Week 1 Challenge
1. Answer:
Water cement ratio = Weight of water / Weight of cement
Assuming 1 liter of water = 1 kg
W/C ratio = 50/250 = 0.2
Cement weight = 250 kg
Sand weight = 250 kg
Aggregate weight = 500 kg
So, the mix ratio 250 : 250 : 500 which is 1:1:2
1:1:2 mix ratio is equivalent to M25
We can not allow this mix to proceed the execution because of two reasons. One is poor w/c ratio. The nominal w/c ratio shall range 0.4 to 0.6.
Secondly, nominal concrete strength to be used for bridges ranges M30 to M50. Because of these reasons, we can not allow such poor mix.
2. Answer:
Distributed load due to surfacing = 0.25 x 19 x 0.5 = 2.375 kN/m
Distributed load due to concrete selfweight = (800x500)-(500x300) x 24 = 6 kN/m
Total load of structure = 8.375 kN/m
Bending moment due to the total load = wl2/8 = 8.375 x 10^2 / 8
= 104.68 kNm
Concrete strength = M50 = 50 N/mm2
Characteristic tensile strength (stress to develop crack in concrete) = 0.18 Fck^2/3 = 2.44 MPa
Bernoulis theory
M/I = s/y
M = (s/y) x I
s = stress of concrete = 2.44
y = neutral axis depth = 400
I = moment of inertia of solid section = (500x800^3/12) - (300x500^3/12) = 21333333333 - 3125000000 = 18208333333
M = 111.07 Knm (cracking moment)
3. Answer:
Limit state of serviceability quasi permanent combination for selfweight and surfacing load as per IRC-6 codebook Table B.3
Factor is 1 because variable load impact should be relived (in question, it is mentioned to ignore temperature load)
So load due to selfweight plus surfacing should be multiplied by 1 only.
So, the service Quasi permanent combination moment of section is 104.68 kNm, which is lesser than cracking moment. So, the section will not crack.
4. Answer:
Number of bars = 4
Dia of bar = 16mm
Tensile strength of rebar = 500 n/mm2
Area of steel = 4 x 201 = 804 mm2
Design tensile strength of rebar = Fu/1.15 = 435 N/mm2
Tensile force (T) = Fy x Ast = 349.7 kN
Since we are calculating ultimate capacity, we have to consider it as cracked section and therefore Strain compatibility C = T
Concrete area under stress (Ac) = Force / Design compressive strength
= 349.7x1000 / (0.67x50/1.5)
= 15658 mm2
Depth of section under stress (h) = Ac / b = 15658 / 500 = 31.3
Converting this rectangular section to actual parabolic section (x) = 0.8 x 31.3 = 25.04 mm
Consider cover = 50mm, effective depth (d) = 750mm
Lever arm (z) = d - (x/2) = 750 - (25/2)
z = 750-12.5 = 712.5 mm
Ultimate Bending moment = T x z = 349.7 x 0.712 = 249 kNm
Reverse calculation from Bending moment to distributed load = 249x10^6 x 8 / (10000^2)
UDL = 19.92 kN/m
So, the given UDL is 25 kN/m which is greater than the resisting capacity of section. So the section will fail.
5. Answer:
Among the three beams, the central beam will cover the maximum slab area. So the design bending moment for ventral beam shall be calculated now.
The UDL of selfweight of beam = 6 kN/m (Permanent)
The UDL of deck = width of deck x density x thickness = 1.25 x 20 x 0.12 = 3 kN/m (Permanent)
The UDL of traffic = 5 x 1.25 = 6.25 kN/m (Live variable)
The UDL of wind = 0.5 x 1.25 = 0.625 kN/m (Wind variable)
The strength combination (B) under ULS adding to the effect of variable loads,
Load factor for Selfweight = 1.35
Load factor for deck = 1.75
Load factor for Traffic (Live variable) = 1.5
Load factor for wind (Live variable) = 1.5
Total factored ultimate load UDL = (1.35x6) + (1.75x3) + (1.5x6.25) + (1.5x0.625)
= 23.66 kN/m
Maximum bending moment = wxl^2/8 = 23.66 x 10^2 / 8 = 295.75 kNm
Assume Fe500 rebar in design,
Required Ast = x
Tensile force (T) = 500/1.15 x X
T = 434x
Concrete area under stress (Ac) = 434x / (0.67x50/1.5)
Ac = 19.43 x
Depth of extreme compression fiber = 19.43x/500= 0.03886 X
Converting this rectangle block into actual parabolic stressblock = 0.8x0.03886 X = 0.031088 X
Lever arm (z) = d - (x/2)
z = 750 - (0.031088 X/2)
z = 750 - 0.015544X
Moment = Tz
295.75 x 10^6 = 434X(750-0.015544X)
295750000 = 325500x - 6.7461x^2
Solving this quadrtic equation, we get x = 927 mm2 is the required reinforcement
6. Answer:
The span of the bridge mentioned is 40 meter. So as per the recommended systems for more than 40 meter, e can go with Ribbed slabs or Precast beams.
Between these two systems, Ribbed slab system may require temporary formwork which will definitely create hindrance for the existing motorway. Since the precast beam system will not require formwork or propping, it is more suitable to go with precast beam system for this kind of situation.
7. Answer:
The span of bridge is 40 meter. The least possible span to depth ratio for a concrete bridge element shall be 1:20
Which will give us 40000/20 = 2000 mm
Taking this as the Precast girder depth = 2000mm
Clearance required for the existing motorway = 5700mm
Assuming slab depth = 150mm
Asphalt surfacing = 120mm
Top level of the new bridge from existing motorway level = 5700 + 2000 + 150 + 120 = 7970 mm
Nearly 8 meter of height is required to connect the bridge from ground level. But the existing Ramps are at the height of 7 meter only. Since the ramps are not level matching with the top of new bridge, they can not be used.