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Re: "Big Column and small beam" Rule

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Dear Alex,

Thank your for compliments. It has been a real pleasure to be of some
assistance.You have now raised a very interesting question; let me think
how best can I answer your question:

1) A flange acts as a part of a slab as the web can not deform alone; the
slab must deform with the web since the two are integral. In the direction
of analysis, the deformation of the slab is exactly equal to the
deformation of the beam at the beam-slab interface and gradually decreases
as the distance of slab fibers increases from the beam center-line.

2) In the light of the above behaviour, the slab contributes to the
stiffness of the beam. If my memory does not fail me, the stiffness of a
flanged beam vary between 1.5 times to around 2.0 times that of the
stiffness of the web alone. The average may be around 1.75 times. But one
can always compute the specific value in each case.

3. The stiffness values of various members meeting at a joint determine the
distribution coefficients, which are the proportions in which the
unbalanced moments are distributed amongst them.

4. Coming back to your specific case, as a part of the slab acts as the
flange of the beam the stiffness of the beam is increased and it attracts
proportionately greater gravity moment.

5. When we consider the mechanism of the beam for resisting the applied
moment, we must recall that the moment is resisted by a concrete-steel
couple (except for an additional contribution from a steel-steel couple,
for beams with compression reinforcement also). The tensile reinforcement
is placed within the web in spans while at the supports, you might wish to
place some reinforcement within the web while a part may also be placed
outside the web and within the slab. But no matter where you place the
reinforcement, the argument that I am building, would still be valid.

6. Now let us see how a hinge is formed. Recall the stress-strain diagram
that is characteristic of ductile steel. Plotted on a graph with the
horizontal axis as strain and the vertical axis as stress, the diagram
ascends in a straight line and then, a little below the yield stress, forms
a second-degree curve and then becomes horizontal. This horizontal part is
called the yield plateau and it results from what is known as yielding.
This plateau is fairly prolonged and continue for a considerable strain
(perhaps by as much as ten times the strain at which yielding was
initiated) and then assumes an upward curve, marking the end of yielding
and the start of what is called strain-hardening.

7. The phenomenon of yielding is such that the the strain continues to
increase without any increase in stress. If we are performing a tensile
test of a reinforcing bar on a machine, the load cell reading comes to a
halt while the elongation continues to take place, while the bar is
yielding.

8. Let us now extend this knowledge to bending. We would design a beam for
a specified moment, M. If you are using the current codes, you must be
using the strength design method (SDM) and would tend to design for an
enhanced moment, Mu, using various multipliers for loads and capacity
reduction factors for flexural capacity. The actual bar stress, at the
service state, may be something like 60% of the yield stress.

9. Now start increasing the moment beyond the design value, M. The bar
stress would gradually increase and beam would continue to offer increasing
resistance. A time would come when the bar stress would reach yield stress.
The moment at point would be around Mu. After this point, the beam would
start rotating without offering increasing bending capacity. The capacity
would remain the same as it was when the yielding started, i.e. Mu. This is
when we say that a hinge has been formed. It is a rusted hinge of sorts as
its capacity to resist moment is not zero but has been locked at a certain
value, which is around Mu.

10. You would, thus, note that a hinge is declared to have been formed and
beam starts rotating, at the onset of yielding of reinforcing steel,
whether the reinforcing steel has been placed entirely within the web or
even when it is partly placed in the flange. As such, whatever was the
flange width you adopted in the analysis, the hinge would be formed due to
yielding of reinforcement wherever it has been located. The beam web, as
well as the flange attached to it, would rotate, when a hinge is formed.

I hope this addresses the question, but if does not, please do not hesitate
to ask if you have further questions.

With kind regards,

Sincerely,

Rizwan Mirza, CEO
Rizwan Mirza, Consulting Engineers
Lahore, Pakistan

On 30 March 2015 at 00:23, Alex C. Nacionales <acnacionales(--nospam--at)gmail.com>
wrote:

> Rizwan,
> 
> Thank You for your comments. It is very enlightening. I have a follow up
> question. If the beam is connected to a RC slab,  then part of the slab
> width
> could be considered as part of the beam- ( T-beam or half T-beam for
> exterior
> frames). How could a hinge form in the beam with the RC slab connected to
> it?
> 
> Thanks again.
> 
> Alex Nacionales
> Vancouver, BC
> 
> -----Original Message-----
> From: seaint-seaosc(--nospam--at)mail-list.com [mailto:seaint-seaosc(--nospam--at)mail-list.com] On
> Behalf Of Rizwan Mirza Sent: March 28, 2015 8:14 AM
> To: seaint-seaosc(--nospam--at)mail-list.com
> Subject: Re: [SEAINT-SEAOSC] "Big Column and small beam" Rule

Truncated 1268 characters in the previous message to save energy.

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