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In Doubly Reinforced beams using WSM, why we get Pc greater than Pt?

 
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Dr. N. Subramanian
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PostPosted: Fri Nov 19, 2021 4:44 pm    Post subject: In Doubly Reinforced beams using WSM, why we get Pc greater than Pt? Reply with quote

Someone asked this question. In tables 72 to 75 of SP 16, Pt is greater than Pc, where sigmast is 140 (fy is 250 MPa) and sigma cbc is  5 to 10.0. Whereas, In tables 76 to 79, when sigmast is 230 (fy is 415 MPa), Pc is greater than Pt. Why?

I asked Prof. H.J. Shah as I forgot WSM, as it is not used for design for a long time (may be eliminated in the next version of IS 456- WSM may
still be used for serviceability checks like deflection, vibration, etc.). I also found the reason independently, by the time Prof. Shah answered.

In the working stress method, permissible stress in compression reinforcement is taken as (1.5m-1) times stress in adjoining concrete.
This value is very small resulting in a large required compression area.

I think that the same doubly reinforced beam will become economical if we design it by LSM, anyone may check it out by taking a specific example.
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vikram.jeet
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PostPosted: Sat Nov 20, 2021 8:59 am    Post subject: Reply with quote

Doubly  Reinforced Beam

WSM vs LSM
For same BM , WSM requires higher depth for a section to be singly reinforced vis a vis LSM.
For same section size  (B,D) WSM requires higher Compression steel vis a vis LSM in doubly Reinforced beam

The main reason lies in an overall safety factor :
In WSM , safety factor for stresses are :
Concrete Bending compressive stress  -  3
Steel bending stress (tensile)-  about 1.80
Load factor = 1.0

In LSM the safety factors are :
Load Factor = 1.5.     (Under  Normal conditions on gravity loadings)
Partial safety for bending comp stress in concrete = 1.5
Hence overall computed safety factor for  cbc = 1.5*1.5 = 2.25
For steel partial safety factor bending stress  tension = 1.15
Hence overall computed FOS for tensile  steel  = 1.5*1.15= 1.725
Overall


From above it is seen that computed overall safety factors are :
WSM = 3.0 Whereas LSM = 2.25   (for concrete bending stress  cbc)
WSM = 1.80  whereas LSM = 1.725 (for steel tensile stress Sigma st)

Thus computed overall safety factors in LSM are  25% lesser than WSM in concrete and  AND  marginally lesser (about 5℅)  for tensile steel stresses .

Compression steel :
Compressive steel stress in WSM is  1.5*m* c at compression steel location.
WSM  M20  value ( 1.5*m*c) = 1.5*14*7.0 =147 Mpa*
            M25                                 = 1.5* 11* 8.5 = 140 Mpa*
            M30.                                = 1.5*9*10 = 135 Mpa*
* Stress will be slightly less depending on c at compression steel.

Referring to SP-16 , based on d'/d comp stress in steel is given in table F.
In LSM compressive  stress in steel is based on a stress of 0.87 fy.
The net compressive stress for comparison with WSM is 0.87/1.5=0.58 fy. (Slightly less as per d'/d )
It is seen that the  steel compressive stress in WSM is about 60% the steel compressive  stress in LSM. (Slightly vary as per d'/d).

Designs are therefore economical in LSM.
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vikram.jeet
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PostPosted: Sat Nov 20, 2021 2:06 pm    Post subject: Reply with quote

Further : Stress in compression reinforcement

Since the permitted bending comp  stress in conc under WSM is  Fck /3 and same in LSM is Fck/2.25 (on computed overall FOS basis),  hence a  higher stress in compression reinforcement in LSM  to the extent = (3/2.25  ) is expected .
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pvgraju
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PostPosted: Mon Nov 22, 2021 10:29 am    Post subject: In Doubly Reinforced beams using WSM, why we get Pc greater than Pt? Reply with quote

Dear Sir,

Permissible working stress in compression steel is 1.5xmxstress in concrete around steel.

stress in concrete around steel is mx2/3 fc = 280/3fc x 2/3 fc = 280/9 fc

so permissible stress in compression steel will be

= 1.5xmxmx280/9 fc.

so as long as fc is not changing permissible working stress in compression shall not change.

Regards,


P.V.Gavarraju.
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vikram.jeet
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PostPosted: Mon Nov 22, 2021 10:51 am    Post subject: Reply with quote

In WSM , stress in compression steel depend on stress in concrete around it. But permissible stress in concrete in bending  is Fck/3.

Same seems true in LSM , but overall FOS (computed) for conc bending stress  = 2.25
Hence stress in compression steel depend upon Fck/2.25.

This could be the reason that compression steel computed in WSM will be higher on two counts :
(I) Lesser Moment capacity as singly reinforced
(ii) Lesser stress available around compression steel
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