Question

Consider a concrete beam in an interior environment as follows: a. The beam is simply supported with a design span of 4.8 metres. b. The dead load in the beam is uniformly distributed. c. Dead load on the beam, in addition to its own weight, is 6.9 KN/m of which 75% is permanent (long term). d. The beam carries a single concentrated live load at mid-span of 70 KN. e. 20% of the live load is considered short term. f. The dimensions of beam are: b = 280 mm, d = 490 mm, cover c = 45 mm g. The beam only has tension reinforcement provided by 3 N20’s. h. f’c = 25 MPa fy = 500 MPa γc = 24 KN/m3 Determine: 1. The design ultimate moment in the beam M*. 2. The ultimate moment capacity of the beam øMuo. Is the beam ductile? 3. The serviceability moment in the beam ?? ∗ . 4. The Code allowable mid-span deflection. 5. The long term deflection at mid-span of the beam. Does it comply with the Code requirements?

Answer 1

Fig.1

Fig.1 shows the max load acting on the part

During serviceability conditions when the beam is no longer bear these loads we would decrease the loads to as per in the question

Fig.2 shows the serviceability condition of the beam

From the law of equilibrium

Ra+Rb=70+(6.9x4.8)=103.12 kN

Considering Ma=0

-Rbx4.8+(70 x 2.4)+6.9x4.8x4.8x0.5)=0

Rb=51.56kN

Ra =51.56kN

Since simply supported Ma=Mb=0

Mc=51.56x2.4-(6.9x2.4x2.4x.5)=103.872kNm

Qn1

Design Moment = 103.872kNm

Similarily we would get

Mc = 31.704 kNm at serviceability conditions

d=490 -45(cover)-(20/2)=435 mm

Qn2

Mu,lim==182.74kNm

Qn 3

Serviceability Moment in beam =31.704 kNm

Qn 4

As per IS 456 Cl23.2 the deflection shall not exceed Span/250,Span/350 or 20 mm whichever is greater

4800/350=13.7 mm

Qn 5

The deflection for uniformily distributed load is

from Sp16 Table 98

Deflection for pointed load

from Sp16 Table 98

During serviceability condition only the permanent loads and reduced live loads shall be taken

But lets take the extreme case

=0.69 mm=2.35 mm

Thus Total deflection=0.69 + 2.35 mm=3.04 mm

Thus it complies with the codal cases even at maximum loads

Mu/bd2=1.94

From Table 2

pt=0.515

Thus 2,20# bars is required

Provided is 3,20#

Thus Safe