My inferiority complex is coming back to haunt me.

t_n_k,

If you are inferior, what chance do the rest of us have?

 

Then WBahn tells me most electrical engineers are lazy and sloppy - more inferiority.

That engineers, of all kinds, are inherently lazy stems from our base motive in becoming engineers -- we want to make lots of things the primary purpose of which is to make our lives "easier"! By the time we discover that designing those "things" requires far more work that the work they could ever save us, personally, it is too late; we are hooked. But at least we can sell those things to others to make their lives easier.

I note the OP's problem is a special case of a more general problem as one might observe in the attachment [see one of my earlier posts] - case 6 is a special case of case 7. This addresses studiot's comment in part, about needing another more general formula to cover single point loading at some arbitrary location along the beam.

It seems to me one could therefore use this more general relationship to determine the location and value of maximum deflection. It would also include the special case of the loading being exactly at the beam center.

Oh sure. Just as in electrical engineering one could always begin with KVL and KCL and the constituitive relations for the devices and hammer away. But we develop lots of special case approaches that apply to special case situations that arise on a regular basis. We like to use simple models and then go out of our way, when possible, to design things so that the simple models are reasonable approximations.

The same is true, on steroids, for mechanical and civil engineering. Very seldom will a practicing civil engineer be called upon to compute the area moment of inertia of a beam. Why? Because the field has long since tabulated them for special case cross-sections and the engineer will, whenever possible, design their structures using those cross-sections. Why? Because it is quicker, cheaper, and less error-prone.

The same with the tables of beam deformations. Sure you could leave out ones that are special cases of others. You could simply expect them to be rederived from the same set of first principles directly each time. But using a more complex formula that is for a concentrated load anywhere on the beam for a load that is concentrated in the center introduces a greater risk of error that also providing an additional, simpler equation that just covers that common, special case. Why it is common? Because structures are deliberately designed so as to make it a common case.

 

t_n_k,

If you are inferior, what chance do the rest of us have?

Thanks - will you be my "shrink"?

 

Yes, t_n_k, you can have more complicated formulae in tables.

But the ones you posted would not do even for calculating a lintel over a doorway in real engineering.

The mech sciences have superposition too and will combine a bunch of such formulae for simple cases like house beams. They would not do for more complicated structures such as Sydney Harbour Bridge.

I see no reason for electrical engineers to feel inferior, they routinely use a higher level of math for everyday calculations than do the mech sciences. The mech sciences only start to compete when there is a need for something like FE analysis or perhaps in fluid dynamics. Chenmical engineers also operate a fairly sophisticated math level.

Back to beams.

It is just wrong to talk of negative deflection. All deflection is positive and the minimum deflection is always zero. This value must occur somewhere unless the supports are not rigid.
A deflection may be upwards or downwards and we use signs to indicate this, but cannot be 'negative'.

I have shown an equation that can be applied to and solved for the entire beam by using the step function.

I thought this might be of interest to electrical engineers who make considerable use of it.
I am sorry this doesn't seem to be the case.

 

It is just wrong to talk of negative deflection. All deflection is positive and the minimum deflection is always zero. This value must occur somewhere unless the supports are not rigid.
A deflection may be upwards or downwards and we use signs to indicate this, but cannot be 'negative'.

So we can have a beam that has a deflection of +1cm at one point and -1cm at another point but both deflections are positive?

I suppose that both compressive and tensile strain are positive, too?

 

So we can have a beam that has a deflection of +1cm at one point and -1cm at another point but both deflections are positive?

Yes indeed you can.

The deflection is always positive but the direction varies.

In case you think I am splitting hairs consider this:

A negative deflection would be if you took a rubber band and stretched it and the band got shorter.
I do not know of any occurrence of this in Nature.

The simplest example I can think of from electrical engineering is the negative Hall coefficient of Be, Mg, In, and Al.

 

Yes indeed you can.

The deflection is always positive but the direction varies.

In case you think I am splitting hairs consider this:

A negative deflection would be if you took a rubber band and stretched it and the band got shorter.
I do not know of any occurrence of this in Nature.

The simplest example I can think of from electrical engineering is the negative Hall coefficient of Be, Mg, In, and Al.

If I stretch a rubber band, it gets narrower. This is a positve deflection?

I squeeze a sponge, this is a positive deflection?

I stretch a sponge, this is also a positive deflection?

Strain, which is all deflection is, is the ratiometric change in the dimensions of an object. In the simple case of the length of a rod, the total length would be the sum of the nominal length plus the strain under stress. Since that strain can be in either direction, how can it be reasonably said that both strains are positive. Are the deformations on both the top and bottom surfaces of the beam both positive?

 

Since that strain can be in either direction, how can it be reasonably said that both strains are positive.

To continue the rubber band example.

I take the band in both hands and stretch it from a length L to 2L.

The left hand moves to the left in a negative direction, the right hand moves to the right in a positive direction.

The deflection is (2L - L) = L which is positive

Again I repeat a negative deflection would be negative (at 0.5L) if the overall length of the rubber band became say 0.5L after stretching.
The directions of movement of opposite ends of the band would still be edit: opposite as before.
This negative number can be imagined and put into the equations, but never realised in the real world.

Incidentally strain is not the same as deflection and should not be substituted for it.

 

Yes, t_n_k, you can have more complicated formulae in tables.

But the ones you posted would not do even for calculating a lintel over a doorway in real engineering.

Aha - now I know why the brickwork above the lintel in one of my doorways is cracking!

 

A negative deflection would be if you took a rubber band and stretched it and the band got shorter.
I do not know of any occurrence of this in Nature.

This statement made absolutely no sense to me and struck me as a red herring until I was laying in bed last night.

This is basically the same as saying that acceleration must always be positive because, if it weren't, it would be if you pushed on an stationary object to the left and it started moving to the right. Or, for a more applicable example to yours, if you pulled on a spring and it got shorter or compressed a spring and it got longer.

But the problem with all of these isn't that it would involve a negative deflection, but rather a negative coefficient, be it mass, spring constant, Young's modulus, whatever is relevant to the situation.

Take a spring of length L with a spring constant of k. dL=kF. F is the tension applied to the spring. If F is positive, dL is positive. If F is negative, dL is negative. The total length of the deformed spring is L+dL. There is no contradition with nature -- the spring deforms in the direction of the force acting on it, whether it be positive or negative. Now, if k were negative, you would have something very different.

 

This is basically the same as saying that acceleration must always be positive because, if it weren't, it would be if you pushed on an stationary object to the left and it started moving to the right. Or, for a more applicable example to yours, if you pulled on a spring and it got shorter or compressed a spring and it got longer.

Yes it indeed would. But this does not mean it does not make sense. What it means is that a negative acceleration/deflection/whatever is imaginary.

We can however put the negative number into the equations of virtual work and achieve useful results in less trivial cases.
Virtual work is not much used in electrical engineering, I have used it to perform magnetic calculations.

To try to show why the simple idea of negative deflection makes no sense I have drawn t_n_k's doorway as a frame.

The dashed line shows the greatly exaggerated deflected form so A moves to A' (slightly to the right and down) and B to B' (slightly to the left and down).

What would be a negative deflection at A or B in your opinion?

The diagram also neatly illustrates the difference between displacement and deflection that often coincide in simpler examples.

The displacement is measured from the original position

The deflection is measured from the deflected head line B'A'