A W CONCEPT IN INC§EA§iNG
STABILITY OF METAL STRUCTURES

mm?! for Hm Degree of M. S.
MICHIGAN STATE WHERE???
A. G. Patwardhan

196.3

THESIS

 

LIBRARY

Michigan State
University

 

A NEW CONCEPT IN INCREASING STABILITY OF
METAL STRUCTURES UNDER PRESTRESS

B ‘i '
:Y/ 'rYXhl

’ ‘\_\ I ‘v ‘
i- ‘0” ._
A: G‘ Patwardhan

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE

Department of Civil Engineering

1963

ACKNOWLEDGMENT

I wish to record my sincere thanks to Dr. C. E. Cutts,
Dr. R. K. Wen, Dr. W. A. Bradley and Dr. J. L. Lubkin for
their very valuable guidance and all the help given to me in
preparation of this thesis.

Mr. Don Childs and the staff of the Division of Engineer-
ing Research did some excellent work in preparation of the
experimental models. Their help is gratefully acknowledged.

***************

ii

TABLE OF CONTENTS

Page
SYNOPSIS O O O O O O O C O O O 0 O O O O O O O 0 O O O 00000 1

I. INTRODUCTION ........ . ..... . ...... 2

II. EXPERIMENTAL VERIFICATION OF THE BUCKLING
LOAD O O O O O O O O C O O O O O O O O O O O O O 0 ° 12

III.APPLICATIONS.. ..... 19

BIBLIOGRAPHY OOOOOOOOOOOOO O O O O O O O O O O O 29

iii

SYNOPSIS

This paper concerns itself mainly with the stability of a metal
structural member under prestress. A very simple method to increase
the stability under high compressive prestressing loads has been
discussed and a mathematical .analysis based on linear elastic and
small deflection theory has been worked out. Results of experiments
on an aluminum and a steel model are given in Part II of the paper.
Part III is devoted to a few examples of practical applications, illustrat—
ing the economy in materials and fabrication costs.

It is shown that by maintaining the prestressing forces on both
sides of the member under prestress, at fixed equal distances from
the center line of the member, and at a number of points in the span,
the buckling strength of the member can be increased many-fold. This

fact is also demonstrated experimentally.

I. INTRODUCTION

During the last twenty years, great strides have been made in
the use of prestressed concrete. The low tensile strength of concrete
has been offset by inducing high compressive stresses in regions,
where,under service loads, tensile stresses would develop. An extension
of this idea to metal structures is a natural corollary and several
excellent papers on this subject have already been published (references
1, 3, 4, 5 and 6 of bibliography). Starting with simple units such as a
member carrying a direct tensile force these papers illustrate how the
prestressing technique can be effectively employed for more complex
structures. Prestressed steel construction has not been widely adopted,
mainly because of the instability of relatively thin sections under high
prestressing forces. Some of the other reasons are the availability of
high, strength steels in the form of usual structural sections, the difficulty
in protecting steel prestressing cables from the elements and the :highcost
of prestressing (at the site of construction.

The problem of instability is not a consideration in prestressed
concrete construction, where the prestressing elements are embedded
throughout their lengths within the section itself and are symmetrically
located about the weaker axis of bending. In the case of metal structures,
however, the cables must be placed outside the section and the buckling
of the member becomes one of primary concern.

The method to improve stability proposed in this paper is both
simple and effective. The equal prestressing forces on the two sides of
the member are maintained-”at fixed equal distances from the axis about
which buckling would occur. This is accomplished by passing the cables

through spacers fixed to the member at a number of points throughout the

span. By this means, considerable resistance to buckling is obtained.
With the buckling of the member, the prestressing wires must follow
the bent form at all points where they pass through spacers fixed to' the
member. - In this deformed state, the tensions in the wires on both

the sides of any spacer a'ndion both the sides of the member have
lateral components of the same sign. These lateral loads produce a
bending moment in the member which tends to straighten the bent form.
This stabilizing effect is in addition to that provided by the internal
bending stresses produced in the member. The closer the spacing of
the spacers, the greater is the stability of the member. A minimum
distance between the spacers can be found so that a given axial prestress-
ing force does not cause instability in the member being prestressed.
This is the problem which is considered in detail in the following

paragraphs.

Buckling Load Under Prestress

 

Notation:

L = Total length of the member.

9/ : Length between adjacent spacers (assumed equal).

n = Number of equal panels, L = nfl,

A = Area of prestressing wires on one side.

E = Modulus of elasticity of the wires.

Eb = Modulus of elasticity of the member.

1b = Moment of inertia of the member about the axis about
which buckling pccurs.

T = Tensile force in the wires on one side of the member.

T0 = Tensile force in the wires on one side at buckling.

6 T, 6 To = Change in tension in the wires on one side as a
result of bending of the member.

(:1 : Distance from the center line of the wires to the longitudinal
axis of the member, at the points where the wires pass
through spacers.

yx = Deflection of any point from the original straight axis.
. . t
yk = Deflection of the pomt at the k h spacer.
y§< , yk’ = First and second derivatives of y with respect to x.

 

 

.1.‘ .1.
fi:. . i. .1... ___... v—_'-—._—..‘..j .‘{F_
-+4 +—l 7' i i LGl T fifi

Figure l

 

 

 

 

 

 

 

 

 

 

 

 

J
L
1!

 

 

 

Figure 1 shows the member under the action of prestressing
tensions '1", symmetrical about the longitudinal axis of the member.

At some tension T the member will be in a state of neutral equi-

O,
librium, when both the straight and the bent configurations are static.
The straight form constitutes the trivial solution. An equation of the
elastic curve in the buckled state, which satisfies the. conditions of

equilibrium and continuity must then be found. Figure 2 shows the

member in this latter state when the prestressing tension is at its

critical value.

 

 

Figure 2

In the following derivation axial shortening of the member is ~
neglected and where terms such as 9. and L occur in the derivation,
the effect of axial shortening is considered negligible.

The tensions in the wires on the two sides of the member will
undergo changes as a result ofjbending of the member. The changes
will depend on the changes in the original lengths of the wires on the
two sides.

Consider one of the segments of the member between the (k-l)t
and the kth spacer (Figure 3). In this segment, the shortening of the

wire on the concave side of the curve is 62 = d { sin( Vile-1) - sin

('y'k) } see OBIS

 

 

 

 

h

 

 

 

Figure 3

For small deflections and consequently small angles, sine of

the angle = tangent of the angle = the angle in radians and sec 6 r: 1,-

k

 

... 6e = d(Y'k_l'y'k)ooooooooo(a-)
n
Total change in length = 6L = d E (y'k__l - y'k) . . . . . (b)
n k=l
but ‘23 (y'k_1 - Y'k) = Y'o ' Y'n
k=1
... 6L=d(Y'O'y'n)ooooeoooooe(1)
. . A E 6 L __
The change in tensmn - 6T0 = —--I-:-—- —
. A Ed
0 0 6T0 -' L (y'o " y'n) o o o o o o e (2)

The sign of 6 To will depend on the sign of the term (y'o - y'n).

By similar reasoning, the change in tension in the wires on the
other side will be of the same magnitude, but opposite in sign.

It will now be shown, that neglecting friction, the tensions in
the wires on both the sides of any spacer have equal magnitudes for

small deflection of the member.

 

 

Figure 4

Referring to Figure 4, the components of tensions at right
angles to any spacer must balance.
If T2 and T1 are the tensions on the two sides of any spacer,

then,

T; cos (y'k - 9k“) = T1 cos (9k - y'k)

are all small angles,

k

Since .y'k, . 6k and 0k+1

T2 ’3: T1-
Thus in the buckled state, the tension-5' in the wires on one side
will be To + 6 TO while that in the wires on the opposite side of the .

member will be TO — 6 To. The tensions in the wires on any side will,

for all practical purposes, “be uniform throughout their lengths.

To+<5To

 

 

 

 

Yk .x Y}: I

 

 

 

 

 

 

Figure 5 shows the free body diagram of the segment, obtained
after making a cut at any point between the (k- l)th and the kth spacers.
All other forces are self equilibrat‘ed' and the only external forces
acting on the member are To + 6To and To - 6T0. ’

The axes chosen are shown in the'figure. Bending moment at

any point distant x from the (k-l)th spacer is

Mx : (To - 5T0) AB COS 6k - (To + 6T0) AC cos 9k

Again assuming cos 9k r: l

 

MX= (To— 6TO)AB- (TO+6TO) AC ....... .(c)
AB : Yx - SLYR“ - d +(y1< - Yk-i) %} . . ..... . ((1)
Ac .-. )Yk-x + d + (yk - yk_l) 7?} - yx . . ...... (e)
A E d ..
5T0 '3 T (y'o -y'n) . .......... . . . (2)
A E" d ~.
Mx = [TO - ' (Y‘o - Y'n)] [yx - in.-. - d + (Yk ' yk-l) “:E'} l

L

'l
A E d .
- [To + L (Y‘o-Y'nH [ 3.le1 + d + (Yk’Yk-i) 35'} - Yx]

i

 

which upon simplifying gives,

ZdZAE
Mx‘ : ZTo { Yx’Yk_1 " (Yk-Yk-1)3ZC—} ' T (Y'O'Y'n) - ~ ' (f)
but M); = 'EbIb y');

. x ZdzAE
- " Eblb Y)? = 2To {yx'Yk-l " (Yen—1’2"} - T (Y'o'Y'n)

 

2T0 2T0 x ZdzAE
n+...._._ 3......— +( -y_)-----} +-——-—(y'-')..(3)
yx EbIb yx Eblb iyk-z yk k I e EbIbL 0 Y n
. 2T 2 .' ., ‘.
Settingi-io— = p leads to the standard form of a second order linear
b b
differential equation,
x dZAE
Y"x + pzyx = p2 lyk‘l + (Yk‘Yk-1)T} + p2 ToL (y‘o-y'n) . . . (4)

Solution of this equation is

dZAE
TOL

 

. ~ x
yX = C1 cos px + C2 Sin px +{Yk-1 + ”Kirk-QT} + (y‘o-y' ‘) . -. (5)

n

The constants C; and C2 are the two constants which must be determined

from the boundary conditions. Setting x = O in equation (5)

 

 

dZAE ,
Yk_1: C1 + TOL (y'O-yn) + yk_l from which,
dZAE
C1: - TOL (y‘o-y'n) . . . . (g)

Setting x = C in equation (5))

yk = C1(cos p8 -1) + yk + C2 sin p2. from Which,
(l-cos p2) i i
C = —— , h

Thusrthe equation of the deflected curve of any segment is;

_ (l-cos pQ ) i x
yX " C1 (C03 pX ' l) + Cl sin pl + VIC-1+ (yk ‘ Yk_1)_2'_ o . . (6)

If C, :t: O, the deflection will be indefinitely large for sin p2 = 0
or pH, 2 n17 where n is odd. For even values of n the second term

be'cornes indeterminate. Differentiating (5) withhreSpect to x

Y‘x:'C1PSinPX+CzPCOSPX+’(3%) ....(i)

Substituting x = O in the above equation
Y'k_i : Cap + (ELLE-35k“) . . . .(j)
and substituting x = Z in equation (i)
Y'k '-‘ -C1psinp£’ +CzpCOSPC +(1k_!:_21m) ...(k)
Subtracting (k) from (j)
Y'kq‘Y'k2C1PSinIfl JFC,2p(1-cosp€).....(l)

and summing over the 'n' panels

10

n
E y'k_l-y'k= y'O-y'n=psinp?. 2C1+p(1-cos pQ)Z‘.C2 . . . (m)
k:l

where 2 Cl‘ and 2‘. C2 represent the sums of the two constants obtained
for each of the n panels.
For the case when C1 if 0 CI- is independent of any properties
peculiar to any segment and hence is the same for all the segments.
So is C2 because C2 = C, (M ). We have also seen that in this case,

sin p9.
the lowest buckling load is that given by p2 = 1r.

 

If C1 = 0, then C; is indeterminate if p2 = 1r
Al ' C “ dZAE ( ‘ ‘ ) the la t ter ' the
so Since 1 — T L y O-y n , s m in

o
' ‘ u _ a 2
parentheSis must be zero, 1. e. , y o y n 0

The equation of the deflected form is, in this case,
- C ' + ) x )
yX- zsin px yk-, + (yk- yk-l T— ....(n
and setting x = e in equation (n)
Ykz C2 Sinpe +ykorCz sian :0... .(7)

Now if C1 = 0, C2 cannot be zero for a nontrivial solution. . '. sin pa = 0
or pi. = nn

The lowest buckling load is again, therefore, given by

z
pQ=w or ZTO=‘Tr—£E7:blb-......(8)

Referring again to equation (6), we find that for sin pi ,._._ O, Y); becomes
infinite if C, is not zero. Since this is impossible, sin p1 = 0 necessarily
iimplies that C1 = 0 because only in that case, can C; have an indeterminate
value. We thus arrive at the condition that in the buckled state
y'o- y'n=0. . . . (A).

From equation (m) we have,

y'O - yh 2' 0 = p (l-cos p9.) 2 C2 and since (.;I - cos p9. ) is
notzeroforpf, = 1r , BC2=0. . . . . (B)

11

Thus we arrive at two important conditions governing the
buckled configuration of the member. They are: l) The slopes at
the two ends of the member have equal magnitudes of the same sign.
This follows from condition (A) above} 2) the sum of the amplitudes
of the sine curves of deflections must add up to zero. This latter
condition can be satisfied in a variety of ways. The buckled shape
can consist of complete sine waves of alternately-positive and negative
signs but equal magnitudes, when the number of panels is even. The
magnitudes of the amplitudes could be different but must add Up to zero.

In an experimental verification of these results the panel which
has the largest initial eccentricity will first bend and then buckle as
the buckling load is approached. ‘Other panels bend in such a way that
they satisfy conditions (A) and (B) found above.

Results of the two loading experiments which are given in
Part II of this paper, show that even with the unequal amplitudes of

defleCLions in the first test and the unsymmetrical buckling phenomena

which occurred in the second, conditions A and B were satisfied.

II. EXPERIMENTAL VERIFICATION OF THE
BUCKLING LOAD

Two experiments were conducted to verify the foregoing

derivation. The object was to verify both the buckling load and the

buckling mode shape of the member.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Sc\rew jack 0.124"dia. high tensile l”xl"x%-"Angles
. ® fl©§ steel wire . i—"Anchor
” f F: k " 1 ..-fi plate
.’ 3:1— “ - ‘* — — r' I f? Anchor
i 31
B C .
Anchor ‘fle ,' o" '30,, ‘ ‘50" l , -o” fiil Sc rjeaxzk
Dummy Strain gage Plan
gage I 4? st
ffi' ‘L- ' .
a s —-—. . o ...—.... a _—- . —-—- g 0 -
Panel 1 i Panql Z | Panel 3 Panel 4 ; Anchor
Screw jack]
Elevation

Figure 6' (not to scale)

1. Loading Experiment on an Aluminum Model

The test member was made of 3003-H-i14 aluminum, which

has a modulus of elasticity of 10 x 106 p. s. i. The member had a
length of 4‘0", a width of f” and was 4" deep, Spacers were made
from 1" x 1" x 133-" aluminum angles and. were fixed to‘ the member by

two i—" diameter brass bolts. The spacing between'the centers of out-

standing angle legs was 1'0". High tensile steel wires, 0.124" in

12

13

diameter and with an ultimate strength between 2.70, 000 and 300, 000
p. s. i.3were used as prestressing elements. One wire was placed on
each side of the member at if" from the center line of the member.
Strain gages were fixed at mid-depth on both the sides of the member
at positions A, B and C shown in Figure 6. Gages at A and B were
connected through a switching unit to the strain meter, with tempera-
ture compensating gages as dummy gages, fixed on the end anchor plates
of the same material as the member. Gages at C were connected in the
adjacent arms of the bridge, so that they recorded only the strains due
to bending, the equal axial compressive stresses being balanced out.
The center lines of all the gages at A and C, coincided with the center
lines of the bolts connecting the angles to the member. It was proposed
to ensure equal tensions on the two sides, by adjusting the jacks so that
the gages at C gave no‘output. It was found that the bolt center lines
on which the gages were fixed did not truly represent the points through
which the resultant of the lateral components of the tensions on the two
sides of the spacers passed, since the wire on the convex side of the
curve pressed on to the spacer and that on the concave side pulled on
the spacer. The compressive load exerted on the spacer was trans-
Initted through the outstanding leg of the angle while the tensile load
was transmitted to the member through the brass bolts. This caused
a difference between the gage center line and the line on which the
resultant of the lateral loads acted. As a result of this difference,
when the bending moment at C was maintained zero, the beam was
found to deflect on one side. This showed that the tensions on the two
sides were unequal. It was decided to obtain symmetry and zero deflection
at the center by adjusting the tensions on the two sides.

The first noticeable deflection occurred in panel 1, which was
later followed by a similar deformation in panel 4. Gage readings were

taken at frequent intervals to ensure that the strains did not attain very

14

large values. At the buckling load, however, the bend in panel I
began to increase and all the strain gages showed continuous straining
without corresponding increase in the tension in the wires. The

I
I I I I I i . O /
tenSions in the Wires were immediately reduced until the strains were

 

 

 

 

 

steady.
The strain meter readings for all the gages just before buckling

took placeJare tabulated below.
Gage No. Strain (micro inches per inch) . Average Station

l -372

2 _270 -321 A

3 -485

~33Z. 5 B

4 -180

5

6 110 diff. between 5 and 6

321 + . .

The average strain just before buckling was 2 332 5 = 326. 75 micro

inches per inch.

 

 

6‘ 2 326.75x10: 3267.5 p.s.i.
Area of member 2 4” xi—" = 1 sq. in.
Buckling load 2 326.7. 5 x 1 = 3267. 5 lb.

2 Z 6 -
. . g 11’ E1 _‘ Tl’ x 10 x10 4 1
Theoretical buckling load — _2'— — 12 x 12 x 12 x 64
for Q. = 1'0".
: 35801b.
, 513 ' ,
- Percent difference = 3580 x 100 = 8. 75%. between the eXperimental and

the. theoretical values .
To find the deflection curve, dial gages were used at positions

shown in Figure 7. Two readings were taken at each dial gage.

15

 

 

.3'2. L21" 2‘ 1-42.; 3L2‘L2‘: 5' 5'1... 10, .5"
New 31 WT. —r'-‘”'r ...-1

 

 

 

5 ’ 9,
6.; 6* ‘6? 6 i
g 4 r) 6/ 8 10
Dial gages

Figure 7 (not to scale)

One reading was taken in the maximum deflected. state and the other
when the tensions in the two wires were completely released. The
pressures exerted by the dial gage spindles were sufficient to bend
this slender member and to minimize this effect, the dial gages were
fixed to the table on both sides of the member in a symmetrical
manner. . Because ofthe permanent deformation which occurred in
panel 1, the deflection curve does not truly represent the ideal curve
which is, shown by dotted lines in Figure 8.on the following page.

The buckling load, however, agrees well with the theoretical value.

2. Tests on Steel Model

 

The steel member had a section 3" x i— " and a length of 4‘6"
between end anchors. Two intermediate spacers 1" wide and i- " thick
were welded to the member on both sides, thus dividing the member into
three equal panels, each 1'6" long. Prestressing was accomplished by
means of a high tensile steel wire placed on each side. . Strain gages
were fixed on both sides at positions A, B, C, D and E shown in
Figure 9. Equal tensions on both sides were obtained by making the
bending stresses at the point B equal to zero. Gages at the other
stations recorded the strains at various positions in the span. 7 They also

served as a. check on the total compressive load on‘the section.

l6

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é." Anchor - Strain gages D my
/ plates 4 ' . gage
' 3 / Misti: ”\‘6 .Ls 10
631, i .3 4+ —5 ' 3'" "7 a i
l .
file 1'6” ‘ 1'6” . ‘3': 1'6" 2]]
Dummy .
Edge

Figure 9 (not to scale)

The actual strains just before buckling were:

W

 

 

Gage No. Strain ( P ~inch/inch) Average Pdsition
-l

1 58 -155 A
Z -152 ‘

3 $-

4 } Difference maintained zero g __ B
5 +290 _159 C
6 -608

7 '150} -152. 5 D
8 - 155

9 '595 .-152.5 E
10 +290

Average strain = 154. 75 micro inches per inch

Corresponding stress = 154. 75 x 30 x 106 = 4642 lb per-"inz.

 

Area = 3" x i—" =' 0.75 inz.
Load = 4642 x 0. 75 = 3480 lb.

. 17" x 30 x 106 3 1 _
Theoretical Load — 18 X 18 X 1—2- X 2,—4- - 3560 lb.

18

Deflections were not measured but the approximate deflected

shape at buckling is sketched in Figure 8(a), page 16.

Conclusion

 

The results of the two experiments show that, 1) the buckling -
load is that obtained in the theoretical derivation, i. e. ,. Per 2 122%,
where g is the length of the member between the adjacent spacers,‘
Z) the slopes at the ends of the member are equal and ofthe same
sign, and 3) the amplitudes of the sine curves of deflectionsgare also
equal but opposite in sign, so that their sum is equal to zero.

The buckling load found experimentally agreed-well with the
theoretical value . The buckled mode shapes were, however, not
exactly what one would expect in a model having similar properties in
all the panels. This discrepancy can be explained by the fact that,
all the panels did notinfact have identical properties. Some of the
panels may have had a greater built-in eccentricity than others.

Bending of these panels may have dictated the mode shape,- of the whole

member in itsnfinal buckled form.

18a

0 o----¢‘na..o_.o ‘_ .‘ j
9.1-0.»..v0. .... ,.‘
...—.-.-..P .0. .0. -

 

GeneralArrangement of the Apparatus for Experiment with
the Aluminum Model.

 

General Arrangement of the Apparatus for Experiment with
the Steel Model.

III. APPLICATIONS

From the foregoing analysis and its experimental verification,
it is apparent that the capacity of any structural member to sustain
large compressive stresses, under prestress can be increased to
any desired level,within the elastic range)by the use of spacers.

With a sufficiently close spacing, the full working stress in compres-
sion can be induced in a member without causing instability.

Several applications to practical problems are possible. . Simple
tension carrying members can be effectively prestressed andtheir load
resisting capacity increased. An extension of this idea to indeterminate
trusses will be more economical since prestressing tension carrying
members will induce stresses of opposite kind in other members as
well. However, with the availability of high strength steels in the
form of the usual structural sectiOns, such-applications lose their
significance. As in the prestressed concrete.construction prestressing
can be effectively utilized in steel construction as well. Steel can
resist both the tensile and compressive stresses equally well and hence
there is no limitation to the maximum eccentricity. Thus with a
smaller prestressing force, large moments can be induced in the
girder. Full advantage of the prestressing technique cannot, however,
be realized in steel construction becauSe of the reasons mentioned
below:

With an eccentric prestressing force, the tension flange can be
prestressed to the full allowable stress in compression. ‘But on
account of the presence of the direct compression, uniform over the
whole section, which has to be superimposed on the bending stresses
due to the eccentricity of the prestressing force, the maximum tensile

stress in the compression flange is much lower than the permissible

1‘?

20

tensile stress. Under the action of external loads, then, the con-
trolling design consideration ‘is the compression. in the compression
flange, while the tension flange remains understressed. This is
particularly true in the case of symmetrical sections such as rolled
I beams. With an unsymmetrical section having a larger tension
flange, it may become possible to have a greater range of stress in
the compression flange between the loaded and the unloaded states of
the beam. With a small compression flange, however, the problem-
of its buckling under full working load, then is of primary concern.

This drawback is automatically overcome in the composites
construction in steel and concrete. The addition of a large area on
the top flange, by way of the concrete slab connected to the steel
girder by shear connectors, increases the modulus of the composite
section with respect to the top, compression flange. The stress
variation due to the superimposed loading is, therefore, much
greater in the tension flange than it is in the compression flange.

Full working stresses under maximumzload, can then.be simultaneously
reached in both the flanges, by suitably-proportioning the section.

In the following few paragraphs, the various. advantages possible
in prestressed composite censtruction are enumerated.

Many of these observations rare with particular reference to
large span bridge girders, in which use of unsymmetrical sections'is
possible. No attempt is made to present detailed design. calculations
and anyone interested is referred to serial nos. 5 and 6 .of,the
bibliography.

Figures 10(a) to 10(d)(on the following page) show diagramatically
the various loading stages, section properties-and the stress variation
across the section at each of the loading stage. Different moduli of
elasticity for concrete undenpermanent and nonpermanent loads have

been assumed. These assumptions are. based on those made in

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22

”Composite Cbnstruction in Steeland Concrete” (reference 1 in the
Bibliography). The same textbook may be referred to for “other
particulars such as choice of trial section, calculation of (section
properties of composite and noncomposite sections and stresses.

The many advantages resulting from using prestressing
technique are enumerated below. Some of these observations are
based on the stress variations at the different states of loading,
sketched in Figures lO(a) to lO(d), to which reference is made.

1. For the bottom flange, the change in stress is from -f's to
+fS where fS is the maximum permissible stress in steel. - The effective
stress utilization is therefore twice that for normal structural steel
construction.

2. The top flange, under prestress, carries a small tensile
stress. The working stress range in this region though not as large
as in the case of the bottom flange is still significant.

3. The top flange is adequately supported along its entire length
by its rigid connection to the concrete slab by shear connectors. The
compi'essive stress can therefore be as high as the full allowable
stress in steel.

’4. The utilization of the full working stress in both the flanges
is made possible by the addition of concrete slab. Whereas, the
stress in the bottom flange varies from -fS to +fs,the variation in the
t0p flange is not so large. However, the addition of further area on
the top flange by way of the concrete slab, while increasing the inOment
'of inertia of the section considerably, shifts the neutral axis of the
composite. section very near to the top flange. This results in a large
increase in the modulus of section with respect to the t0p flange. The
resulting stresses on account of the composite deadsload bending moment
and the composite live load bending moments in the top flange are much

1

lower than those in the bettom flange. A judicibus selection of the

23

section therefore makes it possible for the .full working stress to be
reached simultaneously in both the flanges.

5. A majOr portion of the compressive strength of the slab in
the longitudinal direction is utilized in resisting momentsdue to
superimposed loading. This results in further economy.

6. Shear resistance of the girder is increased. rThe vertical
components of the prestressing forces offer part of the resistance to
external shears, thus relieving the girder web of this part of the total
shear.

7. Loss in prestress due to creep is much less significant than
in prestressed concrete construction.

8. Gain in prestress due to deformation of the girder under
load is significantly larger than in prestressed concrete.ii This is so
because of the proportionately larger area of the prestressing steel.
This large gain offers a greater resistance to moments.

9. The construction being light, the dead load moments are
very much smaller.

10. Girders can be fabricated and prestressed iii the shops and
erected directly on the superstructure. The erection may’ require
little or no shoring. 1

ll. Slab formwork can be directly-‘asupported on the steel girder.
This saves considerable shoring. needed for supporting the slab form-
work.

The design of prestressed composite structures is much more
complex than similar unprestressed structures or non-composite
structures. This is so because the design depends on several para-
meters. Proportions of dead load, composite dead load and live load
moments will dictate the initial choice of trial section. For example,
if the proportion of non-composite dead load moment is high, the

girder section will have to be comparatively heavy and unsymmetrical,

24

with “a larger tension flange. The relative magnitudes of steel and
concrete areas will decide the position of the neutral axis for the
composite section. Since the slab thickness is decided on (strength
requirements in a direction transverse to the length of the beam, any
variation required in the slab thickness will necessitate changes in girder
spacing. The prestressing cable profile will largely be decided on the
type of loading. Shear resistance offered by the prestressing cables
will in turn depend on the cable profile. The external shear and part
resisted by the prestressing cables decide the web area required.

On the proportions of web area to the total steel area and to the top and
bottom flanges depend the properties of the section.

All these interdependent factors make the design Cofi’c‘omposite,
prestressed structures a very complex problem. With any improvement in
the technique, such complexities are inevitable.

The prestressing technique, using spacers, can be advantageously
used for prestressing thin plates and shells. While the more complicated
two and three dimensional problems are outside the scope of this paper,
the principle can be illustrated for a line structure such as a plate

simply supported on two opposite edges.

   
 

Figure 11

X ‘\
Y T
/yx
. ’x/?‘ T
T

Figure ll(a)

25

Figure II shows such a plate under- the action of prestressing ,
farce, applied on one side ofthe plate, by means of high tensile steel
wire, which passes through a number of spacers. The spacers main-
tain the distance 'd' between the wire and the center line of the plate
at those points where the wire passes through the spacers. Figure ll(a)
shows one panel of the plate between any two adjacent spacers. Each
such panel is acted upon by two collinear forces T acting at a fixed
eccentricity d at the two ends.

Let yx be the deflection of any point in the plate panel. The 'x'
axis is the chord joining the ends of the. panel...

The bending moment at this point is

-EIY"x= T(d+yx)........(i)

01‘

T T
" _£____ __ = . .
p2

T
Letting “ii—I- : solution of this equation is,

yx = C1 cos px + C2 sin px — d -- (I)
For x = 0
0 = C1 - d from which C1 = d . . . . . (iii)

and for x = Z

0 = C1 cos p9. + C2 sin pg. - d from which

__ (l-cospl) .
Cz- d sinpfl ........(1V)

 

1- cos p2

 

yx=d(cos px-l + sinpfl. sinpx). . . . (II)
yx becomes infinitely large for pQ, : I‘ITT where n is
odd and'in particular for p9, = Tr
2
E
The critical load is therefore T : If; I as before,

where El refer to the properties of the plate.

2.6

The spaciiig can always be so adjusted, that the design consideration
underi‘the prestressing load, is the maximum compressive stress due
to bending and direct‘force, and not instability. The distance d. can be
suitably adjusted for the kind of plate being prestressed. Thus for a
simply supported plate, the distance d would be the minimum practicable at
the ends increasing to some predetermined value at the centre, linearly
or parabolically, depending on loading. The significantly large-gain in
pres-tress, as the plate deforms under loading, further increases
resistance tomoment. For a plate material with a modulus ofl'elasticity
considerably lower than that of steel such a gain can be really significant.

The following simple example illustrates the increaseain load
carrying capacity of a simply supported plate. The Engineer‘s Theory
of Bending is used and although it is not strictly accurate because of
- large deflections involved, it can be a rough guide for comparison.

Consider an aluminum plate, f" thick and 12" wide,simply sup-
ported at the ends of a span of 3'0".

The plate is prestressed by high tensile steel wire 0. 03 sq. in.
in area and placed through spacers,at an eccentricity of §-" at the ends
increasing linearly to %-" at the centre of the span, at the bottom of the
plate.

Let T be the initial tension.

Max. stress at the bottom fibre

 

T 3 16x6 T .
.. m + Tx-é- x 12 — '3-‘1- 3T -— 3.33T lb. per sq. in.

If the maximum permissible stress in the plate is 10, 000 p. s. i.
3. 33 T = 10,000 and T = 3,0001b.
Let W be the maximum concentrated central load on the plate

and let 6 T be the increase in prestressing force in the wires.
.25. .
18

.Mt—a

W . 1
Change in Bending moment at any section = Tox - (6 T)(§ +

Rotation of the elementary length dx =

.1... (1x - (6T)(l +3.);
EI/ 2 8 72 (
‘ J

27

Assuming E = 10 x106 p.s.i.

64 W 1 x
66—‘1—67)‘ TX- (6T)(-8-+-7_E) dX.

Change in length of a small element dx of the wire =

Ei-gTai-gf'x» (6T)(8+—)} (-8- +—) dx

Total change in length of the wire

18
2 64 W 1
X337 I i157" ‘5T)<5+%’}‘§+%1(dx°

O

1
= 1%?- 23.62W — 1.2261‘}

Change in the tension in the wires

1 . 107
1773-?- (23.62W -1.22. 5T)xo.03x3——’3‘—6———

7.55W-0.39 6T : 6T

7.55w
: —-————-—-—-— :: 0 3' W.
6T 1.39 5 4’

Max. stress at the top fibre in the loaded state

36 . . 3 16X6 3000+5.43W
&W.7 -(3000+5.43 W) ‘8 }x 12 + 3

 

 

57.52 W - 8000 210,000 p. s.i.

18 000
W: -——i———- : 1 .
57.52 i—S-lb

For an unprestressed plate if W' is the max. central concentrated

 

 

36 16x6
1 d, '
oa (W.-—- 4 ) 12 10,000

10 000
w': = 1
72 391b
. . . 315
Additional load capaCity 2‘ -—- x 100 - 226%.

139

28

Total load in the wire = 3000 + 5.43 x 315 = 47101b.
4' 10
Stress in the wire 2 010”? = 157, 000 p. s. i.

Max. stress at the end supports = 1. 33 x 4710 = 6260 p. s.i .

Conclusion

 

The pr: stressing technique, using spacers, opens up immense
possibilities for designing light, economical structures at a fraction of
their present costs. While only very simple structures have been
discussed in this paper, the idea can obviously be extended to more
complex structures such as plates and shells. Further research in
this field, it is hoped, will pave the way for very light and stiff

structures to meet our future needs.

BIBLIOGRAPHY

. Composite Construction in Steel and Concrete by Viest, Fountain
and Singleton, McGraw Hill Book Co., 1958.

. Theory of Elastic Stability, S. Timoshenko and J. Gere, McCraw
Hill Book Co., 2nd Ed., 1961.

. Prestressed Steel Structures, G. Magnel, The Structural Engineer,
Vol. 28, No. 11, Nov. 1950,

. Prestressed Truss Beams, R. L. Barnell, Transactions, .ASCE,
Vol. 124, 1959.

. Design of Prestressed Composite Steel Structures, R. Sziland,
Proceedings, ASCE, Vol. 85, No. St.9, Nov. 1959.

. Behaviour of Prestressed Composite Steel Beams, Peter G. Hoodley,
Journal of Structural Division, Proceedings ASCE, June 1963.

29

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