Prandtl Membrane Analogy Group III The elastic membrane analogy, also known as the soap-film analogy, was first published by pioneering aerodynamicist Ludwig Prandtl in 1903. It describes the stress distribution on a long bar in torsion. In this project, we demonstrated the Prandtl Membrane analogy and related it to the stress distribution in the beam of similar cross section. [Pick the date]

Negi, Piyush Y9413 Jain, Sambhav Y9508 Kumar, Navin Y9361 Meena, Abhijeet Y9 Boyina, Prudviraj Y9

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“Dedicated to our parents and gurus”

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Table of Contents

Acknowledgements ................................................................................................................................. 3

The Prandtl Membrane Analogy: Introduction ....................................................................................... 4

The Theory .............................................................................................................................................. 5

The Setup ................................................................................................................................................ 7

The Procedure ......................................................................................................................................... 8

Data Presentation ................................................................................................................................... 9

Conclusions ........................................................................................................................................... 11

Problems encountered ......................................................................................................................... 11

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Acknowledgements

Unselfish and noble actions are the most radiant pages in the biography of souls.

-David Thomas

We are grateful to Prof Rajesh Kitey, our instructor in-charge, who provided us this

wonderful opportunity to enhance our experimental skills. This project was one of its kinds,

which we experienced for the first time. We would also like to extend our thanks to Mr

Singh, the lab staff of the structures laboratory, who helped us by suggesting better

methods for our setup of the equipment. We are thankful to David, the student in the

structures lab, who helped us with handling of the sensors and the setup. Though this

project has been made by the five of us, it would not have been close to what it is now

without the experienced guidance and assistance of the various tutors and lab staff present

there at all times. Last but not the least we extend our gratitude to all of the friends and to

each and every person who made contribution to our project directly or indirectly.

--Group 3

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The Prandtl Membrane Analogy: Introduction

Shafts with noncircular sections are not uncommon. Although a circular shape is optimal

from a stress analysis view, square or prismatic shafts may be easier to produce. Also, round

shafts often have keyways or other geometrical features needed in order to join them to

gears. All of this makes it necessary to be able to cope with noncircular sections. We will

outline one means of doing this here, partly for its inherent usefulness and partly to

introduce a type of experimental stress analysis.

In an elegant insight, Prandtl pointed out that the stress distribution in torsion can be

described by a Poisson differential equation, identical in form to that describing the

deflection of a flexible membrane supported and pressurized from below. This provides the

basis of the Prandtl membrane analogy, which was used for many years to provide a form of

experimental stress analysis for noncircular shafts in torsion. Although this experimental use

has been supplanted by the more convenient computer methods, the analogy provides a

visualization of torsional induced stresses that can provide the sort of design insight we

seek.

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The Theory

We define the Prandtl stress function using

We can easily check that this definition satisfies equilibrium.

It can easily be shown that the traction-free boundary conditions are satisfied if

where, s is a coordinate system that is tangent to the boundary.

If the cross section is simply connected, then the boundary conditions are even simpler:

From the compatibility condition, we get a restriction on ϕ

where, C is a constant.

Using relations for stress in terms of the warping function ψ, we get

Therefore, the twist per unit length is

The applied torque is given by

For a simply connected cylinder,

The projected shear traction is given by

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The projected shear traction at any point on the cross-section is tangent to the contour of

constant at that point.

The relation between the warping function and the Prandtl stress function is

The equations

are similar to the equations that govern the displacement of a membrane that is stretched between the boundaries of the cross-sectional curve and loaded by a uniform normal pressure.

This analogy can be useful in estimating the location of the maximum shear stress and the torsional rigidity of a bar.

The stress function is proportional to the displacement of the membrane from the plane of the cross-section.

The stiffest cross-sections are those that allow the maximum volume to be developed between the deformed membrane and the plane of the cross-section for a given pressure.

The shear stress is proportional to the slope of the membrane.

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The Setup

Materials Used:

Plastic sheets: to use as membrane

Buckets: to mount the plastic sheets to get the membrane

Water: to apply pressure on the membrane

Hot Air Blower: to fix the plastic membrane firmly

Equipment Used:

Laser Displacement Sensor (LDS) and LDS holding frame

Software Used:

Lab View

Matlab

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The Procedure

The experiment gives the analogy of the stress distribution in a triangular cross section

beam with triangular shaped membrane.

The base of the bucket is cut to give the desired shape of a triangle.

The plastic sheet is fixed on the cut portion to give the membrane.

Hot air blower is used to give smooth surface to the plastic membrane.

Different points are marked on the membrane with the help of reflecting paint, so as to

use the Laser Displacement Sensor to measure deflections.

The position of membrane, which is uniform throughout the membrane due to its

smoothness, is noted by the Laser Displacement sensor.

Another bucket is filled with water, and the bucket with membrane at the base is dipped

inverted in the water up to a certain height.

The air inside the membrane bucket, which has been trapped between water and the

membrane base, builds up pressure inside the membrane.

The membrane deflects due the pressure build-up inside it, and the deflection of the

membrane at different points marked on it is stored with the help of the Laser

Displacement Sensor.

The initial reading of the Laser Displacement Sensor is subtracted from the final one to

give the deflection due to pressure inside the membrane.

The data thus received is used to relate and demonstrate Prandtl Membrane Analogy.

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The Data Presentation

The steps named above are performed for a membrane of quarter-circle shaped and the

displacements are noted at various points.

The data points obtained are:

-1.45805 0 5 -1.28786 1 5 -0.89353 2 5 -0.92117 3 5 -1.01222 0 4 -0.35945 1 4 -0.23435 2 4 -0.32634 3 4 -0.03808 4 4 -0.65639 0 3 -0.08654 1 3 0.167736 2 3 0.300479 3 3

0.43033 4 3 -0.15393 5 3 -0.46431 0 2 -0.05418 1 2 0.242494 2 2 0.439006 3 2 0.463684 4 2 0.234806 5 2 -0.15067 6 2 -0.26821 0 1 -0.02904 1 1 0.256063 2 1 0.395891 3 1 0.424112 4 1 0.517837 5 1 0.174966 6 1 -0.50205 0 0 -0.21741 1 0 0.050827 2 0

0.37114 3 0 0.255243 4 0 0.878758 5 0 0.355355 6 0

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Shown below is a plot of the displacement at different points vs. the co-ordinates of the

points.

Using this plot of displacement vs. the coordinates of different points we try to estimate the

shear stresses at different points.

From the plot its evident that shear stresses are lesser towards the centre of the

cross-section and is more towards the boundaries.

We then try to estimate the shear stresses by measuring the maximum slopes at the

different points.

Slopes:

First Point Second Point Slope

(2,0) (2,1) 0.205

Bottom Side (3,0) (3,1) 0.371

(4,0) (4,1) 0.169

(0,4) (1,4) 0.653

Left Side (0,3) (1,3) 0.570

(0,2) (1,2) 0.410

Curved Side (4,4) (3,3) 0.240

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Conclusions This experiment was instrumental in understanding the basic principles of Prandtl’s membrane

analogy and its possibilities in actual torsional stress determination in a closed and uniform section.

This analogy can now be used in determining shear stress without actually solving the complicated

governing partial defferential equation. This method can be extended to much complex shapes.

Problems encountered

Membrane - Theoretically the best membrane suited for this experiment is soap bubble due to its

uniform surface tension in all directions. But due to practical constraints we came down to two

choices. First was rubber balloon and second was plastic membrane. The rubber balloon had to be

ruled out due to its spherical built and difficulty in application to the expansion frame (the shape

having the same cross section as the cross section of the beam under torsion). Finally, the plastic

membrane was chosen and was made tight by blowing the hot air gun to make the surface tension

uniform.

Pressure vessel – Initially the plan was to use an air pressure device (air pump) but it was ruled out

due to fluctuations in air pressure. After that water pressure mechanism was selected in which a

pressure vessel with a tube protruding out to adjust water was to be built but due to complexities

involved the method was discarded for a new indigenous design. After that we decided to make the

things simple by pouring the water on the vessel itself to generate constant pressure. But due to

safety of the Laser Displacement Sensor it was decided to invert the pressure vessel and pour the

water on the surrounding to generate the pressure.

Readings – The method used to generate pressure didn’t allow for the longer time for observations

due to possible leakage and water vapour condensation. There were induced oscillations in the

water column inside the pressure vessel causing periodic variation in pressure on the membrane.

Also the problem with shorter reading time could have been avoided if manual reading process can

be replaced by some high speed profile scanner or holography.