CONTRIBUTION TO THE ANALYSiS OF STRESSES IN PRESTRESSED CONCRETE BEAMS

CONTRIBUTION TO THE ANALYSiS OF STRESSES IN PRESTRESSED CONCRETE BEAMS

By

T. KLATS}L-i.:'-\YI and G. TASSI

Department of RC'inforr·pd Concrete Structnres, Technical rniyersity. Budapest (ReceiYCd April 8. 1970)

Prt'spntpd by Prof. Dr. E. Bi:iI.C~1'-EI

1. Introduction

Design specificatilJll~ for prestressed concrete beams prescrilw the determination of stresses ^rhH' to any load in the concrt't(' and prf:stressing steel. The calculation should, according to certain specifications. confirm the re"triction of thc rC8idual deformations, and aceonling to other ones. assure th, load capacity hy limiting the stresscs. Calculation:, may be cnrri,'d out in order to check the cracklessne3s. to restrict cracking and to deter- mine the (Iuantit~- "f r)1"(1inary reinforcement rerIuirp!1 to with;;tand t f'l1sile stresses.

Proyidecl the materials are linearly dastic, tlw stresses in a hOillogelleuus Crf,ss-section may readily 1)1' calculated hy applying the elemcntary methods of the strength of matPrial~. Determination of a bending moment associated with the clefonnation of a concrete fihre let onh- by the method of successiye approximations is rather ~imple eyen in the ease of a cracked heam. Ho\\·- eyer, the evaluation of ~t rc~s('" caused by a mom(~n t higher than that producing cracks hy taking into account thp cracking. requires tedious calculation work [1]. Therefore, the approximation to consider a cracked beam to be homo- geneous is a current practice in stress analysis. This approximation permits to check particularly easily the stress induced in the extreme fibre. For the application of approximate procedure [2] it is required to justify this basic assumption. The approximation may be justified by the limitation of the calculated (so-called fictitious) tensile stress or by the application of ordinary reinforcement to withstand the resultant of the calculated concrete te113ile stresses [3].

The problem becomes especially difficult in the case where the restriction of the compressive stresses in the concrete and steel stresses should be pointed out for a moment significantly greater than the cracking moment. This is the case also with the Hungarian Code for Highway Bridges of 1968 [4,5]

which, nevertheless, permits to calculate the stresses eyen in t he cracked beam by making use of the relationships yalid to crackless beams.

Here, it is intended to investigate on what conditions the assumption of a homogeneous cross-section in analysing the stre8ses i8 permissible without

312 T. KLATS.\LLYYI and G. TASSI

making a gross errOL i.e., ,,-hat a correction should be applied ¹¹¹ the case of approximate pyaluation to obtain a more exact result.

b [cm]

b, [cm]

b~ [cm]

v [cm]

II [cm]

h [cm]

x [cm]

Fs [cm"]

'Y_e [kp]

Ts [kp]

Pp [kp]

.If [kpicm ]

Gc [kp/cm'}

a5 Ikp.Jcm']

r;j [kp,'cm']

Gc [kp cm~]

11

% [

%.,

/' -, [

Svmhols - width of flange of T or I beams:

"eb width o(rectangular_ T or I beams:

width of bottom fla;ge of I beam:

thickness of top flange of T or

i

beam:

thickness of bottom flange of I beam (without concrete coyer):

effectiye (approximately total) depth of beam:

... distance from neutral axis to top fibre:

cross-sectional area of prestrcssing "teel:

resultant of the cOlnpres,.iyc stres,,:s in concrete due to pre"tre""ing force and external loads:

resulting tensile force due to prestress and external loads (for cracked beams only in the prestressing :-teel: for a crackle"" heam both in prestressing steel and concrete):

effectiyc prestressing force:

calculated bending moment due to external load:

stress calculated i;l top fibre:

surplus part of stres" in prestressing steel above pre,ue8S:

stre" in concrete at leyel of the centre of gravity of prestres"ing steel:

Pp . 0, h '

ratio of moduli of elasticity of prestressing ,.tee] to concrete:

x

T: II

b;h:

Fs JJ

2. Basic assumptions

The anah-sis of stresses is based Oll the following assumptions:

a) Plane cross sections remain plane in bending (validity of the Ber- nnu1li - ~ ayier hypothesis).

b)

The stress-strain diagram", for hoth concrete and steel are lilwar (yalidity of Hooke's law).

c)

There is ^et perfect bond betweell prestressing hars and concrete.

d) The cross-section of the heam is symmetric, with the axis of symmetry

1Il the planc of bending.

ASAL YSIS OF STRE::::SES 313

e) There is no or negligible ordinary reinforcement.

f) Analysis of thf' cracked cross-section is done with the tensile strength of concrete omitted.

g) Analysis of th(' non-cracked Cl"O;:s-section is done with the concretc cover omitted or with the assumption of all prestressing wires aligned in the extreme fibre :::uhjeet to tension by an external load.

It should he noted that in our investigations the problem of losse~ of prestress is not treated of. thf'reforc the variation or the number ¹¹ derivahle' according to our assumptions ,\-ill not be discussed ('it11e1".

3. Analysis of the cracked cross-section

For analyzing the stresses in the cracker} cross-section, the depth of the neutral axis should he determined. In stress ::'tate II following cracking, this is achievcd for the general cas(, hy finding a straight line normal to the symllletry axis in the plane of hending and hordf'ring a conerete area.

straight line along "which the value of stresses is z('ro as calculated with the areas of concrete in compression and prestressing stt'el [6].

Stresses written for simple cross-sections ]pad to rdationships iIlyolving the distance of the neutral axis frolll the top fihre hut this is giyen in an implicit form only (e.!!. in [7]). In the following ,n' shall proceed this way.

3.1 Choice of the cross-section to be ana(..,.;;;ed

For (,yaluation of the stresses it is practicahle to write the relationships for the T section shown in Fig. 1. Thu:,. in the cas\' uf b b₁ or t· 0 they are "valid for a rectangular s('ction, The T section is PtI'liYalf'llt to the I or hox 5f'cti41n in the case if the l1r~utral axis illt('rs<'cts the weh.

3.2 Relationships for the determination of stresses

Stress(';; will he det('rminec1 according to tIll' ha"ic aSSUlllptions l!l

Section 2. The stresses arf' represented in Fig. l.

The stre:;;:;; ('xces;; in the pre-:trcs:;;ing steel ahoye the prestre:,,,, con'e- :;;ponding to the joint def41rmation of steel and cOllcrete IS:

c H (J~ h x

f]s 71 (J~" (1)

x the cOlllpn'SSlye force:

1- b ^--

(x-

t·FJ,

."Ye Ibx

')

L

^{x (2)}

314 T. KLATS.1LiSYI and G. TA,;"'I

and the tensile force:

From the projection equilibrium condition ~'Yc

extreme fihre of the concrete is gn'cn bv

a~. (b

b

Fig.

(3)

(4)

From the condition of the equilibrium of l1l0111ent~ ahout the llf'utral ^aXIS, bv substitution of the expression (4) we obtain

bx:^l b -

3 3

2P ^p --:--;;----;-:---:;--:-;---:-:---::--=:-:-:;----"7" = J1

x) .

By introducing thc parameters using svmbols ^III Section 1. the :::tress ^III the extreme fibre of the concrete will he given in the form

2np(1- ~)

The value of ~ in thi8 equation may he determined hy the implicit exprf'ssion 2

3 271,u(1

- - - -- (1 - ~) .

A.YALYSIS OF STRESSES 315

In the particular case of a rectangular cross-section. for

PI

= 1 and

% = 0, the relationships become simplified to:

and

4·. Stresses in the crackless cross-section

In the ca::'e of a crackles;; cross-section, the place of the neutral aXI::'

has a significance different from that of a cracked beam. The stresses may be evaluated by the elementary methods of the strength of materials, making allowance for the basic assumptions in Section 2. Naturally, the same values are obtained by introducing ideal cross-sectional characteristics usual for t he calculation of crackless prcstressed beams.

·1.1 Assumption of the cross-section

Since in the case of a crackless cross-section also the bottom flange

IS of importance for the development of the stresses, the investigations will concern the general ca,,~~ of I sections symmetrical ahout a single axis (Fig. 2).

For b = b₁ h~ or v u = 0, this cross-section is a rectangular one. For

b~ b_{l ,} or u =.= 0, it is a T section. A box girder suhject to symmetric bending may he calculated in the same 'wav as an I hemn.

4.2 Relationships for the calculation of the stresses

If the beam is subject to an external hending moment JI, and to a prestressing force Pp (Figs. 1 and 2) by taking into account the stress a~

acting in the prestressing steel [1], and proy-ided the neutral axis intersect;;

the web, the force Se may he given hy Eg. (2) where, as a matter of course, x is still unknown. The tensij!, force is:

O"~ h -- x [ 2 x

3 Periodica Polytechnica Civil 1-1/4-.

b~(h - x)

x

b]) (11 - x

h-x

316 F. KLATS.1Lf.\·YI alld G. TASSI

The concrete top stress may he obtained from the condition of projection equilibrium:

CJ~.

- - - ,

)~

- - - I

'E:~_! _

Fig. 0

From the equilibrium of the moments about the neutral aXIS:

2P

[bX:i

(~~=--~)(x --

p

3 3

bx~-(b---

= JI - Pp(1z x) .

Introducing the parameters using symbols 111 Section 1:

% -).,

and the relationship implicit for • 2 ;:l

3

/11

f~

PI

=;; (1

n.

(1 --;

%y

2n,u(1 _.- ;) (5)

(6)

A_Y_·1LYSIS I)F STRESSES 317

For an I section sYmmetric about two axes r3~ 1: % = %~

and thus (J~. ==

2np(1 - ;) and

3~1 [;::l

(1 - ;)" --

~

- - - - 1

[;~

_ (l ;)"]

;;) ,

Pl

;' (1--

n .

111 the ease of a T section

hence

3

Substitution of

1 :

( ~ ~-^" % -)"

-=--

') ⁽¹

3 (1 -

;:r

- )3]' ') (1

; . - % T :"np

%~ 0,

(1 -

;y -

271,ll(1 - ;)

=Y (1-0·

2n,1l(1

;)

into Eqs (5) and (6) delivers for the stresses in the top fibre of the rectangular cross-section:

;~ - (1 -

;F-

2n,l/(1 ;) and

2n,1l(1 -;:)"

(1 ;)

.

The steel stresses may be determined in the same way.

3*

318 T. KLATS.uI\TI and G. T.I"';SI

5. Comparison of stresses ohtained hy considering the heams crackless and hy taking cracks into acconnt~ resIJectively

The yery aim of our inn'stigation i8 to establi8h the differences in stresses when, according to the theory of elasticity, the cracking of the ten8ile zone is neglected and the ·whole cross-section is taken into account. Comparative calculations of a general validity being almost impos8ible by elementary means, our inyestigations are restricted to some particular. typical ease8.

5.1 Cross-sections and parameters analyzed

The case of the rectangular section in Fig. 3 has been investigated for

Hp 0.05: 0.10: O.ZO and

aT

=

50: 70: 100 kp/cm~.

Fig. 3 The assumed yalues correspond to pre8trf>ssing "tl'el percentages of 0.5 to 1.5.

commonly applied in practice.

The parameters of the T 8ection 8hown in Fig. I are:

np 0.10: O.ZO: 0.30

.,

_0.1

i)l

% 0.1

aT 100: 150; ZOO kpicm'-'.

For the I section symmetric about two axes (Fig. 4):

np.

%

0.10; 0.20: 0.30 0.1

=

0.1

100: 200: 300 kpjcm2 •

5.2 Relative depth of the compression ;:.one and relationship between the relative moments and concrete stresses for different prestresses and different percentages

of prestressing steel

For the com:idered cross-sectional parameters and concrete stresse8 due to prestressing. the values of ~ have been determined in dependence of I',

T. KLATSJIA.'·YI "nd G. T.-IS:;I 319

b

cracked - - - - crackless

o

0,4 0,8 1,2 1,6 2,0 2,4

0,1

0,2 t--._,--.. _-;-___ -L..

0,3 t--- -,-.-... _-

0,7 t---'-·----;-T+l--j-~¥A-_+_-+_-l__+-~

0,8

_I i

_I

I

r:

^{I I}

0,9 i

'gl I

V

I

1,0

Fig, 4

320 A.YALY~IS OF STRESSE."

::r:

^{- - - - cracked crackless}

o 40 80 120

0,1 l--f---r----+---t----i---+--t-

160 200

--

6~ ²⁴⁰

0,4 t--'---+--+---+---+---f--T7''i--r--j;;;r''''--+---r---r.:----;tt,--

0,5 j---;---;----t---j---t-r---r-

0,7 j---;---j----j---r-t1t-H-

0,8 t---j----+-+--+---fii---fi.If--;---r---ti ~+--~r--

5

., I

^0,9 f---l---t--t---;fj---i-+-HI'--t---r---t--r--.r-r-i Fig . . J

.. LYALYS'IS OF STRE:::'SES 321

for 0 ~ 1 and 0 - 2.6, togC'ther with the relationships hetween ; and in the ranges 0 c~ ~ <' 1 and 0 u~ 260 kpicm" 01 practical intereFl.

The analysis has concerned a sYmmetric I section and tlw results are represented in Figs -1 and .S.

5.3 Beam stress error duc to ncglecting the cracliing

The stre,,:"es acting in cracked or crackles;: cross-sections may be com- pared by reacling the yalues belonging to tlw cracked and the crackle"s Cr05:3- sections. respectiyely. OH the ;. ~! ClUyeS at the i' yalues corresponding to the giyen external moment anel pre;:tress. and applying the ;.

u:

^curn's

at the corresponding UT to determine th(' differenc(' I>('tw(>en th(' u~ values for the t\,'O cases (Fig. 6).

, ,.

I",'"

'g1

---Jr----

/ I

I :

52 ---,-

I I I I I

cracked

---

..

crackless

Fig. 6

The error made by assuming cracklessness 1Il the calculation of stresses

111 the prestressing :-teel is:

;- cr2cked t 1 ~ CraCkJessJ' - a c crackk~s -~.-~ ~~~ •

~ ^crackecl ~ crackles::;

322

t.)(5~ %

100 I

1--.

AXAL 1"SI5 OF STRESSES

I

150

62

^{= 160 kplcm2}

- - - -

62

= 220 kp/cm2

f31 =0,1

:x: = 0,1

200

131 = 0,1

f32 = 1 X-X2 =0,1

i 6r kp/cm2

~~ _ _ - L _ _ L - - L _ _ ~~L--L _ _ ^~~ _ _ ^~ _ _ ^~ __

100 200 300

Fig. 7

T. KLATS.u~i.'TI and G. TASSI

I

200~-''--'---'--'---.---r

I I

100 ^I

-

^{... I} '

"

"~ ^I ,

i

^{i i}

- ..J

^1)j..J ~O ^{10 ...} i

... ' -

,- ..

^{' , ~ :}

--1'-~'20 I-_~""-j...

,

---- - - .1_" ~

+:: -- _-

~.::t::-~

-, __ ---- ~J

50 60 70 80 90

400

100 200

Fig. 8

6i

= 160 kp/cm2

62

= 220 kp/cm2

----

6r k 100

~1 =0,1

J( = 0,1 p/cm2

P1 =0,1

~2 = 1 X=JC2=0,1

323

324 T. KL1TS.\L1.\TI "nd G. TASSI

5.4 Ez-alllatioll of the comparatiz:e analysis

The method in Seetion 5.3 has been applied to three typieal cross-,;r-ctions to determine the error made by considering the cracked cross-section to he crackless in the calculation of stresses in the compressed extreme fibre of concrete and in the prestres:"ing steel.

Calculated stresses arc plotted in Figs "{ and 8 for concrete and steeL respectiyely. The error due to the neglection of cracking is seen to decrease 'with the increase of the prestressing steel percentage and the prestrcss.

It should be noted that the mean concrete strcss due to prestre5sing

a a,

IS a r for a rectallgular seetion. 'fflr a T section and for an I section.

~ . 1,9 ~,8

It can he concluded that at or oyer half of the design (permissiblf" or limit) stress, the error in the mean yalne of the concrete stress due to pre- stressing ealculated from a sound cross-section is le"5 than 10 per cent. The erro,' decreases by increasing the prestressing force using a higher percentage of prestressing steel, rather than by applying a higher prestress using a higher tensile prestressillg steel.

When the mean eoncrete eompressiyl' stress caleulatpd from th(' eff'ecti\'e

pre~tressing force reaches the half of design stress, the error made in calculating the ~tress in the prestressing steel ,rill be less than ^{11 •} 1.50 to ^{11 .} :200 kp/cm~.

Thi~ i~ a speeial justification of ",ctting a minimum for the prestressing force.

It should be noted that other yiewpoints, fOT example the (·limination of the risk of hrittle failure as upper limit leach to similar conclusions [8] and thus, requirements for the effectiye prestressing force or mean str.ess in conerete due to prestressing are likely to Le adyantageous not onh- with a ^YleW to sim plified calculation.

Summary

Front the allaly~i~ of c~rlaill typical~ generalizable ca~e~ it i:-:: ~Lated that :;tre~~e~ in bonded-wire prestressed concrete beams may closely be approximated 1>y assuming crack- lessness eYen at moments higher than that of cracking when about half yalue of the design (permi,.sible or limit) concret; stress is reached. Otherwt,e. a more exact calculation is requir~d for which the hasic relationships abo are presented in this paper.

References

1. TASS!. G.: Limit Analvsis of Prestressed Concrete Beams. Periodica Polytechnic a C. E. 13 (1969) 83-92. . .

·1 BOLCSKEI, E.-TASSI~ G.-KLATS~il.·\';:,\YL T.: Reinforced Con~rete Structures. Design of Prestressed Beams. * Tankonyvkiad6. Budapest. 196·t

.3. Betonkalender 1967. W-. Ernst. Berlin. 1967.

ASALYSIS OF STRESSE::' 325 -1. :.'IIinistry of Transport and POH. HI/I 68. Code for Highway Bridges. '" Kiizlekedesi

Dokumentaci6s yallalat. Budapest. 1968.

;). TASSL G.: Specifications for Pre5tressed Concrete Structures of the new Code for Highway Bridges.'" :.'IIelyepitestudomanyi Szemle XYIII 502-50~ (1968).

6. BOLCSKEI. E:- TAS'I. G.: Reinforced Concrete Structures. Prestre:;sed Beams." Tankonvy-

kiad6. Budapest, 1970. .

7. ILDIPE. E.: Yorgespannte Konstruktioncu. Band 1. YEB Yerl"g ffir Bauwesen. Berlin 196,1.

8. \X'I?>DISCII. c'l..: Safety to Brittle Failure in Prestressed Concrete Structures ,,-ith Bond.

Periodica Polytecimica C. E. 14 (1970) '" In Hungarian.

Assoeiat .. Professor Dr. Gpza TAsSI

Sr. Assistant Professor Dr. Tihor KLATS}I:\~YI

Budapest XL, Sztoczek u. :2, Hungary