COPY NO.

TECHNICAL REPORT 4965

COMPONENT PARTS ASSEMBLY WITH JOINTS,

ADHESIVE-MECHANICAL

PART 4. ANALYSIS AND TEST OF BONDED AND

WELDBONDED LAP JOINTS

RICHARD A. MITCHELL RUTH M. WOOLLEY

SAUL M. BAKER

NATIONAL BUREAU OF STANDARDS

WASHINGTON, DC 20234

MARCH 1976

WILLIAM C. TANNER, PICATINNY ARSENAL

PROJECT OFFICER

APPROVED FOR PUBLIC RELEASE, DISTRIBUTION UNLIMITED

PICATINNY ARSENAL

DOVER, NEW JERSEY
The findings in this report are not to be construed

as an official Department of the Army position.

DISPOSITION

Destroy this report when no longer needed. Do not

return to the originator.

\
Unc

1

assif led

SECURITY CLASSIFICATION OF THIS PAGE CWTion Data Entered) REPORT DOCUMENTATION PAGE READ INSTRUCTIONS

BEFORE COMPLETING FORM

1. REPORT NUMBER

Technical Report 4965

2. GOVT ACCESSION NO. 3. RECIPIENT'S CATALOG NUMBER

4. TITLE (and Subtitle) COMPONENT PARTS ASSEMBLY WITH JOINTS,

ADHESIVE-MECHANICAL

PART 4 ANALYSIS AND TEST OF BONDED AND

WELDBONDED LAP JOINTS

5. TYPE OF REPORT & PERIOD COVERED

Final Report-March 1976

6. PERFORMING ORG. REPORT NUMBER

NBSIR 76-1053

7. AUTHORfsJ

Richard A. Mitchell, Ruth M. Woolley,

Saul M. Baker

8. CONTRACT OR GRANT NUMBERfe;

9. PERFORMING ORGANIZATION NAME AND ADDRESS

Engineering Mechanics Section, Mechanics Division

Institute for Basic Standards

National Bureau of Standards, Washington, D.C.

10. PROGRAM ELEMENT, PROJECT, TASK

AREA & WORK UNIT NUMBERS

11. CONTROLLING OFFICE NAME AND ADDRESS

Picatinny Arsenal

Dover, New Jersey 07801

12. REPORT DATE

March 1976

13. NUMBER OF PAGES

70

14. MONITORING AGENCY NAME & ADDRESSfU di//oren< from Controlling Office) 15. SECURITY CLASS, (of this report)

15a. DECLASSIFI CATION/ DOWN GRADING

SCHEDULE

16. DISTRIBUTION ST AT EMEN T fo/ Uiis Roporf;

Approved for public release, distribution unlimited.

17. DISTRIBUTION STATEMENT (of the abstract entered In Block 20, If different from Report)

18. SUPPLEMENTARY NOTES

19. KEY WORDS (Continue on reverse side if necessary and Identify by block number)

Adhesive-bonded joints; bonded joints; cyclic loading; debond analysis;

double-lap-joint analysis; fatigue tests; finite element analysis; joints;

nonlinear analysis; single-lap-joint analysis; single-lap-joint bending;

spotwelded joints; weldbonded joints.

20. ABSTRACT ("Cotrtfotie on rvvers* sftto If n*ce^aaiy aad Identify by block number)

Finite element computer techniques were used to study the linear and

nonlinear structural response of bonded and weldbonded lap joints. Although

the techniques used are applicable to either single-lap or double- lap joints,

the emphasis was on the single-lap joint problem with the attendant complication of joint bending. Nonlinear algorithms were developed to account for

nonlinear stress-strain characteristics of the adhesive, and the joined metal

.^heet, weld-heat softening of the metal sheet, progressive debonding of the

DO , j^^Ta 1473 EDfTlOM OF » MOV 65 IS OBSOLETE

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20. Abstract (continued)

adhesive, and nonlinear cyclic loading. The nonlinear modes of response were

simulated by sequences of linear solutions.

Eight different single-lap joint configurations, designed so as to con- stitute an experimental parameter study, were studied in a laboratory testing

program. Representative specimens were subjected to quasi-static tensile

strength and cyclic-load tests and to tensile fatigue tests. The quasi-static

and fatigue data generally plot into clear S-N patterns that are in a reasonable relationship to the lap joint design parameters. For the most part,

strains measured on the surfaces of the test specimens were in reasonably good

agreement with those computed by finite element analysis, provided out-ofplane bending effects were accounted for. The dominant failure modes were

consistent with the computer analyses.

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TABLE OF CONTENTS

Page

1. INTRODUCTION 2

2. LINEAR ANALYSIS ^

2.1 Planform Analysis 5

2.2 Longitudinal Cross-Section Analysis 5

2. 3 Bending Effects 8

3. NONLINEAR ANALYSIS 13

3.1 Nonlinear Adhesive Shear 13

a. Planform Algorithms 16

b. Longitudinal Cross-Section Algorithm 20

3.2 Nonlinear Metal Algorithm 20

3.3 Numerical Examples 23

a. Monotonlcally Increasing Load 23

b. Cyclic Load 27

c. Adherend Yield 31

4. LAP JOINT TENSILE TESTS 31

4.1 Lap Joint Specimens 31

4.2 Quasi-Static Tests ^0

4.3 Fatigue Tests

4.4 Failure Modes ^9

5. ANALYTICAL-EXPERIMENTAL STRAIN COMPARISONS

6. CONCLUSION 61

7. REFERENCES 65

ANALYSIS AND TEST OF BONDED AND WELDBONDED LAP JOINTS

Richard A. Mitchell, Ruth M. Woolley, and Saul M. Baker

ABSTRACT

Finite element computer techniques were used to

study the linear and nonlinear structural response of

bonded and weldbonded lap joints. Although the techniques used are applicable to either single-lap or double-lap joints, the emphasis was on the single-lap

joint problem with the attendant complication of joint

bending. Nonlinear algorithms were developed to ac- count for nonlinear stress-strain characteristics of

the adhesive and the joined metal sheet, weld-heat

softening of the metal sheet, progressive debonding

of the adhesive, and nonlinear cyclic loading. The

nonlinear modes of response were simulated by sequences

of linear solutions.

Eight different single-lap joint configurations,

designed so as to constitute an experimental parameter

study, were studied in a laboratory testing program.

Representative specimens were subjected to quasi-static

tensile strength and cyclic-load tests and to tensile

fatigue tests. The quasi-static and fatigue data generally plot into clear S-N patterns that are in a reasonable relationship to the lap joint design parameters. For the most part, strains measured on the

surfaces of the test specimens were in reasonably good

agreement with those computed by finite element analysis, provided out-of-plane bending effects were ac- counted for. The dominant failure modes were consistent

with the computer analyses.

Key Words: Adhesive-bonded joints; bonded joints;

cyclic loading; debond analysis; double-lap-joint

analysis; fatigue tests; finite element analysis;

joints; nonlinear analysis; single-lap-joint analysis;

single- lap-j oint bending; spotwelded joints; weldbonded

j oints.
1. INTRODUCTION

The structural response of a bonded or weldbonded lap joint is an extremely complex mechanics problem. These joints are characterized by

material discontinuities, nonlinear material properties, and out-of-plane

bending, each of which can have a first order effect on a critical stress

state. In order to properly interpret laboratory tests of these joints,

and in order to more nearly optimize their designs, it is necessary to

have some understanding of the highly nonuniform stress and strain fields

throughout the joints. The finite element techniques described in this

report are intended for use in developing such an understanding.

Techniques for the linear analysis of these joints were described

in an earlier report [1]. In the present report, algorithms are described for approximating the nonlinear response of these joints by a sequence of linear solutions. Because of the complex and nonlinear

nature of this problem, there is a high potential for obtaining finite

element solutions that deviate far from the physical joint responses

they are intended to simulate. Laboratory testing of lap joints representative of those simulated on the computer is one of the most effective

methods for detecting excessive errors in the finite element model. The

laboratory testing program reported here was intended to serve this purpose within the limited range represented by the specimens and tests.

2. LINEAR ANALYSIS

Because they form the basis for the nonlinear analysis to follow,

the linear planform and longitudinal cross-section analyses reported

earlier [1] will first be briefly described. Figure 1 shows a singlelap weldbonded joint, similar to one studied in the laboratory testing

program, and a comparable double-lap joint. Figure 2 is a schematic

(not to scale) detail of a spotweld region in such a single-lap joint.

In Figure 2 the solid circle represents the visible mark at the edge of

the surface of contact between the spotwelding electrode and the metal

sheet. The inner dashed circle outlines the weld nugget. The area

between the two dashed circles, sometimes referred to as a "halo", is

effectively unbonded due to the displacement and heating of the adhesive during the spotwelding process. Beyond the halo is a region of

transition to full adhesive thickness. The precise shape and dimensions of these spotweld features are functions of several variables,

including the thickness and stiffness of both the metal sheet and the

uncured adhesive and such welding parameters as pressure, current,

resistance, time, and electrode shape. For purposes of describing

their general configurations, bonded joints and spotwelded joints can

be thought of as special cases of the weldbonded joint.

2
NUGGET DIA. 6.3

HALO DIA. 12.2

TRANSITION DIA. 13.2

1

T

3.17

0.13 ADHESIVE z

[a| SINGLE-LAP 3.17

3.17 '

\ \ \

0.13 ADHESIVE —7. I

(b| DOUBLE-LAP

8.35

Figure 1 . COMPARABLE SINGLE-LAP AND DOUBLE-LAP WELDBONDED JOINTS.

DEMINSIONS ARE IN MILLIMETERS (1 in = 25.4 mm).

3
SCHEMATIC DESCRIPTION OF A SPOTWELD IN A SINGLE-LAP WELDBONDED JOINT.
2.1 Planform Analysis

The general approach used here parallels one used earlier for the

planform analysis of composite-reinforced cutouts and cracks [2, 3].

Figure 3(a) shows the network of triangular finite elements used for

the analysis of the weldbonded joint shown in Figure 1 and a similar

spotwelded joint. Figure 3(b) shows the network used for the analysis

of a similar bonded joint. Because of symmetry about the x axis, only

the upper half of a joint is analyzed. The joined sheets are each divided into separate networks of triangular, constant strain (linearly

varying displacement), plane stress elements that are congruent within

the overlap region. Within a bonded and/or welded region the two con- gruent networks are coupled together by an array of special shearstiffness elements linking conjugate pairs of nodal points. External

normal and shear loads are assumed to act only at the edge of a sheet,

in the midplane, and out-of -plane deflections are ignored.

The direct stiffness matrices of the triangular elements are com- puted in the usual way (see, for example, Zienkiewicz [4]), but a different formulation based on the following assumptions is used to compute

the shear stiffness coupling elements. Within the area of a spotweld

nugget (Figure 2) there is no adhesive and the two metal sheets are assumed to be perfectly joined (continuous). There is also no adhesive

in the halo region surrounding a nugget but here the two metal sheets

are assumed to be unbonded. Within the shear-stiffness element the

shear stress is assumed to vary linearly through the metal sheet thickness. That is, the shear stress is assumed to have a maximum value at

the adhesive layer, or at the midplane of a weld nugget, and to decrease

uniformly to zero at a free surface or at the midplane of a double-lap

joint. The effective area of a shear-stiffness element in the x-y plane

is assumed to be equal to one-third the sum of the triangular areas meeting at an overlap nodal point. The effective material thicknesses

within an element are assumed to be the thicknesses at the location of

the conjugate nodal points.

The stiffness matrices of the traingular and the shear-stiffness

elements are superposed to form the stiffness matrix of the entire structure. This latter matrix [K] relates the external forces applied to the

joint {F} to the resulting nodal point displacements {w} according to

the equation

{F} = [K] {w} (1)

This equation can be solved for nodal point displacements {w} throughout

the joint. Then, strains and stresses within the separate elements can be computed directly by matrix multiplication [1, A].

2.2 Longitudinal Cross-Section Analysis

Figure 4 shows finite element networks used for the cross-section

analysis of the comparable single-lap and double-lap weldbonded joints

described in Figure 1. Thickness dimensions and computed vertical deflections are exaggerated. In the double-lap case, symmetry is imposed

5
a] WELDBONDED AND SPOTWELDED

lb] BONDED

Figures. FINITE ELEMENT MESHES USED FOR PLANFORM ANALYSIS OF A

WELDBONDED JOINT, ASPOTWELDED JOINT, AND A BONDED JOINT.

6
[a) SINGLE-LAP

iiimmiz /llhN

/IImini, miiimi m .

(bj DOUBLE-LAP

Figure 4. FINITE ELEMENT MESHES USED FOR LOGITUDINAL CROSSSECTION ANALYSIS OF COMPARABLE SINGLE-LAP AND DOUBLELAP WELDBONDED JOINTS. THICKNESS DIMENSIONS EXAGGERATED 4 TIMES. VERTICAL DEFLECTIONS EXAGGERATED 20

TIMES.
about the horizontal (x) axis. In each case the sheet mid thickness point

at the left end was constrained with respect to x displacement and the

load was applied to the midthickness point (of the finite element network)

at the right end. In the cross-section analysis, linearly varying strain

(quadratically varying displacement) elements are used to better approximate out-of-plane bending. The direct stiffness matrices of the triangular

elements are computed using the area coordinate formulation described by

Zienkiewicz [4]. The spotweld nuggets are approximated by triangular bondline elements of metal, rather than adhesive, equal in area (in plan) to

the circular area of the spotwelds. A nugget region is bounded on each

side by a region of transition to full adhesive thickness. The bondllne

thickness in a nugget or transition region is reduced to approximate the

average adhesive thickness.

2.3 Bending Effects

The importance of bending effects will be demonstrated by a comparison

of numerical results from both the planform and the longitudinal-crosssection computer programs. These results are for the joints shown in

Figure 1 subjected to an applied tensile stress of 69 MPa (10,000 Ibf/in^).

The joint materials were assumed to be linear elastic and isotropic with

the following elastic constants:

Metal sheet: E = 68.4 GPa (9.92 x 10^ Ibf/in^), v = 0.318

Adhesive: E = 4.67 GPa (0.677 x 10^ Ibf/in^), v = 0.35

Figure 5 compares the adhesive shear stresses, t, for the single-lap

weldbonded joint as computed by the planform analysis with those computed

by the cross-section analysis. The difference in shear stress distribution is largely due to joint bending which is included in the cross-section

analysis but not in the planform analysis. This explanation is supported

by a similar plot for a comparable double-lap joint (Figure 6) which shows

much better agreement between the planform and the cross-section analyses.

Figure 7 compares the adhesive normal stresses (peel component, a ) computed by the cross-section analysis for both the single-lap and com- parable double-lap weldbonded joints. The peak tensile stress values at

the right end differ by less than 6 percent. The symmetry constraint

imposed in the double-lap case causes the normal stress peak to be com- pressive at the left end. Peel stresses are not computed in the planform

analysis

Figure 8 compares the strains on the upper surface of the single-lap

weldbonded joint as computed by the planform analysis with those computed

by the cross-section analysis. The plotted points were obtained by applying a bending correction to the planform results. The bending curvatures

(evident in Figure 4) were determined from the cross-section results by

fitting a second degree curve through groups of five adjacent nodal points

8
30

20

10

PLANFORM

CROSS-SECTION

SPOTWELD LOCATIONS

40 80

X, mm

Figure 5. ADHESIVE LAP SHEAR STRESS IN SINGLE-LAP WELDBONDED JOINT,

FROM PLANFORM ANALYSIS AND CROSS-SECTION ANALYSIS.

9
10
30

- SINGLE-LAP

DOUBLE-LAP

1

20

10

r-SPOTWELD LOCATIONS—,

'

0

r

—

N

10

1

/

1

1 J 11 1 1 0 20 40 60 80

X,nrim

Figure 7. ADHESIVE NORMAL (PEEL) STRESS IN COMPARABLE SINGLE-LAP AND

DOUBLE-LAP WELDBONDED JOINTS, FROM CROSS-SECTION ANALYSIS.

11
_2 I I I I I I I L 0 20 40 60

X,mm

Figure 8. LONGITUDINAL STRAIN ON SURFACE OF SINGLE-LAP WELDBONDED

JOINT.

12
along the length of the joint. The components of surface strain due to

bending were then computed directly from these curvatures by assuming a linear variation in bending strain through the joint thickness.

Figure 9 compares the surface strains for the double-lap weldbonded

joint as computed by the planform analysis with those computed by the

cross-section analysis. Although bending is prevented by symmetry along

the X axis in the cross-section analysis, there is some bending of the

outer sheet at each end of the overlap, and there is considerable bending

beyond the overlap (Figure 4). The plotted points were obtained by applying a bending correction (from the cross-section analysis) to the

planform results.

3. NONLINEAR ANALYSIS

Nonlinear algorithms were developed to account for nonlinear stressstrain characteristics of the adhesive and the metal sheet, nonlinear cyclic loading, adhesive debonding, and weld-heat softening of the metal

sheet. The nonlinear modes of response are simulated by a sequence of

linear solutions. Between successive linear solutions, the linear material constants and/or bond couplings of the different finite elements are individually adjusted according to their stress state and, in the case of

cyclic loading, according to their stress history.

3.1 Nonlinear Adhesive Shear

The nonlinear adhesive algorithms assume the shear stress-strain

characteristics described schematically in Figure 10. The single valued

function represented by the smooth curve passing through the origin is,

in the first quadrant, defined by —-, T > T - A exp [-B(t - T^) ^] P

Y

= f (T) =1° P

(2)

G; ' 0 1 ^ 1 ^P

in which G , A, B, C, and x are empirical material constants. For a material with an initial liKear elastic range, G is the linear elastic

shear modulus and x is the proportional limit stress. The relationship

plotted in Figure l8 was determined by fitting a curve to data from three

similar "napkin ring" torsion tests on samples of the adhesive used to

fabricate the lap joint test specimens.* The empirical material constants.

*These adhesive shear stress-strain data were furnished by Feltman Research

Laboratory, Picatinny Arsenal, Dover, NJ. The torsion tests were conducted

by the Singer Co., Little Falls, NJ, using the measurement techniques

described in Reference 5.

13
16

14

12

10

1-2

8

6-

I hi

I

I

I

OVERLAP

PLANFORM

CROSS-SECTION

O PLANFORM (PLUS BENDING)

^ Q

20 40 60

Figure 9. LONGITUDINAL STRAIN ON SURFACE OF DOUBLE-LAP WELDBONDED

JOINT.

14
15
where modulus and stress are expressed in MPa, are

G = 1.729 X 10^

A° = 468. A6 X 10^

B = 2949.3

C = .35

T = 0

P If modulus and stress are expressed in Ibf/in , the constants are

G

T

.2508 X 10^

67.942 X 10^

133.73

.35

0

P

The elastic modulus of the nonlinear adhesive is assumed to be given by

in which v is Poisson's ration, which is assumed to be constant, and G is

a variable secant modulus. The effective shear modulus of any finite

element depends on both the stress and the stress history of that element.

Specifically, an algebraic increase in stress to a point below an "upper

linear limit" results in a linear algebraic increase in strain; an algebraic increase above an "upper linear limit" results in a nonlinear algebraic increase in strain. An algebraic decrease in stress to a point above

a "lower linear limit" results in a linear algebraic decrease in strain;

an algebraic decrease below a "lower linear limit" results in a nonlinear

algebraic decrease in strain. If a change in stress results in an intersection with the single-valued stress-strain curve passing through the

origin (defined by equation (2) in the first quadrant) further change in

stress results in a change in strain along that curve. All linear stressstrain segments have a slope defined by the constant shear modulus G . The nonlinear stress-strain segments, other than equation (2), are programmed as explicit analytical functions with several coefficients read

in as data. The coordinates of the discrete points that define the segmented "linear limits" are also read in as data.

Nonlinear algorithms in the planform analysis account for three different nonlinear modes of response in the adhesive. One nonlinear mode

occurs when the adhesive shear stresses are increased monotonically to a level above the proportional limit stress; a second mode occurs when these

stresses are increased and decreased cyclically; a third mode occurs when

these stresses exceed a prescribed limiting value and the adhesive debonds.

E = 2G(1 + v) (3)

a

Planform Algorithms

16
The monotonic loading algorithm searches for a finite element solution at each of a sequence of levels of Increasing applied load. For each

solution the adhesive shear stresses and strains throughout the joint are defined by a single-valued function such as equation (2). The algorithm

for finding a solution for a particular level of applied load is:

1. Obtain a finite element solution for nodal point displacements throughout the joint. (This may be either the next

previous solution at the current load level or a linearly

scaled solution for a lower load level.)

2. Adjust the adhesive shear modulus of each shear coupling

element. Perform steps a through d for each shear coupling

element

a) Find the shear stress and strain for the element

from the nodal displacements by matrix multiplication (point 1 in Figure 11)

b) Find the point on the nonlinear stress-strain curve with the same strain value as the point found in

step a (point 2 in Figure 11).

c) Compute a secant modulus G* which is the slope of

the line from the origin to the point found in

step b (line 0-2 in Figure 11).

d) Given G as the previous shear modulus for the element,

compute a new shear modulus

G = G + 6(G* - G)

where B is the same for all elements and acts as a multiplier for the change in shear modulus. This

is used to speed convergence.

3. Repeat steps 1 and 2 until converged.

The cyclic loading algorithm searches for finite element solutions, at

each of a sequence of levels of cyclically changing applied load. For each

solution the adhesive shear stresses and strains throughout the joint are defined by nonlinear cyclic loading and unloading relationships of the type

described schematically in Figure 10. The initial finite element solution

of a sequence of cyclic loading solutions is obtained by the monotonic

loading algorithm. Subsequent finite element solutions for nodal point

displacements at particular levels of applied cyclic load are obtained by

the following algorithm.

17
18
1. Solve the linear problem using the element material properties of the final solution at the previous load.

2. Adjust the adhesive shear modulus. Perform steps a through g for each shear coupling element.

a) Find the shear stress and strain for the element

from the nodal displacements by matrix multiplication (points 3 and 5 for the two examples in

Figure 11)

b) Find the nonlinear stress-strain curve for this

element. This is based on the solution at the

last maximum or minimum load and whether this

is an increasing or decreasing portion of the

cycle. (The solution for the previous maximum

or minimum is point 7 or 8 respectively in

Figure 11.)

c) Find the point on the curve with the same strain

value as the point found in step a (points 4 and

6 in Figure 11)

d) Find the point where the curve found in step b crosses the strain axis (points 9 and 10 in

Figure 11).

e) Compute a secant modulus G* which is the slope

of the line from the point found in step c to

the point found in step d (lines 9-4 and 10-6

in Figure 11). This is equivalent to translating the coordinate system and forces G* to

be positive, better approximating the nonlinear

curve

f) Given G as the previous shear modulus for the

element, compute a new shear modulus

G = G + 3(G* - G)

where 3 is the same for all elements.

g) Compute an "initial strain" for the element to

correct for the fact that the lines found in

step e do not pass through the origin. The

amount of this "initial strain" is equal to the

strain value of the point found in step d.

3. Repeat steps 1 and 2 until converged.

19
The adhesive debond algorithm effectively uncouples the shear-stif fnes

elements for which the next previous finite element solution indicated adhesive shear stresses above a prescribed limiting value. This algorithm

can be used in conjunction with either the monotonic loading algorithm or the cyclic loading algorithm.

b. Longitudinal Cross-Section Algorithm

The only algorithm included in the longitudinal cross-section analysis to account for nonlinear adhesive deformation is a monotonic loading

algorithm. This algorithm is the same as the monotonic loading algorithm

in the planform analysis except for the following modification to step 2:

b) Find the point on the nonlinear stress-strain curve with

the same stress value as the point found in step 2a.

3.2 Nonlinear Metal Algorithm

The nonlinear metal algorithm assumes a single-valued normal stressstrain relationship of the type represented by the solid curve in Figure 12

This curve is of the form

e = f(a) =

a

- A exp [-B(a - a^) ^] ' °

^

in which E , A, B, C, and a are empirical material constants. For a material wxth an initial liRear elastic range, E is the elastic modulus

and a is the proportional limit stress. The relationship plotted in

Figuri 12 was determined by fitting a curve (equation (4)) to a portion

of a stress-strain curve for 7075-T6 aluminum alloy given in Reference 6.

For that curve fit, the empirical material constants, where modulus and

stress are expressed in MPa, are

E = 68.74 X 10^

o A = 3.7914 X 10^^

B = 271.82

C = .1 a = 275.8 X 10^

P

20
Figure 12. STRESS-STRAIN PLOT FOR METAL SHOWING OFFSET DUE TO WELD^

HEAT SOFTENING.

21
If modulus and stress are expressed in Ibf/in^, the constants are

E = 9.97 X 10^

o A = 5.4988 X 10^3

B = 112.31

C = .1

a = 40.0 X 10^

P

The shear modulus of the metal is assumed to be given by

in which v is Poisson's ratio, which is assumed to be constant, and E is

a variable secant modulus.

A single algorithm is used for either monotonic or cyclic loading in

both the planform analysis and the longitudinal cross-section analysis.

The algorithm below searches for a finite element solution for which the

normal stresses in the metal are defined by a single-valued function such

as equation (4)

1. Obtain a finite element solution for nodal point displacement throughout the joint. (This may be either the next

previous solution at the current load level or a linearly

scaled solution for a different load level.)

2. Adjust the metal elastic modulus. Repeat steps a through

d for each metal element.

a) Find the longitudinal (x axis) normal stress and

strain for the element from the nodal displacements and matrix multiplication (point 1 in • Figure 12).

b) Find the point on the nonlinear stress-strain

curve with the same stress value as the point

found in step a (point 2 in Figure 12).

c) Compute a secant modulus E* which is the slope

of the line from the origin to the point found

in step b (line 0-2 in Figure 12).

d) Given E as the previous elastic modulus for the

element, compute a new elastic modulus

E = E + 6(E* - E)

22
where 3 is the same for all elements and acts as

a multiplier for the change in elastic modulus.

3. Repeat steps 1 and 2 until converged.

A variation of the above algorithm is used to account for weld-heat

softening of the metal sheet. Figure 13 illustrates the variation in

metal hardness as a function of distance from the weld nugget for a weldbonded specimen of 7075-T6 aluminum alloy.* The plotted curve is a cubic

polynomial fitted to the data points. The weld-heat-softening algorithm

is based on two assumptions: (1) that there is a percentage reduction in

the proportional limit stress equal to the percentage reduction in microhardness number; (2) that the shape of the nonlinear portion of the stressstrain curve is not changed. These assumptions are illustrated by the

dashed curve in Figure 12, corresponding to a 40 percent reduction in the

microhardness number. For finite elements affected by weld-heat softening,

the nonlinear algorithm uses the appropriately translated single-valued

stress-strain curve.

The following numerical examples illustrate some types of results

that can be obtained using the nonlinear algorithms. The nonlinear stressstrain relationships used are those shown in Figure 10 and 12. To conserve

on computer costs for these numerical examples, the single-valued stressstrain curve for the adhesive was assumed to be linear, with a slope of G , up to a shear stress level of 13.8 MPa (2000 Ibf/in^). The linear elastic

material constants used are:

A sequence of solutions involving adhesive yielding, progressive debonding, and weld-heat softening were obtained for the single-lap weldbonded joint shown in Figure 1, using the planform computer program and

the finite element mesh shown in Figure 14. Two solutions were first obtained for an applied longitudinal tensile stress of 138 MPa (20 000 Ibf/

in^) and then two solutions were obtained for twice that applied stress.

At each load level, one solution assumed linear elastic material properties throughout and the other solution assumed the nonlinear stress-strain

relationships. Figure 15 gives the adhesive shear stress variation along

the longitudinal centerline of the joint for the four solutions. These

*These Knoop microhardness data were furnished by Feltman Research

Laboratory, Picatinny Arsenal, Dover, NJ.

3.3 Numerical Examples

Metal: E

Adhesive: G

o

68.74 GPa (9.97 x 10^ Ibf/in^), v = 0.318

1.729 GPa (0.2508 x 10^ Ibf/in^), v = 0.35

a. Monotonically Increasing Load

23
200

O

J I I I I 0 2 4 6 8 10

DISTANCE FROM NUGGET, mm

Figure 13. VARIATION IN HARDNESS ADJACENT TO SPOTWELD NUGGET.

24
Figure 14. FINITE ELEMENT MESH USED FOR iMONLINEAR PLANFORM ANALYSIS

OF WELDBONDED JOINT.

25
26
results indicate a 19 percent reduction in peak stress due to adhesive

nonlinearity at the lower load level and a 51 percent reduction at the

higher load level. Figure 16 shows contour plots of lap shear stress for

the two solutions at the higher load level. The adhesive nonlinearity ac- counts for the great reduction in the lap shear stress magnitude and stress

gradient near the end of the overlap as well as the increase in stress

gradient near the spotwelds.

Two additional nonlinear solutions were then obtained for an applied

tensile stress of 276 MPa (40 000 Ibf/in^) with the assumption that any

pair of conjugate nodal points in the metal adherends were uncoupled (debonded) where the adhesive shear stress exceeded 55 MPa (8000 Ibf/in^).

Thus, there was a sequence of three nonlinear solutions at this load level.

In the first of these solutions, the stresses at the nodal points at each

corner of the overlap region were slightly over the debond stress. Therefore these corner points were uncoupled for the second nonlinear solution

at this load level. In the second nonlinear solution, three additional

nodal points near each corner were slightly over the debond stress, and

therefore they were uncoupled for the third solution. In the third solution, a total of seventeen nodal points in the upper half of the joint were either already debonded or over the debond stress limit. These three

solutions illustrate progressive debonding without any increase in applied

tensile stress. Figure 17 shows contour plots of lap shear stress for

two of these partial-debond solutions. The dark regions are dense contours

linearly interpolated by the computer program between uncoupled points of zero stress and points of peak stress.

The applied tensile stress was then increased to 345 MPa (50 000 Ibf/

in^) for a solution well within the nonlinear range of the metal sheet. The

previous solutions in this sequence had involved no nonlinear deformation

of the metal. For this solution it was assumed that the proportional limit

of the metal in the vicinity of the spotwelds was reduced as described by

the curves in Figures 12 and 13 due to weld-heat softening. It was also

assumed that the seventeen nodal points that were debonded according to the

previous solution at the 276 MPa (40 000 Ibf/in^) applied stress level re- mained uncoupled. Contour plots of the resulting lap shear stress and

longitudinal tensile stress in the metal are given in Figure 18. No significant perturbation of the longitudinal tensile stress in the metal sheet

due to weld-heat softening is evident in Figure 18(b). This is understandable, because the tensile stress levels indicated in the vicinity of the

rightmost spotweld represent relatively small deviations from linearity,

although they are above the proportional limit of the weld-softened metal.

b. Cyclic Load

A sequence of four solutions representing two cycles of loading into

the nonlinear-adhesive range were obtained for the single-lap weldbonded

joint shown in Figure 1 (but with an adhesive thickness of 0.066 mm

(0.0026 in)) and the analysis mesh shown in Figure 14. An initial linear

27
iMiniimin I J I I—I 1 1 iiiiiiiiiiM 14864 2 - 2 46816

(a) LINEAR ANALYSIS

(b) NONLINEAR ANALYSIS

Figure 16. CONTOUR PLOTS OF LAP SHEAR STRESS AT HIGHER APPLIED LOAD

LEVEL. CONTOUR INTERVAL IS 500 Ibf/in2(3.45 MPa )•

28
(a) TWO POINTS DEBONDED

8 6 4 2 2 4 6 8

8 6 4 2 2 4 6

(b) EIGHT POINTS DEBONDED

Figure 17. CONTOUR PLOTS OF LAP SHEAR STRESS SHOWING PARTIAL DEBOND.

CONTOUR INTERVAL IS 500 Ibf/in^ (3.45 MPa).

29
8 6 4 2 2 4 6 8

8 6 4 2 2 4 6 8

(a) LAP SHEAR STRESS

CONTOUR INTERVAL IS 500 Ibf/ln2(3.45 MPa).

5101520 25

0 5101520

3035 4045

25 30354045

(b) LONGINTUDINAL NORMAL STRESS IN RIGHTMOST SHEET

CONTOUR INTERVAL IS 1000 Ibf/in2 (6.9MPa).

Figure 18. CONTOUR PLOTS OF STRESS IN JOINT LOADED WITHIN THE NONLINEAR

RANGE OF BOTH ADHESIVE AND METAL.

30
elastic solution was obtained for an applied longitudinal tensile stress

of 20.7 MPa (3000 Ibf/in^). The applied tensile stress was then increased

to 207 MPa (30 000 Ibf/in^) for a second solution, decreased back to

20.7 MPa (3000 Ibf/in^) for a third solution, and increased again to

207 MPa (30 000 Ibf/in^) for a fourth solution. The latter three solutions

in this sequence involved extensive nonlinear deformation of the adhesive.

Figure 19 gives the adhesive shear stress variation along the longitudinal

centerline of the joint for the four solutions. These results indicate the

residual stresses due to the nonlinear adhesive deformation.

c. Adherend Yield

Two solutions were obtained for a single-lap bonded joint similar to

the weldbonded joint shown in Figure 1 (but with an adhesive thickness of

0.371 mm (0.0146 in)), using the longitudinal-cross-section computer program and the finite element mesh shown in Figure 20. The sheet mid thick- ness point at the left end was restrained and a mean tensile stress of

345 MPa (50 000 Ibf/in^) was applied through the midthickness point at the

right end. For one solution all material properties were assumed to be

linear elastic. For the other solution the nonlinear-metal stress-strain

relationship shown in Figure 12 was used, but the adhesive was assumed to

be linear elastic. Figure 21 shows contour plots of the longitudinal com- ponent of normal stress for these two solutions. The linear-elastic solution gave a peak tensile stress at the end of the overlap 8 percent greater

than that given by the nonlinear-metal solution. Figures 22 and 23 give,

respectively, the variations in the adhesive shear stress and the normal

(peel) stress along the length of the joint for both solutions. Although

the yielding of the metal reduces the peak metal stress, the greater strain

in the metal forces a significant increase in both components of adhesive

stress

4. LAP JOINT TENSILE TESTS

Eight different single-lap joint configurations, designed so as to

constitute an experimental parameter study, were subjected to quasi-static

tensile strength and cyclic load tests and to tensile fatigue tests. The

quasi-static loads were applied in a 50 000 Ibf capacity screw-powered

universal testing machine and the fatigue loads were applied in a 50 000

Ibf capacity servo-controlled electrohydraulic testing machine. These

machines are described in Reference 7. The end fixtures shown attached to

a broken specimen in Figure 24 were used in all tests. The single-pin

fixtures and the flexibility of the long thin specimens assured relatively

low eccentricity of the load applied at the specimen end tabs.

4.1 Lap Joint Specimens

Figure 25 is a drawing of the weldbonded joint that was the central

reference design, a broken specimen of which is shown in Figure 24. The

other seven test specimen designs were nominally the same as the central

reference design except for one dimension. The eight designs were:

31
32
Figure 20. FINITE ELEMENT MESH USED FOR NONLINEAR CROSS-SECTION

ANALYSIS OF A BONDED JOINT. THICKNESS DIMENSIONS EXAGGERATED

4 TIMES, VERTICAL DEFLECTIONS EXAGGERATED 20 TIMES.

f

33
0 1 2 3 4 4323

(a) LINEAR-ELASTIC ANALYSIS

0 1 2 - 3 4 4 3 2 3

(b) NONLINEAR-METAL ANALYSIS

Figure 21. CONTOUR PLOTS OF LONGITUDINAL NORMAL STRESS IN A BONDED

JOINT. CONTOUR INTERVAL IS 5000 Ibf/in^ ( 34.5 MPa).

34
120

Ol I I I I —L ^ 1 I I 0 20 40 60 80

X,mm t

Figure 22. ADHESIVE SHEAR STRESS IN BONDED JOINT FROM LINEAR ANALYSIS

AND NONLINEAR-METAL ANALYSIS.

35
100

80

60-

40

20-

-20

-40

LINEAR

NONLINEAR

20 40

X,mm

60 80

Figure 23. ADHESIVE NORMAL (PEEL) STRESS IN BONDED JOINT FROM LINEAR

ANALYSIS AND NONLINEAR-METAL ANALYSIS.

36
37
1

1 .60 THICK BONDED TAB 6.75 DIA.

• • loo

1 o o

o 6

e o 70

(a) LAP JOINT SPECIMEN

Y

Z

3.17 A

T

NUGGET DIA. 6.3

HALO DIA. 12.2

TRANSITION DIA. 13.2

57.2

0.13 ADHESIVE

(b) OVERLAP REGION DETAILS

^

X

z i

I 3.17

Figure 25. CENTRAL REFERENCE DESIGN OF SINGLE-LAP WELDBONDED JOINT

DIMENSIONS ARE IN MILLIMETERS (1 in = 25.4 mm).

38
1. Central reference joint - 57.2 mm (2.25 in) wide by 3.7 mm

(0.125 in) thick 7075-T6 bare aluminum sheet weldbonded in

a 57.2 mm (2.25 in) long overlap with 0.076 mm (0.003 in)

thick modified epoxy paste adhesive and four 6.3 mm

(0.25 in) diameter spotwelds. The spotwelds were symmetrically located so that each was centered in one quadrant

of the square overlap area.

2. Short joint - The same as design 1 except the overlap was approximately half as long, that is, 28.4 mm (1.12 in),

with two spotwelds.

3. Long joint - The same as design 1 except the overlap was

twice as long, that is, 114.3 mm (4.50 in), with eight

spotwelds

4. Welds-near-end joint - The same as design 1 except the

four spotwelds were centered only 9 . 1 mm (0.36 in) from

the ends of the overlap. The spotwelds were symmetrically

located, the same as design 1, in the transverse direction.

5. Wide joint - The same as design 1 except the overlap was

twice as wide, that is, 114.3 mm (4.50 in), with eight

spotwelds

6. Thin joint - Similar to design 1 except for the aluminum

sheet which was 1.6 mm (0.063 in) thick. Also, the spotweld diameter was only 4.7 mm (0.19 in), which is more

appropriate for this sheet thickness.

7. Spotwelded joint - Similar to design 1, but without

adhesive bond.

8. Bonded joint - Similar to design 1, but without spotwelds.

Also, because of a difference in the specimen fabrication

process, the adhesive thickness ranged from 0.18 mm

(0.007 in) to 0.71 mm (0.028 in).

Young's modulus and Poisson's ratio for the metal sheet were determined from resistance strain gage measurements with back-to-back gages

located midway between the joint and an end tab; the average values obtained from three specimens are those used in the numerical examples

(Sec. 3.3). The shear stress-strain relationship for the adhesive, as determined by three "napkin ring" torsion tests, is given by equation (2)

and the empirical constants G , A, B, C, and t in Section 3.1.

o p

39
The lap joint specimens of a particular design were fabricated* by

first joining two large sheets of the aluminum alloy in a seam joint, and

then sawing the joined sheets, at right angles to the seam joint, to form

the individual specimens. The seam widths were equal to the overlap

lengths of the specimens. The aluminum end- tab material was also bonded

to the large sheets before sawing the individual specimens. The overlap

surfaces were degreased and then chemically etched before joining. The

bonded joints were fabricated by applying the paste adhesive to

the large sheets, clamping the sheets together during a 40-minute curing

cycle in a larg,e 121 °C (250 °F) oven, and then sawing the individual

specimens. The weldbond joints were fabricated by applying the paste adhesive to the large sheets, spotwelding through the uncured adhesive,

sawing the individual specimens, and then curing in a smaller 121 °C

(250 °F) oven for 40 minutes. The weldbonding process gave significantly

greater uniformity of bondline thickness than did the bonding process.

4.2 Quasi-Static Tests

One specimen of each of the eight designs was tested to failure in

tension. Load was applied very slowly over a period of 5 to 20 minutes.

These results are plotted as single-cycle points in Figures 26 through 33.

One specimen of each of seven designs and two specimens of the bonded

joint were instrumented with from 11 to 27 resistance strain gages and

subjected to from 58 to 140 cycles of quasi-static tensile load. Multiple

cycles of load were applied at progressively higher peak load levels until

failure. The different peak load levels are indicated by the cross symbols

connected by dashed lines in Figures 26 through 33. Loads were cycled between a peak value and 10 percent of that value. Surface strains were

monitored throughout each loading program. These strain measurements indicated a relatively linear loading range, a relatively stable nonlinear

range, and an unstable nonlinear range for most of the specimens. Within

a stable nonlinear loading range, successive cycles to the same load level,

although clearly involving nonlinear deformation, resulted in convergence

to a sequence of roughly similar load-strain hysteresis loops. Within an unstable nonlinear range, successive loading cycles resulted in progressive strain growth with no indication of stable hysteresis. An indication

of these loading ranges, as subjectively interpreted from the strain data,

is given on Figures 26 through 33.

4.3 Fatigue Tests

Several specimens of each design were fatigue tested at different load

levels within the linear range and the stable nonlinear range. Essentially

all of the fatigue results (Figures 26 through 33) fall into a clear pattern that is also in a reasonable relationship to the quasi-static data.

*A11 of the lap joint specimens were fabricated by Feltman Research

Laboratory, Picatinny Arsenal, Dover, NJ.

40
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48
Figure 34 is a summary plot of the fatigue results. The thin sheet joint

(design 6) had the greatest fatigue strength. Ranking second in fatigue

strength was the long joint (design 3). These two joints were probably

over-designed in terms of weight and material utilization. The spotwelded

joint (design 7) was the weakest in fatigue, followed by the short joint

(design 2). There was no significant difference in the fatigue strengths

of the other four joint designs; their fatigue strengths all plot within

the cross-hatched region in Figure 34.

4.4 Failure Modes

All specimen failures can be broadly classified into one of three

distinct fracture modes: (1) lap-shear fracture through the entire bonded

and/or spotwelded overlap area; (2) transverse fracture through one metal

sheet, at the outer edge of a row of spotwelds, after adhesive debond to

that point; and (3) transverse fracture through one metal sheet at or near

the end of the overlap. Figure 35 shows examples of each of these failure

modes. Figures 26 through 33 indicate which of these three failure modes

was present for each test by a numeral 1, 2, or 3 at the data point. There

is an apparent correlation between failure mode and load level for all

eight specimen designs. With few exceptions, lap-shear failures (mode 1)

occurred in relatively high-load, low-cycle tests; transverse fracture near

the overlap end (mode 3) occurred in relatively low- load, high-cycle tests.

Transverse fracture at a row of spotwelds (mode 2) usually occurred in tests run at intermediate load levels.

5. ANALYTICAL-EXPERIMENTAL STRAIN COMPARISONS

To evaluate the finite element analyses, strains measured on the surfaces of the test specimens were compared with strains computed by the

planform and cross-section computer programs. Figures 36 through 43 give

strain gage locations and comparable experimental and analytical results

for each of the eight specimen designs for a load level in the linear elastic range; Figure 44 gives similar results for a load level in the stable

nonlinear range. The linear-range test data were recorded the first time

a specimen was loaded to a nominal level of 8900 N (2000 Ibf) and then

normalized to the level of an applied tensile stress of 69 MPa (10,000 Ibf/

in^) . Stable-nonlinear-range test data were recorded at nominal load levels

of 17,800 N (4000 Ibf) and 26,700 N (6000 Ibf) and then linearly interpolated to 25,130 N (5650 Ibf), corresponding to an applied tensile stress of

138 MPa (20,000 Ibf/in^).

Each of the instrumented test specimens was analyzed in the linear

elastic range by the planform computer program and all except the spotwelded

joint were analyzed by the cross-section program. For each of the seven

joints analyzed by both computer programs, bending corrections, based on the

cross-section analysis, were computed and added to the planform analysis

strains. A reasonable bending correction was suggested by the linear nature

49
BdlAI 'SS3«1S 31ISN3±

50
Figure 35. EXAMPLES OF THE THREE FAILURE MODES. FROM TOP TO

BOTTOM: MODES, 1, 2 and 3.

51
1

1

1 - kJ (

C 1 1 17 91 11

/ I^O O

\

iO o ! I

1

j_ Ji Ji iL

o ! j

GAGE

NO.

MEASURED ANALYTICAL

PLANFORM CROSS-SECTION PLANFORM

(PLUS BENDING)

5 1 996

6 2035 996

7 1982 996

8 36 996

9 654 604 628 716

10 301 401 284 311

11 280 383 237 191

12 685 628 720 820

13 660 616 628 727

14 287 384 284 294

15 722 621 619 722

16 202 384 284 294

17 600 509 624 637

18 370 486 379 356

19 620 512 624 639

20 371 486 379 357

21 432 489 523 510

22 462 490 484 468

23 458 500 523 520

24 498 498 484 477

25 -165 -197

26 -195 -196

Figure 36. LINEAR MICROSTRAIN COMPARISONS FOR THE CENTRAL REFERENCE

JOINT.

52
1 I'O

\ -L 11

^

1 o

] 1 ~

10 1

1. 11 ±

^ !

1

GAGE

NO.

MEASURED ANALYTICAL

PLANEORM CROSS-SECTION PLANFORM

(PLUS BENDING)

1 505 591 630 712

2 439 412 331 299

3 377 402 350 289

4 567 611 603 721

5 500 610 620 719

6 421 405 331 292

7 532 550 591 634

8 451 484 423 402

9 409 494 436 426

10 618 550 582 625

11 536 499 510 506

12 508 499 498 490

Figure 37. LINEAR MICROSTRAIN COMPARISONS FOR THE SHORT JOINT.

53
GAGE MEASURED ANALYTICAL

NO. PLANFORM CROSS-SECTION PLANFORM

(PLUS BENDING)

5 781 584 706 751

6 210 455 320 290

7 177 417 281 224

8 807 620 736 813

9 638 504 501 504

10 349 504 507 505

11 473 504 502 504

12 498 504 501 504

Figure 38. LINEAR MICROSTRAIN COMPARISONS FOR THE LONG JOINT.

54
1

"6° o

J_ 5 7 3

O O

o o

_8 J_

o o

I

1

GAGE MEASURED ANALYTICAL

NO. PLANFORM CROSS-SECTION PLANFORM

(PLUS BENDING)

1 775 629 690 804

2 189 405 248 226

3 422 369 360 320

A 571 649 526 690

5 701 508 590 577

6 257 492 412 420

7 604 494 468 455

8 353 495 541 534

9 797 548 685 766

10 294 216 313

11 -211 -191

Figure 39. LINEAR MICROSTRAIN COMPARISONS FOR THE JOINT WITH SPOTWELDS

NEAR OVERLAP ENDS.

55
o

1- 14—y 11—

|5 V_/ \^

o oV X

|7

1 o o 1 3

O O i« o o

8

o o o o

GAGE

NO.

MEASURED ANALYTICAL

PLANFORM CROSS-SECTION PLANFORM

(PLUS BENDING)

1 620 613 546 680

2 408 412 364 326

3 288 364 279 259

4 679 654 624 782

5 525 515 594 605

6 475 502 411 412

7 447 506 514 516

8 561 506 492 480

9 374 590 625 696

10 424 249 273

11 -127 -216

12 656 579 632 689

13 254 236 270

14 -169 -202

Figure 40. LINEAR MICROSTRAIN COMPARISONS FOR THE WIDE JOINT.

56
1

\-(3 o

1

1 ^ 11 15 7 3 /

1 o o

o

1 O 1 1

6 12

10^ o

1

GAGE

NO.

MEASURED ANALYTICAL

PLANFORM CROSS-SECTION PLANFORM

(PLUS BENDING)

1 666 996

'2 1308 996

3 1310 996

4 670 996

5 797 535 672 684

6 185 477 344 332

7 169 460 334 300

8 811 554 685 712

9 784 542 674 690

10 163 470 345 322

11 670 498 517 517

12 295 496 495 488

13 665 501 517 520

14 281 498 495 491

15 477 488 498 485

16 477 488 504 491

Figure 41. LINEAR MICROSTRAIN COMPARISONS FOR THIN JOINT.

57
GAGE MEASURED ANALYTICAL

NO. (PLANFORM)

1 -482 880

2 123 108

3 0 737

4 283 293

5 428 311

6 617 703

7 674 505

8 333 518

9 -391 862

10 100 201

11 -316 -398

Figure 42. LINEAR MICROSTRAIN COMPARISONS FOR THE SPOTWELD JOINT.

58
[ 11 / l\

1 12

13

5 _9 7 3

GAGE

NO.

MEASURED ANALYTICAL

PLANFORM CROSS-SECTION PLANFORM

(PLUS BENDING)

1 617 680 565 677

2 344 351 284 347

3 331 304 268 351

4 648 710 524 663

5 584 527 621 644

6 377 477 383 362

7 382 468 375 345

8 581 527 626 651

9 479 496 480 490

10 47 3 498 508 506

11 686 579 694 736

12 280 235 302

13 -161 -205

14 284 425 334 270

15 36 113 54

16 -168 -180

Figure 43. LINEAR MICROSTRAIN COMPARISONS FOR THE BONDED JOINT.

59
\ 1 1C

/ ' —

( iir^i 23^

(5 1 J_ JL A JL 7 )

1 0 0

(

< > / 1

GAGE

NO.

MEASURED ANALYTICAL

PLANFORM PLANFORM

(PLUS BENDING)

5 853 1992

6 3188 1992

7 3147 1992

8 918 1992

9 1489 1245 1555

10 434 799 548

11 413 741 206

12 1524 1288 1823

13 1473 1262 1572

14 394 757 502

15 1545 1276 1557

16 283 859 608

17 1324 1030 1384

18 626 977 412

19 1330 1037 1391

20 628 969 404

21 886 986 1074

22 916 987 892

23 915 1003 1091

24 970 1003 908

25 -410 -399

26 -321 -394

Figure 44. NONLINEAR- RANGE MICROSTRAIN COMPARISONS FOR CENTRAL

REFERENCE JOINT.

60
of the bending strain evident in Figure 8 and the bending strains indicated

by back- to-back gages located just outside the overlap region on two specimens (Figures 36 and 41). Figure 45 illustrates the application of this

correction for the central reference specimen (design 1). The test data

shown are the strains indicated by gages 10, 12, 18 and 22 (Figure 36). The

two interior gages (18 and 22) were assumed to define a linear variation in

bending strain, with respect to position, for most of the overlap length.

The variation in bending strain was assumed to be proportional to the effective eccentricity of the tensile stress distribution in the metal sheet at

the end of the overlap. The cross-section solution was obtained using the

mesh shown in Figure 46 with the boundary loads acting through the eccentric

interior mesh points as indicated. The amount of eccentricity was determined

by extrapolating a straight line through the test data (gages 18 and 22) to

the end of the overlap and computing the eccentricity that would account for

the linear strain variation. The rationale for this solution is not rigorous,

but it is believed to give more realistic boundary conditions than the midsheet boundary loading applied to the meshes shown in Figures 4(a) and 20.

For two of the seven joints analyzed for the linear-elastic range by

both computer programs (designs 2 and 5) the best agreement with test data

was obtained by the planform analysis. For the other five of these seven

joints, there was better agreement with test data by the cross-section

analysis than by the planform analysis. For three of the joints (designs 3,

4 and 6) the best agreement was obtained by applying the bending correction

to the planform analysis. The above rankings are based on the mean of the

percentage differences between test data and analysis for all gages located

in the overlap region.

For the case of the central reference specimen loaded in the stablenonlinear range. Figure 44, a nonlinear planform analysis was performed.

The bending corrections applied to the nonlinear planform results, however, were obtained by linearly scaling the results of a linear cross-section analysis. Nevertheless, these scaled linear corrections do significantly improve

the agreement between analysis and test data.

6. CONCLUSION

Both of the linear analyses, planform and longitudinal cross-section,

are expected to be useful in the study of bonded and weldbonded lap joints.

The planform analysis alone may be useful for some purposes in the study of

either double-lap joints or single- lap joints that are constrained to prevent excessive bending. Where out-of-plane bending or peel stresses are important, however, the planform analysis should be supplemented by the

cross-section analysis. When using the linear analyses, one should remain

aware that these analyses do not account for nonlinear modes of response

that may be significant, even at moderate load levels.

61
12

10

8

O

<3>

CROSS-SECTION ANALYSIS

PLANFORM [PLUS BENDING]

TEST DATA OO

O

Figure 45. LONGITUDINAL STRAIN ON SURFACE OF CENTRAL REFERENCE JOINT

62
Figure 46. FINITE ELEMENT MESH USED FOR ANALYSIS OF CENTRAL REFERENCE

JOINT, WITH COLLINEAR ECCENTRIC BOUNDARY LOADS INDICATED.

63
The nonlinear analysis algorithms show promise of adequately simulating

several modes of nonlinear response. They do have practical limitations,

however. The solutions are relatively expensive in computer time and storage because each nonlinear solution consists of a sequence of linear solutions and because a part of the results of previous linear solutions must be

stored for use in obtaining later solutions. The nonlinear solutions ordinarily require multiple computer runs interspersed with subjective evaluation of intermediate results and adjustment of convergence parameters by the user of the computer program.

The laboratory test results support a reasonable degree of confidence

in the validity of the computer programs. Apart from this, the test results

for eight different lap joint configurations constitute a significant experimental parameter study, within the limited range represented by the specimens

and tests.

64
7. REFERENCES

1. Mitchell, R. A., Woolley, R. M. , and Baker, S. M. , Finite Element

Analysis of Spotwelded, Bonded, and Weldbonded Lap Joints, NBSIR 75-957

(Dec. 1975).

2. Mitchell, R. A,, Woolley, R. M. , and Chwirut , D. J., CompositeOverlay Reinforcement of Cutouts and Cracks in Metal Sheet, NBSIR

73-201 (Feb. 1973),

3. Mitchell, R. A., Woolley, R. M. , and Chwirut, D. J,, Analysis of

Composite-Reinforced Cutouts and Cracks, J. AIAA, Vol. 13, No. 6,

744-749 (June 1975),

4. Zienkiewicz, 0. C. , The Finite Element Method in Engineering Science , McGraw-Hill, London (1971).

5. Hughes, E. J., and Rutherford, J. L, , Study of Micromechanical

Properties of Adhesive Bonded Joints, Technical Report 3744, Picatinny

Arsenal, Dover, N.J. (Aug. 1968).

6. Metallic Materials and Elements for Aerospace Vehicle Structures,

MIL-HDBK-5A (Aug. 1972).

7. Chwirut, D. J, (coordinator). Research and Testing Facilities of the

Engineering Mechanics Section, National Bureau of Standards, Washington,

D.C., NBS Spec Publ 370 (1973).

65
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