Pai., D. N. C.

English

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Lehigh University
Lehigh Preserve
Fritz Laboratory Reports Civil and Environmental Engineering
1960
Limiting deflections in plastic design by limiting l/d
ratios, C.E. 406 Report, Lehigh University,
(February 1960) M.S. thesis
D. N. C. Pai.
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Recommended Citation
Pai., D. N. C., "Limiting deflections in plastic design by limiting l/d ratios, C.E. 406 Report, Lehigh University, (February 1960) M.S.
thesis" (1960). Fritz Laboratory Reports. Paper 1723.
http://preserve.lehigh.edu/engr-civil-environmental-fritz-lab-reports/1723
LIMITING DEYLECTIONS IN PLASTIC DESIGN
BY"
LIMITING L!dRATIOS
David H. C. Pai
Submitted to Protessor G. C. Driscoll as
fulfillment of the course requirement of C. E.
406 "Special Problems in Civil Engineer1ng~
DEPARTMENT OF CIVIL ENGINEERING
FRITZ ENGINEERING LABORATORY
LEHIGH UNIVERSIty
BETHLEHEM, PENNSYLVANIA
February. 1960
---------
TULE OF CONTENTS
1
• • • • • • • • i1
1. INTRODUCTION • • '. • • '. • '. 1
2. d\$SUMPTIONS AND FUNDAMENTAL CONCEPTS •
.' •
2
METHODS OF CALCULATING DEFLECTIONS •
4. CASE I;: BEAM FIXED AT ONE END, SUPPORTED AT
THE OTHER .. UNIFORMLY DISTRIBUTED LOAD. • • • 5
•
5. CASE II. BEAM FIXED AT ONE END SUPPORTED AT 'mE
OTHER ... CONCENTRATED LOAD AT lUD-SPAN. • • 8
6.
8.
CASE III' aEAM FIXED JAT BOTH ENDS .. UNIFORMLY
DISTRIBtJTm LOADS.. • • '. •
CASE IVe CONTINUOUS BEAM ... TW'O EQUAL SPANS ...
UNIFOmU DISTRIBUTED LOADS" " " •
/
CiASE V:,IN...BASED RECTANGULAR FRAME. UNIFORMLY
DISTRIBUTlID VERTICAL LOADS.. .. .. •
•
•
•
•
•
10
12
14
10.
DISCUSSION • •
•
•
•
•
• •
•
•
16
11
11.
12.
NOMENCLA1'URE
REFERENCES "
•
•
•
•
•
•
•
• • •
•
•
• 18
19
FIGURES. .. • • • • .. .. 20
"-- -- -----
ii
ABSTRACT
Defieetion calculations at working load are time oonsuming
in elastic design. Specifioations of the Amerioan Institute of Steel
Construction and other specification" writing bodies have made it
more or less unnecessary to oalculate deflections b.Y limiting length
of span to depth of beam (LId) ratios to 24. And when beams are sub-
ject to shock or dynamic loading, the LId ratio is reduced to 20.
It is proposed, therefore. that a study could show that by
limiting LId ratios. beams designed plastioally would not have a
maximum deflection at working load more than a certain specified
amount. These studies would help write specifications limiting tId
ratios in plastic design.
, "
.1
1• INTRODUCTION
--
Plastic analysis and design of steel structures is a good tool
for steel construction. It has not. however, eliminated the necessity
of defleotion calculations. In elastic design, deflection calculations
are tedious and are much to be avoided if possible. Therefore, the
AISC partially circumvented the necessity of defieotion calculations
by specti'ying length of span to depth of beam (LId) ratios. Deflection
calculations in plastic analysis is just as necessar,y as that in elastic
design, although there are approximations whioh do s~.mpli1)r the calcula-
tions to a limited extent. But still, t~e amount of work involved
remains objectionable. It is the purpose of this reporttD investigate
into the feasibility of determining appropriate LId ratios for plastic
design. In this report. five cases will be investigated:
(I.) fA beam fixed at one end and. simply supported at the other
with uniformly distributed loads.
(II) Same beam as (I). but loaded with a concentrated load
at mid-span.
(III) A fixed~ended beam with uniformly distributed load.
(IV) JA two-equal-spaned continuous beam l-lith uniformly
distributed load throughout.
(V) fA pin...based rectangular frame with uniformly distributed
<J
vertical loads.
-- ---------------
\" -.'
2. ASSUMPTIONS AND FUNDM1ENTAL CONCEPTS
!Aside from assumptions made in simple plastic theory. CI this
report is based on the followS.ng concepts I
del) The)lIJ;~ relationship is idealized as shown in
Fig. 1.
(2) As a consequence of assumption ~, eaoh span
retains its nexural rigidity EI for the whole·
length between binge sections.
(3) ~nlimited rotation is possible at hinge sections
.. at a moment value of M :::: ~.lt 1
--~------~----_. __ .~~---,----------~-
~See pp. 25. 60, 98 of Reference 1 ..
3. METHODS OF CALCULATING DEFLECTIONS
Def'lebtion oalculations in plastio design fall into two types:
(1) The magnit.ude of deflection at ultimate load I this
is sometimes desirable because the load factor 01'
safety does not guarantee absolute~ against over-
load on rare ocoasions.
(2) 'lhfil magnitude of deflection at working load t this
is of value since a great majority of structures
td1l function at working load most ot the time.
In th~ rsport interest is oentered exclusivoly around the
second type or deflection calculations. i.e. magnitude of deflections
at working load. J\ sufficiently aocurate approximation 1-TUl be used
to oalculate deflections at working load. This consists or a plastic
analysis to obtain the ultimate load. which is then divided by the
load factor (1.8S) to reduce the loading down to working load. For
the types of structures to be investigated. the structures are all in
the elastio range at working load. Theretore. elastic deflection
equationo t-dll be used to calculate defleotions.
The scheme tor the investigation here is to oxpress LId as a
function of SIt (the ratio of deflection to span length.). In
generalt
---------
-4
£' =It ~r (p & Ware interchangeable)
PL c X ~
t:=K (}t ~)
1. =KX .fz . Zd (L!d)L E r
or I 1:1 ra uil - . 2.£ (L!d) ••••• (1)
where K and X are the tactors which depend on the loading and the
geometry of the stNcture, (;"'Y is the propertis ot ASTM tA-7 Steel
E
the material assumed in this report, ¥-= 2t, f is the shape factor
which governs how severely the beam will detleet.
. --...
-5
4. CASEl
BEAM F~ AT ONE END. SUPPORTED AT OTHER - UNIFORMLY. DISTRIBUTED LOADS'
STRUCTURE
400)
LCMDIN(1
MECHANISM
MOlmtT
DIAGRAM
f-- -----'-J...=-----'--f
Following the procedure outlined in the previous chapter. we
have
We = lrli
Wfi./ L c "p fi [ 1 + 1 +
~c [~+ X:x ]"p ••••• (2)
l~ =11.73 ~ (by trial and error)
::e...
L •
F = 1.85 ••••• (:3>
--=~.-=-- ~~= =e-. ---~.~__
'.
.6
Since the stxucture is elast.ic at working load, the defiect10n is
then given as
185 EI
(Ref. 2) ...... (4)
substitutiYlg the value of t-Tw in Eq. :3 into Eq. Ito, we have
de 6.3.5 ~ L3,
L 185 EI
But Mp ::: Uy Z
. [::: 6.35 QyZ (V/185 EI). . ,.,..............
I.
Since Oy::: 33 kat and E ::: 30 x 103 kai, vIe have:
zt2
I
Divide through by L and multiply the right hand side of the equa-
tion by dId we have
f· ::: 3.78 x 10-5 Zd.-I (Lid) I' ---~- -- --[\ '
'-., .-.---_. - ----
or ~ ::: 3.78 x 10.5 (2f) (LId) •• • •• (5)
Using f/L::: 1/360 as an arbitrary value, computations are made tor
the lightest w1de-~lange ot each group in the AISO handbook starting
with the 12" members through the 36 11 shapes. The result is plotted
on a LId va S Ix. curve. (See Figure 2) It is observed that all
---- =~------------ -- ----
',- '
' .. _...-
-7
curves tall within a narrow boundary. The oritical member having
a limiting LId:: 33.0 for a [/L = 1/360. Incidentally, the ratio
d/L = 1/360 is specified by the AISO as the maximum recommended
live load deflection for members Sl1pportingplastered cellings.
Therefore,- tor this particular type of beam and loading plastically
designed, the LId ratio will be 33 which is considerably greater than
a reoommended 24 in elastic design.
5. CASE II
BEAM FIXED AT ONE END, SUPPORTED AT THE OTHER .-
CONCENTRATED LOAD A.T MID-SPAN
STRUCTURE
tAND
LOADING
MECHANISM
MOMENT
DIAGRAM
p
j ~);if
.j.I J;i 1i • I
f-
-i,L
~9
I... ~ ~I
Prooedure here is simUar to that of Case II
••..• (6)
----i
1.
-9
Since the structure is still elastic,
[ = .009371 .pt,3EI
(From p. 370, Ref. 2)
substituting Fq. 6 into Fq. 7,
r ; ~L20= 9.317 x 10- x 3.24 :E
EI
d = 3.02 x 10.2 Uy· ZL2
E I
d=3.33 x 10-; ZL2T
~/L = 3.33 x 10·; . 2t (tId)
..... (7)
,; •• ,; (8)
After the LId V8 [It is plotted, (See figUre 3) it is observed that
the critical LId ratio for a itt = 1/360 is 40 which is even better
than Case I.
6. CASE III
BEAM FIXED AT BOTH ENDS - DISTRIBUTED LOADS
..10
STRUCTURE
!AND
.LOADING
1l1ECHANISM
MOMENT
DIAGIW'1
Similarly
We = Wi
(J e
~ ze
• % ·1·
W L 0. L ='2 v 2'
W\1::: 16 ~
1m:::: ~ = 8.65 !!e
1'" L
The structure is elastic at working load,
.11
(p. 371, Ref. 2)
substituting Eq. 9 into Eq. 10 we have;
••• ,; (10)
d = 8.6.5 or
ZL2
-I
dlL == 2.lf8 x 10~.5 . 2f. (LId) ..... (11)
The critical LId ratio for a d/t =1/360 is 48.3 (See Figure 4)
which is greater yet. This 15 logioal since there is more continuity
in this beam than in previous one.
,
~~---I
-12
7. CASE IV
CONTINUOUS BEAM .. TWO EQUAL SPANS - UNIFORMLY DISTRIBUTED LOADS
STRUCTURE
AND
LOADING /: 1.
MECHANISM
MOMENT
DIAGRAM
~ an elastic ana~sls. the expression for deflection is found
to be J = wx (L.3 + .3LX2 + 213). Maximum'deflection occurs at
48EI
X = 0.4215L; thus giving the expression d =.005416 wt3 ••• (12)
max.. EI
Advantage will be taken of the symmetry of the structure in
calculating for the ultimate load. Consider the lett span:
We = Wi
t-b~L (1-X) e = ~O
with X =.4215L
2L( T - 1)
~ = 11.65 ~
L
,"
, .
• .• (13)
. Since the maxi.mu.m defleotion 1s
-1)
6 tilL)
= .00,5416 -. the value ofEI
•••• (14)
Ww will be substituted into the above equation giving the maximum
defleotion at working load as d= .00,5416 (6.)0 .~') .ii or
f = 3.78 x 10"',5 2f (LId)
'l'he LId ratio for this oase has a critioal value. of 3J. whioh
is identioal to that in Case I. (See Figure S)
o
.14
8. CASE V
PIN-BASED REOTANGULAR FRAME - UNIFORMLY DISTRIBUTED VERTIOAL LOAD
5frudUre
cme;!
loacl,/).7
/!fomenT"
iJ/4frt:7/'YJ
, '.,
T ~::=:::::::::'-_.===lTI
r J. . -IBj?1B
JAn elastic analysis was carried out for a three span continuous
beam with side spans 41.. and side span loads 0( w. The deflection
at the center line of the main span is given as /
By letting d • the side span l~d factor equab zero, a denection
expression for all pin.based rectangular frames with vertical loads
is obtained. Thus.
.1.5
••.• (15)
is the deflection expression for a rectangular frame with variable
column height ~I- •
As for the plastic anelysis to obtain the ultimate load, similar
procedure will be followed noting that hinges at the knees form simu1-
taneously. Thus the ultimate load is
W ::: 16
u
. and ••.• (16)
have
Substituting this value into the expression for deflection, we
or
J _8.6j ~L2 I ( a 2 6 £P )
, -3840+2 ).2 f I 20lJ '"' +3 jJ +9
'j. =2.48 x 10"5 (2f) J,fd (~~d2 ~Ji9L3+2,4 r;r- ••.• (1'7)
This equation shows that the severity of defleo"tion 1s directly pro-
portional to the column height and the magnitude of the shape faotor.
In figure 6, graphs of SIt vs tid tor several factors of colunln
heifJht <J3') have been plotted. The graphs show that the critical
L/q rat:los are 31., 24, 21 and 19. for a {3 value of t. i, 3/4 and 1
respectively.
-16
9. DISCUSSION
From this investigation, it is observed that LId ratios could
be specified in plastic design in order to circumvent the necessity
of deflection calculations. Before discussing suitable LId values,
it should be borne in mind that the investigation here is limited
to the particular cases of frequently occurring structures only. Also,
a SIL =1/)60 represents the dene~tion limitation 1n the elastic
AISC specUioations due to live load. Further. the structural shape
assumed in this report is the standard American wide-flange made of
ASTM A.7 stael-Keeping tha above conditions in mind one may safely
say that the critical LId ratio for a plastically designed beam is
)) as compared with ,24 in an ~lastically designed one. !As for a
rectan~r trame wi,th uniformly distributed vertical loads, the
~/d ratios vary accor4ing to the Qolumn height, 1. e., the taller the
columns, the more defieotion (see Figure 6). Therefore, the general
deri.ation in case V ser~es ets a guide to obtain limiting Lid ratios
for any factor of column height to span length.
iAs was mentioned previously, this report assumes the wide-flange
as the structural shape. The values of 2f or Zd/I. varies from 2.1)
to 2.)2 (see table in back of report) and these .alues establish the
boundaries for the ['It vs tId curves. For other structural shapes
such as the I beams, the 2f values are higher, being between 2.28
and 2.46. Therefore. there will be a corresponding reduction in limit-
ing Lid ratios. However, the prooedure for obtaining critical Lid
ratios remains the same regardless of the type of structural shapes
used.
-~=_._. -----
-17
. 10. ACKNOWLEDGEMENTS
The author is deeply indebted to Professor G. C.
Driscoll, supervisor of this study. His most valuable
suggestions andcttticisms are sincerely appreciated.
\ \ ' , .,
Professor W. J.' Eney is the director of Fritz
Engineering Labor.atory and Head of the Department of
Civil Engineering.
-18
11. NOMENCLATURE
SYMBOLS
d depth of beam
e external when used as a sUbscript
E Young's modulus
load factor of safety = 1'.85
means internal when used as a subscript,
length of span
plastic hinge moment =6"'y z
concentrated load
f
F
.1, i
L
l~
P
Shape factor, t =z/s 2£ =Zd/I
s
w
w
Z
E
V-
a-;
f
section modulus
means ultimate when used as a SUbscript
weight per unit length, or "working" when used as a subscript
total uniform load
ful~ plastic secti.on modulus
strain
stress
yield stress of steel
vertical deflection
slope of deflection curve
-~-,------
2.
4.
..19
12. REFERENCES
Beedle, L. S. PLASTIC DESIGN OF STEEL FRAMES.
John Wiley & Sons, .1ew York,
1958. pp. 184-204. .
Steel Construotion AMERICAN INSTITUTE OF STEEL
C011STRUCTION, NeTtI York. 5th
Fdition, 1951.
Driscoll, G. C., Jr. ROTATION CAPACITY OF A THREE
SPAN CONTINUOUS BEAM, Fritz
Labora.tory Report No. 268.2,
·Lehigh.Universitr. 1957.
Drif,lcoll; G. C.. Jr.. ROTATION CAPACITY REQUIREMENTS
FOR BEAMS AND FRAlJI.ES OF STRUCTURAL
STEEL, Dissertation, Lehig~
University, 1957. PP. 186, 196.

rY
t
tv(
t
--~- e
(a) IDEALIZED STRESS-STRAIN CURVE
/
IL- . . _
--_. c;zf
(b) IDEALIZED MOMENT-CURVATURE CURVE
FIGURE 1
CURVE OF LId VB 6/L FOR CASE I
LId
50
40
I.. -L--~
30
20
10
0lL.... L- l...- ..L- -L- -.L,,-_--'-_
1/2000 1/1000 .: 1/666 1/500 1/400 1:/360
----~ .6/L
FIGURE 2
CURVE OF L/d VB O/L FOR CASE n
rt.....: -----)
.. L/d 40
!
I
!
I
!
30
20
10
OIL-__~L-__~.L-__--:-..l...- :L----_-:-L-_~---:--
1/2000 1/f-000 1/666 1/500 1/400 1/360
-------- J/L
FIGURE 3
CURVE OF LId VB d/L FOR CASE III
o L-.__----,L- ~----Ll ---1-1 -:---LI__LI:---
1/2000 1/1000 1/666 1/500 1/400 1/360
20
30 l.I
10
I
I L
·1I
50 ~
L/d II
I
I
40 ~
----~ ~/L
FIGURE 4
CURVE OF tid vs lit FOR CASE IV
50
L/d
r-i- -+--L --1
40
1/10001/2000
o J<:...-~__-'--__~-'----'-'-_--'-'-'--'--L-I --'--I_~'_'___Il___ _'_I_
1/666 1/500 1/400 1/360
20
10
30
'-
FIGURE 5
40
CURVES OF LId vs oil FOR CASE V
T
C3i
1
(3_1--;r
L/d
30
20
10
o
1/2000
//.J-- --/-'/-2
;J=!
1/1000 1/(;,66
aIL
FIGURE 6
1/500
I L
-'1/400 1/360


Limiting deflections in plastic design by limiting l/d ratios, C.E. 406 Report, Lehigh University, (February 1960) M.S. thesis

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Limiting deflections in plastic design by limiting l/d ratios, C.E. 406 Report, Lehigh University, (February 1960) M.S. thesis

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