٣
אאא אאא
٢٠١
א
א
אאא ١٠א ١٠Zא
١٠ Ø٦Z א
٨J ٤ E١
KL
١٢א ١٢Zא ١٢ Ø٧Z א
KL
١٤א ١٤Zא
E٢
E٣
١٤ ø٨Z א
KL
١٦א ١٦Zא
١٦ Ø٩Z א
E٤
KL١٠Ø٥אKL
א٩J ٤ אK ٢٠א אEאFE١ WאK٩{٠٤{٢Z
KאאE
KאאאאE KאאE
א אE١٠J ٤FאE١FאאאE٢
אK אK ٦{٥ ٤{٥ZאK ٢٠ WאK٢{٠Zאאא١٢Zא
KאאאE
KאאאאE
KאאE - ٦١ -
אאא אאא
٢٠١
א
א
6.50 m
6.50 m
1 0.2
m
4.50
0.2
m
4.50
6.50 m
m
m
6.50 m
E١٠J ٤F
KE١١J ٤F א א E٣J ٢J ١F א א א E٣ WאK٢٠אאאאא
KאE٣،٢،١FאE KאאאE
א א א E Kאא
- ٦٢ -
אאא אאא
٢٠١
א
א
5.00 m
5.00
m
1
5.00 m
6.00 m
0.2
m
2
0.2
m
5.00
0.2
6.00
6.00
5.00
6.00 m
E١١J ٤F
- ٦٣ -
m
m
m
5.00 m
m
3
m
0.2
5.00
m
5.00 m
m
אאא ã¹]<gè…‚jÖ]æ<ËÖ]<Üé×Ãj×Ö<íÚ^ÃÖ]<퉉ö¹] אאאא
א
אאאא
אאאא
٥
אא אאאא
٢٠١
א
א
Wאא אאאאאאאאאא
א א א א א ،א א
א אאאא אאאKא
Kאאאא
אא אKאאאאאא E١ א א א א א א א א E٢ Kאאאאא
אא אא א א א א E٣ Kא
אאאא אאאא א E٤
Kא
Kאא אאאאאאאאאאא ٪١٠٠
Kאאאא Kאא Kאא
אאאאאא
- ٦٤ -
אא אאאא
٢٠١
א
א
WW١J ٥ אKאאאאא KKK א א א א א א KאEאFאאאא
WאאאאW٢J ٥
Wאאאאאא
א אאאאא E١ אא א K א א
K٢٠٢٠WKKא
KE١J ٥FאF٢٥٢٥
T 12 cm b
12 cm b
b< T
20 cm
25 cm
12 cm b
25 cm b< 25 cm
20 cm b< 20 cm
E١J ٥F
א (Lintling)אאאא E٢ אאKאא
Kאאאא
אאאאאאא E٣ K
א אא אא א E٤ Kאאא - ٦٥ -
אא אאאא
٢٠١
א
א
(To reduce buckling length of columns)א E٥ Kאאאאאאאא
אאאאאאW٣J ٥
Wאאאא Main girders אאא
E١
Secondary beamsאאא
E٢
Eא Fא אאאא F א אא א א א אא KE٢J ٥FאאEאאא 4.00 m
4.00 m
Beam
0.3
0.2
Seconadry
m Seconadry
3.00
m
Beam Main
0.3
Beam m
m
3.00
E٢J ٥F
- ٦٦ -
m
אא אאאא
٢٠١
א
א
Fא א אא KEא אא א
Kא
K E٣J ٥F א א א א ،א
א (G1) א א (B1→B2→B3) א
KC4אB1אאאאK 4.00
4.00
4
G1 C3
C4 3 3.00
m 2
C5
B2 1
3.00
m
B3 B1
C2
C1
E٣J ٥F
א KE٣J ٥F א א אא א
אאאאאאאG1א
C1אא G1אאאB3،B2،B1
KאאאC4،C3،C2،
KאאאאW١J ٣J ٥ א Eא א א אא א F א E١ (Embedded beams) א (Dropped beams)א א
Kאא - ٦٧ -
אא אאאא
٢٠١
א
א
אא (Loop of loading)E ٢ Kאאאא אאא א א א א
Kאאאאאאא
Eא FאEFאEJ ٤J ٥Fא
א aאא א d c b a א Kאא
א א א א א א EJ ٤J ٥F א aאאEJ ٤J ٥FאאKEJ ٤J ٥F אKא א Dא e d c b Kאאא
m
m D
m S
m D
S
m D
m D
m
b
c
a
d
S
m
b
c
a
d
S
m S
m D
m D
e m D
m
D
m D
S
E٤J ٥F
- ٦٨ -
m S
m S
אא אאאא
٢٠١
א
א
א Eא אF א א
E٣
J ٥F،EJ ٥J ٥FאKEJ ٥J ٥Fאאאאא
KEJ ٥J ٥Fאאאא،EJ ٥ X m
X m
ks ac Cr
Y
m
Y
Cr ac ks
X
m
Y
m
m
E٥J ٥F
אאאKW٤J ٥ א א אא Wאא
- ٦٩ -
אא אאאא
٢٠١
א
א
Wאא (Uniformly distributed load) א א א אא KE١J ٦J ٥FאאK٥{٠אא
Loads L
b< 5.00 m
E١J ٦J ٥F
Wאא K א א א א אא KE٢J ٦J ٥FK٧،٠אאא
- ٧٠ -
אא אאאא
٢٠١
א
א
w t/m
Loads
L
b<7.00 m L
E٢J ٦J ٥F
Wאא אאאאא אא١٠{٠אאK
KE٣J ٦J ٥F
P
w t/m Loads
L
b< 7.00 m
E٣J ٦J ٥F - ٧١ -
אא אאאא
٢٠١
א
א
Wאאא אאאאאא K٧{٠
Wאא אאאאא
K٨{٠
Wאא א אאאא K
Wאא F א א אא א Kא אא א א א א א Eא KE٤J ٦J ٥Fא KאאאאאאאאW
w t/m
w t/m
E٤J ٦J ٥F - ٧٢ -
אא אאאא
٢٠١
א
א
Kאאאא :٥-٥
אאאEאאFאאא
אאאאא אאאאא
א א א א K אאא
אאאאKאאא
אאא E٤J ١Fאא Allowable working stress אאאאאאאא
Kאאאאא
KאאאW١J ٥J ٥
Wאאאא
א א א Kאא א (Slabs) א
(١
EJ ١J ٢J ٢J ٢Fאא
EJ ٢J ٢J ٢J ٢FאאEאFאאא
(٢
J ٢J ٢Fאא EאFאא א אא
(٣
EJ ٢J ٢
KאאאW٢J ٥J ٥ א א א אא א א א א E٧J ٥F אא
WאKאאאאאאא
- ٧٣ -
אא אאאא
٢٠١
א
א
L Y
X
45°
Beam A
2X
Beam B
KאאאאE٧J ٥F EF א K EF א K
א א אאא KE١J ٥Fאאאאאא
E١J ٥F
אאאאאאאβ،αא
L/2x 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 α 0.667 0.725 0.769 0.803 0.829 0.853 0.870 0.885 0.897 0.908 0.917 β 0.500 0.554 0.582 0.615 0.642 0.667 0.688 0.706 0.722 0.637 0.750
KאאאאאZW KאאZL
KZX - ٧٤ -
אא אאאא
٢٠١
א
א
W (Equivalent coefficient for moment)אאZα (Equivalent coefficient for shear)אאZβ
W
EאאאFאאאZα ×w ×x אאאאא Kאא
EאאאFאאאZβ ×w ×x א א F אא א א KEαאאא
KE٢J ٥FאאאאW
E٢J ٥F
0.5 0.667
L/2 L
1
0.5
0.5
2
=
From table
3
0.5 0.667
4
0.5 0.333
5
=
0.5
6
=
From table
7
Take = 1/2x for one trabizoidal
- ٧٥ -
אא אאאא
٢٠١
א
א
KEאאאE،،٣J ٣FאאFאאאW٣J ٥J ٥ KEאאE٤J ٣FאאFאאאאW٤J ٥J ٥
KEאאE٥J ٣FאאFאאאW٥J ٥J ٥ E١J ٥F אאאא E٨J ٥Fא
אK אאEJ ٨J ٥FאKEJ ٨J ٥F
W
KE٣L١{٢אF٢٠אאאא
WאKEJ ٨J ٥FאאK٣{٠Zאאא
KG1אאB2،B1אאאא
Kאאאאאא 1
2
3
4
A
A
m 6.00
B
6.00
B m
C
C 5.00 m
5.00m
4.00 m
1
2
3
EJ ٨J ٥F
- ٧٦ -
4
E١
E٢
אא אאאא
٢٠١
א
א
1
2
4
3 G2
A
A S2 m 6.00 B2
S1
S2
B B1
S2
B
1
B1
0.4 m
S1
5.00 m
G2 4.00 m
C
2
C 5.00 m 3
EJ ٨J ٥F
h2 ( clear height )
EJ ٨J ٥F - ٧٧ -
B2
S2
B2
m
G1
m B2 6.00
0.2
4
אא אאאא
٢٠١
א
א
א
G1،B2،B1אאאW
E٩J ٥FאאB1אא 2.00
G2
2.50
2.00
G1
G2
2.50
6.00 m
6.00 m
B1 w t t/m G1
G2
G2
B1 E٩J ٥F
Wt on beam B1 = loads from slab + o. w. of beam Loads on slab:
Assume t s = 12 cm = 0.12 m O.W. of slab = 0.12 × 2.5 = 0.30 t/m2 = 3.0 KN / m2 O.W. of flooring
= 0.15 t/m2 = 1.5 KN / m2
L.L
= 0.30 t/m2 = 3.0 KN / m2
Total load on slab
= 0.75 t/m2 = 7.5 KN / m2 Slab s1
is 4 × 6 m
r1 = 6/4 = 1.5 α1 = 0.853
and
β1 = 0.667 (from table 5-1) Slab s2
is 5 × 6 m
r2 = 6/5 = 1.2 α1 = 0.769 - ٧٨ -
and
β1 = 0.582 (from table 5-1)
אא אאאא
٢٠١
א
א
Beam B1 has two equal span = 6.0 m Assume And
b =20 cm
t = span / 10 = 60 cm
W
WאB1אא(working loads)א W α = total working load for moment
= 0.853 × 0.75 × 4/2 + 0.769 × 0.75 × 5/2 + 0.2 (0.6 – 0.12) × 2.5 = 2.96 t/m
WאB1אא(working loads)א W β = total working load for shear
= 0.667 × 0.75 × 4/2 + 0.582 × 0.75 × 5/2 + 0.2 (0.6 – 0.12) × 2.5 = 2.33 t/m
KאB1אאאאאW Kאאא B2אא
KאG1(Main girder)אEאFא
E١٠J ٥FאאB2אא
2.50
2.50 col
col
col 6.00 m
6.00 m B2 w t/m t col
col B2
E١٠J ٥F - ٧٩ -
col
אא אאאא
٢٠١
א
א
Loads on slab: Assume t s = 12 cm = 0.12 m O.W. of slab = 0.12 × 2.5 = 0.30 t/m2 O.W. of flooring
= 0.15 t/m2
L.L
= 0.30 t/m2
Total load on slab
= 0.75 t/m2
Slab s2
is 5 × 6 m
r2 = 6/5 = 1.2 α1 = 0.769
and
β1 = 0.582
Beam B2 has two equal span = 6.0 m Assume And
b =20 cm
t = span / 10 = 60 cm Loads from wall:
= 3.0 – 0.6 = 2.4 mאאH wall = W wall = γ b × t w + o.w. of plaster = 1.2× 0.2 + 0.05 = 0.29 t/m2
W
WEא)B2אא(working loads)א W α = total working load for moment
= 0.769 × 0.75 × 5/2 + 0.2 (0.6 – 0.12) × 2.5 + 0.29 × 2.4 = 2.38 t/m
WEאFB2אא(working loads)א W β = total working load for shear = 0.582 × 0.75 × 5/2 + 0.2 (0.6 – 0.12) × 2.5 + 0.29 × 2.4 = 2.38 t/m
- ٨٠ -
אא אאאא
٢٠١
א
א
KE١١J ٥FאאG1אא
R
R B1
2.50
B1
2.00
2.50
col
2.50 2.50
2.00
5.00 m
4.00 m
5.00 m
G1
E١١J ٥F
٠{٤Z٤٠ZG1א
W٠{٩Z٩٠Z
KאאLאZאאאא
Kאאא
אא W (אא ) אאKא
E٢J ٥FאאK (α = β = 0.5). ﺣﻴﺚ.W β
Wt = total working load on G1 = ∑ loading area / span + o.w. of girder i.e. W α = W β = ( 2 × 0.75 × 0.5 × 4 × 4/2 + 2 × 2 × 0.75 × 0.5 × 5 × 5/2 ) / 14 + 0.4 × (0.9 – 0.12) × 2.5 = 1.77 + 0.78 = 2.55 t/m - ٨١ -
אא אאאא
٢٠١
א
א
WאG1אאB1אאא B1 א (working reaction) א א א א א
E١
، א EאF א אא א אא
E٢
KRB1G1אא
(load for א א ، א א
WKKKKKKKKshear coefficient) (w β)
RB1 = R w + R m KאאאZR wW
KB1אאאZR m
KB1אאאאE١٢J ٥Fאא
- ٨٢ -
אא אאאא
٢٠١
א
א
w t/m or w1 Beam B1 Loads
w = 2.33 t/m
G1
G2
G2
6.00 m
6.00 m
w L2 2.33 * 6 * 6 = 10.485 m.t M= = 8 8
B.M.d G1
G2
10.485
Reaction 10.485 6
10.485 10.485 *2 = Rm 6 6 6.00 m
G2
10.485
6.00 m
E١٢J ٥F
Wאא
R w = (wβ×L/2) ×2 = (2.33 ×6/2) ×2 = 13.98 ton
R m = (M - ve /L) ×2 = (wβ×L2/8L) ×2 = (2.33 ×6×6 / (8×6))×2 = 3.495 ton i.e. RB1 = 3.495 + 13.98 = 17.5 ton
KE١٣J ٥FG1אאאא
- ٨٣ -
אא אאאא
R
٢٠١
א
א
= 17.5 t B1 w = 2.55 t/m
R B1
= 17.5 t
col
col 5.0
4.00 m
5.0
14.00 m
E١٣J ٥F E١J ٥Fא G1،B2،B1אאאאW E١٤J ٥FאאEאFB1א
E١
E١٥J ٥F
2.33 t/m Load for shear 6.00 m
0.4
6.00 m
0.6 0.6
0.4
K ( Shear factor) q
8.4 t 5.6 t S.F.D Q=K * w L q
B
5.6 t 8.4 t
E١٤J ٥F
- ٨٤ -
אא אאאא
٢٠١
א
א
2.96 t/m
Load for moment
6.00 m
-24
-9 +11
-4.44
6.00 m
-24 K m ( moment factor)
+11
2 w L= 11.84 9
- 4.44
9.7
2 B.M.d M= wL /Km
9.7
E١٥J ٥F
WEאFB2אE٢
W β (for shear) = 1.74 t/m W α (for moment) = 2.22 t/m
KB2אאאאאW KE١٦J ٥FאאG1 אE٣
- ٨٥ -
אא אאאא
٢٠١
א
א
17.5 t 2.55 t/m
Loads 35.35 t
17.5 t
5.00 m
4.00 m
5.00 m
35.35 t
35.35 22.6 5.1 S.F.D 5.1 22.6 35.35 B.M.d
144.88
8 mt
w L2 = 8 mt 8 Mmax= 144.88 + 5.1 =150 mt
5.1
E١٦J ٥F
Design of R.C. Sections of BeamsאאאW٦J ٥
א א א א א א א א Kאא א א K א
אאKאאאא
KאאאאאאE١٧J ٥F - ٨٦ -
אא אאאא
٢٠١
א
א
E١٧J ٥F
א(Deflection)אA-BאE١٨J ٥Fא א א א K א א
WEB-BאFאאEA-AאF
E١٨J ٥F - ٨٧ -
אא אאאא
٢٠١
א
א
אאאאא
E١
א E١٩J ٥F א א KE١٩J ٥F E١٩J ٥F א
Kאאאא (Neutral Axis) אאאKאא
Tא E١٩J ٥Fא אאאאא
E٢
KE١٩J ٥Fאא
b
B-B
A-A
A-A T
E١٩J ٥F
Wאאאאאא
،E١٩J ٥FאאFאאא E EEF،(F
אא F T אא (T-shaped) T E E١٩J ٥ א (L-shaped)Lאא W K(L-Section)>L
- ٨٨ -
אא אאאא
٢٠١
א
א
אאאאW١J ٦J ٥ אאאאאאאא
Wאאאא
אא (Linear) א
KE٢٠J ٥F
E١
ft
d
+
d
fc Stress diagram
Strain diagram
E٢٠J ٥F אאאא אאאאאאאא
א f
c
E٢
א א
Kf sאאאא
E١J ٥J ٥F א א א א
א (M) א E٤J ٥J ٥F ،E٣J ٥J ٥F ،E٢J ٥J ٥F، Kאא(Q)אאא Wאאdאא - ٨٩ -
E٣
E٤
אא אאאא
d
=
k
1
٢٠١
א
א
M b
(4-1) W
אא(Depth of R.C. Sec.)אאאZd EE٢٠J ٥F&E١٩J ٥FאאF
(f c) א א Z k1 KאאאאE١J ٤Fאא(f s)אא
KאאאZB
f s,bאf c f cuE٥ WאW f cu = 250 kg/cm2
f c = 80 kg/cm2
,
So, k1 = 0.253
and f s = 1400 kg/cm2
,
k 2= 1185
E١J ٤Fאאאdאא
E٦
WאאאAsא
E٧
(4-2)
A
s
=
k
M × d 2
From figure (5-21)
t = d +d′ Where; d′ = 3 + Ø /2 + (2.5 + Ø /2) × (n-1)
KאZnW
W٢٠ZØ،אZ(n)אא d′ = 3 + Ø /2 ≈ 4 cm. KE٢١J ٥FאאKא
- ٩٠ -
אא אאאא
٢٠١
א
א
fc
t
d
+
d
fs / n
b
E٢١J ٥F
EאאאאF LTאאאW٢J ٦J ٥ (Flanged Beam) LאTאIא∏אKאאא KE٢٢J ٥אFLאTאאאא
א א א אא Kאאאא
א K T א א א א א (Floor)Kא אא KLאא
אא אא (B)אא WLאT
- ٩١ -
אא אאאא
٢٠١
א
א
WאאאאאBאא T B = 12 ts + b Or
B = L/3
(for simple beams)
Or
B = L/4.5 (for continuous beams)
E
(4-3)
LE
B = 4.5 t s + b Or
B = L/6 (for simple beams)
(4-4)
Or
B = L/9 (for continuous beams)
t sK٨KאZ LKאאZ
M
M B
B ts
d
ts d
As
As
bO T - section
bO L - section
E٢٢J ٥F
אאא(Maximum B)אאW
KE٢٣J ٥FK
- ٩٢ -
אא אאאא
٢٠١
א
א
C
L
C
L
S
S
S
6.30 L sec
m
T sec bo
L sec
C
L
C
L
E٢٣J ٥F
LTא
J WT EאFא (Z)א KאאE٢٤J ٥F
EאFאא (Z)א
KTאאKE٢٤J ٥F
- ٩٣ -
E١ E٢
אא אאאא
٢٠١
א
א
B
B z
ts
ts
z d
t
t
bO
bO
E٢٤J ٥
Wאא (neutral axis distance (z))אא
Z = 0 . 14
M B
(4-5)
אאאא BK Bא (r)א
WKאאBr
(4-5)
Br= r × B
B/b0&ts/zאK(r)אE٣J ٥Fא
- ٩٤ -
אא אאאא
٢٠١
א
א
(Br=r.B) (r)אE٣J ٥F B/b0 2.0 2.5 3.0 3.5 4.0 5.0
1.0 1.0 1.0 1.0 1.0 1.0 1.0
0.9 1.0 .99 .99 .99 .99 .99
0.8 .98 .98 .97 .97 0.97 0.97
0.7 .96 .95 .94 .94 .93 .93
ts/z 0.6 .92 .90 .89 .89 .88 .87
0.5 .88 .85 .83 .82 .81 .80
0.4 .82 .78 .76 .74 .73 .71
0.3 .76 .70 .67 .65 .63 .61
0.2 .68 .62 .57 .54 .52 .49
Wאא d =
k
M 1
B
r
:k1אW f c′= 0.75fc
′
T = d + d
Kאאא f cW
K(t)אא
WאTאאא
A
s
=
k
M × d 2
Kf s , (f c′)אE١J ٤Fאk2W
אא (shear stress)אא (Check)א Kאא
- ٩٥ -
אא אאאא
٢٠١
א
א
Wא אLTאאE١ Kאאא
אא (L)אE٢ אאאEאאFא Kאאא٪١٥
Shear stressאW٣J ٦J ٥
(q) א א א א (shear forces (Q)) א
אא (Diagonal Tension) (T) (shear stress) KE٢٥J ٥Fאא
Wאאqא
q =
Q 0 . 87 × b × d
Kg / cm2
T q q
q q
T E٢٥J ٥F
Wא - ٩٦ -
אא אאאא
٢٠١
א
א
K٢L٧אאא E١ ١٨ E F E٢ KאאאאאHאאאK٢L
.אאאאאW٤J ٦J ٥
אאאא אאאא
Wא J ٥ ١٤J ٥Fאא K(Shear force diagram) (S.F.D)א E١ KE١٦
WאאאאאE٢
q
max
=
Q 0 . 87 × b × d
q maxK (Linearity)א Q q W K(Face of column)אא
K(7kg/cm2)אאאאאאE٣ KE٢٦J ٥Fאאא
- ٩٧ -
אא אאאא
٢٠١
א
א
Q Q
By concrete
q
S.F.D
st
7 kg/cm q = 1: 1 q st 3 2 max
Ab
q max
E٢٦J ٥F
Kq max½⅓אאאאאאE٤
n × Ast × b× s
f
s
I.e.
q stirrups =(⅓: ½) q max =
K(No. of branches of stirrups)אאאZnW K(Area of bar of one branch of stirrups)אZA st
(A st = 0.503 cm2)W٨א
K٢L١٤٠٠אאZf s (breadth of beam)א Zb KאZs
- ٩٨ -
אא אאאא
٢٠١
א
א
Wאא(A sb)אאE٥
A
sb
A
=
b
f
Where;
× b s
A b = area from diagram of shear stress.
אאW٥J ٦J ٥ אאE١J ٥FאאB1א א
KE١٥J ٥&١٤J ٥Fא
Design data: L= Effective Span =6.0 m Q max. +ve = 8.4
ton
Q max. – ve = 5.6
ton
M max. –ve = 11.84 t.m. M max. +ve = 9.7
t.m. kg/cm2
Assume f cu = 250 f c = 90
kg/cm2
f s =1400
kg/cm2
Therefore: Q Design =
8.4
ton
At Sec. 2-2 M –ve =
11.84 t.m.
At Sec. 1-1 M +ve =
9.7
t.m.
From table k1 = 0.233 k2 = 1171 Design of Sec. 2-2
(Slab lies in Tension zone)
So design as a rectangular Sec. - ٩٩ -
אא אאאא
٢٠١
א
א
d =
k
1
M
× 10
2
5
= 0 . 233
b
11 . 84 × 10
5
20
= 56 . 7 cm
Take:
t = 60 cm d act. = 56 cm
A
s
=
M × 10 k ×d
5
=
2
2
act
11 . 84 × 10 1171 × 56
5
= 18 . 06
cm
2
Take 6 Ø 20 mm (18.8 cm2) Check on section 1-1 (slab is in compression zone) So design as a T section B (breadth of flange) is taken the least of: B = 12 t s + b = 12 × 10 + 20 = 140 cm Or B = L/4.5 = 600/4.5 = 133.33 cm Or B = ¢ :¢ =450 cm I.e. B =133.33 cm Z = 0 .14
M
+ ve
B
9 . 7 × 10
= 0 .14
133 . 33
6
= 11 .94 cm
And
Z > t s > 10 cm So: the sec. is actually T sec. f c′ = 0.75 f c = 0.75× 90 = 67.5 kg/cm2 Take f c′ = 70 kg/cm2 I.e.
k1= 0.279 k2= 1200 d act.= 56 cm
As at sec .1 = Take 5 Ø 20 mm (15.7 cm2)
- ١٠٠ -
M × 10 k ×d
5
1
2
act
=
9 .7 × 10 5 = 14 .43 cm 2 1200 × 56
אא אאאא
٢٠١
א
א
8.4 t
sec 2
q
255 q
max
8.62 kg/cm
Ab
st
45°
3.6 m
x 2.4 m
5.6 t
7 kg/cm
E٢٧J ٥ Check of shear:
> 7 kg/cm2
q
2 max .
=
8 . 4 × 10 3 = 8 . 62 kg / cm 2 0 . 87 × 20 × 56
Take stirrups 2 branches 5 Ø 8 mm/m
q
st
=
n × Ast × b× s
f
s
=
2 × 0.503×1400 1 1 = 3.52 = ( : ) q 20× 20 3 2 2 max.
A
bent
=
A ×b = × b A f f
= (45 ×
b
b
s
s
8.62 − 3.52 + 7 − 3.52 20 2 )× = 2.28 cm 2 1400
Take 2 Ø 20 mm (6.28 cm2 for more safety)
- ١٠١ -
אא אאאא
٢٠١
א
א
KB1אאE٢٨J ٥Fאא
B1 B1אE٢٨J ٥F KאאאאאאW٧J ٥ WאאE١ K٣٥٠H٣٠{٤H٣٠{٨ KאאאאאאאE٢
א ٢٨א א א א E٣ ٢L٢٥٠
K٢٤אאאE٤ אF ٣٥L٢٤ אאאאE٥ KEא
Kאא٤L١א٥L١אאאE٦ Kאא٧L١אאאE٧
K٣٥אאאE٨
אEFאאאאאאא E٩ Kאא٤L١אא
- ١٠٢ -
אא אאאא
٢٠١
א
א
אאאאE١٠
E٣٥L٢٤Fא
אא
אא
L٨Ø٥
١٦Ø٢
J
١٦Ø٢
١
L٨Ø٥
١٨Ø٢
٢٠Ø٢ ٢٠Ø٢
٢
L١٠Ø٦
٢٠Ø٢
٢٥Ø٣
٢٢Ø٣
٣
L١٠Ø٦
٢٥Ø٢
٢٢Ø٥ ٢٢Ø٥
٤
- ١٠٣ -
אאא ã¹]<gè…‚jÖ]æ<ËÖ]<Üé×Ãj×Ö<íÚ^ÃÖ]<퉉ö¹] אאאא
א
אא
א אא
٦
אא אא
٢٠١
א
א
אא אא אאא אא אא
Kא אאא א ،אא
אאאאאאא Kאאאאאא אא E١ אKאאאאא Kאאאאא
Kאאא
E٢
E٣
Kאאאא
E٤ Kאא אאאאאאאאאאא ٪١٠٠ Kאאאא • Kאא Kאא אאאאאא
- ١٠٢ -
אא אא
٢٠١
א
א
W١J ٦ Kאאאאאאאאא א א א א א
א א א K א א
Kא
א א א K א Kאאאאא KאאE١J ٦FאאKאאא
KאאW٢J ٦ אא א א Wאאאא (i.e 0.05t or 0.05b) Kא٠{٠٥ K٢٠
E
E
אאאאאאאאא
KEאאאאF
- ١٠٣ -
אא אא
٢٠١
א
א
(Braced and un-braced column) אאאW٣J ٦ אא א אאאאא א א א (Shear walls)אא א
א א אKא K Kאאאא
WE٢J ٦FאW
אאא (Shear wall A and B)א Yאאאא
EאF ٥אK ٨ ← ١ Xא KXא(un-braced)Yא(Braced)א
E٢J ٦F - ١٠٤ -
אא אא
٢٠١
א
א
אאW٤J ٦ (λ i = He/ i)א(λ b=He/b)אאא א KE١J ٦Fאאאא
WKאאאאZiW
i = 0.3b for rectangular section i = 0.25 D for circular section KאאאאD،אאאbW
אאאE١J ٦F א אאא אא λi λb λ b ٥٠ ١٢ ١٥ ٣٥
٨
١٠
א
Kאאאאא
•
אאאW٥J ٦ Wאאא Kאאא E١
א F Kא א אא א
(٢
KEא KאאאW١J ٥J ٦
Wאאאאאא KאEאאFאאא E١ KE٣J ٦FאאK(Y)א(X)אאאא
א(B-3)אאאאEאאFא E٢ K(A)אאאא - ١٠٥ -
אא אא
٢٠١
א
א
E٣J ٦F Wאאאא(B-3)אאPא אE٣
KEAאאאFא אE
Kאאאא E
KאאאאאאE Kאאא E
- ١٠٦ -
אא אא
٢٠١
א
א
Therefore; total load P of each floor is equal to: P = W slab ×A + weight of beams + weight of walls + own weight of column Where; W slab = t s × 2.5 + weight of flooring + Live Loads. t s = 12 cm,
If Weight of flooring = 150 kg/m2
and Live Loads = 300 kg/m2
So, W slab = 0.12 × 2.5 + 0.15 + 0.3 = 0.75 t/m2 Weight of beams = b × t × 2.5 × ∑ L beams
KE٢٠ZFאZ b W K(t = span / 10: 14) אZ t אאא (A)אאאאאאאZ ∑ L beams K
Weight of walls = γ wall × h wall ×t wall × ∑ L walls KאZγ wallW
KEFאאאZ h wall
KEאאאFאZ t wall א א F א א אא אא א Z ∑ L
For example; If γ wall = 1.2 t/m3; t wall = 0.2 m
;
∑ L walls = 6.0 m h wall = 2.4 m
;
;
And, own weight of plaster = 50 kg/ m2 = 0.05 t/m2 i.e.,
Weight of wall = (1.2 × 0.2 + 0.05) × 2.4 × 6.0 = 4.176 tons / floor
Own weight of column / floor = b c × t c × 2.5 × h c Where; - ١٠٧ -
walls
KEא
אא אא
٢٠١
א
א
E٠{٦F٠{٦←٠{٢ZאZb c ٢{٠←٠{٢٥ZאZt c אאZh c
WPcאאא Pc = total vertical load on column = N × Pc/floor KאאZNW KאאאאאאZPc/floor
W
א Eא א א א F א א א ٪ ١٠ ± א
א
KאאאאאKא אאW٦J ٦ Wאא Wאאאא E١ t / b ≤ 5
and
h/b ≥ 5
אZbW
אZt אאZh
Wאאא E٢ andEאFλ b = he / b ≤ 15 for braced columns EאFλ b = he / b ≤ 10 for un-braced columns
(buckling length of column)אאZhe W EE١J ٦FאאF אZλ b
- ١٠٨ -
אא אא
٢٠١
א
א
אאW١J ٦J ٦ Kא א א א א
Kא
Wא(Fixed end)אאWE١
KE٤J ٦FאאKא
א אאאא Kאאאאא
E٤J ٦F
אא אאאא WE٢ K(Partially restrained)אאאאא
- ١٠٩ -
אא אא
٢٠١
א
א
אאא WE٣ Kא
אאאאא WE٤ Kאא
E٢J ٦Fא β = He / H0 βאאאא אאאא βאאאE٣J ٦Fא (Braced column)אβאE٢J ٦FאKאא
K(Un-braced column)אβאE٣J ٦FאK
(Braced column)אβ = He / H0E٢J ٦F
אא
אא
٣
٢
١
٠{٩٠
٠{٨٠
٠{٧٥
١
٠{٩٥
٠{٨٥
٠{٨٠
٢
١{٠
٠{٩٥
٠{٩٠
٣
(un-Braced column)אβ = He / H0E٣J ٦F
אא
אא
٣
٢
١
١{٦٠
١{٣٠
١{٢٠
١
١{٨
١{٥٠
١{٣٠
٢
J
١{٨
١{٦٠
٣
J
J
٢{٢٠
٤
- ١١٠ -
אא אא
٢٠١
א
א
EFאאאאW٧J ٦ אאEFאא K א א EF Ke min.אא
e min = 0.05 t or 20 mms (Whichever is bigger)
א א א א א א
Wאאאאא
P = fc0 × Ac + 0.44 f y ×A sc
(6-1)
W
KEE٤J ١FאאKf cu FZfc0 KאאאZAc
KאאאאאZf y KאאאאZA sc
A sc = 1% ×AcWאµ = A sc/Ac = 1%א
E١J ٦F אאאאא W Kא
WW١J ٧J ٦
WאK١٠٠Zאאא K(Mild steel 24-35)א• א
K٢L٢٥٠Z(f cu)א • Assume:
µ (total steel ratio) = 1% ; f co = 60 kg/cm2
א
(for f cu =250 kg/cm2)
And
f y = 2800 kg/cm2
i.e.,
applying in equation (6-1)
So
100×1000 = 60×Ac + 0.44×0.01×Ac×2800 = 72.32 Ac - ١١١ -
אא אא
٢٠١
א
א
Ac = 1382.27 cm2 b = breadth of column = 25 cm; Take t =
1382 . 27 = 55 . 3 cm So 25
Take
t = 60 cm A sc = 1% ×25×60 = 15 cm2 Choose 8 Ø 16 mm
KאE٥J ٦Fא
אE٥J ٦F
אאW٨J ٦ Wאאאאאאא Kא אE١ KE٦J ٦FאאKאאא אE٢
KE٧J ٦Fאאאאאאאאא E٣
- ١١٢ -
אא אא
٢٠١
א
א
אאאE٦J ٦F
- ١١٣ -
אא אא
٢٠١
א
א
E٧J ٦F
KE٨J ٦FאאKאאאאא
E٤
- ١١٤ -
אא אא
٢٠١
א
א
E٨J ٦F KE٩J ٦FאKאאא
E٩J ٦F
- ١١٥ -
E٥
אא אא
٢٠١
א
א
א א א
E٦
WE١٠J ٦FאאKאאאאא D=
b
2
2
+ t Minimum diameter
KאאאZDW KאאZ b KאאZt
E١٠J ٦F אאW٩J ٦ א ٪ ٠{٨ א א א א א א א E١ אאא א٪ ٠{٦א
KE٤J ٦FE١J ٦Fאאא(λ i)א(λ b)
א ٪ ١ אאא א אא אE٢ Kאאא٪١{٢א
אאא א אאE٣ Wאא
Kאאאא٪٤ • - ١١٦ -
אא אא
٢٠١
א
א
Kאאאא٪٥ •
Kאאאא٪٦ •
KאE٤ K١٢אE٥
K٢٠אאאאאאE٦ K ٣٠אאאE٧ אאאK ٢٥ KE١١J ٦F א א F ١٥ א א א א א
KEF٦אאאאא
E١١J ٦F
WאאאאאאE٨ KE١٥FE
K٢٠אאאE
K٨¼E٩
KאאאאאאE١٠
- ١١٧ -
אא אא
٢٠١
א
א
א K ٣ K ٨ א E١١ אאא
Kאאא١٠١٠אאא Kאא٤٠אאE١٢
W١J ٩J ٦
WאאE٦FאאאE١٢J ٦Fא (C5 ; C6)٦W٥אאא
אאאK (C5 ; C6) ٦W ٥א
Kא
אאאE١٢J ٦F
- ١١٨ -
E١
E٢
אא אא
٢٠١
א
א
א W
(L.L = 200 kg/m2)א• א
KאאE١٣J ٦Fאא •
٧٠ZאאאK٢{٩٠Zאא •
EאאFA5א(C5)٥אא EאאFA6א(C6)٦אא
E١٣J ٦F
- ١١٩ -
אא אא
٢٠١
א
א
For column (C5): • Slabs:
W slab = t s× γ c + flooring + L.L = 0.1×2.5 +0.15 + .200 = 0.6 t/m2 Area (A5) = (3.0 + 1.0) ×(3.5/2) = 7 m2 ∑ loads from slabs = 0.6×7 + (0.12 -0.1) × 2.5× (3.5/2) ×3 = 4.463 tons • Beams:
∑ L (of Beams) = (3.5/2) + 1 +3 +1 = 6.75 ms O.W. of Beams = 0.2× (0.7 – 0.1) ×2.5 = 0.3 t/m ∑ loads of Beams = 6.75 ×0.3 = 2.025 tons. • Walls:
γ wall = 1.2 t/m3 O.W. of wall = 0.2 ×1.2 +0.05 (Plaster) = 0.29 t/m2 ∑ L (of Walls) = 1 + 3 + (3.5/2) +1 = 6.75 ms h wall = 2.9 – 0.6 = 2.3 ms ∑ loads of walls = 6.75 ×0.29×2.3 = 4.5 tons. • Columns:
assume column dimension = 20 × 60 cm O.W. of column = 0.2 ×0.6 ×2.5 ×3 = 0.9 ton Total load P on column C5 = (Pc5) Pc5 (per one floor) = 4.463 + 2.025 + 4.5 + 0.9 =11.888 tons Pt = 11.888 × 6 = 71.328 tons = 72.0 tons For column (C6):
• Slabs:
W slab = t s× γ c + flooring + L.L - ١٢٠ -
אא אא
٢٠١
א
א
= 0.1×2.5 +0.15 + .200 = 0.6 t/m2 Area (A6) = (2.5/2 + 2.0) × (3.5/2 + 3.5/2) = 3.25 × 3.5 = 11.375 m2 ∑ loads from slabs = 0.6×11.375+ (0.12 -0.1)×2.5×(3.5/2)×3.25 = 7.11 tons • Beams:
∑ L (of Beams) = (2.5/2) + 2 +3.5/2 +2.5/2 +3.5/2 = 8.0 ms O.W. of Beams = 0.2× (0.7 – 0.1) ×2.5 = 0.3 t/m ∑ loads of Beams = 8.0 ×0.3 = 2.4 tons. • Walls:
γ wall = 1.2 t/m3 O.W. of wall = 0.2 ×1.2 +0.05 (Plaster) = 0.29 t/m2 ∑ L (of Walls) = 8.0 ms h wall = 2.9 – 0.6 = 2.3 ms ∑ loads of walls = 8.0 ×0.29×2.3 = 5.336 tons. • Columns:
assume column dimension = 20 × 70 cm O.W. of column = 0.2 ×0.7 ×2.5 ×3 = 1.05 ton Total load P on column C5 = (Pc6) Pc6 (per one floor) = 7.11 + 2.4 + 5.336 + 1.05 =15.896= 16 tons Pt = 16 × 6 = 96 tons Design of column C5: Assume: f c0 = 60 kg/cm2
for
f y = 2800 kg/cm2 So;
f cu = 250 kg/cm2; and A sc= 1 % Ac
Pc5 = Ac × fc0 + 0.44× A sc× f y
72×1000 = 60 × Ac + 0.44×0.01×Ac×2800 = 60 Ac +12.32 Ac = 72.32 Ac Ac = 72000/ 72.32 = 995.6 cm2 Take i.e. - ١٢١ -
b = 20 cm
t = 995.6 /20 = 49.8 cm
אא אא
٢٠١
א
א
Take column section = 20 × 50 cm Area of required steel reinforcement (A sc) = 1/100 × 20×50 = 10 cm2 Take 6 Ø16 mm (A sc = 12.1 cm2) Use stirrups 5 Ø 8 mm/m
. C5 ( واﻟﺬي ﻳﻮﺿﺢ ﻗﻄﺎع ﻋﺮﺿﻲ ﻓﻲ اﻟﻌﻤﻮد١٤-٦) اﻧﻈﺮ اﻟﺸﻜﻞ رﻗﻢ
C5 ( ﻗﻄﺎع ﻋﺮﺿﻲ ﻓﻲ اﻟﻌﻤﻮد١٤-٦) ﺷﻜﻞ رﻗﻢ Design of column C6: Assume: f c0 =60 kg/cm2 f y = 2800 kg/cm2 So;
f cu = 250 kg/cm2;
for
and A sc= 1 % Ac
Pc6 = Ac × fc0 + 0.44× A sc× f y
96×1000 = 60 × Ac + 0.44×0.01×Ac×2800 = 60 Ac +12.32 Ac = 72.32 Ac Ac = 96000/ 72.32 = 1327.4 cm2 Take i.e.
b = 20 cm
t = 1327.4 /20 = 66.4 cm ≈ 70 cm Take column section = 20 × 70 cm
Area of required steel reinforcement (A sc) = 1/100 × 20×70 = 14 cm2 - ١٢٢ -
אא אא
٢٠١
א
א
Take 8 Ø 16 mm (A sc = 16.1 cm2) Use stirrups 5 Ø 8 mm/m
. C6 ( واﻟﺬي ﻳﻮﺿﺢ ﻗﻄﺎع ﻋﺮﺿﻲ ﻓﻲ اﻟﻌﻤﻮد١٥-٦) اﻧﻈﺮ اﻟﺸﻜﻞ رﻗﻢ
C6 ( ﻗﻄﺎع ﻋﺮﺿﻲ ﻓﻲ اﻟﻌﻤﻮد١٥-٦) ﺷﻜﻞ رﻗﻢ
W١٠J ٦ WאאE١٢J ٦Fא
(C1-C2-C3-C4-C7) ٧– ٤– ٣– ٢– ١ א א א
Kא
E١
Kא(C1-C2-C3-C4-C7)٧–٤–٣–٢–١א
E٢
Kאאאאא
E٣
Kאא
Kאא
- ١٢٣ -
אאא ã¹]<gè…‚jÖ]æ<ËÖ]<Üé×Ãj×Ö<íÚ^ÃÖ]<퉉ö¹] אאאא
א
א
א
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א
א
א
א• א אאא אאא אא
א א א א א א ،א א אא KאאאאאKאאאא
א• א E١ אKאאאאא אאאאא
E٢
אאאKאא אא אא
E٣
א אא א א א
E٤
Kא
א Kא א א א אא Kאא
א א א א א א א
E٥
Kאאאאא
Kאאא • אאאאאאאאאאא ٪١٠٠ Kאאא• א • Kאא • Kאא אאא
אאאאא
- ١٢٤ -
אא
٢٠١
א
א
א
W١J ٧ א Kאאאא א א א א א א א
א א א א א א K K Kאאאאאאא
Shallow א א א א
א א אF א א א K foundation א ،אאא Eא
KאאאKDeep foundationא
אאאW٢J ٧ אא א א K א א א א
WאאאאאאKאא،א KאאאאאאKKא אאאW٣J ٧ אFאאאאאא
אאאאKEאא،אא
Wא
א א K (Strip footings) א א
(١
אא אא אאא
K
א א K (Spread footings) א א
(٢
Kאאא K (Combined footings)אא א א K ، א
KKא
- ١٢٥ -
(٣
אא
٢٠١
א
א
א
KאK(Raft foundations)אא
(٤
Kאאאאאא، (Design of Strip footings)אאW٤J ٧ א אאאKאאאE١J ٧F אא א אאא אK Bא Kאא K B א b א א א א
K(Reinforcement)(Thickness)א
،א אאא אא
א א א אא
Kא(D f)א(q all.)אאא
Kאאאאאא
(7-1)
PT = 1−
γ
P
a
×
q
D
f
all
KאאאZPW
Kאא،אאאZPT
KE٣L٢٠Z٣L٢ZFאאZγ a KאZD f
(Gross allowable bearing stress of the soil)אאאZq all
אאאאאא
KאאPTא B×1.0 = PT / q all
(7-2)
- ١٢٦ -
אא
٢٠١
א
א
א
Kא،BאE٢J ٧Fא
E١J ٧F
- ١٢٧ -
אא
٢٠١
א
א
א
WאאW١J ٤J ٧ אאאKאא אאאא
((PT-P)/B)אא (PT/B)אאא WאF netאאאאK F net = P /B
(7-3)
K Double Cantilever אאא Wא
Wאא
(7-4)
S = (B-b) / 2
Wאאא S = (B-b/2) / 2
(7-5)
E٦J ٧Fאאא 2
M = F net × S / 2
K
(7-6)
Kאאאאאא WאאQ bא
Q b = F net × S
(7-7)
W א א א אא א א א K(Bond stress)אאאאא - ١٢٨ -
E١
אא
٢٠١
א
א
א
א K א א א
E٢
Kא
Kאאא
Eא א אF א א א
א א א א א
E٣
אאאאאFKא אא א א K Eא א א
אKאKא א א א א א
א א E٣J ٧F E١J ٧F Kאאא W = Pc / Sc
(7-8)
אאKאאאZWW KאZPc KאאZSc
(Strip אאאא
אKאאK footing) Kאא،אאאא
אאW٢J ٤J ٧ ١
KL٢٠אא٣٠
אKאאאK٢L١{٠א١{٠אK Kא
- ١٢٩ -
אא
٢٠١
א
א
א
א
E٢J ٧F
P
T
P
= 1−
γa × D q all
= f
20 = 25 t / m 2 × 1 .0 1− 10 B= 25/10 =2.5 m Concrete Section: F n = 20/2.5 = 8 t/m
S = (B –b/2)/2 = (2.5 - 0.3/2)/2 = 1.175 m M = F n ×S2/2 = 8 × (1.175)2/2 = 5.523 t.m. /m Q b = F n ×S = 8 × 1.175 = 9.4 t/m Assume: f c = 60 f s = 1400
kg/cm2 kg/cm2 k1 = 0.313 k2 = 1217
- ١٣٠ -
אא
٢٠١
א
א
א
d = k1
M 5.523 × 10 5 = 0.313 = 23.26cm b 100
d min = b Take
or 25 cm d = 30 cm
t = 30 + cover = 30 +5 = 35 cm
A
s
2 5 .523 × 100000 M cm = = = 15 .127 × 1217 × 30 d k2
Take 8 Ø 16 mm/m (16.1 cm2) 2
< 10 kg /cm
9.4×1000
Q
= = 8.96 (safe) q = 0.87× d × ∑o o o o 0.87×30×(8π ×1.6) b
As′ ≥ 0.2 % Ac = 0.2 % ×30×100 = 6 cm2 As′ = 5 Ø 16 (min. Reinforcement)
E٣J ٧F - ١٣١ -
אא
٢٠١
א
א
א
٢ ٥{٠٠א ٤٠א
F٢L٠{٧٢אאKא١{٢א KE١٤ Ø٨ وﺗﺴﻠﻴﺤﻪ. ٣٠ ×٣٠א
א
E٤J ٧F
Design of long beam Load per meter = 40/5 = 8 t/m
M
max
=
W ×l
2
12
=
8×5 12
↑
2
= 16 .67 m.t
Choose b = 40 cm
d = k1 Take
- ١٣٢ -
M 16.67×100000 = 0.313 = 64.3cm b 40 t = 75 cm
and
d = 68 cm
אא
٢٠١
א
א
א
As =
2 16.67 × 100000 M = = 20.13cm 1217 × 68 k2 × d
Choose
8 Ø 18 mm
Shear
Q = 20 ton
kg /cm2 q = Choose
20 × 1000 = 8 .45 0 .87 × 40 × 68
4 branches Ø 8 mm @ 15 cm Stirrups 2
kg /cm
q
stirr
=
4 × 0.503×1400 = 4.67 15× 40
The rest to be resisted by four (4) bent up bars Ø 18 mm Design of slab:
P
T
=
8 8 = = 12 . 0 t / m 1 .2 × 2 2/3 1− 7 .2 B = 12 / 7.2 = 1.67 → 1.7 m F net = 8 / 1.7 = 4.7 t/m S = (1.7 – 0.4) / 2 = 0.65 m M = 4.7 (0.65)2 /2 = 0.993 m.t/m Q b = 4.7×0.65 = 3.055 t/m
d = k1
M = 0 . 313 b
0 . 993 × 100000 100
= 10 cm
For rigidity requirements take t = 20 cm d act = 17 cm
A
s
=
M 0.993 × 100000 = = 4.8cm 2 / m k2 × d 1217 × 17 Take
kg./cm2(safe)
q
b
=
A s min = 6 Ø 14 mm / m
Q 3.055 × 1000 = = 7.83 < 10 0.87 × d × ∑ o o o o 0.87 × 17 × (6π × 1.4) - ١٣٣ -
אא
٢٠١
א
א
א
As′ ≥ 0.2 % Ac = 0.2 % ×20×100 = 4cm2 As′ = 5 Ø 12 mm/m (min. Reinforcement)
KאאE٥J ٧Fאא
E٥J ٧F - ١٣٤ -
אא
٢٠١
א
א
א
Design of Spread Footings אאאW٥J ٧ אאאאאאאא Kא א אא Kא
אאאE٦J ٧FאKאאא
Kאא
Two Way Footings אאאE One Way FootingsאאאE
אאאאאE٦J ٧F
אאאKאאאא
אא אא Kא אKאאאאאKאאאא WKאאא - ١٣٥ -
אא
٢٠١
א
א
א
(L - l) = (B - b)
(7-9) Wאאא
A = L × B = PT/q all
(7-10)
q all = P T / (L × B) (7-11) א B ، L E١١J ٧F ، E١٠J ٧F Or
E١J ٧F א PT KE ٥F א K
W FKאE١١J ٧Fאqallאא E١ KEאא
אא א E١J ٧F א א PT E٢
א א PT K
K(B,L)
אאאE٧J ٧F - ١٣٦ -
אא
٢٠١
א
א
א
אאאאא Kאאאאאאאא
Kאאאאאאאא
KאKE٧J ٧F
Wאא Section a -a:
M
=
(7-12) M
=
P A
⎡ ⎛ L − l ⎞2 ⎟ / 2 + ⎢b ⎜ ⎢⎣ ⎝ 2 ⎠
(2
P 24 × A
B + b
)(L
⎛ L − l ⎞ − b )⎜ ⎟ ⎝ 2 ⎠
(B − l
2
⎤ / 3⎥ ⎥⎦
)2 Section a′- a′:
(7-13) M
=
P 24 × A
(2
L + l
)(B
− b
)2
Wאא Section a -a: (7-14) M
=
P L
(L
−
l
)2
/ 8
(7-15) M
=
P B
(B
−
b
)2
/ 8
א אKE١٤J ٧F،E١٢J ٧Fאאאא
Kאאאאאא
א אאא
KאאאKאאא WQ bא - ١٣٧ -
אא
٢٠١
א
א
א
Fn=P/A
(7-16)
WאאQ b For section a -a: Q b =¼ (B + b) (L - l) × F n
(7-17) For section a′- a′:
Q b =¼ (L + l) (B - b) × F n
(7-18) Punching Depth אאW١J ٥J ٧
(L × B) א א (l × b) א א
א א K (Direct Shear) א א א א אא אאאא K٢L٨q PאאאאאKא
Kאאאאאאא KE٨J ٧Fאאאא א
EאאFא˚٤٥א WאאKאא
(7-19)
Q
P
2 ⎞ ⎛ 2 ⎞ ⎛ = P − Fn ⎜ b + d ⎟ × ⎜ l + d ⎟ 3 ⎠ ⎝ 3 ⎠ ⎝
- ١٣٨ -
אא
٢٠١
א
א
א
Wאאאא (7-20) q P =
QP 2 d (l + b + 1 .33 d )
E٨J ٧F
אאW٢J ٥J ٧ ١
K ١{٠אK ١٠٠ ٤٥ × ٤٥ E١٨ø٨אF٢L١{٧٥א
- ١٣٩ -
אא
٢٠١
א
א
א
א
E٩J ٧F
P
T
Footing Dimensions:
= 1 −
γ
P a
× D q all
= f
100 = 112 . 9 ton 2 ×1 1 − 17 . 5
A = 112.9 / 17.5 = 6.452 m2
B=
A = 2.54 = 2.55m Concrete Sections:
M=
P (2B + b)(L − l )2 = 100 2 (5.1 + 0.45)(2.55 − 0.45)2 = 15.683m.t 24 × A 24 × (2.55)
Q b =¼ (B + b) (L - l) × F n =¼ (2.55 + 0.45) (2.55 -0.45) × 15.38 = 24.221 ton
Q - ١٤٠ -
2 ⎞ ⎛ 2 ⎞ ⎛ = P − Fn ⎜ b + d ⎟ × ⎜ l + d ⎟ P 3 ⎠ ⎝ 3 ⎠ ⎝
אא
٢٠١
א
א
א
d = 0 . 313 ×
15 . 683 × 10 5 = 49 cm But (45 + 20 )
Take d =50 cm
Q P = 100 − qP
100 2.55 2
&
t = 55 cm
2 2 ⎞⎛ ⎞ ⎛ ⎜ 0.45 + × 0.5 ⎟⎜ 0.45 + × 0.5 ⎟ = 90.564 ton so, 3 3 ⎝ ⎠⎝ ⎠
QP 90 . 564 × 10 3 = = = 5 . 78 kg / cm 2 d (l + b + 1 . 33 d ) 2 × 50 (45 + 45 + 66 . 7 )
2
M 15.68 × 10 5 2 As = k2 × d = 1217 × 50 = 25.75cm Choose 13 Ø 16 mm
q
b
=
Q 24.221× 1000 = = 8.57 < 10kg / cm2 0.87 × d × ∑ o o o o 0.87 × 50 × (13π × 1.6) Footing is 2.55 ×2.55 ×0.55 (13 Ø 16 in each direction)
אאאאאאE١٠J ٧Fאא Kא
- ١٤١ -
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A = P / F all.net = 150 / 20 = 7.5 m2 Choose L = 3.0 m
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B = 2.5 m
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M a−a =
P (2 B + b )(L − l )2 = 150 (5 . 3 )(2 . 25 )2 = 22 . 36 t .m 24 × A 24 × 7 . 5
M a ′− a ′ =
P (2 L + l )(B − b )2 = 150 (6 . 75 )(2 .2 )2 = 27 .225 t .m 24 × A 24 × 7 . 5
Q b a-a =¼ (B + b) (L - l) × P/A = ¼ × 150/7.5(2.8) (2.25) = 31.50 ton Q b a′-a′ =¼ (L + l) (B - b)× P/A = ¼ × 150/7.5(3.75) (2.2) = 41.25 ton Concrete Sections; d = 0 .313 × - ١٤٣ -
22 .360 × 10 5 = 66 . 16 cm (30 + 20 )
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t = 70 cm
M 22.360 × 10 5 = = 28.24cm 2 Choose 15 Ø 16 mm As = k2 × d 1217 × 65 M 27.225 × 10 5 = = 35.5cm 2 Choose 18 Ø 16 mm As′ = k2 × d 1217 × 63
Q 41.25 × 103 qb = 0.87 × d × ∑ o o o o = 0.87 × 63 × (18π × 1.6) = 8.36 < 10kg / cm2 (Safe) Q p = 150-20(0.3+0.44)(0.75+0.44) = 132.4 ton
QP 132 . 4 × 10 3 qP = = = 5 . 04 < 8 kg / cm 2 ( Safe ) 2 d (l + b + 1 . 33 d ) 2 × 65 (75 + 30 + 97 )
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(References) אא 1) BS 8110 – 1985
2) CP 110 , 1972 3) Jack C. McCormack ˝Design of Reinforced Concrete˝ Fourth Edition Copyright © 1998 by Addison Wesley Longman, Inc. 4) James G. Macgregor ˝Reinforced Concrete – Mechanics and Design˝ Third Edition © 1997 Prentice-Hall, Inc. אאאא˝אK E٥ KאJ א˝א K١٩٩٥א١٩٨٩˝אאאא ˝אE٦ 7) M. Hillal ˝Fundamentals of reinforced and pre-stressed concrete˝ Dar ElNashr Cairo 1992. 8) Reynolds, C.E and Steadman, J.C ˝Reinforced Concrete Designers˝ Handbook 9th Edition. -1981. 9) Robert Park and William L. Gamble. ˝Reinforced Concrete Slabs˝ Second Edition John Wiley & Sons Inc 2000. 10) Shaker El-Behairy, ˝Reinforced Concrete Design Handbook˝ Part I , II and III Fifth Edition Dar-El-Nashr-Cairo 1998.
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