1

THEORY FOR THE CALCULATION OF THE TANGENTIALRESIDUAL STRESS DISTRIBUTION IN CURVED BEAMS

J. Lo REMBOWSKI,Research Staff, General Motors Corporation,

Detroit, Mich.

ABSTRACT

A method is developed for the determinationof tangential residual stress distribution incurved beams of rectangular cross- section.The determination of such distributions isbased upon the fact that when a residuallystressed and thin layer is removed from theperiphery of the beam, the radius of curvatureof the beam changes. By knowing this change,·it is possible to calculate the stress in thelayer removed.

The method developed may be handled on adesk calculator, although it may be extremelylaborious. The method is more easily handledusing machine computation facilities.

INTRODUCTION

This report presents a theory for the cal­culation of the tangential residual stressdistribution in curved beams of constant rec­tangular cross-section. The theory is basedon the fact that if such a beam is residuallystressed and a thin layer is removed from itsperiphery the radius of curvature of the beamwill change. By knOWing this change, it ispossible to calculate the stress that was inthe layer removed. The assumption is madethat the residual stresses in the radial andaxial directions are negligible.

Pr.esented by title at the Spring Meeting of the Society forExperimental Stress Analysis in Cleveland, Ohio, May, 1958.

The equation developed in this report forthe residual stress in a curved beam is com­plicated and would be extremely laborious toevaluate with a desk calculator. However, theequation can easily be evaluated by a digitalcomputer. If the digital computer is not avail­able, the theory used for straight beams canbe used on curved beams without a prohibitiveerror. For curved beams in which the ratioof the radius of curvature of the centroid ofthe beam to the radial thickness is 10: 1, thetheory used for straight beams may be appliedwith an error of less than 4 percent. For aratio of 5: 1, the error is less than 10 per­cent[l]*. However, for smaller ratios and forgreater accuracy, it is necessary to usecurved beam theory. The theory presentedhere is not exact, but the results obtainedusing it are in very good agreement with thoseobtained using the exact solution,which isconsiderably more complicated. The error isless than one-half percent for a radius tothickness ratio of 4: 1 [2] .

THEORY

First it is necessary to UI}derstand whathappens to a residually stressed curved beamwhen a thin layer is removed. Consider thebeam of unit width shown in Fig. 1.

If in the infinitesimal thickness dz there isa stress s, the removal of the layer dz willremove a force s·dz. Now this force at thesurface of the beam can be resolved into a

* Superiors in brackets pertain to references listed at theend of the paper.

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(1)

(2)

(3)

N.O. 1

de/>M::-AeE~,

t - zM " - S(z,z)dz -.. 2

b b---de/> (e/>+d¢)-¢ R+dr-e R-e

~ :: ¢ b

R-e

R - e - R - dR + e R - e dR

(R-e)(R+dR-e)' -1- ,,- R-e'

VOL. XVI

where A is the area of the cross-section, e isthe distance from the centroid to the neutralaxis, and E is Young's modulus. The minus'sign is needed if M is defined as above.

Substituting (1) in (3), equating the right handsides of (2) and (3), and rearranging termsgives

The bar is assumed to be of unit width. M isdefined to be positive if the curvature of thebeam decreases. But the bending moment isalso give~ by [4]

if dR is very much less than R.

Finally consider the curved beam shown inFig. 3. Let the tangential residual stress dis­tribution in the beam be represented by S(Z,x)

at any time during the dissection.The variable z refers to the thickness of

metal removed from the outside surface, andthe variable x refers to the location of thestress below the original outside surface. Atany time during the metal removal x lies be­tween z and t, or is equal to z or t. The ini-

. tial stress distribution in the beam before anymetal has been removed is S(O,X), and theini­tial stress in the outside surface S(O,O)' Thestress in the 'new outside surface after a thick­ness z has been removed is S(z,Z). The radiusof curvature of the centroid of the beam is afunction of the amount of metal that has beenremoved. Let this function' be representedby R(z).

Now suppose that a thickness z has been re­moved from the outside of the curved beam,Fig. 3, and ~n additional elemental thicknessdz is being removed. The bending moment re­sulting from the removal of the layer dz isgiven by

PROCEEDINGSS.E.~.A.t.

FI G. I.

FIG.2.

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I-+-

I

pure bending moment and a force that pro­duces a direct stress uniformly distributedover the cross- section. For the latter theforce must be applied at the centroid of thecross-section [3]. If this is done, the momentarm o~ the pure bending moment is t/2 as in­dicated in the figure, and its magnitude iss· dz' t/2. The magnitude of the direct stressis s' dZ/t.

Now consider the beam shown in Fig. 2 asthe radius of curvature of the centroid changesfrom R to R + d R due to pure bending. Theneutral axis is denoted by n - n and the centroidby c-c. The angle.ep subtends an arc length b

on the neutral axis as does the angle ep + d¢

and e is the radial distance between the neu­tral axis and the centroid.

Now consider the ratio de/> Ie/> • This ratioappears in many of the equations of curvedbeams. To measure either quantity, however,would be difficult. But d¢/¢ may be expressed'in terms of the radius of curvature of the cen­troid of the beam which is easily determined.

THEORY FOR THE CALCULATION

Substituting Eqs. (9) into (8) gives

ER'(z)[(t-z) - 2e - 2m]dzdS b = (10)

[2R(z) - t + z + 2m][R(z) - e]

The total change in stress at section s- sdue to the removal of the layer dz is, adding(7) and (10),

s

FIG.3.

2AeE dR(z)S(z,z) : - ---

(t-z)dz R(z)-e'

But A: (t - Z), and d R (z) / d z = R' (Z), so

2EeR'(z)5(z,z) = - --­

R(z) - e

s

(4)

(5)

-2EeR'(z)dz

dS=dSd+dS b = [R(z)-e](t-z]

ER'(z) [(t - z) - 2e - 2m] dz

+ [2R(z) - t + z + 2m][R(z) - eJ' (11)

Now suppose that a layer z= t - m has beenremoved from the beam. The change in stressat s:ection s-s will be the sum of all thechanges from each elemental thickness dz re­moved. Therefore,

The cut dz produces a change in stress atthe section s-s, which is m units outside theinside diameter, due to a change in directstress and a change in bending stress. Thechange in direct stress is

Z i 2EeR'(z) dz

1 dS = S(z,z) - 5(0,z): 1o - o..[R(z)-eJ(t-z]

1i 'ER'(z)[(t - z) - 2e - 2m] dz

+ 0 [2R(z)-(t-z)+2m][R(z)-e]' (12)

The change in bending stress at section s - sis [4]

S(z,z)dzdSd= t-z

Substituting (5) in (6), we get

2EeR'(z)dZd 5 d : - -[-R-(z-)---e-J-(t---z)

My

dS b = Ae(r -y) ,

(6)

(7)

(8)

However, S(z,Z) is known from (5), and m = t- Z.Making these substitutions in (12) and re­arranging terms gives

2EeR'(z) i eR'(z)dzS(O, z): - ,+ 2Ef ...,.------=-=-----=-

R(z)-e o [R(z)-e][t-z]

i R'(z)dz+2E(t-Z)1 -

o [R(Z) - eJ[2R(z) + t + Z - 2z]

fZR'(z>[(t - z) - 2e] dz

- E 0 [R(z) - e][2R(z) + t + Z _ 2z]· (13)

Now as a first approximation [5],

where M is given by (2), y is the distance fromthe neutral axis to section s - s and is taken aspositive toward the center of curvature, A thearea of the cross-section, e the distance fromthe neutral axis to the centroid, and r the ra­dius of curvature of the neutral axis.

r = R( z) - e.

_ 2E(t-Z)2 R,(Z) Z (t-z)R'(z)dz5(O,z) = - _ _ + 2E1--~-----='

J2 [R (z )] 2 - (t - z) 2 0 12 [R (z )] 2 - (t - z) 2

_ i . J2R(z)R'(z)dz

+ 2E(t - z)~ {12 [R(z)] 2 _ (t _ Z)2}[2R(Z) + t + Z - 2z]

_ 1% [12(t-z)R(z)-2(t-z)']R'(Z)dZ (15)E 0 {12[R(z)J2 - (t - z)2} [2R(z) + t + Z - 2z] •

t - z~ = - S(Z,Z) dz -2-

t - zY=-2-- e - m ,

A= f-z,

Ee R'(z) dz (t - z)

R(z) - e

(9)

(t - z) 2e=--.

J2R(z)

Making this substitution in (13) gives

(14)

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S.E.S.A. PRoe EEDINGS VOL. XVI NO.1

CONCLUSiON

It may finally be seen that Eq.· (15) gives thestress at the depth X" i before any metal hasbeen removed. Therefore, the equation givesthe original tangential residual stress dis­tribution in a curved beam, if the radius ofcurvature of the centroid of the beam is knownas a function of the thickness of metal re­moved( Unfortunately, this will not be ananalytic function so numerical methods areused to evaluate the equation. In order to de­termine R(z), the radius of curvature of thecentroid of the beam, D(z), the inside diameterof the beam, is measured. Then R(z) for anyvalue is given by

INSIDE DIAMETER MEASUREMENT

It is essential that the inside diameter, 0 (z) ,

of the curved beam be measured with care;particularly when the derivative of D(z) is .small. At the Research Staff, the inside di­ameter is measured by means of a commer­cially available remote printing comparatorhaving a minimum division reading of 0.00005inch.

For example, if a curved beam having anominal inside diameter of 0.5 inch and a wallthickness of 0.05 inch experiences a di­

. ametral change of 0.00005 inch when a layer0.001 inch thick is removed, a stress changeof about 4,000 psi is indicated..

where (t - z) is the radial thickness of thebeam. By inspecting the equation for R(z) itmay be seen that R(z) will change if D(z) and/or(t - z) change.· The first term is due to bendingand the second is due to a change in thickness.The only change in R(z) that can be consideredhere is the change due to bending, so when thederivative of R(z) is taken, (t - z) must be con...sidered a constant. Then R'(z) for any valueof z is given by

D(z) t-zR(z) :: -2- + -2- ,

0' (z)R'(z):: --2 .

(16)

(17)

ACKNOWLEDGEMENTS

The author wishes to thank Mr. M. Simpsonfor review of the paper and Mrs. D. Jimisonfor typing of the manuscript.

81 BLiOGRAPHY

[1] Timoshenko, S., "Strength of Materials", (Part IT), D.Van Nostrand, New York, 2nd edition, 1941, p. 7l.

[2] Timoshenko; S. and Goodier, J. N., "Theory of Elas-ticity", McGraw-Hill, New York, 2nd edition, 1951, p. 64.

[3] Timoshenko, S., "Strength of Materials", p. 69.[4] Ibid, p. 67.[5] Ibid, p. 70.

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