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CONTINUOUS BEAM EXAMPLE 3.2
Figure above shows a plan view of a school building. It is constructed using cast in-situ method and the slab spans 3 x 8 m each. Prepare a complete design of beam 2/A-D.
Design data:
Slab thickness = 125 mm Loads from finishes, partitions etc. = 15 kN/m Characteristic live load, qk = 10 kN/m Nominal cover, c = 25 mm Concreteβs characteristic strength, fck = 30 N/mm2 Steel characteristic strength (main), fyk = 500 N/mm2 Beam size, bw x h = 250 x 450 mm
A
A
CONTINUOUS BEAM EXAMPLE 3.2
1. Calculate the loads acting on the beam.
Characteristic permanent action on beam, gk
Self-weight of beam = 25 x bw x h = 25 x 0.25 x 0.45 = 2.81 kN/m
β΄Total charac. permanent action on beam 2/A-D, gk = self-weight of beam + finishes
= 2.81 + 15 = 17.81 kN/m
β΄Total charac. variable action acting on beam 2/A-D, qk = 10 kN/m
Therefore, design load acting on beam 2/A-D, w = 1.35 gk + 1.5 qk
= 1.35 ( 17.81 ) + 1.5 (10) = 39.04 kN/m
Solution
(1.35 gk + 1.5 qk)
CONTINUOUS BEAM EXAMPLE 3.2
2. Draw the shear force (SFD) and bending moment diagram (BMD).
The coefficients from Table 3.5 can only be applied to continuous beam analysis when all this provisions are fulfilled.
a) Qk β€ Gk = 10 < 17.81 β΄OK! b) Loads should be uniformly distributed over 3 or more spans = 3 spans β΄OK! c) Variation in span length should not exceed 15% of the longest =same span β΄OK!
Beam 2/A-D
8 m 8 m 8 m
w = 39.04 kN/m
0.45F
0.6F
0.55F 0.6F
0.55F 0.45F
0.09FL
0.11FL
0.07FL
0.11FL
0.09FL
+ + +
- - -
+ + +
- -
CONTINUOUS BEAM EXAMPLE 3.2
3. Calculate the design moments and shear force values.
F = wL = ( 39.04 ) x ( 8 ) = 312.32 kN
4. Design the main reinforcements. i) Calculate the effective depth, d.
Assume Οbar = 20 mm
Οlink = 8 mm
d = h β c - Οlink - Οbar/2 = 450 β 25 β 8 - 20/2
= 407 mm
8 m 8 m 8 m
w = 39.04 kN/m
140.54
187.39
171.78 187.39
171.78 140.54
224.87
274.84
174.9
274.84
224.87
+ + +
- - -
+ + +
- -
h d
CONTINUOUS BEAM EXAMPLE 3.2
β’ At mid span A-B and C-D (design as flange section)
M = 0.09 FL = 224.87 kNm
Section A-A
Lo values
Mid span A-B & C-D => lo = 0.85 (8000) = 6800 mm
Mid span B β C => lo = 0.7 (8000) = 5600 mm
3000 mm 3000 mm
bw = 250 mm bw = 250 mm bw = 250 mm
CONTINUOUS BEAM EXAMPLE 3.2
Calculate the effective width of flange, bf
ππππ = οΏ½ππππ,π + ππ β€ π
ππππ,π = π.π ππ + π.πππ β€ π.π ππ π1 = π2 = (3000 β 250)/2 = 1375 ππ ππππ,1 = ππππ,2 = 0.2 (1375) + 0.1(6800) = πππ ππ β€ 0.2 (6800) = 1360 ππ ππππ,1,2(πππ) = π1,2(1375) => ππΎ! ππππ = οΏ½ππππ,π + ππππ,ποΏ½ + ππ = (955 + 955) + 250 = ππππ ππ < π = 1375 + 1375 + 250 = 3000 ππ β΄ πΆπ²!
Design the main reinforcement.
ππ = 0.567ππππβπ οΏ½π ββπ2οΏ½ = 0.567(30)(2160)(125) οΏ½ 407 β
1252οΏ½ = 1582 πππ
Compare Mf with the design moment, M.
M =224.87 kNm < Mf = 1582 kNm
β΄ Neutral axis lies in flange / below flange.
β΄Design the beam as rectangular / flanged beam.
πΎ =π
ππ2πππ=
224.87 π₯ 106
2160 π₯ 4072 π₯ 30= 0.02 < 0.167
β΄ compression reinforcement is required / not required
π§ = π οΏ½0.5 + οΏ½0.25 βπΎ
1.134οΏ½ = οΏ½0.5 + οΏ½0.25 β
0.021.134
οΏ½ = 0.98π > 0.95π
β΄ use = 0.95d
Flanged beam
CONTINUOUS BEAM EXAMPLE 3.2
π΄π πππ =π
0.87ππ¦ππ§=
224.87 π₯ 106
0.87π₯ 500 π₯ 0.95 π₯ 407= 1337 ππ2
β΄Provide: 5H20 (Asprov = 1570 mm2)
Check the area of reinforcement.
π΄π ,πππ β₯ 0.26 πππ‘πππ‘π
ππ¦π β₯ 0.0013 ππ‘ π
π΄π ,πππ = 0.26 (2.9)(250)(407)
500= 153 ππ2 β₯ 0.0013 (250)(407) = 132 ππ2
β΄ ππΎ!
π΄π πππ₯ = 0.04π΄π = 0.04 π₯ 250 π₯ 450 = 4500 ππ2
π΄π πππ < π΄π ππππ£ < π΄π πππ₯ β΄ ππΎ!
β’ At support B & C (design as rectangular section)
M = 0.11 FL = 274.84 kNm
πΎ =π
ππ2πππ=
274.84 π₯ 106
250 π₯ 4072 π₯ 30= 0.22 > 0.167
β΄ compression reinforcement is required / not required
π§ = π οΏ½0.5 + οΏ½0.25 βπΎβ²
1.134οΏ½ = 0.82π
Calculate dβ
πβ² = π + β ππππ +β πππ
2= 25 + 8 +
202
= 43 ππ
CONTINUOUS BEAM EXAMPLE 3.2
π΄π β²πππ =(πΎ β πΎβ²)πππππ2
0.87ππ¦π(π β πβ²) =
( 0.22 β 0.167)(30)(250)(407)2
0.87 ( 500 )(407 β 43 ) = 416 ππ2
β΄Provide: 3H16 (Asβprov = 603 mm2)
π΄π πππ = πΎβ²πππππ2
0.87ππ¦ππ§+ π΄π β²πππ =
0.167(30)(250)( 407 )2
0.87(500)( 0.82 π₯ 407)+ 416 = 1845 ππ2
β΄Provide: 3H25 (Asprov = 1963 mm2)
Check the area of reinforcement.
π΄π ,πππ β₯ 0.26 πππ‘πππ‘π
ππ¦π β₯ 0.0013 ππ‘ π
π΄π ,πππ = 153 ππ2
π΄π πππ₯ = 4500 ππ2
π΄π πππ < π΄π ππππ£ < π΄π πππ₯ β΄ ππΎ!
β’ At mid span B-C (flanged section)
M = 0.07 FL = 174.9 kNm
Calculate the effective width of flange, bf
ππππ = οΏ½ππππ,π + ππ β€ π
ππππ,π = π.π ππ + π.πππ β€ π.π ππ π1 = π2 = (3000 β 250)/2 = 1375 ππ ππππ,1 = ππππ,2 = 0.2 (1375) + 0.1(5600) = πππ ππ β€ 0.2 (5600) = 1120 ππ ππππ,1,2(πππ) = π1,2(1375) => ππΎ! ππππ = οΏ½ππππ,π + ππππ,ποΏ½ + ππ = (835 + 835) + 250 = ππππ ππ < π = 1375 + 1375 + 250 = 3000 ππ β΄ πΆπ²!
CONTINUOUS BEAM EXAMPLE 3.2
Design the main reinforcement.
ππ = 1582 πππ
Compare Mf with the design moment, M.
M = 174.9 kNm < Mf = 1582 kNm
β΄ Neutral axis lies in flange / below flange.
β΄Design the beam as rectangular / flanged beam.
πΎ =π
ππ2πππ’=
174.9 π₯ 106
1920 π₯ 4072 π₯ 30 = 0.02 < 0.167
β΄ compression reinforcement is required / not required
π§ = π οΏ½0.5 + οΏ½0.25 βπΎ
1.134οΏ½ = οΏ½0.5 + οΏ½0.25 β
0.021.134
οΏ½ = 0.98π > 0.95π
π΄π πππ =π
0.87ππ¦ππ§=
174.9 π₯ 106
0.87π₯ 500 π₯ 0.95 π₯ 407= 1040 ππ2
β΄Provide: 4H20 (Asprov = 1271 mm2)
Check the area of reinforcement.
π΄π ,πππ β₯ 0.26 πππ‘πππ‘π
ππ¦π β₯ 0.0013 ππ‘ π
π΄π ,πππ = 153 ππ2
π΄π πππ₯ = 4500 ππ2
π΄π πππ < π΄π ππππ£ < π΄π πππ₯ β΄ ππΎ!
Flanged beam
CONTINUOUS BEAM EXAMPLE 3.2
5. Design the shear reinforcement.
VEd = V max = 0.6 F = 187.39 kN
Calculate VRd,c
π = 1 + οΏ½200407
= 1.7 β€ 2.0 π ππ ππ
ππ =π΄π πππ€π
= 1963
250 π₯ 407= 0.02
ππ π,π = 0.12π(100πππππ)13ππ€π β₯ ππππ
= 0.12(1.7)οΏ½100(0.02)(30)οΏ½13(250)(407) = 81.26 kN > Vmin
ππππ = οΏ½0.035 π3/2πππ1/2οΏ½ππ€ π = 43.24 kN
β΄VRd.c = 81.26 kN
Compare VEd with VRd,c
VEd (187.39) > VRd,c (81.26) => shear reinforcement is required
Calculate VRd,max
@ 22Β°,
ππ π,πππ₯ = 0.36 ππ€π οΏ½1 β πππ250οΏ½πππ
(πππ‘ π + π‘ππ π) (22Β° β€ π β€ 45Β°)
= 0.36 (250)(407) οΏ½1β 30
250οΏ½ (30)(πππ‘ 22Β° + π‘ππ 22Β°)
= 335.88 ππ
CONTINUOUS BEAM EXAMPLE 3.2
Compare VEd with VRd,max
VEd (187.39) < VRd,max (335.88)
Design shear reinforcement
π΄π π€π
= ππΈπ
0.78ππ¦ππ cot π
= 187.39 π₯ 103
0.78(500)(407)(cot 22Β°)= 0.48
Try H8, Asw = 2Ο(β link)2/4 x 2 legs = 101 mm2
π = 1010.48
= 210 ππ < 0.75π = 0.75 (407) = 305.25 ππ
β΄Provide: H8 β 200 c/c
6. Check beamβs capacity against deflection.
Check only at mid-span with maximum moment.
π =π΄π ,πππ
ππ€π=
1337250 π₯ 407
= 0.013
π0 = οΏ½ππππ₯ 10β3 = 5.48 π₯ 10β3
Ο > Οo
ππ
= πΎ οΏ½11 + 1.5 οΏ½πππππ
π β πβ²+
112οΏ½
ππποΏ½πβ²πποΏ½
From table 7.4N, K = 1.3 (end span of continuous beam) β where the moment is maximum
= (1.3) οΏ½11 + 1.5 οΏ½(30)(5.48 π₯ 10β3)
0.013 β 0+
112
οΏ½(30)οΏ½0
(5.48 π₯ 10β3)οΏ½ = 18.8
CONTINUOUS BEAM EXAMPLE 3.2
(i) Calculate the modification factor
a) Modification factor of tension reinforcement,
310ππ
=500
ππ¦π οΏ½π΄π ,ππππ΄π ,ππππ£
οΏ½=
500
500 οΏ½13371570οΏ½
= π.ππ
b) Modification factor for flange section,
bf/bw = 2160/250 = 8.64 > 3, therefore the modification factor (flange) = 0.8
c) Modification factor for span length more than 7 m
Effective span length 8 m > 7 m, therefore modification factor span length = 7/l eff
= 7/8 = 0.88
d) Calculate (L/d)allowable
οΏ½πΏποΏ½πππππ€ππππ
= οΏ½πΏποΏ½πππ ππ
π₯ πππππππππ‘πππ ππππ‘ππ
οΏ½πΏποΏ½πππππ€ππππ
= 18.8 π₯ 1.17 π₯ 0.8 π₯ 0.88 = 15.5
a) Calculate (L/d)actual
οΏ½πΏποΏ½πππ‘π’ππ
=ππππππ‘ππ£π π πππ πππππ‘β
ππππππ‘ππ£π ππππ‘β =
8000407
= 19.66
b) Compare with (L/d)actual with (L/d)allowable
(L/d)actual > (L/d)allowable
CONTINUOUS BEAM EXAMPLE 3.2
Therefore, the deflection check fails !
Try to increase from 5H20 to 6H20 (As prov = 1890 mm2)
Modification factor for tension reinforcement = Asprov/As req = 1890/1337 = 1.4
Recalculate (L/d)allowable = 18.8 x 1.4 x 0.8 x 0.88 = 18.5
Try to increase from 6H20 to 5H25 (As prov = 2450 mm2)
Modification factor for tension reinforcement = Asprov/As req = 2450/1337 = 1.83
Recalculate (L/d)allowable = 18.8 x 1.83 x 0.8 x 0.88 = 24.2
Compare with L/d actual
(L/d)actual = 19.66 < (L/d)allowable = 24.2
Therefore, the deflection check passes !!
Therefore, beam is safe / not safe against deflection.
i) (L/d)actual β€ (L/d)allowable β Beam is safe against deflection (OK!) ii) (L/d)actual > (L/d)allowable - Beam is not safe against deflection (Fail!)
CONTINUOUS BEAM EXAMPLE 3.2
7. Check the beam for cracking.
Check only at mid span with maximum spacing.
i) Calculate the clear horizontal distance between bars in tension, S1.
π = (π β 2πππ£ππ β 2β ππππ β 2β πππ)
π = 250 β 2( 25) β 2(8) β 2( 20)
= 144 ππ
ππ = ππ¦π
1.15 π₯
πΊπ + 0.3 ππ1.35πΊπ + 1.5 ππ
= 5001.15
π₯ 17.81 + 0.3 (10)
1.35(17.81) + 1.5 (10) = 232 πππ
Taking wk = 0.3 mm
Maximum allowable clear spacing = 230 mm
β΄OK!
β΄Crack check passed / failed !!
The value must not exceed its maximum allowable clear
spacing
4H20
250 mm
S1
CONTINUOUS BEAM EXAMPLE 3.2
Minimum bar spacing (check for the closest bar spacing)
Minimum bar spacing between reinforcements
= max {k1. Bar diameter, dg + k2, 20 mm}
i) 1.25 = 25 mm ii) dg + k2 = 20 + 5 = 25 mm iii) 20 mm
Minimum bar spacing = 25 mm
Compare with actual bar spacing = 54.5 mm (5H25) > 25 mm β΄Ok!!