# Calculate the sag (deflection) of a membrane

• dykuma
In summary: The surface is spherical in the limit of small deformationsIn summary, the conversation is discussing the problem of finding the sagitta, or depth, of a membrane that is stretched over a boundary and subjected to a constant force. The equations used in this case make assumptions about the shape of the deflection and the surface tension of the material. The conversation also delves into the issue of applying additional tensions to the membrane and the shape of the deformation. Ultimately, it is determined that surface tension is a separate property from any horizontal tensions and that the shape of the deformation will not be a perfect sphere, but for small deflections, it will be very close. The conversation suggests solving Poisson's equation to answer the question.
dykuma

## Homework Statement

Not sure if this is the appropriate place to ask this question, but I didn't know where else to ask.

Anyway.

Let's assume that I have a film (or membrane) material that is stretched over some boundary, (lets assume a thin hollow cylinder). Then some constant force is applied over the surface (a pressure). Assuming the deflection is small, what is the sagitta (or depth) that the membrane is deflected by for a give material?

## Homework Equations

In this case, I've made several assumptions. First, the deflected part of the membrane is actually a very tiny slice of a sphere. As such, for small deflections I'm able to treat one surface of the membrane like the inside of a bubble (specifically the concave side of the deflection).

As such, my equations are:

Where P is some difference of pressure on either side of the membrane, Rc is the radius of curvatures, r is the radius (or half the span) of the un-deflected membrane,

For my purposes, I have no need to solve for the "sag" at the moment, as that information is encapsulated in the radius of curvature and can be easily calculated.

## The Attempt at a Solution

Obviously, with the conditions I have set, the solution to the problem is:

Anyway, the issue I am having is being able to solve this for a specific material. From what I understand, T is the surface tension of the material, but I am not sure this is always available. My guess was that:

T=E⋅t

where E is the Young's modulus, and t is the thickness of the material. However, I am not sure that is correct.

This issue is further complicated when I assume that some form of tension is already being applies to the membrane in order to stretch it. Do I need to take that into account as well?

Basically, my question is, if I have the produce sheet of a given material (like a rubber sheet), how do I obtain the value for T?

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dykuma said:
where E is the Young's modulus, and t is the thickness of the material
No, that's not how surface tension works. It really is a force within the surface, so is unrelated to the thickness, and quite distinct from Young's modulus.

dykuma said:
This issue is further complicated when I assume that some form of tension is already being applies to the membrane in order to stretch it. Do I need to take that into account as well?
No. You cannot apply additional tensions. The surface would simply stretch (without any increase in tension) to reach the wider range. It is not like elastic. The tension is entirely dictated by surface tension.

The surface will not be spherical in shape. For small deviations the surface deviation is going to satisfy Poisson’s equation, which is a second order differential equation. I would not call this an introductory physics problem.

haruspex said:
No, that's not how surface tension works. It really is a force within the surface, so is unrelated to the thickness, and quite distinct from Young's modulus.

I figured. So the surface tension is a separate property then. Is there a name that this property commonly goes by (besides surface tension)?

haruspex said:
No. You cannot apply additional tensions. The surface would simply stretch (without any increase in tension) to reach the wider range. It is not like elastic. The tension is entirely dictated by surface tension.

So the surface tension is entirely separate from any horizontal tensions I might use to "flatten" out the membrane under a pressure that previously deformed it. I was afraid that was true too. Where would I take such a situation under consideration then?

Thinking about it, I feel like stretching it would increase the overall restoring force over the membrane (naively thinking of this like a hooke's law problem). Is this true.

Orodruin said:
The surface will not be spherical in shape. For small deviations the surface deviation is going to satisfy Poisson’s equation, which is a second order differential equation. I would not call this an introductory physics problem.

I felt that it might have been an introductory problem given the assumptions made. I also know that the shape of the deformation will not be a sphere (I believe that the actual shape should be something like a catenary), but for very small deflections (much smaller than my very exaggerated image), it should be very close to a sphere, no? For me to answer my question, should I just go for solving Poisson's equation instead?

dykuma said:
I feel like stretching it would increase the overall restoring force over the membrane (naively thinking of this like a hooke's law problem). Is this true.
No, as I wrote, it is not like an elastic membrane. Suppose you have it as a flat rectangle bounded by four wires. If you slide a pair of wires apart or together the membrane will get correspondingly thinner or thicker, but the tension will not change.

Orodruin said:
The surface will not be spherical in shape.
It is given as stretched over the end of a cylinder and we are ignoring gravity. The pressure difference is the same at all parts of the membrane, and that dictates the local product of curvatures. By symmetry, would it not be spherical?

haruspex said:
It is given as stretched over the end of a cylinder and we are ignoring gravity. The pressure difference is the same at all parts of the membrane, and that dictates the local product of curvatures. By symmetry, would it not be spherical?
Yes. Sometimes I think too fast for my own good.

## 1. How do you calculate the sag (deflection) of a membrane?

To calculate the sag of a membrane, you need to know the membrane's tension, thickness, and length, as well as the force applied to the center of the membrane. The formula for sag is: sag = (F x L^2) / (8 x T x d^2), where F is the force, L is the length, T is the tension, and d is the thickness of the membrane. This formula assumes that the membrane is uniform and has fixed ends.

## 2. What is the significance of calculating the sag of a membrane?

The sag of a membrane is an important factor to consider in the design and engineering of structures such as bridges, tents, and suspension cables. It determines the amount of deflection or deformation that will occur under a given load, which can affect the overall stability and safety of the structure.

## 3. Can the sag of a membrane be negative?

Yes, the sag of a membrane can be negative, which is also known as uplift. This occurs when the force applied to the center of the membrane is greater than the tension in the membrane, causing it to curve upwards instead of downwards. This can be a concern in structures such as bridges, where uplift can cause instability and additional stress on the supports.

## 4. What are some common factors that can affect the sag of a membrane?

The sag of a membrane can be affected by various factors such as temperature, humidity, wind, and the weight of any objects resting on the membrane. Changes in these factors can cause the membrane to expand or contract, which can alter its tension and ultimately affect its sag.

## 5. Are there any limitations to the sag formula for membranes?

Yes, the sag formula for membranes has limitations and assumptions that may not apply to all situations. It assumes that the membrane is uniform, has fixed ends, and is only subjected to a single force at the center. In reality, membranes may have varying thicknesses, supports, and multiple forces acting on them, which can affect their sag. Therefore, it is important to use the formula as a guide and make adjustments based on the specific characteristics of the membrane and its application.

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