Prestressed Concrete Deep Beams With Openings

Prestressed ConcreteDeep Beams WithOpenings

Michel SargiousProfessor of Civil EngineeringThe University of CalgaryCalgary, Canada

Walter DilgerProfessor of Civil EngineeringThe University of CalgaryCalgary, Canada

Based on a finite element analysis, the authorspresent design charts and nomographs fordetermining the reinforcement in prestressedconcrete deep beams with two openings ofdifferent sizes.

n this study, a single span concretedeep beam with two openings and

prestressed with straight tendons isanalyzed using the finite elementmethod.

The beam considered is of unitheight to span ratio, and the prestress-ing force is applied at two locationswith regard to the position of the

opening. Openings of various sizes,relative to the size of the beam, arecovered, and the maximum stressesand forces are determined at the criti-cal locations due to the beam's ownweight, external load and prestressingforces.

The results are plotted in the formof charts and nomographs which can

64

COMMERCIAL ORRESIDENTIAL AREA

DEEP BEAM ZONE:;.'.:( PARKING AREA)::

OFFICES

BEAM TOWER

DEEP/BEAM

MAIN STORE

STREET LEVEL

SECTION A — A

44 4eA

PLANa) MEDIUM RISE OFFICE BUILDING

SECTION B—B

B

4-B

b) PARKING IN AN APARTMENTBUILDING

Fig. 1. Typical examples of deep beams with openings.

be used easily by the designer for de-termining the critical stresses andforces due to loads and calculating therequired prestressing forces. A fullyworked numerical example is in-cluded to demonstrate the use of thecharts.

Background

Deep beams with openings occurfrequently in multistory buildings.For example, they can be used as ex-terior walls supported on a few col-

umns in the lower floors and carryingcentral columns in the upper floors(Fig. 1a).

In addition, in buildings where oneor more floors above street level areutilized for parking, deep beams areoften used as exterior and interior car-rying walls (Fig. 1b). Deep beamswith openings also occur in silos andpower generating stations.

Concrete deep beams can either bereinforced with mild reinforcing steelor prestressed with high tensilestrength tendons. A major disadvan-tage in conventional reinforced con-

PCI JOURNAL/May-June 1977 65

crete deep beams is the formation oflarge cracks unless the beams are de-signed for stresses of such low mag-nitude that efficient use of material isnot possible.'

Prestressing of deep beams al-leviates this problem, makes betteruse of the concrete and usually pro-duces a more economical design.Also, cracks which are normally ex-pected at the corners of large open-ings in deep beams can be avoided byprestressing.

Stress distributions in deep beamsdue to loads as well as due to pre-stressing are non-linear. 2 , 3 For thisreason, numerical methods, such asthe finite element technique, areneeded for investigating how the ten-sile stresses and forces due to loadscan be reduced or eliminated by pre-stressing. Nomographs and designaids for determining the criticalstresses and forces due to variousloading and prestressing conditionscan be used advantageously for thispurpose.

Study Objectives

The major purpose of this study isto determine the:

1. Magnitude and location ofmaximum stresses in deep beams withopenings due to member's ownweight, external loads and prestress-ing, taking into account the effect ofopening size.

2. Values and locations of prestress-ing forces which would counterbal-ance the stresses and forces due toloads in an effective manner.

3. Magnitude and distribution ofthe splitting tensile stresses in the an-chor zones of the prestressing tendonsand the steel reinforcement requiredto resist them.

4. Comparative effect of prestress-ing on a deep beam as opposed to amild steel reinforced member.

Research Program

In this study, a single span deepbeam with two openings subjected toa central top concentrated load isanalyzed (Fig. 2). The height of thebeam is equal to its effective span,and the loads to which the beam issubjected are: beam's own weight, ex-ternal concentrated load and pre-stressing.

Only straight tendons are consid-ered in representing the prestressingforce, since polygonal tendons are lessefficient in resisting edge tensilestresses.4 Also, the study has concen-trated on the analysis of beams withtwo openings, since a preliminary in-vestigation of the stresses and forcesin beams with one opening has shownthat they are less critical in the lattercase.

The following three variables areconsidered in the analysis:

(a) Location of prestressing ten-dons: (i) Prestressing force, PA,at 0.05H above the bottom sur-face of the beam, and (ii) Pre-stressing force, P B, at 0.05Habove the top surface of theopening.

(b) Width of opening, D, with re-spect to the beam's effectivespan, L:

DIL = 0.2, 0.3, and 0.4

(c) Depth of opening, E, with re-spect to the beam's height, H:

E/H = 0.4 and 0.50

The critical stresses and forces aredetermined for each of the variablesin Conditions (b) and (c) above and forthe following cases of loading:

Case 1: Beam's own weight, yL2B(neglecting the increase inweight due to the totalspan of 1.05L and decreasein weight due to the pres-ence of the openings).

66

D

-^PB0.05H CT__

B

PA0.05 H AT

0.1 L

Fig. 2. Dimensions and locations of sections for deep beams with two openings.

Case 2: Prestressing force, PA,applied at 0.05H above thebottom surface of thebeam.

Case 3: Prestressing force, PA,applied at 0.05H above thetop surface of the opening.

Case 4: Vertical downward centralexternal load, V, dis-tributed over 0.1L at thetop of the beam.

Method of Analysis

The finite element methods is cho-sen for analyzing this problem since itis well suited for application to deepbeams. Due to symmetry, only one-half of the deep beam is analyzed

after dividing it in its plane by imagi-nary lines into 140 quadrilateral ele-ments. Each element has four nodalpoints and has three degrees of free-dom at each node.

Each supporting column has threenodes with no vertical displacementallowed at the central node. The twoother nodes are free to move but eachis subjected to a known reaction equalto 25 percent of the total vertical reac-tion at the support.

Both horizontal movement and rota-tion are prevented along all nodesalong the vertical axis of symmetry ofthe beam.

The equations used in the analysisand the method of solving them canbe found elsewhere4 ' 5 and will not berepeated here.


a)BEAM'S OWN WEIGHTMULTIPLIER = 6LZBSL

b) EXTERNAL LOADMULTIPLIER=BL

B

C) PRESTRESSING PAP

MULTIPLIER= BLd) PRESTRESSING PB

MULTIPLIER=-

Fig. 3. Horizontal stress distribution along Sections A, B, C, D, and E for atypical case with D/L = 0.3, EIH = 0.5.

STRESS DISTRIBUTIONALONG SECTION E'-E'

E E'

a) EXTERNAL LOAD, MULTIPLIER = V/BL

Analytical Results

Fig. 2 shows the deep beam consid-ered in the analysis, the appliedforces and their locations as well asthe locations of the various sections.The distribution of the horizontalstresses due to the beam's ownweight, external load and prestressingalong Lines A-A, B-B, C-C, D-D, andE-E are plotted in Fig. 3.

This figure indicates that the upperpart of the deep beam, above theopening, acts in bending under theweight of the beam and the externalloads. The lower part below the open-ing acts as a tie subjected to bending.

The same figure indicates also thathigh horizontal tensile stresses de-velop along the lower edge of thegirder and along the upper edge of theopening under the beam's own weight

E IE'

10.8 I++I V 11.6

and external loads. These stresses arenot highest at Section E-E, but ratherreach their maximum values at a dis-tance of about O.1L from the centerline of the beam.

Fig. 4 compares the horizontalstress distribution above the openingalong Sections E-E and E'-E' due toexternal loads and due to prestressing.The values of the stresses and theirdistributions at both sections are al-most identical except for the highlocalized stresses at Point S.

A nomograph has been developedin Fig. 5 to determine the value of themaximum tensile stress above theopening, along Section E-E, due tothe beam's own weight and externalloads, and the corresponding com-pressive stresses due to prestressing.

A similar nomograph is presented inFig. 7 for determining the criticalstresses at Point S at the upper inner

E E'

2.1 1.8

5.3 + 5.23;•:...... .'::'; PB

8.2 S

b) PRESTRESSING PB , MULTIPLIER=PB/BL

Fig. 4. Comparison between horizontal stress distribution above the openingalong Sections E-E and E'-E' for the case D/L = 0.4 and EIH = 0.5.


corner of the openings. Fig. 5 can beused to determine the values of theprestressing forces required to elimi-nate the tensile stresses due to loadsalong Section E-E above the opening,while Fig. 7 helps in determining theresidual localized tensile stresses atPoint S due to loads and prestressing.

It would be uneconomical to coun-

terbalance the full tensile stress,localized at Point S, by prestressing. Abetter design would be to check theremaining tensile stress at Point S,after determining the prestressingforces according to the nomograph inFig. 5.

The nomograph of Fig. 6 can beapplied to determine the values of the

MULTIPLIER:

`P/BL = FOR PRESTRESSINGV/BL = FOR EXTERNAL LOAD6 L = FOR BEAMS OWN WEIGHT

6.4 STRESS STRESS FACTORx MULTIPLIER

Cg gE 1 eE

AM`

_v4i xx

5.6 `^`S_ EIGHT 0 w

4.8CASE 2 PRESTRESS ING

W

BELOWOPENING E=0.4 H

4.0— C\4 WO

Vic,ASH04^

W

W

U. x?FR F q4 `a' au, 3.2 <040 o x

fACASE 3\ PR^S W

o;

2.4 Q^ E\jW\U

W

uWi l

1.6

I90

w

W

U

0W

vW

U2

OWnW

U

0.8

01

0.2 0.3 D/L 0.4

Fig. 5. Nomograph for determining the maximum horizontal stress above theopening along Section E-E, due to loads or prestressing.

70

tensile forces due to loads in thelower part of the beam below theopening and the values of theprestressing forces required tocounterbalance those tensile forces.

As shown in Fig. 3, the horizontalstress distribution in the lower part ofthe beam, along Axis E-E, due to thebeam's own weight and external loadsis not identical to that due to pre-stressing, P A , below the opening. Thismeans that balancing the tensile forcein this part of the beam due to its own

weight and external loads by an equalcompressive force due to prestressing,P A, does not eliminate all the tensilestresses near the bottom edge of thebeam.

The resultant of these residual ten-sile stresses can be resisted by addi-tional mild reinforcing steel close tothe lower edge of the beam. Thenomograph shown in Fig. 8 can beused for this purpose since it gives theexcess tensile force near the bottom ofthe beam.

0.2CMULTIPLIER:

-IOPA = FOR PRESTRESSING BELOW OPENING

-PB = FOR PRESTRESSING ABOVE OPENING

V FOR EXTERNAL LOAD

S L2B =FOR BEAMS OWN WEIGHT

FORCE FORCE FACTOR x MULTIPLIER

91

W

CASE 3 PREP E NG

--7V/ yPRESTRESSING

0.20 0.30 D/L 0.40

Fig. 6. Nomograph for determining critical forces due to loads and balancingforces due to prestressing in the lower part below the opening.

N

cr

II-- 0.1.U

tWU

O 0.1W

t •

O

owuW NN WW QQ UU


MULTIPLIER-P/BL = FOR PRESTRESSINGV/BL = FOR EXTERNAL LOAD

3.0 5L=FOR BEAMS OWN WEIGHTSTRESS = STRESS FACTOR x MULTIPLIER

N

CASE 2 E=0.5H ^-. `CASE

PRES RES 4 N - /OPENING BELOW /

CA SE I E = 0. 4 H - _

5.0 CASE I E = O.5 HBEAM'S OWN WEIGHT

0I-C)

W

///

3 m$

0.PREST RESSING A BOVE CASE=/ / a^i'OPENING

o,

I.0 j CASE Q^^ ?/EXT. LOAD '

AD

Q20

0.30 D/L 0.40

Fig. 7. Nomograph for determining critical stress at Point S.

x

0

WN

W

U

xIQ

0

W

WU)

C)

41h

xn

The values of stresses and forces inall nomographs are normalized, sothat they can be determined for deepbeams with two openings of differentsizes. In this study, a unit height tospan ratio is considered as a reason-able representation for deep beamswith height equal to or greater thanthe span. The stresses in deep beamswith heights greater than spans areusually negligible in the upper partbeyond H = L.

Stresses in theAnchor Zones

Prestressing forces of large mag-nitude produce high tensile stressesin the anchor zones. These stressesand the corresponding splitting ten-sile forces due to prestressing shouldbe determined accurately so that suf-ficient vertical or spiral reinforcement

72

0.16

0.15

0.14

0.13

0.120

Q 0.11 NLL

W 0.100

LL 0.09

0.08

0.07

0.06

0.05

0.04

5E / N%

MULTIPLIER = 5 L2 BFORCE=FORCE FACTOR x MULTIPLIER

0.20 0.30 D/L 0.40

Fig. 8. Nomograph for determining the excess tensile force in the lower partbelow the opening.

can be used to prevent horizontalcracking.

Fig. 9 shows the tensile stress dis-tributions in the anchor zones for thetwo prestressing locations, PA and PB.The figure also gives the values of themaximum tensile stresses and resul-tant tensile forces when the size of theopening is D = 0.3L and E = 0.5H.

Figs. 10 and 11 can be used in de-

termining the values of the maximumtensile stresses and the magnitudeand location of the resultant splittingtensile forces in the anchor zones fordeep beams with openings of varioussizes. In particular, Fig. 10 can beused when the prestressing force isapplied below the opening, while Fig.11 is for the case of a prestressingforce at 0.05H above the opening.


1.45 P/BL

PB

x T =0.11 L

2.14 P/BL

PAxT =0.I L

Fig. 9. Typical distribution of the splitting tensile stresses in anchor zones.

Application of Results

The designer of prestressed con-crete deep beams with openingswould be interested in obtaining an-swers for the following items:

1. Values and locations of the criti-cal stresses and forces due to thebeam's own weight and externalloads,

2. Magnitude and locations of theprestressing forces required tocounterbalance the criticalstresses and forces due to loads.

3. Amount and location of addi-tional mild steel reinforcementrequired to resist tensile stressesat critical locations such as theanchorage zones of the prestress-ing tendons.

4. Advantages of prestressing overconventional reinforcement, bothfrom the viewpoint of cost andbeam performance.

The example given below providesanswers to these questions.

Design Example

A concrete deep beam with twoopenings has the following properties:Width, B = 1 ft (0.30 cm)Effective span, L = 60 ft (18 m)Height, .tJ = 60 ft (18 m)Distance between center lines of

opening = 30 ft (9 m)X, for openings = 21 ft (6.3 m)Opening length, D = 18 ft (5.4 m)Opening height, E = 24 ft (7.2 m)Volume weight of concretey = 150 lb/ft3 (23.56 KN/m3)Compressive strength of the concrete

= 5000 psi (34.50 MN/m2)Allowable stress in prestressing steel

= 150 ksi (1034 MN/m2)Allowable stress in reinforcing steel

= 26 ksi (179 MN/m2)Vertical external load V = 2yL2B, applied

over a length of 6 ft (1.8 m) at the topsurface of the beam.

74

22 E=0.4

a 1.8E_m 1.4 H - 0.5

1.0

0 0.10 0.20 0.30 0.40 D/L

(a) MAXIMUM TENSILE STRESS, f

0.26

0.22

0.18a_

0.14

• e

0

0 0.10 0.20 0.30 0.40 D/L

(b) MAXIMUM SPLITTING TENSILE FORCE, T

0.20

0.14I-

0.10

0.06

0' 1 1. 1. 1 a.

0 0.10 0.20 0.30 0.40 D/L

(c) LOCATION OF T WITH RESPECT TO BEAM EDGE, X T

Fig. 10. Tensile stresses and forces in the anchor zone due to prestressing,PA, below the opening.


2.0

1.6 H = 0.5

1.2 0.4xw

0 1 1

0 0.10 0.20 0.30 0.40 D/L

(a) MAXIMUM TENSILE STRESS , ft

aH

0.24

02

0.10 0.20 0.30 0.40 D/L

(b) MAXIMUM SPLITTING TENSILE FORCE, T

0.20

0.14 H= 0.4

E_

0.10 H -05

0

0 0.10 0.20 0.30 0.40 D/L

(c) LOCATION OF T WITH RESPECT TO BEAM EDGE, XT

Fig. 11. Tensile stresses and forces in the anchor zone due to prestressing,PB, above the opening.

76

To make a sound judgment as to thestructural merits of using prestressing,let us make two designs for the beam:the first using prestressed concreteand the second employing reinforcedconcrete, and then compare the re-sults.

1. Prestressed concrete beamBeam's own weight (approx.):

yL 2B = 0.15 (60)2= 540 kips (2402 KN)

External load:

V = 2yL2B= 1080 kips (4804 KN)

D/L = 18/60 = 0.3E/H = 24/60 = 0.4

Use the nomograph (Fig. 5) to de-termine the critical tensile stress inthe upper part of the beam above theopening. Then:

Maximum tensile stress along Sec-tion E-E due to beam's own weight is0.62 y L.

Maximum tensile stress along Sec-tion E-E due to external load:

V = 2.12 V/BL = 2.12 x 2 yL2B/BL=4.24yL

Maximum compressive stress alongSection E-E, due to P A is 0.48 PA/BL.

Maximum compressive stress alongSection E-E due to PB is 3.68 PB/BL.

Maximum tensile stress along Sec-tion E-E above the opening, due tobeam's own weight and external load:

0.62 y L + 4.24 y L = 4.86 y L

Since prestressing below the open-ing is not very effective in counter-balancing the tensile stresses (orforces) above the opening due toloads, only prestressing, PB , above theopening will be used for this purpose.

3.68 (P B/BL) = 4.68 y L

or

pB= 4.86yL2B3.68

__ 4.86x150x602x13.68 x 1000

= 715 kips (3180 KN)

Use Fig. 6 to determine the criticalforces and the required prestressingbelow the opening. Then:

Tensile force due to beam's ownweight:

0.097 x 540 = 52.4 kips (232 KN)

Tensile force due to external load:

0.175 x 1080 = 189 kips (839 KN)

Compressive force due to PB:

0.122 x 715 = 87.23 kips (388 KN)

Compressive force due to PA:

0.117x10PA=1.17PA

or

PA = 52.4 + 189 - 87.231.17

= 132 kips (585 KN)

Total area of cross sections of theprestressing tendons:

(715 + 132)/150 = 5.65 sq in. (36 cm2)

Excess tensile force near the bottomedge of the beam (from Fig. 8):

0.119yL 2B = 0.119x540= 64.26 kips (286 KN)

Additional steel reinforcement to beplaced close to the bottom edge of thebeam:

64.26/26 = 2.47 sq in. (16 cm2)

Maximum tensile stress in the an-chor zone of P A (from Fig. 10a):

2 P A __ 2 x 132,000BL 12x60x12

= 30.6 psi (211 KN/m2)

Maximum splitting tensile force inthe anchor zone of P A (from Fig. 10b):

0' 19 PA= 0.19x132= 25.1 kips (111 KN)


Maximum tensile stress in the an-chor zone of P, (from Fig. 11a):

1.4 PB _ 1.4x715,000

BL 12x60x12= 115.8 psi (798 KN/m2)

Maximum splitting tensile force inthe anchor zone of P 5 (from Fig. lib):

0.23 PB = 0.23 x 715

164.45 kips (731 KN)

Cross-sectional area of steel rein-forcement in the anchor zone of PA:

24.13/26 = 0.93 sq in. (6.0 cm2)

Cross-sectional area of steel rein-forcement in the anchor zone of Ps:

164.45/26 = 6.3 sq in. (40.8 cm2)

The steel reinforcement in each ofthe anchor zones can be in the form ofvertical stirrups extending 2 ft aboveand below the prestressing tendons.

The residual tensile stress at thecorner point, S, of the opening can bedetermined from the nomograph (Fig.7).

Tensile stress due to dead weight ofbeam:

2.35 y L = 2.35 x 150 x 60/144= 147 psi (1013 KN/m2)

Tensile stress due to external load:

5.65 V/BL = 5.65 (1080)/[(1)(60)(144)]

706 psi (4867 KN/m2)

Compressive stress due to pre-stressing force PA:

1.85 P A/BL = 185 x 132,0001 x 60 x 144

= 28 psi (195 KN/m2)

Compressive stress due to prestress-ing force PB:

7.36 PB/BL = 7.36 x 715,0001 x 60 x 144

= 609 psi (4202 KN/m2)

Excess tensile stress at Point S

147 + 706 - 28 - 609 = 216 psi(1490 KN/m2)

This stress can be resisted by theconcrete.

2. Reinforced concrete beamThe resultant tensile force in the

upper part of the beam above theopening due to the beam's ownweight and external Ioads (from Figs.3 and 5) is:

0.62 y L [0.25 x 2 (LB)/3] +2.12 V/(BL) [0.30 x 2 (BL)/3]

= 55.8 + 441 = 496.8 kips (2209 KN)

Required area of steel reinforce-ment:

496.8/26 = 19.1 sq in. (122.1 cm2)

Resultant tensile force in the lowerpart of the beam below the openingdue to beam's own weight and exter-nal loads (from Fig. 6):

52.4+.189=241.4 kips(1072 KN)

Required area of steel reinforce-ment:

241.4/26 = 9.28 sq in. (60.1 cm2)

Comparison between Methods 1and 2

From an economical viewpoint,prestressing tendons of 5,65 sq in. (36cm2) cross-sectional area will save:

19.1 + 9.28 - 2.47 - (0.93 + 6.3)/10= 25.19 sq in. (162 cm2)

of reinforcing steel.

Only one-tenth of the cross-sec-tional area of the steel required in theanchor zones is considered in thecomparison since this reinforcementwill not extend the full height, butrather around the prestressing ten-dons only for a height less than 0.111.

From a performance viewpoint,prestressing of deep beams preventsthe development of cracks in thelower part of the beam and above theopening.

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Conclusion

Prestressing of deep beams whichhave openings produces a crack-freestructure and provides an efficientand economical use of materials.

Thinner concrete deep beams canbe used if prestressing is appliedwithout danger of having cracks at thecritical locations. Also, the cross-sectional area of the prestressing steelneeded is much smaller than that re-quired if mild reinforcing steel wereused.

The above advantages must beweighed against the unit cost of pre-stressing steel and prestressing opera-tions which are usually higher thanthe unit cost of mild steel reinforce-ment.

In deep beams with openings, it isimportant to limit the height of theopening and have it located closer tothe bottom edge of the beam. Bydoing this, the top part of the beamabove the opening will have sufficientdepth to resist the bending momentsand shearing forces due to appliedloads, since this part carries most ofthe bending moments and shearingforces in the beam,

Acknowledgment

The authors wish to thank HughHawk, graduate student in the De-partment of Civil Engineering at TheUniversity of Calgary, for running thecomputer program and plotting mostof the curves.

Discussion of this paper is invited.Please forward your discussion to PCIHeadquarters by November 1, 1977..

Notation

B = width of deep beamD = length of openingE = height of openingH = total height of deep beamL = span of deep beam measured

between center lines of sup-ports

P = prestressing forceR = external central load acting at

top of beamT = resultant splitting tensile force

in anchor zoneV = vertical external loadfz = maximum tensile stressxpOS = distance between center line

of openings and bottom edgeof beam

x T = distance between outer verti-cal edge of beam and resultantsplitting tensile force

y = volume weight of concrete

References

1, Goodkind, Michael, Discussion of"Stresses and Forces in Deep BeamsPrestressed with Straight Cables," byM. A. Sargious and G. Tadros, PCIJOURNAL, V. 20, No. 2, March-April1975, pp. 92-94.

2, Geer, E„ "Stresses in Deep Beams,"ACI Journal, Proceedings V. 56, No. 7,January 1960, pp. 651-661.

3, Leonhardt, F., and Walther, R., "Wan-dartige Trager," Deutscher Ausschussfur Stahlbeton, Heft 178, Berlin, 1966.

4. Sargious, M. A., and Tadros, G.,"Stresses and Forces in Deep BeamsPrestressed with Straight Cables," PCIJOURNAL, V. 19, No. 4, July-August1974, pp. 86-99.

5. Zienkiewicz, O. C,, The Finite ElementMethod in Structural and ContinuumMechanics, McGraw-Hill PublishingCompany Ltd., New York, N.Y., 1971.


Prestressed Concrete Deep Beams With Openings

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