NASA/TM-97-206218

Stress Analysis of B-52 Pylon Hooks for

Carrying the X-38 Drop Test Vehicle

William L. Ko

Dryden Flight Research Center

Edwards, California

National Aeronautics and

Space Administration

Dryden Flight Research Center

Edwards, California 93523-0273

October 1997
NOTICE

Use of trade names or names of manufacturers in this document does not constitute an official

endorsement of such products or manufacturers, either expressed or implied, by the National

Aeronautics and Space Administration.

Available from:

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CONTENTS

ABSTRACT ....................................................................... 1

NOMENCLATURE ................................................................. 1

INTRODUCTION .................................................................. 2

X-38 HOOK DESCRIPTIONS ......................................................... 3

FINITE-ELEMENT ANALYSIS ....................................................... 4

Finite-Element Modeling .......................................................... 4

Constraints ..................................................................... 4

Loadings ....................................................................... 4

EQUIVALENT CURVED BEAM ANALYSIS ........................................... 5

Curved Beam Geometry ........................................................... 5

Stress Equations ................................................................. 5

RESULTS ......................................................................... 6

Finite-Element Analysis ........................................................... 7

Stresses and Deformations ...................................................... 7

Load-Stress Equations ......................................................... 8

Equivalent Curved Beam Analysis ................................................... 9

Stress Distributions ............................................................ 9

Load-Stress Equations ........................................................ 10

Practical Approach ........................................................... 10

CONCLUDING REMARKS ......................................................... 11

APPENDIX A ..................................................................... 21

APPENDIX B ..................................................................... 24

REFERENCES .................................................................... 28

iii

ABSTRACT

The finite-element structural analysis method was used in a two-dimensional stress concentration

analysis of a new pylon hook for carrying the X-38 lifting body atmospheric drop test vehicle on the B-52

carrier aircraft. The stress distributions in the hook were obtained, and the critical stress points were identified. The functional relationships between the applied hook load and the induced maximum tangential

and shear stresses were established for setting the limit hook load for test flights. By properly representing the X-38 hook with an equivalent curved beam, the conventional curved beam theory predicted the

values of the maximum tangential and shear stresses quite close to those calculated from the finiteelement analysis. The equivalent curved beam method may be of practical value during the initial stage of

the hook design in estimating the critical stresses and failure loads with reasonable accuracy.

a

b

C

CRV

DAST

E

g

gl

h

JLOC

l

M

P

PFE

s

Pf

t

Pf

q

qFE

NOMENCLATURE

radius of hook inner circular boundary region, in.

radius of hook outer circular boundary region, in.

inner to outer radii ratio, c = a/b

crew return vehicle

drone for aerodynamic and structural testing

Young's modulus, lb/in 2

curved beam depth parameter for end moment loading

curved beam depth parameter for end force loading

thickness of hook, in.

joint location

moment arm of hook load, in.

end moment for equivalent curved beam, M - Pl, in-lb

hook load or end force for equivalent curved beam', lb

partial hook load for finite-element model, PFE = 0.01 (P/h), lb

ultimate hook load in shear failure, lb

ultimate hook load in tensile failure, lb

distributed hook load, lb/in.

nodal force for finite-element model, Y-'qFE = PFE ' lb/node
r S

x,y

0

Oc

v

of

(_r

M

fir

P

0 o

M

0 o

P

(_0

O0 0 = 29.25 °

O0 0 = 0 c

"of

ZrO

M

"CrO

P

'r,rO

T'xy

radial distance, in.

radial location of maximum shear stress "Crolmax, in.

rectangular Cartesian coordinates

angular coordinate measured from horizontal x-axis, deg

angular location of critical stress point for Ot[ma x , deg

Poisson's ratio

tensile failure stress, lb/in 2

total radial stress in the curved beam, lb/in 2

radial stress in the curved beam due to end moment M, lb/in 2

radial stress in the curved beam due to end force P, lb/in 2

tangential stress in the hook from finite-element analysis, lb/in 2

total tangential stress in the curved beam, lb/in 2

tangential stress in the curved beam due to end moment M, lb/in 2

tangential stress in the curved beam due to end force P, lb/in 2

value of 00 at 0 = 29.25 °, lb/in 2

value of 00 at 0 = 0 c , lb/in 2

shear failure stress, lb/in 2

total shear stress in the curved beam, lb/in 2

shear stress in the curved beam due to end moment M, lb/in 2

shear stress in the curved beam due to end force P, lb/in 2

shear stress in the hook from finite-element analysis, Ib/in 2

maximum value of ( )

INTRODUCTION

The vehicle designated the X-38 is a lifting body flight vehicle that will develop the technologies for

the emergency crew return vehicle (CRV) for the International Space Station. The operational CRV will

be 28.5 ft long and 14.5 ft wide. The structure will be aluminum with a shell of graphite-cyanate ester

2
epoxy.The thermalprotectionsystemwill consistof improvedSpaceShuttleOrbiter-derivedblankets

andtiles. The vehiclewill havefour controlsurfaces:two body flapsfor pitch androll control,and two

ruddersfor yaw androll control.TheoperationalCRV will reenterfrom orbit andglide unpoweredlike

theSpaceShuttleOrbiter,butunliketheOrbiter,it will notemployasteephigh-energyapproachlanding.

Becauseof potentiallyexcessivelandingspeedsforthesmallbut heavyvehicle,theCRVwill usea steerableram-airparachutebelow Mach0.25andflare to asoftlanding.Theentiresequencefrom reentryto

landingis to befully autonomouswithoutanyneedfor humanintervention.

TheexperimentalX-38 vehicleshownin figure 1is anearlyfull-scalevehicleandis 24.5ft long and

11.5ft wide, weighing 16,557lb. The X-38 will becarriedunderthewing of the NASA DrydenB-52

aircrafton two newL-shapedhooksona speciallydesignedpylon up to analtitudeof 40,000ft to conductnonpiloted,free-flight droptests.Themainpurposesof thetestsareto evaluatetouchdowndynamicsof thevehicle-parachutecombination,to studyfree-flightvehiclestability andcontrol characteristics

in transonicand subsonicflight, to assessparachutesystemreliability, and to demonstratefree-flight

vehicleenergymanagementandpositioningcapabilities.

TheL-shapedhookswill inducestressconcentrationin theinnercircularboundaryregionand,therefore, the new hook mustbedesignedin sucha way asto reducethe stressconcentrationandeliminate

fatigueconcernsduringtheX-38 program.Thus,anexaminationof the inducedstressfield in thenew

hookis importantto validateits structuralintegrity.

This reportconcernsthefinite-elementstressanalysisof anewlydesignedhookfor carryingtheX-38

vehicleanddocumentstheanalysesperformed.Thereportpresentsthe analyticalstressdistributionsin

the new hook to locatethe critical stresspoints in tensionand shear.Certainfunctional relationships

betweenthe hook load andthe inducedmaximumtangentialstressand the maximum shearstressare

established,andanequivalentcurvedbeamtheoryis developedto analyzethehook.

X-38 HOOK DESCRIPTIONS

Figure 2 shows a sketch of the B-52 carrying the X-38 vehicle on new identical front and rear hooks

(called X-38 hooks) on a dedicated pylon. The X-38 vehicle rolling motion during captive-carry flight is

restrained by two sway braces as shown in the figure. Figure 3 shows the geometry of the L-shaped X-38

hook. The X-38 hook is made of 4340 steel that was heat-treated and has the following material properties

E = 29 x 10 6 lb/in 2

v = 0.32

(_f = 101,000 lb/in 2

"cf = 60,600 lb/in 2

The hook has thickness h = 2.8 in. The inner circular arc boundary of the hook has a radius of 0.5 in.,

and the outer circular arc boundary has a radius of 2 in. The centers of the two radii do not coincide.

Because of the nonconcentric nature of the inner and outer circular arc boundaries, the hook is clearly not

a conventional curved beam.
TheX-38 designhookload,P, is 42,027 lb. This hook load consists of four components:

• Static load due to X-38 weight

* Static load due to tightening of sway braces

• Aerodynamic load generated from aerodynamic force exerted on X-38 during captive flight

• Inertia load arising from X-38 vertical inertia load (2.2 times gravitational acceleration) during

captive flight

The hook load P is located at a distance l = 0.75 in. from the y-axis (fig. 3). To avoid local point loading,

the hook load is distributed over the flat horizontal surface of the hook through a rounded triangular sleeve

that can rotate freely around a circular pin (fig. 3) when the hooks swing to launch the X-38 vehicle.

FINITE-ELEMENT ANALYSIS

The Structures Performance and Resizing (SPAR) finite-element computer program (ref. 1) was used

in the stress concentration analysis of the X-38 hooks.

Finite-Element Modeling

Because the critical stress points are in the lower L-shaped region of the hook', only this region was

modeled. Figure 4 shows the finite-element model generated for the X-38 hook. The model has 1212

joint locations (JLOC) and consists of 40 triangular combined membrane and bending elements (E33 elements) and 1125 quadrilateral combined membrane and bending elements (E43 elements). To maintain

proper element length to thickness ratios of the two-dimensional elements (E33, E43) for computational

accuracy, the thickness of the model was chosen to be 0.01 in.; namely, the model is actually a thin slice

cut out from the full thickness of the hook.

Constraints

The upper left corner point is fixed, and the upper horizontal straight boundary is allowed to move

freely in the horizontal direction (x-direction) with the vertical motion (y-direction) constrained.

Loadings

Two hook-loading conditions were considered: a concentrated load or a distributed load (fig. 4). The

purpose of these dual analyses is to compare the stress concentrations under different loading conditions.

The loads shown in figure 4 are the actual hook loads and not the partial loads for input to the finiteelement model. For the 0.01-in. thick finite-element model, the applied partial hook load PFE was calculated from

P 42, 027

PFE=h XO'O1- 2.8 X0.01 = 150.101b (1)

For the concentrated load case, because there is no node coincidental to the actual loading point, the

applied load PFE was divided into two parts and applied at two adjacent nodes between which the actual

loading point lies, preserving the effective loading point location of l = 0.74375 in.
For thedistributedloadcase,theappliedload PFE is equally distributed over the nodes lying in the

hook flat horizontal boundary, and the load intensity qFE is applied at each node where YqFE = PFE"

The span of this distributed loading zone is the length of one flat side of the rounded triangular sleeve

(fig. 3), again preserving the effective loading point location.

EQUIVALENT CURVED BEAM ANALYSIS

The shape of the inner rounded boundary of the hook is a circular arc (fig 3). However, the corresponding outer curved boundary is a nonconcentric circular arc (i.e., the inner and outer circular arcs have

different centers). The hook curved region, therefore, is clearly not of a conventional curved beam geometry. The purpose of this section is to represent the hook with an equivalent curved beam, and to explore

whether the curved beam theory might roughly estimate the critical stress values. As will be seen later,

this simple approach turned out to be of great practical value in estimating the critical stress values and

the hook failure loads by using a desk calculator (or a simple Fortran program) before conducting the

time-consuming finite-element stress analysis.

Curved Beam Geometry

For analysis, the curved region of the hook is represented by an equivalent curved beam of one

quadrant (0 < 0 < re/2; fig. 5). The inner radius, a, of the equivalent curved beam has an identical radius as

that of the hook (a = 0.5 in.). The outer radius, b, of the equivalent curved beam was so chosen that its circular boundary will circumscribe the outer boundary of the hook, which resulted in a radius b = 4.28 in.

The equivalent loading condition on the curved beam may be considered as the summation of two

loading cases: bending under the end force P and bending under the end moment M = Pl (refs. 2--4; fig. 5).

Stress Equations

The induced bending stresses for end force loading (ref. 5) can be expressed as

_r (r, 0): b_g I c 2 (1 <bl r._ 0) - +c )r]S,n_ _

(2)

_o(r, O) = bPgl[3b-c2(b) 3-(l+c2_bl (_-0])rJS,n_ "

(3)

c2 -I''c2 cos/;_O) (4)

where c is the radii ratio of a/b, and g l is defined as

gl = 1-c2+(l+c2)ln(c) (5)

5
For endmomentloading,theinducedbendingstressescanbeexpressedas

- b2hg -c l nl l (6)

M M

G 0 (r) =

b2hg 1-c2+c21n(c)[1 + (_)2]-(1- c2)ln(!)) (7)

M

"lTr0 = 0 (8)

where g is defined as

_c (9)

The total stresses in the equivalent curved beam are then obtained by the summation of the above two

similar stress components.

P M

(_r = t_r + t_r (10)

P M

G 0 = G0 + t_0 (11)

P M

XrO = XrO + XrO (12)

The maximum tangential stress t_Olrnax is located at point (r = a, 0 = 0°), but the maximum shear

stress "CrOmax is located at another point (r = rs , 0 = 90 °) where rs is given by

rs = Z a +b 2 +12a2b 2- a 2+b 2

1/2

(13)

Equation (13) was obtained by differentiating equation (4) with respect to r and then setting the

resultant equation equal to zero. Notice that rs is also the radial location of the maximum radial stress

_Pr max (tangential location is 0 = 0 °) induced by force P (eq. (2)). Fortran programs for calculating the

distributions of G 0 and "_rOin the equivalent curved beam are attached in appendix A.

RESULTS

The following sections present the results of both the finite-element analysis and the equivalent

curved beam analysis.

6
Finite-Element Analysis

The deformations, stress distributions, and load-stress equations for the X-38 hook are presented in

the following.

Stresses and Deformations

Figures 6 and 7, respectively, show the deformed shapes of the X-38 hook under concentrated and

distributed loadings. The vertical deflections at the horizontal tip of the hook are 4.04 x 10-3 in.

and 4.11 x 10-3 in., respectively, for the two loading cases. Notice that these deflections are close to

each other.

Figures 8 and 9 show the distributions of tangential stress fit along the inner boundary of the hook for

the concentrated load and distributed load cases, respectively. For the concentrated load case (fig. 8), the

element with the maximum tangential stress gt[max = 53,510 lb/in 2 is at the inner boundary with angular

location 0 = 0 c = 24.75 °. For the distributed load case (fig. 9), the element with maximum tangential

stress o t max -- 53,964 lb/in 2 is at the inner boundary with angular location 0 = 0 c = 29.25 °.

For the above L-shaped X-38 hook, just as for other B-52 hooks analyzed in the past (figs. B-1

through B-8 in appendix B; ref. 6), the critical stress points are not located exactly at 0 c = 0 ° as in the

case of the classical curved beam (eq. (3)), but are located within the region 11.25°< 0 c < 36.25 °.

Figures 10 and 11 show the distributions of tangential stress (_t and shear stress Xxy along the critical

hook cross-section lines for the concentrated and distributed loading cases, respectively. For both of the

loading cases, the Xxy distributions profiles are saddle shaped with the Xxylmax points located relatively

near the upper hook boundary. For the concentrated load case (fig. 10), the maximum shear stress

"_xy max -- 11,004 lb/in 2 lies in the vertical cross-section slightly to the right of the loading point. The

maximum shear stress Xxylmax - 9,093 lb/in 2 for the distributed load case (fig. 11) lies in the vertical

cross-section passing through the right-hand termination point of the distributed loading zone. Table 1

compares the angular locations of the critical tangential stress points and the maximum stresses induced

in the X-38 hook under concentrated and distributed loadings calculated from finite-element analysis.

Table 1. Comparison of finite-element results of concentrated and distributed loadings

of the X-38 hook.

Loading case 0 c , deg ot max' lb/in2 "Cxy max' lb/in2

Concentrated load 24.75 53,510 11,004

Distributed load 29.25 53,964 9,093

The results of the distributed load case (actual loading situation, fig. 3) in table 1 will be compared

with the results using the equivalent curved beam theory.

7
Load-Stress Equations

From the results of the stress analysis (figs. 8-11), the functional relationships between the hook load

P and the maximum stresses may be established as follows:

Concentrated load:

_t[max

P

= 1.2732 lb/in2/lb (14)

Distributed load:

Zxy Imax

P

- 0.2618 lb/in2/lb (15)

P

- 1.2840 lb/in2/lb (16)

"_xy [max

------_-- = 0.2164 lb/in2/lb (17)

If _t]max and "Cxylmax in equations (14) through (17) reach their respective failure stress levels

(_t max -- _f and Xxy max = "Cf), the load-stress equations (14) through (17) can then be used to calculate the hook failure loads in tension and in shear as

Concentrated load:

Distributed load:

t

Pf = 0.7854 fff lb

$

Pf = 3.8192 xf lb

(18)

(19)

t

Pf = 0.7788 fff lb

s

Pf = 4.6219 xf lb

(20)

(21)

Using the failure stress values of the hook material given earlier ((yf = 101,000 lb/in 2 and

• f = 60,600 lb/in2), the hook failure loads in tension and in shear may be calculated from equations (14)

through (17). Table 2 shows the results.

8
Table 2. Failure hook loads in tension and shear

(cf = 101,000lb/in2,'lTf = 60,600 lb/in2).

Loading case P , lb P Sf, lb

Concentrated load 79,325 231,444

Distributed load 78,659 280,087

Table 2 shows that the X-38 hook will definitely fail in tension rather than in shear. The limit hook

load for test flights may then be established by applying a proper safety factor.

Equivalent Curved Beam Analysis

The results of the stress analysis of the equivalent curved beam and the established functional relationships between the hook load and the critical stresses are presented in the following sections.

Stress Distributions

Figure 12 shows the distribution of the total tangential stress along the inner boundary of the equivalent curved beam, induced by the combined loading of the end force P (where P = 42,027 lb) and the end

moment M (where M = Pl = 42,027 × 0.74375 = 31,258 in-lb). The magnitudes of the bending stress

P M

components (G 0 and G0 ) are indicated in the figure for comparison. At 0 = 29.25 °, which is the location

of C t max point in the X-38 hook for the distributed load case, G O has the value of c010 = 29.25 ° =

54,636 lb/in 2. This value is close to the values of C t max = 53,964 lb/in 2 calculated from the finiteelement analysis of the actual hook (fig. 9). This discovery is significant and of great practical value

because with only hand calculations using the curved beam stress equations, one can quickly estimate the

intensity of the critical stress in the L-shaped X-38 hook during the initial design phase before initiating

the time-consuming finite-element analysis.

Figure 13 shows the distributions of G O and XrO along their respective critical cross-section lines.

The maximum shear stress XrO max = 9,327 lb/in 2 lies in the 0 = 90 ° plane and is located at a radial

distance of rs = 0.8441 in. (0.3441 in. from the hook horizontal boundary). This value is also close

to "Cxy max = 9,093 lb/in 2 calculated from the finite-element analysis (fig. 11). Table 3 summarizes

these results.

Table 3. Comparison of critical stresses in X-38 hook calculated from the

finite-element analysis and the equivalent curved beam theory.

Mode of analysis Ctlmax' lb/in2 "Cxy max' lb/in2

Finite-element analysis 53,964 9,093

Equivalent curved beam theory 54,636 9,327

Percent error 1.25 2.57

9
Notice that the errorsareamazinglylow. This finding implies the powerful practicalvalue of the

equivalentcurvedbeamtheory.

Load-Stress Equations

From figure 13, the relationships between the hook load P and the tangential stress o 0 at 0 = 29.25 °

and the maximum shear stress are XrOlmax established as follows:

130 0 = 29.25 °

P

= 1.3000 lb/in2/lb (22)

"_rO max

P

= 0.2219 lb/in2/lb (23)

Substitution of (if0 0 = 29.25 ° and "CrOImax) in equations (22) and (23), respectively, with their failure

stresses (Cry and xf ) yields the following hook failure loads in tension and in shear.

t

Pf = 0.7692 13f lb (24)

s

Pf = 4.5065 '_f lb (25)

Applying the given values of (of - 101,000 lb/in 2 and '_f = 60,600 lb/in2), the failure load equations

(24) and (25) become

t

Pf = 77,689 lb (26)

s

Pf = 273,096 lb (27)

Notice that the levels of the hook failure loads are close to those shown in table 2, indicating the great

practical value of using the equivalent curved beam theory in the initial stage of hook design for quick

estimation of the hook failure loads.

Practical Approach

The practical way to use the equivalent curved beam theory may be summarized as follows. During

the early stage of hook design, for a trial hook geometry and the loading condition, an equivalent curved

beam of one quadrant may be created. The inner circular boundary of the equivalent curved beam should

match the inner circular boundary of the hook. The outer circular boundary of the equivalent curved beam

should circumscribe the outermost curved boundary of the hook. The curved beam theory is then used to

calculate stress distributions in the equivalent curved beam. The maximum shear stress lies in the 0 = 90 °

plane and at a known radial distance rs given by equation (13). However, the angular location 0 c (which

is configuration dependent) of the critical tangential stress point in the hook is still unknown. A reasonable guess is to use 0 c = 26 °, the average value of the known 0 c of past B-52 hooks (appendix B) and of

the X-38 hook, to obtain the critical tangential stress at r = a. Through this approach, the approximate

hook failure loads in tension and shear may be estimated.

10
In addition to the X-38 hook case,the validity of the equivalentcurved beamtheory was further

testedforthepreviousB-52hooksshownin appendixB (ref.6).ThetangentialstressesG0 at thealready

known 0c locations, and at the averaged angular location 0 c = 26 °, were calculated. The results are compared with the finite-element solutions and are shown in table 4.

The relatively low solution errors indicate the powerful practical value of the equivalent curved beam

theory in estimating the critical stress levels during the preliminary hook design.

Table 4. Comparison of critical tangential stresses in B-52 hooks calculated from the finiteelement and the equivalent curved beam analyses.

X-38 Front Old rear DAST New rear

Hook / Item hook hook hook rear hook hook

0 c , deg 29.25 26.25 36.25 11.25 28.75

(Yt[max' lb/in2 53,964 73,522 172,200 100,700 100,400

g0[0 = 0c ' lb/in2 54,636 64,088 165,567 96,557 94,138

Percent error +1.25 -12.83 -3.85 -4.11 -6.24

Averaged 0 c , deg 26 26 26 26 26

(Y0 0 = 26 o, lb/in 2 55,945 64,212 179,721 89,550 95,992

Percent error +3.67 -12.66 +4.37 -11.07 -4.39

CONCLUDING REMARKS

Finite-element stress concentration analysis was performed on the X-38 hook subjected to concentrated and distributed loadings. The locations and the magnitudes of the tensile and shear critical stresses

were identified. The key findings are as follows:

.

2.

The X-38 hook was found to fail in tension rather than in shear.

The critical tensile stress point is located on the X-38 hook inner boundary at 0 = 24.75 ° and

0 = 29.25 °, respectively, for the concentrated and distributed loadings, where 0 is an angle measured from the horizontal x-axis.

.

.

The maximum shear stress point is located in the cross-section near the root of the X-38 hook

horizontal arm and is relatively closer to the upper horizontal boundary.

By representing the X-38 and other B-52 hooks with equivalent curved beams, the classical

curved beam theory could be used to quickly estimate the critical stresses in the hooks and the

hook failure loads with reasonable accuracy, before performing the more detailed finite-element

analysis.

Dryden Flight Research Center

National Aeronautics and Space Administration

Edwards, California, April 11, 1997

11
EC96 43737-13

Figure 1. The X-38 lifting body atmospheric test vehicle to be launched from the B-52 aircraft.

_-- B-52

t

_ Sway brace /- Pylon

Rea_ Wing

v16,557 Ib k_t J

= 24.5' '_

,/- Ground

970738

Figure 2. Sideview of the X-38 lifting body atmospheric test vehicle mated to the B-52 pylon

through two new hooks.

12
1.50

R

23.40

12.00

4.75

2.90

,t = 0.75 --_ Rounded

I_ I /r triangular I

Circular pin .__ I sleeve

I

_. _._o--_

R

30°

30°

hook

970739

1.00 R 2 PL

_vX

Figure 3. Geometry of the X-38 hook for carrying X-38 atmospheric test

vehicle; dimensions in inches, except as noted.

13
,t = 0.74375 "

Concentrated load

P = 42_027 Ib

or

Distributed load

,T..,q= e = 42,027 Ib

q

() () []

IIIltlllll]

Illllll[

,,,,,,,,11 I

IIII1111

Illlllll}t [

IIIIIIll

IIIIIIII

1212

4O

1125

Joint locations (JLOC)

Triangular elements (E33)

Quadrilateral elements (E43)

Figure 4. SPAR finite-element model for the X-38 hook.

970740

14
p _\_\_\\\_\\\_\\_

Actual cas X-38

Bending under

end force

hook

P

I

I

I

I

I

I

L

__ _ i) a ` 0"5in"

I / b=4.28in.

,l = 0.74375 in.

l

i Equivalent

"_ curved beam

+ P

I

I M--P_

Bending under

end moment

_\\\\\\\\\\\\\\'_

I : I a = 0.5 in.

,p_-'-a / i / b=4.28in.

"" _0 i/ ,t=0.74375in.

;/\

_ k_ Equivalent

"_ curved beam

970741

Figure 5. Representation of the X-38 hook geometry with an equivalent curved beam.

15
Concentrated load

P = 42,027 Ib

4.04 x 10-3 in.

970742

Figure 6. Deformed shape of the X-38 hook under concentrated load; P = 42,027 lb.

.I

4.11 x 10-3 in.

IIIIIIIIIIIII I

III IIIII

IIIII

tNI""'"'" illlillllll

l!i Ill[llllll

IIIIIIIIIll

I1111111111 IIIIIIIIII

970743

Figure 7. Deformed shape of the X-38 hook under distributed load; Zq = P = 42,027 lb.

16
Concentrated load .L

P=42,0271b I'

(_t, Ib/in2

-2-1 0 1 2 3 4 5 6 7x 10 4

= 24.75 °

O'tlmax = 53,510 Ib/in2

O'tlmax

- 1.2732 Ib/in2/Ib

P 970744

Figure 8. Distribution of tangential stress _t along inner boundary of the

X-38 hook; concentrated load P = 42,027 lb.

o't, lb/in 2

-2-1 01 2 3 4 5 6 7x104

Distributed load

,_1 = P = 42,027 Ib

= 29.25 °

(ftlmax = 53,964 Ib/in2

(_tlma------Ex = 1.2840 Ib/in2/Ib

P

970745

Figure 9. Distribution of tangential stress c_t along inner boundary of

the X-38 hook; distributed load Zq -- P = 42,027 lb.

17
74" 70,_

O'tlmax = 53,510 Ib/in 2

Concentrated load

P = 42,027 Ib

0 c = 24.75 °

'_xy

1;xylmax = 11,004 Ib/in 2

O'tlma_x= 1.2732 Ib/in2/Ib .8

P

1;xylmax 17xy' Ib/in 2

= 0.2618 Ib/in2/Ib

P

1.4 x 104

970746

Figure 10. Distributions of tangential stress a t and shear stress x , along x)

their respective critical cross section lines; concentrated load P = 42,027 lb.

(_tlmax = 53,964 Ib/in 2

Distributed load 0 c = 29.25 °

Tq = p = 42,027 Ib

q

O"tlmax

"_xylmax= 9,093 Ib/in 2

- 1.2840 Ib/in2/Ib

P

.8 1.4 x 104

'_xylmax

= 0.2164 Iblin211b '_xy, Iblin2

P 970747

Figure 11. Distributions of tangential stress G t and shear stress "r,x,,,along their

respective critical cross section lines; distributed load ,v_z/= p = 42,027 lb.

18
O.e,Ib/in2

0 = 29 25° 0 2 4 6 8 x 104

- • , . .xx\\\\\\\\\\\\\\\\\\_,

,_....,/ /==:::t{,---L--__L_t t 3/ I i I I

P = 42,027 Ibl _ ;__--------_ I |

p_ I_._ _Nk,'__ O'olo= 29 25o = 54,636 ,b/in2 /

°°)1 -'_ - " i/

,v,_- P.,= f/ _/

970748

Figure 12. Distributions of tangential stress c o along the inner boundary of the

equivalent curved beam; P = 42,027 lb; M = 31,258 in-lb.

19
O

P = 42,027

i

rs = 0.8441 in.

% = 29.25 ° = 54,636 Ib/in 2

M=P_=

. 31,258 in-lb

1;rOImax = 9,327 Ib/in 2

(_ele = 29.25 ° -..

= 1.3000 Ib/in2/Ib

P

't_rOImax= 0.2219 Ib/in2/Ib

P

0 .8 1.4 x 104

_rO, Ib/in2 970749

Figure 13. Distributions of tangential stress c o and shear stress 17r9 along their

respective critical cross section lines; P = 42,027 lb; M = 31,258 in-lb.

2O
APPENDIX A

Fortran Programs for Equivalent Curved Beam Analysis

PROGRAM A

c EQUIVALENT CURVED BEAM THEORY FOR X-38 AND B-5 HOOKS STRESS ANALYSIS

c i. Tangential stress distribution along equivalent curved beam

c inner boundary

c Equivalent curved beam outer boundary circumscribes hook

c outer boundary

c Programed by Dr. William L. Ko, March 13, 1997

PROGRAM B52HOOKS

DIMENSION SIGP (27)

DIMENSION TAUP (27)

DIMENSION SIGT (27)

DIMENSION SIGM(27 )

DIMENSION PP (27)

DIMENSION PPP (27)

DIMENSION SS(27)

DIMENSION SSS (27)

DIMENSION XMM (27)

DIMENSION Y(27)

DIMENSION X (27)

REAL X

DATA X/0. ,4.5,9., ii.25,13.5,18., 22.5,26. ,26.25,27. ,28.75,

+29.25,31.5,36., 36.25,40.5,45., 49.5,54., 58.5,63., 67.5,72.,

+76.5,81., 85.5,90. /

c X-38 hook

A=0.5

B=4. 276880

e=0.74375

H=0.01

P=I50. 096429

c B-52 front hook

c A=0. 3125

c B=I. 9

c e=0.1365

c h=0.01

c P=91. 4077

c B-52 old rear hook

c A=0.16

c B=I. 9

c e=0.42

c H=0.01

c P=185.1242

c B-52 DAST hook

c A=0. 4375

c B=2.25

c e=0.253

21
c H=0.01

c P:143.2665

c B-52 new rear hook

c A:0 .5

c B=2.5

c e=0.46

c H=0.01

c P=156. 1775

C=A/B

DO I0 I=l, 27

X(I) = X(I)/57.29578

Y(I) = X(I)'57.29578

GP=I. - (C*'2) + (i. + (C*'2)) *ALOG (C)

G=0.25" ( (i.-C*'2) **2) - (C*'2) * ((ALOG(C)) **2)

SS (I) :COS (X(I))

SSS (I) =SIN(X(I) )

PP(I) = (P/(B*H*GP) ) *SS (I)

PPP (I) = (P/(B*H*GP) ) *SSS (I)

XMM (I) = (P'e) / ((B**2) *H'G) * (SS (I) /SS (I))

c Tangential stress at inner boundary due to force P

SIGP (I) =PP (I) * (3. * (A/B) - (C*'2) * ((B/A) *'3) - (I.+C*'2)

+* (B/A))

c Tangential stress at inner boundary due to Moment M = Pe

SIGM(I) =-XMM(I) * (I.-C*'2+ (C*'2) *ALOG(C) * (i. + (B/A) **2)

+- (i.-C*'2) *ALOG (B/A))

c Shear stress at inner boundary due to force P

TAUP (I) =-PPP (I) * (A/B+ (C*'2) * ((B/A) *'3) - (i. +C*'2) *

+ (B/A))

c Total tangential stress at inner boundary

SIGT(I) =SIGP(I) +SIGM(I)

PRINT i00, Y(I), SIGP(I), SIGM(I), SIGT(I), TAUP(I)

i00 FORMAT(/IX, F14.5, 4F14.4)

10 CONTINUE

END

22
PROGRAM B

c EQUIVALENT CURVED BEAM THEORY FOR X-38 HOOK STRESS ANALYSIS

c I. Distribution of tangential stress along 29.25 degree radial line

c 2. Distribution of shear stress along 90 degree radial line

c Programed by Dr. William L. Ko, March 13, 1997

PROGRAM DEPTH

DIMENSION SIGP (19)

DIMENSION TAUP (19)

DIMENSION SIGT (19)

DIMENSION SIGM (19 )

DIMENSION SIGTH (19)

DIMENSION R (19 )

REAL B

DATA R/0.5, .625, .75, .844102, .875,1. ,1.25,1.5,1.75,2. ,2.25,

+2.5,2.75,3. ,3.25,3.5,3.75,4. ,4.14453/

A=0 .5

B=4.276880

P=150 .096429

H=0.01

e=0.74375

DO i0 I=i,19

C=A/B

GP=I. - (C*'2) + (i. + (C*'2)) *ALOG (C)

G=0.25* ( (i.-C*'2) **2) - (C*'2) * ( (ALOG (C)) **2)

PP= (P/(B*H*GP) )

XMM: (P'e)/ ((B**2) *H'G)

c Tangential stress on 0 degree radial line due to force P

SIGP(I)=PP*(3.*(R(I)/B)-(C**2)*((B/R(I))**3)-(I.+C**2)

+* (B/R(I)) )

c Tangential stress due to Moment M = Pe

SIGM(I) :-XMM* (I.-C*'2+ (C*'2) *ALOG(C) * (i.+ (B/R(I)) **2)

+- (i.-C*'2) *ALOG (B/R (I)) )

c Shear stress on 90 degree radial line due to force P

TAUP(I)=-PP*(R(I) /B+(C**2)*((B/R(I))**3)-(I.+C**2)*

+(B/R(I) ) )

c Tangential stress on 29.25 degree radial line due to force P

c Cos(29.25 degree) = 0.872496

SIGTH (I) =SIGP(I) * (0. 872496)

c Total tangential stress along 29.25 degree radial line

SIGT (I) =SIGTH (I) +SIGM(I)

PRINT I00, R(I), SIGTH(I), SIGM(I), SIGT(I), TAUP(I)

i00 FORMAT(/IX, F12.5, 4F12.4)

10 CONTINUE

END

0_3
APPENDIX B

Stress Distributions in B-52 Original Pylon Hooks

h = 1 188 in.

P_ _i _0.3125_

_h= 1.0 in._

f Front hook

Region

- _ modeled

-- Groove

boundary

970750

Figure B-1. Geometry of the B-52 original pylon front hook.

0 .5 1.0 x 105

i,

Thickness

",'_'- transition line

-loxlo ",, [

-.5 _ __, 1.0_.\1.5 /

ib/in_t, 2 0 _ff 0c=26".25° /

.5 _ (+) t max = • X 1 Ib/in 2

/\ application

1.0 _- "- point 970751

Figure B-2. Distribution of tangential stress (Yt along inner boundary of the front hook; P = 10,000 lb.

24
rear hook

Region .... _._

modeled h = 0.928 in._

J 970752

Figure B-3. Geometry of the B-52 original pylon old rear hook.

O't, Ib/in2

105 x 2.0 1.5 1.0 .5 0

rear hook

inner boundary

(-)

-2.0 x 105

-1.0

Gc = 36.25°

C+)

Load

application

point

Gt,

0

Ib/in 2

970753

Figure B-4. Distribution of tangential stress tJ t along inner boundary of old the rear hook;

P = 17,179.53 lb.

25
DAST rear

hook

Region

modeled

h = 0.698 in.

0.4375

970754

Figure B-5. Geometry of the B-52 original pylon DAST rear hook.

.5 0 -.5 x 105

I I

II0c = 11.25 °

atlmax = 1.007 x 105 Ib/in:

(4 -1.5 x 105 1.0

-.5

at,

Ib/in2

0

Load application

.5 .5 point

:) 1.0

970755

Figure B-6. Distribution of tangential stress a t along inner boundary of the DAST

rear hook; P = 10,000 lb.

26
New rear

hook

h=l in.

Thickness

transition

Region

modeled

0.5 R

h = 1.1 in.

970756

Figure B-7. Geometry of the B-52 original pylon new rear hook.

0 -.5 x 105

Oc = 28.75 °

Thickness

transition

point

-1.0 x 105

-.5

(_t,

0 Ib/in 2

(_tlmax = 1.004 X 105 Ib/in 2

Load

application

970757

Figure B-8. Distribution of tangential stress a t along inner boundary of

the new rear hook; P = 17,179.53 lb.

27
REFERENCES

.

.

.

,

,

6.

Whetstone, W. D., SPAR Structural Analysis System Reference Manual, System Level 13A, Vol. 1,

Program Execution, NASA CR-158970-1, Dec. 1978.

Ko, William L., Delamination Stresses in Semicircular Laminated Composite Bars, NASA

TM-4026, Jan. 1988.

Ko, William L. and Raymond H. Jackson, Multilayer Theory for Delamination Analysis of a Composite Curved Bar Subjected to End Forces and End Moments, NASA TM-4139, Sept. 1989. Also

published in Composite Structures 5, I. H. Marshall, ed., Elsevier Applied Science, London, 1989,

pp. 173-198.

Ko, William L. and Raymond H. Jackson, Open-Mode Delamination Stress Concentrations in Horseshoe and Elliptic Composite Curved Bars Subjected to End Forces, NASA TM-4164, Jan. 1990.

Lekhnitskii, S. G., Anisotropic Plates, Gordon and Breach Science Publishers, New York, 1968.

Ko, William L., Stress Analyses of B-52 Pylon Hooks, NASA TM-84924, Oct. 1985.

28

REPORT DOCUMENTATION PAGE Form Approved

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1. AGENCY USE ONLY (Leave blank) 2. REPORT DATE 3. REPORTTYPE AND DATES COVERED

October 1997 Technical Memorandum

4.TITLE AND SUBTITLE 5. FUNDING NUMBERS

Stress Analysis of B-52 Pylon Hooks for Carrying the X-31 Drop Test

Vehicle

6. AUTHOR(S)

William L. Ko

7.PERFORMINGORGANIZATIONNAME(S)ANDADDRESS(ES)

NASA Dryden Flight Research Center

P.O. Box 273

Edwards, California 93523-0273

9.SPONSORING/MONITORING AGENCYNAME(S)ANDADDRESS(ES)

National Aeronautics and Space Administration

Washington, DC 20546-0001

WU 250 88 00

8. PERFORMING ORGANIZATION

REPORT NUMBER

H-2187

10. SPONSORING/MONITORING

AGENCY REPORT NUMBER

NASA/TM-97-206218

11. SUPPLEMENTARY NOTES

12e. DISTRIBUTION/AVAILABILITY STATEMENT

Unclassified--Unlimited

Subject Category 39

12b. DISTRIBUTION CODE

13. ABSTRACT (Maximum 200 words)

The finite-element structural analysis method was used in a two-dimensional stress concentration

analysis of a new pylon hook for carrying the X-38 lifting body atmospheric drop test vehicle on the

B-52 carrier aircraft. The stress distributions in the hook were obtained, and the critical stress points

were identified. The functional relationships between the applied hook load and the induced maximum

tangential and shear stresses were established for setting the limit hook load for test flights. By

properly representing the X-38 hook with an equivalent curved beam, the conventional curved beam

theory predicted the values of the maximum tangential and shear stresses quite close to those

calculated from the finite-element analysis. The equivalent curved beam method may be of practical

value during the initial stage of the hook design in estimating the critical stresses and failure loads with

reasonable accuracy.

14. SUBJECTTERMS

Equivalent curved beam theory, Finite element stress analysis, Stress concentrations,

X-38 hooks

17. SECURITY CLASSIFICATION 18. SECURITY CLASSIFICATION 19. SECURITY CLASSIFICATION

OF REPORT OF THIS PAGE OF ABSTRACT

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34

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