experimental stress analysis-chapter 8

Experimental Stress Analysis

Department of Mechanical Engineering

Unit 8: Moire’s Methods Moire’s fringe produced by mechanical interference:

Moire’s fringes are produce by arrays, these arrays are series of straight parallel lines, a series of radial lines coming from a point, a series of concentric circle, a pattern of dots.

Arrays consisting of straight parallel lines having nontransparent bars with transparent interspaces of equal width are most commonly used for experimental work. Such arrays are called grids, gratings or grills.

Circular grating parallel gratings

Crossed grating Radial gratings



Arrays having up to about 4 lines per cm are called grids. Arrays having from 20 to 400 lines per cm are called gratings. Two mutually perpendicular line arrays are termed as cross gratings. Generally having up to 200 lines per cm are most commonly used in this method. Two lines grating used in the strain analysis are, the model or specimen grating and

master or reference grating. Model grating can be applied by coating the specimen with a photographic emulation.

Master grating by bonding, ruling or etching. The model grating deforms with the test specimen where as the master grating doesn’t

deforms during test. The distance between the master grating lines is called the pitch P. The center to center distance on the model grating after deformation will be denoted by

P’.



By interfacing the two gratings a pattern of fringes is produced. The distance between the fringes is called fringe spacing and is denoted D. Any line perpendicular to master grating line will be called the principal direction and a

line parallel to master grating line will be called secondary direction. When the grating are superimposed one against the other that produces Moirés fringes,

either their having a difference in pitches, or difference in orientation. The dark fringes will appear at the point where the non-transparent strip falls over a

transparent strip. When two non transparent strips overlap it produces light fringe. For identification purpose, we shall denote the center lines of the dark lines of gratings

with number 0 to m for one and 0 to n for the other, and the center lines of the light fringes will be denoted by number o to N, where N is the fringe order.

The fringe order N and the parameters m and n identifying the lines of the two interfering gratings must satisfy an identical equation which is,

푚 ∓푛 = 푁 The family of Moire’s fringes for which the equation m-n=N is valid is the subtractive

Moire’s pattern, while the family of Moire’s fringes for which the equation m+n=N is valid is called the additive Moire’s fringe.

Moiré’s fringe analysis techniques Two different approaches are used in order to obtain strains by the Moire’s method. One is called the geometrical and other consists in relating the fringes to the displacement field.

Geometrical approach

Moire’s fringes are formed by two interfering lines screens or gratings. One is the printed in the model(model grating) is subjected to deformations produced by applied loads. The second( master grating) is applied on the top of the first. The pitch of the model and master gratings is assumed to be the same before application of the load. When the load is applied to the specimen the model grating deforms and gives rise to Moire’s interference fringes which can be studied for strain analysis. Knowing the distance between the master lines and measuring the distance between the fringes, it is possible by geometric analysis of the intersections of the two system of lines to compute the distance between the model grating line at a point, and the corresponding change in direction. With these data , normal and shear strains can e computed.



Let p= pitch of the master grating which is also equal to the pitch of the model grating before deformation.

P’= pitch of the model grating after deformation.

q= specimen grating orientation angle measured from the reference grating line to the specimen grating line positive if counter clockwise.

d= interfringe spacing, i.e the perpendicular distance between two neighboring fringes.

It is assume that before deformation orientation of specimen grating is same as master grating.

From triangle ABM

푐표푠 ∅ −휋2 =

푝퐴퐵

퐴퐵 =∅

=∅ ……(a)

From triangle ABC

푐표푠 ∅ −휋2 − 휃 =

푝′

퐴퐵

퐴퐵 =′

∅=

′

(∅ )……(b)

From equation a and b



푝′

sin(∅ − 휃) =푝

sin∅

푝′ = (∅ )∅

…………… (1)

We know that BF

From triangle BDE

sin휃 =푝퐵퐷

퐵퐷 = …….(c)

From triangle BDF

푐표푠 ∅ −휋2 − 휃 =

δ퐵퐷

퐵퐷 =δ

sin(∅−휃)…....(d)

From equation c and d

푝sin휃

=δ

sin(∅ − 휃)

δ = 푝sin 휃 sin(∅ − 휃)………(2)

δ푝 =

sin(∅ − 휃)sin 휃

δ푝 =

sin∅ cos휃 − sin휃 cos∅ sin 휃

δ푝 = sin∅ cot휃 − cos∅

δ푝+ cos∅ = sin∅ 1

tan휃

tan휃 = ∅

∅….....(3)



Equation 2 can be written as

sin(∅ − 휃) = 푝 sin휃……(4)

Substitute eq 4 in eq 1

푝′ =푝

sin∅δ

푝sin휃

푝′ = δ ∅

sin 휃….(5)

푝′ =δ

sin∅ sin휃cos휃cos휃

푝′ =δ

sin∅ tan휃1

sec 휃

푝′ = δ ∅

tan 휃√

……(6)

Substitute eq 3 in eq 6

푝′ =δ

sin∅sin∅

δ푝 + cos∅

1

1 + sin∅δ푝 + cos∅

푝′ =δ

δ푝 + cos∅

δ푝 + cos∅

δ푝 + cos ∅+ 2 δ푝 cos∅ + sin ∅

푝′ =δ

δ푝 + 1 + 2 δ푝 cos∅



Once the deformed specimen pitch p’ has been determined the component of normal strain in a direction perpendicular to the lines of the master grating can e computed as

휀 =푝′ − 푝푝

Displacement approach A Moire’s fringe is a locus of points having the same magnitude of displacements in the principal direction of master grating. Such a locus is called an isothetic. Therefore, a Moire fringe, an isothetic pattern, can be visualized as a displacement surface where the height of a point on the surface above a reference plane represents the displacement of the point in the principal direction of master grating. Two isothetic patterns are obtained using gratings perpendicular ti x-axis and y-axis, respectively, on the surface of a specimen under investigation. From these moiré gratings u and v displacements are determined by noting down the order of fringes Nx and Ny

Then

푢 = 푁 푝

푣 = 푁 푝……….. (1)

The Cartesian components of strain can be computed from the derivatives of displacements as follows:

휀 =휕푢휕푥

휀 = …………………. (2)

훾 =휕푢휕푦 +

휕푣휕푥

The slope of displacements as above are obtained by drawing tangents to the displacement curves of u and v fields along x and y axis.

Fig 1(a) shows the Moire fringes when the model grating is perpendicular to x-axis. Order of the fringes Nx, are marked as shown(fig.1a).



Lines along x and y axis say AB and CD are drawn. The displacement u along AB and CD are plotted by noting that

푢 = 푁 푝

Where p is pitch of the master grating.

Now fig shows the moiré fringes when the model grating is perpendicular to y-axis. Order of the fringes Ny are marked as shown. Line AB and CD along x and y axes are drown. The displacement v along AB and CD is plotted by noting that

푣 = 푁 푝

Where ‘p’ is the pitch of the master grating. From the plots of u versus y, v versus x, and v versus y, strains at any point are determined by using the relationships given by eq 2.

When the strains are large, then

휀 = 1 + 2휕푢휕푥 + +

휕푢휕푥 +

휕푣휕푥 − 1

휀 = 1 + 2휕푣휕푦 + +

휕푣휕푦 +

휕푢휕푦 − 1

훾 = sin

⎣⎢⎢⎡휕푢휕푦 + 휕푣

휕푥 + 휕푢휕푥

휕푢휕푦 + 휕푣

휕푦휕푣휕푥

(1 + 휀 ) 1 + 휀푦푦⎦⎥⎥⎤



Out of plane displacement measurement In certain plane stress problems and in a wide variety of problems involving laterally loaded plates, out of plane displacements ω becomes important considerations. A Moire’s method for determining out of plane displacements has been developed.

For out of plane displacement measurements, a master grating is employed in front of the specimen, and a collimated beam of light is directed at oblique incidence through the master grating and onto the surface of the specimen, as shown in fig. the shadow of the master grating on the surface of the specimen serves as the specimen grating. When the specimen is viewed at normal incidence, Moire’s fringes form as a result of interference between the lines of the master and the shadows. Use of a matte surface to ensure distinct shadows improve the quality of the Moire’s fringe patterns.

From fig it can be seen that the difference in distance between the master grating and the specimen surface at two adjacent fringe locations can be expressed as

푑 − 푑 =푝

tan훼

Where p is the pitch of the master grating and 훼 is the angle of incidence of the collimated light beam.



In practice, the master grating is located a small distance away from the specimen to accommodate any surface displacements toward the master grating and to serve as a datum plane for the measurement of load-induced, out of plane displacements. Any distribution of Moire’s fringes appearing with the master grating in this initial position will represent irregularities in the surface of the specimen. The presence of any irregularity must be accounted for in the final determination of the out of plane displacement.

If a point of zero out-of-plane displacement is known to exist at some point in the specimen, the master grating can be positioned to locate a Moire fringe over this point. At all other fringe locations, the out-of-plane displacement ω can then be expressed as

휔 =푛푝

tan훼

Where n is the order of the Moire’s fringe at the point.

Out of plane slope measurement/ Ligtenberg method The out of plane displacement measuring technique provide the required curvatures for a solution to the stress problem. In practice, the double differentiations cannot be performed with sufficient accuracy to provide suitable values for the curvatures. To overcome this experimental difficulty, Ligtenberg has developed a Moire method for measuring the partial slope 휕휔 휕푥⁄ and 휕휔 휕푦⁄ . A single differentiation then provides reasonably accurate values for the required curvature.

The essential features of the Ligtenberg method are as shown in fig. the equipment consists of a fixture for holding and loading the plate, a large cylindrical surface with a coarse line grating, and a camera for recording the Moiré’s fringe patterns. The surface of the plate is made reflecting since the camera views the image of the grating on the surface of the plate. Since the image does not depend on the angle of incidence of the light, a collimated beam is not required for this method. The moiré fringe pattern is formed by superimposing grating images before and after loading.

From the fig it can be seen that the location on the grating being viewed by the camera, as a result of reflections from a typical point P on the surface of the specimen, shifts as the plate deflects under load. the Moire fringe pattern formed by the superposition of the images provides a measure of this shift. Shift can be expressed in terms of the local slope of the plate as

푠 = 2휃푑

Where s= magnitude of shift

휃= local slope of plate in a plane perpendicular to un-deformed plate and lines of grating



d= distance between plate and grating

A Moire fringe will form upon superposition of the two images if the shift s is equal to the pitch p of the grating. Thus the order of the Moire fringe can be expressed as

푛 =2휃푑푝

휃 =푛푝2푑

The distance d should be large to minimize the effects of plane displacements 휔 on the shift distance s.

The angle 휃 given by eq 2 is the partial slope 휕휔 휕푥⁄ or 휕휔 휕푦⁄ depending on the orientation of the grating. Two Moire patterns of this slope will be needed to solve a plate problem completely. The two patterns can be obtained by rotating the grating 90 deg after the first pattern is recorded.