# Bending of plates

1. Jul 17, 2011

### cve60069

Hello. First time on the forum.

I have a plate being supported off a series of beams (see pdf). The beams are close together and restrained. The plate is simply supported on all sides and the beams simply supported. The plate is to be used to distribute a point-load over the beams and is free to rotate. I need to analyse the deflections and stresses in the beams and plate.

To analyse the structure, I have assumed the beams to act as a plate with an EI equal to the EI of a single beam multiplied by the number of beams; the EI of the plate is known. The whole structure is allow to deflect 1mm say, so the deflection in the plate = deflection in the beams. I then work out P for a 1mm deflection. Apply the actual load and from the resulting deflection, calculate the stresses, etc.

Am I right in my thinking please?

Regards cve

#### Attached Files:

• ###### Plate.pdf
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2. Jul 17, 2011

### Studiot

First, welcome to Physics Forums.

I am having a little difficultly reconciling some aspects of your description, perhaps yopu would clarify further.

First if the plate sits on the ribs how is it supported at the edges?

Second what do you mean by the ribs are 'restrained but simply supported'.

3. Jul 17, 2011

### cve60069

Thank you for the response.

The ribs are prevented from moving laterally. The ribs are set into an opening and the slab cast upon the ribs. The slab is cast in such a way that all four sides are supported. I have a picture of the concept (attached). The ribs shown are much closer than drawn.

The slab is concrete and I am hoping to design it to be able to spread the load over the steel. The deflections expected will not exceed 3mm over 1200mm.

#### Attached Files:

• ###### Slab.jpg
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4. Jul 17, 2011

### afreiden

Two-way slab with point load at center?
I think you first need to know the actual area over which this "point load" acts. You need to know this in order to check punching shear in the slab.

Then, you want to know how this load is distributed to the ribs. The easiest way to accomplish this would be to find the "effective width" as described here: http://www.eng-tips.com/viewthread.cfm?qid=112794

I believe that you then have a way of distributing the load to the beams, using simple tributary area methods. Since I assume the ribs are composite with the deck, then just check their shear stresses, bending stresses, deflections, in the normal way?

You should check with someone else that the above is valid, as I am not a practicing structural engineer. I believe that what I wrote is conservative. There are other, more precise, ways of tackling the problem, of course. I think that your method may be flawed because in order to take the total "EI" of the ribs equal to simply the "EI" of one rib multiplied by the # ribs, don't you have to assume that the slab is rigid?

5. Jul 18, 2011

### cve60069

afreiden

The ribs are close together 200mm centres with ribs cut from a 76x127UB. If I assume that the slab (be it a perfect plate) deflected, then the ribs would "mould" to the shape of the slab. The ribs are not able to distort the shape of the slab, only support. The ribs would bend to the shape of a deformed plate. As the depth of the ribs are constant, I have assumed the EI to be a multiple of the ribs.

Thank you for pointing me to the effective width of a slab. This tip I think may have answered my problem concerning the load distribution and may make the analysis we are discussing unnecessary.

6. Jul 18, 2011

### Studiot

I am still troubled by your structural arrangements.

"simply supported on four sides"

So what is the purpose and connection of the ribs?
How are they supported?
How is the slab simply supported on the two sides perpendicular to the run of the ribs.

The whole concept of effective width applies to slabs supported on two sides only or very wide (theoretically infinitely) slabs so that support at the other two sides is irrelevant.

Westergaard et al assumed composite action in his slab analyses.
You description has specifically excluded this vis a vis the ribs.

Incidentally, this type of question would be best discussed in one of the engineering foums here.

7. Jul 19, 2011

Thank you.