why plastic design

8/10/2019 Why Plastic Design

1/28

Y

PL STIC

DESIGN

by

Lynn S

Beedle

Prepared

for delivery to

ISC USC Conference on

PL STIC

DESIGN IN STRUCTUR L STEEL

Los

Angeles California

CONTENTS

1

INTRODUCTIONS

2

STRUCTUR L

STRENGTH

3 HISTORIC L

DEVELOPMENT

4 FUND MENT L CONCEPTS

5 JUSTIFIC TION

6

IN DEQU CY OF

STRESS S DESIGN CRITERION

7 CONCLUDING

REM RKS

Fritz Engineering

Laboratory

Department

of

Civil

Engineering

Lehigh University

Bethlehem

Pennsylvania

November 1956

Fritz Laboratory

Report

No 205 52


2/28

1. I N T

R O D

U T ION

S te e l

possesses

d u c t i l i t y - - A unique

property t hat no o t h e r

s t ruc tura l

m a te ri al e x hi b it s

i n

q u ite

the

same way

Through

.ductil i ty

structural

s teel

is

able to absorb la r g e deformations beyond th e e las t ic

l i m i t

w itho ut th e d a n g e ~

o f f r act ur e.

Although

t h e r e ar e a few i n s t a n c e s where

conscious

use

has

been made

o f this p r o p e r t y by

an d

la r g e

the

engineer ha s n ot been a b l e to ful ly e x p l o i t

this ~ e a t u r e

o f

duct i l i ty i n s t r u c t u r a l s tee l s a result o f these ~ i m i t a t i o n s

i t

turns

out

t h a t

we

have been

making

a

considerable

sacr i f ice

o f

economy

Engineers have k0

wn o f

t hi s duct i l i ty

for y e a r s an d s in c e

the

1920 s

have been attempting

to

see i f some conscious

use

c o u l d n t be made o f

t hi s

property

in

design.

P l a s t i c

design

is

th e

real izat ion

o f

t h a t

goal.

e

a re

now equipped

to

apply t hi s new design concept to s ta t ica l ly loaded

frames

o f

s t r u c t u r a l s tee l for

continuous

beam an d s i ngl e

s t o r y

building

frames continuous

over on e

or

more

spans.

This

category

accounts

for

a

very

s u b s t a n t i a l

p o r tio n

o f

th e

to ta l

tonnage o f f a b r ic a te d structural s t e e l . p r o d u c e d

annually

in the

United ~ a t e s Not only are we equipped these techniques

have

a l r e a d y been

applied

in Europe

an d

during

this

p a s t summer a s t r u c t u r e was designed

according

to th e

plast ic

methods an d a c t u a l l y e r e c te d in

Canada.

e have only been able to

achieve

t h a t goal--namely t he p ro du ci ng o f

a p r a c t i c a l

design

method--because

two

important

conditions

were

sa t is f ied

Firs t the theory concerning

the plas t ic behavior

o f

continuous

s tee l frames

has

been systematized an d

reduced to

simple design

pr oc e dur e s . Secondly

every

con-

ceivaQle f a c to r t h a t might tend to l i m i t th e l o a d - c a r r y i n g c a p a c ity to something

l es s than

tQat

p r e d ic te d by th e simple plast ic theory has been

in v e s tig a te d

an d

r u l e s

have been formulated

to

safe-guard

a g a in s t such

f act or s .


3/28

205.52

-2

t .is our objective in th is conference to present to

you

the a v a i l a b l e

methods o f p l a s t i c a n a l y s i s th e d es ig n p ro ce du re s t h a t

have

been worked

out

using these

methods

t he e xp er im en ta l

v e r i f i c a t i o n

of

these procedures

by

a c t u a l

tes t resul ts an d .t o cover in the

time

t h a t permits

the

s ec on da ry d e si gn consid

e r a t i o n s t h a t so o f t e n are

s tu mb li ng b lo ck

to an y

new

design technique. I n

.short we

hope

to stimulate

your f u r t h e r study o f th e topic.

-If

we

do

n ot

suc

ceed

in

demonstrating

th e

sim plicity

o f the p l a s t i c method please remember t h a t

you d i d n t

master

a l l you know

about

present design methods

in

few one-hour

l e c t u r e s .

t is the purpose o f

t hi s

p a rt ic ul ar t al k to describe the fundamental

c on ce pt s i nv o lv e d i n p l a s t i c

design

to just i fy i t s a p p l i c a t i o n to

s t r u c t u r a l

s t e e l

frames an d t o d em on st ra te that

some o f

t he c on ce pts a re a c t u a l l y

p a r t

of

our

present

design

procedures.

o

some

o f

you

these concepts

m y

be

new

Others

of

you

have had

some

experience in

solving

problems. t won t h u r t an y

of

u s h o w e v e r t o have

definitions cl ear l y

in mind

an d

these

a re

included in

a n a pp en di x.


4/28

205.52

2.

S T R U C TU R A ST R EN G T H

Functions

of Structures)

1. Limits

of Usefulness

The

design of any engineering structure,

be

ita bridge

or building,

is sat isfactory

i f i t

can

be bui l t

with the needed

economy and

i f throughout

i t s

useful

l i fe i t

carr ies i t s i nt ended loads

and

otqerwise performs i ts in

tended function s already mentioned, in th e p roce ss of selecting sui table

members

for such

a

structure, i t 1s necessary to

make a general analysis

of

structural

strength

and

secondly t o

examine

ce rt ai n de ta il s

to assure

that

local

fai lure

does not

occur.

The

abi l i ty to carry the load

may be

termed s tructural strength .

Broadly

speak ing, th e

s t ructural strength or

des ign load of

a s teel frame may

be

determined or controlled

by a number of

factors,

factors that have

been

c al le d l im it s of structural usefulness . These

are:

f i r s t

attainment

of yield

point

.stress

conventional

design),

bt i t t l e

fracture,

fa t igue instabi l i ty

de

f lect ions, and finally the attainment of

maximum

plast ic

strength.

Plast ic Design s n Aspect of

Limit

Design

trict ly speaking,

a

design

based

on

anyone

of the

above-mentioned

six

factors

could

be referred

to as

a

l imit

design ,

a lthough the

term usual

ly

has

been

applied

to the

determination

of

ultimate

load

as l imited by buck

l ing

or

maximum

s t r n g ~ h

1)

PL STIC

DESIGN

as

an

aspect

of

l imit

design

and

as

applied

t o con tinuous

beams and frames embraces,

then, the las t of the l imits

-- the

attainment

of

maximum plast ic

strength.

Plast ic design, then,

is

f i r s t a design on

th e

basis of

the

maximum

load

the structure

wil l carry as determined from an analysis o f stre ng th

in

the

plast ic

range

that

i s

a plast ic

analysis) ,

Secondly

i t consists

of a


5/28

205.52

-4

consideration by r ul es q rf ormu la s of certain factors that

~ m i g h t

otherwise

tend to prevent

the

s t ructure

from attaining

the computed maximum load. Some

of

these

factors

may be

present

in

convent ional design.

Others

are associated

only

with th e plas t ic behavior o f t he -s tr uc tu re . But the unique feature of

plast ic

design is

that

the

ult imate

load rather than the

yield s t ress

is re-

garded a s th e d esign

cr i te r ion .

I t has long

been

known

that whenever

members are

r ig id ly connected

the structure has a much greater

load-carrying

capacity than

indicated

by th e

elast ic

stress

concept.

Continuous

or

r igid

frames

are

able to

carry

in -

creased loads above : f i r s t

yield

because

-structural steel has

th e capacity

to

yield in

a ductile manner with no loss

in

strength;

indeed

with f requen t

increase in

resistance.

Although the phenomenon will be

discribed

in

complete

detai l

la ter ,

in general terms what happens is this: As

load is

applied

to

th e

structure the cross-section with

th e greatest

bending moment

will

event-

ually reach the

yeld

moment Elsewhere the

structure is e la st ic

and

the

peak

moment values are less than

yield.

As

load is

added a

zone of

yielding

develops a t the f i r s t

cr i t ica l sect ion;

but due

to

the ducti l i ty of s teel , the

moment a t that

sect ion

remains about constant . The structure therefore cal l s

upon i t s l e ss -heav ily s t ressed portions to carry the increase in load.

v e n t u a ~ l y

zones

of

yielding

are formed a t

o th er s ec ti on s

unti l

th e moment capacity has

been

used up a t a l l necessary cr i t ica l sections.

After

rea ch in g th e maximum

load

value

the

structure

would

simply

deform

a t

constant load

3. Elast ic Versus Plast ic

Design

We

can t repeat too

often

th e dist inct ion

between

elas t ic design and

plast ic design. In conventional elast ic design a member is selected

such

that

As shownrobably the most common equation we now use is S

orking

load.

th e maximum allowable stress

i s

equal to 20 000 pounds

per

s qu are in ch a t the

M

0

.i n

Figure

such

a beam

has

a

re se rv e o f

strength of

1 65 i f

the

yield

point


6/28

205.52

-5

s t r e s s i s

33,000

pounds per square

inch.

Due

to

the ducti l i ty

of s t e e l there

i s a b it more reserve (14

for

a wide

flange

shape). So th e to ta l

inhe r e nt

overload

fa cto r o f

s a f e t y

is equal

to

1.88.

In p l a s t i c

design, on the

othe r

hand;

we s t a r t . w i t h t h e ultimate

load Fo r

i f we analyze

an indeterminate s t r u c t u r e

we

w i l l find

t h a t we can

compute

the ultim a te

load much e a s i e r than we

ca n

compute

the

y i e l d load. So

we

m ultiply

the working load, P

w

by the same load factor 1.88) an d then se

l ec t a member t hat w i l l

reach

t h i s factored

load.

I f

.wetook

the

tro ub le to

draw

the

l oad. -vs-

e f l e t ~ o n u r v e

for the

r e s t r a i n e d beam we would

g et the

curve shown in Figure

1.

I t ha s th e same

ultimate load

as

the

conventional

design of

th e simple

e ~ n d the member is

elas t ic

at

working

load. The i mp or ta nt t hi ng

to

note is t h a t the fact or

of

safet y is the same in the p l a s t i c design

o f

the indeterminate

s t r u c t u r e

as i t

is

in

t he c o nv en ti on a l

design of

the simple

beam.

While

the r e

are

o th er fe atu res h ere, t he i mp or ta nt thing

to

g e t

in

mind a t this

stage

is t h a t in conventional procedures we find

the

maximum

moment under the working

load

an d

s e l e c t

a member such t h a t

the

maximum s t r e s s

i s not

great er than 2 k si

: the

fact or o f s a fe ty

inhe r e nt

in

t h i s procedure

i s

1.88 i n . t h e case

o f

a simple beam); on th e

othe r

hand in p la sti c design we

multiply the

working

load

by F = 1.88 an d se lec t a memberwhichwill

just

support the

ultim a te

load.

Already

we have used two new terms: l i m i t

design

an d p l a s t i c

design.

L e t s

include them in .our l i s t

o f d efin i t ions

see appendix).


7/28

205.52

_

~ I ~ S T R I C

AL

DE V E

LO

P MEN T

-6

The

concept

of

de.sign

based

on

ul t imate

load

as

the

cri ter ion

i s more

than 40 years old The application of

plast ic

analysis

to

s tructural design

appears to

have

been in i t ia ted by Dr. Gabor Kazinc.zy, a H.ungarian, who

published

results

of his te.sts of clamped

girders

as

early

as 1914. (2 ) He also

suggested

analyt ica l procedures s imil ia r to thoSE: now current,

and

designs of

apartment-'

type

buildings were

actual ly

carr ied

out .

In

his Strength

of Ma te ri al s(3 )

Timoshenko

refers

to

early

suggest-

ions to ut i l i ze ultimate load capacity in

the

plast ic range and s ta tes

such a

procedure appears

logical in the case .o f s teel structures

sub

mitted to the action of stat ionary lo ad s, sin ce in such cases a fa i lure

owing to the fatigue of metal is excluded

and

only fai lure due to the

y ie ld in g o f metals has

to

be considere.d.

Early tes ts in Germany

were

made by Maier-Leibnitz(4) who showed

that

the

ult imate

capacity was not affected by s ett lemen t o f supports of

continuous

beams. so

doing he corroborated

the

procedures previously

developed

by others for

the

calculation

of

maximum load capacity. The effor ts of Van

den

Broek(l) in this

country

and J.

F.

Baker(5) and his

associates in

Great

Britain to actual ly ut i l i ze

th e

plas t ic reserve strength as a design cri . terion

are

well know (Prof, Van den

Broek was teaching about duct i l i ty in 1918). ProGress in theory

of plas t ic

s t ruct

ural

analysis (par t icular ly

that at Brown

University) ha s

been

sunnnarized

by

Symonds

and

Neal(6).

A

survey

of

design

t rends,

by

Winter (7 ), d is cuss es

brief ly

many

of the

factors

germain .to plas t ic design.

For

more

than ten years

th e

American nst i tu te of

Steel

Construction,

th e Welding Research

Council ,

th e Navy

Department, and

the

American

Iron

and

Steel nst i tu te

have

sponsored studies a t Lehigh Universi ty.

These

studies

have

f ea tu re d not

only the

ver if ication of this method o f a na ly sis

through

appropriate


8/28

205 52

tes ts

on

large st ructures but have given

part icular

attention to the conditions

that must bernet to

s tisfy

important

secondary

design

r e q u J r e m e n t s { ~ .

u h

of th is w ill

be discussed

l t r in

the confe rence


9/28

20.5 52

4. F U N D A

ME

.N T L 0 N C E P T S

With this background of the

functions of

a s t ructure ,

le t

us s t a r t

with the

fundamentals

of

the

simpleplast : ic

theory,

-8

1.

Mechanical

Properties

outs tanding property of s tee l which as

already

m e n t i o ~ e d sets

it apart from

other

struc.tural materials ,

is the amazing

ducti l i ty

whd.chi t

possesses,

This

is

characterized

by Figure 2.

In

F1gure 3 are shown

par t ia l

tensi le

s t ress s t ra in

curves for a number of different

s tee ls

Note

that when

the

elastic.

l imi tis

reached,

elongations

from

8

to

15

times

the

elas t ic

lim

t

take place without any decrease 1n load,

Afterwards

some increase in strength

1s exhibited

as the

material strain

hardens.

Although the f i r s t applicati.on of plas t ic design i s to structures

f ab ri ca te d o f

s t ructural

grade s tee l it is no less applicable to s teels

of

higher

strength

as

long

as

th ey poses s

t he neces sa ry

duct i l i ty

Figure

3 a t-

tests to the

abi l i ty

o fa wide

range of

steels to deform

plastically

with

c h r c t ~ r i s t i c s simili .ar

to

A-7 s teeL

I t

is

important to bear

in

mind that the strains shown

in

this

fig

ure are real ly

very smal l,

As

shown

in

Figure

4,

f or o rd in ary

structural

s tee l fin al f ai lu re bY,rupture occurs only af ter a

specimen

has

stretched

some 15 to 25 times

the

maximum

strain

that is encountered

in

plas t ic

design.

Even

in plast ic analysis ,

a t

ultimate

load tb e

cr i t i ca l strains

wil l not have

exceeded

aboutL 5 longation. Thus the use of ultimate.

strength

as th e d esign

cri ter ion

s t i l l l eaves ava ilab le .

a major portio n o f the

reserve

duct i l i ty

of

s teel

which

can

be

used

as

an added margin of safety, Be.ar

in mind that th is

maximum

strain

of 1 5

i s

a

strain

a t ultimate load

in the

structure--not a t

working

load. In most cases

under

working

load

the s t rains

wil l

s t i l l be below


10/28

205.52 -9

the elast ic l imit . We must dist inguish also between

the

term u1ti.mate

10a.d

as we u.se i t

here

to mean. t:he maxi mum load a st.ructu re

.will

carry,

as

dist inct

from th.e ultimate

strength

exh.ibited by

an

acceptance te.stcoupon

So ,

we

wil l

add t ha t d e fi ni ti on to

our

l i s

to

2.

Maximum Strength of Some Elements

On the basis

of

t.he duct:i.lity of s tee l (characterized by Figure 4)

we

can now quickly calculate the

maximum carrying

capacity of

certain

elementary

structures .

.

As our f i r s t example take a tension .member suc h as an eye bar

(Figure

5) .

I f we

compute

the ,stress

we

find t4at

r

P

I f we

draw th.e

l o d v e r s u s d ~ f l e c t i o n r e l t i o n s h i p i t

wil l be

elast ic

unti.l

th e yield point is

reached.

As

shown in Figure 5 the

dq:flection a t

the

elast ic

l imi t i s given by

y ~

o

;Since

the

stress

dis t r ibut ion

i s

uniform

across

the

section, unrestricted

plast ic flow ,will se t in when

the

load r eaches

the

value given by

This

i s

therefore,

the ultimate load. I t s th e m a x imum l oad . the structure

wil l

carry

w ith out the onset

of

unrestricted plast ic ftow.

As

a second

example we

wil l

consider

the

three-bar

st.ructureshown in

Figure 6.

I t

i s

selected

because

i t

is an

indeterminate

structure

since

the

s ta te of

stress

cannot

be

determined

by

s ta t ics

Consid.er

f i r s t th e

elasUc

sta te .wewil l f i r s t

take

equilibrium and obtain:


11/28

205.52

Next we would consider continuity

and

use the condition

that

with

a

r ig id

cross

bar

the to ta l displacement of bar I will be equal to that of bar

2.

Therefore

-10

With t hi s r el at ionsh ip

between T

I

and

T

obtained

by

continuity condition,

we

then find

from

th e equil ibr ium condi tion

that

The

load t

which

the structure will f i r s t yield

may

then be determined

by

substi tuting

.in

this

exp re ss ion the maximum load whichT

.can reach, namely OyA

Thus

P T

x A

y r ry

f we were interested

in

the

displacement

t

the yield.

load

would

bedetermin-

ed

from

Oy

= y L

=

E

ow

when th e member is p rt i l ly plastic

deforms

as

i f

were a two-bar

structure except

a

constant force .equal to

Oy A

is

supplied by Bar the member

is

in

th e plastic range . This

si tuat ion continues

unt i l

th e load re aches th e

yield value in the

two oute r b ars .

Notice how easi ly

we

can compute th e ultimate

load:

p = 3cr A

u y

The continuity

condition need

not

be considered

when

the

ultimate

load in

th e

plastic range

is being

computed

The

load deflection relat ionship for

the structure shown

in Figure

6

is

indicated

t the

bottom.

Not unt i l

the

l oad reaches

that

value

computed

by

a plastic

analysis

did .the deflections commence to increase rapidly. The de -

flection .a t ultimate load can be computed from:


12/28

205.52

The three essential

features

to

this

simple plastic analysis are

as fo llows:

-11

1. each

portio n o f

th e structure each bar) reached

a

plas t ic yield

condition

2. th e equilibrium

conditiQnwas

sat isf ied

a t

ultimate

load

and

3. there was unrestricted plas t ic flow

a t the ultimate

load

These

same

features

are

a ll that are required to complete th e plastic analysis

of an indeterminate

beam

or

frame,

and in fact, th is simple example i l lus t ra tes

a l l

of

th e essential

fe atu re s o f

a plas t ic analysis.

3. Plast ic

Bending

.Let

us now trace th e stages of

yield

stress penetration in

a

rectang-

u1ar beam

subjected

to a

progressive increase

in

bending

moment. At

the top

of

Figure 7 is

replotted

th e s t ress s t ra in

relationship

We

retain th e assumption

that

bending

strains

are

proportional to the

distance

from the neu tr al axi s.

At Stage 1, as shown in

the

next l ine

of

Figure 7 the

strains have

reached th e yield strain

When more moment

i s app li ed say

to Stage

2), the

extreme

fiber strains

are

twice the elas t ic l imit value. The

si tuation is

sim-

i lar

for Stage

3

max 4y). Finally

a t Stage

4

th e extreme

fiber has s t ra in

.

ed

to

Est.

What

are

the

stress d is tr ibu ti on s t ha t

cor re spond to

these

s t ra in

diagrams? These are

shown in

the next l ine of Figure

7

s long as the strain

is greater

than the

yield value, notice from the s t ress s t ra in curve that th e

stress

remains a t

cry The

stress

distr ibut ions,

therefore follow

direct ly

from

th e assumed strain distr ibut ions.

s

a

l imit we obtain the stress b1ock --

a

rectangular pattern.which

is

very

close

to

the

stress

distr ibut ion

at-Stage

4.


13/28

205.52

-12

One term contained in

Figure

7 and

not

as

yet defined

.is

th e

curva

ture, 0. This

is the rel t ive rotat ion of

two

sections

a

uni t

distance

aparto

Just as i t is basic to the fundamentals of el s t ic analysis ,

the

moment-curvature

M-0 relat ionship

is

a basic concept in plast ic

analysis .

40

Moment-Curvature Relationship

In

the upper

portion

of Figure

8

is

shown, then, the

moment-curvature

relat ionship

for

the

rectanglar

beam

jus t

considered

in

Figure 7.

The numbers

in the circles

cor respond to

the

stages of

Figure

7. Notice that-Stage 4

approaches

a l imit and represents the

complete

pl st ic yield

of

the cross

section

o

There i s a 50 increase in

strength

above the computed

el s t ic

l imit (Stage 1

due

to

pl st if ic t ion

of th e cross-section. This represents one of

th e s ou rces

o f re serv e

strength

beyond th e

el s t ic

l imit of r igid frame.

(We ve

used

another new term that we

can

now add to

our l i s t

of defini

t ions. P last i f icat ion

is the

development of

ful l

pl st ic yield of

the

cross

section.

t

is

th e p roce ss

that

takes

place

as

we

move

from

th e

el s t ic

l imi t

to

ful l

yield.)

The moment-curvature relat ionship for a

typical

wide flange beam i s

shown in

the lower

portion in

Figure

8 (the

s t ress

~ s t r i u t i o n s cor respond to

th e let tered

points

on the M 0

curve) t

should be noted that the

reserve

strength

beyond

the el s t ic

l imi t

is smaller than for the rectangle. The

aver

age

value

for

al lWF

beams

is

1.14.

Correspondingly

there

is

a more

rapid

approach to when

compared

with th e rectangle. As a

matter

of fact , when

the curvature is

twice

th e

el s t ic

l imit value (Stage c) the

moment

has reached

wi

thin

of the ful l

plas

t ic moment.

5.

Plast ic

Modulus

h Shape Factor

f and

Plast ic

Hinge

When

the

yield

point is

reached

in

.bending,

the

to t l resistance

pro-..fided by the

cross-sect ion,

as shown in.Figure 9, is given

by

(Stage 1):


14/28

205.52

M S

y Y

where.S is the

sect ion

modulus. When

t h e ~ e c t i o n e c o m e s

completed plas t ic

-13

the

resist ing

moment

approaches

a maximum .valuecal led

the

plas t ic

moment, Mp

The s t ress block

for

this

case

is

also

shown in

Figure

9.

,The

magnitude of

this moment may

be

computed direct ly from this s t ress dis t r ibut ion.

t s

. A

equalto the

couple

created by the

tensi le and

compressive forces

. .

. 2

Thus,

.M

p

a ~ y 2

y 2

The

quant i tyAy

is

cal led

p last ic modulus and

is

denoted

by

Z;

therefore

The

plast ic

modulus is

equal

to the combined s ta t ica l moments of the

cross-

section areas above and below

the

neutral

axis ,

since

the

s t ress

a t

every point

on these

areas is

the

same. We have two additional terms to add to our d i c t i o n ~

ary: . the plas t ic moment and

the

plas t ic

modulus ).

The ra t io

of maxi mum

.moment

of resistance

M)

to

the

elas t ic

moment

of resistance

M

is dependent

only upon the form

of

the

cross-sect ion. t s

y

therefore cal led the shape fac tor , f, and may be

computed

from

~ = f

M S

Y

For

wide

flange shapes the

average

value of f is 1.14, vary ing f rpmalow

1.09

to

a

high

of

1.

22.

In

Figure

10

are

shown a number

of

shapes

with th e

corre;;.

spPriding

value of f .b eing t abu la ted .

You wil l remember from Figure 8 that when the moment approached the

limi

ting

v alu e th en rotat ion increased wi thou t l imi t . . In other words the

member was

free

to

rotate through

many times the angle that the

samelength.of

beam could

bend

elas t ica l ly . Actually the sect ion under maximum moment is

act-

ing jus t

l ike a

hinge except

that

t

is

res t rained

by

a

constant

moment,

~

We


15/28

205.52 -14

cal l

this

phenomen.on a

plastic

hInge..

and in our

later

calculations

we

wi

11

as-

sume

that

the

hinge

forms a t discrete points of

zero

length. Thus th e member

under bendi.ng

behaves e1asUcally

up

to the

ful l plasti.c moment, M ,

and then

is free to rotate

at

thatconsta.nt

moment.

6.

Redis tr ibution of Moment

Now

i f

the shape factor were

a ll

. that were

involve.din p s t i ~

de-

sign .we

wouldn t

be discussing

th e

topic a t al l A second factor contributing

to the

.re.serve

o f s tr engt h

(and

usually to

a

greater

extent

than the

shape

factor) is

called

redistrlbpti.on .of moment t is que to th e action of

plas

ti.c

hinges. As load is added to

a s

truc ture,

eventually

th e

plas

t ic

moment

is

reached

at a cr i t ical

secti.on. That

cr i t ical

section

is

the one that

is

most

highly stressed

in th e

elast ic

range. As further load

is added,

this

plastic

moment value

is main ta ined while the

.section

rotates . Other less-

highly-stressed sections maintain equ il ib rium with

th e

increased load by a pro-

porti.onate increase. This process of redistr ibut ion of

moment due

to succes. sive

formatiol1

of

plastic

hinges

continues unti l . theult imate load is

reached.

Tbi.s s exactly

th e

process

that

took place

in

the case of the th ree-

bar truss except that

forces

in ste ad o f

moments

were involved. When

the

force

in.Bar 2 reached

the yield

condi.tion i t remained

constant

there while

the

forces

continued

to

in.crease in

Bars 1 and

3.

The

ultimate load

was

reachedwhen.all

c d t ical bars became plas ti.c

.Let s

i l lus t ra te th e

phenomenon

now

with

th e

case shown

in.Figure

11 ,

a fixed ended beam with .a concentrated load

off-center.

As the loadP is in-

creased the

beam

reac.hes i t s elast ic l imit a t th e

lef t

end (Stage 1).

The

mo-:

menta a t s ~ c t i o n s B au, dCare le ss t.han

the

maximum moment. .Note

that in

this

example we

will

consider

th e idealized M 0 relationship as

shown

in th e

lower

lef t hand portion.

The dotted c,urve shows

th e

more .precise . behavior) .


16/28

205.52 -15

As the

load is

further increased, a

plas t ic hinge

eventual ly forms

a t

-SectionB

The formation of th e plas t ic hinge a t A wil l p erm it the beam to

ro-

ta te there without absorbing any

more

moment.

Referring

to

the load-deflection

curve immedia te ly

below

the moment diagrams the deflection is

increasing

a t a

greater rate,.,

Eventually,

a t Stage 3, when the load

is

increased suff icient ly to

form a

plast ic

hinge a t section C, a l l

o f

the avai l .iblemoment

c apac it y o f

the

beam wil l have been exhausted and th e

ult imate

l oad r eached .

I t

is

evident

from

the

load-deflection

curve

shown in

the

lower

par t

of the

figure

that t he fo rmat ion

of

each plas t ic hinge acts to remove

one

of

the

indeterminates

in the problem

and

the

subsequent

load-deflection

relat ion

ship wil l be

that of

a

and

simpler) s t ructure . In the

elas t ic

range,

the

deflection under

load

can be determined for

t he compl et el y elas t ic beam. Star t

ing from point 1 the segment represents the load-def lec t ion .curve of the

beam in sketch b loaded within the

elas t ic

range. Likewise the load-deflection

curve of the beam in ske tchc looks simi

l ia r

to the portion 2-3.

Beyond

Stage

3 th e beam will continue to deform without an increase in

load

jus t l ike a

l ink

mechanism i f the

plas t ic hinges were

replaced

by

real

hinges

.. We add

the word

mechanism

to our l i s t of terms because we wil l

be

referr ing ,t o i t frequently.)

The fundamental

concept s involved in

the

simple plas t ic

theory may be

summarized

as follows and indeed

are

demonstrated by Figure :

1.

Plast ic

hinges wil l

form in s tr uc tu ra l

members a t

points

of

maximum

momen

t .

2. A plas t ic hinge i s characterized by

large

rotat ion a t the

plast ic moment

value

= cry Z

3.

The formation of plast ic h inge s a llows a subsequentredis

tr ibut ion of moment

unt i l

is

reached

a t each

c r i t i ca l

maximum )

section.


17/28

205.52 -16

4. The maximum load

.will be

reached when a sufficient number

of plast ic hinges have formed to

create

a mechanism.

6. Introduction to Methods of Analysis

.Onthe basis of the

principles

we have jus t discussed

perhaps

you

have already

visualized

how

to

compute the ultimate load: Simply sketch a

moment diagram

such

that

plast ic

hinges

are formed

a t a

sufficient

number of

section

to allow mechanism motion. Thus

in Figure

12 ,

the

bending moment

diagram

f or t he uni fo rm ly -loaded, fixed-ended

beam

would

be drawn such

that.M

p

was reached

.at the

two ends and . thecenter .

In

this way a

mechanism i s formed

Byequilibrium,

WuL

8 =

2M

P

6M

p

L

How does this compare

with

the load a t f i r s t y ie ld t

the

elas t ic

l imit

see dotted

moment-diagram

in

Figure

12

we

know

f rom.aconsiderat ionof

continuity

that th e center moment

i s one-half

the end moment Thus,

WL

= My M

y

=3

My

2 2

W

Y

Therefore

th e

reserve

strength due

to redis t r ibut ion

of moment

is

16 Mp t

=

12

My/t

4 M

p

3

My

Considering the average shape factor of WF

beams, the

to ta l reserve strength

=j 1.14

= 1.52

For

this

part icular

problem,

then, the

ultimate

load

was 52 greater than

the

load a t f i r s t yield,

representing

a considerable margin that is disregarded

in


18/28

205.52

convent ional design.

=

There

are

other

methods

for

analyzing

a

structure

for

t s

ultimate

load, in

part icular

th e

Mechanism Method

to bedesc.ri.bed later ,

which s t r ts

out w ith an assumed mechanism in ste ad o f an assumed moment diagram.

But in

every

method,

there

are

always

these

two important features: 1

th e

formatlon

of plastic

hinges

and 2 the

development

of

a mechanism.


19/28

205.52

5. J l lS T

I F I C

A

T I O N Q F

P L .AS T I C D E ~ GN

The

reasons now

should be

evident as

to why

theappl ica . t ionof

plast ic

analysis to structural

design

i s just if ied theyare

l .t:conomy

2 SimpHci ty

3 . Rationali ty .

A word, now about each of these:

Since

there

is considerable

reserve o f s tr engt h beyond the elas t ic

-18

l imit and

since

the corresponding ultimate load

may be computed qu it e accurately,

then

atructural

members of

smaller size .will

adequately

s uppo rt th e

working

loads

when

design is

based on

maximum

strength.

n analysis

based

upon ultimate

load

possesses an inherent simplicity

because

the

elas t ic .condition

of

continuity need no

longer

be

considered.

This

was

evident

from ou r

cons idera tion of the three-bar

truss

Figure

6 and

th e

fixed-ended beam Figure 12 .

Final ly t he concep t is more rat ional .

y

plast ic analysis t he eng inee r

can determine with

an accuracy that

far exceeds h is p re se ntly a va il ab le t ~ h n i q u s

the

real maximum

strength

of

a

s t ructure . Thereby

the factor of

safety

has more

real

meaning than

a t present.

I t

is

not unusual

for the factor of safety to

vary

from

1.65

up

to

3

or

more

for structures designed acco rd ing to convent iona l

elastic methods.


20/28

205.52

6.

INAD,E

gu A Cy

A S T HE D E S I G N

OF

S

T R B S S

C R r T ER I ON

-19

The question immedi ate1y

ar ises

can t. one

simply ch ange the

allowable

stress

and

retain

the

present stress concept.? W h ~ l e the.oreti,cal1y

possible,

the pract ical

answer is n o .

I t would

mean

dLfferentworking stre.ss for

every

type

of structure

and would vary

for dIfferent loading conditions.

It, is true that in

simple

struc.tures the concept of the .h.ypothetical

yield

point

as a

l imi t

of usefulness is rat ional . This is

because

the

ultimate

load c apac ity o f

a simple beam

i s but

10

to 15

greater than

the

hypothetical

yield

point, and deflections s t a r t

increasing r at he r r ap id ly a t

such.a load.

Whi le

t would

seem .10gica1 to extend

elas t ic

s t ress analysis to indeterminate

s t ructures ,

such procedures

have

t ended to over- emphas ize

the

importance of

s t ress

rather than strength as a guide

in engineering

desi,gn

and have

introduced

a complexity that now seems unnecessary for large number

of

s t ructures ,

Actually

the

idea

of

design on the basis

of

ultimate load rather than

allowable

stress is a return to

the rea l i s t ic

point of view that

.had

to

be adopt

ed

by our forefathers

in

a

very

crude way because

they did not posse ss

knowledge

of mathematics and s ta t ics that would allow

them

to compute stresses.

As a matter of

fact ,

to a greater

extent

than we may real ize,

the,maxi

mum

strength

of

a

structure

has

always been

the

dominant

design

cr i te r ia

When

the

permissive .working

stress of 20 ksi has led to designs that were consistent ly

too

conservat ive,

then

that

stress has been

changed

Thus t he b en efits o f

plas t ic i ty have been

used

consciously or unconsciously

inde.sign.

I t

is

also

patently evident to most

engineers

that present

design

proc:dures completely

disregard l oca l over -s tr e ss ing

a t points

of stress-concentrat ion,

etc .

Long

experience

with

similar

structures so designed shows that this is a safe proce-


21/28

205.52

-20

dure. Thus, the stresses w calculate for design purposes are not true maximum

stresses a t a l l

they

simply

provide

an

index

fa ' struiC,tural

design,

In , the remainder

of these remarks a number of examp1 s

will be given

in

which ,the ducti l i ty of s tee l has bee.n

counted upon

(knowingly or not) i.n

present design,

,Plast ic

analysis ha.s not

gener.ally

been used as a basis for

determining these pa rt icu la r q e si gnru le s

and

as

a r e s ~ l t

so-called

elast ic

stress

formulas

have been devised i.n a rather haphazard fashion. A rat ional

bas is ' fo r

th e des ign

of a complete

s tee l

frame. (as well as

l.ts detai ls)

can

only be

attained

when the. maximum strength:i.n the plast::Lc

range is

selected as

th e de sign

r i t e r i o n ~

Some examples

in

which w rely upon th e ducti l i ty of s tee l for

sa t i s -

fac tory performance

are

the

following

and are

l i s

ted i.n two categorie,s:

1,

Factors that are.Neglected

(1 )

Neglect o f re sid ua l stresses

in

th e

case

of

flexure.

(2)

Cambering

of

beams

and

the

neglect of

th e

resul t ing

residual

stresses.

Neglect o f e re ctio n stresses.

(4 )

Neglect

of foundation ,settlements.

(5) e g l ~ t of

over-s t ress

a t

points of s t ress -concent ra t ion

(holes, etc .

(6)

Neglect

of bending

,i n angles connected in tension

by

one

leg only.

(7 ) NegleCt

of over

stress a t points

of

bearing.

(8) Design of connections on the assumption of a

uniform

distr ibut ion of st.resses among t.he r ive ts

bolts ,

or

welds.

(9)

Neglect

of the ,difference in s t ress-dis t r ibut ion arising

from

the. canti lever

as

compared

with

the

portal

method

of

wind stress analysi.s.

, I I . . Revisions in working stress due to

reserve

plas t ic strength

(10)

Bending

stress of

30

ksi in

round

pins.

(11)

Bearing

s t ress

of

40

ksi

on

pi.ns

in double shear.

(12) Bending ,stress

of

24

ksi in

framed

s

true

t.ures a t

poi.nts of

in ter ior supports.

Figures

13- 18 are i l lus t rat ive of

items

1 3

8, 10, and 12, above. InFigure

13

i t

is demonstrated,

for

example, that cool.i.ng residual stresses ~ t J h o s e in -


22/28

205.52

-21

f luencewe

n e g l ~ t

and yet wh.ich a re p re se nt in . a l l ro l l ed beams cause

yielding

in the

flange

t ips

even a t

the working load

Structural .members experience yield while being straightened in

the mil l

fabricated

i n a shop

or

forced i nt o po si ti on during erect ion.

Actually t during

these

three operat ions that duct i l i ty of s tee l

beyond

the

yield

point

is

called

upon to th e greates t degree. ijaving permit ted suchyle ld-

ing in

the mill ,

shop

and

f ie ld

there

i s no valid

basis

to

prohibi t

t

there

af te r ;

provided such

yield has

no adverse

ef fec t

upon

the s t ructure .

s

an

example,.

Figure

14

shows how

erect ion

forc.es Vlltroduce bending

moment

. into

a

s tr uc tu re p ri or to the applicat ion

o f e xte rn al

load.

Altboughthe

yield-point load

i s r educed a s

a result

of

these erect ionmoments ,

there

is no

ef fec t

whatever

on the maximum strength.

Consider, next , th e de sign of a r iveted

or

bolted jo in t .

Th e

com-

mon assumption is made that each

fastener

carr ies the same

sh.ear

force. This

i s

true

only for the case

of

two

fasteners . When

more

are

added

(Figure

15),

then

as long

as the jo in t remains elas t ic the outer

fasteners

must carry the

greater proportion of the load. For the example with

four

r ive ts i f each r ive t

transmitted

the same

load, then,

between

r ive ts

C and D one plate .would

carry

perhaps three times

the force

in the other. Therefore t

would s tre tch

three

times

as

much and

would

necessari ly

force

the outer r iver D

to

carry

more

load. The t ~ ~ l forces

would

look something l ike those shown under the heading

uF,:lastic . What eventually happens is

that the

outer

r ive ts )'lield,

redis t r i

but ing for ce s

to the inner r ive ts . There fo re the basis for design of a riveted

jo in t

is

real ly

i t s

ultimate

load and not the attainment

of

f i r s t

yield .

Figure

16 shows a l ine

of

bolts demonst ra ting the d i ff er en t ia l p l as ti c action

in

the

.various

fasteners .


23/28

205.52

~

Figure 17

is

a

further

example and is

concerned

w:i.th the desi.gn of

a round pin. In a simple beam ofWF shape when.the maximum s t ress due to bend

ing reaches the y ie ld p oi nt most of th e u sa ble strength has been exhausted.

However for some cross sect inal

shapes

muchaddi. tion.al load may be carried

wi

t hout excess ive deflections.

The relat ion betwee.n

bending

moment

and curva

ture

forWF and ro und

beams is shown

in Figure 17 .

Tneupper curve is

for

the

pin

the lower

for

a

typicaLWF

beam the

non dimensi.onal

plot

b ei ng su ch

that

the two curves coincide

in

the

elast ic

range. The maximum bending

strength

.of

the

wide flange

beam

is 1.14M

whereas

that

of

the

pin 1.70Myo

The

y

permissible design s t resses according

to AISC are 2 kst for the

:WF and 30

ksi

for the

pin.

Expressing these

stresses

as

rat ios

of yield point s t ress

WF:

i z = 2 =

0.61

cry

33

Pin:

iaz

= 30

=

o

91

cry 33

For a simply supported beam the stresses

m o m n t s ~

and loads a l l bear a

l inear

relationship to one another in th e elast ic range and thus

M

My

=

r

rr

y

Therefore the moment

a t allowab1e.working

stress Mw

in theWF

beam is .61 My;

1.87

.14 My

61

y

for

the

pin on

the

other hand Mv = .91My.What is

t.hetr:ue load facl:tt.t

of safety for each case?

P

inaX

~ x

W F F =

=

Pw

.

Pin:

F

=

P

max

=

~ x

= 1. 7

:Y

=

P

w

.91 My

1.87

The exact agreement between

the

tru e f ac to rs

of safety with

respect

to ultimate

load in the two .cases while somewhat

of

a coincidence is indicat ive of the

fluence of

long years

of experience on the part of engineers which has resulted


24/28

205.52

i n d if fe re nt

permitted working

stresses

for vari.ous

conditions result ing

pract ice

Probably no such analysis as

t he forego ing

. influenced

th e c ho ice

of

d if fe re nt u ni t s tr es se s that

give

ident ical

factors of

sa fe ty wi thva rio ll s

sect ions, nevertheless,

t he choice

of such .stresses

is ful ly

jus t i f ied

on this

basis . h n

years

of

experience and

conunon

sense

have led to

cer tain empirical

pract ices

these

practices

can usually be

jus t i f ied on s ient i f i

basis

The final

example

Figure

18

demonstrates

the

use of 4 ksi t

points of in ter ior

support in

continuous beams and snows how this i s a safe

procedure according .to

pl st i

analysis.


25/28

205.52

.7 .

c a N C L D

D IN

G

R A

RK

-24

As

a

background

for

the

talks that

arE

to follow,

t

has

been

my

purpose to introduce to

you the

simple plas t ic theory, to define

some

terms,

and.to

demonstrate that our

dependence

upon the

duct i l i ty

of s tee l

has

always

been very real Perhaps we can

do no bet ter

than to refer to Figure

1 again,

as

a basis for reviewing what we have

learned

thus far:

1

.Whereas

i ne l a s t i c

design a

sect ion

is selected on the

basis

of a maximum allowable stress at.working .10ad we do not deal

a t

a ll with

th e ove rlo ad

factor of ~ f e t y ~ in plastic design

we

compute

f i r s t

the

ultimate

load P

w

x F

and

select a mem-

ber th at w ill fa i l a t

that

load.

2 . Since eve ry beam

has

a

l imiting

plast ic moment of resistance

~

the

ultimate

load may be computed on the basis that a suf

f ic ient number of plastic hinges

wil l

have formed to create a

mechanism.

3 Dse of maximum strength

as

the

design

cr i ter ion

provides a t

least

the same

margin of re se rv e strength

as is

presently

afforded in the conventional design of simple beams.

4

.

At work ing

load

th e .structure

i.s

s t i l l in the so called elast ic

range.

5 In

most

cases ,

a s t ructure

designed

by.

th e plas t ic methodwill

deflect no more

at working

load

than

wil l a simply-supported

beam designed by conventional methods to support same load.

6 Plast ic des ign g ives promise of

economy

in the

use

of stee l of

savings in th e design

off ice by

virtue

of

l t s

simpli.city,

and

of building frames

more logical ly designed for g re at er over -a ll

strength.

Plast ic design

has

come of age.

Cons,iderable i

terature is available

in

the

r ~ of lecture notes 8 , reference books 9, 10 , and various technical

11

proceedings

.Mr. Higgins and Mr. Estes of th e

American

Inst i tu te of Steel

.Construction are nearing completion of a manual

on

plast ic . designwhlch will

a f ~

ford the des igner with even more specific examples and

tec.hnlques.

Thus engineers

in this country wil l . be able to join

wi

th

those

i.nEngland and in Canada who have

a lready applied

plastic

analysis

to their design

problems.


26/28

205.52

LIMITDESIGN

PLASTIC DESIGN

ULTIMATE.LOAD (P

u

or

MAXIMUM STRENGTH

PLASTIFICATION

PLASTIC

MOMENT Mp

Novembe.r, 1956

Plast ic Design In Steel

SOME.DEFINITIONS

A

design based

on

any chosen l i .mitof

s t ructural

usefulness,

A

design

based upon the ultimate load-carrying capacity

maximum

s tr ength ) o ft l ,e

structure, The term plastic

i s

derived from th e fact that the

ultimate

load is computed from

a knowledge of the strength.of .s teel in th e plas t ic range,

The highest load .a

structure.will carry,

I t is not to he

confused with the

term

as applied

to

th e

ultimate-r;ad

carried by an ordinary tensi le coupon,)

The development of ful l plast.i.c yield of the cross-section,

~ x i m u m

moment

of

r es is tance o f a fu lly-y ielded cross -sec tion ,

PLASTICMODULUS

Z Combined s tat ical moments of th e cross -sect ional . a reas above and

below

the neutral .axis,

PLASTIC HINGE

.SHAPEFACTOR

f

,MECHANISM

. REDISTRIBUTION OF

MOMENT

A

yielded

section

o f a

beam

which

acts as i f i t were

hinged,

excep t w ith a cons tan t rest ra in i.ng momenL

The

ra t io

of the maximum resis t ing moment of a

cross-section

~

to th e yield moment

My)

hinge system -- A system of members than

can

move without

an

increase

in load .

A

process which results

in

the successive

formation

of

plas t ic

hinges

unt i l

th e

ultimate

load

is

reached,

.

By

i t

the

less

highly

stressed

portions of a structure also may

reach the

~ - v a l u e .


27/28

205.52

SELECTED

REFERENCES

1.

.Van

den Broek

2. Kazinczy,

Gabor

3. Timoshenko

4.

Maier-Leibni tz

5

.

Baker, J . . F.

6 . Symonds &.Neal

7.

Winter

T ORYOF LIMIT DESIGN

Wiley,

N. Y., 1948.

KISERLETEK

.BEFALAZOTT

.TARTOKKAL

( Exper:ime.nts

wi

th

Clamped

Girders ) ' ,

Betonszem1e,

Vol.

2, No .4 p. 68,

Apri l

1, 1914; Vol.

2No.

5, p. 83, May 1, 1914; and

,Vol. 2, No .6 , p. 101, June

1, 1914.

STRENGTH OF MATERIALS Par t I I ,

DIVan

Nostrand, New

York

1956.

CONTRIBUTION

TO THE

PROBLEM OF ULTIMATE

.CARRY

CAPACITY

OF

SIMPLE

AND

CONTINUOUS

B E ~

OF

STRUCTURAVSTEEL .

AND

TIMBER

Die

Bautechnik1 , 6 (1927).

A REVIEW OF.RECENT INVESTIGATIONS INTO THE BEHAVIOR OF STEEL

FRAMES

IN

THE

.PLASTIC RANGE

Jn l . Ins t i t u t e of Civi l

Eng in ee rs, 3 , 185-240, 1949.

~ E C E N T P R O G R E S S

IN

THE PLASTIC METHODS OF STRUCTURAL

ANALYSIS Journa l of ,Franklin Ins t : i t u t e ,

252

5 , p.

383-407, Nov., 1951, and

252 6 , p.469-492, Dec. 1951.

TRENDS IN STEEL DESIGN AND RESEARCH

Corne l lUnivers i

ty ,

Sept . 1951

8. Beed le , Thurl imann , PLASTIC DESIGN IN STRUCTURAL STEEL

SunnnerCourseLecture

Ket ter Notes, AISC, Lehigh Univers i ty , September 1955.

9.

Baker

Horne,

Heyman

10.

Neai

.11. AISC

STEEL SKELETON

Vol. 2,

Plas t ic Behavior and Desi.gn,

Cambridge

Univers i ty , 1956.

THE PLASTIC

METHODS OF

STRUCTURAL ANALYSIS (Text , to be

publ ished)

PLASTIC

DESIGN IN STEEL Proceedings Nat ionaLEngineer

ing .Conference, AISC, 1956.


28/28

why plastic design

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