Marshall S White, Jiyoun Yoo, and Joseph R

LoferskiApril 3, 2012

Modeling Compressive Stress Modeling Compressive Stress Distributions at the Interface Distributions at the Interface Between a Pallet Deck Between a Pallet Deck and Distribution Packaging and Distribution Packaging

Components of the unit load portion of consumer

and industrial product supply chains

This research

focuses on

Compression stress concentrations at the

interface between the pallet deck and distribution

packaging

Stress concentrati

on

Pallet deck

Packaging

Pallet deck deflection

“Non-uniform stress distribution”“Stress concentrations”

The beam on an elastic foundation is an inversion of the

actual situation within the unit load

Beam

Elastic foundation

Packaging

Pallet section

General Solution for Beam Deflection supported by a

deformable elastic foundation

k packaging

stiffnessEI pallet deck stiffness

0y

0

0M

0Q

deflection at x=0slope at x=0

moment at x=0shear force at x=0

y

x0 L

P P

General solution for beam deflection y at any point x

4

(4

(3

(2

1

(40(302010

4

),cossinhsin(cosh4

1)

),sin(sinh2

1)

),cossinhsin(cosh2

1)

coscosh)(

:

)1

)1

)(1

)()(32

EI

k

xxxxxF

xxxF

xxxxxF

xxxF

Where

xFQEI

xFMEI

xFxFyxy

Jiyoun Yoo, PhD candidate 6

P P

Three Levels of pallet connection

fixity are modeled

P P

Free ends

P P

Semi-rigid (Nailed joint)

Fixed ends

• Same Pivot Point • Two equal

concentrated forces (p) applied to two pivot points

Free End Model

L2

y

x

PP

PP

A B C D

-L1 0

A

P

BP

P

MB MB

B CC D

P

MCPMC

P

2PM0

=

Boundary Conditions

(a) at x2 = 0; Q0 = P

(b) at x1 = 0 and x2 = 0; M(x1) =

M(x2)= M0

(c)at x1 = 0 and x2 = 0; y(x1) =

y(x2)= y0

(d)at x2 = L2; M=M0

(e) at x2 = L2 ; Q = -P

Semi-Rigid Joint Model

L2

y

x

PP

PP

A B C D

-L1 0

A

P

BP

PB C

C D

P

P

P

2P

M0=

d

FN

MD

MB MBMA

MA

MC MDMC

Boundary Conditions

(a) at x2 = 0; Q0 = P

(b) at x1 = 0 and x2 = 0; M(x1) =

M(x2)= M0

(c)at x1 = 0 and x2 = 0; y(x1) =

y(x2)= y0

(d)at x2 = L2; M=M0

(e) at x2 = L2 ; Q = -P

MA = FN x dFN = force exerted by nails d = distance between nail and stringer

FN = kyk =rotational modulus (lbs. /in.) y = the vertical displacement of the deck at the nail location (in.)

Fixed End Model

L2

y

x

PP

PP

A B C D

-L1 0

A

P

BP

PB C

C D

P

P

P

2P

M0=

MD

MBMBMA

MA

MC MDMC

Boundary Conditions

(a) at x2 = 0; Q0 = P

(b) at x1 = 0 and x2 = 0; M(x1) =

M(x2)= M0

(c)at x1 = 0 and x2 = 0; y(x1) =

y(x2)= y0

(d)at x2 = L2; M=M0

(e) at x2 = L2 ; Q = -P

(f) at x1 = 0 and x2 = 0; θ(x1) = θ(x2)= θ0

Model Inputs and Outputs

Compressive Stress Distribution

Models: Free Ends

Bottles (k=1345 lbs./in.)

Applied stress

(6.5 psi)

Empty box (k=854 lbs./in.)

Applied stress

(6.25 psi)

Flour sacks (k=615 lbs./in.)

Applied stress

(6.82 psi)

Compressive Stress Distribution Models :

Semi-Rigid Joint

Bottles (k=1345 lbs./in.)

Empty box (k=854 lbs./in.)

Flour sacks (k=615 lbs./in.)

Applied stress

(6.5 psi)

Applied stress

(6.25 psi)

Applied stress

(6.82 psi)

Compressive Stress Distribution

Models : Fixed Ends

Bottles (k=1345 lbs./in.)

Empty box (k=854 lbs./in.)

Flour sacks (k=615 lbs./in.)

Applied stress

(6.5 psi)

Applied stress

(6.25 psi)

Applied stress

(6.82 psi)

4.625” 4.625”

Load head

Load Applicator

Packaging

PressureSensor

Pallet Section

I-Beam

0.5” LVDT

1” LVDT

2” LVDT

15.75”

20”

Experimental Validation of Model Compression Stress Distribution

Load cell

Packaging

Pressure sensorPallet section

LVDT

Testing setup

Fixed Ends (Epoxy Glued)

Top View

Semi-rigid joint (nailed)

Free Ends

Front View

Wood Pallet Sections

Corrugated Container with Contents

Measurement of Design Variables

Modulus of Elasticity

Rotation Modulus

Packaging Stiffness

• Pallet deck stiffness

• Third point bending test

• Nailed joint stiffness

• Moment-rotation curve

• Vertical Compression stiffness of packaging

• Load-deflection curve

Input Values for the Beam Models

MOE (lbs./in.2)

Rotation Modulus(in.-lbs./radian)

Packaging stiffness (lbs. /in.)

3/4" 3/8" 

3/4" 3/8" 

Bottles in a box

Flour sacks

Empty box

A-1 1339611 B-1108059

6  6493.7

15921.1

4  1344.8

7618.44 853.84

A-2 1235658 B-2107726

0

A-3 1245681 B-3108247

2

A-4 1118084 B-4102298

7

A-5 1177946 B-5162595

0

A-6 730658 B-6140778

8A-7 854455 B-7 716771A-8 1238123 B-8 822750

A-9 1172598 B-9113112

4

Free end

Semi-rigid joint

Fixed end

Strain Gage Pressure Sensor Mats

Sensel

Pressure Images: Free Ends

5psi

30psi

5psi

30psi

Bottle Empty Flour

3/4”

3/8”

Pressure Images: Semi-Rigid Joints

5psi

30psi

5psi

30psi

Bottle Empty Flour

3/4”

3/8”

Pressure Images: Fixed Ends

5psi

30psi

5psi

30psi

Bottle Empty Flour

3/4”

3/8”

Raw Pressure Data

Location (in.) 0.8 1.2 1.6 2 2.4 2.8 3.2 3.6 4 4.4 4.8 5.2 5.6 6 6.4 6.8 7.2 7.6 80.4 4.422 4.422 4.422 4.422 4.422 4.422 4.422 4.422 4.422 0.525 0.025 0 0.025 0.056 0.025 0.035 0.025 0 00.8 4.422 4.422 4.422 4.422 4.422 4.422 4.422 4.422 4.422 1.095 0.025 0 0 0.056 0 0 0 0 01.2 4.422 4.422 4.422 4.422 4.422 4.422 4.422 4.422 4.422 1.503 0.035 0 0 0 0.035 0 0 0 01.6 4.422 4.422 4.422 3.138 4.422 4.422 4.422 4.422 4.422 0.295 0.067 0.09 0.056 0.035 0.056 0.067 0 0 02 4.422 4.341 3.1 1.331 2.107 1.607 4.422 4.422 1.52 0.126 0 0 0.025 0.045 0.295 0.138 0.228 0.176 0.067

2.4 4.422 4.422 4.422 3.96 4.422 4.422 4.422 4.422 4.422 0.282 0.035 0 0.268 4.422 1.046 3.624 0.025 0 1.6422.8 4.422 4.422 4.422 4.422 4.422 4.422 4.422 4.422 4.422 0.323 0 0 0.101 0.309 0 0.056 0.067 0 03.2 4.422 4.422 4.422 4.422 4.422 4.422 4.422 4.422 4.422 0.74 0.241 0 0.067 0.025 0 0.025 0.035 0.15 03.6 4.422 4.422 4.422 4.422 4.422 4.422 4.422 4.422 4.422 4.422 0.228 0 0.056 0.045 0 0 0 0.067 0

Location (in.) 0.8 1.2 1.6 2 2.4 2.8 3.2 3.6 4 4.4 4.8 5.2 5.6 6 6.4 6.8 7.2 7.6 80.4 24.094 22.491 19.802 21.493 21.294 7.688 20.299 10.819 0.529 0 0 0 0 0 0 0 0 0 00.8 24.094 11.011 24.094 24.094 22.092 3.547 18.515 23.092 13.131 0 0 0 0 0 0 0 0 0 01.2 16.938 8.914 24.094 24.094 24.094 2.044 13.325 24.094 14.102 0 0 0 0 0 0 0 0 0 01.6 24.094 10.819 16.545 3.907 0 0 7.220 16.938 15.858 0.529 0 0 0 0 0 0 0 0 02 16.152 9.293 0 0 0 0 10.723 24.094 22.992 2.394 0 2.044 1.699 1.442 3.727 0.449 0.449 0.215 0

2.4 24.094 17.627 7.033 2.569 9.959 1.958 3.727 8.725 19.604 5.641 1.699 2.394 2.746 0.449 0.855 2.394 1.613 1.527 0.8552.8 22.491 24.094 21.693 5.273 8.630 9.388 24.094 11.972 2.569 0 0 0 0 0 0 0 0 0 03.2 19.010 24.094 23.392 3.727 1.958 2.481 16.250 12.454 9.769 0 0 0 0 0 0 0 0 0 03.6 24.094 24.094 24.094 5.549 8.630 4.088 15.467 11.395 23.994 0 0 0 0 0 0 0 0 0 0

0.8” 8” 15.2”

0.8” 8” 15.2”

2-D Pressure Distribution on

5 psi sensor

Tabulated Actual Pressure Values

2-D Pressure Distribution on 30 psi sensor

Plot of Raw Pressure Data

Location (in.) 8.4 8.8 9.2 9.6 10 10.4 10.8 11.2 11.6 12 12.4 12.8 13.2 13.6 14 14.4 14.8 15.20.4 0 0 0 0 0 0 0 1.713 0 4.633 3.727 3.100 0 0 0 0 3.100 00.8 0 0 0 0 0 0 0.511 1.642 12.164 22.691 21.393 24.094 22.192 24.094 22.791 14.394 20.696 11.2981.2 0.996 0 0 0 0 0 0 0 6.288 4.451 0 8.536 8.820 6.474 24.094 20.895 24.094 5.1811.6 0 0 0 0 0 0 0 0 0 0 8.253 0 16.447 11.683 0 9.483 18.120 10.6282 0 0 0.787 0 0 0 0 0 8.914 12.261 0 12.937 9.769 14.783 12.841 19.604 24.094 11.683

2.4 3.368 0 0.724 3.368 1.095 3.546 0 3.100 5.641 7.970 3.997 0 0 0 0.000 4.088 14.588 13.0342.8 0.819 0 0 0 0 0 0 0 9.483 3.458 5.826 0 0 0 7.126 11.972 21.593 8.9143.2 0 0 0 0 0 0 0 0 5.641 17.135 13.810 5.733 8.536 4.542 24.094 24.094 24.094 24.0943.6 0 0 0 0 0 0 0 0 7.688 22.092 20.696 15.761 24.094 24.094 24.094 24.094 24.094 9.578

Average 1.73 0 0.76 3.37 1.1 3.55 0.51 2.37 8.104 13.51 12.18 14.31 15.57 14.25 19.32 16.97 21.72 12.57

Location (in.) 0.8 1.2 1.6 2 2.4 2.8 3.2 3.6 4 4.4 4.8 5.2 5.6 6 6.4 6.8 7.2 7.6 80.4 24.094 22.491 19.802 21.493 21.294 7.688 20.299 10.819 0.000 0.525 0 0 0 0 0 0 0 0 00.8 24.094 11.011 24.094 24.094 22.092 3.547 18.515 23.092 13.131 1.095 0 0 0 0 0 0 0 0 01.2 16.938 8.914 24.094 24.094 24.094 0.000 13.325 24.094 14.102 1.503 0 0 0 0 0 0 0 0 01.6 24.094 10.819 16.545 3.907 0 0 7.220 16.938 15.858 0.000 0 0 0 0 0 0 0 0 0.6162 16.152 9.293 3.1 1.331 2.107 1.607 10.723 24.094 22.992 0.000 0 0 0 0 0 0 0 0 0

2.4 24.094 17.627 7.033 0.000 9.959 0.000 3.727 8.725 19.604 5.641 0 0 0 1.046 3.624 0 0 1.642 02.8 22.491 24.094 21.693 5.273 8.630 9.388 24.094 11.972 0.000 0 0 0 0 0 0 0 0 0 03.2 19.010 24.094 23.392 3.727 0.000 0.000 16.250 12.454 9.769 0 0 0 0 0 0 0 0 0 03.6 24.094 24.094 24.094 5.549 8.630 4.088 15.467 11.395 23.994 0.74 0 0 0 0 0 0 0 0 0

Average 21.67 16.94 18.21 11.18 13.83 5.26 14.4 15.95 17.06 1.9 0 0 0 1.05 3.62 0 0 1.64 0.62

Predicted vs. Measured Results: Free

EndBottle

3/4”

3/8”

Empty Flour

Predicted vs. Measured Results:

Semi-Rigid JointsBottle

3/4”

3/8”

Empty Flour

Predicted vs. Measured Results:

Fixed Ends

3/8”

Bottle

3/4”

Empty Flour

• A modification of the principles of an elastic beam supported by an elastic, deformable, foundation can be used to estimate the compression stress distributions between the pallet deck and packaged products. The model inputs are the compression stiffness of the packaged product and the bending stiffness of the pallet deck.

• The compression stresses at the interface between the pallet deck and packaging are not uniformly distributed. Stress concentrations occur over the pallet stringers or blocks.

• Average applied compression stress levels of 6 to 7 psi resulted in compression stresses on packaging within a unit load from 0 to 50 psi, for the specimens tested. Maximum compression stresses can be five to ten times greater than the average applied stress.

Summary

SummarySummary

• Stiffer pallet decks and stiffer connections between the pallet deck and stringers or blocks, significantly reduces the maximum compression stress on the packaged product in unit loads.

• Pallet design can be used to reduce the compression stresses on packaging and reduce packaging cost.