ENGINEERING RESEARCH INSTITUTE THE UNIVERSITY OF MICHIGAN ANN ARBOR Final Report A PRELIMINARY REPORT ON TIE DETERMINATION OF STRESSES IN THIN PROPELLER BLADES Part 'I: T:h'eo~re tcali Stud'y C, N D;,eSilva Part II~ Ixper:imntal Study S. K. Clark Project 2528 WRIGHT AIR DEVELOPMENT CENTER WRIGHT- PATTERSON AIR FORCE BASE, OHIO CONTRACT NO AF 155(616)-5590 September 1957

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TABLE OF CONTENTS Page PART I. THEORETICAL STUDY 1 ACKNOWLEDGEMENT 2 PREFACE 2 L1o INTRODUCTION 3 2. PART A: THE AIRFOIL SECTION 3 Ao The Geometry of the Section 3 B. Formulation of the Problem C. Solution of the Homogeneous Differential Equation (2.11) 7 D. Loading of the Elliptic Cylinder 10 3. PART C: THE ROUND SHANK 11 A. Geometry and Formulation of the Problem 11 B. Solution of the Homogeneous Differential Equation (3.4) 12 Co Loading on the Round Shank and a Particular Solution of Equation (3,4)) 14 40 THE TRANSITION REGION B 16 A.' Introductory Remarks 16 B. Construction of a Developable Surface 16 C. The Orthogonal Geodesic Coordinate System 18 APPENDIX 23 Loading of the Elliptic Cylinder 23 REFERENCES 30 PART IIo EXPERIMENTAL STUDY 31 PREFACE 32 GENERAL STATEMENT OF BACKGROUND 33 OBJECTIVES OF TEST PROGRAM 33 METHODS OF LOADING 34 Direct Tensile Loads 34 Pure Bending Moment 35 INSTRUMENTATION 37 RESULTS 37 APPENDIX 39 Blade Geometry 39 ii

PART I 'THEORETICAL STUDY by C. N. DeSilva and P. M. Naghdi

ACKNOWLEDGEMENT The authors gratefully acknowledge the assistance of Mrs. B. T. Caldwell in the various parts of this report. The Appendix pertaining to the load on the propeller blade was prepared entirely by her. PREFACE In order to absorb the enormous power of modern aircraft engines, the propeller blade must be so designed that its airfoil section merges into the round shank at the spinner within a very short length. This region from the airfoil to the shank - the so-called transition region - becomes highly stressed during operational conditions, The bounding surface of this region is not simple; indeed, its mathematical specification becomes very difficult. The ordinary simple methods of stress analysis fail to predict even approximately the stresses developed in such a propeller blade. Thus, it becomes necessary to employ more stringent and accurate theories to determine the stress distribution. In the present report, we confine ourselves: to the "hollow blade" design of such a propeller and apply the bending theory of shells in an attempt to find the stresses within the propeller. This is no easy task especially for the transition region, where, as was mentioned, the bounding surface is mathematically difficult to prescribe. The present report summarizes only the theoretical work accomplished so far, work which is substantial indeed but still requires considerable effort to bring to completion. Hence, this report must be regarded as containing only the preliminary work on the project. 2

1. INTRODUCTION In order to apply the theory of thin-elastic shells to find the stress distribution in the hollow blade propeller, we have shown in Fig. 1 a simplified middle surface of the blade. It is evident by inspection that this middle surface can be subdivided into three component parts as follows.: Part A is the airfoil region. The transverse section of Part A (transverse implying perpendicular to the axis of the blade) is an ellipse. The airfoil region will therefore be treated as an elliptic cylindrical shell. Part B is the transition region. This iz the region in which the elliptic cylinder of Part A changes into Part C which is the round shank. The transverse section of Part C is a circle, allowing this region to be treated as a circular cylindrical shell. Before proceeding to a detailed examination of these component parts of the propeller, a word about the loading is in order. From physical tests, it has been found that the lift coefficient may be assumed, with negligible error, to exist only on the airfoil region. Thus, Parts B and C may be assumed free of surface loads while a variable surface load acts on Part A. All three blad regions, however, will be subjected to a centrifugal force due to rotation of the blade. For the sake of generality, however, no assumption will be made here about the character of the loading. 2. PART A: THE AIRFOIL SECTION A. TEE GEOMETRY OF THE SECTION The airfoil section is of constant thickness h = hA and will be treated as a thin elliptical cylindrical shell. The transverse section is an ellipse ro represented by x = al sin 0 (2.1) y = bl cos (2.2) where a1 and b1 are the lengths of the semi-major and semi-minor axes of the ellipse, respectively. The angle O - 0A, as well as XA, the arc length of o0, are measured clockwise from the vertical axis. Let a3 be the unit normal vector to the middle surface M of the shell and let ~ be the distance along a3 of a point from M, such that ~ = ~ hA/2 defines the two surfaces which are the boundaries of a shell of thickness hA. Then the position vector of a point of the shell is R LA R1 + ~ a3, '5

PA Ri_ PART C B PART A ~~~~mia~ ~ ~ ~ ~ ~~a B~~~~~~l r ~ ~~ ~~~~ p ____ I _____ j LE LCL- LA a-cz Fig. I. Ro~~~~~~~~~

where LA is a characteristic length and LA Ri denotes the position vector of a point on M. The equation of the line element of the shell is given by ds2 = gl 52 + dz2 + (2.3) where gll = a1 312 (1 + k12 cos2,) b (2.4) kL = (1- 1)/1 and xA = al B1 (1 + k12 cos2 t)/ dt (2.5) which can easily be expressed in terms of elliptic integrals. We also note that RA, the radius of curvature of the ellipse, is given by RA = a, ~ (1 + k12 cos2 S)3/ (2.6) and that RHi: RA(/2) = al o 1<. 1 (2.7) R~4:a RA(O) = B. FORMULATION OF TEE PROBLEM Following the theory of cylindrical shells, as given in Ref. 1., the determination of the stress distribution is reduced to establishing a stress function X and a dimensionless displacement w normal to the middle surface. With the introduction of dimensionless coordinates _ X=A -: =A (2.8) s ~A hA we define

LA LA Po P, PP - e, p LAP (2.9) Ag =r P dt, A = p dt 0 O In the above equations, Pg, P, and P are the components of the load intensity in the G, 9, and T directions, respectively LA s LA LA (A (2.10) LA = min s YBAmin LA is the span of the shell (see Fig. 1) and S is the circumference of the elliptic section To The differential equations for the deflection w and the function cD may be written as (Ref. 1, p. 422) LL(r) - i c2 'i,, = Q (2.11) where the comma denotes partial differentiation and L( ) c o?( ),oo + P2( ),: (2.12) = w+i K (2.13) Q = q + i Kq - = m =12(12 -p) E): E LA q = ~P ( p AQ +p) j (2.14) K E LA Y M 412(177~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~-)~~~~~~~

q = C4 A,Q + 4 A' gg v ~a22 (p, + PG, ) (2.14) p= _ =, (l + k. os2 )-/2 RA =alP1 Also the stress resultants and the stress couple may be expressed as N =, Ng = N ng, Ng - 0 ng M0 = LA 2mOe, Mg = - LA 0m MgG = - M = C LA m1 5(2.15) %Q = P %, Qg =a qg where B i v mg = - 2. w, + 2 w P a (2.16) =_ _ B(l-v)w nQ - =,( - A A; ng =,G ' A.; n@ = -'n, qg = mgO + mog, ' qg = mogO +mgg k3E LA hA 2(lv2) L. - (2.17) 12 (1 -v2) jP LA and E and v are Young's modulus and Poisson's ratio, respectively. C. SOLUTION OF TEE HOMOGENEOUS DIFFERENTIAL EQUA1TION (2.11) Assuming that, has the form

= eiY g(Q) (2.18) where y is some constant then with the use of (2.12), the homogeneous equation (2.11) may be written as d4g + C1 a + [C2 +c3 E 2()]g = O (2.19) do4 do In (2.19), Cj are constants defined as follows: C1 = -2, 22 C = i ' C3 i (2.20) Since from (2.5) and (2.8), - is a function of ~, being defined for 0 < ~ < 2x, we now transform (2.19) so that 0 becomes the independent variable. As a result of this transformation, (2.19) can be now written in the following form: d4g +2 + +9 +i Ag = s0 (2.21) dr d03 dod2o In (2.21),a the following definitions hold: = 3 k12 sin 20 A-1 ~ = k12 [4(1 +k12) + sin2 0 (7 k12 - 8) - 11 k-2 sin4 0] A.2 2 A4 4 ~o= kl2 cos 0 sin 0 [- 4 + 5 k12 + 9 kL4 - 18 k12 sin2 0 - 3 k14 sin2 0 - 6 k14 sin4 01] A (2.22) (?= t 04 (2 + i A28 a1r3 A1/2 A = (1 + k12 cos052 0) Recalling from (2.4) that kZ = 12 7 8

then under the restriction 0 < P <_ 1, we note that A is analytic and nonvanishing in I0: < 0<2A. It follows, therefore, that c, ', a, and are analytic -in I1o Since ~ appears only as trigonometric functions in the coefficients of (2.21), we introduce a new variable X as follows TX. = cos into the homogeneous equation (2.21) which becomes (1 _X2)2 giv + [_ 6X(1 _42) _ (1 -2)3/2a] g' + [- + 7h + + 3 aX(l _-%2)1/2+ (-l i_2)] g" + [X+. _y2) _/2( _2)(1 -:t)1/2] g, + 4 =g.O Here primes denote differentiation with respect to 7. After manipulation, this equation may be written in the form (_- 1)4 gi +() ( gj + Q( + a ) (- 1) g" + 63(Z) (- 1) gt + a4(%) g = o (2.23) where a =cT) = (1 + k122) X (1 + k12l2) (XJ+ 1) (1 +) [- 4 + (7 + 15 k2) 1 2 ) - 12 k2 4] (1 + k12,.2)2 (X + 1)2 2 32r2 ( o 2 (1 + k12) I (X- 1) a3 ( 1+ 1)2 (1+.k' ),(, 1) [(1 13k)-15k2 )-k1+k-) X + ~kl-4] ( (2.24) 9

a ) r4 (A 3 (1 + k 2X2)2 (- 1)2 + 2i 3 a \3 (1 + ki2X2)1/ (,L 1) + I It will be noted that the aj are analytic at 1= 1 and have poles at;= -1. It follows then, from the form of (2.23), that 76 = 1 is a regular singular point of the differential equation and solutions may be found which are valid in the circular region with = 1 as center and extending to the pole -T = -1. Assume a solution of the form 00 g(M) = Z cn (%- l)n+rj (2.25) n=O where rj are the solutions of the indicial equation r(r-l)(r-2)(r-3) +6Ll(l) r(r-l)(r-2) +a2() r(r-l) +a3(l)r +a4(1) = o From (2.24), at -= 1, we have a.(1) = 3 63(1) = l2(1) = 3/4 d4(1) = o Hence, the indicial equation reduces to the form r(r-1)(r-1/2)(r-3/2) = 0 the roots of which are (0 rj 1/2 (2.26) 3/2 Solutions of the form (2.25) will exist for r = 3/2 and r = 1. The other two solutions will probably contain a term in log X. Da LOADING OF THE ELLIPTIC CYLINDER The loading is described in detail in the Appendix. 10

3- PART Co THE ROUND SHANK A. GEOMETRY AND FORMULATION OF THE PROBLEM This component of the propeller blade will be treated at this time because it is a circular cylindrical shell, i.e., it is a special case of the elliptic cylindrical shell the theory of which was given in the previous sections. We have al = b, = R0 = 1 (3.1) It follows then that s = 2t Ro C = lc/Lc p = 2 = 2xRo/Lc (3.2) ICf mn {ek = hc/Lc Lc.= min o where 2c is the length of the round shank measured along the axis of the propeller (see Figo 1). The dimensionless coordinates Q, g, and r are now defined as xc Z 1B 1A e = - - S x =., Xc=~~~~s Ica~ ~ hc (3,3) c: o oC where the length xc and the angle C are both measured fromthe vertical axis, and he, which is a constant, is the thickness of the shell. All the previous expressions for the displacements and stresses still hold. The differential equation to be solved is again given by LL(*) - i c2 ~,g = 2 (3.4) where the definitions of (2.12), (2.13), and (2.14) hold and it must be noted that E2 is a constant because of (352), viz., c2 = 2m c~3X/ (3/5) 11

Bo SOLUTION OF THE HOMOGENEOUS DIFFERENTIAL EQUATION (3.4) Consider the homogeneous equation (354) which, by virtue of the definition of L(l), may be written as a4 +, 2 2 *2 + 4 -a4 C2 22 - i c62 - (35.6) Let = n(~) eint9 (357) Then (305) becomes:iv (2 ~n2l2 E fi n4lr4Gtf4 fiV(2 n2ir2 +. fn 4 fn = 0 (3-8) where primes denote differentiation with respect to ~. Assuming solutions of (357) in the form fn = eb (3-9) we have from (35.8) ~2X2a2 n4g4Ct4 54 - 2 (na + i to ) + = 0 (3.10) where 2 mP. 2m Ro =o = = (3-11) At once, then =2 =2C2 [(n+ i Lo2) ~ lo (-Lo2 + 2i n2)1/2] + 2 n/2 [(2n - 2 2)1/2 + i (e2 + _ o) 1/2] Q4 = 4 + 4 n4 12

and hence 52 = n * (10 1/2 i 012 + (12 + l02) /2: 21 (Note that plus sign goes with plus sign, minus with minus, from the previous equation. ) Let rl2 = n4 + l04 + 1oA Q2 n + o 10 2 1 r2: 4 + lo4 + o2 -J l {n (2 _ 102)1/2 + o 2 (72 + Ao2)/} then (F to 12%1/21/ +- - n2 0 (2 2)1/21 1/2: cn + i dn (3.12) C1 2/1/ = ~ IL+ n..2 * — 1 ) - i [r2 -n+ (n - 2)1/2: (hn + i kn) 313 Then +(ecn + i dn)g ~n(~) e+(hn + i kn)~ (3.14) 15

and the homogeneous solution ' may, therefore, be written from (3.7) as H = E [An (Cn+idn)g B (cn+idn)g n=l + Cn e(hn+ikn)g + Dn e (hn+ikn)] in (3-15) C. LOADING ON THE ROUND SHANK AND A PARTICULAR SOLUTION OF EQUATION (314) Consider the centrifugal loading -defined by pA) = po ohA (QT - ZA) 0 < ZA <_ A pB) p( = po 2hB ()e + C + B ZB) O < Z<B (3.16) (C) P) = po D 2hc (ai + C - ZC) 0 < ZC < lC where w is the angular velocity and the superscripts identify the component of the propeller. We rewrite (3.16) in the form P(A)() = Po mhA OA (A P( )() p cO 2hA 2B (1 + bt) ( - (3-17) P pO hC I) C= and note that aS a consequence of this and using (2.9) )A P d5 AgB)() = a OpB) dg + A(A(3.18) A (C) o P(c) d + (B) (1) A~ (g) = -- P d~+A In (3,17), we have assumed that hA and hC are constants while hB varies. linearly 14

hB = hA (1 + ) (3.19) b = ~-1 hA and we have defined (see Fig. 1) LeC ~= c + aC (3,20) ~ECB = I + B = AT iA Integrating (3.18) after substitution of (3o17), we have A(A) = a h A Li A Q) 'iPo po hACA O - A(B)() LB ECA B E - CB +A (A ) (3.21) Ag pow =2 L S 1W b- 2 j +A~ (i) +321 (C) LC (B) Az (C) '= Po:h.cLTE~a j.A (1) a C I,.c Assuming Po = P = 0 then from (2.9), (2.13), and (2.14) Ag = 0 q = 0 L q = v O 2 hC ' = qo (3.22) Q= i E K% and (304) may be written as a + 2 Aa~,, + -4 i 2 = i K q A particular solution of this equation is 15

*p =K q 2 ha Substituting the expressions for K, qo, and c2, we have IP = _v (jc2 Ro) L O a52 a (3~23) We have finally *H * = PJ r(3. 24) where IH -is given by (3 15), *p by (3.23). 40 THE TRANSITION REGION B A.o INTRODUCTORY REMARKS The transition region, as its name imprier, refers to the transition from the airfoil section to the round shank of the propeller blade. Consequently, at one boundary of this region, the transverse section is circular while at the other boundary, the transverse section is elliptical. Considering the transition region as a shell, its middle surface is arbitrary, satisfying only the requirements that one -edge is circular, the other elliptic. Because of the complexity of the equationns governing the deformation of an arbitrary shell, some simplification is necessary. Rather than omit terms from these equations whose effect cannot be truly paredict-ed, it is proposed to require that the middle surface of the transition region be developable. This means that the expressions for the Christoffel symbols of such a middle surface will be simplified so that on the surface the square of the line element may be expressed in the form ds2 = du2 + dv2. As a consequence, the membrane equations of equilibrium can be solved in terms of a stress function and its first and second derivatives. To summarize, therefore, we must find (i) A developable surface such that its boundary transverse sections are circular and elliptical. (ii) Orthogonal geodesic coordinates u,v such that ds2 = du2 + dv2.o Bo COJNSTRUCTION OF A DEVELOPABLE SURFACE Let z be the axis connecting the centers of an ellipse and a circle which are parallel to each other, 2B be the length from center to center along z 16

(see Fig. 1), and let the center of the ellipse be the origin of our coordinate system. Consider a point Po on the circle given by P: (Ro sin -c, Ro Cos )e ~ aA + QB) (4.1) and a point Pe on the ellipse specified by Pe: (a1 sin OA. b-l cs 1 A, A) (4,2) where Ro is the radius of the circle and al, bl are the semi-principal axes of the ellipse (see Fig. 1i) By projecting the line P P onto the x-z and y-z planes, it can be shown that the position vector P PoT is given by r RV 0 sin + sin 0eA Rcos c+(1 Z* blCOs A;z + LA] or equivalently r: [ Ro sin 0c + (l-) al sin PA: Ro Cos Xc + (1-) bl eos A i 1Bt + A] (4.3) where Z* = Z 1 aA = z*/IB Equation (43.) defines a ruled surface whose Gaussian curvature K must vanish if the surface is required to be developable, i.e., LN M M K L.MF = 0 H2 (4.4) LN- M' = 0 where, in the notation of Ref. 7, L = - 11, M- = n - r12, N - n.r22. +. - - rlxr2 x H = Irl x ral (4.5) + a+ a__r rC -=., r~ - a'~1 17

Sl, 52 being the parametric curves. We next choose z and OA as the two independent variables, and note that since rll = 0, L vanishes and with the use of (4.5 ), our condition for developability (4.4) becomes ra (rl x r) = 0 (46) Evaluation of (4.6) using (4.3) yields d C — A[bl sin OA cos 0C - al cos OA sin C] = 0 d OA whence we have the developability condition tan OA = 11 tan OC (4.7) where a, The required developable surface, after substitution of (4.7) into (43 ), is therefore given by x = sin C [Ro* + (1-) aiij 1+ tan2 C = cos C joC + ( 1-R) b1 1 + tan2c / (4 RO b 1 + r tan2 OC z = QB + QA Co THE ORTHOGONAL GEODESIC COORDINATE SYSTEM In finding an orthogonal geodesic coordinate system, we adopt the procedure given in Ref. 8, pp. 136-145. Since any straight line on a surface is a geodesic, it follows that OC = constant represents a system of geodesics. To this end, we define ul = 1 - v; V1 = C R =,,; b = -b (4.10) bl LB ~r = e1 tan2 vl } ' = ( J5

and therefore rewrite (4.9) as x y z rD -LB LB X = b cos vl [R + ul(-R + Ti2 f)] LB (4. l.) Y = b sin vi [R + u1 (-R + f)] LB z_ -_ Q1/2 ul +B A LB LB Introducing a new coordinate u2 and assuming ul = Ul(u2,V1), we can write ds2 = E2 du22 + 2 F2 du2 dv1 + G2 dv12 (4.12) where E2 = a rD a r AD A o ( au 2 a U a2 A aFv i avl \1/ = v —rD. - (~ +B +2(-i) and Ao(V2) = b2 [RZ + 1 - 2 R f _ 2 (1 2) cos2 V1 f2(v1) + I/b2] B(vl) = b2 [R + u (-R + 2 f)] (4.14) C(v!) = b2 (1 - 2) cos v1 sin vl f [R + u1 (-R + 12 f3)] It must be pointed out that use has been made of the identity 19

-ft= d - - (1 - T12) sin v1 cos vl f3(vl) dv1 Requiring u2 and v1 to be orthogonal, i.e., F2 = 0, we have ___aul~~~ C ~(4 15) )vl AO or equivalently, by using (4014) U + ql ul = q2 (4.16) Cv) where ql(vl) = (1 - r12) cos vl sin v1 f (-R + Ai2 f3) Ao (4.17) q2(v1) = -R (1 - 1h ) cos v1 sin vl f Ao Noting that qi dvl = -log A we can formally write the solution of (4.16) as U1 = _1/ - G*(v3) A (vI) (4.18) RG*((1-2) ' cos V1 sin vI f(vl) G*(Ml) = dvl fi72 1/2 With ul as defined by (4.18), (4.12) may now be written after substituting for E2 as given by (4.13), as ds2 ' Ao (u du2 + G2 dvl2 (4.19) We now set u3 = S 1o/2 Ul ua 20

so that (4.19) becomes ds2 = du32 + G2 dVl1 (4 19a) From (4.18) auz 1 dg(u2) au2 A1'/2 du2 U3 = df d dU2 = g(Uand (4.18) becomes U3 ul G ''*(v) (4.20) O /2 from which u3 may be expressed in terms of u1, and v1. Applying the orthogonality condition (4.15) to the-expression for G2 in (4.13), we have G2: B - = B [1 - (1-q2)2 cos2 v1 sin2 vj f2(v1)Ao ] AO Substituting for B from (4.14) and for ul from (4.20), we finally obtain 4 = C1(v1 U3 + C2(vl) (4.21) where ( 2)2 cos2v1 sin v2 c1(v1) - in vl{.2 f( [R + 1/2 3(Vl) Ao (v) Ao / (vI) (4.22) {1 (l.-r12.)2 cos2 Vl sin2 vL f 1 G*(vl)) C2(V.) = {,- -.I(v11/2R. _~ R+iT2f3(V.1) -o(vi) If Ca(v*) = O, then if we let V = f C1(v1) dv1 (4.23) 21

then (4l19a) becomes ds2 = du3 + u32 dv22 (4.24) Finally, if we define U = U3 COS V2 (4.25) v = u3 sin v2 Equation (4.24) reduces to the form ds2 = du2 + dv2 (4.26) 22

APPETNDIX B. T. Caldwell LOADING OF TEE ELLIPTIC CYLINDER In the i-direction, there is an effective loading component Pg() = Po a) hA A - (Al) where po is the density of the material, w is the angular velocity, and IT = IA + ~B + IC + 1c as shown in Fig. 1. In the G-direction, the loading component is negligible, i.e., P, = 0 In the ~-direction, we assume P to have the following form: P( 1) = po [1.- S(G)] PIl() (A2) where po is a constant, and 00 Pi(I) = S an+l sin (2n+l) sq (A3) n=O In (A3), the coefficients am will be so chosen that pl(t) varies as an ellipse of major axis IA and of arbitrary minor axis. In order to find a representative form for S(Q) in (A2), we select a typical pressure distribution for the airflow around a thin airfoils To this end, we make use of the data given on p. 328 of Ref. 2 for an airfoil which sufficiently approximates the elliptic section so that the general nature of the results will hold for our study. The formula used for the determination of S is taken from p. 77 (loc. cit.) as are the following quotations with our comments in parentheses: "The velocity distribution about the wing-section is thus considered to be composed of three separate independent components as follows: 1a The distribution corresponding to the velocity distribution over the basic thickness form at zero angle of attack (v). 2. The distribution corresponding to the load distribution of the mean line at its ideal angle of attack (Av; in this particular case Av = 0 because the airfoil and the ellipse are both symmetric about the horizontal axis). 23

30 The distribution corresponding to the additional load distribution associated with the angle of attack (Avg). "The local load at any chordwise position is caused by a difference of velocity between the upper and lower surfaces. It is assumed that the velocity increment on one surface is equal to the velocity decrement on the other surface?" And from p. 79 comes the remark "although this method of superposition of velocities has inadequate theoretical justification, experience has shown that the results are adequate for engineering uses." If V is the velocity at infinity, Po the pressure at infinity, and Pa the mass density of air, then the pressure Pe at any point on the surface of a cylinder of infinite length is (cf. p. 77, Ref. 2) e= Hp Pa V2S (A4) Pe = p - p Pa VS where - p = Po + PAV (A) and S {v I + av' (A6) the plus sign being used for the upper portion of the external boundary surface, the minus sign for the lower portion. S is readily found since v/V and Ava/V are tabulated on po 328 of Ref. 2 for different values of percent of chord where the chord is the line of symmetry in this case (chord has the usual aeronautical definition). This means that the S computed for a given percent of chord is the value at the upper or lower intersection of the boundary surface with the perpendicular to the chord at the given percentage of chord length. We now assume that the S of the airfoil at a given percent of chord is identical with the S of the elliptic middle surface at the same percent of chord. Knowing the percent of chord of the ellipse, we can find the length of arc corresponding to it. In particular, we have { (X FAM), XA >XAM o< 0< <1: (A7) AM X A < x24

- 100 - al[E(O,,1/2)+E(Qt,I] 1/2 < X. < 1 < < i 100- '2 3xA (A8) ai[5E(,xj/2)-E(:', ], 1 > ' > 1/2, 0 < <-100 - 2 aj[3E(Q',,/2)+z+E(o,1 1/2 > x~ > 0 o,<_ 2 -100 2 0<XA <g E(e'i,) = f [1 - sin2 0' sin2 t]1/2 dt s' = 4E (Qt,ic/2) (A9) O' = are.'in (1/2 -' = are Sin (1 - _i2)/ 100 In the above equations, x, is the percent of chord, xA is the arc length of the ellipse measured counterclokwise from the point (.a1,0), and @ is a dimensionless coordinate whose origin occurs at XAM, the point of maximum pressure on the boundary surface. 'This point is determined from the condition [see Equation (A4) ]: V V Evaluating the elliptic integrals E(.0-,~) by means of Refs. 4 and 5 and using Equations (A7) to (AlO), we compute S(Bj) at which are discrete values of @. This computation is given in Table I. The values of the constants u;sed are. a1 = 5-3, bl - 0.2 (All) e:= 87.8573820 whence, from (A10O) and po 328 of Ref. 2 XAM = 0.012042 (A12) 25'

TABLE I CALCULATIONS LEADING TO THE FUNCTION S(Oj) AND TE FNAL RESULTS FOR S(Q) XA v A v v v E cQ; xA/al Qj - + S ( 9j) X1 x2/al E(;.) XA/5 v - s- 0%-+ - s (ej) s(o) 0 -1 90~ 1.00296504 0 0.987958 0 5.471 -5.471 29.93 29.93 1.25 - ~ 975 77.161432~.97585945.02710559.006756.994714 1.029 1.376 -.347.120 25.45 2.3 -.954 72.5540120.954654.530.04831051.012042 0 1. o40 1. 040 o o o 2.5 -.95 71.805128~.95062706.05233798.013046.001004 1. 042.980 +. 062.00oo38.008 5. -.9 64.158067~.90040715.10255789.025564.013522 1.047.689.358.128.109 7.5 -.85 58.211669~.85028905.15267599.038056.026014 1.051 ~ 557.494.244.207 10 -.8 53.130102~.80021254.20275250.050538.038496 1.053.476.577 335.301 15 -.7 44.427004Q.70011909.30284595.075488.063446 1.055.379.676.457.470 20 -.6 36.869898~.60006632.40289872.100427.088385 1.05'1.319.738.545.614 30 -.4 23.578178~.40001683.60294821.150291.138249 1.060.241.816.666 -747 40 -.2 11. 536959~.20000195.80296309.200147.188105 1.064.196.868.753.75 s50 0 0~ 0 1.00296504.25.237958 1.066.160.go906.821 ~705 60.2 11.536959~.20000195 1.20296699.299853.287811 1. o68.130.938.880.712 I1) 70.4 23.578178~.40001683 1.40298187.349709.337667 1.0 o64. 104.960.92.822 (3\)~ ~ 80.6 36.869898~.60006632 1.60303136 ~.399573.387531 1.051 '~.077.974.949 1.001 90.8 53.1301020.80021254 1.80317758.449462.437420 1.017.0 9.968.937 1.160 95 ~. 9 64.1580670.90040715 1.90337219.474436.462394.981.032.949.901 1.204 100 1 90~ 1.00296504 2.005938.5.487958 0 0 0 0 1.217 95.9 64.158067~.90040715 2.10848795.525564.513522.981.032 1.013 1.026 1,198 go90.8 53.130102~.80021254 2.20868256 ~ 550538.538496 1.017.049 1.066 1.136 1.156 80.6 36.869898~.60006632 2.40882878.600427.588385 1051.77 1.128 1.272 1.071 70.4 23.578178~.40001683 2.60887827.650291.638249 1.064.104 1.168 1.364 1.O57 60.2 11. 536959~,20000195 2.80889315.700147.688105 1.068.130 1.198 1.435 1.241 50 0 0~- 0 3.00889510.75 ~ 737958 1.066.160 1.226 1.503 1.595 40 -.2 11.536959~.20000195 3.20889705s.799853.787811 1.064.196 1.260 1.588 1.974 30 -.4 23.578178~.40001683 3.40891193.849709.837667 1.060.244 1.304 1.700 2.155 20 -.6 36.8698980.60006632 3.60896142.899573.887531 1.057.319 1.376 1.893 1.945 15 -.7 44.427004~.70011909 3.70901419.924512.912470 1.055 ~ 379 1.434 2.056 1.672 10 -.8 53.130102~.80021254 3.80910764.949462.937420 1.053.476 1.529 2.338 1.372 7.5 -.85 58.2116690.85028905 3.85918415.961944.949902 1.051.557 1.608 2.586 1.099 5. -.9 64.158067~.90040715 3.90930225.974436.962394 1.047.689 l. 736 3.014 2.508 2.5 -.95 71.805128~.95062706 3.95952216.986954 ~974912 1.042.980 2.022 4.088.748 1.25 - ~ 975 77. 161432~.97585945 3. 98475455.993244.981202 1.029 1.376 2.405 5.784 5.792 0 -1. 90~ 1.00296504 4.01186014 1..987958 0 471 5.471 29.93 2993 he data in this column are taen from the experimental data of Ref., p. 328. ~The data in this column are taken from the experimental data ofw Ref. 2, p. 328.

There remains the task of finding a simple functional relationship between S(Oj) and -. We note from the results of our previous computation that S has a high peak occurring.at =. =1 - AM J o s S(jo) - Sj o = 29.665 (A13) We assume, therefore, that S has the form 00 S(Q) = Z Bn sin n~O + F1(@). (Al4) n=l F1(q) = A1 [sin2 2 m3ce] +( + (A15) and impose the following conditions for the determination of F1(@): (i) Fl(lo) = A1 + 8, 8 > o (ii) =1( o Jo (iii) F.(0joz) = -So-4 ) = S5.73 Condition (i) implies that.XAM sin 2 mi - - sin - s 2 m = 21 (A16) sin2 m2 l )= 0.9996 From (A16) and ( ii ) A1 sin 2 mt (1 - = 29.665 27

i.e., A1 - 29.68 (A17) From (A15) and (A17), and from the computed data noting that jo-1 = 0.981202 (0.993244) k = 0.4801 i.e.,, ko = 108 (A_8) where the nearest integral value of ko is used. The values of Bn are evaluated in the usual manner yielding B1 = 1.451 B3 = 0.5676 B2 -0.4966 B4 = -0.3070 (A19) B5 = 0.53280 Five terms of the Fourier series are sufficient for satisfactory convergence. We have, therefore S(-) = n Bn.sin._n + 29.68 [sin2 42 xe] (@ + 0.012042) (A20) n —l The plot of S(G) together with that of the function S (j) is shown in Fig. -2. 28

30 \I, (1) w268 0. Z z Z r S(j) From the experimental data of Ref. 2 c>.0 S(8) ---- Assumed functional representation 0 cn 4 z 0o 0 0 ~~,EI 'm — i 0 0.2 0.4 0.6 0.8 1.0 DIMENSIONLESS PARAMETER B AS DEFINED BY EQ. (A7) Fig. 2.

REFERENCES 1. A. -E. Green and W. Zerna, Theoretical Elasticity, Oxford Univ. Press, New York, 1954. 2. Ira H. Abbott and Albert E. von Doenhoff, Theory of Wing Sections, McGrawHill Book Co., New York, 1949. 35 Mlthematical Tables Project, Table of ArcSin X, Columbia Univ. Press, 1945.5 4. Tables of the Complete and Incomplete Elliptic Integrals, Reissued from,,,,i-.,,,,m.. _, __,. -.,..:_'_.:' _.... Legendre.'s Traite des Fonctions Elliptiques, Paris, 1825, with an Introduction by Karl Pearson, F.RR.S. Cambridge Univ. Press, New York, 1934. 5 N Ma-hematical Tables Project, Tables of Lagrangian Interpolation Coefficients, Columbia Univ. Press, 1944. 6. Edoardo Storchi, "Integrazione delle equazioni indefinite della statica dei veli tesi su una generica superficie," Atti della Accademia Nationale dei Limei, Rendiconti, Classe di Scienze fisiche, Mathematiche e naturali, Volume 8, 1950, p -326. 70 C. E. Weatherburn, Differential Geometry of Three Dimensions, Cambridge Univ. Press, New York, 1927o 8. D. J. Struik, Iectures on Classical Differential Geometry, Addison-Wesley Press, Cambridge, 1950.

PART II EXPERIMENTAL STUDY by S.O K. Clark

PREFACE Models of three geometries. of thin propeller blades were cast of aluminum alloy 3555 and used for an experimental stress analysis comparing the effect on stresses of various transition sections' The blades were identical in both the hub and outboard (airfoil) sections, and differed only in the transition regions. Strain -measurement in the transition region of each blade was accomplished by the use of stress-coat and electrical resistance strain gauges. Loading:fixtures were constructed which permitted the application of pure tensile loads and of pure bending moment about either the flat-wise or chordwise aaxis of the blade. Two blades were loaded in tension and one in bending about the flat-wise axis in order to-demonstrate the feasibility of the loading fixtures and the strain-measuring techniques. 52

GENERAL STATEMENT OF BACKGROUPi Work on the experimental phase of this program began in April, 1956, and approximately five months were consumed in planning tests and in the design of propeller blades and loading fixtures, 'Serious difficulty was encountered in finding acceptable methods of manufacturing the blades, since machining proved to be too costly and since only one foundry was willing to attenmpt direct casting on the blades~ In view of the limited. financial resources of the project, it -was decide-d to attempt casting, and a contract for this was let to Pressure Cast Products, Inc., 1028 Vermont Ave. Detroit 16, Michigan, in November, 1956, with delivery promised for February 1, 1957X Considerable difficulty was experienced by this vendor in obtaining any kind of a casting, and after several months of unsucceassful trials a casting consultant was engaged by the project to provide advice to the vendor, Following this, complete castings were obtained, the last of the series being delivered July 20, 1957. Thz castings as received were not acceptable from the standpoint of: dimensional tolerances and contour shape -They were, howomever, sound and free from obvtious flaws, and in light of this and considering that the project was now seriously short of time for completion of any testing it was decided to accept the Castings. These castings were then machined and assembled in the fittings, and trial tests were run on two blades in tension and on one blade in bending. All fittings operated satisfactorily, and the method of applying loads and moments appears to be a.sound one. OBJECTIVES OF TEST PROGRAM The general.objective of the test program was to provide experimental. data which could be used aS verification of certain calculations to be made in the second year of theoretical work on this problem. It was decided in conference with Prof. P. M, Naghdi, director of the theoretical group, that it would be des.irable to calculate from theory and to measture experimentally the stresses produed in an actual aerod.ynamic loadbing and by actual centrifugal loads. The experiment was originally designed to include loads as shown in Fig. 3. The result of the loadts is that each cross section of the transition region is acted upon by the following forces: (a) A constant tensile force due to the mrw2 of all material outside of the traxsVition regiono (b) A variable tensile force due to the body force mrw2 acting throughout the transition region. (c) A bending moment which is variable and which is dlue to the ipresence of the aerodynamic pressure distribution. This bending moment has

T lInk x PRESSURE CENTRIFUGAL FORCES I /y Fig..3. components in both the flat-wise and long-wise directions of the oblade. (d) A shear force due to the presence of the aerodynamic pressure distribution o (e) A twisting moment or couple due to the possible eccentricity of the resultant of the aerodynamic forces, It wasa decided to attempt to piece the complete loading together by performing a series of tests and by combining the results by superposition. In order to do this effectively, it was planned to request funds for a.centrifugal test (cov ering loading "b" in the list above) and for an actual shear and bending loading test (items "c" and "'d" above) during the anticipated second year of operation of the project. When it.became evident that funds were not available for the second year of effort, it was decided to complete that part of the testing which would demonstrate the feasibility of the proposed loading schemes in so far as they had progressed. For this reason tensile and pure bending tests were chosen, since fittings were already manufactured for these tests and since the data obtained might be useful in propeller design. The methods of loading are explained in detail in the following sections. METHODS OF LOADING DIRECT TEISE LJOADS D iract tensile loads were applied to the specimen by means of a conventional Riehle testing machine. Loading was accomplished through chain links as intermediate members, to insuire that eccentricities did not introduce unwanted beending 34

moments' A blade loaded in this fashion is shown in Fig. 4. By means of these fittings loads of 40,000 total pounds could be easily applied in tension to the specimen Fig. 4. PURE BENDING MOMENT A means for the application of pure bending moments without shear or tension was devised. In the form used here it is most easily described by the photograph of Fig. 5. Basically a single tensile load is split by means of cables and pulleys so that the specimen is subjected to 1/4 of this total load as each of four points as shown in Fig. 6. The fittings for the blade models were constructed in such a way that this pure bending moment could be applied about either of the axes of the propeller blade o 35

~qml ';J. 0L 0o~~~~~~~~~i ~~or 7536l - | | |~~~~:9;; ~l*ll - - - E - E - | |~~~~~~~~~B B~~BI ~ rz:ii- - - - s l::': ~ -|| | - l W~~~~~~~~~i ---- W l I_ |-_lil b: l~~~~~~~: ____911I111 _1111 * J l _l l li _l ~~~~~~:: __ _ a _D E _ w

INSTRUMENTATION Instrumentation of the propeller-blade models was performed by qualitative use of stress-coat and by electrical resistance strain gauges. Stress-coat was utilized todetermine the directions of principal strain so that gauges might be properly oriented. it was not found possible to photograph the stress-coat pattern with enough clarity to present in this report, Type A-18 strain gauges were placed on all models, both inside and outside, at 0%, 25%,v 50%, 75%, and 100% of the transition length, measured from the root end of the transition and at other points as appeared desirable from the stresscoat tests. This particular type of gauge was chosen since its short length made installation in a restricted region somewhat easier, while the averaging effect of its length was kept.small. At each of these sections along the transition length, a band of strain gauges was placed, These gauges were attached in pairs, one circumferential and one aligned with the longitudinal axis of the blade since the str-ess"oat tests indicated these to be the principal strain directions. RESULTS Due to the limited resources in time and funds it was not possible to perform as complete an interpretation of the measured strains as was desired. For this reason the results obtained are presented in summary form only, while the complete data are to be transmitted to WADC for possible further interpretation. Tensile load strains were measured on the 12-inch and 6-inch transitionlength models 'at loads of 10,000 and 20,000 pounds. The modulus of elasticity used in the calculations was determined by test to be 10 x 106 psi. Assuming that the gauge pairs are oriented in the principal strain directions, and assumeing that the stress component through the thickness of the blade shell is zero, the -stresses may be immediately computed in terms of the measured strains and the physical properties. A very simple method of presenting these data is used, in which the ratio of maximum measured shear stress to nominal shear stress is tabulated. Nominal shear stress is defined as 1/2(P/A), where P - load and A = cross-sectional area. The maximum value of this ratio for each of the two transition sections tested is given in Table II. TABLE II Transition Section Length, in. ( Tmeas /Tnom )max 6 2.35 12 1.62 37

For the bending test a couple of 2000-inch-pounds was applied about the flat axis of the 12-inch transition-length propeller blade and strains were again measured. Using the same assumptions as before, the maximum value of the measured bending strain may be converted to stress and divided by the nominal beam bending stress to yield a simple dimensionless ratio. Data of this nature from the 12 -inch blade are given in Table III. TABLE III ({bmeas TraSition Length, in. \bnom/imax 12 1.89 Here, the nominal bending stress is taken to be Me/I, and does not imply anything other than a convenient quantity for use in manufacturing a dimensionless ratio. 38

APPENDIX BLADE GEONMETRY The transition-section region was in all cases designed on the basis of one half cycle of a cosine curveo As an example, the two lines of Fig, 7 may be joined by a curve, shown as a dotted line, of the equation y _ b(1 cos i 2 which satisfies both ordinate and slope continuity,. Curves of this type were fitted to each of the two plan views (side and front) of each of the three lengths of transition section, for 'both.inside and outside profiles The cross section of each blade., again both inside and outside, was taken to be an ellipse whose ma~jor and minor axes, were chosen to conform, with the half-cosine curve previously diScussed. The constant outboard section was also defined by ellipses. Thus, the shell in all cases is defined as the material lying between two concentric.ellipses, with the cireular shank merely a special case of this, It should be pointed out that this definition of shape is almost identical with that which would be obtained in the usual manner of fairing with splines. Blade drawings are given in Plates I through IVT 7,. Figo. 7 39

UNIVERSITY OF MICHIGAN 3 9015 02844 0512 I 3 9015 02844 0512