examplesoffullyintegratedshearheadsareallaci-typeshearheads, … · 2019. 11. 11. · cruciform and...

Examples of fully integrated shearheads are all ACI-type shearheads, Tobler Walm, Compositecruciform and so forth. The behaviour of such systems is governed by composite action whichagain is strongly influenced by concrete fracture. Although fully integrated configurationsoffer a high punching shear capacity, they also suffer from a significant conceptional drawback.According to the requirements of seismic design, the column needs to remain in the elasticregion in order to avoid local storey mechanisms. On the other hand, plastic deformationwithin the connection is strongly required to reduce the forces acting on a structure duringseismic excitation. This makes the formation of cracks in the slab inevitable to allow theflexural reinforcement to yield and to activate shear reinforcements. In other words, themain energy dissipative mechanism of conventional assemblages strongly relies on concretefracture. Whilst the cracking process under monotonic (i.e. static) loading conditions iscontrolled, under reversed loading conditions it is not. For these reasons fully integratedsystems usually have a rather low ductility as well as limited rotation capacity and exhibithigh degradation of shear strength.

Partially integrated shearheads, which can be used to describe several forms including thenovel development proposed in this work, are partially embedded in the slab. This can beachieved by isolating the column from the slab by an opening around the column in such away that only the structural steel shear arms establish the connection between the slab andthe column. The philosophy behind partially integrated shearheads is to utilise the salientfeatures of steel in terms of strength, ductility and hysteretic behaviour; the exposed parts ofthe shear arms are designed to act like fuses and lend itself to controlled plastic deformation.

This concept utilises the salient features of steel in terms of the response to seismicallyinduced loads and leads to a very ductile behaviour with large column drift capacities. Inthis thesis it will be demonstrated that the fuses (i.e. exposed parts of the shear arms) shouldpreferably be designed for shear yielding. Passive steel shear yielding devices are renownedfor their high ductility, robust resistance to degradation under cyclic loading, high initialstiffness and significant energy dissipation capabilities. The latter absorbs a large amount ofseismic input energy and reduces the plastic deformation demand to the existing structurewhich helps to protect it from severe structural damage. Similar shear fuse concepts appliedfor eccentrically braced frames have been published by Mazzolani (2008) and Chan et al.(2008). Other, similar applications can be found in coupled shear walls published by Harrieset al. (1993) and Fortney et al. (2007).

Clearly, the philosophy of a partially integrated system is based on the condition that thesteel capacity is consistently lower than surrounding concrete mechanisms. The concretecomposite capacity of the embedded parts needs to be ruggedised in such a way that limitedstresses are induced into the concrete slab, which mitigates degradation effects; concretestrength is then widely maintained under cyclic loading conditions. Moreover, the residentcomplexity in the D-region in vicinity of the column is significantly reduced by means of anaperture around the column. The critical force flow from the slab into the column of such an

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isolated system is more definite. Additionally, the plastic failure mechanism of the shearhead(fuses) can be accurately tuned for specific demands.

3.11 Concluding Remarks

In this chapter different shear head details and concepts have been introduced with the aimof determining which is most suitable for a partially integrated shearhead. According tothe literature presented, the availability of shearhead systems suitable for steel columns islimited, and those discussed before appear mostly restricted to gravity load cases only.

When it comes to shearhead performance under lateral loading conditions, the literaturebecomes scarce. The reason for this might be the poor performance of fully integratedshearheads compared to conventionally reinforced slab-to-column connections, and researchobviously focused on the latter. However, as there is a dearth of experimental data on theperformance of shearhead systems under seismic loading, the code provisions are also rathertacit about shearhead design under combined loading.

While the control perimeter approach in shear head reinforced cases for combined loadingis relatively straight forward, the complex composite behaviour of the shearhead inside thecontrol perimeter is non-distinctive; this suggests that the boundary conditions for an ana-lytical model for the shearhead are vague. Furthermore, since the force distributions are notsymmetric under combined loading conditions, the shearhead behaviour strongly depends onthe load eccentricity.

When it comes to cyclic lateral loading conditions mainly conventional reinforced concretecolumn-to-flat slab connections have been examined and data on the cyclic performance ofshearhead connections are hardly available. This proves that accurate predictions concerningcolumn drift ratios or the capacity of conventional shearhead reinforced connections undercyclic loading conditions cannot be made.

However, the following conclusions could be drawn from this chapter:

• Lateral loading causes mirror-inverted load reversals and therefore requires a line ofsymmetry in the axis of the slab

• The shearhead needs to enable the load bearing mechanism to act in both directions,which requires it to be centrally located within the slab depth. Some cruciform shear-head details show that the bottom flanges are directly resting on the form work inorder to increase the effective control perimeter, which is counterproductive in com-bined loading conditions.

• In this context, the systems Tobler Walm, Sheardome, NUUL-system as well as bothpatents mentioned in Section 3.6 are unsuitable for gravity loads in conjunction withlateral loading.

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Although the activation of composite action is generally desired, the application of shear con-nectors such as headed shear studs raise several issues related to fatigue failure and concretedegradation effects when subjected to cyclic loading. Therefore, the Composite cruciformcould be susceptible to low cycle fatigue failure. Low cycle fatigue issues also concern thefracture toughness of welded connections which typically appear in the welding seem of headedstuds. Thus, the adoption of headed shear studs to improve the composite behaviour of theshear arms was discarded.

Finally a new shear head concept has been introduced which will be thoroughly discussed indue course. Representatives of such partially integrated systems are the Geilinger mushroomwith discontinuous reinforcement (see Fig. 3.14) and the other ones focused on in this work.

The concept was developed to address the shortcomings of conventional details described inthe previous and the present chapter. Compared to fully integrated shearheads, the omissionof concrete in the area inside the shearhead lowers the punching capacity. This results fromthe lack of composite action and discontinuous reinforcement within the perimeter betweenthe shearhead and the slab. On the other hand, this offers favourable inelastic behaviour andalso contributes to a reliable and efficient design. Finally, it is an additional positive sideeffect that services can be conducted through the slab without the requirement of additionalholes.

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Chapter 4

Experimental Methodology

4.1 General remarks

Referring to the novel shearhead concept in Section 3.10, the testing programme presentedin this thesis primarily aimed to investigate and to compare the performance of a fully- anda partially integrated shearhead. Therefore, a multifunctional test rig first had to be builtwhich was suitable for conducting large scale slab tests on slab-to-column assemblages.

An ACI-type shearhead has been adopted initially. The secondary aim of this testing pro-gramme was to further optimise the partially integrated shearhead detail. The large scaletests typically involved one test under gravity- and one test under combined cyclic loadingconditions for each shearhead configuration.

In the course of testing, firstly a shearhead in a fully integrated configuration was tested. Thisconventional configuration will be referred to as ’Type-A’ detail. The Type-A detail servedas a benchmark in terms of the comparative assessments of the other shearhead configura-tions. Thereafter, the same shearhead was tested in a partially integrated configuration. Thepartially integrated configuration will be referred to as ’Type-B’ detail. The ’Type-B’ detailwas designed in accordance with Concept C in Section 7 of EC 8 (2005) which requires mea-sures for preventing the contribution of concrete in the main dissipative zones under seismicconditions.

After the general applicability of the partially integrated Type-B detail was verified, the test-ing programme focused on further improvement of the composite behaviour of the latter. Asmall scale test rig had thereafter been built which enabled the investigation of the compositebehaviour of the embedded parts of the shear head. The small scale tests were not intendedas substitutes for large scale tests. However, in due course a higher number of tests couldbe conducted. Hence, these tests eventually provided more information on the compositebehaviour of the shear arms than a significantly smaller number of large scale slabs wouldhave done.

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These tests referred to as ’slab panel tests’ consisted of two series. Series-1 investigated theeffectiveness of edge reinforcement to prevent localised concrete damage. Series-2 was usedfor further optimisation of Series-1 details with the purpose of increasing both the capacityand the durability of the shear arm.

On the basis of the slab panel test results, an improved shearhead detail, ’Type-C’ wasdeveloped. The results showed that Type-C detail offered significantly improved behaviour.When compared with Type-A detail, the Type-C detail proved to have superior behaviouralcapabilities especially under cyclic loading conditions.

This chapter aims at furnishing the reader with detailed information on the experimentalmethodology of this testing programme. This chapter hence contains the material propertiesof the concrete, reinforcement and structural steel parts as adopted in the tests. Furthermore,the large and small scale testing rigs will be explained in detail. Accordingly, a detailed de-scription of the large and small scale specimen geometry, reinforcement layout and shearheaddetails is given. Finally the adopted instrumentation setup for large and small scale tests ispresented. It needs to be pointed out that the shearhead details presented in this chapterevolved in the course of the testing process. Some of the information presented in this chapterwill be thoroughly discussed in Chapter 5.

4.2 Material tests

4.2.1 Concrete properties

Ready mixed concrete was used with a nominal strength of C35/40. Cement type CEM-Iwas used and the maximum aggregate size was da = 10mm. The consistence class was S3(pump mix) with an average slump of 150mm. No additives were added to the concrete andthe same mix design was used for all tests. Concrete control specimens were always curedboth alongside the slabs and in water. The average water temperature measured 20◦C.

The concrete compressive strength was measured on 100mm cubes and on cylinders with anominal diameter of D=100mm and a nominal length of L=255mm. Top and bottom surfacesof the cylinders were ground before testing. The concrete tensile strength was measured onsplit cylinders (Brazilians) with a nominal diameter of D=150mm and a nominal length ofL=230mm. The split cylinder strength was obtained according to σs = 2P/ (πLD), where Pdenotes the splitting force. The actual dimensions of the control specimens were measureddirectly before testing. All control specimens were tested in a load controlled testing machinewith a loading rate of 300MPa/min. The mean concrete material properties are given inTable 4.1. Cylinder- and split cylinder strength were not obtained for the SP specimens.Nevertheless, the deviations of the concrete strengths are small which shows that the deliveredconcrete quality was fairly consistent.

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Specimen fcube fcyl fsplit

- [MPa] [MPa] [MPa]

Type-A grv. 48.16 43.48 2.9

Type-B grv. 48.73 43.17 2.6

Type-C grv. 47.60 41.28 3.0

Type-A lat. 48.16 43.48 2.9

Type-B lat. 48.90 43.97 3.0

Type-C lat. 47.60 41.28 3.0

SP1 44.40 - -

SP2 43.00 - -

Table 4.1: Averaged concrete properties

where Type-A, B and C refer to the large scale test details, the abbreviations grv. and lat.stand for gravity- and lateral loading respectively and SP1 and SP2 refer to the first andsecond slab panel (SP) test series.

4.2.2 Rebar properties

The reinforcement type used was Class-B with a nominal yield strength of fy = 550N/mm2.The tensile strength was typically measured on 400mm offcuts of the same batch of rein-forcement used in the test specimens. The direct tensile tests were displacement controlledwith a constant loading rate of 5mm/min. The elongation was measured by means of a clipextensometer with a gauge length of 100mm. The stresses were evaluated for the nominalcross section. A typical stress-strain curve of a rebar sample is depicted in Fig. 4.1.

Figure 4.1: Typical stress-strain diagram for T12 reinforcement

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The averaged reinforcement material properties are given in Table 4.2.

Specimen E0 σ02 σu εL

- [GPa] [MPa] [MPa] [%]

T12 200 552 617 14.5

Table 4.2: Averaged rebar properties

where E0 denotes the modulus of elasticity, σ02 denotes the 0.2% proof stress, σu representsthe tensile strength and εL represents the elongation after fracture.

4.2.3 Structural steel properties

In order to evaluate the steel material properties, the testing procedure followed the provisionsof BS-EN-10002-1 (2001). Necked tensile coupons were cold machined from the RSC-51x38channel section, where samples were taken from the centre of the flanges and the web. Neckedtensile coupons were cold extracted from the RHS-100x60x6 and SHS-180x10 sections eachin the centre on all four sides. The dimensions of the coupons are given in Table 4.3. Thehot rolled channel section had a nominal grade of S235 and the hot finished tubular sectionshad a nominal grade of S355.

Coupon Ltot wtot Lgauge A0

- [mm] [mm] [mm] [mm2]

RSC 51x38 flange 400 24 150 113

RSC 51x38 web 400 24 150 110

RHS 100x60x6 205 25 75 91

SHS 180x10 300 25 90 160

Table 4.3: Coupon dimensions

where Ltot is the total coupon length, wtot is the total coupon width, Lgauge is the gaugelength measured between the necked edges and A0 represents the nominal cross section.

The tensile tests were displacement controlled with a loading rate of 1.5mm/min. Elongationswere measured contactless by means of a optical extensometer (see Fig. 4.2). Therefore, twowhite marker points with a distance of 50mm (gauge length) were each administered to thesurface of the necked area. The axial elongation was measured electro-optically based on therelative displacement of the marker points.

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Figure 4.2: Coupon testing machine with video extensometer

The coupons after fracture for the RSC-51x38, RHS-100x60x6 and SHS-180x10 section aredepicted in Fig. 4.3, 4.4 and 4.5 respectively.

Figure 4.3: RSC 51x38 coupons after testing

Figure 4.4: RHS 100x60x6 coupons after testing

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Figure 4.5: SHS 180x180x10 coupons after testing

Typical stress-strain curves in the same order are presented in Fig. 4.6, 4.7 and 4.8.

Figure 4.6: Typical stress-strain diagram for RSC 51x38

Figure 4.7: Typical stress-strain diagram for RHS 100x60x6

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Figure 4.8: Typical stress-strain diagram for SHS 180x10

The averaged coupon material properties are given in Table 4.4.

Coupon E0 σ02 σu εL εf

- [GPa] [MPa] [MPa] [%] [%]

RSC 51x38 210 300 401 41 111

RHS 100x60x6 200 405 484 27 96

SHS 180x10 198 562 503 34 89

Table 4.4: Averaged steel properties

where E0 denotes the modulus of elasticity, σ02 denotes the 0.2% proof stress, σu representsthe tensile strength, εL represents the elongation after fracture and εf represents the tensileductility. The tensile ductility was obtained by εf = ln (A0/Au), where A0 denotes the initialcross sectional area and Au denotes the cross sectional area after fracture.

4.3 Large scale slab testing rig

A purpose-built test-rig was designed and constructed in order to enable a realistic experi-mental assessment of the behaviour of tubular column-to-flat slab connections under lateralcyclic loading conditions in conjunction with co-existing gravity loads. The layout of the testrig basically follows the test rig used by Pan & Moehle (1992). The prime condition for thedesign of the rig was to enable gravity- as well as uniaxial lateral loading either separatelyor in any combination of both. Furthermore, flexibility in terms of slab size, slab depth andcolumn height was considered for a more versatile application of the test-rig.

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Fig. 4.9 shows an elevation and the plan view of the large scale slab test-rig.

Figure 4.9: Schematic setup for large scale testing rig

As illustrated in Fig. 4.9, the test rig consists of the following parts:

• The 25t reaction frame connected to the strong floor via nine pre-stressed bolts

• The 25t horizontal actuator connected to the reaction frame and the top of the columnstub via hinges

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• The ’tie member’ connected with the reaction frame and the bottom of the column viahinges

• Eight floor beams each connected to the strong floor with two pre-stressed bolts

• Eight ’pin ended ties’ connected to the floor beams via one way hinges and attached tothe slab specimen by spherical bearings

• The 100t vertical hydraulic jack connected to the bottom of the column stub, sharingthe same hinge with the ’Tie member’

• The square-shaped reinforced concrete slab specimen

• The shearhead system cast in the slab and connected to the tubular steel column stubsvia bolted head plates

The rig was constructed in such a way that the slab could move horizontally in the directionof lateral loading. This causes a rotation of the column relative the slab which induces anexternal bending moment. The mechanism employed in this rig works independently of theapplied gravity load (see Fig. 4.10).

Figure 4.10: Simplification of the test rig system

In order to consider different slab depths and column heights, a height-adjustable connectionwas employed. Fig. 4.11 shows a detail of the height-adjustable connection which nips to thevertically aligned flanges of the reaction frame. This clamped connection is achieved by 16prestressed high strength M24 bolts.

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Figure 4.11: Hight adjustable clamped connection

Two of these connections were designed to connect both the horizontal actuator and the tie-member with the reaction frame. In this context the tie member balanced the lateral forceof the horizontal actuator in order to minimize horizontal (transverse) forces acting onto thevertical jack and the load cell.

Eight pin ended ties were symmetrically arranged around the edges of the specimen whichconnected the latter with the strong floor. Four ties were each located at the corners whereasfour ties were located at the midspans of the slab.

The bottom ends of the ties were equipped with one-way hinges, whereas the top ends com-prised spherical bearings. These bearings were centrally pressed into a milled groove in thecentral plate of the hinge (see Fig. 4.12). These spherical bearings enabled the hinge to alsoswivel perpendicularly to the main axis of rotation which allowed for the small out of planerotation of the slab when loaded vertically.

The bottom hinges of the pin ended ties were bolted to floor beams which were made ofpairs of parallel flange channel (PFC) sections. The top hinges were connected to the slabspecimen as shown in Fig. 4.12.

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Figure 4.12: Detail of spherical bearing

Fig. 4.13 shows two details of the pin ended ties. The left detail shows a normal strut, whereasthe right detail shows a spliced strut. The spliced strut was located in front of the reactionframe and resolved the clash with the horizontal tie member.

Figure 4.13: Pin ended ties/struts

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Taking advantage of the square grid of strong bolts in the strong floor, four different specimensizes can be accommodated in the rig. This is possible by rearrangement of the floor beams.However, only the arrangement shown in Fig. 4.9 was adopted for all tests.

In terms of loading, the lateral load was applied either monotonically or cyclically througha 25t actuator with a stroke of ±125mm. The vertical load could be applied independentlyby a 100t hydraulic jack operating in a ’force-controlled’ mode. Fig. 4.14 shows the reactionframe, the height-adjustable connection with the horizontal tie member and the eight pinended ties as built. The grillage sitting on the top hinges of the vertical ties was used foralignment purposes to facilitate the emplacement of the slab specimen.

Figure 4.14: Reaction frame and pin ended ties with horizontal tie member

Fig. 4.15 shows a readily assembled slab specimen which is shown to be connected to thepin ended ties by eight pairs of M24 all-thread bars. Beams made of pairs of 1300mm longPFC-sections and 50mm thick square plates sitting on top of the slab were used to distributethe tie forces onto larger areas. Fig. 4.15 also shows the horizontal actuator connected tothe clevis on top of the column head. The scaffolding poles shown were part of the external

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frame which was used to measure the slab displacements.

Figure 4.15: Testing rig with slab specimen in cyclic loading configuration

Fig. 4.16 shows a close-up of the three-way hinge which connected the vertical jack with thecolumn end and the horizontal tie member. A load cell was located between the jack and thebottom plate. Top centre of Fig. 4.16 shows the bolted joint between the column stub andthe cast-in shearhead. This joint was designed to alleviate casting of the slab as the mouldcould be placed directly on the floor.

Figure 4.16: Close up of vertical hydraulic jack with three-way hinge and bottom columnstub

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4.4 Slab panel testing rig

This section will highlight the specifications of the purpose-built test-rig in order to enablean experimental assessment of the composite behaviour of the embedded shear arms. Thegeneral layout of the test rig follows the requirements of reflecting in-situ conditions of thelarge scale tests.

Figure 4.17: Schematic setup for slab panel rig

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A simplification of these conditions was necessary to ease the handling of the specimen aswell as to reduce the complexity for the benefit of behavioural interpretation.

The rectangular slab panel (SP) specimen was held within a specially designed frame. Asillustrated in Fig. 4.17, this frame consisted of two side walls made of 15mm thick steel plates.These were stiffened with two transverse walls in order to form a rigid box. The frame waswelded to 50mm thick base plates which were stressed to the strong floor by 4 floor bolts.

The specimen was simply supported along its shorter sides by means of channel sections.Therefore, PFC-sections were welded to the side walls. Additionally, the flanges were re-inforced with triangular stiffeners. The specimen was wedged between the flanges of thechannel by using pairs of steel wedges which prevented the specimen from loosening duringcyclic loading. Fig. 4.18 shows the slab panel wedged between the flanges of the PFC-sectionas described before.

Figure 4.18: Close up of installed specimen

A 25t actuator was accommodated within an externally braced testing frame which wasdirectly located above the steel frame that held the specimen.

Fig. 4.19 shows the general assembly as built, with the actuator being connected to thespecimen via a double hinge. Out-of-plane forces arising from out-of-plane displacements ofthe double hinge were transferred to the external frame by a horizontal guide beam. Thisbeam was designed to move with the actuator ram to prevent the induction of transverseforces into the interconnected load cell.

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Figure 4.19: Slab panel rig with specimen and external frames for gauging

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The bottom base plate of the hinge was bolted to an interface member which connected it tothe slab panel. Fig. 4.20 shows a close up of this clamped connection. The clamp consisted ofa top and bottom plate with a M20 all thread bar either side which enabled it to be clampedonto the sheararm extension of the slab panel.

Figure 4.20: Slab panel with shear arm connection and gauging system

4.5 Large scale slab specimens

4.5.1 Determination of slab size

The large scale specimen represented a small part of a greater structural system. Ideally, thetraction forces along the boundary of such a specimen are exactly the same as those actingon the equivalent virtual boundary in the domain from which the latter has been removed(Cut-principle).

The determination of the specimen size in a concentric gravity loading case is reasonablystraight forward. The circular boundary follows the geometric locus where the radial bending

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moments are zero. The diameter of this circle is D ≈ 0.44L in a square flat slab system withthe span L. The shear-traction forces which are uniformly distributed along the edge aretypically replaced by concentrated tie-forces which are evenly distributed in a polar arrayalong the slab edge. Determination of the specimen size in a combined loading situation withvariable eccentricity however requires a compromise.

For the determination of the specimen size ’St-Venant’s principle’ was applied. The engi-neering interpretation of St-Venant’s principle states that: ’Two statically equivalent forcesystems that act over a given small portion S on the surface of a body produce approximatelythe same stress and displacement at a point in the body sufficiently far removed from theregion S over which the force systems act’ (Boresi & Chong, 1987).

Hence, it was to show that the disturbances caused in the specimen by the simplified boundaryconditions are sufficiently far removed from the examined region of interest. In other words,within the greater (superordinate) structural system the conditions in close proximity of thecolumn must not deviate too much from the conditions in the slab specimen.

The specimen size was based on a typical two-storey reference building (i.e. superordinatestructural system) with a storey height of 3.0m and with 5.0m square panels. Linear elasticfinite element analysis was conducted. Columns were modelled with beam elements and thefloor slabs were modelled isotropically with shell elements. Two different load cases wereconsidered separately. Load case one (LC1) involved gravity loading and load case two (LC2)involved lateral loading only. The lateral load was applied horizontally (in plane) along thespans of every floor.

Fig. 4.21 compares the displacement contour plots in the reference building (dashed boundary)with that in the slab specimen (solid boundary). The superscripts ’a’ and ’c’ representgravity loading conditions whereas ’b’ and ’d’ represent lateral loading conditions. The blackbold dots along the boundary of the specimen represent pin supports as considered in thearrangement of pin ended ties in the test rig (see Fig. 4.9). The optimum specimen size andaspect ratio for both load cases was found iteratively on a trial and error basis. Fig. 4.21shows that despite the huge deviation of displacements near the boundary, the displacementcontours close to the column are similar. Hence, the depicted slab geometry was consequentlyadopted for the large scale testing scheme.

It is noteworthy that due to St-Venant’s principle, punching shear tests on square slabs givevery similar results to equally sized circular slabs.

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Figure 4.21: Elastic displacement field contours of reference building and specimen

4.5.2 Specimen details

All slab-to-column assemblages were constructed at 60% of the full scale. The spans of theslabs measured 1440x1220mm between the centrelines of the pin-ended ties. The slab depthof 155mm was chosen to minimise size effects without exceeding the maximum allowable selfweight which was limited by the crane capacity. The column measured 1185mm between themidpoints of the clevises located at the top and bottom ends of the column.

Type-A specimen reflected a conventional shearhead configuration (see Fig. 4.22). Type-Band Type-C specimen reflected a novel, partially integrated configuration. Fig. 4.23 showsthat both Type-B and Type-C specimens featured a sqare hole around the column whichmeasured 500x500mm.

All specimens were cast in oiled plywood moulds. They were then covered with plastic sheetsin the first week after casting. In total six large scale specimens were employed, and two ofeach type were cast simultaneously.

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4.5.3 Reinforcement details

The tensile reinforcement ratio of ρ = 1.06% was chosen to ensure that the flexural capacityof the slab was higher than its expected punching shear capacity.

The tensile reinforcement consisted of T12 bars at 100mm centres in each direction. Thecompressive reinforcement consisted of T12 bars at 200mm centres all with a nominal coverof 20mm.

The Type-B and Type-C specimens had bundled reinforcement bars adjacent to the edges ofthe hole, to compensate for the bars that were discontinued across the width of the opening.

Additional U-bent bars were used to trim the edges of the opening as well as the slab edges.There was no difference between the Type-B and Type-C reinforcement layout and the rein-forcement ratio was the same for all specimens. Fig. 4.22 and 4.23 illustrate the reinforcementlayout of Type-A, Type-B and Type-C specimens.

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Figure 4.22: Type-A reinforcement layout

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Figure 4.23: Type-B and Type-C reinforcement layout

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All shearheads were centrally embedded between the four reinforcement layers. Fig. 4.24shows a close-up of the reinforcement arrangement of the fully integrated Type-A specimen.The concrete, in this case, was cast up to the column section.

Figure 4.24: Type-A reinforcement and shearhead configuration

Fig. 4.25 shows the reinforcement detail of the partially integrated Type-B specimen. Theconcrete, in this case, was cast up to the edge of a plywood box in such a way that afterremoval of the box a 500x500mm hole was left around the column.

Figure 4.25: Type-B reinforcement and shearhead configuration

Fig. 4.26 shows the Type-C shearhead-reinforcement detail. The concrete was cast up to theedge of the collar (edge beams) in such a way that bottom and top surfaces were both flushwith the flanges of the collar. A detailed description of the shearheads used are given in thefollowing subsection.

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Figure 4.26: Type-C reinforcement and shearhead configuration

4.5.4 Shearhead details

The cruciform Type-A and Type-B shearheads used measured 2Lv = 1000mm from tip totip. They consisted of a column made of a square hollow section SHS-180x10. The samecolumn section was adopted for all shearhead details. Rectangular, 30mm thick head plateswere through-welded to the ends of the column section which were bolted to the columnextensions using six pre-stressed high strength M24 10.9 bolts. That facilitated casting andhandling of the specimen as well as enabling the reuse of the column extensions.

The shear arms were made of pairs of hot rolled back to back welded channel sections RSC-51x38. Two 4mm V-welds were therefore continuously provided along the splice. The layoutof the shear arm-to-column connection followed the need to prevent yielding of the columnsection. The shear arms could therefore not be directly welded onto the column walls. Thus,the shear arms were connected to the column via 8mm thick polygonal shaped gusset plates.These were located on top and bottom of the shear arm flanges like a sandwich. These gussetplates measured 360x360mm and were destined to transfer the forces from the shear arms tothe column smoothly and reliably.

The shear arms were connected to the gusset plates by fillet welds. In the Type-A and theType-B version fillet welds were provided parallel to the flanges underneath the gusset plates.Additional head fillet welds were provided along the onset of the gusset plates. The gussetplates themselves were continuously fillet welded (weld thickness a=6mm) all around thecolumn section. Fig. 4.27 depicts the Type-A and Type-B shearhead geometry.

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Figure 4.27: Type-A and Type-B shearhead detail

Fig. 4.28 shows a close up of the gusset plates and the welding detailing as used for Type-Aand Type-B shearheads.

Figure 4.28: Type-A and Type-B shear arm-to-column connection

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Fig. 4.29 depicts the Type-A shearhead as built.

Figure 4.29: Type-A shearhead as built

Fig. 4.30 shows the Type-B shearhead detail as built.

Figure 4.30: Type-B shearhead as built

Fig. 4.31 shows that three shear dowels in form of 75mm long T12 offcuts were each weldedonto the embedded parts of the sheararms in order to improve their composite behaviour.

Figure 4.31: Close up of Type-B shear arm as built

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Fig. 4.32 illustrates the geometry of the Type-C shearhead which measured 2Lv = 1300mm

from tip to tip. The adopted shear arm sections were the same as those used in Type-A andType-B versions. The thickness of the gusset plates was increased to 12mm and the gussetplates consisted of eight symmetric parts which were arranged in such a way that the weldingseems were aligned parallel to the axial shear arm directions.

Figure 4.32: Type-C shearhead detail

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Fig. 4.33 shows the fatigue detail category 56* according to EC 3 (2005a).

Figure 4.33: Detail category 56* (from EC 3, 2005a)

Fig. 4.34 shows the detail category 125 with a different welding detail.

Figure 4.34: Detail category 125 (from EC 3, 2005a)

Both figures shows that the expected crack initiation stress level is more than twice as highwhen head fillet welds are omitted. Although the Eurocode provisions for fatigue are basedon high cycle fatigue tests, the welding detail shown in Fig. 4.35 was expected to also behavefavourably under low cycle fatigue conditions.

Figure 4.35: Type-C shearhead welding detail without head fillet weld

A square collar in form of parallel flange channels PFC-150x90x24 was provided around thecolumn. The collar measured 500x500mm between the internal web edges. The flanges ofthe collar pointed outwards in such a way that they encompassed the edges of the opening.The shear arms passed through rectangular apertures which were provided in the webs of thePFC-collar. The shear arms were connected to the collar by means of 15mm thick weldedstiffener plates. These were located between the flanges of the shear arm and the collar.

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Fig. 4.36 shows the Type-C shearhead as built. It shows 150mm long anchor pieces made ofRSC-51x38 offcuts which were perpendicularly welded to the tips of the shear arms. Addi-tional pairs of U-bent T16 bars were provided at either side of the shear arm. These barswere welded to the webs of the collar. Therefore, holes with a diameter of 20mm were drilledinto the collar’s web. The ends of the U-bent bars were circumferentially weld-connectedfrom both sides.

Figure 4.36: Type-C shearhead as built

4.6 Instrumentation for large scale slab tests

Loads were measured with load cells positioned between the hydraulic rams and the hingesat the respective column ends. Fig. 4.37 gives an overview of the instrumentation of the largescale slabs.

Displacements of the horizontal actuator ram were measured in the actuator itself. Horizontaldisplacements of the slab were measured with LVDT’s (transducers) on two points along theshorter span of the specimen.

The central vertical column displacement was measured with a transducer that was locatedbetween the strong floor and the bottom hinge. Vertical displacements of the slab were mea-sured with LVDT’s located above two pin ended ties located at two mutually perpendicularedges. Furthermore, vertical displacements of the edge of the hole or the collar were measuredwith draw wire transducers which were located below the slab.

During cyclic testing, vertical displacements of the slab edges were measured by means oftwo laser interferometers. This contact-less method was necessary to prohibit disturbancescaused by sliding of the transducer.

Compressive concrete surface strains were measured with electrical resistance strain gauges.These were attached to the bottom surface with a two component polyurethane resin. As de-picted in Fig. 4.37, two pairs of strain gauges were located below two mutually perpendicularshear arms in order to measure tangential- and radial strains.

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Figure 4.37: Test setup and instrumentation of slab specimen

The top surfaces of all slabs were white-washed for better contrast when marking the crackpattern. Tensile concrete surface strains were measured manually at various load stagesadopting the Demec system. Therefore, a 150mm wide marker grid was attached to thesurface with plastic padding. The grid enabled the measurement of tangential- and radialstrains directly above one longitudinal and one transverse shear arm.

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Fig. 4.38 depicts the location of the adopted Demec marker scheme.

Figure 4.38: Location of strain gauges and Demec markers

Fig. 4.38 also shows the location of strain gauges at the shear arms. These were surfacemounted with cyanoacrylate adhesive after adequate preparation of the designated area. Thestrain gauges and feed cables were then sealed with a thin layer of polyurethane resin. Axialstrains of the extreme fibres were measured in the top- and bottom flanges of two mutuallyperpendicular shear arms by means of electrical resistance strain gauges.

In Type-A detail strains were measured at the critical section next to the onset of the gussetplates and at the ends of the shear arms. In Type-B and Type-C details strains were measuredat the onset of the gusset plates and close to the shear arm-collar intersection.

Additionally, shear strains were measured by means of electrical resistance strain gaugerosettes. These were centrally located at the web at the critical section.

4.7 Slab panel specimens

The geometry of the SP was chosen to reflect the local conditions of the embedded sheararm of Type-B detail. The SP quasi represented one quarter of the large scale slab; that

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is, it represents one embedded shear arm of a partially embedded shearhead detail. The SPtests therefore provided valuable information on the composite behaviour of the shear armsin the large scale tests, especially on the effects of local concrete failure and concrete strengthdegradation.

The specimen measured 900x600mm on plan with a depth of 155mm. Fig. 4.39 shows thegeometry of the slab panel with Series-1 shear arm details. The location of Series-2 sheararms within the SP was the same as in Series-1.

Figure 4.39: Series-1 and Series-2 slab panel with shear arm position

All specimens were cast in oiled plywood moulds. Series-1 and Series-2 specimens were eachcast at the same time. The specimens were covered with plastic sheets in the first week aftercasting. In total 10 slab panel specimens were tested. Series-1 consisted of 4 slab panelswhereas Series-2 consisted of 6 slab panels.

4.7.1 Reinforcement details

The reinforcement detailing was chosen to reflect the reinforcement layout of the large scaletests as closely as possible. Fig. 4.40 shows the adopted reinforcement layout which consistedof T12 bars at 100mm centres on the top and T12 bars at 200mm centres at the bottom.The nominal cover was 20mm. The reinforcement consisted of closed loops to achieve therequired anchorage. In the second test series, additional stirrup triples made of T8 bars wereemployed at both sides next to the supports at the front edge. This enhanced the shearcapacity of the panel to achieve plastic deformation in the shear arm extensions.

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Figure 4.40: Slab panel reinforcement layout

4.7.2 Shear arm details

The shear arms of all series were centrally positioned in the slab panel between the four rein-forcement layers. The shear arm extensions which projected out of the front edge measured150mm. The shear arm extensions were stiffened with 8mm thick welded end plates. Theload was applied at an eccentricity of 85mm from the slab edge. This eccentricity was chosento make shear action dominant in the shear arms as this was the governing case in the largescale tests.

In Series-1 two different cross sections were adopted. Firstly, a rectangular box section RHS-60x100x6.3 was used. Secondly, the same section was sawn in half and welded back-to-backin order to form an I-section configuration with the same capacity as the one before.

Two of the specimens involved a 500mm long parallel flange channel PFC-150x90x24 whichwas trimming the edge of the slab. This channel section, which will subsequently be referredto as ’edge reinforcement’, was cast flush with the face side. The flanges therefore encom-passed the edge of the opening. The shear arms intersected the edge reinforcement throughrectangular apertures which were provided in the webs.

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Table 4.5 gives the embedded length L and other details of shear arms used in Series-1 tests.

Specimen Length Section Edge reinf.

SPT1-1 250mm Box No

SPT1-2 250mm I No

SPT1-3 250mm Box Yes

SPT1-4 250mm I Yes

Table 4.5: Properties of Series-1 shear arms

Fig. 4.41 and 4.42 show a close up of the connection between the shear arm and the edgereinforcement. In the case of the SPT1-3 and SPT1-4 detail, 15mm thick stiffeners werewelded between the shear arm section and the flanges of the edge reinforcement. These werelocated above and below the webs of the shear arm section to ensure an efficient shear forcetransfer. This connection detail was adopted in all edge reinforced specimens including theType-C shearhead detail.

Figure 4.41: SPT1-3 shear arm-to-edge reinforcement detail

Figure 4.42: SPT1-4 shear arm-to-edge reinforcement detail

Fig. 4.43 and 4.44 show SPT1-1 and SPT1-2 shear arm configurations without edge rein-forcement. Fig. 4.45 and 4.46 show SPT1-3 and SPT1-4 shear arm configurations with edgereinforcement.

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Figure 4.43: SPT1-1 shear arm assemblage




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In Series-2, six different edge reinforced shear arm details were investigated. All adopted sheararms consisted of pairs of hot rolled back-to-back welded RSC-51x38 sections (the same asthose adopted as shear arms in the large scale tests). The edge reinforcement was the sameas used before but in one case consisted of a cold formed C-150x90x5 channel section with aconstant thickness of 5mm.

Additional features involved 500mm long T16 U-bent bars which were located at both sides ofthe shear arm. These were fillet welded to the edge reinforcement and fully tied into the mainreinforcement. Some details also comprised anchors in the shape of 150mm long RSC-51x38offcuts which were perpendicularly welded to the ends of the arms (like in Type-C detail).Table 4.6 gives the embedded length L and other details of shear arms used in Series-2 tests.

Specimen Lv Edge reinf. U-bars Anchor

SPT2-1 150mm PFC 150x90x24 Yes No

SPT2-2 340mm PFC 150x90x24 No No

SPT2-3 250mm PFC 150x90x24 Yes Yes

SPT2-4 250mm PFC 150x90x24 Yes No

SPT2-5 250mm C 150x90x5 Yes No

SPT2-6 250mm PFC 150x90x24 No Yes

Table 4.6: Properties of Series-2 shear arms

where Lv represents the embedded length of the shear arm.

Figs. 4.47 through 4.52 show all Series-2 shear arm configurations as specified in Table 4.6.


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4.8 Instrumentation for slab panel tests

Fig. 4.53 gives an overview of the instrumentation setup used for the slab panel tests. Thetop- and the front faces of all specimens were white-washed for better contrast. The actuatorload was measured with load cells that were positioned between the hydraulic ram and thedouble-hinge.

Figure 4.53: Instrumentation setup for slab panel tests

Displacements of the ram were measured within the actuator as all tests were displacementcontrolled. The central vertical shear arm displacement was measured with a transducerlocated between the strong floor and the bottom plate of the clamped connection. Verticaldisplacements of the slab panel were measured on the top surface with LVDT’s locatedadjacent to the supports and at midspan. Horizontal displacements of the slabs were measuredwith LVDT’s on two points along the back edge of the slab panel. End rotations of the sheararm extension and those of the edge reinforcement (if present), were measured by means ofinclinometers.

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Typically for all adopted shear arms, Fig. 4.54 shows the location of the electrical resistancestrain gauges at the shear arms. These were surface-mounted with cyanoacrylate adhesiveafter adequate preparation of the designated area. Axial strains in the extreme fibres weremeasured in the top- and bottom flanges. Additionally, shear strains were measured by meansof electrical resistance strain gauge rosettes. These were centrally located at the web at thecritical section.

Figure 4.54: Strain gauge location at shear arms

4.9 Concluding remarks

This chapter gave an overview on the experimental methodology adopted for the tests pre-sented in this work. The test series consisted of large scale and small scale tests. Materialproperties of concrete, reinforcement and steel used for the specimens were presented.

Thereafter, the needs for development and design of the large scale test rig were discussed.It was shown that the rig is capable of simulating gravity- and cyclic lateral loading eitherseparately or in any combination of both. Subsequently, the reinforcement detailing andthe adopted shearheads were explained. Regarding the latter, firstly a conventional ACI-type shearhead was adopted which was then further developed. Furthermore, the employedgauging systems were explained which included the position of load cells, transducers andstrain gauges.

Following the need to investigate the local shear behaviour, a slab panel test rig was devel-oped. It was shown that this rig was capable of simulating uniaxial cyclic shear loading.Subsequently, the reinforcement layout and the adopted shear arm details were presented.Series-1 shear arm details were presented and discussed. Their modification for Series-2 testswas subsequently discussed. Finally the gauging system for slab panel tests was presented.

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Chapter 5

Large- and small scale test results

5.1 General remarks

This chapter presents the results of large- and small scale tests in the order conducted. Theresults presented include loading protocols, crack patterns, crack widths, load displacementresponses, measured strains, a detailed description of the observed mechanical behaviour andthe failure mechanisms.

This chapter starts with gravity loading tests conducted on Type-A and Type-B shearheadconfigurations. It continues with Type-A and Type-B results of cyclic loading tests. Smallscale test results of Series-1 and Series-2 slab panels follow. Ensuing tests of Type-C detailunder gravity- and combined loading conditions are then being presented. Finally the testresults are compared and assessed and the conclusions drawn are summarised.

In the large scale tests the expression ’longitudinal’ is defined as any direction parallel tothe lateral (horizontal) loading direction. This corresponds to the X-direction as shownin Fig. 4.22 and 4.23 in Chapter 4. Consequently, the expression ’transverse’ denotes anydirection perpendicular to the lateral loading direction (Y-direction). The expression ’vertical’is defined as an upwards direction perpendicular to the slab or perpendicular to the XY-plane.The term ’left hand side’ in conjunction with the longitudinal shear arm is defined by lookingin the positive Y-direction.

In the small scale tests the term ’longitudinal’ is defined as a direction parallel to the sheararm axis, whereas ’transverse’ means perpendicular to the shear arm axis.

The terminology ’fuse’ denotes the dissipative zone or dissipative element in the shearhead,which refers to the shear arm length between the gusset plates and the edge of the hole inthe slab.

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5.2 Type-A gravity loading test results

The gravity load was applied vertically from the bottom of the column in 100kN steps. Ateach load step the cracks were marked, crack widths were measured and a set of Demecreadings was taken. At a load level of 100kN radial hair width cracks formed at the topsurface at the corners of the column. Tangential cracks were first observed at 200kN wherethen a regular crack pattern developed as the load was further increased. Failure occurredin punching at a load level of 450kN.

The load displacement response is depicted in Fig. 5.1 which shows a distinct brittle responsewhich is typical for punching shear failure in flat slabs.

Figure 5.1: Load displacement response of Type-A detail

The crack pattern and the failure perimeter are shown in Fig. 5.2. The continuous yellowline represents the visible perimeter, whereas the dotted line is the assumed extension of theperimeter, based on acoustic examination. The visible perimeter is relatively large since theshear crack ran along the top reinforcement. This caused the tensile reinforcement layersand the cover to separate from the concrete below. It is very likely that a smaller andsteeper punching cone formed closer to the column. The location of the shear arms andthe associated control perimeter according to the ACI-building code (dashed line) are alsodepicted in Fig. 5.2. It appeared that the actual failure perimeter was much larger thanpredicted.

The largest radial crack width measured was 1.04mm at a load level of 400kN. The highestradial tensile strain measured directly above one shear arm was 0.21% at the same load level.The maximum tangential strain across the shear arm measured 0.51%. No significant yieldingof the reinforcement occurred when considering these low strain levels and the additionalreduction due to back projection of the strains from the surface to the reinforcement level.

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Figure 5.2: Failure perimeter and crack pattern of Type-A gravity loading test

The shearhead was extracted from the slab after failure to investigate the post failure de-formation in the arms and in the column. Fig. 5.3 shows the permanently deflected sheararm which indicates the occurrence of minor plastic deformation at the onset of the gussetplates. From this it can be inferred that the critical section of Type-A detail is located atthe intersection of the shear arm with the gusset plate.

Figure 5.3: Plastic deformation of Type-A shear arm

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For investigation the inside of the column, the bolted head plates were removed. It showedthat the gusset plates prevented plastic deformation of the walls and therefore avoided localbuckling of the column.

Fig. 5.4 depicts the axial strain development in top and bottom flanges of a longitudinal anda transverse arm. The strains are shown for the critical section next to the onset of the gussetplates. It shows that the compressive strain levels in the bottom flanges are lower than thosein the top flanges which indicates composite action. The strain levels exceed the yield strainlevel of 0.15% but significant plastic deformation did not occur (see Fig. 5.3).

Figure 5.4: Axial strains in flanges at critical section

5.3 Type-B gravity loading test results

The gravity load was applied vertically from the bottom of the column in 30kN steps. Radialcracks were first observed at a load level of 60kN. These cracks occurred at the corners ofthe opening and above the embedded parts of the shear arms. Extensive formation of radialcracks occurred at 150kN when the surface cracks started to penetrate through the depth ofthe slab along the sides of the opening. At 210kN local shear cracks developed at the bottomflanges and propagated at an angle of ≈ 45◦ along the sides of the opening towards the tensilesurface.

Fig. 5.5 shows a close-up of the shear arm intersection. It shows such a typical shear crackat a load level of 240kN and two surface cracks penetrating the slab directly above the sheararm.

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Figure 5.5: Crack formation in the side face of the opening

The specimen failed at 385kN due to local punching shear failure first at the transverse- andsubsequently at the longitudinal arms. Punching was triggered by the previously describedshear cracks. The load displacement response is depicted in Fig. 5.6 which shows a ratherductile response. The quasi-horizontal plateau was characterised by yielding of the reinforce-ment bars adjacent to the edges of the opening which was caused by direct bearing of theshear arms.

Figure 5.6: Load displacement response of Type-B detail

Fig. 5.7 shows the cracking pattern and the failure perimeter (yellow) above the shear arms.Looking at the marked cracks it appears that no tangential cracks occurred and that only aregular radial crack pattern developed. Maximum tensile strains at 360kN measured abovethe shear arms were 0.085% in the radial- and 1.88% in the tangential direction.

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The tangential strains were approximately 22 times higher than the radial strains which inferslow levels of radial bending moments associated with this detail.

Figure 5.7: Failure perimeter and crack pattern of Type-B gravity loading test

Fig. 5.8 shows a close-up of the local failure surface above a transverse shear arm. The U-bent bars trimming the edge of the opening did not enclose the main reinforcement in thisdirection. This prevented the vertical force component from being transferred to the mainreinforcement. This resulted in large crack widths and yielding of the flexural reinforcementdue to high bearing pressure of the shear arm onto the reinforcement.

The shear crack width in the longitudinal side face measured 19mm at ultimate. Significantcracking also occurred in the corners of the opening owing to stress concentrations in theseregions. The bottom surface (compressive side) on the other hand remained intact.

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Figure 5.8: Localised shear failure at shear arm

The shearhead was extracted from the slab after failure to investigate the plastic deformationof the fuses. Fig. 5.9 shows a close up of the shear arm between the gusset plates and theintersection with the slab which is indicated by line (b) shown on the right hand side. It showsthat considerable plastic shear deformation was evident and that plasticity was restricted tothe fuse length.

Figure 5.9: Plastic deformation of Type-B shear arm

Fig. 5.10 depicts the axial strain development in the top flanges of a longitudinal and atransverse arm. These are shown for a section next to the onset of the gusset plates (a)and for a section next to the edge of the hole (b). The strain levels in section (a) indicatedsignificant plastic deformation whereas the strain levels in section (b) remained below theyield point.

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Figure 5.10: Axial strains in flanges front and back

Fig. 5.11 shows the strains in section (b) separately at a larger scale. This shows thatthe strains changed from initial compression into tension which was most likely caused bygeometric nonlinearity effects. When looking at Fig. 5.9 it is evident that the zone of localplastic deformation in the flanges moved inwards as concrete damage progressed. Therefore,the measured strains in section (b) are not really significant.

Figure 5.11: Axial strains in flanges back

Fig. 5.12 shows the strain development measured in one longitudinal and in one transversearm at an inclination of 45◦ to the shear arm axis.

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Figure 5.12: Shear strains in webs

5.4 Type-A combined cyclic loading test results

For this test a combination of a constant gravity load and a cyclic lateral load was considered.The applied gravity load level of Fv = 200kN represented SLS conditions which were deemedto be 45% of the punching capacity of Type-A detail. The cyclic loading regime followedthe provisions of ECCS (1986). The required actuator yield displacement ey = 15mm wasestimated by preliminary numerical analysis. The yield displacement was defined as theintersection of the initial elastic slope E0 with the tangential stiffness of slope E1 = E0/10.Table 5.1 gives the actuator displacement uact. and the number of cycles per interval wheren = 1, 2, 3... denotes an interval counter starting from interval ’f’.

Interval cycles/interval uact.

- - [mm]

a 1 ±1ey/4

b 1 ±2ey/4

c 1 ±3ey/4

d 1 ±4ey/4

e 3 ±8ey/4

f 3 ± (2 + 2n) ey

Table 5.1: Cyclic loading characteristics

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The adopted loading regime is depicted in Fig. 5.13 which shows the actuator displacementversus the number of cycles. It needs to be stressed that the maximum stroke of the actu-ator was limited to ±85mm. This is 5mm less than the suggested displacement of ±90mm

according to the ECCS procedure. A loading rate of 10mm/min applied in a sine mode waskept constant for every cycle.

Figure 5.13: Cyclic loading regime for large scale tests

The loads were applied in two phases. In the first phase the gravity load was applied from thebottom in two steps in order to capture the crack initiation. The gravity load was thereaftermaintained with force control throughout the test. In the second phase the lateral load wasapplied in displacement control. The actuator was put on hold after every completed cycleto allow the specimen to be examined.

Crack initiation was observed at Fv = 100kN and the typical crack pattern developed whenthe load was further increased. After completion of the 1st cycle no additional cracks devel-oped but after the 2nd cycle new cracks started to propagate in the longitudinal direction.Additional radial cracks formed during the 3rd and 4th cycle with some cracks initiating atthe ends of the shear arms.

The slab failed in punching shear without any warning signs during the 6th cycle. Fig. 5.14shows that punching failure occurred at one side only where the forces from gravity- andlateral loading were additive.

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Figure 5.14: Punching perimeter of Type-A detail immediately after failure

Fig. 5.15 shows the crack pattern and the failure perimeter (green). The shape of the failureperimeter was similar to the gravity test but its size was restricted to half of the slab andthe other side showed no damage at all. At failure the crack pattern was purely radial andnot fully developed which shows that the present punching shear mechanism was differentfrom the gravity case. The bottom surface failed adjacent to the longitudinal column face bycausing a wedge-shaped cavity. This was caused by concrete crushing as a matter of directbearing of the column onto the surrounding concrete.

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Figure 5.15: Failure perimeter and crack pattern of Type-A combined loading test

Fig. 5.16 shows the hysteretic loops: significant stiffness degradation occurred in the 5th cyclewhich indicates that punching shear was precipitated by cyclic concrete strength degradation.

The maximum drift ratio reached was 2.5% with a maximum displacement amplitude ofumax = 30mm. The maximum lateral load magnitude was Fh = 76kN. This corresponds toa moment resisting capacity of around Mmax = Fh · 1.185 = 90kNm.

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Figure 5.16: Hysteretic response of Type-A detail

Fig. 5.17 shows the axial strain development during cyclic loading. The denotation (a) meanstop- and (b) means bottom flange which were both measured in a longitudinal arm next tothe gusset plates. It shows that again the tensile strain was higher than the compressivestrain and that the plastic capacity of the shear arms was barely utilised. The alternatingstrains in the transverse arm next to the gusset plates implies warping torsion action withinsignificantly low strain levels.

Figure 5.17: Axial strain development in critical section

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5.5 Type-B combined cyclic loading test results

For this test a combination of a constant gravity load and a cyclic lateral load was consid-ered. The applied gravity load level of Fv = 100kN represented the elastic limit load of theconnection. The same lateral loading scheme as depicted in Fig. 5.13 was adopted for thistest. Crack initiation was observed at a gravity load level of Fv = 80kN .

Radial cracks initiated in the corners of the opening in the slab and above the embeddedshear arms. After application of the 3rd cycle, shear cracks developed in the side faces ofthe opening. These cracks typically initiated at the flanges of the longitudinal shear armsand propagated towards the top surface in an angle between 30◦ and 45◦. With increasinglateral displacement, shear cracks also started to emerge in the opposite direction towardsthe bottom surface as a matter of load reversals which overruled the gravity load. Withevery subsequent cycle, additional radial cracks developed when the shear cracks reached thetop surface. After completion of the 8th cycle, the shear crack width measured 4mm whichindicates significant concrete damage.

Fig. 5.18 shows the development of shear cracks in both directions at the intersection of onelongitudinal shear arm with the slab.

Figure 5.18: Localised shear cracks at longitudinal shear arm intersection

This crack pattern indicates that a strut-and-tie model developed where the tensile componentof which was transferred by the legs of the U-bent bars trimming the edges. The behaviourwas indeed similar to the one observed in the Type-B gravity test. Fig. 5.19 shows thedeflected state of the longitudinal fuses. The plastic deformation of the left-hand side fusewas governed by shear. The increasing damage of the concrete at the shear arm intersectioncaused the arm to debond and therefore increased the effective length of the fuse. This caused

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the stiffness to decrease with increasing concrete damage. However, the local shear failurewas stable and not progressive, as it did not significantly decrease the connection strength.

Figure 5.19: Shear head deflection under lateral loading

Fig. 5.20 depicts the crack pattern at ultimate load which shows that punching shear failuredid not occur.

Figure 5.20: Crack pattern of Type-B combined loading test

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Instead, failure occurred in the 14th cycle as a matter of extreme low cycle fatigue (ELCF).The top flanges of the longitudinal shear arms fractured next to the head fillet weld at theonset of the gusset plate (see Fig. 5.21). This area was amenable for ELCF as a result of microstructural transformation due to welding head exposure, which increased the brittleness ofthe parent material.

Figure 5.21: ELCF failure of head fillet weld on top flange

Fig. 5.22 depicts the hysteretic response of Type-B detail. It shows that the hysteretic re-sponse was stable with very little strength degradation. The maximum drift ratio reachedwas 7.2% with a maximum displacement amplitude of umax = 85mm. The maximum mag-nitude of lateral load was Fh = 57kN, which corresponds to a moment resisting capacity ofaround Mmax = 68kNm.

Figure 5.22: Hysteretic response of Type-B detail

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Fig. 5.23 shows the axial strain development during cyclic loading. The strains are shown fora section next to the onset of the gusset plates (a) and for a section next to the edge of thehole (b).

Figure 5.23: Axial strain development in dissipative zone

The step-growth development of strains was caused by low resolution of the measured datawhich accidentally occurred during testing. However, it shows that the strains at the onsetof the gusset plates were much higher than those at the opposite side. The reason for thisis explained in Section 5.3. The strain levels at the critical section are considerably high asthese nearly reached the hardening range. The strain amplitudes dropped from the 12th cycleonwards which shows the beginning of ELCF failure.

Fig. 5.24 depicts the principal strains under 45◦ as a measure of shear in the web in onelongitudinal arm. It shows that the principal strains in the 5th and 6th cycle are higher thanthe axial strains in the flanges. The sudden drop of strains happened when the strain gaugerosette detached from the web.

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Figure 5.24: Shear strain development in dissipative zone

5.6 Series-1 slab panel test results

The cyclic loading regime followed the provisions of ECCS (1986). The required actuator yielddisplacement was estimated to be ey = 3.5mm based on preliminary monotonic numericalanalysis. The loading protocol used for all SP tests (unless stated otherwise) is depicted inFig. 5.25. The procedure which led to that loading regime is the same as described in Section5.4. The load was applied in displacement control. A loading rate of 5mm/min was used forall tests, and it remained constant for every cycle.

Figure 5.25: Loading regime for SP tests

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5.6.1 SPT1-1

This test involved a tubular shear arm without edge reinforcement. Crack initiation occurredearly into the first cycle. The cracks initiated at the corners of the tubular section andpropagated at an angle of approximately 30◦ towards the surface in the respective directionof loading. These cracks extended in the successive load cycles but no new cracks formed(see Fig. 5.26).

Figure 5.26: Crack pattern at front face of SPT1-1 detail

The stiffness reduced significantly when the shear cracks reached the upper and lower surfaces.The crack pattern on the top surface was not very significant, apart from a few cracksspreading along the shear arm. At the 5th cycle a punching shear cone started to developwhere the face side cracks intersected with the top surface and at the 8th cycle, a local failureperimeter was evident. The perimeter had a width of 600mm and was limited in depth to theembedded length of the shear arm, where the reaction force at the tip of the arm triggeredpunching shear failure. Fig. 5.27 shows the crack pattern at failure. The failure perimetercan be approximated as a semi-circle with a radius of 300mm.

Figure 5.27: Crack pattern at top face of SPT1-1 detail

Owing to the separation of the top and bottom embedment of the shear arm, the hystereticresponse showed extensive degradation and pinching effects from early cycles on. The peakload of 40kN occurred at a displacement of 5mm after which the response softened rapidly

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with increasing displacement. The test was stopped after the 12th cycle where the residualcapacity dropped to 20kN (see Fig. 5.28).

Figure 5.28: Hysteretic response of SPT1-1 detail

The response of the specimen was asymmetric since the response was very sensitive to crackinitiation; this depended on whether a tensile or compressive force was applied in the firstcycle. Therefore, the signs of the initial actuator force were chosen in alternation for everysubsequent cycle. The asymmetry was amplified by the double hinge configuration whichcaused the shear arm to rotate more in the downwards than upwards load cycle. No yieldingof the shear arm section occurred during the test. The end rotation measured at the head plateof the shear arm was ±1.2◦ at peak load and approximately ±4.3◦ at ultimate displacement.

5.6.2 SPT1-2

In this test the sheararm comprised an I-section. Cracks first developed in the 2nd loadcycle. The cracks initiated at the outer edge of the flanges and propagated at an angle ofapproximately 30◦ − 40◦ towards the surface supports in the respective direction of loading.Fig. 5.29 shows that the crack pattern on the front face was symmetric and similar to thatobserved in the Type-B large scale test.

From the 2nd cycle onwards longitudinal cracks formed in the top surface of the slab overthe flange of the shear arm. A fan-like radial crack pattern developed in the subsequent loadcycles.

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A punching cone appeared to develop in the 5th load cycle. The punching cone was boundby the inclined shear cracks that extended from the tips of the lower flange. Punching failureoccurred in the 7th cycle. The perimeter had a width of 700mm and was limited in depthto the embedded length of the shear arm. The failure crack pattern at ultimate load canbe seen in Fig. 5.30 where the yellow line indicates the failure perimeter based on acousticexamination. The failure perimeter can be grossly approximated by a semi-circle with aradius of 350mm.

Figure 5.30: Failure perimeter (yellow) of SPT1-2 detail

The capacity peak of 74kN was reached at a displacement of 7mm and from thereon a strongsoftening behaviour could be observed. The test was stopped after the 11th cycle where theresidual capacity dropped to 27kN (see Fig. 5.31). No yielding of the shear arm sectionoccurred during the test. The end rotation measured at the head plate of the shear arm was±1.1◦ at peak load and approximately ±4.1◦ at ultimate displacement.

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5.6.3 SPT1-3

This test incorporated a tubular shear arm with edge reinforcement. The load was appliedmonotonically in displacement-controlled steps with a loading rate of 2mm/min. The reasonfor this loading scheme was to investigate any significant behavioural differences betweencyclic and monotonic loading. The initial step size was 0.5mm/step up to 107kN which wasthereafter increased to 1.0mm/step up to a load level of 126kN. From thereon a step sizeof 2.0mm/step was adopted up to peak load. From there, two final load steps of 5mm and10mm were applied after which the test was stopped.

Cracking was first observed at 70kN when shear cracks emerged at both edges of the channeland propagated towards the supports (see Fig. 5.32).

Figure 5.32: Crack pattern of front face of SPT1-3 detail

The initial behaviour was governed by transverse beam action between supports. At around90kN the shear cracks at the support penetrated the top surface of the slab. This led to

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the reduction in stiffness which can be seen in Fig. 5.35. At this load level, inclined cracksemerged on the top surface at the corners of the channel section.

At a load level of about 130kN the shear capacity of the transversal beam at the supportsdeteriorated due to concrete degradation and from then on the behaviour changed over tolongitudinal bending action. The longitudinal bending action led to a force redistributionwhere parallel cracks emerged in the transverse direction. In this phase the shear force andbending moment transfer of the shear arm to the slab is of particular importance. Thelongitudinal bending action caused a deep crack along the edge reinforcing channel. Theconcrete at both ends of the channel had failed in shear, which resulted in a large rotation(see Fig. 5.33). The end rotation measured at the head plate of the shear arm was 2.2◦ atpeak load and 5.0◦ at ultimate displacement.

Figure 5.33: Permanent displacement of SPT1-3 detail after failure

For the shear force transfer mechanical interlock is required which can only build up if suf-ficient axial pressure is provided. Since the reinforcement was not connected to the edgechannel the increasing rotation decreased the shear capacity of the cracked section. Herethe purpose of the shear arm became twofold: firstly it activated axial force-induced contactpressure as a matter of bond action and secondly it directly transferred the forces via directbearing action.

With increasing load the bond capacity of the shear arm diminished and therefore the abilityof shear force transfer of the channel section onto the slab reduced. The failure perimeterand orthogonal crack pattern of the top surface are depicted in Fig. 5.34.

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The residual strength of the specimen is governed by direct bearing action of the embeddedshear arm from the edge of the channel onwards. The overall behaviour of the slab panel wasvery ductile with a distinct softening post peak behaviour. The peak load measured 140kN ata displacement of 14mm where the residual strength was 110kN at a displacement of 30mm(see Fig. 5.35).

Figure 5.35: Monotonic response of SPT1-3 detail

5.6.4 SPT1-4

This test involved an I-section with edge reinforcement. Crack initiation occurred during the4th cycle. The cracks initiated at the front face at the corner ends of the channel section andpropagated directly to the supports. A discrete symmetric crosswise shear crack pattern was

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evident inferring concrete compressive struts having developed between the edge supportsand the flanges of the channel (see Fig. 5.36).


With every subsequent cycle with increasing amplitude, significant concrete damage occurredin the cracks near the supports. The initial behaviour of the specimen was governed by beamaction in the transverse direction. In this case the crack pattern showed that the effectivewidth of the beam measured about 200mm from the front face. The beam action diminishedas the concrete was progressively damaged at the supports. Instead, the propagation of cracksfrom front to back indicated force redistribution into intact areas, which activated residualstrength. This shows that biaxial bending action governed the behaviour subsequent to thedeterioration of beam action. A large crack occurred along the flanges of the channel section,which fully penetrated the slab depth. The crack pattern at failure is shown in Fig. 5.37where the yellow line indicates the failure perimeter based on acoustic examination.


The failure perimeter width was defined by the span and measured about 800mm and thedepth of the perimeter measured about 350mm.

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The capacity peak of 130kN was reached at a displacement of 14mm and from thereon astrong softening behaviour could be observed. The test was stopped after the 16th cyclewhen the residual capacity dropped to 35kN (see Fig. 5.38). No yielding of the shear armsection nor visible plastic deformation of the channel section occurred during the test. Theend rotation at the head plate of the shear arm was ±2.4◦ at peak load and approximately±5.1◦ at ultimate displacement.


5.7 Series-2 slab panel test results

In general failure types of the Series-2 specimens were similar to the edge reinforced Series-1specimens. Therefore the Series-2 test results are not presented individually and emphasiswas put on highlighting presentation of the differences. Fig. 5.39 shows the typical crosswiseconcrete failure of the SPT2-1 specimen at the support.

Figure 5.39: Typical failure pattern near support (SPT2-1)

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During cyclic loading the wedge shaped areas which were enclosed by the shear cracks loosenedand came off. Fig. 5.40 depicts the hysteretic response of the SPT2-1 detail which had anembedded length of 150mm and welded U-bent bars but no anchor.


Fig. 5.41 shows the concrete failure of SPT2-2 specimen at the supports. This detail compriseda 340mm long shear arm without any additional features. It shows that the crack pattern atthe supports was neither affected by the presence of U-bent bars nor by the anchors.

Figure 5.41: Typical failure pattern near support (SPT2-2)

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Fig. 5.42 shows the hysteretic response of the same detail.


Fig. 5.43 shows the permanently deformed shear arm extensions of SPT2-3 detail. Thisdetail comprised of a 250mm long shear arm with U-bent bars and anchor. It shows thatthese measures significantly increased the capacity to such an extent that plastic deformationin the shear arm could be achieved. The U-bent bars became increasingly important whenthe shear strength at the supports diminished and the behaviour changed from transverseto longitudinal action. At this stage a deep crack developed along the edge reinforcementwhich fully penetrated the depth of the panel in each case. The U-bent bars prevented theshear arm-edge reinforcement assemblage from detaching from the slab panel. This increasedthe ductility of the detail as the shear force transfer via concrete aggregate interlock couldbe maintained. Furthermore, the U-bent bars enabled the transfer of eccentricity-inducedbending moments into the slab.

Figure 5.43: Plastic deformation of SPT2-3 shear arm extension

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The influence of the anchor in the SP tests was not as significant as the U-bent bars. Theanchor mainly affected the post peak response. Fig. 5.44 shows the failure pattern of thetop surface of the SPT2-3 detail. It shows that failure occurred along the supports due toconcrete crushing below the steel wedges, which highlights the integrity of this shear armdetail.

Figure 5.44: Failure pattern of top surface (SPT2-3)

The hysteretic response of the SPT2-3 detail is depicted in Fig. 5.45.


Fig. 5.46 depicts the hysteretic response of the SPT2-4 detail which consisted of a 250mmlong shear arm with U-bent bars but without anchor. It can therefore be assumed that theabsence of the end anchors decreased the energy absorption capabilities of this detail.

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Fig. 5.47 shows the permanently deformed flanges of the weak-edge reinforcement of theSPT2-5 detail as the flanges could not resist the bearing pressure acting on them. Thiscaused force concentrations to occur at the shear arm-slab panel intersection. The largestplastic deformations arose at the ends of the edge reinforcement because of the higher stiffnessnear the supports.

Figure 5.47: Close up of deformed edge reinforcement SPT2-5

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Fig. 5.48 shows the localised concrete failure caused by concrete strength degradation.

Figure 5.48: Failure pattern of bottom surface (SPT2-5)

Fig. 5.49 shows the hysteretic response of the SPT2-5 detail. In comparison to the responseof the SPT2-4 detail, the positive effect of the U-bent bars could not be utilised to the samedegree, which resulted in lower capacity and ductility.


Fig. 5.50 depicts the hysteretic response of the SPT2-6 detail which consisted of a 250mmlong shear arm without U-bent bars but with an anchor. It shows that the effect of U-barson the capacity was greater than the effect of the anchor.

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Table 5.2 gives an evaluation of the performance of all Series-2 specimens.

Specimen Fp ep Fp/Fu ep/ey Fdeg

- [kN] [mm] [1] [1] [%]

SPT2-1 195 16 1.47 4.85 20

SPT2-2 175 14 1.97 2.07 25

SPT2-3 215 19 1.25 4.83 27

SPT2-4 212 15 1.75 3.02 19

SPT2-5 205 14 1.60 2.46 22

SPT2-6 123 14 1.76 7.58 28

Table 5.2: Evaluation of Series-2 SP tests

where Fp denotes the peak capacity, Fu represents the residual capacity at ultimate displace-ment of 28mm, ey denotes the yield displacement as stipulated in the ECCS provisions, ep

represents the peak displacement and Fdeg is the degradation of capacity at peak.

5.8 Type-C gravity loading test results

The gravity load was applied vertically from the bottom of the column in 30kN steps. Hair-width cracks developed at the top surface as the load was increased to 90kN. These crackspropagated radially from the corners of the collar. At 150kN radial cracks also startedto initiate along the collar edges and propagated into a symmetric and regular pattern.Tangential cracks (parallel to the collar edges) occurred simultaneously above the embedded

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anchors at a load level of 320kN. With increasing load level at about 400kN these crackskinked and continued to propagate radially towards the corners of the slab.

The specimen was unloaded at a load level of 570kN without failure in order not to exceed thesafe working capacity of the test rig. The crack pattern at peak load is depicted in Fig. 5.51which shows that the behaviour was governed by flexure. Signs of pending punching shearfailure were not apparent.

Figure 5.51: Crack pattern of Type-C gravity loading test

Fig. 5.52 depicts the measured load displacement response including the unloading path. Thehighest circumferential strain measured in the top surface was 0.4% at 500kN. It occurredabove the longitudinal arm at a distance of 140mm away from the collar edge. The highestradial tensile strain at the same load level was 0.38% which occurred along the longitudinalarm above its tip. These cracks widely reclosed after unloading, which indicates that nowidespread plastic elongation of the reinforcement was present.

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Figure 5.52: Load displacement response of Type-C detail

At peak load the differential deformation of the fuse between the gusset plate and the collarmeasured 8mm. Fig. 5.53 presents the shearhead after it was extracted from the slab. Itshows the permanently deflected state of the fuses which infers shear deformation similar tothose observed in Type-B detail.

Figure 5.53: Permanent plastic deformation of fuses

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Fig. 5.54 shows that the shear arm extensions remained straight and that plasticity wasconfined to the fuses. Furthermore, also the collar showed no signs of plastic deformation.

Figure 5.54: Collar and shear arm extensions

Fig. 5.55 depicts the axial strain development in the top flanges of a longitudinal and atransverse arm. These are shown for a section next to the onset of the gusset plates (a)and for a section next to the edge of the hole (b). The strain levels in section (a) indicatesignificant plastic deformation whereas the strain levels in section (b) remained below theyield point.

Figure 5.55: Axial strains in flanges front and back

Fig. 5.56 shows the strains in section (b) separately on a bigger scale. This indicates thatthe strains changed from initial compression into tension. Similarly to the Type-B detail,geometric nonlinearity effects interfered with the localised bending strains in the flanges. Itis very likely that the strain levels in the bottom flange in section (b) were much higher andsimilar to the measured strain levels in section (a).

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Figure 5.56: Axial strains in flanges back

Fig. 5.57 shows the strain development measured in one longitudinal and one transverse armat an inclination of 45◦ to the shear arm axis. It shows that shear yielding occurred fromabout 375kN with a distinct hardening response.

Figure 5.57: Shear strains in webs

5.9 Type-C monotonic pushover test results

The specimen of Type-C gravity test was retested under combined monotonic loading con-ditions until failure. First a gravity load of Fv = 250kN was applied. Then the lateral loadwas applied in a first phase with a loading rate of 5mm/min in 10mm steps. The actua-

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tor displacement was stopped at 90mm displacement. Since total failure did not occur, theactuator displacement was reset to zero and the gravity load was subsequently increased toFv = 500kN . The lateral displacement was again increased monotonically in a second phaseuntil failure.

Failure was initiated in the longitudinal left hand side where forces from gravity- and lateralloading were additive, at a lateral displacement of 76mm. The deflected longitudinal sheararms with the governing fuse are depicted in Fig. 5.58.

Figure 5.58: Shearhead deflection at ultimate load (Fv = 500kN)

The head fillet weld which connected the flange with the gusset plates failed. Total failureoccurred in the second phase when the entire flange ruptured, which consequently led toa significant drop in capacity. The right hand side fuse and the transverse arms remainedintact. Fig. 5.59 and 5.60 show the weld failure in the left hand side fuse.

Figure 5.59: ELCF failure of flange (side view)

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Figure 5.60: ELCF failure of flange (top view)

Fig. 5.61 depicts the load displacement diagrams of both phases. Total failure occurred at48mm displacement at 50kN lateral load. The column drift ratio at failure was 6.2%.

Figure 5.61: Load displacement response of Type-C detail (retest)

5.10 Type-C combined cyclic loading test results

For this test a combination of a constant gravity load and a cyclic lateral load was considered.The applied gravity load level of Fv = 200kN was the same as applied in the Type-A test.The same lateral loading scheme as depicted in Fig. 5.13 was adopted for this test.

During application of the gravity load the developing crack pattern was the same as describedin Section 5.8. Subsequently, after every following lateral load reversal, the existing hair-widthcracks propagated radially towards the corners of the slab. The crack pattern was mainly

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caused by flexure and was stable upon application of the maximum amplitude as neither newcracks initiated nor existing cracks extended. The crack pattern after completion of the testis shown in Fig. 5.62.

Figure 5.62: Crack pattern of Type-C combined loading test

Fracture of the parallel aligned fillet welds which connected the top flanges with the gussetplates was visible at the 13th cycle. This led to a drop in capacity after each following cycle.Fracture was progressive, although the amplitude was not increased, and considering the lownumber of cycles which caused failure, this can be interpreted as ELCF failure. Fracturefollowed the fillet welding seams along the flange edges on top and underneath the gussetplates. The parent material remained intact, but the gusset plates eventually completelyseparated from the flanges.

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Figs 5.63 and 5.64 each show a close-up of the fillet weld failure at the connection of the sheararms with the gusset plates. Furthermore, Fig. 5.63 shows that the left-hand side fuse wasgoverned by shear yielding.

Figure 5.63: ELCF failure of fillet welds (side view)

Figure 5.64: ELCF failure of fillet welds (top view)

The hysteretic response of the Type-C detail is given in Fig. 5.65 which was stable throughoutthe test with limited degradation. A total number of 15 cycles was applied and the test wasstopped after a degradation of capacity of 23%. The lateral peak-load was Fh = 70kN whichcorresponds to an induced bending moment of Mmax = 83kNm. The maximum column driftachieved was 6.8%.

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Figure 5.65: Hysteretic response of Type-C detail

Fig. 5.66 shows an elevation of the slab with the permanent column rotation at ultimate.

Figure 5.66: Column drift at maximum actuator displacement

Fig. 5.67 shows the axial strain developments during cyclic loading. These are shown for asection next to the onset of the gusset plates (a) and for a section next to the edge of the hole(b). The figure shows that the axial strains levels are symmetrical in both sections undercyclic loading, because the collar prevented the local deterioration of the concrete strengthin section (b). Furthermore, ELCF effects became apparent at the 11th cycle.

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Figure 5.67: Axial strain development in dissipative zone

Fig. 5.68 depicts the shear strain development during cyclic loading. It shows that shearyielding in the webs of the longitudinal arms occurred prior to yielding of the flanges.

Figure 5.68: Shear strain development in dissipative zone

5.11 Comparative assessment

In this chapter the results of Type-A and Type-B large scale gravity tests were presented.Fig. 5.69 shows that the initial stiffness of Type-A and Type-B details were similar to eachother. Furthermore, the load displacement responses of the gravity tests revealed that thefully integrated detail had a higher capacity when compared to the partially integrated shear-

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head; albeit the response of Type-B detail was more ductile, and failure was less brittle thanthe Type-A detail.

Figure 5.69: Gravity load displacement responses of all details

Comparison of the measured strain levels and the deformed shape of the extracted shearheadsshow that the plasticity level was higher in Type-B detail; that is, the plastic capacity in thelatter case could be utilised to a much higher degree as the fuses distinctly yielded in shear.Nevertheless, the problem associated with Type-B detail was localised shear failure above theembedded parts of the shear arms due to force concentrations in the concrete.

Lateral cyclic tests conducted on the same details showed that the conventional shearheadconfiguration was largely unsuitable for extreme seismic loading. The performance undercyclic loading was poor, as punching shear failure was precipitated at a load level of abouthalf the gravity load capacity. In other words punching shear failure occurred with limitedwarning under gravity load SLS conditions.

Fig. 5.70 compares the hysteretic responses of the Type-A and Type-B details. It showsthat the Type-B detail performed significantly better than the conventional Type-A detail.Although the gravity load capacity of the Type-B detail was lower, the performance undercombined loading conditions was considerably enhanced. The hysteretic loops in Fig. 5.70show that those associated with the Type-B detail were stable which led to increased energyabsorption capabilities. Furthermore, punching shear failure under lateral loading conditionscould be prevented by the novel shearhead configuration.

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Figure 5.70: Hysteretic responses of Type-A and Type-B detail

Generally the behaviour of the partially integrated shearhead system was governed by ductileshear yielding of the fuses. Comparison of the measured strain levels shows that the utilisationof plasticity in the steel was higher in the partially integrated assemblage.

Although the novel concept was vindicated, two issues associated with the Type-B detailneed to be considered. Firstly, the force concentration at the shear arm-slab intersection ledto localised concrete shear failure and strength degradation under cyclic loading which hasa significant influence on the capacity. Secondly, the fuses are amenable for ELCF effectswhich can occur at the connection of the shear arms with the gusset plates.

Slab panel tests were conducted to address the afore-mentioned problem of localised concretedamage. In Series-1 the general behaviour of the unreinforced shear arms (SPT1-1 and SPT1-2) was relatively poor and largely not suitable for dissipative seismic design. All the Series-1specimens tested were governed by concrete fracture as the shear strength of the shear armswas greater than that of the concrete.

The behaviour of the specimen was governed by its ability to form an adequate strut-and-tiemodel to transfer the shear force from the arms into the concrete. The I-section behaved muchbetter than the tubular section and showed a higher capacity because the flanges enabled thedevelopment of compressive struts. These struts could rest on the flanges further away fromthe concrete surface, which increased the failure perimeter. In the case of a tubular section,no efficient strut-and-tie action could be mobilised and the resistance was mainly governedby local bending action of the reinforcement layers between the flanges and the surface.

Fig. 5.71 schematically shows the strut-and-tie model which explains the observed crackpattern in the front face of the specimens. In this context the reinforcing detail of the U-bent bars became important. In the gravity loading test of Type-B detail (see Section 5.3),

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the transverse arms failed before the longitudinal arms; the U-bent bars did not encompassthe main reinforcement in the transverse direction. Hence, the tensile component could notefficiently be transferred to the main reinforcement.

Figure 5.71: Strut-and-tie model for I-section

The edge-reinforced shear arm details behaved better and showed the potential for majorimprovement mainly in terms of capacity. The channel section prevented the concrete fromlocal damage and distributed the force onto a larger area. Therefore the edge-reinforcedspecimens could activate a larger perimeter. The rotational behaviour and its dependence onthe reinforcing detail was of particular importance for the Series-2 tests to mitigate concretestrength degradation.

The Series-2 results that were presented in this chapter showed that the behaviour of theshear arm detail could further be ameliorated by the introduction of welded U-bent barsand anchors. Both measures helped to maintain the contact pressure in the cracked inter-face between the edge reinforcement and the slab, which positively affected the shear forcetransfer. An elongation of the embedded shear arm length alone could not compensate forthe effectiveness of the U-bars. The specimen with the longer arm and without additionalreinforcement (SPT2-2) behaved in a rather brittle manner, and so did the specimen withthe weak channel. Comparison of the SPT2-1 with the SPT2-3 shows that the anchor had apositive effect on the post peak softening behaviour, when the behaviour shifted from trans-verse to lateral action. At this stage the anchored shear arms could more effectively maintainthe contact pressure between the channel and the slab.

It was found that the SPT2-3 detail behaved most favourably of all investigated details (seeTable 5.2). At this point it needs to be stressed that the peak capacity of the SPT2-3 detailexceeded the plastic web capacity of the shear arm extension by a factor of 2. This impliesthat the plastic section capacity was significantly enhanced by tensile action arising fromgeometric nonlinearity. The success of the U-bent bars and end anchors is explained by theefficient transfer of this tensile component.

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Fig. 5.72 compares the hysteretic responses of an unreinforced shear arm SPT1-2 with SPT2-3.The capacity of SPT2-3 was 2.8 times higher than that of SPT1-2 and the peak displacementwas increased three times. Although the SP tests are not perfectly reflecting the conditionsof large scale tests, strong similarities are inherent in terms of bending-shear interaction.

Figure 5.72: Hysteretic responses of SPT1-2 and SPT2-3

Therefore, the adoption of welded U-bent bars in the shearhead system helps to controlcracking whereas the anchor increases the pull-out capacity of the shear arm. An indicationfor significant tensile action in the shear arms can be deduced from the crack pattern depictedin Fig. 5.51, as tangential cracks evolved directly above the location of the anchors. Thisemphasises the importance of tensile action arising from higher order effects. Followingthe SPT2-3 detail, these additional features, along with an edge reinforcement in form ofa continuous collar, were consequently considered in the development of Type-C shearheaddetail.

Finally gravity- and cyclic test results of a partially integrated shearhead (Type-C detail)were presented in this chapter. Fig. 5.69 shows that although the ultimate capacity of theType-C detail remains unknown, the capacity of a conventional configuration was at leastexceeded by 25%. Although subjected to a higher gravity load level, the lateral peak loadcapacity of the monotonic push over test had about the same capacity as its fully integratedpendant. The push-over test of the Type-C detail also revealed a high connection toughnessas the residual resistance maintained two thirds of its initial capacity.

A cyclic lateral loading test showed that the behaviour of the Type-C detail was almostexclusively governed by the performance of the steel fuses. Fig. 5.73 compares the hystereticresponses of a fully integrated shearhead with a partially integrated shearhead. It showsthat the initial stiffness of both details was approximately the same. It also shows that

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the hysteretic response of the Type-C detail was closer to the theoretical bi-linear response.When the absorbed energy of 2.98kJ of the Type-A detail is compared with 15.71kJ of theType-C detail, it shows that the latter offers superior dissipative capabilities.

Figure 5.73: Hysteretic responses of Type-A and Type-C detail

Strain measurements revealed that the highest levels always occurred in the top flanges atthe transition from the shear arm to the gusset plates. With hindsight to ELCF failure,two different welding details as described in Subsection 4.5.4 were tested. In the case of theabandoned head fillet, fracture could only progress parallelly to the flanges without reducingthe effective cross section of the shear arm. In the case of the fully welded detail, the forcescould be transferred more efficiently due to the absence of shear lag, which resulted in a highermoment capacity. On the other hand, ELCF fracture severely affected the entire flange whichreduced the cyclic lifetime of the connection. It can be concluded that the modified weldingdetail had a longer cyclic lifetime but gave a lower moment capacity when compared to thefully welded detail. It needs to be emphasised however that neither of these adopted weldingdetails can be considered as ideal.

Results revealed that the collar successfully suppressed concrete degradation and that thesurrounding concrete remained widely undamaged. It was consequently shown that punchingshear failure under cyclic lateral loading conditions could be prevented.

In conclusion, it could be demonstrated by experiment that the concept of partially integratedshearheads for reinforced flat slabs works. Partially integrated systems offer significantlyimproved capabilities in terms of ductility and energy dissipation and the capacities of suchsystems remain within the range of fully integrated shearheads.

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Chapter 6

Numerical methodology

6.1 Introduction

The Finite Element Method (FEM) was originally invented due to the need for obtaining atleast approximate solutions to complex problems in the physical world. In the field of theFinite Element method the term ’approximate’ means that this method allows any degree ofdesired proximity to the exact solution which is only limited by the computational capacityof the computer used. The FEM is therefore an essential, extremely powerful, and very oftenthe only available tool to solve complex problems in engineering practice.

In the introduction to this chapter the author would like to take the opportunity to raise aspecific issue which needs to be addressed when using the FEM. Nonlinear Finite ElementAnalysis (NLFEA), as conducted in this thesis, predicts theoretical solutions to physicalproblems. A theoretical result can only be falsified but not verified (Popper, 1963). Thismeans that an NLFEA result can only agree with experimental results within acceptablelimits of error. It is therefore crucial for every NLFEA to validate the adopted procedureagainst different sources of experimental data in order to gain confidence in the correctnessof the results.

The source of errors can be split into measurement errors and those caused by the NLFEAprocedure itself, all of which causes ambiguity in the validation process. Regarding thelatter, assuming that geometry and boundary conditions have been considered correctly, thesources can be grossly split into discretisation errors (as NLFEA is an approximative method),software implementation errors and errors arising from the adoption of incorrect parameters.

Discretisation errors are generally tackled by mesh sensitivity studies. Significant deviationsof predicted results can be expected due to the differences of software code implementationsespecially when material nonlinearity is concerned. Apart fom that, probably the highestlevel of uncertainty in NLFEA is caused by the choice of material parameters. In this con-text the most ambiguous material parameters are those which either cannot be obtained byexperiment (ficticious parameters) or those which strongly depend on the measuring method.

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A parametric study is therefore essential when dealing with advanced material models toincrease the objectivity of the results.

For the numerical analysis conducted in this thesis the commercially available software DI-ANA v. 9.3 has been used.

This chapter gives a concise overview of the main concept behind FEM. Firstly the focus isput on the first order finite element analysis of linear elastic materials which includes all theimportant steps involved.

The linear finite element method is thereafter extended to NLFEA procedures. Firstly be-cause concrete shows a highly non-linear material behaviour. Secondly, because the problemstreated in this thesis require the consideration of geometric nonlinearity. Therefore, constitu-tive models for concrete and steel as implemented in DIANA are introduced. In conjunctionwith the concrete compressive behaviour a cracking concept which considers the tensile be-haviour of concrete is discussed in detail. Subsequently the adopted iterative solution strategyand algorithms for solving nonlinear problems is discussed.

6.2 Stress and strain tensors

This section contains the definition of the stress and strain variables which have been usedin this chapter for the presentation of the FEM and some relevant material models. Hereonly a selection of relevant stress and strain definitions is given (Boresi & Chong, 1987).For a concise formulation the ’Einstein notation’ (index notation) has been used. Since theproblems treated in this work are of a three dimensional nature the indices can take theintegers i=1,2,3 according to the axes of the Cartesian coordinate system. The Einstein sumconvention has been consequently adopted which says that two identical indices require asummation of the respective variable.

A stress state in an arbitrary point can be expressed as a symmetric second order tensorwhere the entities of which are vectors and follow the transformation rule:

σij =

⎡⎢⎣ σ11 σ12 σ13

σ21 σ22 σ23

σ31 σ32 σ33

⎤⎥⎦ (6.1)

The stress tensor can be transformed into principal normal stresses in such a way that theshear components vanish. The problem can be written as:

det (σij − σ δij) = 0 (6.2)

where δij is the so called Kronecker delta which is defined to be 1 if i=j and 0 otherwiseand which therefore represents the unity matrix. Equation 6.2 represents an eigenvalue

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problem. The characteristic equation of which is of cubic order and gives 3 solutions namelythe principal normal stresses σ1 ≥ σ2 ≥ σ3:

σ3 − I1σ2 + I2σ − I3 = 0 (6.3)

where I1, I2 and I3 are the stress invariants which represent the trace of σij , the sum of theprincipal minors of the determinant of σij and the determinant of σij respectively:

I1 = σii = σ1 + σ2 + σ3 (6.4)

I2 =12(I21 − σijσij

)= σ1σ2 + σ2σ3 + σ3σ1 (6.5)

I3 =16(2σijσjkσkl − 3I1σijσji + I3

1

)= σ1σ2σ3 (6.6)

The stress tensor σij can be split into a volumetric- (also known as spherical, irrotational ordilational) and a deviatoric part (also known as equivolumnal or distortional). It needs tobe stressed that the volumetric tensor causes a volume change whereas the deviatoric tensoronly causes a distortion of the material which does not alter the volume.

σij = p δij︸︷︷︸V

+ sij︸︷︷︸D

(6.7)

p =13

(σ11 + σ22 + σ33) =13σii (6.8)

sij =

⎡⎢⎣ σ11 − p σ12 σ13

σ21 σ22 − p σ23

σ31 σ32 σ33 − p

⎤⎥⎦ (6.9)

where p denotes the hydrostatic pressure and sij represents the deviatoric stress tensor. Theinvariants of deviatoric stress tensor J1, J2 and J3 can be obtained the same way as for σij

and can be written as:

J1 = sii (6.10)

J2 =12sijsij =

12(s21 + s2

2 + s23

)(6.11)

J3 = sijsjkskl = s1s2s3 (6.12)

where s1, s2 and s3 are the eigenvalues of sij . The invariant J2 is of great importance for theformulation of many constitutive models and can be expressed in principal stress coordinatesas follows:

J2 =16

[(σ1 − σ2)

2 + (σ2 − σ3)2 + (σ1 − σ3)

2]

︸︷︷︸only related to shear

(6.13)

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If σij is transformed into principal stress coordinates it reads:

σij =

⎡⎢⎣ σ1 0 0

0 σ2 00 0 σ3

⎤⎥⎦ (6.14)

As will be seen later in this chapter some material models are expressed in the octahedralstress space. An octahedral stress plane is defined in such a way that its outward normalmakes equal angles with each of the principal stress directions. The normal stress action onan octahedral plane can be expressed as:

σoct =13

(σ1 + σ2 + σ3) (6.15)

and the shear stress acting in the octahedral plane can be written as:

τoct =

√(23J2

)(6.16)

It is convenient to express the invariants as geometrical properties for visualisation purposes.These invariants p, J and θ can such be written as:

p =13

(σ1 + σ2 + σ3) = σoct (6.17)

J =1√6

√(σ1 − σ2)

2 + (σ2 − σ3)2 + (σ1 − σ3)

2 (6.18)

θ = tan−1

[1√3

(2

(σ2 − σ3)(σ1 − σ3)

− 1)]

(6.19)

Fig. 6.1 shows that p is a measure for the distance along the hydrostatic axis (= spacediagonal) between the origin and the deviatoric plane. The latter is defined as any planeperpendicular to the hydrostatic axis. Any stress state within the deviatoric plane can beexpressed in polar coordinates. The deviatoric stress invariant J is a measure for the radiusbetween the current stress state and the hydrostatic axis. Lode’s angle θ defines the radialorientation. It can, depending on the convention, vary between +30◦ (triaxial extensionσ1 = σ2 ≥ σ3) and −30◦ (triaxial compression σ1 ≥ σ2 = σ3) or between 0◦ and 60◦.

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Figure 6.1: Geometrical significance of p, J and θ (from Potts & Zdravkovic, 1999)

The gradient of a relative displacement vector can be split into a symmetric second ordertensor εij and into an anti symmetric second order tensor ωij :

ε′ij = εij + ωij (6.20)

The latter represents the rigid body rotation where εij represents the strain tensor and canbe written as:

εij =

⎡⎢⎣ ε11 ε12 ε13

ε21 ε22 ε23

ε31 ε32 ε33

⎤⎥⎦ (6.21)

The entities of the strain tensor can be expressed as:

ε11 =∂u1

∂x1︸︷︷︸1st order

+12

[(∂u1

∂x1

)2

+(

∂u2

∂x2

)2

+(

∂u3

∂x3

)2]

︸︷︷︸2nd order

(6.22)

ε12 =12

(∂u1

∂x2+

∂u2

∂x1

)︸︷︷︸

1st order

+12

(∂u1

∂x1

∂u1

∂x2+

∂u2

∂x1

∂u2

∂x2+

∂u3

∂x1

∂u3

∂x2

)︸︷︷︸

2nd order

(6.23)

and consequently as

εij = εji =12

(ui,j + uj,i + uk,i uk,j) (6.24)

The principal strains ε1, ε2 and ε3 can be obtained by solving the eigenvalue problem:

det (εij − ε δij) = 0 (6.25)

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As a result of the characteristic cubic polynomial equation the strain invariants I′1, I

′2 and I

′3

are:

I′1 = ε1 + ε2 + ε3 (6.26)

I′2 = ε1ε2 + ε2ε3 + ε3ε1 (6.27)

I′3 = ε1ε2ε3 (6.28)

The strain tensor can be split into a volumetric and deviatoric component:

εij = ev δij︸︷︷︸V

+ eij︸︷︷︸D

(6.29)

ev =13

(ε11 + ε22 + ε33) =13εii =

13εvol (6.30)

eij =

⎡⎢⎣ ε11 − ev ε12 ε13

ε21 ε22 − ev ε23

ε31 ε32 ε33 − ev

⎤⎥⎦ (6.31)

where eij is the deviatoric strain tensor and the invariants of which can be written as:

J′1 = eii (6.32)

J′2 =

12eijeij = − (e1e2 + e2e3 + e3e1) (6.33)

J′3 = sijsjkskl = e1e2e3 (6.34)

where e1, e2 and e3 are the principal values of the deviatoric strain tensor.

6.3 The Finite Element Method for linear elastic materials

The finite element method is a numerical method. In contrast to other methods, such aslimit equilibrium, stress field and limit analysis, the FEM is capable of satisfying all fourfundamental requirements for a complete theoretical solution in continuum mechanics. Thatis satisfaction of equilibrium, compatibility, constitutive laws and boundary conditions. TheFEM therefore provides extensive information which allows detailed insight into every levelof structural behaviour.

According to Potts & Zdravkovic (1999) the following steps are required for solving a problemwith the FEM:

1. Element discretisation

2. Primary variable approximation

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3. Formulation of element equations

4. Definition of boundary conditions

5. Solution of global equations

6.3.1 Element discretisation

This initial step requires the definition of the geometry which is usually called ’modelling’.The geometrical domain is then divided into sub-domains. A number of finite elements arethen assigned to each sub-domain which in total forms the finite element mesh. This processis also called ’discretisation’.

Generally, such a mesh can involve one-, two- and three dimensional elements which arecommonly called beam-, shell- and solid elements respectively. Every element consists ofnodes which in the most basic element formulations are located at both ends in case of abeam element or at the corners in case of 2D and 3D elements. Advanced elements compriseadditional intermediate nodes which are typically located along the boundaries of the element(disregarding the Lagrangian element formulation). These are called ’higher order elements’,and their use is recommended to avoid ’shear locking’ amongst other undesirable effects.

For the sake of accuracy it is crucial that the mesh topology satisfies certain mesh qualitystandards. These depend on the element type and concern the aspect ratio, minimum andmaximum angles between the edges (distortion) amongst other criteria. Furthermore, themixed use of higher- and lower order elements needs to be strongly avoided.

The accuracy of the results can be improved by increasing the number of elements (i.e.increasing the mesh density) or by increasing the order of the elements. Especially areas ofrapid changes of displacements in the mesh (usually the region of interest) must be refined bythe use of smaller elements. Both measures, however, increase the computational costs, whichrequires a mesh optimisation process. This optimisation process involves a mesh-sensitivitystudy which aims to investigate whether the results converge to the exact solution when themesh density is increased.

Numerical analyses of punching shear problems conducted by the author showed that theadopted shell elements with embedded reinforcement in their current formulation were in-cabable of capturing punching shear failure. The response always reflected flexural behaviour.This behaviour can be explained by the incapability of shell elements to reflect complex straindistributions which typically occur in vicinity of the column in a punching shear situation.

For this reason concrete was discretised with 20 noded isoparametric brick (solid) elements(see Fig 6.2). One node in this element has 3 degrees of freedom (3 displacements). Structuralsteel elements were discretised by 8 noded (serendipity class) isoparametric, Mindlin type shellelements (see Fig 6.3). One node in this element has 5 degrees of freedom (3 displacementsand 2 rotations). Reinforcement was modelled discretely with 3 noded cable elements (see

206

Fig 6.4). One node in this element has 3 degrees of freedom (3 displacements). More detailsof the adopted elements are given in the following subsection.

Figure 6.2: 20 noded solid element (from DIANA, 2008)

Figure 6.3: 8 noded shell element (from DIANA, 2008)

Figure 6.4: 3 noded cable element (from DIANA, 2008)

6.3.2 Primary variable approximation

In the displacement-based FEM, the primary variable is the displacement which varies overthe domain giving a displacement field. Other quantities are treated as secondary variablesand can be computed from the primary variable. The key aspect of the FEM is that thedisplacements within an element are approximated by a function of the nodal displacements.These functions are termed ’shape functions’.

In the following section the FEM is briefly presented for linear materials where the influenceof temperature and time dependency are ignored. Considering a 3D analysis, in case of asolid element, the displacement field is characterised by 3 displacements u1, u2 and u3 inthe respective directions of a Cartesian coordinate system. The displacement field within anelement can thus be described by equation 6.35. [N ] represents the quadratic matrix of shapefunctions which interlinks internal- with nodal displacements, where the latter are treated asunknown degrees of freedom. The index ’n’ in equation 6.35 denotes ’nodal’.

⎧⎪⎨⎪⎩

u1

u2

u3

⎫⎪⎬⎪⎭ = [N ]

⎧⎪⎨⎪⎩

u1

u2

u3

⎫⎪⎬⎪⎭

n

(6.35)

207

The adopted element types used for NLFEA as presented in this thesis are higher orderelements. The shape functions of the adopted elements are all second order polynomials(incomplete second order polynomials in case of the shell elements) which can be found inDIANA (2008).

Usually the element as it appears in the finite element mesh is derived from a parent elementwhich is defined in natural coordinates ξ, η and ζ. The natural coordinates vary from -1 to+1. If the same functions which describe the displacement are used to map the geometryfrom natural to global coordinates, such an element is called ’isoparametric’. The globalcoordinates of an arbitrary point within a solid element can be obtained as follows:

x1 =n∑

i=1

Ni x1i and x2 =n∑

i=1

Ni x2i and x3 =n∑

i=1

Ni x3i (6.36)

where n denotes the number of nodes, x1i, x2i and x3i are the nodal coordinates in the globalcoordinate system and Ni are the ’interpolation functions’. These are expressed in terms ofnatural coordinates. For each node in the element such an interpolation function exists whichtakes the value of +1 at the corresponding node and zero for the remaining nodes.

6.3.3 Formulation of element equations

The incremental displacements within an element are expressed as follows:

{Δd} = [N ] {Δd}n (6.37)

where

{Δd} =

⎧⎪⎨⎪⎩

Δu1

Δu2

Δu3

⎫⎪⎬⎪⎭ and {Δd}n =

⎧⎪⎨⎪⎩

Δu1

Δu2

Δu3

⎫⎪⎬⎪⎭

n

(6.38)

The six first order strain-displacement relations can be written according to equation 6.24as:

Δε11 =∂Δu1

x1Δε12 =

12

[∂Δu1

∂x2+

∂Δu2

∂x1

](6.39)

Δε22 =∂Δu2

x2Δε23 =

12

[∂Δu2

∂x3+

∂Δu3

∂x2

](6.40)

Δε33 =∂Δu3

x3Δε13 =

12

[∂Δu1

∂x3+

∂Δu3

∂x1

](6.41)

208

Consideration of the above relations in equation 6.37 gives equation 6.42 which gives theincremental strains within an element as a function of the incremental nodal displacements.

{Δε} = [B]

⎧⎪⎨⎪⎩

Δu1

Δu2

Δu3

⎫⎪⎬⎪⎭

n

= [B] {Δd}n (6.42)

The matrix [B] consists of the derivatives of the shape functions Ni with respect to thenatural coordinates. The Jacobian matrix is used to convert the derivatives from natural-into global coordinates such that:

{∂Ni

∂ξ

∂Ni

∂η

∂Ni

∂ζ

}T

= [J ]{

∂Ni

∂x1

∂Ni

∂x2

∂Ni

∂x3

}T

(6.43)

where [J ] can be written as:

[J ] =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

∂x1

∂ξ

∂x2

∂ξ

∂x3

∂ξ

∂x1

∂η

∂x2

∂η

∂x3

∂η

∂x1

∂ζ

∂x2

∂ζ

∂x3

∂ζ

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

(6.44)

It is however necessary to express the global- in terms of natural coordinates such that:

⎡⎢⎢⎢⎢⎢⎢⎢⎣

∂Ni

∂x1

∂Ni

∂x2

∂Ni

∂x3

⎤⎥⎥⎥⎥⎥⎥⎥⎦

= [J ]−1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

∂Ni

∂ξ

∂Ni

∂η

∂Ni

∂ζ

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

(6.45)

The constitutive laws of a material relate stresses and strains, and therefore interlink equilib-rium with compatibility. The constitutive behaviour of a material can therefore be writtenaccording to equation 6.46.

{Δσ} = [D] {Δε} (6.46)

where {Δσ} represents the incremental Chauchy stress tensor, {Δε} represents the incre-mental strain tensor and [D] represents the constitutive matrix (also known as generalisedHooke’s law) which is a fourth order tensor.

The entities in such a constitutive matrix are called elasticity constants. A triclinic materialwhich is the most general form requires 21 independent elasticity parameters. A monoclinic

209

material requires 13 independent parameters. Orthotropic material behaviour is defined by9 parameters, whereas a transverse isotropic material requires 5 constants. The most basicform is isotropic material behaviour which requires 2 parameters which are commonly definedby the modulus of elasticity and Poisson’s ratio.

The element equations can now be obtained applying the principle of minimum potentialenergy. It asserts that a body deforms into a position in such a way that it minimises itspotential energy. That is a body subjected to static loading conditions seeks an equilibriumcondition which is the position with the lowest potential energy. According to Mang &Hofstetter (2000) this tendency can be explained by the second law of thermodynamics.It states that a system strives to maximise its entropy and that the entropy of a systemmaximises at equilibrium. The incremental total potential energy of a body ΔE can bewritten as:

ΔE = ΔW −ΔL (6.47)

where ΔW is the incremental strain energy of the deformed body and ΔL is the work doneby the body forces and traction forces.

ΔW is obtained by integrating the strain energy density over the volume V of the elementaccording to equation 6.48.

ΔW =12

∫V

{Δε}T {Δσ} d V (6.48)

ΔL is the work done by the loads which consists of the work done by the body forces andsurface or traction forces (see equation 6.49). The first part is obtained by integration overthe volume of the element and the second part is obtained by integration over the surface.

ΔL =∫V

{Δd}T {ΔF} d V +∫A

{Δd}T {ΔT} d A (6.49)

In equation 6.49 {Δd}T = {Δu1, Δu2, Δu3} represents the incremental displacement vector,{ΔF}T = {ΔFx1, ΔFx2, ΔFx3} denotes the body force vector and{ΔT}T = {ΔTx1, ΔTx2, ΔTx3} represents the surface traction vector.

Equation 6.50 represents the principle of minimum potential energy. Substitution of equations6.48 and 6.49 in equation 6.50 and making use of equations 6.37, 6.42 and 6.46 yields equation6.51.

δΔE = δΔW − δΔL = 0 (6.50)

[KE ] {Δd}n,E = {ΔRE} (6.51)

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Equation 6.51 interlinks the nodal displacements {Δd}n,E with the nodal forces {ΔRE} ofone single finite element. The element stiffness matrix KE is given in equation 6.52 and theright hand side load vector is given in equation 6.53.

[KE ] =∫V

[B]T [D][B]d V (6.52)

{ΔRE} =∫V

[N ]T {ΔF} d V +∫A

[N ]T {ΔT} d A (6.53)

It needs to be stressed that the integrals in equations 6.52 and 6.53 are evaluated for the nat-ural coordinate system, which shows the advantage of the isoparametric element formulation.As the values in the natural coordinate system vary between -1 and +1 a standard procedurefor integration can be used. However, a coordinate transformation of the surface and thevolume using the determinant of the Jacobian |J | according to equation 6.54 is required.

dV = dx1 dx2 dx3 = |J | dξ dη dζ (6.54)

The integrals in equations 6.52 and 6.53 can usually not be obtained exactly and thereforeneed to be evaluated numerically. In the adopted software code DIANA, four integrationprocedures, namely the Simpson-, Gaussian, Newton-Cotes and Lobatto integration schemeare available. However, the Gaussian integration scheme was used for the analysis presentedin this thesis because of its high numerical efficiency. Equation 6.55 represents the Gaussianquadrature for a two-dimensional integration. It shows that the integral is replaced by theweighted sum of function values which are evaluated at specific integration points. Theseintegration points are also called Gauss-points where the locations of which are fixed withinthe geometry of the parent element.

+1∫−1

+1∫−1

f(ξ, η) dξ dη =nG∑i=1

nG∑j=1

wi wj f(ξi, ηj) (6.55)

In linear elastic analysis a full Gauss integration scheme gives exact values. The full integra-tion scheme for the adopted solid and shell elements consists of 3x3x3 Gauss points in theξ, η and ζ directions. Especially if material nonlinearity is introduced, and bearing in mindthat the stresses within an element are evaluated at the Gauss points, the number of inte-gration points needs to be increased to sustain the accuracy of the integration. On the otherhand the numerical integration procedure is computationally expensive; that is a decreaseof integration points significantly decreases the running time of a problem. It is thereforeuseful, for large numerical problems, to use reduced integration schemes for elements in lessimportant areas in the domain.

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After the element equations are formulated separately, the assembly of these into a set ofglobal equations is required. The form of these is similar to equation 6.51 and can be writtenas:

[KG] {Δd}n,G = {ΔRG} (6.56)

where [KG] represents the global stiffness matrix, {Δd}n,G denotes the incremental displace-ment vector of all the nodes in the mesh and {ΔRG}n,G is the global right hand side loadvector.

The process of assembling the single element stiffness matrices into a global stiffness matrixis called ’direct stiffness method’. The principle of this method is based on the summationof common nodal contributions of the individual elements at the nodes of the global mesh.This applies for nodal displacements as well as for nodal forces. A detailed description of thisprocedure can be found in Potts & Zdravkovic (1999).

6.3.4 Definition of boundary conditions

The choice of appropriate boundary conditions is of great importance in finite element analy-sis. These can be split into natural- (Neumann) or essential (Dirichlet) boundary conditions.The natural boundary conditions are basically the external loading conditions (e.g. line loads,surcharge pressures or self weight) which affect the right hand side load vector {ΔRG}n,G.The essential boundary conditions are displacement conditions (e.g. prescribed- or fixeddisplacements) which affect the global nodal displacement vector {Δd}n,G.

Sufficent displacement conditions need to be assigned to the mesh in order to prevent ’illconditioning’ in form of rigid body motions. In case of an ill-conditioned mesh the globalstiffness matrix becomes singular (|ΔKG| = 0) and equation 6.56 cannot be solved.

6.3.5 Solution of global equations

In the final step of the FEM procedure the global set of equations is solved for the unknownnodal displacements {Δd}n,G. Hereby usually a huge set of simultaneous equations needs tobe solved. This task has been the focus of research for the past decades and many differentmathematical techniques to tackle this issue are available. One of the most prominent solutionstrategies is the ’Gaussian elimination method’ which is a direct solution method. If the sizeof the stiffness matrix increases, the direct methods become increasingly inefficient. In thiscase ’sparse solver’ techniques can be more efficient where the stiffness matrix is partitionedand these partitions are then solved separately. A detailed description of additional solutionstrategies can be found in Potts & Zdravkovic (1999). However, once the nodal displacementsare obtained, the secondary variables such as stresses and strains can be computed fromequations 6.42 and 6.46.

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6.4 Modelling concrete compressive behaviour

The advances in experimental research on concrete as well as research on modelling concretematerials numerically made great progress during the last three decades. As a matter of factthe available literature on these subjects is vast and still increasing rapidly. For this reasonit is hardly possible to cover the entire field of mechanical behaviour of concrete and concretematerial modelling in this thesis.

This section, therefore, gives a brief description of the basic mechanical behaviour of concretesubjected to different loading conditions. Based on that, some basic constitutive modellingapproaches are thereafter presented and discussed. This section concludes with a more de-tailed presentation of the ’Total Strain’ model which has been adopted for the analysis inthis work. In that respect it needs to be stressed that time dependent material behaviour(e.g. creep, shrinkage and relaxation) has not been considered as these effects are consideredto be insignificant for the present problems.

6.4.1 Continuum considerations

Especially when modelling concrete material behaviour the theoretical assumptions madein treating it as a continuum must be brought in accord with the observed nature of thematerial.

Continuum mechanics is a phenomenological theory. Its purpose is not to describe continuumbehaviour on the basis of the structure of a material such as, for instance, the atomic lattice orthe pore structure. In this context it is noteworthy that the elasticity parameters mentionedin Subsection 6.3.3 which describe material behaviour need to be obtained experimentally.

In continuum mechanics it is assumed that the matter which forms a material is homoge-niously distributed in such a way that it fills the entire space. In other words, a continuumfeatures no voids or cracks which means that no part of a volume can be deformed to zero orto infinity. Furthermore, the continuum can be sub-divided into regions of infinitesimal sizewhere the properties of which are still those of the bulk material.

The material structure of concrete, on the other hand, is complex and its behaviour is stillthe focus of extensive research activity. Concrete material consists of a skeleton of solidparticles (aggregates) which form a pore structure. These pores are filled with cement pastewhich conglutinates the particles, whilst the cement paste itself consists of solid and liquidphases. Concrete naturally comprises microcracks and voids, all of which are violating theassumptions of a continuum as the pore structure and the exact location, shape and sizeof the aggregates are disregarded. Fig. 6.5 shows the tacit assumptions being made whentreating concrete as a continuum.

213

Figure 6.5: Phenomenological conversion of concrete into a continuum

However, continuum mechanics provides a robust and reliable theoretical concept which isessential for structural analyses to be comprehensible and efficient. To get continuum me-chanics and the natural material behaviour into reasonable agreement, the length scales ofthe investigated domain need to be much greater than the size of its constituents. Unlike insteel or other similar materials, in structural concrete analysis the expression ’much’ is notclearly defined. It is therefore important to consider the accuracy of local effects (e.g. crackstrains) in the scale of the discretisation.

6.4.2 Uniaxial compressive behaviour

The unconfined uniaxial behaviour can be divided into four phases where fc represents theunconfined uniaxial peak strength. In the first part, up to fc/3, concrete behaves almost linearelastic and existing micro cracks in the pore structure do not progress. According to Kotsovos& Newman (1978) the stress level at fc/3 has been termed ’onset of localised cracking’. Inthe second phase between fc/3 and fc/2 bond cracks start to propagate but the fractureprocess is stable; under constant load the released energy is still smaller than the requiredenergy to progress fracture. In the third phase between fc/2 and 3fc/4 mortar cracks areinitiated and bond cracks are propagating. The stress level of 3fc/4 has been termed ’onsetof unstable fracture propagation’ where it coincides with the minimum volumetric strain.Progressive damage occurs between 3fc/4 and fc where with increasing compressive strainthe stress-strain curve descends and undergoes compression softening. This phase is governedby the appearance of macroscopic cracks until crushing occurs at the ultimate compressivestrain εu.

According to Chen (2007) the descending branch of the stress-strain curve is difficult tomeasure and strongly depends on the measuring method. Fig. 6.6 depicts a typical unconfineduniaxial compression response. The left hand figure (a) shows the stress-strain responsewhereas (b) shows the development of volumetric strain εv = I

′1 with increasing uniaxial

pressure σ. The volumetric strain decreases rather linearly up to 3fc/4 where upon loadingεv sharply increases which indicates an expansion of the sample.

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Figure 6.6: Typical uniaxial stress-strain curve (a), volumetric strain (b) (from Chen, 2007)

Fig. 6.7 shows the development of Poisson’s ratio ν with increasing uniaxial stress level. Upto 4fc/5 of axial stress the Poisson’s ratio ranges between 0.15 ≤ ν ≤ 0.25. At higher stresslevels ν increases which shows that the value of a constant (elastic) Poisson’s ratio becomesirrelevant.

Figure 6.7: Relation between normalised stress and Poisson’s ratio (from Chen, 2007)

6.4.3 Biaxial behaviour

Fig. 6.8 shows a typical bi-axial strength envelope which indicates that the uniaxial com-pressive strength increases under biaxial compressive loading. According to Nelissen (1972)

215

the strength increase under biaxial loading conditions is widely stress-path dependent. At abiaxial compression stress ratio of σ2/σ1 = 0.5 the concrete strength is enhanced by approx.25%. At an equal biaxial compression state σ2/σ1 = 1 the concrete strength is enhanced byabout 16%. A nearly linear relation between compression-tension loading can be seen whereno strength enhancement appears at biaxial tension conditions.

Figure 6.8: Biaxial test data for different compressive strengths (from Kupfer & Gerstle,1973)

The afore mentioned increase of volume also occurs under bi- and triaxial conditions and istermed ’dilatancy’. According to De Borst & Vermeer (1984) dilation is caused by frictionalsliding, either along particles or along micro cracks. This phenomenon can be mechanicallyexplained by two layers of loose granular material which have to undergo both a vertical anda horizontal displacement if these layers slide along each other. The vertical uplift causes thevolume to increase under plastic deformation.

6.4.4 Triaxial behaviour

Fig. 6.9 shows the axial stress-strain relationship for concrete subjected to different levels ofcompressive confinement pressure. It shows that both the concrete strength and the ductilityincreases with increasing levels of confinement (Mirmiran & Shahawy, 1997). With increasinglevels of hydrostatic pressure the failure of the pore structure shifts from cleavage to crushingof cement paste (Kotsovos & Newman, 1978).

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Figure 6.9: Triaxial stress-strain relationship for concrete (from Chen, 2007)

Fig. 6.10 depicts the basic failure principles of concrete on a material level provided that thestrength of the cement paste is lower than that of the aggregates. Triaxial test data showedthat concrete failure can be described as a surface in the three dimensional principal stressspace. The fact that the shapes of these failure surfaces for different kinds of concrete aresimilar, gave rise to the development of mathematically defined failure criteria. Experimentsrevealed that the deviatoric sections at low p-levels are convex triangles with rounded corners.With increasing volumetric pressures the deviatoric sections are approaching a circular shape,which shows that failure at increasing pressure levels does less depend on Lode’s angle.

Figure 6.10: Basic failure modes of concrete

Fig. 6.11 depicts such surfaces in the 3D stress space. The elastic limit surface denotes theonset of stable micro crack propagation and the failure surface represents the onset of unstable

217

crack propagation (Chen, 2007). Fig. 6.11 also shows that the meridians of the failure surfacesare curved. A meridian in this respect is defined as the intersection of a plane which containsthe hydrostatic axis with any of the afore mentioned surfaces. Generally these meridians aresmooth and convex where the curvature of which depend on I1. (Schutz, 2010).

Figure 6.11: Triaxial failure surface for concrete (from Chen, 2007)

6.4.5 Elasto plastic modelling

A general historical overview of constitutive modelling of concrete can be found in Babu et al.(2005). In this subsection the classical approach to model concrete material behaviour bymeans of elasto-plasticity is briefly discussed. The implementation of elasto-plastic constitu-tive models in the framework of the finite element method allowed a close simulation of thecomplex three dimensional concrete behaviour.

The ’yield function’ determines a three-dimensional stress state at which a material exhibitsnot only elastic- but also plastic strains. In the case of perfectly plastic material behaviour,the yield function describes a 3D surface in the principal-stress space which does neitherchange its location nor its size. Such yield functions generally used for concrete are discussedin the next subsection. The values of the yield function F (σ) have the following physicalmeanings:

F (σ) < 0 → (linear) elastic behaviour (6.57)

F (σ) = 0 → perfectly plastic behaviour (6.58)

A value of F (σ) > 0 for rate independent plasticity is impossible which means that a stresspoint when once touched the surface can only move on the surface. Fig. 6.12 gives a graphicalinterpretation of the yield function and the two conditions. The enclosed area of the yieldfunction is called the ’elastic domain’.

218

Figure 6.12: Flow rules of an elasto-plastic model concept

The ’flow rule’ is required to determine the evolution of the plastic strain increment dεp atevery stress state. This is necessary as the elastic constitutive relations in the plastic regimedo not hold. Hereby ’coincidence of axes’ is assumed in which the directions of accumulatedstress and the incremental plastic strain coincide. To describe the direction of straining a socalled ’plastic potential function (surface)’ P (σ) is required. The outward normal vector ofplastic strain increment dεp can be expressed as:

dεpij = dλ

∂P (σ)∂σij

(6.59)

where the gradient ∂P/∂σ determines the direction-, and the scalar dλ ≥ 0 defines themagnitude of the vector dεp.

If the plastic potential function is identical with the yield function P (σ) = F (σ) the flowrule is said to be ’associated’. If P (σ) �= F (σ) the flow rule is said to be ’non-associated’.

The incremental plastic strain vector dεp is pointing in the direction of the outward normalvector of the plastic potential surface. Considering such a vector in J-p space, the horizontalcomponent of the strain vector is a measure for the dilation. If the horizontal componentof the strain vector is zero, the model is said to be ’non-dilatant’. However, the dilatationalor volumetric behaviour is an important aspect in concrete material modelling. De Borst &Vermeer (1984) report that associated flow rules in conjunction with the failure criterionsmentioned in the next subsection generally overpredict the volume changes (dilation) associ-ated with shear distortion. Therefore the literature provides various forms of non-associatedplastic potential functions which aim to more realistically reflect the volumetric behaviour.More detailed information can be found in Schutz (2010).

As will be seen later in this chapter, constitutive modelling of concrete requires the adoptionof at least two failure surfaces. One surface typically controls compressive failure and theother tensile failure. The intersection of these surfaces increases the complexity and causessingularities which are difficult to be dealt with numerically (De Borst, 1987).

219

Clearly concrete does not behave perfectly plastically. To realistically predict its behaviour,the failure surface is consequently required to change its size and/or its location. Thereforethe yield- and plastic potential functions need to be modified as follows:

F (σ, k) = 0 (6.60)

P (σ, m) (6.61)

A hardening/softening rule defines the state parameters k and m and enables the calculationof the scalar dλ. The expansion (isotropic hardening) of the failure surface can then be limitedby an ultimate surface which defines failure. One example of such a model is the Chen&Chencriterion (Chen & Saleeb, 1982). A more detailed description of hardening schemes for thevon Mises criterion and its application for modelling steel behaviour is given in Section 6.7.

6.4.6 Constitutive models for concrete

Probably the most prominent plasticity model used for concrete is the Mohr-Coulomb (MC)criterion. According to Coulomb’s law of friction, the shear strength σnt of an infinitesimalsurface element depends on any simultaneous acting normal stress σnn:

|σnt| = f (σnn) (6.62)

According to the MC criterion material failure is initiated when the largest of the three Mohrcycles touches an envelope. The simplest form of such an envelope is a line (see equation6.63). The osculation point and the envelope are then defining the orientation of the aforementioned infinitesimal surface element.

|σnt| = c + σnn tan ϕ (6.63)

where ϕ represents the friction angle and c denotes the cohesion.

Hence, the MC criterion is a two parameter model. The parameters ϕ and c are related tothe uniaxial tensile strength ft and the uniaxial compressive strength fc which can be usedfor calibration as follows:

|ft| = 2c cosϕ

1 + sin ϕ(6.64)

|fc| = 2c cosϕ

1 − sin ϕ(6.65)

Substitution of equation 6.64 and 6.65 in equation 6.66 gives equation 6.67.

σmax1 + sin ϕ

2c cosϕ− σmin

1 − sin ϕ

2c cosϕ= 1 (6.66)

σmax

ft− σmin

fc= 1 (6.67)

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