*Plastic theory of bending (Topic 1)

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Advanced Local Structural Design & Analysis of Marine Structures

* Plastic theory of bending (Topic 1 )

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[Part I] Plastic Design of Structures

– Plastic theory of bending (Topic 1) – Ultimate loads on beams (Topic 2)

– Collapse of frames and grillage structures (Topic 3)

[Part II] Elastic Plate Theory under Pressure

– Basic (Topic 4)

– Simply supported plates under Sinusoidal Loading (Topic 5) – Long clamped plates (Topic 6)

– Short Clamped plates (Topic 7)

– Low aspect ratio plates, strength & permanent set (Topic 7A)

[Part III] Buckling of Stiffened Panels

– Failure modes (Topic 8)

– Tripping (Topic 9) + Post-buckling strength of plate (Topic 9A) – Post-buckling behaviour (Topic 10)

[Theory of Plates and Grillages]

Adv. Local Structural Design & Analysis of Marine Structures (Overview)

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Objectives

The aim of this lecture is:

• To equip you with the knowledge of plastic theory of bending.

Picture from:

https://amarineblog.com/2017/06/25/ship-strength-calculation-part-1/

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Learning Outcome

At the end of this lecture, you should be able to:

– Be aware of the assumptions made in the plastic theory of bending.

– Determine the equal area axis of a plastic cross section.

– Calculate plastic section modulus and plastic bending moment.

– Appreciate the shape factor.

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• Ships and offshore structures are designed to withstand both:

– static loads and

– dynamic loads due to extreme environmental conditions

Introduction

• Maximum design stress σ

_d

in

structures is traditionally limited to a prescribed fraction of material yield stress σ

_Y

as shown in the figure.

A: proportional limit.

B: elastic limit.

C: upper yield point.

D: lower yield point.

E: strain hardening point.

F: ultimate strength.

G: breaking point

• Maximum design stress is sought by means of elastic structural analysis

https://www.britannica.com/science/elastic-limit

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Introduction

• The application of the elastic analysis is justified for most structural members, which suffer from repeated loads,

• because yielding under such loads is less acceptable

• However, even if the stress exceeds yield stress locally in the structure, fracture does NOT always occur because of redistribution of stress.

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Introduction

The plastic design may be applied to:

– Deck structure loaded by local force such as wheel load

– Watertight bulkhead in dry cargo hold loaded by water pressure – Bow structure loaded by wave impact

– Tank wall structure loaded by sloshing impact

Therefore, it is desirable to estimate the collapse load of a structure and to apply the safety factor for it.

– Thus, the safety factor for plastic design is defined as ultimate strength divided by design load while it is the ratio of yield load to design load for elastic design.

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* Plastic Theory of Bending OS T

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• Consider a beam of symmetrical cross-section subjected to bending shown

• Assumptions:

Plastic Theory of Bending

– Plane sections remain plane before and after bending, and normal to the neutral axis.

– Ignoreaxial and shear effect.

– Ideal elastic-plastic stress-strain curve.

– Initially stress free (no residual stress).

– Compressive stress σ_yc in yield equals to tensile stress σ_yt in yield. (Note for high strength steel, σ_yc is generally greater than σ_ytand approximates 1.1 σ_yt) – No instability due to compression.

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in the elastic range

Plastic Theory of Bending

• Strain on the strip d y:

e = (l

₁

– l

_o

)/l

_o

= [(R + y) q – R q ]/(R q ) = y/R where, l

_o

= R q

• Stresses:

R

= Ey

  = 

_Y

when e > yield strain e_Y

y R

E I

M 

=

= This is “Simple Bending Theory” or

“Engineers Bending Theory”. M

dx y EI d

² ₂

=

 = E e

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2

2 2 2

&

1

Ey EAy E EI

L R F A M Fy Ay M y dA

R R R R

L M ML

L L

R R EI EI

M d y d w d y

or M EI

R EI dx dx dx

M E

I R y

q   

q



= = → = → = = = → = =

   

= =   =   =

   

= = → =

→ = =



Reminder – Simple Beam Theory

y R

E I

M = =  This is “Simple Bending Theory” or

“Engineers Bending Theory”. M

dx y EI d

² ₂

= Tip: This will be utilised in Topic 9 (Bending strain energy)

See next page

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[Source] http://homepages.engineering.auckland.ac.nz/~pkel015/SolidMechanicsBooks/Part_I/BookSM_Part_I/07_ElasticityApplications/07_Elasticity_Applications_04_Beam_Theory.pdf

(Additional) Moment-Curvature Relationships

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• For ideal elastic-plastic material (assumption 3), strain distribution and stress distribution on the cross-section under plastic bending are shown below.

e_Y

_Y ^e^Y

e_Y

e > e_Y

e > e_Y _Yt

_Yc

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Plastic Theory of Bending

The force on the strip d y thick and b(y) wide F = A =  b d y

Corresponding moment with respect to N.A.

M = Fy =  b y d y

For equilibrium of force,

= 0



−

t

c

y

 b dy

t et ec 0

et ec c

y y y

Y Y

y y y

b dy E b y dy b dy

 R  ⁻

− −





+



−



=

where A_tand A_c are the plastic areas intensionandcompressionrespectively.

A_e is the area of elastic region. y_e is the distance of the N.A. shifted to a new position.

Y t e e Y c 0

A E A y A

 + R − =

(1)

R

= Ey

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Plastic Theory of Bending

The total moment of resistance M

_R

on the cross-section is

2

t t et ec

c et ec c

et ec

y y y y

R Y Y

y y y y

y

Y t t Y c c

y

M b y dy b y dy b y dy b y dy

A y E b y dy A y

R

    

 

−

− − −

−

= = + −

= + +

   



R Y t t e Y c c

M A y E I A y



R



= + +

where I_eis the second moment of area in elastic regionabout the N.A.

are the distances of the centroids of A_t^cand A_c^yfrom the N.A. ^t

y and

(2)

Y t e e Y c 0

A E A y A



+ R −



=

(1)

t 0

c

y

F y  b dy

= − =

 

R

= Ey



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Plastic Theory of Bending

• When first fibre reaches yield, denoted by M

_Y

, following which with further loading, the extent of area still elastic gradually reduces.

• Plasticity will spread through the cross-section, and the opposite fibre becomes plastic.

• Finally, the whole cross-section is plastic when M

_R

= M

_p

, A

_e

= 0 and I

_e

= 0.

M_Y

R Y t t e Y c c

M A y E I A y

 R 

= + +

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Plastic Theory of Bending

Equal Area Axis

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• Hence, for fully plastic section, equation (1) reduces to A

_e

= 0 and A

_t

= A

_c

=A/2

and the neutral axis finally moves to the axis separating the areas in tension and compression. This axis is known as Equal Area Axis.

• Equation (2) becomes

( )

p Y

2

t c

M =  A y + y

where M

_p

is the fully plastic moment of resistance and A is the cross-sectional area.

Y t e e Y c 0

A E A y A



+ R −



=

(1)

R Y t t e Y c c

M A y E I A y



R



= + +

(2)

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• In fact, final bending collapse will occur at plastic state. Thus, the fully plastic moment M

_p

is the limiting bending strength of the cross-section.

where

• Write:

p Y p

M =  Z

(

_t _c

)

p

A y y

Z = +

2 Z

_p

is the plastic section modulus .

• The state of M

_p

, when rotation occurs without a change in the moment, is known as plastic hinge and fully yielded section.

(3) (4)

( )

p Y

2

t c

M =  A y + y

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Shape Factor

• The reserve of strength beyond the yield moment can be described by the ratio of plastic moment M

_p

to yield moment M

_Y

.

p Y p p

Y Y

M Z Z

 

=  = =

where Z is thesection modulus and equals toI_NA / y_max & f is the shape factor and is function ofgeometry only.

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Rectangular cross-section

6 2

12 ²

3 bd

/ d

/ bd y

Z I

max

NA = =

=

( )

4 4

4 2 2

bd2

d d y bd

A y

Z_p _t _c  =



 

 +

= +

=

• Then, f = 6/4 = 1.5, which indicates the section carries 50% more load than yield load.

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Shape Factor

• The safety factor of plastic design is denoted as f

_p

and expressed as:

where M_d is maximum design bending moment f_e is thesafety factorofelastic design.

• Thus, the safety factor of plastic design is the product of shape factor and the safety factor of elastic design.

e d

y y

p d

p

p f

M M M

M M

f = M = = f

[Source] https://www.abbottaerospace.com/whats-new/cozzone-beam-section-shape-factor-k/

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[Part I] Plastic Design of Structures

– Plastic theory of bending (Topic 1) – Ultimate loads on beams (Topic 2)

– Collapse of frames and grillage structures (Topic 3)

[Part II] Elastic Plate Theory

– Basic (Topic 4)

– Simply supported plates under Sinusoidal Loading (Topic 5) – Long clamped plates (Topic 6)

– Short Clamped plates (Topic 7)

– Additional (Low aspect ratio plates, strength & permanent set)

[Part III] Buckling of Stiffened Panels

– Failure modes (Topic 8) – Tripping (Topic 9)

– Post-buckling behaviour (Topic 10)

[Theory of Plates and Grillages]

Adv. Marine Structures / Adv. Structural Design & Analysis (Next class)

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Kam Sa Hab Ni Da

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