SSC-382
REEXAMINATION OF DESIGN
CRITERIA FOR STIFFENED
PLATE PANELS
This document has been approved
for public release and sale; its distribution is unlimited
SHIP STRUCTURE COMMITTEE
1995
The SI-tip STRUCTURE COMMITTEE is constituted to prosecute a research program to improve the hull structures of ships and other marine structures by an extension of knowledge pertaining to design, materials, and methods of construction.
RADM J. C. Card, USCG (Chairman) Chief, Office of Marine Safety, Security
and Environmental Protection
U. S. Coast Guard
Mr. Thomas H. Peirce Mr. Edwin B. Schimler
Marine Research and Development Associate Administrator for
Ship-Coordinator building and Technology Development
Transportation Development Center Maritime Administration
Transport Canada
MILITARY SEALIFT COMMAND Mr. Robert E. Van Jones (Chairman) Mr. Rickard A Anderson
Mr. Michael W. Tourna Mr. Jeffrey E. Beach
AMERICAN BUREAU OF SHIPPING Mr. Stephen G. Arntson
Mr. John F. Conlon Mr. Phillip G. Rynn Mr. William Hanzelek
U. S. COAST GUARD ACADEMY LCDR Bruce R. Mustain
U. S. MERCHANT MARINE ACADEMY
Dr. C. B. Kim
U S NAVAL ACADEMY Dr. Ramswar Bhattacharyya
CANADA CENTRE FOR MINERALS AND ENERGY TECHNOLOGIES
Dr. William R. Tyson
SOCIETYOF NAVAL ARCHITECTS AND MAR INEENINEE RS
Dr. William Sandberg
U. S. TECHNICAL ADIVSORY GROUP TO THE INTERNATIONAL STANDARDS ORGANIZATION CAPT Charles Piersall
The SHIP STRUCTURE SUBCOMMITTEE acts for the Ship Structure Committee on technical matters by providing technical coordination for determinating the goals and objectives of the program and by evaluating and interpreting the results in terms of structural design, construction, and operation.
MARITIME ADMINISTRATION Mr. Frederick Seibold
Mr. Richard P. Voelker
Mr. Chao H. Lin Dr. Walter M. Maclean
NAVAL SEA SYSTEMS COMMAND
Mr. W. Thomas Packard
Mr. Charles L Null Mr. Edward Kadala Mr. Allen H. Engle
DEFENCE RESEARCH ESTABLISHMENT ATLANTIC
Dr, Neil Pegg
LCDR Stephen Gibson
Dr. Roger Hollingshead Mr. John Porter
SHIP STRUCTURE SUBCOMMITTEE LIAISON MEMBERS SHIP STRUCTURE COMMITTEE
Dr. Donald Uu Senior Vice President American Bureau of Shipping
U. S. COAST GUARD CAPT G. D. Marsh CAPT W. E. Colburn, Jr. Mr. Rubin Scheinberg Mr. H. Paul Cojeen TRANSPORT CANADA Mr. John Grinstead Mr. Ian Bayly Mr. David L. Stocks Mr. Peter Timonin
NATIONAL ACADEMY OF SCIENCES -MARINE BOARD
Dr. Robert Sielski
NATIONAL ACADEMY OF SCIENCES -COMMITTEE ON MARINE STRUCTURES
Mr. Peter M. Palermo
WELDING RESEARCH COUNCIL
Dr. Martin Prager
AMERICAN IRON AND STEEL INSTITUTE
Mr. Alexander D. Wilson
OFFICE OF NAVAL RESEARCH Dr. Yapa D. S. Rajapaske STUDENT MEMBER
Mr. Trevor Butler
Memorial University of Newfoundland
Mr. Edward Comstock Mr. Thomas W. Allen Mr. Warren Nethercote
Director, Naval Architecture Engineering Officer (N7) Head, Hydronautics Section
Group (SEA 03H) Military Sealift Command Defence Research Establishment-Atlantic
Naval Sea Systems Command
EXECUTIVE DIRECTOR CONTRACTING OFFICER TECHNICAL REPRESENTATIVE
CDR Stephen E. Sharpe, USCG Mr. William J. Siekierka
U. S. Coast Guard Naval Sea Systems Command
Member Agencies: American Bureau of Shipping Defence Research Establishment Atlantic Maritime Administration Military Sealift Command Naval Sea Systems Command Transport Canada United States Coast Guard
Ship
Structure
Corn m ¡ttee
An Interagency Advisory Committee March 24, 1995
Ssc-
Z
Address Correspondence to: Executive Director
Ship Structure Committee U.S. Coast Guard (G-MI/SSC) 2100 Second Street. S.W. Washington, D.C. 20593-0001 Ph: (202) 267-0003 Fax:(202) 267-4677 SSC-382 SR- 1350
REEXAMINATION OF DESIGN CRITERIA FOR STIFFENED PLATE PANELS The emphasis on reducing vessel weight in the recent generation of ships has lead to increased usage of high strength
steels to allow for thinner scantlings. These designs provided
panels of equivalent overall strength but with less inherent
rigidity. This, in turn, has resulted in unanticipated failures at the intersections of transverse and longitudinal stiffeners with the plating. These failures demonstrated the need torethink some of the assumptions currently used in the design
process.This project analyzed the total stresses at the panel to
stiffening system interface.
After using conventional design
procedures for a panel section,
finite element models of the
panel were subjected to various anticipated panel loads and the
resulting stresses were analyzed. The report concludes with a
discussion of the effect of the less stiff panel structures on failures and adjustments which should be considered in the design procedures. Recommendations for future research are given.
C. CARD
Rear AdmVal, U.S. Coast Guard Chairman, Ship Structure Committee
T.chrtica R.port Documentation Page I. Ripari Ne. S3C--38. 2. Go.rnm.nt Ac.,iuon N0. PB95-188131 3. R.cp.nt s Coi&og Ne. 4. Ttl. b,,tIq
"Re-Examination of Design Criteria for Stiffened Plate Panels"
5. Ripari oat.
June 1994
6. Perform,n9 Qrgc,xo,en Cod.
8. Performing Organi ¡oison Riper' No.
"SR-1350"
7. Author's) .
Dhruba J. Ghose and Natale S. Nappi
9. Perforwnrig Organization Name and Addr.a.
Designers & Planners, Inc.
2120 Washington Blvd., Suite 200
Arlington, VA 22204
10. Wont Unit Ne. (TRAIS)
11. Contract er Grant No.
DTCG2392RE01030
13. Typ. of Ripari and P.rod Co'n.rsd
FINAL
12. Sponsoring Agency Nam. and Add,..,
Ship Structure Comxaittee U.S. Coast Guard
2100 Second Street, S.W.
Washington, D.C. 20593
14. Sponsoring Ag.ncy Cod.
G-M
15. Supplementary Notes
Sponsored by the Ship Structure Committee and its member agencies.
6. Abstruct
Current ship structural design criteria for stiffened plate panels is based upon a
strength of materials approach using either linear plate or beam theory. This
approach neglects the effects of vertical shear (normal to the plate surface), membrane and torsional stress components induced by the flexibility of the panel's
supporting structure. Recent trends towards the use of higher strength materials
have resulted in the design of grillage structures which are more flexible and therefore increase the vertical shear, membrane and torsional stress components
in the plate panel. This report presents the results of a study undertaken to
determine the effect of the stiffness characteristics of the supporting members
of the grillage structure on the plate panel stress. Grillage scantlings were
developed using first principals based approach and then analyzed using finite element techniques to take into account the flexibility of the grillage stiffeners and to quantify the effects of vertical shear, membrane and torsional stress
components.
17. K.y Words
Finite Elements, Stiffened Plates, Stiffeners, Grillage, Design Criteria, Shear Stress, Normal Stress, Hull Structure
18. Qissyibsition Statimint
Available frotu:
National Technical Information Service
Springfield, VA 22161
19. Security Closiif. (of iii,, riper?)
Unclassified
2C. Security Clasail. (of ihi& page)
Unclassified
21. No. of Pegs'
1.1.2
22. Pic,
in2 ft2 yd2 mi2
Approximate Conversions to Metric Measures
Symbol When You Know Mu/tip/v by
To Find Symbol
LENGTh
oz ounces lbpounds short tons (2000 lb)
AREA VOLUME MASS (weight) 28 grams 0.45 kilograms 0.9 meiricton
NISTUnàted
States Oepwthient 01 Comerca
T,chnologv Aánlnisbition National inffijt of Standards and Tehnology Metnc Prorani, Galth&sburg, MD 20899
METRIC CONVERSION CARD
Approximate Coñversions from Metric Measures
cm2 m2 m2
km2 r
ha g kg C) tsp teaspoons 5 milliliters mL Thsp tablespoons 15 milliliters mL in3 cubic inches 16 milliliters mL fi oz fluid ounces 30 milliliters ritt. c cups 0.24 liters L 4 Pt pints 0.47 liters L qt quarts 0.95 liters L gal gallons 3.8 liters L ft3 cubic feet 0.03 cubic meters yd3 cubic yards 0.76 cubic meters rn3 TEMPERATURE (exact) (Ji °F degrees subtract 32, degrees Fahrenheit multiply by 5/9 CelsiusSymbol When You Know Multiply by
To Find LENGTH rom millimeters 0.04 inches cm centimeters 0.4 inches m meters 3.3 feet Ce.) m meters 1.1 yards -km kilometers 0.6 miles AREA cm2 square centimeters 0.16 square inches
°
km2 square kilometers 0.4 m2 square meters 1.2square yards square miles
ha
hectares
2.5
acres
(10,000 m2) milliliters milliliters liters liters liters cubic meters cubic meters
MASS (weight) 0.035 ounces 2.2
pounds lÀ short tons VOLUME 0.03 fluid ounces 0.06 cubic inches 2.1 pints 1.06 quarts 0.26 gallons 35 cubic feet 1.3 cubic yards TEMPERATURE (exact) °C degrees multiply by 9/5, degrees °F Celsius add 32 Fahrenheit Symbol
in in ft yd mi in2 yd2 mi2 oz lb fi oz in3 pt qt gal ft3 yd3
.c -40 .20 0 20 37 60 80 100 I I -40 0 32 80 98.6 160 212 o.' C,' water freezes body temperature water boils -4 g grams kg
kilograms metric ton (1,000 kg)
square inches 6.5 square centimeters square feet 0.09 square meters square yards 0.8 square meters square miles 2.6 square kilometers acres 0.4 hectares in inches 2.5 centimeters cm ft feet 30 centimeters cm yd yards 0.9 meters m
-mi miles 1.6 kilometers km mL mLo
L L L m3 m3Table of Contents
1.0 INTRODUCTION i
1.1 Background 1
1.2 Objective 2
1.3 Approach 3
2.0 DESIGN OF STIFFENED PLATE STRUCTURE 5
3.0 FINITE ELEMENT MODELS OF STIFFENED PLATE STRUCTURE . . 12
3.1 Introduction 12
3.2 Determination of Mesh Size 12
3.3 Modeling of Longitudinals and Transverses . . . 14
3.4 Modeling of Plating using Plate Elements . . . 19
3.5 Modeling of Plating using Brick Elements . . . 22
3.6 Modeling of Plating with Initial Deformations . 22
4.0 RESULTS 27
4.1 Comparison of Stresses using Plate Elements . . . 27
4.2 Comparison of Stresses using Brick Elements . . . 41
4.3 Comparison of Stresses using Initially Deformed
Structure 73
4.4 Discussion of Results 73
5.0 CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE WORK . . . 78
6.0 REFERENCES 83
List of Tables
2.1 DIMENSIONS AND SCANTLINGS OF OSS GRILLAGE 10
2.2 DIMENSIONS AND SCANTLINGS OF HSS GRILLAGE li
3.1 OPTIMUN MESH SIZE DETERMINATION USING PLATE
ELEMENTS is
3.2 COMPARISON OF STIFFENER MODELS USING PLATES
AND OFFSET BEAMS 18
3.3 SECTIONAL PROPERTIES OF LONGITUDINALS AND
TRANSVERSES (OSS) 20
3.4 SECTIONAL PROPERTIES OF LONGITUDINALS AND
TRANSVERSES (HSS) 21
4.1 MAXIMUM DISPLACEMENT AND STRESSES OF OSS
MODELS (USING PLATE ELEMENTS) 28
4.2 MAXIMUM DISPLACEMENT AND STRESSES OF HSS
MODELS (USING PLATE ELEMENTS) 28
4.3 MAXIMUM DISPLACEMENT AND STRESSES OF OSS MODELS
(USING BRICK ELEMENTS) 44
4.4 MAXIMUM DISPLACEMENT AND STRESSES OF HSS MODELS
USING BRICK ELEMENTS) 45
4.5 MAXIMUM STRESSES IN MODEL 3 WITH INITIAL DEFORMATION
(OSS AND HSS) 74
4.6 EFFECT OF VARYING ELEMENT SIZE ON THE RESULTS OF
BRICK MODEL 3 (OSS MATERIAL) 76
List of Figures
3.1 Mesh Size Determination using Plate Elements 16 3.2 Schematic of Quarter Grillage Model for
F.E. Analysis 23
3.3 Finite Element Model of Plate Stiffened Structure
(Using Plate Elements) 24
4.1 Plot of Von Mises Stress for 055 Model 1
(Plate Elements) 29
4.2 Plot of Normal Stress, a» for OSS Model 1
(Plate Elements) 30
4.3 Plot of Von Mises Stress for HSS Model 1
(Plate Elements) 31
4.4 Plot of Normal Stress, a» for HSS Model 1
(Plate Elements) 32
4.5 Plot of Von Mises Stress for ass Model 2
(Plate Elements) 33
4.6 Plot of Normal Stress, a» for OSS Model 2
(Plate Elements) 34
4.7 Plot of Von Mises Stress for HSS Model 2
(Plate Elements) 35
4.8 Plot of Normal Stress, for HSS Model 2
(Plate Elements) 36
4.9 Plot of Von Mises Stress for ass Model 3
(Plate Elements) 37
4.10 Plot of Normal Stress, a» for OSS Model 3
(Plate Elements) 38
4.11 Plot of Von Mises Stress for HSS Model 3
(Plate Elements) 39
4.12 Plot of Normal Stress, a» for HSS Model 3
(Plate Elements) 40
4.13 Comparison of Stresses in ass Plate Model with
Single Panel 42
4.14 Comparison of Stresses in HSS Plate Model with
Single Panel 43
4.15 Plot of Von Mises Stress for ass Model 1
(Brick Elements) 47
4.16 Plot of Normal Stress, a» for ass Model i
(Brick Elements) 48
4.17 Plot of Vertical Shear Stress, ry, for ass Model 1
(Brick Elements) 49
4.18 Plot of Vertical Shear Stress, r for ass Model i
(Brick Elements) 50
4.19 Plot of Von Mises Stress for HSS Model i
(Brick Elements) 51
4.20 Plot of Normal Stress, a» for HSS Model 1
(Brick Elements) 52
4.21 Plot of Vertical Shear Stress, r> for MSS Model i
(Brick Elements) 53
4.22 Plot of Vertical Shear Stress, r, for HSS Model i
(Brick Elements) 54
4.23 Plot of Von Mises Stress for OSS Model 2
(Brick Elements) 55
4.24 Plot of Normal Stress, ai), for OSS Model 2
(Brick Elements) 56
4.25 Plot of Vertical Shear Stress, for
OSS Model 2 (Brick Element"s) 57
4.26 Plot of Vertical Shear Stress, r,, for
OSS Model 2 (Brick Elements) 58
4.27 Plot of Von Mises Stress for HSS Model 2
(Brick Elements) 59
4.28 Plot of Normal Stress, for HSS Model 2
(Brick Elements) 60
4.29 Plot of Vertical Shear Stress, r for
HSS Model 2 (Brick Elements) 61
4.30 Plot of Vertical Shear Stress, r for
HSS Model 2 (Brick Elements) 62
4.31 Plot of Von Mises Stress for OSS Model 3
(Brick Elements) 63
4.32 Plot of Normal Stress, r for OSS Model 3
(Brick Elements) 64
4.33 Plot of Vertical Shear Stress, for
OSS Model 3 (Brick Elements) 65
4.34 Plot of Vertical Shear Stress, r for
OSS Model 3 (Brick Elements) 66
4.35 Plot of Von Mises Stress for HSS Model 3
(Brick Elements) 67
4.36 Plot of Normal Stress, a for HSS Model 3
(Brick Elements) 68
4.37 Plot of Vertical Shear Stress, r,, for HSS Model 3
(Brick Elements) 69
4.38 Plot of Vertical Shear Stress, rJ( for HSS Model 3
(Brick Elements) 70
4.39 Comparison of Stresses in OSS Brick Model with
OSS plate model 71
4.40 Comparison of Stresses in HSS Brick Model with
HSS plate model 72
5.1 Plot of Vertical Shear Stress, in Brick Model
of Single Panel 81
5.2 Plot of Vertical Shear Stress, rain Brick Model
of Single Panel 82
Nomenclature
b = breadth of the panel, m
be = effective width of plating, m
t = thickness of the panel, min
C = Constant depending on the plate material and
location on
the ship
H = Head of sea water, feet
K = Factor depending on the aspect ratio of the panel
E = Young's Modulus, N/mm2
= Yield Stress,
N/mm2A = Total cross-sectional area of the beam,
cm2ASH = Shear area, cm2
= Shear area along 'y' axis,
cm2= Shear area along 'z' axis,
cm2I = Moment of Inertia, cm4
I,. = Moment of Inertia about 'y' axis, cm4
I
= Moment of Inertia about 'z' axis,
cm4= distance of neutral axis from plating, cm
= distance of neutral axis from stiffener flange, cm
SM = Sectional Modulus to the plating, cm3
SMf = Sectional Modulus to the stiffener flange, cm3
J = St. Venant's Torsional Constant, cm4
r = radius of gyration, cm
FULT
= Ultimate Strength of the plating,
N/mm2FCR
= Critical Buckling stress due to axial compression,
N/mm2FBCR
= Critical Buckling stress due to bending, N/mm3
FSCR
= Critical Buckling stress due to shear, N/mm
= Normal Stress in longitudinal direction, N/ramn2
= Normal Stress in transverse direction, N/mmn
zz
Normal Stress in vertical direction, N/mm'
vi
stiffened plate in the
stiffened plate in the
stiffened plate per longitudinal stiffened plate per transverse
(A/B) (iT/iL)°25 = Virtual Aspect Ratio (Measure of stiffness
of cross stiffened panels)
= Vertical shear Stress, N/mm = Vertical shear Stress, N/mm vin = Von Mises Stress, N/mm2
Initial Deformation at x and y, 'rim
L = Length of Beam, m s = stiffener spacing, m d = web depth, mm
t, = web thickness, rrìm
bf = flange width, rim
t = flange thickness, min
p = normal pressure, N/mm2 IL = moment of inertia of the
longitudinal direction IT = moment of inertia of the
transverse direction iL = moment of inertia of the iT = moment of inertia of the
1.0
INTRODUCTION1.1 Background
Due to the emphais on increasing ship lengths and reducing
structural weight to increase cargo capacity, there has been
an increase in the use of high strength steels on commercial
vessels, especially tankers. Recently, tankers constructed of
high strength steel have experienced damage to their side
shell and bottom structures. This damage is characterized by the initiation of cracks in plating at the intersection of transverses and longitudinal stiffeners.
The use of high strength materials has resulted in the design of grillage structures that are lighter and more flexible.
But, even though the material's strength is greater, its
stiffness is the same. Therefore, reducing the area of the stiffener increases its flexibility. The result of this increased flexibility is an increase in secondary stresses due to the increased deflection of supporting structure.
The stiffened plate panel, which is intended to provide
watertightness and contribute to a major portion of hull girder longitudinal and transverse strength, must be designed
to withstand primary stresses due to hull girder bending,
secondary stresses due to bending from local loading of the pláte-stiffener combination and tertiary stresses due to bending of the plate panel itself from local lateral loads. Current design criteria for plate panels of grillage structure are based upon a strength of materials approach using either
linear plate or beam theory. Acceptance is based upon
comparing calculated stresses with allowable stress levels
certification bodies (i.e. regulatory bodies and U.S. Navy)
have allowed higher levels for both primary and secondary
design stresses, which in turn means larger values of actual cyclic as well as static stresses. Vertical shear (normal to the plate panel), membrane and torsional stress components are not accounted for in the selection of the plate panel scantlings when using a strength of materials approach.
Based upon this, it is assumed that the damage found in
stiffened plate panels is due to stress levels which either exceed the yield stress of the material or induce fatigue
failure. In practice, fatigue failure is avoided through
quality control of welds, careful design of connection details
and limiting the allowable stress levels.
The design of
plates based upon first principals approach does not take into consideration the stresses induced by the flexibility of its supporting structure. Therefore, the emphasis of this study will be on determining the added stresses in plate panels due
to increased flexibility of grillage structure.
1.2 Objective
The objective of this task was to evaluate the combined
effects of vertical shear (normal to the plate panel)
membrane stress and torsion on the total stress of a stiffened plate panel as a result of the effects of the overall grillage
response Specifically, the impact of the error introduced by
ignoring grillage behavior, such as the effects of vertical
shear and membrane stresses, was determined. In order to
quantify the additional stresses induced by grillage behavior on stiffened plate panels, a series of finite element model were developed and parametric studies performed varying key
design parameters for a typical grillage structure designed to current industry recognized practice. For this task, the U.S. Navy design practice was chosen since it is based upon a first principals approach versus emprically based design equations.
1.3 Approach
The purpose of this study was to determine the effect of the
stiffness characteristics of the supporting members. of
grillage structures on the plate panel stress. The first step of this process was to design a grillage using strength of materials - first principals based approach. Grillage scantlings designed were then analyzed using FEM techniques to take into account the flexibility of the grillage stiffeners and to quantify stress componenets.
Six grillage designs were developed, three of Ordinary
Strength Steel and three of High Strength Steel. The designs represent the bottom structure between transverse bulkheads
fore and aft and longitudinal girders on either side. The
overall length of the grillage and the plate panel aspect
ratios were kept constant, and the breadth of the grillage was varied to modify the stiffness characteristics.
Finite element analysis of the grillages were performed to determine the stress components and variation of stress through out the grillage structure. Two sensitivity studies were performed to determine simplifications that would allow for results that are within the acceptable level of accuracy and the hardware resources and computational time available.
First the optimum mesh size was determined. Second, a
comparison of the results of a grillage model using plate
model using plate elements to represent plates and beam elements to represent stiffeners was accomplished. Finite element models of the six grillages developed made use of the results of these simplifications. In order to quantify the
stress components, the following types of models were
developed:
Plate Models - used as the base model for
comparison.
Brick Models - used to quantify the vertical shear stress component
Plate Model wIth initial deflection - used to
2.0 DESIGN OF STIFFENED-PLATE STRUCTURE
Current U.S. Navy structural design practices [l,2,3]1 were
used to calculate the scantlings for six (6) stiffened panel
structures. These stiffened panels are comprised of 3 OSS and
3 HSS systems. The overall length, A, of the grillage was
maintained at 15.24 m (50 ft), while the overall breadth, B,
was varied, e.g. 10.06 m (33 ft), 6.4 m (21 ft), and 2.74 m (9
ft) . The resulting geometrical grillage aspect ratio are 1.52,
2.38, and 5.56. The plate panel aspect rajo remained the same, namely 3.05 in x 0.9 m (10' x 3'), for all six systems.
The panel was loaded with an uniform lateral pressure of 0.107
N/mm2 (15.56 lb/in2),
which is equivalent to a head of sea
water of 9.14 m (30 ft) . The plate panel thickness was
selected using Navy formula [2] of:
b
CWhere H is the head of sea water, in feet. C is a constant
which is a function of plate material and its location on the
ship. The C values takes into account the acceptable
deformation of the structure. C values have been established for the following locations:
Topside Plating
Lower shell and tank boundaries
Boundaries for the control of flooding.
1
And K is a factor that takes into consideration the aspect ratio of the panel, b/a>0.5 (i.e., short panels)
For these studies, C was taken as 350 for OSS plating and 400 for HSS plating which corresponds to topside structure where the minimum amount of deformation (no permanent set) is allowed for and hence would result in a conservative design.
The resulting plate thicknesses were 15.88 mm (0.6250") and
14.29 mm (0.5625") for OSS and HSS respectively.
The longitudinal beam stiffeners were designed as continuous beams over non-deflecting supports, while the transverse girders were designed as clamped beams. The following steps were used to develop these designs:
Calculate plating thickness, t, for normal loads due to a uniform pressure from a head of sea water using the equation above.
Determine the effective width of plating, b, based on shear lag approach (post-buckling response),
Select a beam size (i.e., a tee beam attached to
the plate) E
be = 21
Determine plate/beam section properties, including the cross sectional area, A; shear area, ASH; moment
of inertia, I; distance of the neutral axis from
the plating, yp; distance of the neutral axis from
the stiffener flange,
yf;
sectional modulus to theplating, SMp; sectional modulus to the stiffener
flange, SMf; and radius of gyration, r.
Determine the secondary bending moments and shear forces.
Determine the shear stress t the supports and the
secondary bending stresses at two locations; one
at the support and the other at the mid span of the
beam. At each location, compute bending stresses
at the plate and at the flange of the stiffener.
Check plating for ultimate strength (FULT)
Check plating for buckling under in-plane compression, bending and shear, (FCR, FBCR, FSCR)
Check composite plate-tee beam for yielding in tension/compression due to bending.
1O.Check composite plate-tee beam for maximum web shear stress.
ll.Check tee stiffener for Tripping.
l2.Check tee stiffener flange for local buckling.
The results of the above design procedures have been
summarized in Tables 2.1 and 2.2 for OSS and HSS grillage
designs respectively. These scantlings were used to develop the FEN models described in section 3.0. The symbols used in the first column of tables 2.1 and 2.2 are listed below:
dimension of the stiffened panel in the long direction
dimension of the stiffened panel in the short direction
dimension of the unsupported span of plating in the long direction
dimension of the unsupported span of plating in the short direction
uniform lateral pressure plate thickness
moment of inertia of the stiffened plate in the longitudinal direction
thoment of inertia of the stiffened plate in the
transverse direction
moment of inertia of the stiffened plate per longitudinal
moment of inertia of the stiffened plate per transverse
'(A/B) (iT/iL)'0.25' = Virtual Aspect Ratio. (Measure of
stiffness of cross stiffened panels)
The virtual aspect ratio, (A/B) (T/iL)°5, is derived from the
fact that the stiffness of an unstiffened gross panel, of
constant thickness, is usually a function of the length to
breadth ratio (i.e A/B) . This ratio is commonly refered to as
8 'A' = 'B' = 'a' = 'b' = 'p' = 't' = 'IL' = 'IT' = 'iL' = 'iT' =
the panel aspect ratio. Therefore A/B would represent the
ratio of the stiffness in one direction to the other
direction. If the panel is supported by mutually perpendicular intersecting beams, whose stiffnesses are different, then the ratio of the gross pane]. stiffness would have to be modified to account for the stiffness provided by the moment of inertia
of those beams. When this is accomplished the quantity called
"virtual aspect ratio" is used. Hence, the virtual aspect
ratio is equal to (A/B) (T/L)°25, assuming that the Young's
Modulus, E, is the same in both directions.
Detail calculations for one grillage, are provided in Appendix 1.0 to illustrate the process used. It should be noted that
the original calculations were performed using the English
system, therefore the plate thickness selected in the appendix is not a standard metric plate size.
TABLE 2.1
-DIMENSIONS AND SCANTLINGS OF OSS GRILLAGE
lo Model No. 1 2 3 A (ra) 15.24 15.24 15.24 B (m) 10.06 6.40 2.74 a (m) 3.05 3.05 3.05 b (m) 0.91 0.91 0.91 t (mm) 15.88 15.88 15.88 Long'l Size WT 205x110x23 WT 205x140x23 WT 205x140x23 Trans Size 914x457x15.9/28.6 T W-T 690x250x123.5 W-T 410X180X53 p (N/mni) 0.107 0.107 0.107 IL (cm4) 6984.36 6984.36 6984.36 IT (cm4) 717,011.14 255,524.21 48,157.93 ASL (cm) 14.06 14.06 14.06 AST (cm) 149.68 100.65 39.87 iL=IL/b(cm4/cm) 76.36 76.36 76.36 iT=IT/a(cm4/cm) 2,352.53 838.36 157.97 A/B 1.52 2.38 5.56 (A/B) (iT/iL)"1/4 3.57 4.33 6.66
TABLE 2.2
-DIMENSIONS AND SCANTLINGS OF HSS GRILLAGE
11 Model No. i 2 3 A (m) 15.24 15.24 15.24 B (m) 10.06 6.40 2.74 a (m) 3.05 3.05 3.05 b (m) 0.91 0. 91 0.91 t (mm) 14.3 14.3 14.3 Long'l Size WT l80x130x19.5 WT 180x130x19.5 WT 180x130x19.5 Trans Size W-T 920x305x167.5 W-T 610x230x101 W-T 410x175x38 p (N/mm2) 0.107 0.107 0.107 IL (cm4) 4,578.55 4,578.55 4,578.55 IT (cm4) 444,701.66 156,461.39 34,630.45 ASL (cm2) 11.48 11.48 11.48 AST (cm) 144.58 80.77 30.19 IL=IL/b(cm4/cm) 49.98 49.98 49.98 iT=IT/a(cm4/cm) 1,458.94 513.41 113.56 A/B 1.52 2.38 5.56 (A/B) (iT/iL) t1/4 3.52 4.26 6.82
3.0 FINITE ELEMENT MODELS OF STIFFENED PLATE STRUCTURE
3.1 Introduction
This section discusses the finite element analysis of the
stiffened-plate structures designed in section 2.0. The P.C. based finite element software ALGOR was used to analyze the plate and brick models using linear elastic theory. The VAX based NASTRAN software was used to analyze the brick model with an initial deformation. All analyses in this study were originally performed using english units. The final results
presented in this document are in metric units, therefore
standard metric sizes were not .selected. For each of the six
configurations designed in section 2.0, two models were developed, one using plate elements and the other using brick
elements. Additional models were developed to; determine
optimum mesh size, study the effect of using offset beams, study the response of single panels under uniform load with clamped boundary conditions and study the effect of initial deformation of the plating. The procedures adopted for each of these studies are described in detail below.
3.2 Determination of Mesh Size.
It is a well known fact that the results of a finite element
analysis are predicated by the mesh size adopted in the
analysis. For accurate results finer mesh sizes are
desirable. But at times the price to be paid in terms of
hardware resources and computational time is too enormous to justify such refinements. Therefore, to ascertain mesh sizes which would yield results within established limits of
accuracy without straining available computer resources, a
study was undertaken to determine the optimum mesh size.
The usual procedure is to study the convergence of finite
element results obtained using various mesh sizes to
available theoretical results. For a single unstiffened panel that is part of a grillage, the edge f ixity will depend upon its location within the grillage and the stiffness of its supporting structure. Currently, theoretical solutions of stiffened-plate structures under normal pressure loads
with various edge fixities are not available. Hence, the
convergence study was performed using one single unsupported panel with fully fixed end conditions for which theoretical results exist.
Performing a convergence study on a single panel has two
advantages. First, the results of a finite element analysis
of a panel under uniform pressure could be compared to
available theoretical solutions of rectangular plates with
all edges fixed under uniform pressure. Second, using a
single panel of unsupported plating for the mesh size
determination, enormously reduces the size of the model.
The single panel chosen for the mesh size determination was the unsupported span of plating, 3.05m x 0.91m (lO'x3'), from the stiffened-plate structure designed in section 2.0. The plating thickness used was as determined in section 2.0 for OSS steel panels, which is 15.88mm (0.625") . The plating was
subjected to the same lateral pressure of 0.107 N/nim (15.56
psi) as used in section 2.0 and clamped on all sides. For
this configuration, five models using plate elements were
developed having uniform square meshes. The element sizes
76.2mm (3"), 38.1mm (1.5"), and 19.05mm (0.75"). The
displacement and the two normal stresses, a and a , were
compared with theoretical results given in Reference [4]
based on elastic small deflection theory. These results are
summarised in Table 3.1 which also contains the in-plane
shear,
t,,
Von-Mises and the maximum principal stresses. Forillustrative purpose, the time and the memory required to run
each model are also included in the table. In Figure 3.1,
results from Table 3.1 are plotted as percentage differences between the finite element analysis and theory. As expected,
with a finer mesh, the results converge to the theoretical
values. For element sizes less than 152.4mm (6"),
displacement converge exactly to the theoretical value while for element
sizes less than 76.2mm (3"), stresses are within 10% of the theoretical values. Each refinement produces results closer to theory, but the amount of time and storage space (memory) required increases. While model 5, with
a mesh size
of19.05mm (0.75") produces results which are almost within 2.5% of the theoretical result, it takes 75 times longer and about
70 times more memory than model 3 having a
mesh size of
76.2mm (3") . Hence, for the present study models having
element sizes of 76.2mm (3") square were chosen.
3.3 Modeling of Longitudinals and Transverses.
Full 3-dimensional models of cross stiffened panels using
plate elements to model the plating as well as the
longitudinals and transverses are complex. As mentioned in section 2.0, when designing stiffened-plate structure, the longitudinals and transverses have been checked explicitly for their various modes of failure. Therefore for the
TABLE 3.1. OPTIMUM MESH SIZE DETERMINATION USING PLATE
ELEMENT S
s Normal Stress in 'x' direction
s = Normal Stress in 'y' direction
t = In-plane Shear Stress
Vm = von mises Stress
max.prin. = maximum principal Stress
Note % Differences are with respect to Elastic Small
Deflection Theory 15 Model 1 2 3 4 5
Elas.
SmallDefl.
ElementSize
304.8 152.4 76.2 38.1 19.05 (mm x mm) x X x X X 304.8 152.4 76.2 38.1 19.05 Dispi. (mm)2.04
2.59
2.59
2.59
2.59
2.57
(% Diff)-20.82
0.45 0.45 0.45 0.45 (N/mrn2) 51 97 109 115 118 121 (% Diff)-58.13
-19.86
-10.29
-5.12
-2.41
a,»N/mrn2) 119 149 164 171 175 178 (% Diff)-33.17
-16.40
-7.99
-3.78
-1.68
t(N/rnm2) 18 24 22 23 23 vm (N/rnm"2) 119 149 164 171 175max. prin.
119 149 164 171 175 2 (N/mmtime(min)
0.23 0.42 1.51 11.03 117.28 memory (Mb) 0.024 0.095 1.001 8.8 71.310 o -40 -50 -60 16
,/
I
/
/
a xx N4
o 2 3 4 MODELS FIGURE 3.1Mesh Size Determination using Plate Elements
-lo
V) w o C w
-w w
present study, the area of interest was the behavior of the
plating in a cross stiffened structure under normal loads.
To economise on the size of the models and still maintain
accuracy, the longitudinal and transverse stiffeners were
modeled using beam elements. Since FEM grids combine elements at their neutral axis, the beam elements were offset from the plating by an amount equal to the distance of their neutral axis from the plating. Sectional properties used to
represent the stiffeners include total cross-sectional area
(A), the shear areas (A and Ar), moment of inertias
about the
strong and weak axes of bending
(I and I) and the St.Venant's torsional constant (J) . The properties were
obtained by treating the longitudinals and transverses as
stand alone beams with no effective plating.
To validate the use of offset beams to represent the
longitudinal and transverse stiffeners, two finite element
models were developed. In one, the stiffeners were modeled
using discrete plate elements, in the other offset beam
elements were used. For this comparative study, the second ordinary strength steel (OSS) stiffened-plate model, 15.24m x 6.4m (50' X 21'), designed in section 2.0 was used. For the
model in which the stiffeners were represented by discrete
plate elements, the webs of the longitudinals were two
elements deep and those of the transverses were four elements
deep. The flanges of both the longitudinals and transverses
were two elements wide. In both models, the plating was
represented by 152.4mm (6") square elements. Due to symmetry
in loading and boundary conditions,
only a quarter of the
structure was modeled. Results of the comparison are shown in Table 3.2. In addition to displacement, Table 3.2 also contains the computational time taken and the memory required
TABLE 3.2. COMPARISON OF STIFFENER MODELS USING PLATES AND OFFSET
BEAMS. (PLATING MODELED USING PLATE ELEMENTS>
= Normal Stress in 'x' direction
=
Normal Stress in 'y' direction
=
In-plane Shear Stress
v.mises
=von mises Stress
Note
:% Differences are with respect
tothe model in which the
stiffeners are modeled using plates.
18 MODEL STIFFENERS AS PLATES STIFFENERS AS BEAMS DIFFERENCE
Displacement(mm)
9.73
9.99
2.69
(N/mm2)140.53
139.85
0.48
a, (N/mm2)
190.34
192.92
1.36
. (N/mm2)54.28
55.55
2.34
v.mises
(N/mm2)169.31
171.87
1.51
Time (mm)
21 4-81.00
Memory (Mb) 25 6-76.00
while the differences in the displacements and stresses between the two models are less than 3%, the saving in time
and storage (memory) is as high as 75%.
3.4 Modeling of Plating using Plate Elements
Six stiffened-plate structures, three made of ordinary strength steel (OSS) and three of high strength steel (HSS)
were designed in section 2.0. To study the effect of deflection of the supports (longitudinals and transverses) on the plate panels of a stiffened-plate structure, linear elastic finite element models were developed for each of the above six structures using 76.2mm (3") square plate elements to model the plating and offset beams to represent the longitudinals and the transverses. The beam elements used to model the stiffeners were offset at the connection points to the plating by an amount equal to the distance of the neutral axis of the stiffeners from the bottom of the plate surface.
The thicknesses of the plating used were 15.88mm (0.625")
for the OSS models and 14.29mm (0.563") for the HSS. The
sectional area properties for the longitudinals and
transverses used for the six models are summarised in Table 3.3 and 3.4. For all six models, Young's modulus, E, of 206,850 N/mm2 (30 X 106 psi) and Poisson's ratio of 0.3 were
used as material constants, an uniform pressure load of
0.107 N/mm2 (15.56 psi) was applied normal to the plane of the plating, in the direction of the stiffenrs. Due to symmetry of the structure and the load, only a quarter of the
structure was modeled. These stiffened-plate models
represent bottom structures between transverse bulkheads on
the forward and aft ends and longitudinal girders on either
TABLE 3.3. SECTIONAL PROPERTIES OF LONGITUDINALS AND TRANSVERSES (OSS)
A Total Area
= Area along 'y' axis
= Area along 'z' axis
= Moment of Inertia about 'y' axis
I = Moment of Inertia about 'z' axis
J = St.Venant's Torsional Constant
YT1a = Distance of the neutral axis from the plating
20
MODEL i MODEL 2 MODEL 3
LONGL. TRANSV. LONGL. TRANSV. LONGL. TPANSV.
A (cm2) 29.006 271.270 29.006 157.322 29.006 6.O58 A(cm2) 14.090 145.161 14.090 100.322 14.090 39.964 A(cm2) 14.916 126.109 14.916 57.000 14.916 27.193 I(cm4) 258.65 22,787.01 258.65 3,310.95 258.65 775.23
I(cm4)
1,136.35 233,612.22 1,136.35 80,869.90 1,136.35 12,741.53 J(crn4) 8.70 473.71 8.70 180.10 8.70 36.42 Y (cm) 14.983 66.309 14.983 46.772 14.983 28.699TABLE 3.4. SECTIONAL PROPERTIES OF LONGITUDINALS AND TRANSVERSES (HSS)
A Total Area
Ay = Area along 'y' axis
A = Area along 'z' axis
= Moment of Inertia about 'y' axis
I-. Moment of Inertia about z' axis
J = St.Venant's Torsional Constant
= Distance of neutral axis from plating
21
MODEL i MODEL 2 MODEL 3
LONGL. TRANSV. LONGL. TRTANSV. LONGL. TRANSV.
A (cm') 24.387 214.445 24.387 129.387 24.387 48.761 A(cm2) 11.452 145.671 11.452 81.458 11.452 30.187 A(cm2) 12.935 68.774 12.935 47.929 12.935 18.574 I(cm4) 185.76 5,622.25 185.76 2,255.60 185.76 510.76 I(cm4) 718.50 192,408.89 718.50 52,509.55 718.50 8,499.36 J(cm4) 6.66 259.02 6.66 128.45 6.66 13.19 na (cm) 13.246 59.731 13.246 41.849 13.246 27.607
side. For the quarter models, two of the edges are fully fixed representing fixities at the transverse bulkhead and longitudinal girder. On the other two edges, symmetry boundary conditions are applied. A schematic of the structure is shown in Figure 3.2.
Figure 3.3, shows the actual
finite element model of the
first OSS plate-stiffened structure, 15.24m X 10.06m (50' X33') designed in section 2.0.
The other five models are
similar except for their overall dimensions and scantlings.
3.5 Modeling of Plating using Brick Elements
To study the effect of vertical shear on the plate panels of a stiffened-plate structure, linear elastic finite
element models were developed by substituting
the plateelements used in the previous sub-section with 76.2mm (3U)
square brick elements to model the plating.
One layet of
brick elements was used through the thickness,
thereforethe thickness of the brick elements was the same as
thethickness of the plating. The thicknesses of the plating
used were 15.88mm (0.625") for the OSS models and 14.29mm
(0.563") for the HSS model. As in the previous
sub-section, offset beams were used to represent the
longitudinal and the transverse stiffeners. The sectional
area properties for the stiffeners, material properties,
load and boundary conditions were the same as those used in the plate models, see Table 3.3 and 3.4.
3.6 Modeling of Plating with Initial Deformation
It
has been observed that
in stiffened-plate structures,due to the shrinkage of welds at the attachment of the
23 /
\\ \\
\
\
\/
\\
\ \\
,/\\
\
\
\\
\\
\
\/\
\
\\
\
/\
X\
\
\
s(\\
\\
\
/
\
\
\\
\
\
\/
\\
\
\/
\
\/\
\)
\\ ,I
\\
\
\
'(\
\
\
\/\
\
\
\
\
\
\
\\
\/
\
\\
\/
\
\
\,
X\/
\
\
\\,/\
\/\
\
\
\
\
\
\)¼
\\//\\
\\
\\
\\
\
\
/\
\
/
\
\
\
/\
\
\
\
\,
\
\ /
\
\\
\\ /\
\\
\
\
X X\/
\
\
\
\
\
\
\
/\
\
/\
\\
\
\(
\
\
\
/\
\
\
\
\
\
\
\
\
\
\
\/\\
\
\
\\
//\\
\
\
\
\\
\\\ \\\
\
V \./
\
FIXED D.0
z
PUITING MODELED USING FUITE ELEMENT5
(MODEL i
: 5D'x]' ;
055)
FIGURE 3.3
FINITE ELEMENT MODEL OF PLATE STIFFENED STRUCTURE
stiffeners to the plating, the unsupported span of the plating bounded by the stiffeners develops an initial deformation. As mentioned in [4] depending on the amount
of initial deformation present in the plating and the degree to which the edges are restrained from pulling in,
the plating under normal (lateral) loads, develops membrane tension. This membrane effect which, is
absent when the
plate is perfectly flat, produces a normal (or lateral) component of membrane tension which can carry a portion of the normal pressure load and hence can be beneficial. Tostudy such membrane effects
on stiffened-plate structurewith deflecting supports (longitudinals and transverses),
two finite element models of the third stiffened-plate structure 15.24m x 2.74m (50' X 9') designed in section 2.0
were developed. For both,
the OSS and HSS models, brick
elements were used to model the plating and as before offset beam elements were used to represent the stiffeners.
The initial deformation in the unsupported span of
theplating between the stiffeners
was defined by
a doublesine function as given below
( ,'
I
" .iryA(,)tS1fl -
Sin-'\aJ\b)
where Initial deformationt = thickess of the plating
a,b length and breadth of the unsupported span
of plating
Using this expression the maximum deflection always occurs at the middle of the panel. The deflection value was chosen to
be equal to one plate thickness. For this study, 38.1mm
(1.5") square brick elements were used and as before their
thicknesses were the same as that of the plate. The same material constants, loading and boundary conditions were used as in sub-section 3.4 and 3.5.
4.0 RESULTS OF FINITE ELEMENT ANALYSIS
4.1 Comparison of Stresses using Plate Elements
Tables 4.1 and 4.2 summarize the results of the analysis
performed using plate elements to represent plate panels of the six stiffened-plate models designed in section 2.0. The output includes maximum displacement, normal stresses o and in-plane shear stress and Von-Mises stress for the OSS and HSS models respectively. The normal stress, is in the longitudinal direction or the long direction of the structure
while a is the normal stress in the transverse direction.
The Von-Mises stress represents the Hecky - Von-Mises failure
criteria and takes into account all the normal stress
components as well as the the shear stresses. In the tables, locations of the maximum values of displacements and stresses are given in parenthesis below each quantity.
For comparison, the last column of Tables 4.1 and 4.2 (with
the heading 'SP') contains results of a single panel with
fully fixed edges. This single panel is the same as an
unsupported panel of plating of models 1, 2, and 3. This
panel has the same thickness as that of the stiffened-panel
(15.88mm) (0.625") for OSS structures and 14.3mm (0.563") for
HSS structures) . This single panel represents a case where
the longitudinals and transverses are excessively stiff and hence are undeflecting.
Figures 4.1 to 4.12 show the stress patterns of the normal, a,, and the Von-Mises stresses for models 1, 2 and 3. Stress
TABLE 4.1. MAXIMUM DISPLACEMENT AND STRESSES AT THE BOTTOM SURFACE OF OSS MODELS
(USING PLATE ELEMENTS; t =
15.88 mm; a235
N/rnm)TABLE
4.2.
MAXIMUM DISPLACEMENT AND STRESSES AT THE BOTTOM SURFACE OF HSS MODELS(USING PLATE ELEMENTS; t = 14.29 mm; ty1 35O N/mm)
28 MODEL 1
(15.24x10.06)
MODEL 2(15.24x6.40)
MODEL 3(15.24x2.74)
S.P.
(3.05x0.91)
Disp. (mm) 14.45 10.42 7.442.59
(cm,cm) (457,503) (457,320) (457,137> (152,46) xx (N/ram2) 157.28 151.97 151.94 108.57 (cm,cm) (0,137) (0,137) (610,137) (0,46)c, (N/mItt2)
241.70 209.13 195.92 163.76 (cm,cm)(610,0.0)
(518,0.0)
(457,0.0)
(122,0.0)
ty (N/mia)
60.9953.26
37.54 22.48 (cm,cm) (587,107) (587,99) (587,99((23,15)
v.mises
(N/mItt2) 230.64 189.87 184.05 145.55 (cm,cm)(610,0.0)
(610,0.0)
(457,0.0)
(122,0.0)
MODEL 1(15.24x10.06(
MODEL 2(15.24x6.40)
MODEL 3(15.24x2.74)
S.P.
(3.05x0.91)
Disp. (mm) 18.68 14.46 10.32 3.55 (cm,cm( (457,503> (457,320) (457,137) (152,46) (N/mm2) 192.31 187.65 188.01 134.03 (cm,cm) (0,137) (0,137) (610,137) (0,46) (N/mm( 281.23 258.09 241.85 202.18 (cm,cm((610,0.0)
(518,0.0)
(457,0.0)
(122,0.0)
r, (N/mm2)
73.54 67.07 46.5127.76
(cm,cm) (587,107) (587,99) (587,99) (23,15) v.m.ises (N/mItt) 250.50 232.66 227.43 179.70 (cm,cm)(610,0.0)
(610,0.0)
(457,0.0)
(122,0.0)
as.u.u
a.uu.I.uIIUu..
SU
IUUUU.UUIUUUaSSj uuIluueuIuIuIuuuauu.uIumauuuuuas.l.us...eapuu.u.ssuuu.a...a.s.m...s.ss iIsuIuuuuuuuivauSi
IIUSISui'I i iii iw I.SSlUUIeUUUUUUuJ j...
uIu.I...I...I.u..
uuuuuuguaeIuSuuuuUuuuu..uuuSSUISUUUUPUUUSSU.lIUu.s.uulIaeÍal.IUUUUUUUIUIUIP slu.. u....ueiuiuIa.euuuuusu i.u.a..u....l.u.u.auuu...a.u.u...u.u.0I U SU UUSU U U S U SUI Ulul I III U I UN II UI USUIU I U UUIUIUIUSU I UI U I
U I UU URSS U S U IØIUIUI U IS e i
i UIU U U U U S UIUS UIU i
UUSSUÌUUSUIUSUUS UIUUUUSSINI.S.PUSUU.S.IUØ.UUUUSIIUUU UIUIUUSUU UIUIUUUUUSU
U
UiUIUSUSUSUISUSiSUUUIUUIUIUi5UU5USUUiU5UUUIiU5UIU5UUiUiUUS
IUIV1*i
1US
UIUIUUUIUSUIUUUSUI iIUUUSUUUISIUSUSIIUIUIUUIUIIUSIUUSUSSUI.IUUIUIUIUIUIUIIUUI.IUSUSUUIUUUUU.UII IUUUUUUUUIUUIUIUIUI iUIUIiU
U UIUIUIUSUSUUUIUUSUIUUUIIIUIUSUIUUUIUUUIUIUSUIUIUIUIII ISIUUUU iiUIUIUIUIUiUIISUUUISSUIUIUlUIUSUUUIUIaISmI..UIUI.iUI.U.iUi.IeIUI.SUIUiU5UIFL IUSSUUUIUIUISUUUUUIUIUIUiUSUuIUIUUSth IUIUIUUUUUUUIUSUUUU UiUiI*iUiUIUIUIUiiUiUiUUUIIIUiUSUiUU UUSUUiUUUIUIUiUiUiUliUIUi.U.iUIUiUSUS USUIU UIUuIU IUIUIUSUUUUUUUIUSUSUSUIUUeUUSSUUIUUUSUIUUUSUIIUIUIUSUUUIUIUIUI.
SU1UUUIUSUUUSU1U1UII IIUUUUSUUSUISUIUUUIUUUUUUIUUIUUUSUSUUUIUIUUSUIUUUUUUIUUUUUUUUUIUSUIUU IUIlII$IUIUIi.IUuU,pUiUuIUIItI.I.
I
SUIS
SIUUUIt
iUiUIUUUUUUSUUUiUIIIlSUSIUlUUUiUSUiUIUiUIUIUiUUIiUIUIUIU5Uj UIUUUUIUSUSSUIIUUUUUUUIUSUSUIUSUSUI iUiUSUUUIUUUUUiUU5JaUUUIIU5UUiUiU
U
2ISUUUUUUSUIUIU
1U1UUUUUIUIUUUIUSUUSUUU1UUIUUUSUSUSUSUSUUUUUSUSSIUSUIUIUSU IIUSUUUUUUUUUUIUIUIUUUUUSUIUUUUIUILUUUUISUUUUUSUUUIUIUUUUUIaUUSUIU
UUSIUUUIUIUIUIUIUUSUUUUUIU..IUSUIUUUI
SUUUUUIUUUIUIUIUI V UUUIUUIUSUSUSUIUUUUaIUUUSUSUUUUUUUSUUU rIUIIUIUUIIUUIIUSUUUUIS..UUI.IUIUS
IS
SU SU SU .IUSUUUI USUS USUUU lUaU
SUIS USU UUUUIU UUSU SUS UUU U UIU SeSUSUS USU SW
SUIUIUIUSUUUISUUI UIUIUIUSUUU5U5UU5A USUUSUUUUUIUIUSUUUUUUUUIIUUUSIUSUUU USUUUUSU.UIUIUSUSSSUU..U.UI.UUSUII SUIUUUUUUUUUSUIUIUU UIUUSSUUUIUIUSUUUUUUUUUUIUUIUIIUUUIUUUUSUUUU.IUUUSSUUUUUU.IU$.UUIU..ISI.II SUIU
-IUSUUUIUIUSUSUUIUSUS$IUUUIUIUIUUUIU5USSUISI.IUUUIUSUUUUUIUI.,.IUU...IUSUIP
....w...d*UUSUje5
IlIUSIUIUb.d...,d..
SUSUUIUUUIUSUIUSSI ISUUUSUUUUIUSUSUSUSUIIUUUØSSUIUUUUUIUIUSUSSSSUIUIUUUUSIUSSUUUIUS.SUS.IUUUIt IUSUUUUUUUUUIUSUIIII UUUIUUIUUUSUUIUIUUUIUUUSUUSSSUUUUUIP.UIUUUUSIUSUUUUSUUUUSU,.I.IU..I...I.SUIP SiUUSSUSUlUSUSSUU IUISUIUSUSIUUIIUUUUUIUIUUUIUUUSUIUUr1I.SS.SUBUSUSUI..UI.I..UIUIUI.IU..IUUUU SlUUUSUUUSUIUlUSU$U
UIUIUUU
IUIUUUIUIU5USUISIIIP UUSIU
IUIUIUSUUUIUUUUUUUISIUUUiUIUSSUUUUIUUUIUIUSUIUSUIUaU..S.UU.USUU
SUSUUUIUIUUUSUSSUS
USUUUUSUIUSUIUSUUUIUSUSUUUUUSUSUSUUUUi JIUUSU
SUIUUUIUUUIUSUIUSUS S USSIUSUUSUIUSUSUUUUUSUSUUUUSUUUUUUUD iISSSUIUiUSUSUUUIUIUlISUSUUU5UUSUSUSS UUUSU
SU SU UI UUUUS USU SU UUSUSUUUSU SU U USU SU i U Uai USUI SUIU SUUUUUUUSUUUS USUSUUUSUI ISUUaSk
UUSUSUSUUUSUSUIUSUI S UIISISUSUSUUUUSUIUIUUUIUUIUSUUUUIUIISUSIUIUSSSUUUIUSUIUSUIUSUUUIUIUUUSUUUSa ss.s.p.r.p.U5piiIis.pp,ppa.
I.l.. UUUiUU!JI .mtwujw i1meuTIru 1 .dSJaS4 SUU UUUSUSUStUaIUIUUUUSUSUUU IIUSSUSU5UUUUUIUuUIUSUUUUUUUSUIUSUSU5ft UUSUUUIUUUIUIUIUISIUIISUIIUSUIUSUUUIISUUUUUUUUSUIIUUUUUIII i UISUIUSUIUIUUUUUUUSUSUUUUUUUUUUUIUUUr SUSUSUSUUUSUSUSUSI ESISUISUSUIUSS I W Ui USSUSUIUSUUUSUSUUUISUUUUUSSUUUUUUUIP
SUUUUUUUSUUUUUIUUU. SSIUSSUSSUIUIUUUUUUUSUSUUUSUSUSUUUUUSgWUiUUUUIUIUUUUUUUIUSUIUUUUCUUUIIUIUIUUUUI SSUSUUUSUUUUUUUUUit SØIUSUUSUUUSUIUSUSUSUUU5U5IIU5USUUIUliWIU5IUIU5UIUUUUUIUlUlUIUUUUU5UIUUU5UI5 UU5UIUUUIUUUUUSUUUI SIIUUSUSUSUSUUUUUSUIUUUUUIUIUIUUUUUIUSSILWUUISUSUUUUUUUUUUIIUIUSUIIUIIUUSUUUIUS UIUSSIUUUSUUUIUStU UIUUSIUUUSUSU
i IUISU5U5U5UUU5US
SUUUSUUUUUSUSUUUISSISWISUSSUUUSUUUUUUSSUSUSUSIUUIUIUSUSUSU.FS ISSUIUUUUIUIIUSUUUIUIUSUUUUUSUUIUIUUUSU SUSUUUUUUUIUUUUUSSSSWIIUISUUUUUSUSUUSSUSUI.IUUUSUSUSUUUSUIWI LIUUIUSUIUSUUUSUSUUUSUUUUUUUUIUSSUUU
--BWIWVUSUUU FIGURE 4.1 OSS PLATE MODEL 1;
VONMISES STRESS
(VIEW FROM THE BOTTOM) -psi N/rnm 51]H.51E1E
L15] 5L
Liii 231 173 116 58 i..u.uu.uuu.uuuu..U,Fusuula lnUUaUSSIUUIaSIUUUUUIUSUlalUIUIUUUl .uuuuluuuu.u.uuua.r usauI .S..I.I.iuU.S.i.uul.IUI.U.U.a.I.i.i.N u I I U W I U $ Jb UI I I I U W I W r W W l S UI * 1W j I h J b W I'I I I tI ¿ I I -.uItIn
.e.Iuuw---iu.s..Iu -.___ ._.wIUUILFmi.... m p u s u i r i i q j s w I s ji u w u n u i M I I I I il iji i l (E U UI aUS I I L m i I R I im iii ii w I i . . i i i e iiU.UlUUUUUSSlUUUUu$.IUuUUUUlUSUUlUIUlUUUSNIUUUIIUUUUUUIUUUUIUIUUUUUSR
..u.u...u.l..uI.lU
uu..I.I.uuIuuIuuIu IUUUUIIIIUUUUUUUIUUUUIUUU1SUUIUUIIUIEilÍIUIUIIUUUUUIIIIUIIIUIUlUUUIISlSIul .uu.s.s..ui...u.s...u.uuuuuuuu.iuuuUUiuiUUiaLuuUuiIuauuUUUuiUuuuUUUSuIas
u .u. a s. u. u...
I UUSUIa a.. .i.u.s. a s u i a
a...s.au....a.i.ióiiai.aa.,...,.u...i.u...a.aai...i..uIui.i.ai.ia..uiai..
ssauIuauauuagauuauasuuauuuauusuIasuaauuaulalaaa,IaalaIaa u fl--:. --.a.aaa.uri.ip.. ...-dauSIIUIiu .iaa.isaam II I u a ' i I a m SS as i a a a a a n i a a a i N i i m i im i a ma i n p i k i ir ! w w u m i a i * ii m iiiiuì..e.a.a. i i irn tj ti a.u.s...uua... a aisIuaaaauaua... a.a.u.s.u.a.a. i a.a.u...a... i maaaaa.0 ...aa...a.uuusUaaaia.iu....i.s.a.aauuuslaIa.uaaauaut.u.a.u.a.a.suauls .a .a.aausuuuuuu.a.a.a.u.u.ua..aaa...u...a.aaaI. s.u.aUa.I.a.U.UU.el .aa.l..s.u.m.suu.u.u.uus.aaa.auu.I.uua.aaa.auuu..a.uuuuu.Iu.as.auuua.u.u.u.ti,at u...
.l.i.a.sai.a.,u.I.a.Iau.a.auIaaul.au.aaei.R.I.uu.eIuaautsuaa.a.s.aua.i , -uI.egataa.. .
-
p-. . ..aau,aw..._ a.a.auaa..uuaua.uIauta.su...s.a.a.a.s.l.a.au uuuuaau tivaasssIuaauauuuuuSIaUauau auu.uua'napnlluuuuauaaaea.a.a.ua....u.a.a.e.i.a.arnnu a uaUai.IUUUIUI. ..a..ua..u.uu.a nlunlhagauaraa.umaaauaauauuasa.a.auainauut uaInl..lau.InlaapiiiaaIahJnlaLuuaI.a..aaa.alulaaal. ..aiaIeunlaolaaawtanuIlanaunuauuualua ..anla...lsaaiahllauaaaauualaa.aa a.au.a.aaa..uuI.laaaMaauan.nlu.aaaala,aumia.11li,.arIaaai.,
-'ça.a.an,ip-'u..a.i.i
-.uaia uauUaaiIi dUlInlIelUl.. a a a a i a a.. a.u.---,, .IOIU ua a ru .i alalia . a w. .0 -.a... s tana.a.u.a...a...anliaa ua.uLe.uaaauuauauaUuuaauuuuauaauaeaJuuaurlsaaua...s.a.auus.auaasuaaaaaaanl.., au..l.a.u.IUI.aUa.aa.uiaa..ul.l.auaUl
u..
...aaaaw.laa..a
sua
a...u.. 1u1 aura a a ni a aia... sui... ua.e. u.. aa aaiai a a a a lu lU u.u.. ...a.i U.u... sui... uueaan la a
aivau.aiar...UI.a
u..
a.s.aninumar.anna.s...auuu.uuuuua.s...suuua....uag.a
..a.auu....r.nuni aina..aan*uu ianaa,aua.a..ua.susu..aaueau,,aauanunlaaanaa...a.a.a.a.a.a.aaauuu..aa.a.aaua ua ini e ai au a.. a nu i Leasau va
val a at u vu a ana t aia a a I a unan au n uva a ama la u uni. uva a il'
anas ava av van avviai nati un a nani na a na a na aaa a anni ana rusa an aras ana
aaauav,,1---.laaana,,aaØavay,,,guii,uu.annlnuas tuaniatsaut
al.atutiaaaa:-
iliUU..I -.-liaI.
- --i
..alu.t -...aaa. 'aauI,. ..asuLiiaani. ---,..uau.vwua.iuuaiaa iu,u.a.ua.i.a.ilasp u.iu.tølwia.uaa..lauaelniianvatuaalu,.ia.. ni.euIgia.a..*mnvnlauuanammanaaaaaainaavaalataaaaa.aaanuultiavla ami a..,.-. uauaauaaLano.uauaaauasa.a.aaaaa.ga,
a.a..,.a...asuu.aau..aua.auauu...uemuas aa..a.a.l.uuavuaaiu t ua.sanIauauuuuuisiuluas a .a.Iuaa.a.i.Iuuua.a.a.a..uaU..lua..uaala .a.a..a..a..u.aas -a nutu...I.uu.Ua...u.su ..aaii. -Itnlll 1 UUaU.aiia .a.a.a.i.a.pa.Ua, auatas.uuauUU.ulauuuu,.uuIaalUlaaiaa . vaanvana.l.u.a.u.a...u.a...u.uua..u...aau Ul auauaialUUaulaaUaalui... Sulla naan;tua.ua..a.a.a.auaaasaUa.aauauuavua u..vaaaa.I.aaanhaP aaaua,avaavulaaaahluIll.nannvuauaanlalua5lU i uaanataanlaanusprurp.auauauuusuu.aaaaeuanan a. ..a.aa..a.a. ---v.pvnlassuaaaanamaavajiajtav,avauuaui.1 -tln aan.uauvuuv anua animi unas a a a a. a FIGURE 4.2
OSS PLATE MODEL 1;
STRESS (VIEW FROM THE BOTTOM)
257
144
3181
194
psi
-E
91
881
UIUJ/Ntsd
HIJ
ILLIE
6EE9 L
V L62L2
R' BhE 9E(OJLO
HJ WO'dMIA) SS1LS SSIN-NOA
!I
IUOW UVnId
SSH£t
flfI
Ii,iri,iplSaÓl$ asauuuuiuIuIu$I,a...u...u..
RIUS UUUSUIUUUUUUUIUUUUUUUUUUUIUUUL $UUIUUUUUUUUU,pu..i...suu...
IIs.auIuu...U..IIaiuu.u.u.,.
1mI.I.I.I.I.,.I.I.U.UUp...l.IUIUU.II.ISIUIUIIaI
4rIIIUI
rISIUUUIUIN....IPU.IUI.I...IPPU..IUUSItaI.!Ir.ePUUUI---'---'.-'..-.-'--'.
'--.- ---1!lI5I
IIIP---.II1eWIII...
.ia
i. . . ....uie.ui.Ii IusaI.Isuu a.a iuuut IP UIU eI a e s.,.... ucui e s. s. u ei.. cii iBi e Cucul u. cpi
i u. u i... i
I Piui.iui.i.iii.i
.l...u....u...u...u,.u..u.uiui.e.t 'ueiu.uuuuuuu.u....u.u..u.usuiu..i.ui.g iuu.up.se.e.s.uuuiuu.i.iiiuui.uuui.iuuiiui.iui..uu.u.u,
.u.u.isi..a.i.u.0 i .iiiuiuiuiuiuiuuuiuuBIuiuiuiiuiuuiui seuuu.uIuu.iiuuuuiusuuu.uiuuuuussup. :iiu.,.u.uu.uiuuuuu. iiuuuIUiuiiiui.iui.ueIuiuiei.Iuiuiuiiuiel
i.Ir
uuu..u...puu.i.i., iiuiuIUuuiiIuiUuui.iuii.iuiRiaiuIuiiuleUIUiuia.uiuuui.i.ivaeiiuIuuiU.uiiUiiiI .uiuiu..,aiui.i.u.i Iwiuiuiuiaiiuiiei.uuiueii.iuiiiuiuiuiiiraaeieiuiui.i.,.,..u..u..u.u.u,.i..u,iu e $c,iuuuiiieiuiuisiui iiisuiuuiiuiiuuuuuiuiiuueiiiuiifleIeesiuiuiuuuuuii.ieuui.i.sui..i.pi.lJca.i auiuuiuuuuuauuui iuuuuuipiee.m.sm.siuiuuui.u.uupuiiui. u.u.... -- '-''- ---r
.'.. ----'.,iuI ,ueu.p
'---IiiiUiqliCi$..,.
ii.i.Odi.i.i.s.. ...
i
I u ii. ululi U i uuiu.ui.u.. u ici a... i. u..
.iu.
iuu.iu.uiuuuuu.u.ui..uiuu.i..u..u..uec ru.uuuuui.,u..su....u...s..uiuuuuuiuu... iuiuu..u..iuiuu..0
luuuuuuuiuuuuuuuius..u.ui.iu.i.uuuuuu,u
upu.. u
u.uuiui.i..ui
.uuuuuuui.uuuuiei.iiiui.iepup.uuu.,iuiFei.uuiuiu,uu.iu.u.ua.uuuui.iu,ii.u,ui.0 iiuiiuueuuuuiuuuuuu4uuuuiuu.uiu.uuiuiuua..u.!M.uuuuuuiuus,tp.iui.u.s.u.u...iuuu,uuuiuuuu.,.uuuuu. .uu.uu...i.ui.euu
iiuiuuuuuuiiuuieu.ia.e.m.e.m..uuuuuu..el.uuiui.uu,u,.i.u.uui.,u.ui..uu.,uuu.uc. Iuuuuuuuiuiiiuiuu
. iuuiu.uuuuiueiuu....eeeui.i.u.uuu.Iu.iliiiuuui.iuiu..,.iuiu,u.uuuiuu.iuu..ir. iuuui.uuiiuiia..i.Iuuusuiuu.uu.ui.iuuetIuiui.uuu.aiui...iuiu.u..iuiuuuiuu.uu,, .uuuuu..uuiuuuiu.ui
''-'-
--«.-. te'--'p--«.-.--«.-.iijt--«.-.,uuu, ieuur:t,u--«.-.--«.-.N' "--«.-.
..auN*up.p...uuhuiusscjieueiape..,.u.u.ua. uuuuiuuuuuiuuuUuiiuee.uuu.uuu.uii.iuuuuuuuu..u.iei.i eiiieiuiuieuuuei.i..u.pu iu.uiIeuuivauuuuu 'u.uu.uuuuuu.u.uiuiuuii.i..uuuu.uasu.u. uuiuuuuØupiuuuuuup icuuiuieu.u,c.uu..0i.ui..u..
s, eiuiuiIIeuuu uiuivaluuulu.i..iiuuuuu.iu.e.uuu.uuuUj,ailiuu.i.,..ece.die.u.u..i....e.i.u.ii.u, u..iuu.uuuu.a...ii 'uuu.ul...i.i.ui.u.up..u.u..uu.... aeu.uiuuulu.uuu..suuuuu...eiuiui.uuu.i.a...u.u.uu..iiuiuu.iuuululuuuuuuuuuiluuuuuusupLi uiu..u.uuuuiu.uu.iuu.ie.lu.uu.uu.iu...i .u.u.u.uueii.u.iue
sui .U.I. u.u.uu.. usia uu..iuluuuuuu..i.e. ' u u, ululi u.i...s.0 u i .iui.iu lisis... ilUiI heu ...uiiui.iui
.iu..iiu.u,,,,-.-p-e-mr:,ui.s.s aueulu,;u.
duflLhai.uluUmuluiuie
u.uul.sdheuuiu..auIeuw..i.m
u u.u... .iui uhusu su... u.u... ..uur r .ii UU e iu....u.u.aa. uluea ue..h.u. ecu. saisi u ...ui ..0 use
.ulur
ieuuuauuhuui;I,,euI.us.Iv,uuisurausuuuIu. u,...uu.uu.h..uuciui.uc lIuu.isnivaUul.uluiIshIiluiuuuir u
ciIe.mauluu.uusa3uu.ujensnasiusecispiuiucucussivaIuuis iuuIihu..seueseu. iuiiuuupuuuuuiiuluaehu.eiacaueuuuuuniu.uis.la.$gssuLuseIaauIaiaaasussusuu.s d liiuiSali$us*piliLiu.uuss..usu.ii.iu.uu.uiu.uuuiu...n.uuuua.u...uuuu.uauhiuu....u..:u.uuu..iu....i.i.iuuuuuu.iuu.cu....u.i ui....uuu....i..ui.u.IuusIiiupup.l,l.0 u ....uusu.uuuuu.u..0
suuuuuPu1mii.m .iuuiu..huuuuu...uuuu.u.uu.uu.uu.u...uu...u..uuuuuu....suuuuus.a.u.uu..ur ....uuuu.uiuiu.uuuirsuuu.uu.suuu...uuuu.i..u...uuui.u,uug.uuuu.us..uuuuuuu.u..suu.suu.,uuu,iuu. u...uuuu.uuu..uuuiu iiuU.li.u.
sii
pusisuiuu'wi
,uuiui, 1iuuuaim,N,,u,a.u.i.u.u.uu..u...
sIusIsIaIu.uuuussuuuuIuuuIuu.Iusa..sI
I IUIRUUIUUUIUIUSuspuuIlluaIii
iuIaI..uauu..i.i. flususu.uuu.I...I.uI.uuu.uuI.u.s...ueu ii.ia.I.I
I .pPI
i.iui.i...i.u.i.i.
UlUISIUUIUlUIUIUI$UUIIUlUUISIISiI I auli
I I I S SU S U I I I IU SU S U SUI S I UU U l U UI UI SI ØI U UU I I I Uil UI S IUSN 1ff lt M ! t li; ii i ,i ì Hib II II I W w U li a i IL I U I 17 I l II I I I I I I I I liii J W l I II 11 I $ b m i iu a '* ' u w LI U m I Id - I I U IlL II I. _ . ..- Un UI Uf J i . s ii l.a_I UIUIuUUUuIUUaUIUuUauuUSSuUUUJI IUIIUIUUUUUUUUUUUÓIUUUUUUIIlIIUI Iv UUU.SU..UU..I. IIIIUIUUUUIUUUUUUIUIU IUUCUUIUUUIIIItIUIISlIUUUUIIUIUUUIUPUUSIIUIUlgUImiI, IUUUUIUPUUUUIiCSUUUUIUIeUUSUIUUUUPIUIØIUtUIUIUIUUUIIUIIUSIU r -. UIUI&ÌUS .UUUII
._IIUIIIII_.
4I$IIINIUh_ : . mIUIUIILUUIUIUIUIIIUIUIUlI a IIIU S UUU I II$ UI
I U I I III USI Uil I UI UIU I I IU UI U U lU I I I U I I .i. IUISUS
S U S U U UI U IUUU U SI SIIIIIU liii I E
I lUIS U laU S U I UIUU U I
II I I UUU I UU alU I UU U U I U U U I U U USI SU UI I I II m
U I I Uil nl I UUUUIUUUUUIUUUUUUW&IiIIUIIIUUIUIUUUUUIUUUIUUUUIUUUUUUSUUUIIIIIIUIUUIU 11111 UIVUIIF ¿ -. IUIltlIU -- .'.
.-. UIEIIIS .III
., .".,-
.i$UIUIUIU.-. .dSIIIIUIUU.. ...aUUIgUUIUIliSUUIUIIU IUIUUOiUUUIUIUILub.N. II II I I I I U I U I U I U I U il I Sill U I I U I U U U I U A U I U I S I I U S I I I W I 0 I S i ¡I I U I WI I U il U I U A I S I I i il i lI I I T 91 fl I I I rn ml fl S I S II IUUIUlUUUUUUUCUiClSIdUllIUiUIUUUUUSlUSlilUUUUUUUUlUUSIIlU IUUUUUUSIIUUUIUUU UUUUIUIUUU.U.UIUUJIUISIUUISIIUIUUU.UIUISIUIUIIUIUU.IUU1IflUIaS..I IUUUIUSUUIUUUUUSSISISIUIIUSU UIUIIUIUIUIUIIUIUUSUIUIUIUIUUUPUUUUUIUUUIIIUIIS UUIIUIUUUIUIUUUIIISIIaISIUI UlA*lSiUILlUUiUlUlUUUiUlUI5UUUIUUUIl5UiUlEL5iiI U 111111111510151515 I lU 11011 i U SI i UI UIUUIIII S Sill 1111515 lUIS 1011*1115 SISI Il I Ill I Il 11151111 IlS
i 1111111111 SIll -ValUgUISIIlIUjlr IUISU1UIVPP IJliUlU55l
UWlU--"""----'lielUill - --¿SIUlIWIl.. ---. -.4iSISlS liPIlSUllllLIUIIIw. ,... ISVIIImIllUSI PSIlIJIlUISISIUl IIlWlI5IIiIlI 111111 IPUlfl 111111111 IIIISIUPIPIIIIÌIISÍISSIIIINIIftOIII III
111111
fflhIIIISIIl*llll11lL1Ulll IIIlSlSmWhIflbl
IIltIlI11I11lljöiUSeI5iU5UiUiUISiIÌ*lUiUII5IIllUIIlIUl5lUIUIUUUlUlUiUlUIÓ$UIUUUIftlJlI
I UU SU SUI U 11111515 II VII S U U Sill U SU SU S UUUUUIUUUUU UI U SU S SI III UPS li
SUie UUU USaI U1UUU1 U UUUUUISI U UUUUJ Ill
I
IIIUIUUUIUIU1UUSISII VIISP1UIUUUUUIU1IUUUUUU11UUUUUUUIUIOIIISUIUPUUIUSUIUUUIUIUUUUUIUUUUUSU5UIU5USIPSIIUISIUUUIUUUIU&SII.SIIIMSUUU
IUIUIIUISS
IIUUUUIRIUUUIUIVI
IUUUPUUUUUSIUIIVIIUILI IIVIISSUUIUSUUSSUIUSUUUIUIUIUUUUUII*UIPSISIUIU1ISSISUUISUIUUUUUS$IUUUUUIIIU5SIUIUv,s
ISSUS
SSIUIUUIIVIIIDIUI IINftSIIIUIUUISSIUSS 111511111F 11505P511055r1511,UISlIllSIilS
-
«." -IllS - --Urss--
.--.--,-..--' -JIIPSI --. -aUIlI __. 55..._ S -.555 150.. .... SUISPSIU*SISUUUUIV -ICUPIUSUI IUEUSSIUSUPUIUSUIIVIUIUU )UIIUUIuii0IlP55uSIS5Ul1FlIISIPl1ftlPiPSIUIUF IUS11VIIIUtUPUIUIS -ISIlUISIS á11iSiSIUISUlIIIS5UU S UUlUIUII5iS5S5SUiUUUlU5 515W IUSUPUSIIUSSIIIUIP -IPSSUSISUIUIUSUISUUIUUUUUUUUUUSIPUISI UIIU5l5IU55SUIUiUU5l55UIU5UIU5UU55 U 5Si -ISIIIISIUUSUIUSUUUUUUUUIUIUSUSS1 0IILUUUIUIUSUIUUUIUSUSUUUUUSUIUSUI0IFV UUUUSUSUUUIUPIIIIVS -,VWI,IFIUSUIUIUSUIUUUU.UUIUUUSUUI.SISr VIUIVSSIUIUSU5UIUIUUUUUIUSUIUIU1UUIISIII! IUIUIUSSIUIIPUIVFUI IllIlIUUtSSU5UI5IUUUISUUUUUIIIIU5UIS SIÍIIvSI.U.SU5. 11111 IIIIIU*IUØUISIIWIVIP -'iitVlllSWISlIillIIlVS11iSlllIIVllIUSUI5i? '11F 01*15111 11111U110101 -11151F -aS*s*S11IUlsøulSIIIsIrIv' -51155151 VISSISIWFI 115505 IWF--- T4F
,.--i. -S, III Ill FIGURE 4.4 HSS PLATE MODEL 1;STRESS (VIEW FROM THE BOTTOM)
'-1EI85E1
D1H.2
-1fl9.
_]L11
psi
N/mm2282
152
23-107
-236
T
8tì
IT
061
z/N
tsd
5111BE 5959
L29EL
V9R9E1E'9h5L2
(NOEi LOHL NOL
MIIA)SSLLS S3SIN-NOA
f TUON LV1T SSO SflfL
.-u, a ...0 i.. . i. u.. u u.... u.. u .... . i u u . n
i J I U U ISI ISU IIII IUSUI SIIlII Iu I iU I III U U I
...,...u...us...,.I...uI..Mm ni ..ui.e...i..uuuu.,...,...u.u.su.i iuis
i.u..uu...i.uu....u.u.uu..e.u.u.uuu..u.
.u.I.I.U.iu.e....u.u.i.i...i.i...ui.i. I I...I.,.U...I....
I.U.IUIulueu....R.uuuuI.I.I...I.IUIUuUI1u.IU.Uu.uuU.U..UuUUu,.I.uUUuu.u.u.U., UIUIUeU uaai...ie....u.u..uu.ii...ua.u.u.uil..uieu...aeuuuuuu..u.s.uus.peus, ...m.u.us....0 UaUUUISUuuI ...u...uss.i.ies .u...,ui.u.s.i.u.uu,e...uuauuI.muaua...s...a..uiise...
mI.mt'ml..
-mpI
-'--ø.UlIt4 iSu ...
iu...u.u.i.ui....ii..u.su...i.i..
.u.i.i....u.u...ai...s...a..,r ue.e..mseiu.u...
îp.p...u.epuuuu.,uuuIep.e.l.u.p,ur .u.u..u..u...u..u...eauu.,....u.u...s...u....0
pUUSIUU.IUlRUIISUUUUSRUuIUUI.uUuIInwUUUIUIIUUU.IIIUIUIuu.IUpUUuI..uUIuipep..uuuIuuIIuUIII...u...
auUppusnp.iI.I..mNp.u.susu Illuissul.
uwtiuiI.,w..s.iui.s,r.sussusU I.s...,.çII. u.pa,.t lUUIUUUkUuUI U 5URU U IUIUUI a...s...s...u.Iusu.uuus.s....u...s.uu ai!eUUIUUUSUUUUUUI5UsUupUUu..UIu5p P...I...U..U.U,..UUUIU5U
s.s.s.s.u.uu.pUuuIusIIUuu..s.uuus.ttU..us.s.I.suusu.ua.susuu...u..s.sds s... UuUusUUUi5Ususus tir1lr,m5U5ulI.
---;sUss.0 ---. -.i.sUUuusUsUsUsuIuuuuuuUuuusssues..r .u...s.s.s...s.s.u.s....u....u..s..uo. e...s...
.u.s.u...u....u...l...u... ...u....
uu...uu.suU...s.s.a.aap...aas..sus....'t.uuu.Iu..s.pap...pa,U...us...su...s .ss...s.s...,...s.0
t
UII
6Ut m.UM.IIUUI5UU.. IUIUUUUIUUUI IUSUUUU.UmIUInIU
IIUUUUUUUIIIItIPUUII
UUiIUIUUSUIII
,UsSIUs.u,.Iutr,,,.
.i...uu.ua....s...u...s..t ttI.usu..Iu....I...s...a...,...s.s... I UuuUSUluesuseus
UU5U5UUUUUIÓUUUI
U SU5UUUSUUUUUUUSU I5UUUU UU PU$U UU U
ti UUUUaIUe UUUU UUUUU UiuuuiuUUUus USU J UsIUSU U U5U5UUU
!U5UPUIIl' mEIa.dIlUI eudd.daiadd.da.1.gUuIU5Ub1 5a5UUU5IU5UUIUS ...U..UU...I.UU...U.I..UU.,1tUUUI...U....UUUUUU.UI..U5.U.,U...tt, 5UUUUU5UIUIUUlUI U 15UUU5UIIUUUSUUUUIUUUUUUSUUUUSUUUI1IUSUsU5UUUUUsUUUUUU 'Pu. .s.sSU.Jri..U...
IuUJ UU5sUII'!F
IIUUUUUUU. i,-riN ii It ti.i ilII ....UUUSt-Itrntirtmir nitttn' tittttmlt, tntr'U1I.UU5I t
r
