Marine_Structural_Design Episode 5 potx

Part I1 Ultimate Strength Chapter 8 Buckling/Collapse of Columns and Beam-Columns 8.1 This Chapter does not intend to repeat the equations and concept that may be found from exiting bo

138 PART I Strucntral Design Principles

Minimize degradation at system level so that when local fatigue degradation occurs, there are no significant effects on the system’s ability to perform satisfactorily Here good fatigue design requires system robustness (redundancy, ductility, capacity) and system QA Inspections and monitoring to disclose global system degradation are another strategy to minimize potential fatigue effects

Cyclic strains, material characteristics, engineering design, specifications, and life-cycle QA (inspections, monitoring) are all parts of the fatigue equation This is the engineering equation

of “fail safe design” fatigue may occur, but the structure can continue to h c t i o n until the fatigue symptoms are detected and repairs are made

The alternative is “safe life design” no significant degradation will occur and no repairs will

be necessary Safe life designs are difficult to realize in many long-life marine structures or elements of these structures This is because of the very large uncertainties that pervade in

fatigue design and analysis Safe life design has been the traditional approach used in fatigue

design for most ocean systems The problems that have been experienced with fatigue cracking in marine structures and the extreme difficulties associated with inspections of all types of marine structures, ensure that large factors of safety are needed to truly accomplish safe life designs For this reason, fail-safe design must be used whenever possible Because of the extreme difficulties associated with inspections of marine structures and the high likelihood of undetected fatigue damages, it is not normally reasonable to expect that inspections will provide the backup or defenses needed to assure fatigue durability

NTS (1998), “NORSOK N-004, Design of Steel Structures”, Norwegian Technology

Standards Institution, (available from: www.nts.no/norsok)

API (2001), “API FV 2A WSD, Recommended Practice for Planning, Designing and

Constructing Fixed Offshore Platforms - Working Stress Design”, American Petroleum Institute, Latest Edition

API (1993), “API Rp 2A LRFD - Recommended Practice for Planning, Designing and Constructing Fixed Offshore Platforms - Load and Resistance Factor design, First Edition 1993

API (2001), “MI F 2FPS, Recommended Practice for Planning, Designing and Constructing Floating Production Systems”, First Edition

API (1997), “API FV 2T - Recommended Practice for Planning, Designing and Constructing Tension Leg Platforms”, Second Edition

IS0 Codes for Design of Offshore Structures (being drafted)

Part 11: Ultimate Strength

Part I1 Ultimate Strength

Chapter 8 Buckling/Collapse of Columns and Beam-Columns

8.1

This Chapter does not intend to repeat the equations and concept that may be found from exiting books on buckling and ultimate strength, e.g (Timoshenko, 1961 and Galambos, 2000) Instead, some unique formulation and practical engineering applications will be addressed

Figure 8.1 Coordinate System and Displacements of a Beam-Column

with Sinusoidal Imperfections

Let's consider the case in which the initial shape of the axis of the column is given by the following equation, see Figure 8.1 :

m

w, = w,,, wn-

1

Initially, the axis of the beam-column has the form of a sine curve with a maximum value of

worn in the middle If this column is under the action of a longitudinal compressive force P,

an additional deflection w, will be produced and the final form of the deflection curve is:

The bending moment at any point along the column axis is:

142 Part XI Ultimate Strength

Then the deflection w, due to the initial deformation is determined from the differential equation:

To satisfy the boundary condition (w, = 0 for x = 0 and

B = 0 Also, by using the notationa for the ratio of the longitudinal force to its critical value:

x = 1)for any value of k, A =

This equation shows that the initial deflection w,,, at the middle of the column is magnified

at the ratio - by the action of the longitudinal compressive force When the compressive

force P approaches its critical value, a approaches 8.0, the deflection w increases infinitely Substituting Eq (8.9) into Eq (8.3), we obtain:

Chapter 8 BucklingKollapse of Columns and Beam-columns

= Area of the cross section

= Distance from the neutral axis to the extreme fiber

r

W

S = Radius of the core: s = -

A

= Radius of gyration of the cross section

By taking the first term of the Fourier expansion

A simple method to derive the ultimate strength of a column is to equate oMAX in Eq (8.12) to

yield stress cry :

(8.16)

144

The above equation may be written as:

OiLT -[ +(l++)OE]Ou" +UEQY = o

in European steel structure codes

Chapter 8 Buckling/Collapse of and Beam-columns 145

The Johnson-Ostenfeld approach was recommended in the first edition of the book "Guide to Stability Design Criteria for Metal Structures" in 1960 and has been adopted in many North American structural design codes in which a moderate amount of imperfection has been

implicitly accounted for The Johnson-Ostenfeld formula was actually an empirical equation

derived from column tests in the 1950s It has since then been applied to many kinds of structural components and loads, see Part 2 Chapters 10 and 11 of this book

8.2

8.2.1 Beam-Column with Eccentric Load

Buckling Behavior and Ultimate Strength of Beam-Columns

Figure 8.3 Beam-Column Applied Eccentrical Load

Consider a beam-column with an eccentricitye, at each end, see Figure 8.3 The equilibrium

equation may be written as:

146 Part II Ultimate Strength

Eq (8.26) is called the secant formula Taking the first two terms of the formula expansion:

and substituting Eq (8.27) into Eq (8.28), we obtain:

n 2 e, P

(8.27)

(8.28)

8.2.2

The deflection for a beam-column in Figure 8.4 may be obtained easily by superposition of Eq

(8.9) and Eq (8.23), the total deflection is:

Beam-Column with Initial Deflection and Eccentric Load

The maximum deflection occurs at the center of the beam-column:

The bending moment at any section x of the beam-column is:

Chapter 8 BucklingKoIlapse of Columns and Beam-columns 147

Figure 8.4 An Initially Curved Beam-Column Carrying Eccentrically

Applied Loads 8.2.3 Ultimate Strength of Beam-Columns

For practical design, a linear interaction for the ultimate strength of a beam-column under combined axial force and bending is often expressed as:

where PuLT and MuLT are the ultimate strength of the beam-column under a single load

respectively Based on Eq (8.34), the maximum moment in a beam-column under combined

axial forces and symmetric bending moments M , , is given by:

148 Part 11 Ultimate Strengrh

For beam-columns under combined external pressure, compression, and bending moments, the

ultimate strength interaction equation may be expressed as:

(8.39) where the ultimate axial strength PuQ and the plastic moment capacity M , (considering the

effects of hydrostatic pressure) are used to replace the parameters in Eq.(8.37) in which the effect of hydrostatic pressure has not been accounted for in calculating PuLT and M,,

8.3 Plastic Design of Beam-Columns

8.3.1

When a beam cross-section is in filly plastic status due to pure bending, M,, the plastic

neutral axis shall separate the cross-sectional area equally into two parts Assuming the

distance from the plastic neutral axis to the geometrical centers of the upper part and lower part of the cross-section is y u and y L , we may derive an expression for M , as below:

Plastic Bending of Beam Cross-section

Chapter 8 Buckling/ColIapse of Columns and Beam-columns 149

150 Part II UItimate Strength

8.3.2 Plastic Hinge Load

Let’s consider a fully clamped beam under laterally uniform pressure p, the work done by

external load p may be calculated as,

We = [ p d y = 2 p [ A &&= B PI2

where 1 is the beam length and B denotes the rotational angle at two ends where plastic hinges

occurred The work done by the plastic hinges at two ends and the center is

This sub-section derives the plastic interaction equation for a beam-column due to the action

of combined moment and axial load, for two most used types of cross-sections

Rectangular Section

The rectangular section is characterized by its width b and height h When it is in filly plastic status, the stress in its middle will form the reduced axial load N The stress in upper and lower parts will contribute to the reduced plastic moment M Assuming the height of the middle part that forms reduced axial load N is e, we may derive,

Plastic Interaction Under Combined Axial Force and Bending

Chapter 8 Buckling/CoNapse of Columns and Beam-columns 151

The general solution of Eq (8.68) is:

w = Asin kx i Bcoskx -I- Cx + D

(1) Columns with Hinged Ends

The deflection and bending moments are zero at both ends:

(2) Columns with Fixed Ends

The boundary condition is

(3) Columns One End Fixed and the Other Free

The boundary condition at the fixed end is:

X ~ E I

P, =-

412

(4) Columns with One End Fixed and the Other Pinned

Applying the boundary conditions to the general solution, it may be obtained that:

coeficients and effective length for columns with various boundary conditions A general buckling strength equation may be obtained as below:

Chapter 8 Buckling/Collapse of Columns and Beam-columns

Figure 8.5 End-Fixity Coefficients and Effective Length for Column

Buckling with Various Boundary Conditions

8.4.2

Problem:

Compare type diffkrent types of ultimate strength problems in a table: buckling vs fracture

Solution:

Normally ultimate strength analysis is inelastic buckling analysis of beam-columns, plates and

shells with initial imperfections However, it should be pointed out that final fracture is also part of the ultimate strength analysis The assessment of final fracture has been mainly based

on BPD6493 (or BS7910) in Europe and API 579 in the USA, see Chapter 21 In fact there is a similarity between buckling strength analysis and fracture strength analysis, as compared in the table below:

Example 8.2: Two Types of Ultimate Strength: Buckling vs Fracture

154

Loads

Imperfection

Buckling Strength Compressiodtorsiodshear force

Geometrical imperfection and residual stress due to welding, impacts etc

Design criteria Curve fitting of theoretical

equations (Perry-Robertson, Johnson etc) to test results Analysis Objectives (1) Determine buckling load,

(2) Determine allowable imperfection,

(3) Determine dimensions such

as stiffness, wall-thickness etc

Pari XI Ultimate Strength

id Fracture Strength Analysis

Linear fiacture mechanics

Curve fitting of theoretical

equations(interacti0n equation between ductile collapse and brittle fracture) to test results

(1) Determine fracture load, (2) Determine allowable defect size,

(3) Determine dimensions such as wall-thickness etc

8.5 References

1

2

3

Timoshenko, S P and Gere, J (1961), “Theory of Elastic Stability”, McGraw Hill

Galambos, T.V (2000), “Guide to Stability Design Criteria for Metal Structures”, 4th

Edition, John Wiley & Sons

Hughes, 0 (1988), “Sh, Structural Design, A Rationally Based, Computer Aided Optimization Approach“, SNAME, (previously published by John Wiley & Sons, in 1983)

Part I1 Ultimate Strength

Chapter 9 Buckling and Local Buckling of lhbular Members

In the past 40 years, many kinds of offshore structures have been built and are in service for

drilling and production in the oil and gas industry Semi-submersible drilling units are one of the most commonly used offshore structures owing to their high operation rate and good performance in rough sea However, this type of offshore structure has no self-navigating systems, and cannot escape from storms and rough sea conditions Therefore, the structure

must have enough strength to withstand extreme sea conditions (100 years storm)

Consequently, no buckling and/or plastic collapse may take place under ordinary, rough sea conditions if the structural members are free of damages

On the other hand, the bracing members of drilling units are ofien subjected to accidental loads such as minor supply boat collisions and dropped objects from decks Furthermore, a fatigue crack may occur after a service period Such a damage will not only cause a decrease

in the load carrying capacity of the damaged member, but will also change the internal forces

in undamaged members Consequently, under rough sea conditions, buckling and/or plastic collapse may take place in the undamaged members as well as in the damaged members This can cause a loss of integrity of the structure system From this point of view, the ultimate strength limits and the load carrying capacity of tubular bracing members in serni-submersible drilling units should be assessed carefully

Many studies have been performed during the last decade regarding the ultimate strength of tubular members For example, Chen and Han (1985) investigated the influence of initial

imperfections such as distortions and welding residual stresses on the ultimate strength of

tubular members, and proposed a practical formula to evaluate the ultimate strength Rashed (1980) and Ueda et al (1984) developed the Idealized Structural Unit (element) for a tubular member, which accurately simulates its actual behavior including overall buckling and

plastification phenomena They showed that accurate results are obtained within very short

computation time when applying this model

However, these results can only be applied to tubular members with small diameter to thickness ratios, e.g D/t less than 30-50, which are typical bracing members in jackets and jack-ups Local shell buckling need not be considered in these members On the other hand,

156 Part II Ultimate Strength

bracing members in semi-submersible drilling units have large D/t ratios, e.g between 70 and

130 For such tubular members, local buckling may take place before or after the ultimate

strength is attained as Smith et al (1979) and Bouwkamp (1975) observed in their experiments Therefore, the assessment of the load carrying capacity of such bracing members, both ultimate strength and strength reduction due to local buckling, must be considered However, a systematic study of this phenomenon has not been performed yet

In this chapter, a series of experiments are first carried out using large scale tubular test specimens, which model a bracing member in an existing semi-submersible drilling unit Axial compressive loads are applied with eccentricity Small-scale tubular test specimens are prepared, of which D/t ratios are between 40 and 97, and tested under the same loading conditions Then, based on experimental results, an analytical model is proposed to simulate the actual behavior of a tubular member considering the influence of local buckling Furthermore, the Idealized Structural Unit is developed by incorporating this model The validity and usefulness of the proposed model is demonstrated by comparing the calculated results with the present and previous experimental results

9.1.2

The basic safety factors in offshore structural design are defined for two cases:

-

Safety Factors for Offshore Strength Assessment

Static loading: 1.67 for axial or bending stress The static loads include operational gravity loading and weight of the vessel

- Combined static and environmental loads: 1.25 for axial or bending stress The static loads are combined with relevant environmental loads including acceleration and heeling forces For members under axial tension or bending, the allowable stress is the yield stress divided by the factor of safety as defined in the above

9.2 Experiments

9.2.1 Test Specimens

Dimensions of a typical bracing member in an existing semi-submersible drilling unit are shown in Table 9.1 The slenderness ratio is not so different from that of a bracing member in fixed type jackets or jack-up type drilling units

Table 9.1 Dimensions of Existing Bracing Member and Test Specimen

Length Outer Diameter Thickness

27840 Existing Bracing

Chapter 9 Buckling and Local Buckling of Tubular Members

From this exercise, it may be concluded that local buckling takes place before the hlly plastic condition is satisfied at the cross-section where internal forces are most severe A welded tube

on the market is selected as test specimens, whose collapse behavior is expected to be close to

the above-mentioned bracing member The dimensions of the test specimens are shown in Table 9.1 Their diameter is 508 mm, and their length is taken to be 8,000 mm so that their slenderness ratio will close to that of the existing bracing member The scale factor is 113.5,

and this specimen is referred to as a large scale test specimen The D/t is 78, which is small

compared to that of the existing one However, it is still larger enough for local buckling to be

Figure 9.1 Large Scale Test Specimen and Its End Fixture

The large-scale test specimen is illustrated in Figure 9.1 The tube’s wall thickness is 6.4 mm However, within 750 mm from both ends, the thickness is increased to 10 mm to avoid the occurrence of local collapse near the ends