On the Reduction of Shear Reinforcement from Stuttgart Shear Tests (1961-1963)

On the reduction of shear reinforcement as derived from the Stuttgart shear tests 1961 1963 Autoren Leonhardt Fritz Objekttyp Article Zeitschrift IABSE congress report Rapport du congrès AIPC IVBH Kongressbericht Band Jahr 7 1964 Persistenter Link httpdxdoiorg105169seals7972 PDF erstellt am 15022015 Nutzungsbedingungen Mit dem Zugriff auf den vorliegenden Inhalt gelten die Nutzungsbedingungen als akzeptiert Die ETHBibliothek ist Anbieterin der digitalisierten Zeitschriften Sie besitzt keine Urheberrechte an den Inhalten der Zeitschriften Die Rechte liegen in der Regel bei den Herausgebern Die angebotenen Dokumente stehen für nichtkommerzielle Zwecke in Lehre und Forschung sowie für die private Nutzung frei zur Verfügung Einzelne Dateien oder Ausdrucke aus diesem Angebot können zusammen mit diesen Nutzungshinweisen und unter deren Einhaltung weitergegeben werden Das Veröffentlichen von Bildern in Print und OnlinePublikationen ist nur mit vorheriger Genehmigung der Rechteinhaber erlaubt Die Speicherung von Teilen des elektronischen Angebots auf anderen Servern bedarf ebenfalls des schriftlichen Einverständnisses der Rechteinhaber Haftungsausschluss Alle Angaben erfolgen ohne Gewähr für Vollständigkeit oder Richtigkeit Es wird keine Haftung übernommen für Schäden durch die Verwendung von Informationen aus diesem OnlineAngebot oder durch das Fehlen von Informationen Dies gilt auch für Inhalte Dritter die über dieses Angebot zugänglich sind Ein Dienst der ETHBibliothek ETH Zürich Rämistrasse 101 8092 Zürich Schweiz wwwlibraryethzch httpretrosealsch IVal On the Reduction of Shear Reinforcement as Derived From the Stuttgart Shear Tests 19611963 Sur une reduction de larmature de cisaillement basee sur les essais de Stuttgart 19611963 Über die Verringerung der Schubbewehrung auf Grund der Stuttgarter Schubversuche von 19611963 FRITZ LEONHARDT o Professor Technical University of Stuttgart Notation t total depth of section d distance from extreme compression fiber to centroid of tension rein forcement z distance from T to C leverarm of internal forces b width of web b width of compression flange s spacing of stirrups or bent up bars in a direction parallel to the longitudinal reinforcement h amount of horizontal movement of the Mzline to get the Tline I span length a distance of single loads from support M bending moment C internal compression force in compression chord of truss due to bending Cs compression force ofthe inclined struts ofthe truss Tx tension force of the longitudinal reinforcement tension chord of truss due to bending Ty tension force of stirrups P load on test beam Pp prestressing force V total shear force Vt shear force carried by the compression members of the truss V shear force carried by the web reinforcement shear stress V b z Vi b z 324 FRITZ LEONHARDT IVa 1 tensile stress in web reinforcement fc compressive cylinder strength fp compressive stress in p c due to prestressmg force after all losses at centroid of cross section fy yield strength of reinforcement t t2 prineipal stresses according to bending theory of beams Av area of web reinforcement within the length s of the beam Ac gross area of section x angle between web reinforcing bars and longitudinal axis of beam ß angle between compression strut and longitudinal axis of beam in truss analogy Av r Ty b s v factor of shear coverage related to shear reinforcement according to Mörschs analysis k second reduction factor for shear coverage for short beams or loads close to support 1 Introduction The Stuttgart shear tests which have been published in seven reports 1 2 3 4 5 6 and 7 have brought considerable new knowledge due to the fact that the many parameters which have influence on the ultimate shear load have been studied each separately It has been found e g that the most unfavourable shear condition is given by one or two concentrated loads M per span with a moment shear ratio of yj 24 to 35 All other load patterns can be considerably more favourable The following paper deals only with this M unfavourable loading short beams or loads near the support with 2A are treated separately It must also be mentioned that all the Stuttgart tests have been made with high tensile steel as reinforcement with a yield point of fy 60 000 psi with deformed bars 2 The Variabüity of the Mechanism in the Cracked State The classical shear analysis of W Ritter and E Morsch explain the shear strength in the cracked state with the truss analogy using a truss with parallel chords compression struts at an angle of 45 and vertical or with 45 inclined tension bars fig 1 It is assumed that the shear force V will be fully carried by the web members which leads to the following percentage of shear reinforcement in the web for the length s v V Av r r TT t with r rf fv b zsmyfv b s ON THE REDUCTION OF SHEAR REINFORCEMENT 325 This amount of shear reinforcement may be caüed füll shear coverage a reduced amount will be defined by the reduction factor n 1 referring to this füll coverage with vertical stirrups n Sold truss n fS t c T ein CT M stirrup forces Ty l related to the beam diagonal compression forces Cs J length Af z with inclined stirrups PSP Z rL A CUT CT M J5 diagonal forces 5 7 related to the beam length Atz Fig 1 Assumptions of classical truss analogy according to E Morsch With this classical truss analogy the equilibrium conditions for the inner forces are fulfilled but not the compatibility conditions which are badly hurt because the compression members of concrete are 3 to 20 times stiffer than the tension members of thin steel bars Nature does not know this truss analogy and acts according to the law of the minimum of deformation work and therefore the inner forces in reinforced concrete beams change considerably as soon as cracks appear strongly depending upon the stiffness ratio between the concrete and the steel members A beam with a thick web will show con siderably different internal forces compared with a beam of equal chord members but with a thinner web Even the crack pattern changes with this stiffness ratio and has influence on the mechanism We shall try to explain these mechanisms by our test results using again trusses as an analogy 21 Mechanism for Slabs and Beams with Thick Webs fig 2 In beams or slabs with rectangular crosssection we have the mechanism of a tied arch and the shear forces are carried totally by the inclined upper chord arch or strut The tie force decreases towards the support by the unreliable bending resistance of the concrete teeth between the cracks as shown by G Kani in 17 unreliable because this resistance decreases to 326 FRITZ LEONHARDT IVal almost zero when the ratio between crack spacing and crack depth becomes small in beams with t 15 For ultimate load this decrease of the tie force can become small therefore the tie should contmue to the supports with füll area and be well anehored As soon as cutoff bars are used in rectangular crosssections with their Tied archaction i 1 rjgmn i rtTf Total shear force in C A Decrease of tie force due to bending resistance of concrete teeth Htoiüisw Fig 2 Tied arch action in beams with thick webs kplcrri 50 ksi J i PI2 PI2 3500 lpp i B4B7 3000 6 FT 2500 3T 3 2000 VA 1500 0 i 1000 500 500 12 load P 24 30 Mp I 50 kips Fig 3 Average stresses in stirrups measured in beams with the one variable bfb width of compression zone to width of web with equal and constant shear reinforcement i 038 compared to stresses calculated according to Morsch ON THE REDUCTION OF SHEAR REINFORCEMENT 327 length determined by the moment diagram then the shear strength wül be reduced considerably 6 A shear reinforcement in such crosssections does not change this mechanism under working load stirrups remain without tension up to very high load intensities and they begin to carry only after shear cracks enter the inclined compression chord G The test results as shown in fig 3 prove that the stresses in stirrups are strongly influenced by the thickness of the web and in beams with rectangular crosssection they remain very small or are even negative compression up to 07 Pu ultimate load 22 The Change of the Mechanism by the Reduction of the Webthickness If in Tbeams the compression and tension chords are kept equal and only the thickness of the web is reduced then the tensile stresses of the equal stirrups grow almost proportional to the ratio bjb fig 3 The measured tension forces in stirrups however remain far below those calculated with Morschs truss analogy even for webs as thin as 2 with bjb 16 This means that even in beams with very thin webs a part of the shear force must be carried by the chords and not by the web members In fact we have measured tensile stresses on top of the compression flange near the supports Therefore even in Tbeams the resultant compression force C which can be considered as the top chord of an imaginated truss must be inchned Fig 4 shows that a slope of 112 to 120 between the load and the upper end of the last shear crack is easily possible and has been observed This slope means that 25 to Fig 4 In Tbeams the resultant force G in the compression chord can be inclined by 1 12 to 1 20 in spite of shear cracks covering the whole depth of the web ET t T 1 Fig 5 Crack pattern of rectangular beam ET 1 with bb 1 compared with pattern of Tbeam ET 4 with bjb 3 Both beams have equal longitudinal and shear reinforcement Average angles ß of shear cracks differ considrably Nr truss System a3z PV 4 v T5 PV 1 16 05z X34 BV I 24V BV PV P45 1 z 1 r 7z t fFT hPbv KW LZ 71 15 V 25 BV zV T MSV 1 BV T l PV y P34 s f 1 Z P 15z 15z 1 15 V i i l PV 30 BV 173z 173z T 173 346 V MM tension forces in the web related to z 060 V 10 V 45 ifFv 071 V 067 V 057 V in the chord near ihe support 30 V 15 V 05 V 15 V 173 V Fig 6 The influence of an inclination of the top chord or of the angle ß j 45 of the com pression struts in simple trusses on the vertical tension force Ty and on the tie force Ty near the support ON THE REDUCTION OF SHEAR REINFORCEMENT 329 15 of V is carried by the compression chord in a beam with a shear span of 3 In the bottom chord longitudinal reinforcement a small part of V might be carried by the dowel effect of the reinforcing bars however this effect is surely overestimated in 15 as soon as stirrups are used Further we have observed that the inclination of the shear cracks is not simply 45 but varies considerably with the stiffness ratio expressed by bjb and can be as low as 30 beam ET 1 in 2 flg 5 This means that the dia gonal struts of our imaginated truss are less inclined and therefore a smaller quantity of vertical tension bars the stirrups will be sufficient for the equi librium Simultaneously the tension force in the bottom chord near the support will be larger than in a truss with 45 struts Fig 6 explains by simple truss analysis how the vertical tension forces Ty and the chord tension Tx are influenced when the inclination of the top chord varies from 1 3 to zero 70 ksi 4 8 g 4 500 p 4 ups 7 load PIMpl shear stress vkplcm2 50700 psi Fig 7 Average stresses in stirrups measured in beams with constant 6 6 and As variable only the percentage of shear coverage q compared with calculated stresses according to Morsch 330 FRITZ LEONHARDT IVal or the angle ß of the diagonal struts varies from 45 to 30 Ty can be between zero and V related to the beam length Al z The inclination of the shear cracks can also be influenced to a smaller extent by the percentage of shear coverage expressed by n We have found that for equal bfb 16 the angle ß of shear cracks can vary between 37 for 7 025 and 44 for 77 10 the average Variation however is small On the other side n has no influence on the cracking load or on the distance between lines of calculated and actual steel stresses in stirrups as shown in fig 7 for T beams with constant bb only v variable 60 ksi 1 p i 4000 LU T2 3000 7t vV 2000 6597 fo Zb3A 000 Lj9721 20 160 40 200 Mp load 2P 400 kips Fig 8 Measured and calculated stresses in stirrups of beams with extremely thin webs 66 15 shear stresses v reaching 2500 psi Beam T 1 had vertical stirrups T 2 inclined stirrups Even in large beams with extremely thm webs and a ratio of bjb j5 as we have tested in our beams T 1 and T 2 1 5 where the angle of the cracks was about 45 the forces in the stirrups do not reach the values of Mörschs truss analogy fig 8 This means that even for highest shear stresses and in Ibeams a part of the shear force is carried by a slight inclination of the resultant compression force in the top chord and by frame action between the webmembers and the chords Fig 9 shows how the portion of the shear force which is carried by stirrups is varying with increasing load for different ratios bb and two grades of shear coverage These lines show that the load intensity has considerable influence ON THE REDUCTION OF SHEAR REINFORCEMENT 331 on the mechanism the webmembers increase their portion of the shear with increasing load intensity and only ultimate load conditions teil the safety and give the design criteria 10 Oß 06 04 02 V V inclined stirrups 09 vertical stirrup 03t 1 PJ ultimate load for bending Us Fig 9 Portion of shear force V carried by webreinforcement for different widths of webs expressed by 66 and for different shear coverage n plotted against load intensity PPu frflbw ultimalt kn4 for workirg load compreuer PL Fig 10 Distribution of the shear force V 1 on web members stirrups and struts 2 on compression chord carrying Vc by inclination 3 on frame action due to stiffness of junetion between web and chord members depending on 66 for working load and shortly below ultimate load 332 FRITZ LEONHARDT IVal In fig 10 the influence ofthe ratio between web stiffness and chord stiffness expressed by bb on the mechanism is shown in a different way the lines give the distribution of the shear force on the webmembers the compression chord and on the frame action due to fixed connection between struts and chords In the normal T beams with bb between 3 and 6 the web members carry less than half of the total shear force even under ultimate load con ditions The result is that the mechanism changes with decreasing web thickness and increasing load intensity from the tied arch to a truss with a curved or inclined top chord and with diagonal struts of which the inclination varies between 30 and 45 r r 30 b36 Jäto5 Fig 11 The actual mechanism of internal forces can be compared with such trusses with inclined compression chords and diagonal web struts with angles ß 45 the angles of chord and struts mainly depending on 66 extended truss analogy M0 Ig iL kß g wrong mechanism rPP mp ata true mechanism Fig 12 Common idea of the mechanism of continuous beams compared with the findings by tests for rectangular cross sections ON THE REDUCTION OF SHEAR REINFORCEMENT 333 This means that the trusses which we must have in mind in order to under stand the shear problem must have variables as shown in fig 11 23 Mechanism of Continuous Beams The system of internal forces in continuous beams is generally assumed to have a simple beam between the points of inflection M 0 suspended from a cantilever beam fig 12 The Stuttgart tests and UStests 8 9 showed that for rectangular crosssections or Tbeams with thick webs the inclined compression chord extends directly to the support so that the bottom tie reinforcement must also here be extended partially tili to this support Cor respondingly there are very flat shear cracks near the inner support with HHS r 1 t v iV tt ÄjSrf n f fl Pzr 4 i 9nftyafatjiaa ffijjtfrfcafrp 3 XZ s HS rjii n jri S fai SS Mö y CT1 QU b MI 7 0Ä I a jsfe bx I jao ßD SA 2 ootioßi iaoi HO out yw 250t an 5 2001 äct0 fm 3001 2001 Fig 13 Small inclination of shear cracks of continuous beams near inner support if bbgS 334 FRITZ LEONHARDT IVal angles as low as 30 fig 13 As a consequence stirrups are more necessary near the support than in the region of the small moments where we have no cracks up to high load intensities so that stirrups in this region did show only small stresses Also in continuous beams the mechanism changes with the stiffness ratio bfb in a similar way as we have shown for single span beams In beams with thin webs there is truss action under high loads with a smaüer influence of the inclination of the chords but a larger influence of the angle ß of the struts near inner supports The ratio of VjV is larger there than near outer supports 3 Calculation of the Reduction of Shear Reinforcement 31 Slender Beams and Slabs The described mechanisms make it clear that a füll shear coverage according to Morschs theory is not necessary because the tensile forces in the web are considerably smaller and their magnitude depends primarily upon the stiffness ratio bjb The sound design principle for reinforced concrete to carry all tensile forces by steel can therefore be fulfüled by a reduced shear coverage v 1 corresponding to the requirements of the ACI Code 31863 The measured stresses in stirrups showed in all tests a characteristic pattern as drawn in fig 14 The line of the stresses in the stirrups as a function of the load runs almost parallel to the one calculated with Morschs truss analogy however in a horizontal distance equal to the load Pcrack This load Pack is the one under which a shear crack has reached the stirrup We have defined Pack as the load which is found by continuing the measured line straightly towards the abscissis To this load corresponds to a shear stress v which we iaccording lo Morsch lvy Li ü critical zone depends on b and d VV observed f mi Fig 14 Characteristic line of stresses in web reinforcement compared with classical analysis of Morsch ON THE REDUCTION OF SHEAR REINFORCEMENT 335 found to be different from vc as found for beams without shear reinforcement The US results give vc Vfe influenced by the tensile strength of the concrete The value vi however is influenced by the portion of Vi Pcrack which represents the portion of V which is carried by the compression members of the truss and therefore shows a straight line function with the compressive strength fc fig 15 We have found this value to be different for single and continuous beams 1 for single span beams v fc for continuous beams 1 22 CIO Ic Pcracj increases with increasing 6 and z because the corresponding shear force is Vi vibz The thicker the web the larger the distance between the actual stress line of the stirrups and the calculated one according to Morsch This is caused by the inclination of the compression chord and the angles ß 45 for the diagonal struts From the characteristic line of fig 14 we derive the formula for the stresses of the stirrups V V4 2 kptcnf 4 uniform ootf uniform toad WO iic P slabs y jKcn 30cm t Kern 20cm 075yZZ 4 4 xp 30cm 32cm 50 cm m Vj beams with web reinforcement vu beams without web reinforcement 14 0 100 00 300 tOO S00 kpzrr2 Fig 15 Shear stress Vi due to portion of shear force carried by compression members of single span beams with shear reinforcement plotted against the cube strength of concrete compared with the ultimate shear stress Vu of beams without shear reinforce M ment for different depth d Only beams with 32 j25 336 FRITZ LEONHARDT IVal Consequently the factor for the necessary percentage of shear coverage can be written V vvDLVi wdl 3 with D L working load v factor of safety This formula corresponds in principle to the ACI Code 31863 and the Stuttgart tests confirm fully the soundness of this new USCode concerning shear If we plot the vjfc of all our test results for the mentioned ränge of ad and the results of 9 against the percentage of shear coverage n we get fig 16 The two lines for 116 and 122 are below all corresponding points with two exceptions three values of continuous beams which did not fail by I US Cod I ACI 3IS631 will I cuYiTiri3eapii ai T h Of Single span beams Stuttgart contlruous beams Stuttgart t continuous beams Stuttgart HC wb crushing failure Tube I srtly bent 49 bars oond fallurtt over support fir lor 1 075 OSO factor of shear coverage n Fig 16 The ultimate shear stresses of test beams with shear failures plotted against percentage of shear coverage compared with curves r ON THE REDUCTION OF SHEAR REINFORCEMENT 337 shear in the web but by bond failure of the top chord bars which have only about half the bond strength of bottom bars due to the Sedimentation of the concrete after compaction One value concerns a simple span beam which was webreinforced with bent up bars only which is not allowed The graph proves that the percentage of shear coverage ij calculated with eq 3 using vt according to 1 gives sufficient safety vDL refers to the maximum value of V of the beam and n should be constant for the shear zone belonging to this max V An upper Hmit of v must be chosen in order to avoid failure by diagonal compression in the web In our tests we found that the diagonal compression is not only influenced by the direction of the web reinforcement as can be derived from the truss analogy but it is further influenced by the angle of the shear cracks We found that for vertical stirrups the principle diagonal com pression stress for high shear intensity can reach i2 24 v for low shear stresses it can be t2 35 v For inclined stirrups a 45 closely spaced it reaches t2 17 v Therefore using a factor of safety of 21 and a reduction for sustained load the upper limit of the shear stresses due to the working load can be chosen oi Single Spans continuous beams DL f 9 V max y013 fc for inclined stirrups Jfiwj v OS fc lor vertical stirrtps and bent up bars OS Of 06 OX W tactor ol shear coverage n Fig 17 The required shear coverage for working load design and upper limits of allowable shear stresses 338 FRITZ LEONHARDT IVal for vertical stirrups and stirrups combined with bent up bars for inclined stirrups a 45 to 55 maxtjr 015 maxvnjr 019 UDL since compression is critical and not tension we refer v to fc and not to Vfe The maximum shear stresses for working load give the necessary percentage of shear coverage v for single span and continuous beams according to fig 17 The upper limits are governed by web compression the lower limits show the low values which can be allowed without web reinforcement and with constant ties For T beams with bb 13 the minimum requirements for shear rein forcement must be observed 32 Further Reduction of the Shear Coverage for Short Beams Short Slabs or Brackets The Stuttgart tests have confirmed the results of other investigators 12 concerning the influence of or of ad for single loads and of the slenderness ratio ld for distributed loads as shown in fig 18 According to these test 10 0 ki ff 1 concentrated loads 1 umfcrm load slenderness ralio moment swcr ratio Fig 18 The increase of shear carrying capacity without shear reinforcement in beams with the decrease of the slenderness ratio ld or of the shear span ratio ad for single loads ON THE REDUCTION OF SHEAR REINFORCEMENT 339 results the shear strength of beams without shear reinforcement increases rapidly for concentrated loads as soon as ad 3 or for distributed loads as soon as Ijd 12 For deep beams with ljd 1 such high shear stresses had been obtained that a shear reinforcement would make no sense as several tests have actually proved 18 7 This fact can be fully explained by the arch action of the concrete it allows us to further reduce the shear coverage by the factor k which can be taken from the straight line given in fig 19 The necessary shear reinforcement will finally be calculated by A K Fssina zfv or with the percentage of shear reinforcement r kv K n T s sin oc s os Z t c L orf QS W 2ß Fig 19 Reduction factor k for the portion of the shear force to be carried by web rein forcement or for the shear coverage depending on Ijd for uniform load or on ad for contribution of concentrated loads 4 Structural Conditions for Reduced Shear Reinforcement 41 Preference of Stirrups The reduced shear coverage should preferably be made with stirrups in close spacing the spacing decreasing with increasing values of the shear stress from a t2 to t6 Bent up bars of high strength steel are less suitable and lead to larger width of shear cracks Inclined stirrups show the best efficiency fig 20 Bent up bars are acceptable in continuous beams in the region of 340 FRITZ LEONHARDT IVal smaü moments for changing some bars from bottom to top avoiding herewith the large anchorage length especiaüy in the top region The stirrups must be well anehored preferably with hooks fig 21 Stirrups made of a welded wire fabric with a wire spacing of 2 to 6 and with a top anchorage by welded longitudinal wires are especiaüy suited for ribs joists and small beams in buildings 060 060 i 040 i PL 6 020 OOkibs load PIMpl t A 1 A I 1 1 A 1 1 llllllll iniiiiii i 1 NNN 1 Fig 20 Maximum crack width of shear cracks for average types of shear reinforcements Comparison for equal crosssections Ac and A Ul tarnet farmt of Vkrvpa mlöttvi ttmti U TT DTT üu hooks hr dekrmed es wet as smootti bars Fig 21 Value of different forms of stirrups the upper anchorage with a short length is very important 42 The Determination of the Points for Cut Off and Bent up Bars It is known that the tie force T is not proportional to the bending moment as soon as the shear stress is low For higher shear stresses the distribution of the tie force depends also upon the direction of the shear reinforcement For low values of v and rectangular cross sections or slabs the tensile force in the M longitudinal reinforcement can be as high as T Il5 7 near the support z Morschs truss analogy gives for vertical stirrups a tie force T MV w This value increases according to the described mechanisms The length of longitudinal bars can therefore not be found with the Mjz line but with a Tline which is found by moving the Mzline horizontally by the amount of h fig 22 This h must be made depending upon v Since the ON THE REDUCTION OF SHEAR REINFORCEMENT 341 shear stress v is connected with the factor of shear coverage 77 according to fig 17 it will be sufficient to determine h from v and the direction of the web reinforcement The two following values cover practical needs fig 23 h 12 09 7j d 05 d for vertical stirrups or vertical stirrups combined with bent up bars h 12127702 for inclined stirrups a 45 55 large anchor lenglh for lop bars r IZ U Fig 22 The determination of the length of eut off bars must be made for a tie force diagram with the Tline found by moving the Mz line horizontally with the length h found by the diagram of fig 23 Vd k s JA1209d for vertical Stirn vertical stirrups with bent up bi DS IS or 06 pN N rs 06 Jl 1215 n for 45 to 50 inclined Stirn 04 neorly in direction of the N 02 s 0 l 02 04 05 oa 10 Fig 23 Values of h for Tline depending on the factor of shear coverage n and the type of web reinforcement 342 FRITZ LEONHARDT IVa 1 Top bars for negative moments over intermediate supports of continuous beams must get a larger length of anchorage than bottom bars if the beams are concreted in normal position because the bond strength of top bars is low due to the Sedimentation of the compacted concrete The bond stress of top bars should always be calculated and it will be found that often small dia meters must be used and well distributed horizontally into the slab fig 24 undesired concentration of bars recommended distribution of bars Wi V r4H Z72Y Fig 24 Top bars for negative moments of continuous beams should be distributed sidewards into the slab using medium diameters of bars 43 Anchorage In beams and slabs without shear reinforcement especially in beams with a slenderness of ld 8 or with concentrated loads close to the supports with ad 2 a safe anchorage of the longitudinal bars must be provided In many cases the bond length of straight deformed bars is here inadequate and hooks may be necessary preferably hooks horizontally placed or loops or special anchor pieces 5 Torsion and Prestressing Two tests on large prestressed hollow box girders of which F Leonhardt and R Walther have reported in 16 have shown that similar relations for reduced web reinforcement as shown here can be established for the shear forces due to torsion and for the reduction by prestressing forces It was found however that the prineipal compression stress can become as high as 35times the value calculated by usual analysis and therefore can become critical earlier than generally suspected One must therefore be careful about the diagonal compression in webs On the other side for torsion also prineipal tension has not to be limited for the design because it is easily possible to take care of the tensile forces by reinforcement in the webs Fig 25 shows a first tentative proposal for the determination of the neces sary coverage of shear in prestressed concrete girders depending upon the shear stress v due to loads and prestressing force and depending upon the average normal stress due to the final prestressing force fp PpAc These lines need further confirmation by tests ON THE REDUCTION OF SHEAR REINFORCEMENT 343 Vu fcube 1 ote ai6 an 001 006 004 002 oaio 015 unsofe due ta diagonal compression in web above horizontal line Q20 Q25 030 QJ5 z tt 3 tä OÄ 05 oS a05 Pp J2 fcubt cube 04 OS Oi percentage of shea coverage n Fig 25 The necessary shear coverage for prestressed concrete girders considering the degree of prestressing by the average compression stress due to Pp depending on the maximum shear stress due to loads and P Limits to prevent web crushing failures Literature 1 Leonhakdt F and Waither R Versuche an Plattenbalken mit hoher Schub beanspruchung Heft 152 des DAfSt Berlin 1962 2 Leonhardt F and Walther R Schubversuche an einfeldrigen Stahlbetonbalken mit und ohne Schubbewehrung Heft 151 des DAfSt Berlin 1962 3 Leonhardt F and Walther R Schubversuche an Plattenbalken mit unter schiedlicher Schubbewehrung Heft 156 des DAfSt 4 Leonhardt F and Walther R Schubversuche an Durchlaufträgern Heft 163 des DAfSt 5 Leonhardt F and Walther R Beiträge zur Behandlung der Schubprobleme im 344 FRITZ LEONHARDT IVa 1 Stahlbetonbau Beton u Stahlbetonbau 1961 Heft 12 1962 Hefte 2 3 6 7 u 8 1963 Hefte 8 u 9 7 Fortsetzungen 6 Leonhabdt F and Walther R Schubversuche an Platten mit geschweißten Bewehrungsmatten Beton u Stahlbetonbau 1964 Heft 4 u 5 8 Fortsetzung zu Beiträge zur Behandlung der Schubprobleme im Stahlbetonbau 7 Leonhardt F and Walther R Untersuchungen an wandartigen Trägern mit unterschiedlicher Bewehrung und Belastung Heft 172 des DAfSt 8 Bryant Robert H Bianchtni Albert C Rodrigttez Jose J and Kesler Clyde E Shear strength of twospan continuous reinforced concrete beams with multiple point loading ACIJournal Sept 1962 Proc V 59 9 Rodrigttez Jose J Bianchini Albert C Viest Ivan M Kesler Clyde E Shear strength of twospan continuous reinforced concrete beams ACIJournal April 1959 Vol 30 10 Ferguson Phil M Some implications of recent diagonal tension tests ACIJournal Aug 1956 Vol 28 11 ACI Buüding Code 31863 June 1963 ACI Detroit USA 12 Laupa A Siess Ch P Newmark N M Strength in shear of reinforced concrete beams University of Illinois Bulletin No 428 13 Gttralnick S A Shear strength of reinforced concrete beams Proceedings of the American Society of Civil Engineering St 1 Vol 85 1959 14 Franz G and Niedenhoff H Die Bewehrung von Konsolen und gedrungenen Balken Beton u Stahlbetonbau 1963 Heft 5 15 Krefeld W J and Thurston C W Studies of the shear and diagonal tension strength of simply supported reinforced concrete beams Report Columbia University in the City of New York Department of Civil Engineering and Engineering Mecha nics June 1962 16 Leonhardt F and Walther R Torsions und Schubversuche an vorgespannten Hohlkastenträgern Festschrift der Beton u Monierbau AG Düsseldorf Okt 1964 17 Kani G N J The riddle of shear failure and its Solution ACIJournal April 1964 18 Chow Li Conway H D and Winter G Stresses in deep beams Proc ASCE Sep No 127 May 1952 Paper 2557 Summary The classical shear analogy of W Ritter and E Moersch using a truss with parallel chords and compression struts at an angle of 45 could be extended on the basis of the Stuttgart shear tests to trusses with an inclined compression chord and struts with angles between 30 and 45 according to the observed inclined compression forces The ratio between width of com pression flange and width of web is of a particular influence on these inclina tions of the truss members and therefore on the amount of forces in the tension bars This allows to reduce the shear reinforcement by subtracting a stress tod from the classical shear stress t0 The consequent increase of the tensile force in the bottom chord between load and support can be found by moving the Mjzline horizontally by a portion of the depth depending on the percentage of shear coverage for which formulas are given On this way the tensile force is also found for which the longitudinal bars must be anehored at end supports ON THE REDUCTION OF SHEAR REINFORCEMENT 345 Practical recommendations are given for the design of the shear reinforce ment in single span and continuous beams The future development of the extended truss analogy on prestressed beams and beams with torsion load is indicated Resume Lanalogie classique pour le calcul au cisaillement selon W Ritter et E Morsch qui admet des treillis ä membrures paralleles et bielles de com pression ä 45 peut etre etendue a des treillis ä membrure superieure inclinee et bielles de compression inclinees de 30 ä 45 Le rapport de la largeur de la dalle de compression ä celle de Tarne de la poutre influe fortement sur les efforts dans les armatures tendues Lanalogie elargie du treillis conduit ä une diminution de larmature de cisaillement parce quelle permet de reduire la contrainte de cisaillement classique t0 dune constante t0J On tient compte de laugmentation des efforts de traction longitudinaux qui en resulte dans la zone deffort tranchant en deplacant horizontalement la ligne Mfz dune fraction de la hauteur utile d fraction dependant du pour centage darmatures de cisaillement En meme temps on obtient aussi leffort de traction qui doit etre ancre ä lappui Lauteur donne des regles pratiques pour le dimensionnement de larmature de cisaülement dans des poutres simples et des poutres continues il montre le developpement futur de cette theorie pour des poutres sollicitees ä la torsion et des poutres precontraintes Zusammenfassung Die klassische Schubanalogie mit parallelgurtigen Fachwerken mit 45 Druckstreben von W Ritter und E Morsch konnte auf der Grundlage der Stuttgarter Schubversuche zu Fachwerken mit geneigtem Obergurt und Druckstreben in Neigungen von 30 bis 45 erweitert werden Das Verhältnis der Druckplattenbreite zur Stegbreite ist von besonderem Einfluß auf die Kräfte in den Zugstäben Die erweiterte Fachwerkanalogie führt zur Ver ringerung der Schubbewehrung indem ein Festwert tod von der klassischen Schubspannung t0 in Abzug gebracht wird Die daraus folgende Zunahme der Zuggurtkraft im Querkraftbereich wird durch eine horizontale Verschiebung der MfzLinie um einen vom Schub deckungsgrad abhängigen Teil der Nutzhöhe d berücksichtigt Man erhält damit auch die am Auflager zu verankernde Zuggurtkraft Für die Ausbildung der Schubbewehrung in Einfeld und Durchlaufträgern werden praktische Hinweise gegeben Die künftige Entwicklung der erweiter ten Theorie auf Träger mit Torsionsbelastung und unter Vorspannung wird angedeutet