3.11 Concluding Remarks

Transcription

1 Examples of fully integrated shearheads are all ACI-type shearheads, Tobler Walm, Composite cruciform and so forth. The behaviour of such systems is governed by composite action which again is strongly influenced by concrete fracture. Although fully integrated configurations offer 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 elastic region in order to avoid local storey mechanisms. On the other hand, plastic deformation within the connection is strongly required to reduce the forces acting on a structure during seismic excitation. This makes the formation of cracks in the slab inevitable to allow the flexural reinforcement to yield and to activate shear reinforcements. In other words, the main energy dissipative mechanism of conventional assemblages strongly relies on concrete fracture. Whilst the cracking process under monotonic (i.e. static) loading conditions is controlled, under reversed loading conditions it is not. For these reasons fully integrated systems usually have a rather low ductility as well as limited rotation capacity and exhibit high degradation of shear strength. Partially integrated shearheads, which can be used to describe several forms including the novel development proposed in this work, are partially embedded in the slab. This can be achieved by isolating the column from the slab by an opening around the column in such a way that only the structural steel shear arms establish the connection between the slab and the column. The philosophy behind partially integrated shearheads is to utilise the salient features of steel in terms of strength, ductility and hysteretic behaviour; the exposed parts of the 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 seismically induced loads and leads to a very ductile behaviour with large column drift capacities. In this thesis it will be demonstrated that the fuses (i.e. exposed parts of the shear arms) should preferably be designed for shear yielding. Passive steel shear yielding devices are renowned for their high ductility, robust resistance to degradation under cyclic loading, high initial stiffness and significant energy dissipation capabilities. The latter absorbs a large amount of seismic input energy and reduces the plastic deformation demand to the existing structure which helps to protect it from severe structural damage. Similar shear fuse concepts applied for 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 Harries et al. (1993) and Fortney et al. (2007). Clearly, the philosophy of a partially integrated system is based on the condition that the steel capacity is consistently lower than surrounding concrete mechanisms. The concrete composite capacity of the embedded parts needs to be ruggedised in such a way that limited stresses are induced into the concrete slab, which mitigates degradation effects; concrete strength is then widely maintained under cyclic loading conditions. Moreover, the resident complexity in the D-region in vicinity of the column is significantly reduced by means of an aperture around the column. The critical force flow from the slab into the column of such an 111

2 isolated system is more definite. Additionally, the plastic failure mechanism of the shearhead (fuses) can be accurately tuned for specific demands Concluding Remarks In this chapter different shear head details and concepts have been introduced with the aim of determining which is most suitable for a partially integrated shearhead. According to the literature presented, the availability of shearhead systems suitable for steel columns is limited, and those discussed before appear mostly restricted to gravity load cases only. When it comes to shearhead performance under lateral loading conditions, the literature becomes scarce. The reason for this might be the poor performance of fully integrated shearheads compared to conventionally reinforced slab-to-column connections, and research obviously focused on the latter. However, as there is a dearth of experimental data on the performance of shearhead systems under seismic loading, the code provisions are also rather tacit about shearhead design under combined loading. While the control perimeter approach in shear head reinforced cases for combined loading is relatively straight forward, the complex composite behaviour of the shearhead inside the control perimeter is non-distinctive; this suggests that the boundary conditions for an analytical model for the shearhead are vague. Furthermore, since the force distributions are not symmetric under combined loading conditions, the shearhead behaviour strongly depends on the load eccentricity. When it comes to cyclic lateral loading conditions mainly conventional reinforced concrete column-to-flat slab connections have been examined and data on the cyclic performance of shearhead connections are hardly available. This proves that accurate predictions concerning column drift ratios or the capacity of conventional shearhead reinforced connections under cyclic 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 of symmetry 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 shearhead details show that the bottom flanges are directly resting on the form work in order to increase the effective control perimeter, which is counterproductive in combined loading conditions. In this context, the systems Tobler Walm, Sheardome, NUUL-system as well as both patents mentioned in Section 3.6 are unsuitable for gravity loads in conjunction with lateral loading. 112

3 Although the activation of composite action is generally desired, the application of shear connectors such as headed shear studs raise several issues related to fatigue failure and concrete degradation effects when subjected to cyclic loading. Therefore, the Composite cruciform could be susceptible to low cycle fatigue failure. Low cycle fatigue issues also concern the fracture toughness of welded connections which typically appear in the welding seem of headed studs. Thus, the adoption of headed shear studs to improve the composite behaviour of the shear arms was discarded. Finally a new shear head concept has been introduced which will be thoroughly discussed in due course. Representatives of such partially integrated systems are the Geilinger mushroom with 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 in the previous and the present chapter. Compared to fully integrated shearheads, the omission of concrete in the area inside the shearhead lowers the punching capacity. This results from the lack of composite action and discontinuous reinforcement within the perimeter between the shearhead and the slab. On the other hand, this offers favourable inelastic behaviour and also contributes to a reliable and efficient design. Finally, it is an additional positive side effect that services can be conducted through the slab without the requirement of additional holes. 113

4 Chapter 4 Experimental Methodology 4.1 General remarks Referring to the novel shearhead concept in Section 3.10, the testing programme presented in this thesis primarily aimed to investigate and to compare the performance of a fully- and a partially integrated shearhead. Therefore, a multifunctional test rig first had to be built which 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 programme was to further optimise the partially integrated shearhead detail. The large scale tests typically involved one test under gravity- and one test under combined cyclic loading conditions for each shearhead configuration. In the course of testing, firstly a shearhead in a fully integrated configuration was tested. This conventional configuration will be referred to as Type-A detail. The Type-A detail served as a benchmark in terms of the comparative assessments of the other shearhead configurations. Thereafter, the same shearhead was tested in a partially integrated configuration. The partially integrated configuration will be referred to as Type-B detail. The Type-B detail was designed in accordance with Concept C in Section 7 of EC 8 (2005) which requires measures for preventing the contribution of concrete in the main dissipative zones under seismic conditions. After the general applicability of the partially integrated Type-B detail was verified, the testing programme focused on further improvement of the composite behaviour of the latter. A small scale test rig had thereafter been built which enabled the investigation of the composite behaviour of the embedded parts of the shear head. The small scale tests were not intended as substitutes for large scale tests. However, in due course a higher number of tests could be conducted. Hence, these tests eventually provided more information on the composite behaviour of the shear arms than a significantly smaller number of large scale slabs would have done. 114

5 These tests referred to as slab panel tests consisted of two series. Series-1 investigated the effectiveness of edge reinforcement to prevent localised concrete damage. Series-2 was used for further optimisation of Series-1 details with the purpose of increasing both the capacity and the durability of the shear arm. On the basis of the slab panel test results, an improved shearhead detail, Type-C was developed. 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 behavioural capabilities especially under cyclic loading conditions. This chapter aims at furnishing the reader with detailed information on the experimental methodology of this testing programme. This chapter hence contains the material properties of 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 description of the large and small scale specimen geometry, reinforcement layout and shearhead details is given. Finally the adopted instrumentation setup for large and small scale tests is presented. It needs to be pointed out that the shearhead details presented in this chapter evolved in the course of the testing process. Some of the information presented in this chapter will be thoroughly discussed in Chapter Material tests Concrete properties Ready mixed concrete was used with a nominal strength of C35/40. Cement type CEM-I was used and the maximum aggregate size was d a =10mm. The consistence class was S3 (pump mix) with an average slump of 150mm. No additives were added to the concrete and the same mix design was used for all tests. Concrete control specimens were always cured both 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 a nominal diameter of D=100mm and a nominal length of L=255mm. Top and bottom surfaces of the cylinders were ground before testing. The concrete tensile strength was measured on split cylinders (Brazilians) with a nominal diameter of D=150mm and a nominal length of L=230mm. The split cylinder strength was obtained according to σ s =2P/(πLD), where P denotes the splitting force. The actual dimensions of the control specimens were measured directly before testing. All control specimens were tested in a load controlled testing machine with a loading rate of 300MPa/min. The mean concrete material properties are given in Table 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 delivered concrete quality was fairly consistent. 115

6 Specimen f cube f cyl f split - [MPa] [MPa] [MPa] Type-A grv Type-B grv Type-C grv Type-A lat Type-B lat Type-C lat SP SP 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 and second slab panel (SP) test series Rebar properties The reinforcement type used was Class-B with a nominal yield strength of f y = 550N/mm 2. The tensile strength was typically measured on 400mm offcuts of the same batch of reinforcement used in the test specimens. The direct tensile tests were displacement controlled with a constant loading rate of 5mm/min. The elongation was measured by means of a clip extensometer with a gauge length of 100mm. The stresses were evaluated for the nominal cross section. A typical stress-strain curve of a rebar sample is depicted in Fig Figure 4.1: Typical stress-strain diagram for T12 reinforcement 116

7 The averaged reinforcement material properties are given in Table 4.2. Specimen E 0 σ 02 σ u ɛ L - [GP a] [MPa] [MPa] [%] T Table 4.2: Averaged rebar properties where E 0 denotes the modulus of elasticity, σ 02 denotes the 0.2% proof stress, σ u represents the tensile strength and ɛ L represents the elongation after fracture Structural steel properties In order to evaluate the steel material properties, the testing procedure followed the provisions of BS-EN (2001). Necked tensile coupons were cold machined from the RSC-51x38 channel section, where samples were taken from the centre of the flanges and the web. Necked tensile coupons were cold extracted from the RHS-100x60x6 and SHS-180x10 sections each in the centre on all four sides. The dimensions of the coupons are given in Table 4.3. The hot rolled channel section had a nominal grade of S235 and the hot finished tubular sections had a nominal grade of S355. Coupon L tot w tot L gauge A 0 - [mm] [mm] [mm] [mm 2 ] RSC 51x38 flange RSC 51x38 web RHS 100x60x SHS 180x Table 4.3: Coupon dimensions where L tot is the total coupon length, w tot is the total coupon width, L gauge is the gauge length measured between the necked edges and A 0 represents the nominal cross section. The tensile tests were displacement controlled with a loading rate of 1.5mm/min. Elongations were measured contactless by means of a optical extensometer (see Fig. 4.2). Therefore, two white marker points with a distance of 50mm (gauge length) were each administered to the surface of the necked area. The axial elongation was measured electro-optically based on the relative displacement of the marker points. 117

8 Figure 4.2: Coupon testing machine with video extensometer The coupons after fracture for the RSC-51x38, RHS-100x60x6 and SHS-180x10 section are depicted 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 118

9 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 119

10 Figure 4.8: Typical stress-strain diagram for SHS 180x10 The averaged coupon material properties are given in Table 4.4. Coupon E 0 σ 02 σ u ɛ L ɛ f - [GP a] [MPa] [MPa] [%] [%] RSC 51x RHS 100x60x SHS 180x Table 4.4: Averaged steel properties where E 0 denotes the modulus of elasticity, σ 02 denotes the 0.2% proof stress, σ u represents the tensile strength, ɛ L represents the elongation after fracture and ɛ f represents the tensile ductility. The tensile ductility was obtained by ɛ f = ln (A 0 /A u ), where A 0 denotes the initial cross sectional area and A u 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 experimental assessment of the behaviour of tubular column-to-flat slab connections under lateral cyclic loading conditions in conjunction with co-existing gravity loads. The layout of the test rig basically follows the test rig used by Pan & Moehle (1992). The prime condition for the design of the rig was to enable gravity- as well as uniaxial lateral loading either separately or in any combination of both. Furthermore, flexibility in terms of slab size, slab depth and column height was considered for a more versatile application of the test-rig. 120

11 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 column stub via hinges 121

12 The tie member connected with the reaction frame and the bottom of the column via hinges 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 to the slab specimen by spherical bearings The 100t vertical hydraulic jack connected to the bottom of the column stub, sharing the 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 stubs via bolted head plates The rig was constructed in such a way that the slab could move horizontally in the direction of lateral loading. This causes a rotation of the column relative the slab which induces an external bending moment. The mechanism employed in this rig works independently of the applied 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 connection was employed. Fig shows a detail of the height-adjustable connection which nips to the vertically aligned flanges of the reaction frame. This clamped connection is achieved by 16 prestressed high strength M24 bolts. 122

13 Figure 4.11: Hight adjustable clamped connection Two of these connections were designed to connect both the horizontal actuator and the tiemember with the reaction frame. In this context the tie member balanced the lateral force of the horizontal actuator in order to minimize horizontal (transverse) forces acting onto the vertical jack and the load cell. Eight pin ended ties were symmetrically arranged around the edges of the specimen which connected the latter with the strong floor. Four ties were each located at the corners whereas four 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 comprised spherical bearings. These bearings were centrally pressed into a milled groove in the central plate of the hinge (see Fig. 4.12). These spherical bearings enabled the hinge to also swivel perpendicularly to the main axis of rotation which allowed for the small out of plane rotation of the slab when loaded vertically. The bottom hinges of the pin ended ties were bolted to floor beams which were made of pairs of parallel flange channel (PFC) sections. The top hinges were connected to the slab specimen as shown in Fig

14 Figure 4.12: Detail of spherical bearing Fig shows two details of the pin ended ties. The left detail shows a normal strut, whereas the right detail shows a spliced strut. The spliced strut was located in front of the reaction frame and resolved the clash with the horizontal tie member. Figure 4.13: Pin ended ties/struts 124

15 Taking advantage of the square grid of strong bolts in the strong floor, four different specimen sizes 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 through a 25t actuator with a stroke of ±125mm. The vertical load could be applied independently by a 100t hydraulic jack operating in a force-controlled mode. Fig shows the reaction frame, the height-adjustable connection with the horizontal tie member and the eight pin ended ties as built. The grillage sitting on the top hinges of the vertical ties was used for alignment purposes to facilitate the emplacement of the slab specimen. Figure 4.14: Reaction frame and pin ended ties with horizontal tie member Fig shows a readily assembled slab specimen which is shown to be connected to the pin ended ties by eight pairs of M24 all-thread bars. Beams made of pairs of 1300mm long PFC-sections and 50mm thick square plates sitting on top of the slab were used to distribute the tie forces onto larger areas. Fig also shows the horizontal actuator connected to the clevis on top of the column head. The scaffolding poles shown were part of the external 125

16 frame which was used to measure the slab displacements. Figure 4.15: Testing rig with slab specimen in cyclic loading configuration Fig shows a close-up of the three-way hinge which connected the vertical jack with the column end and the horizontal tie member. A load cell was located between the jack and the bottom plate. Top centre of Fig shows the bolted joint between the column stub and the cast-in shearhead. This joint was designed to alleviate casting of the slab as the mould could be placed directly on the floor. Figure 4.16: Close up of vertical hydraulic jack with three-way hinge and bottom column stub 126

17 4.4 Slab panel testing rig This section will highlight the specifications of the purpose-built test-rig in order to enable an experimental assessment of the composite behaviour of the embedded shear arms. The general layout of the test rig follows the requirements of reflecting in-situ conditions of the large scale tests. Figure 4.17: Schematic setup for slab panel rig 127

18 A simplification of these conditions was necessary to ease the handling of the specimen as well as to reduce the complexity for the benefit of behavioural interpretation. The rectangular slab panel (SP) specimen was held within a specially designed frame. As illustrated 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 was welded 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 reinforced with triangular stiffeners. The specimen was wedged between the flanges of the channel by using pairs of steel wedges which prevented the specimen from loosening during cyclic loading. Fig shows the slab panel wedged between the flanges of the PFC-section as described before. Figure 4.18: Close up of installed specimen A 25t actuator was accommodated within an externally braced testing frame which was directly located above the steel frame that held the specimen. Fig shows the general assembly as built, with the actuator being connected to the specimen via a double hinge. Out-of-plane forces arising from out-of-plane displacements of the double hinge were transferred to the external frame by a horizontal guide beam. This beam was designed to move with the actuator ram to prevent the induction of transverse forces into the interconnected load cell. 128

19 Figure 4.19: Slab panel rig with specimen and external frames for gauging 129

20 The bottom base plate of the hinge was bolted to an interface member which connected it to the slab panel. Fig shows a close up of this clamped connection. The clamp consisted of a top and bottom plate with a M20 all thread bar either side which enabled it to be clamped onto 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 Determination of slab size The large scale specimen represented a small part of a greater structural system. Ideally, the traction forces along the boundary of such a specimen are exactly the same as those acting on 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 reasonably straight forward. The circular boundary follows the geometric locus where the radial bending 130

21 moments are zero. The diameter of this circle is D 0.44L in a square flat slab system with the span L. The shear-traction forces which are uniformly distributed along the edge are typically replaced by concentrated tie-forces which are evenly distributed in a polar array along the slab edge. Determination of the specimen size in a combined loading situation with variable eccentricity however requires a compromise. For the determination of the specimen size St-Venant s principle was applied. The engineering interpretation of St-Venant s principle states that: Two statically equivalent force systems that act over a given small portion S on the surface of a body produce approximately the same stress and displacement at a point in the body sufficiently far removed from the region 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 boundary conditions 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 the column 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. superordinate structural system) with a storey height of 3.0m and with 5.0m square panels. Linear elastic finite element analysis was conducted. Columns were modelled with beam elements and the floor slabs were modelled isotropically with shell elements. Two different load cases were considered 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 the spans of every floor. Fig compares the displacement contour plots in the reference building (dashed boundary) with that in the slab specimen (solid boundary). The superscripts a and c represent gravity loading conditions whereas b and d represent lateral loading conditions. The black bold dots along the boundary of the specimen represent pin supports as considered in the arrangement of pin ended ties in the test rig (see Fig. 4.9). The optimum specimen size and aspect ratio for both load cases was found iteratively on a trial and error basis. Fig shows that despite the huge deviation of displacements near the boundary, the displacement contours close to the column are similar. Hence, the depicted slab geometry was consequently adopted for the large scale testing scheme. It is noteworthy that due to St-Venant s principle, punching shear tests on square slabs give very similar results to equally sized circular slabs. 131

22 Figure 4.21: Elastic displacement field contours of reference building and specimen Specimen details All slab-to-column assemblages were constructed at 60% of the full scale. The spans of the slabs measured 1440x1220mm between the centrelines of the pin-ended ties. The slab depth of 155mm was chosen to minimise size effects without exceeding the maximum allowable self weight which was limited by the crane capacity. The column measured 1185mm between the midpoints 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-B and Type-C specimen reflected a novel, partially integrated configuration. Fig shows that both Type-B and Type-C specimens featured a sqare hole around the column which measured 500x500mm. All specimens were cast in oiled plywood moulds. They were then covered with plastic sheets in the first week after casting. In total six large scale specimens were employed, and two of each type were cast simultaneously. 132

23 4.5.3 Reinforcement details The tensile reinforcement ratio of ρ = 1.06% was chosen to ensure that the flexural capacity of the slab was higher than its expected punching shear capacity. The tensile reinforcement consisted of T12 bars at 100mm centres in each direction. The compressive reinforcement consisted of T12 bars at 200mm centres all with a nominal cover of 20mm. The Type-B and Type-C specimens had bundled reinforcement bars adjacent to the edges of the 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 reinforcement ratio was the same for all specimens. Fig and 4.23 illustrate the reinforcement layout of Type-A, Type-B and Type-C specimens. 133

24 Figure 4.22: Type-A reinforcement layout 134

25 Figure 4.23: Type-B and Type-C reinforcement layout 135

26 All shearheads were centrally embedded between the four reinforcement layers. Fig shows 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 shows the reinforcement detail of the partially integrated Type-B specimen. The concrete, in this case, was cast up to the edge of a plywood box in such a way that after removal of the box a 500x500mm hole was left around the column. Figure 4.25: Type-B reinforcement and shearhead configuration Fig shows the Type-C shearhead-reinforcement detail. The concrete was cast up to the edge of the collar (edge beams) in such a way that bottom and top surfaces were both flush with the flanges of the collar. A detailed description of the shearheads used are given in the following subsection. 136

27 Figure 4.26: Type-C reinforcement and shearhead configuration Shearhead details The cruciform Type-A and Type-B shearheads used measured 2L v = 1000mm from tip to tip. They consisted of a column made of a square hollow section SHS-180x10. The same column section was adopted for all shearhead details. Rectangular, 30mm thick head plates were through-welded to the ends of the column section which were bolted to the column extensions using six pre-stressed high strength M bolts. That facilitated casting and handling 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 layout of the shear arm-to-column connection followed the need to prevent yielding of the column section. 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 gusset plates measured 360x360mm and were destined to transfer the forces from the shear arms to the column smoothly and reliably. The shear arms were connected to the gusset plates by fillet welds. In the Type-A and the Type-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 gusset plates themselves were continuously fillet welded (weld thickness a=6mm) all around the column section. Fig depicts the Type-A and Type-B shearhead geometry. 137

28 Figure 4.27: Type-A and Type-B shearhead detail Fig shows a close up of the gusset plates and the welding detailing as used for Type-A and Type-B shearheads. Figure 4.28: Type-A and Type-B shear arm-to-column connection 138

29 Fig depicts the Type-A shearhead as built. Figure 4.29: Type-A shearhead as built Fig shows the Type-B shearhead detail as built. Figure 4.30: Type-B shearhead as built Fig shows that three shear dowels in form of 75mm long T12 offcuts were each welded onto 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 139

30 Fig illustrates the geometry of the Type-C shearhead which measured 2L v = 1300mm from tip to tip. The adopted shear arm sections were the same as those used in Type-A and Type-B versions. The thickness of the gusset plates was increased to 12mm and the gusset plates consisted of eight symmetric parts which were arranged in such a way that the welding seems were aligned parallel to the axial shear arm directions. Figure 4.32: Type-C shearhead detail 140

31 Fig shows the fatigue detail category 56* according to EC 3 (2005a). Figure 4.33: Detail category 56* (from EC 3, 2005a) Fig 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 high when head fillet welds are omitted. Although the Eurocode provisions for fatigue are based on high cycle fatigue tests, the welding detail shown in Fig was expected to also behave favourably 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 the column. The collar measured 500x500mm between the internal web edges. The flanges of the 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 the PFC-collar. The shear arms were connected to the collar by means of 15mm thick welded stiffener plates. These were located between the flanges of the shear arm and the collar. 141

32 Fig shows the Type-C shearhead as built. It shows 150mm long anchor pieces made of RSC-51x38 offcuts which were perpendicularly welded to the tips of the shear arms. Additional pairs of U-bent T16 bars were provided at either side of the shear arm. These bars were welded to the webs of the collar. Therefore, holes with a diameter of 20mm were drilled into the collar s web. The ends of the U-bent bars were circumferentially weld-connected from 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 hinges at the respective column ends. Fig gives an overview of the instrumentation of the large scale slabs. Displacements of the horizontal actuator ram were measured in the actuator itself. Horizontal displacements of the slab were measured with LVDT s (transducers) on two points along the shorter span of the specimen. The central vertical column displacement was measured with a transducer that was located between the strong floor and the bottom hinge. Vertical displacements of the slab were measured with LVDT s located above two pin ended ties located at two mutually perpendicular edges. Furthermore, vertical displacements of the edge of the hole or the collar were measured with draw wire transducers which were located below the slab. During cyclic testing, vertical displacements of the slab edges were measured by means of two laser interferometers. This contact-less method was necessary to prohibit disturbances caused 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 depicted in Fig. 4.37, two pairs of strain gauges were located below two mutually perpendicular shear arms in order to measure tangential- and radial strains. 142

33 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 crack pattern. Tensile concrete surface strains were measured manually at various load stages adopting the Demec system. Therefore, a 150mm wide marker grid was attached to the surface with plastic padding. The grid enabled the measurement of tangential- and radial strains directly above one longitudinal and one transverse shear arm. 143

34 Fig depicts the location of the adopted Demec marker scheme. Figure 4.38: Location of strain gauges and Demec markers Fig also shows the location of strain gauges at the shear arms. These were surface mounted with cyanoacrylate adhesive after adequate preparation of the designated area. The strain gauges and feed cables were then sealed with a thin layer of polyurethane resin. Axial strains of the extreme fibres were measured in the top- and bottom flanges of two mutually perpendicular 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 gusset plates and at the ends of the shear arms. In Type-B and Type-C details strains were measured at 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 gauge rosettes. 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 shear arm of Type-B detail. The SP quasi represented one quarter of the large scale slab; that 144

35 is, it represents one embedded shear arm of a partially embedded shearhead detail. The SP tests therefore provided valuable information on the composite behaviour of the shear arms in the large scale tests, especially on the effects of local concrete failure and concrete strength degradation. The specimen measured 900x600mm on plan with a depth of 155mm. Fig shows the geometry of the slab panel with Series-1 shear arm details. The location of Series-2 shear arms 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 each cast at the same time. The specimens were covered with plastic sheets in the first week after casting. In total 10 slab panel specimens were tested. Series-1 consisted of 4 slab panels whereas Series-2 consisted of 6 slab panels Reinforcement details The reinforcement detailing was chosen to reflect the reinforcement layout of the large scale tests as closely as possible. Fig shows the adopted reinforcement layout which consisted of 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 the required anchorage. In the second test series, additional stirrup triples made of T8 bars were employed at both sides next to the supports at the front edge. This enhanced the shear capacity of the panel to achieve plastic deformation in the shear arm extensions. 145

36 Figure 4.40: Slab panel reinforcement layout Shear arm details The shear arms of all series were centrally positioned in the slab panel between the four reinforcement layers. The shear arm extensions which projected out of the front edge measured 150mm. The shear arm extensions were stiffened with 8mm thick welded end plates. The load was applied at an eccentricity of 85mm from the slab edge. This eccentricity was chosen to make shear action dominant in the shear arms as this was the governing case in the large scale 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-back in 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 which was trimming the edge of the slab. This channel section, which will subsequently be referred to as edge reinforcement, was cast flush with the face side. The flanges therefore encompassed the edge of the opening. The shear arms intersected the edge reinforcement through rectangular apertures which were provided in the webs. 146

37 Table 4.5 gives the embedded length L and other details of shear arms used in Series-1 tests. Specimen Length Section Edge reinf. SPT mm Box No SPT mm I No SPT mm Box Yes SPT mm I Yes Table 4.5: Properties of Series-1 shear arms Fig and 4.42 show a close up of the connection between the shear arm and the edge reinforcement. In the case of the SPT1-3 and SPT1-4 detail, 15mm thick stiffeners were welded between the shear arm section and the flanges of the edge reinforcement. These were located above and below the webs of the shear arm section to ensure an efficient shear force transfer. This connection detail was adopted in all edge reinforced specimens including the Type-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 and 4.44 show SPT1-1 and SPT1-2 shear arm configurations without edge reinforcement. Fig and 4.46 show SPT1-3 and SPT1-4 shear arm configurations with edge reinforcement. 147

38 Figure 4.43: SPT1-1 shear arm assemblage Figure 4.44: SPT1-2 shear arm assemblage Figure 4.45: SPT1-3 shear arm assemblage Figure 4.46: SPT1-4 shear arm assemblage 148

39 In Series-2, six different edge reinforced shear arm details were investigated. All adopted shear arms consisted of pairs of hot rolled back-to-back welded RSC-51x38 sections (the same as those adopted as shear arms in the large scale tests). The edge reinforcement was the same as used before but in one case consisted of a cold formed C-150x90x5 channel section with a constant thickness of 5mm. Additional features involved 500mm long T16 U-bent bars which were located at both sides of the shear arm. These were fillet welded to the edge reinforcement and fully tied into the main reinforcement. Some details also comprised anchors in the shape of 150mm long RSC-51x38 offcuts 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 L v Edge reinf. U-bars Anchor SPT mm PFC 150x90x24 Yes No SPT mm PFC 150x90x24 No No SPT mm PFC 150x90x24 Yes Yes SPT mm PFC 150x90x24 Yes No SPT mm C 150x90x5 Yes No SPT mm PFC 150x90x24 No Yes Table 4.6: Properties of Series-2 shear arms where L v represents the embedded length of the shear arm. Figs through 4.52 show all Series-2 shear arm configurations as specified in Table 4.6. Figure 4.47: SPT2-1 shear arm assemblage 149

40 Figure 4.48: SPT2-2 shear arm assemblage Figure 4.49: SPT2-3 shear arm assemblage Figure 4.50: SPT2-4 shear arm assemblage Figure 4.51: SPT2-5 shear arm assemblage 150

41 Figure 4.52: SPT2-6 shear arm assemblage 4.8 Instrumentation for slab panel tests Fig gives an overview of the instrumentation setup used for the slab panel tests. The top- and the front faces of all specimens were white-washed for better contrast. The actuator load was measured with load cells that were positioned between the hydraulic ram and the double-hinge. Figure 4.53: Instrumentation setup for slab panel tests Displacements of the ram were measured within the actuator as all tests were displacement controlled. The central vertical shear arm displacement was measured with a transducer located between the strong floor and the bottom plate of the clamped connection. Vertical displacements of the slab panel were measured on the top surface with LVDT s located adjacent to the supports and at midspan. Horizontal displacements of the slabs were measured with LVDT s on two points along the back edge of the slab panel. End rotations of the shear arm extension and those of the edge reinforcement (if present), were measured by means of inclinometers. 151

42 Typically for all adopted shear arms, Fig shows the location of the electrical resistance strain gauges at the shear arms. These were surface-mounted with cyanoacrylate adhesive after adequate preparation of the designated area. Axial strains in the extreme fibres were measured in the top- and bottom flanges. Additionally, shear strains were measured by means of electrical resistance strain gauge rosettes. These were centrally located at the web at the critical 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 presented in this work. The test series consisted of large scale and small scale tests. Material properties 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 either separately or in any combination of both. Subsequently, the reinforcement detailing and the adopted shearheads were explained. Regarding the latter, firstly a conventional ACItype shearhead was adopted which was then further developed. Furthermore, the employed gauging systems were explained which included the position of load cells, transducers and strain gauges. Following the need to investigate the local shear behaviour, a slab panel test rig was developed. 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 tests was subsequently discussed. Finally the gauging system for slab panel tests was presented. 152

43 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. The results presented include loading protocols, crack patterns, crack widths, load displacement responses, measured strains, a detailed description of the observed mechanical behaviour and the failure mechanisms. This chapter starts with gravity loading tests conducted on Type-A and Type-B shearhead configurations. It continues with Type-A and Type-B results of cyclic loading tests. Small scale test results of Series-1 and Series-2 slab panels follow. Ensuing tests of Type-C detail under gravity- and combined loading conditions are then being presented. Finally the test results are compared and assessed and the conclusions drawn are summarised. In the large scale tests the expression longitudinal is defined as any direction parallel to the lateral (horizontal) loading direction. This corresponds to the X-direction as shown in Fig and 4.23 in Chapter 4. Consequently, the expression transverse denotes any direction 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 looking in the positive Y-direction. In the small scale tests the term longitudinal is defined as a direction parallel to the shear arm 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 in the slab. 153

44 5.2 Type-A gravity loading test results The gravity load was applied vertically from the bottom of the column in 100kN steps. At each load step the cracks were marked, crack widths were measured and a set of Demec readings was taken. At a load level of 100kN radial hair width cracks formed at the top surface at the corners of the column. Tangential cracks were first observed at 200kN where then a regular crack pattern developed as the load was further increased. Failure occurred in punching at a load level of 450kN. The load displacement response is depicted in Fig. 5.1 which shows a distinct brittle response which 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 The continuous yellow line represents the visible perimeter, whereas the dotted line is the assumed extension of the perimeter, based on acoustic examination. The visible perimeter is relatively large since the shear crack ran along the top reinforcement. This caused the tensile reinforcement layers and the cover to separate from the concrete below. It is very likely that a smaller and steeper punching cone formed closer to the column. The location of the shear arms and the associated control perimeter according to the ACI-building code (dashed line) are also depicted in Fig It appeared that the actual failure perimeter was much larger than predicted. The largest radial crack width measured was 1.04mm at a load level of 400kN. The highest radial 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 yielding of the reinforcement occurred when considering these low strain levels and the additional reduction due to back projection of the strains from the surface to the reinforcement level. 154

45 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 deformation in the arms and in the column. Fig. 5.3 shows the permanently deflected shear arm which indicates the occurrence of minor plastic deformation at the onset of the gusset plates. From this it can be inferred that the critical section of Type-A detail is located at the intersection of the shear arm with the gusset plate. Figure 5.3: Plastic deformation of Type-A shear arm 155

46 For investigation the inside of the column, the bolted head plates were removed. It showed that the gusset plates prevented plastic deformation of the walls and therefore avoided local buckling of the column. Fig. 5.4 depicts the axial strain development in top and bottom flanges of a longitudinal and a transverse arm. The strains are shown for the critical section next to the onset of the gusset plates. It shows that the compressive strain levels in the bottom flanges are lower than those in the top flanges which indicates composite action. The strain levels exceed the yield strain level 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. Radial cracks were first observed at a load level of 60kN. These cracks occurred at the corners of the opening and above the embedded parts of the shear arms. Extensive formation of radial cracks occurred at 150kN when the surface cracks started to penetrate through the depth of the slab along the sides of the opening. At 210kN local shear cracks developed at the bottom flanges and propagated at an angle of 45 along the sides of the opening towards the tensile surface. Fig. 5.5 shows a close-up of the shear arm intersection. It shows such a typical shear crack at a load level of 240kN and two surface cracks penetrating the slab directly above the shear arm. 156

47 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- and subsequently at the longitudinal arms. Punching was triggered by the previously described shear cracks. The load displacement response is depicted in Fig. 5.6 which shows a rather ductile response. The quasi-horizontal plateau was characterised by yielding of the reinforcement bars adjacent to the edges of the opening which was caused by direct bearing of the shear 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 a regular radial crack pattern developed. Maximum tensile strains at 360kN measured above the shear arms were 0.085% in the radial- and 1.88% in the tangential direction. 157

48 The tangential strains were approximately 22 times higher than the radial strains which infers low 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 this direction. This prevented the vertical force component from being transferred to the main reinforcement. This resulted in large crack widths and yielding of the flexural reinforcement due to high bearing pressure of the shear arm onto the reinforcement. The shear crack width in the longitudinal side face measured 19mm at ultimate. Significant cracking also occurred in the corners of the opening owing to stress concentrations in these regions. The bottom surface (compressive side) on the other hand remained intact. 158

49 Figure 5.8: Localised shear failure at shear arm The shearhead was extracted from the slab after failure to investigate the plastic deformation of the fuses. Fig. 5.9 shows a close up of the shear arm between the gusset plates and the intersection with the slab which is indicated by line (b) shown on the right hand side. It shows that considerable plastic shear deformation was evident and that plasticity was restricted to the fuse length. Figure 5.9: Plastic deformation of Type-B shear arm Fig depicts the axial strain development in the top flanges of a longitudinal and a transverse 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) indicated significant plastic deformation whereas the strain levels in section (b) remained below the yield point. 159

50 Figure 5.10: Axial strains in flanges front and back Fig shows the strains in section (b) separately at a larger scale. This shows that the strains changed from initial compression into tension which was most likely caused by geometric nonlinearity effects. When looking at Fig. 5.9 it is evident that the zone of local plastic 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 shows the strain development measured in one longitudinal and in one transverse arm at an inclination of 45 to the shear arm axis. 160

51 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 F v = 200kN represented SLS conditions which were deemed to be 45% of the punching capacity of Type-A detail. The cyclic loading regime followed the provisions of ECCS (1986). The required actuator yield displacement e y =15mm was estimated by preliminary numerical analysis. The yield displacement was defined as the intersection of the initial elastic slope E 0 with the tangential stiffness of slope E 1 = E 0 /10. Table 5.1 gives the actuator displacement u act. and the number of cycles per interval where n = 1, 2, 3... denotes an interval counter starting from interval f. Interval cycles/interval u act. - - [mm] a 1 ±1e y /4 b 1 ±2e y /4 c 1 ±3e y /4 d 1 ±4e y /4 e 3 ±8e y /4 f 3 ± (2+2n) e y Table 5.1: Cyclic loading characteristics 161

52 The adopted loading regime is depicted in Fig which shows the actuator displacement versus the number of cycles. It needs to be stressed that the maximum stroke of the actuator 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 was kept 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 the bottom in two steps in order to capture the crack initiation. The gravity load was thereafter maintained with force control throughout the test. In the second phase the lateral load was applied in displacement control. The actuator was put on hold after every completed cycle to allow the specimen to be examined. Crack initiation was observed at F v = 100kN and the typical crack pattern developed when the load was further increased. After completion of the 1 st cycle no additional cracks developed but after the 2 nd cycle new cracks started to propagate in the longitudinal direction. Additional radial cracks formed during the 3 rd and 4 th cycle with some cracks initiating at the ends of the shear arms. The slab failed in punching shear without any warning signs during the 6 th cycle. Fig shows that punching failure occurred at one side only where the forces from gravity- and lateral loading were additive. 162

53 Figure 5.14: Punching perimeter of Type-A detail immediately after failure Fig shows the crack pattern and the failure perimeter (green). The shape of the failure perimeter was similar to the gravity test but its size was restricted to half of the slab and the other side showed no damage at all. At failure the crack pattern was purely radial and not fully developed which shows that the present punching shear mechanism was different from the gravity case. The bottom surface failed adjacent to the longitudinal column face by causing a wedge-shaped cavity. This was caused by concrete crushing as a matter of direct bearing of the column onto the surrounding concrete. 163

54 Figure 5.15: Failure perimeter and crack pattern of Type-A combined loading test Fig shows the hysteretic loops: significant stiffness degradation occurred in the 5 th cycle which indicates that punching shear was precipitated by cyclic concrete strength degradation. The maximum drift ratio reached was 2.5% with a maximum displacement amplitude of u max = 30mm. The maximum lateral load magnitude was F h = 76kN. This corresponds to a moment resisting capacity of around M max = F h = 90kNm. 164

55 Figure 5.16: Hysteretic response of Type-A detail Fig shows the axial strain development during cyclic loading. The denotation (a) means top- and (b) means bottom flange which were both measured in a longitudinal arm next to the gusset plates. It shows that again the tensile strain was higher than the compressive strain and that the plastic capacity of the shear arms was barely utilised. The alternating strains in the transverse arm next to the gusset plates implies warping torsion action with insignificantly low strain levels. Figure 5.17: Axial strain development in critical section 165

56 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 considered. The applied gravity load level of F v = 100kN represented the elastic limit load of the connection. The same lateral loading scheme as depicted in Fig was adopted for this test. Crack initiation was observed at a gravity load level of F v =80kN. Radial cracks initiated in the corners of the opening in the slab and above the embedded shear arms. After application of the 3 rd cycle, shear cracks developed in the side faces of the opening. These cracks typically initiated at the flanges of the longitudinal shear arms and propagated towards the top surface in an angle between 30 and 45. With increasing lateral displacement, shear cracks also started to emerge in the opposite direction towards the bottom surface as a matter of load reversals which overruled the gravity load. With every subsequent cycle, additional radial cracks developed when the shear cracks reached the top surface. After completion of the 8 th cycle, the shear crack width measured 4mm which indicates significant concrete damage. Fig shows the development of shear cracks in both directions at the intersection of one longitudinal 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 component of which was transferred by the legs of the U-bent bars trimming the edges. The behaviour was indeed similar to the one observed in the Type-B gravity test. Fig shows the deflected state of the longitudinal fuses. The plastic deformation of the left-hand side fuse was governed by shear. The increasing damage of the concrete at the shear arm intersection caused the arm to debond and therefore increased the effective length of the fuse. This caused 166

57 the stiffness to decrease with increasing concrete damage. However, the local shear failure was stable and not progressive, as it did not significantly decrease the connection strength. Figure 5.19: Shear head deflection under lateral loading Fig depicts the crack pattern at ultimate load which shows that punching shear failure did not occur. Figure 5.20: Crack pattern of Type-B combined loading test 167

58 Instead, failure occurred in the 14 th 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 the onset of the gusset plate (see Fig. 5.21). This area was amenable for ELCF as a result of micro structural transformation due to welding head exposure, which increased the brittleness of the parent material. Figure 5.21: ELCF failure of head fillet weld on top flange Fig depicts the hysteretic response of Type-B detail. It shows that the hysteretic response was stable with very little strength degradation. The maximum drift ratio reached was 7.2% with a maximum displacement amplitude of u max = 85mm. The maximum magnitude of lateral load was F h = 57kN, which corresponds to a moment resisting capacity of around M max = 68kNm. Figure 5.22: Hysteretic response of Type-B detail 168

59 Fig shows the axial strain development during cyclic loading. The strains 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). Figure 5.23: Axial strain development in dissipative zone The step-growth development of strains was caused by low resolution of the measured data which accidentally occurred during testing. However, it shows that the strains at the onset of the gusset plates were much higher than those at the opposite side. The reason for this is explained in Section 5.3. The strain levels at the critical section are considerably high as these nearly reached the hardening range. The strain amplitudes dropped from the 12 th cycle onwards which shows the beginning of ELCF failure. Fig depicts the principal strains under 45 as a measure of shear in the web in one longitudinal arm. It shows that the principal strains in the 5 th and 6 th cycle are higher than the axial strains in the flanges. The sudden drop of strains happened when the strain gauge rosette detached from the web. 169

60 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 yield displacement was estimated to be e y =3.5mm based on preliminary monotonic numerical analysis. The loading protocol used for all SP tests (unless stated otherwise) is depicted in Fig The procedure which led to that loading regime is the same as described in Section 5.4. The load was applied in displacement control. A loading rate of 5mm/min was used for all tests, and it remained constant for every cycle. Figure 5.25: Loading regime for SP tests 170

61 5.6.1 SPT1-1 This test involved a tubular shear arm without edge reinforcement. Crack initiation occurred early into the first cycle. The cracks initiated at the corners of the tubular section and propagated at an angle of approximately 30 towards the surface in the respective direction of 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 cracks spreading along the shear arm. At the 5 th cycle a punching shear cone started to develop where the face side cracks intersected with the top surface and at the 8 th cycle, a local failure perimeter was evident. The perimeter had a width of 600mm and was limited in depth to the embedded length of the shear arm, where the reaction force at the tip of the arm triggered punching shear failure. Fig shows the crack pattern at failure. The failure perimeter can 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 hysteretic response showed extensive degradation and pinching effects from early cycles on. The peak load of 40kN occurred at a displacement of 5mm after which the response softened rapidly 171

62 with increasing displacement. The test was stopped after the 12 th cycle where the residual capacity 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 crack initiation; this depended on whether a tensile or compressive force was applied in the first cycle. Therefore, the signs of the initial actuator force were chosen in alternation for every subsequent cycle. The asymmetry was amplified by the double hinge configuration which caused the shear arm to rotate more in the downwards than upwards load cycle. No yielding of the shear arm section occurred during the test. The end rotation measured at the head plate of the shear arm was ±1.2 at peak load and approximately ±4.3 at ultimate displacement SPT1-2 In this test the sheararm comprised an I-section. Cracks first developed in the 2 nd load cycle. The cracks initiated at the outer edge of the flanges and propagated at an angle of approximately towards the surface supports in the respective direction of loading. Fig shows that the crack pattern on the front face was symmetric and similar to that observed in the Type-B large scale test. From the 2 nd cycle onwards longitudinal cracks formed in the top surface of the slab over the flange of the shear arm. A fan-like radial crack pattern developed in the subsequent load cycles. 172

63 Figure 5.29: Crack pattern at front face of SPT1-2 detail A punching cone appeared to develop in the 5 th load cycle. The punching cone was bound by the inclined shear cracks that extended from the tips of the lower flange. Punching failure occurred in the 7 th cycle. The perimeter had a width of 700mm and was limited in depth to the embedded length of the shear arm. The failure crack pattern at ultimate load can be seen in Fig where the yellow line indicates the failure perimeter based on acoustic examination. The failure perimeter can be grossly approximated by a semi-circle with a radius 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 strong softening behaviour could be observed. The test was stopped after the 11 th cycle where the residual capacity dropped to 27kN (see Fig. 5.31). No yielding of the shear arm section occurred 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. 173

64 Figure 5.31: Hysteretic response of SPT1-2 detail SPT1-3 This test incorporated a tubular shear arm with edge reinforcement. The load was applied monotonically in displacement-controlled steps with a loading rate of 2mm/min. The reason for this loading scheme was to investigate any significant behavioural differences between cyclic and monotonic loading. The initial step size was 0.5mm/step up to 107kN which was thereafter increased to 1.0mm/step up to a load level of 126kN. From thereon a step size of 2.0mm/step was adopted up to peak load. From there, two final load steps of 5mm and 10mm were applied after which the test was stopped. Cracking was first observed at 70kN when shear cracks emerged at both edges of the channel and 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 around 90kN the shear cracks at the support penetrated the top surface of the slab. This led to 174

65 the reduction in stiffness which can be seen in Fig At this load level, inclined cracks emerged 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 supports deteriorated due to concrete degradation and from then on the behaviour changed over to longitudinal bending action. The longitudinal bending action led to a force redistribution where parallel cracks emerged in the transverse direction. In this phase the shear force and bending moment transfer of the shear arm to the slab is of particular importance. The longitudinal bending action caused a deep crack along the edge reinforcing channel. The concrete 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 at peak 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 sufficient axial pressure is provided. Since the reinforcement was not connected to the edge channel the increasing rotation decreased the shear capacity of the cracked section. Here the purpose of the shear arm became twofold: firstly it activated axial force-induced contact pressure as a matter of bond action and secondly it directly transferred the forces via direct bearing action. With increasing load the bond capacity of the shear arm diminished and therefore the ability of shear force transfer of the channel section onto the slab reduced. The failure perimeter and orthogonal crack pattern of the top surface are depicted in Fig

66 Figure 5.34: Crack pattern at top face of SPT1-3 detail The residual strength of the specimen is governed by direct bearing action of the embedded shear arm from the edge of the channel onwards. The overall behaviour of the slab panel was very ductile with a distinct softening post peak behaviour. The peak load measured 140kN at a 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 SPT1-4 This test involved an I-section with edge reinforcement. Crack initiation occurred during the 4 th cycle. The cracks initiated at the front face at the corner ends of the channel section and propagated directly to the supports. A discrete symmetric crosswise shear crack pattern was 176

67 evident inferring concrete compressive struts having developed between the edge supports and the flanges of the channel (see Fig. 5.36). Figure 5.36: Crack pattern at front face of SPT1-4 detail With every subsequent cycle with increasing amplitude, significant concrete damage occurred in the cracks near the supports. The initial behaviour of the specimen was governed by beam action in the transverse direction. In this case the crack pattern showed that the effective width of the beam measured about 200mm from the front face. The beam action diminished as the concrete was progressively damaged at the supports. Instead, the propagation of cracks from front to back indicated force redistribution into intact areas, which activated residual strength. This shows that biaxial bending action governed the behaviour subsequent to the deterioration 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 where the yellow line indicates the failure perimeter based on acoustic examination. Figure 5.37: Crack pattern at top face of SPT1-4 detail The failure perimeter width was defined by the span and measured about 800mm and the depth of the perimeter measured about 350mm. 177

68 The capacity peak of 130kN was reached at a displacement of 14mm and from thereon a strong softening behaviour could be observed. The test was stopped after the 16 th cycle when the residual capacity dropped to 35kN (see Fig. 5.38). No yielding of the shear arm section nor visible plastic deformation of the channel section occurred during the test. The end rotation at the head plate of the shear arm was ±2.4 at peak load and approximately ±5.1 at ultimate displacement. Figure 5.38: Hysteretic response of SPT1-4 detail 5.7 Series-2 slab panel test results In general failure types of the Series-2 specimens were similar to the edge reinforced Series-1 specimens. Therefore the Series-2 test results are not presented individually and emphasis was put on highlighting presentation of the differences. Fig shows the typical crosswise concrete failure of the SPT2-1 specimen at the support. Figure 5.39: Typical failure pattern near support (SPT2-1) 178

69 During cyclic loading the wedge shaped areas which were enclosed by the shear cracks loosened and came off. Fig depicts the hysteretic response of the SPT2-1 detail which had an embedded length of 150mm and welded U-bent bars but no anchor. Figure 5.40: Hysteretic response of SPT2-1 detail Fig shows the concrete failure of SPT2-2 specimen at the supports. This detail comprised a 340mm long shear arm without any additional features. It shows that the crack pattern at the 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) 179

70 Fig shows the hysteretic response of the same detail. Figure 5.42: Hysteretic response of SPT2-2 detail Fig shows the permanently deformed shear arm extensions of SPT2-3 detail. This detail comprised of a 250mm long shear arm with U-bent bars and anchor. It shows that these measures significantly increased the capacity to such an extent that plastic deformation in the shear arm could be achieved. The U-bent bars became increasingly important when the shear strength at the supports diminished and the behaviour changed from transverse to longitudinal action. At this stage a deep crack developed along the edge reinforcement which fully penetrated the depth of the panel in each case. The U-bent bars prevented the shear arm-edge reinforcement assemblage from detaching from the slab panel. This increased the ductility of the detail as the shear force transfer via concrete aggregate interlock could be maintained. Furthermore, the U-bent bars enabled the transfer of eccentricity-induced bending moments into the slab. Figure 5.43: Plastic deformation of SPT2-3 shear arm extension 180

71 The influence of the anchor in the SP tests was not as significant as the U-bent bars. The anchor mainly affected the post peak response. Fig shows the failure pattern of the top surface of the SPT2-3 detail. It shows that failure occurred along the supports due to concrete crushing below the steel wedges, which highlights the integrity of this shear arm detail. Figure 5.44: Failure pattern of top surface (SPT2-3) The hysteretic response of the SPT2-3 detail is depicted in Fig Figure 5.45: Hysteretic response of SPT2-3 detail Fig depicts the hysteretic response of the SPT2-4 detail which consisted of a 250mm long shear arm with U-bent bars but without anchor. It can therefore be assumed that the absence of the end anchors decreased the energy absorption capabilities of this detail. 181

72 Figure 5.46: Hysteretic response of SPT2-4 detail Fig shows the permanently deformed flanges of the weak-edge reinforcement of the SPT2-5 detail as the flanges could not resist the bearing pressure acting on them. This caused force concentrations to occur at the shear arm-slab panel intersection. The largest plastic deformations arose at the ends of the edge reinforcement because of the higher stiffness near the supports. Figure 5.47: Close up of deformed edge reinforcement SPT

73 Fig shows the localised concrete failure caused by concrete strength degradation. Figure 5.48: Failure pattern of bottom surface (SPT2-5) Fig shows the hysteretic response of the SPT2-5 detail. In comparison to the response of the SPT2-4 detail, the positive effect of the U-bent bars could not be utilised to the same degree, which resulted in lower capacity and ductility. Figure 5.49: Hysteretic response of SPT2-5 detail Fig depicts the hysteretic response of the SPT2-6 detail which consisted of a 250mm long shear arm without U-bent bars but with an anchor. It shows that the effect of U-bars on the capacity was greater than the effect of the anchor. 183

74 Figure 5.50: Hysteretic response of SPT2-6 detail Table 5.2 gives an evaluation of the performance of all Series-2 specimens. Specimen F p e p F p /F u e p /e y F deg - [kn] [mm] [1] [1] [%] SPT SPT SPT SPT SPT SPT Table 5.2: Evaluation of Series-2 SP tests where F p denotes the peak capacity, F u represents the residual capacity at ultimate displacement of 28mm, e y denotes the yield displacement as stipulated in the ECCS provisions, e p represents the peak displacement and F deg 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. Hairwidth cracks developed at the top surface as the load was increased to 90kN. These cracks propagated radially from the corners of the collar. At 150kN radial cracks also started to 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 184

75 anchors at a load level of 320kN. With increasing load level at about 400kN these cracks kinked 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 the safe working capacity of the test rig. The crack pattern at peak load is depicted in Fig which shows that the behaviour was governed by flexure. Signs of pending punching shear failure were not apparent. Figure 5.51: Crack pattern of Type-C gravity loading test Fig depicts the measured load displacement response including the unloading path. The highest circumferential strain measured in the top surface was 0.4% at 500kN. It occurred above the longitudinal arm at a distance of 140mm away from the collar edge. The highest radial tensile strain at the same load level was 0.38% which occurred along the longitudinal arm above its tip. These cracks widely reclosed after unloading, which indicates that no widespread plastic elongation of the reinforcement was present. 185

76 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 collar measured 8mm. Fig presents the shearhead after it was extracted from the slab. It shows the permanently deflected state of the fuses which infers shear deformation similar to those observed in Type-B detail. Figure 5.53: Permanent plastic deformation of fuses 186

77 Fig shows that the shear arm extensions remained straight and that plasticity was confined to the fuses. Furthermore, also the collar showed no signs of plastic deformation. Figure 5.54: Collar and shear arm extensions Fig depicts the axial strain development in the top flanges of a longitudinal and a transverse 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) indicate significant plastic deformation whereas the strain levels in section (b) remained below the yield point. Figure 5.55: Axial strains in flanges front and back Fig shows the strains in section (b) separately on a bigger scale. This indicates that the 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. It is very likely that the strain levels in the bottom flange in section (b) were much higher and similar to the measured strain levels in section (a). 187

78 Figure 5.56: Axial strains in flanges back Fig shows the strain development measured in one longitudinal and one transverse arm at an inclination of 45 to the shear arm axis. It shows that shear yielding occurred from about 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 conditions until failure. First a gravity load of F v = 250kN was applied. Then the lateral load was applied in a first phase with a loading rate of 5mm/min in 10mm steps. The actua- 188

79 tor displacement was stopped at 90mm displacement. Since total failure did not occur, the actuator displacement was reset to zero and the gravity load was subsequently increased to F v = 500kN. The lateral displacement was again increased monotonically in a second phase until failure. Failure was initiated in the longitudinal left hand side where forces from gravity- and lateral loading were additive, at a lateral displacement of 76mm. The deflected longitudinal shear arms with the governing fuse are depicted in Fig Figure 5.58: Shearhead deflection at ultimate load (F v = 500kN) The head fillet weld which connected the flange with the gusset plates failed. Total failure occurred in the second phase when the entire flange ruptured, which consequently led to a significant drop in capacity. The right hand side fuse and the transverse arms remained intact. Fig and 5.60 show the weld failure in the left hand side fuse. Figure 5.59: ELCF failure of flange (side view) 189

80 Figure 5.60: ELCF failure of flange (top view) Fig depicts the load displacement diagrams of both phases. Total failure occurred at 48mm 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 F v = 200kN was the same as applied in the Type-A test. The same lateral loading scheme as depicted in Fig was adopted for this test. During application of the gravity load the developing crack pattern was the same as described in Section 5.8. Subsequently, after every following lateral load reversal, the existing hair-width cracks propagated radially towards the corners of the slab. The crack pattern was mainly 190

81 caused by flexure and was stable upon application of the maximum amplitude as neither new cracks initiated nor existing cracks extended. The crack pattern after completion of the test is shown in Fig 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 gusset plates was visible at the 13 th cycle. This led to a drop in capacity after each following cycle. Fracture was progressive, although the amplitude was not increased, and considering the low number of cycles which caused failure, this can be interpreted as ELCF failure. Fracture followed the fillet welding seams along the flange edges on top and underneath the gusset plates. The parent material remained intact, but the gusset plates eventually completely separated from the flanges. 191

82 Figs 5.63 and 5.64 each show a close-up of the fillet weld failure at the connection of the shear arms with the gusset plates. Furthermore, Fig shows that the left-hand side fuse was governed 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 which was stable throughout the test with limited degradation. A total number of 15 cycles was applied and the test was stopped after a degradation of capacity of 23%. The lateral peak-load was F h = 70kN which corresponds to an induced bending moment of M max = 83kNm. The maximum column drift achieved was 6.8%. 192

83 Figure 5.65: Hysteretic response of Type-C detail Fig shows an elevation of the slab with the permanent column rotation at ultimate. Figure 5.66: Column drift at maximum actuator displacement Fig shows the axial strain developments during cyclic loading. 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 figure shows that the axial strains levels are symmetrical in both sections under cyclic loading, because the collar prevented the local deterioration of the concrete strength in section (b). Furthermore, ELCF effects became apparent at the 11 th cycle. 193

84 Figure 5.67: Axial strain development in dissipative zone Fig depicts the shear strain development during cyclic loading. It shows that shear yielding 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 shows that the initial stiffness of Type-A and Type-B details were similar to each other. Furthermore, the load displacement responses of the gravity tests revealed that the fully integrated detail had a higher capacity when compared to the partially integrated shear- 194

85 head; albeit the response of Type-B detail was more ductile, and failure was less brittle than the 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 shearheads show that the plasticity level was higher in Type-B detail; that is, the plastic capacity in the latter 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 the embedded parts of the shear arms due to force concentrations in the concrete. Lateral cyclic tests conducted on the same details showed that the conventional shearhead configuration was largely unsuitable for extreme seismic loading. The performance under cyclic loading was poor, as punching shear failure was precipitated at a load level of about half the gravity load capacity. In other words punching shear failure occurred with limited warning under gravity load SLS conditions. Fig compares the hysteretic responses of the Type-A and Type-B details. It shows that 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 under combined loading conditions was considerably enhanced. The hysteretic loops in Fig show that those associated with the Type-B detail were stable which led to increased energy absorption capabilities. Furthermore, punching shear failure under lateral loading conditions could be prevented by the novel shearhead configuration. 195

86 Figure 5.70: Hysteretic responses of Type-A and Type-B detail Generally the behaviour of the partially integrated shearhead system was governed by ductile shear yielding of the fuses. Comparison of the measured strain levels shows that the utilisation of plasticity in the steel was higher in the partially integrated assemblage. Although the novel concept was vindicated, two issues associated with the Type-B detail need to be considered. Firstly, the force concentration at the shear arm-slab intersection led to localised concrete shear failure and strength degradation under cyclic loading which has a significant influence on the capacity. Secondly, the fuses are amenable for ELCF effects which 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 concrete damage. 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-1 specimens tested were governed by concrete fracture as the shear strength of the shear arms was greater than that of the concrete. The behaviour of the specimen was governed by its ability to form an adequate strut-and-tie model to transfer the shear force from the arms into the concrete. The I-section behaved much better than the tubular section and showed a higher capacity because the flanges enabled the development of compressive struts. These struts could rest on the flanges further away from the 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 governed by local bending action of the reinforcement layers between the flanges and the surface. Fig schematically shows the strut-and-tie model which explains the observed crack pattern 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), 196

87 the transverse arms failed before the longitudinal arms; the U-bent bars did not encompass the main reinforcement in the transverse direction. Hence, the tensile component could not efficiently 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 major improvement mainly in terms of capacity. The channel section prevented the concrete from local damage and distributed the force onto a larger area. Therefore the edge-reinforced specimens could activate a larger perimeter. The rotational behaviour and its dependence on the reinforcing detail was of particular importance for the Series-2 tests to mitigate concrete strength degradation. The Series-2 results that were presented in this chapter showed that the behaviour of the shear arm detail could further be ameliorated by the introduction of welded U-bent bars and anchors. Both measures helped to maintain the contact pressure in the cracked interface between the edge reinforcement and the slab, which positively affected the shear force transfer. An elongation of the embedded shear arm length alone could not compensate for the effectiveness of the U-bars. The specimen with the longer arm and without additional reinforcement (SPT2-2) behaved in a rather brittle manner, and so did the specimen with the weak channel. Comparison of the SPT2-1 with the SPT2-3 shows that the anchor had a positive effect on the post peak softening behaviour, when the behaviour shifted from transverse to lateral action. At this stage the anchored shear arms could more effectively maintain the contact pressure between the channel and the slab. It was found that the SPT2-3 detail behaved most favourably of all investigated details (see Table 5.2). At this point it needs to be stressed that the peak capacity of the SPT2-3 detail exceeded the plastic web capacity of the shear arm extension by a factor of 2. This implies that the plastic section capacity was significantly enhanced by tensile action arising from geometric nonlinearity. The success of the U-bent bars and end anchors is explained by the efficient transfer of this tensile component. 197

88 Fig 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 displacement was increased three times. Although the SP tests are not perfectly reflecting the conditions of 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 control cracking whereas the anchor increases the pull-out capacity of the shear arm. An indication for significant tensile action in the shear arms can be deduced from the crack pattern depicted in Fig. 5.51, as tangential cracks evolved directly above the location of the anchors. This emphasises the importance of tensile action arising from higher order effects. Following the SPT2-3 detail, these additional features, along with an edge reinforcement in form of a continuous collar, were consequently considered in the development of Type-C shearhead detail. Finally gravity- and cyclic test results of a partially integrated shearhead (Type-C detail) were presented in this chapter. Fig shows that although the ultimate capacity of the Type-C detail remains unknown, the capacity of a conventional configuration was at least exceeded by 25%. Although subjected to a higher gravity load level, the lateral peak load capacity of the monotonic push over test had about the same capacity as its fully integrated pendant. The push-over test of the Type-C detail also revealed a high connection toughness as 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 almost exclusively governed by the performance of the steel fuses. Fig compares the hysteretic responses of a fully integrated shearhead with a partially integrated shearhead. It shows that the initial stiffness of both details was approximately the same. It also shows that 198

89 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 the Type-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 at the transition from the shear arm to the gusset plates. With hindsight to ELCF failure, two different welding details as described in Subsection were tested. In the case of the abandoned head fillet, fracture could only progress parallelly to the flanges without reducing the effective cross section of the shear arm. In the case of the fully welded detail, the forces could be transferred more efficiently due to the absence of shear lag, which resulted in a higher moment capacity. On the other hand, ELCF fracture severely affected the entire flange which reduced the cyclic lifetime of the connection. It can be concluded that the modified welding detail had a longer cyclic lifetime but gave a lower moment capacity when compared to the fully welded detail. It needs to be emphasised however that neither of these adopted welding details can be considered as ideal. Results revealed that the collar successfully suppressed concrete degradation and that the surrounding concrete remained widely undamaged. It was consequently shown that punching shear failure under cyclic lateral loading conditions could be prevented. In conclusion, it could be demonstrated by experiment that the concept of partially integrated shearheads for reinforced flat slabs works. Partially integrated systems offer significantly improved capabilities in terms of ductility and energy dissipation and the capacities of such systems remain within the range of fully integrated shearheads. 199

90 Chapter 6 Numerical methodology 6.1 Introduction The Finite Element Method (FEM) was originally invented due to the need for obtaining at least approximate solutions to complex problems in the physical world. In the field of the Finite Element method the term approximate means that this method allows any degree of desired proximity to the exact solution which is only limited by the computational capacity of the computer used. The FEM is therefore an essential, extremely powerful, and very often the 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 a specific issue which needs to be addressed when using the FEM. Nonlinear Finite Element Analysis (NLFEA), as conducted in this thesis, predicts theoretical solutions to physical problems. A theoretical result can only be falsified but not verified (Popper, 1963). This means that an NLFEA result can only agree with experimental results within acceptable limits of error. It is therefore crucial for every NLFEA to validate the adopted procedure against different sources of experimental data in order to gain confidence in the correctness of the results. The source of errors can be split into measurement errors and those caused by the NLFEA procedure itself, all of which causes ambiguity in the validation process. Regarding the latter, assuming that geometry and boundary conditions have been considered correctly, the sources 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 deviations of predicted results can be expected due to the differences of software code implementations especially when material nonlinearity is concerned. Apart fom that, probably the highest level of uncertainty in NLFEA is caused by the choice of material parameters. In this context the most ambiguous material parameters are those which either cannot be obtained by experiment (ficticious parameters) or those which strongly depend on the measuring method. 200

91 A parametric study is therefore essential when dealing with advanced material models to increase 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 is put on the first order finite element analysis of linear elastic materials which includes all the important steps involved. The linear finite element method is thereafter extended to NLFEA procedures. Firstly because concrete shows a highly non-linear material behaviour. Secondly, because the problems treated in this thesis require the consideration of geometric nonlinearity. Therefore, constitutive models for concrete and steel as implemented in DIANA are introduced. In conjunction with the concrete compressive behaviour a cracking concept which considers the tensile behaviour of concrete is discussed in detail. Subsequently the adopted iterative solution strategy and 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 used in this chapter for the presentation of the FEM and some relevant material models. Here only 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 the problems treated in this work are of a three dimensional nature the indices can take the integers i=1,2,3 according to the axes of the Cartesian coordinate system. The Einstein sum convention has been consequently adopted which says that two identical indices require a summation of the respective variable. A stress state in an arbitrary point can be expressed as a symmetric second order tensor where the entities of which are vectors and follow the transformation rule: σ 11 σ 12 σ 13 σ ij = σ 21 σ 22 σ 23 (6.1) σ 31 σ 32 σ 33 The stress tensor can be transformed into principal normal stresses in such a way that the shear components vanish. The problem can be written as: det (σ ij σδ ij ) = 0 (6.2) where δ ij is the so called Kronecker delta which is defined to be 1 if i=j and 0 otherwise and which therefore represents the unity matrix. Equation 6.2 represents an eigenvalue 201

92 problem. The characteristic equation of which is of cubic order and gives 3 solutions namely the principal normal stresses σ 1 σ 2 σ 3 : σ 3 I 1 σ 2 + I 2 σ I 3 = 0 (6.3) where I 1, I 2 and I 3 are the stress invariants which represent the trace of σ ij, the sum of the principal minors of the determinant of σ ij and the determinant of σ ij respectively: I 1 = σ ii = σ 1 + σ 2 + σ 3 (6.4) I 2 = 1 ( ) I σ ij σ ij = σ1 σ 2 + σ 2 σ 3 + σ 3 σ 1 (6.5) I 3 = 1 ( 2σij σ jk σ kl 3I 1 σ ij σ ji + I 3 ) 1 = σ1 σ 2 σ 3 6 (6.6) The stress tensor σ ij can be split into a volumetric- (also known as spherical, irrotational or dilational) and a deviatoric part (also known as equivolumnal or distortional). It needs to be stressed that the volumetric tensor causes a volume change whereas the deviatoric tensor only causes a distortion of the material which does not alter the volume. σ ij = pδ ij + s ij }{{}}{{} V D (6.7) p = 1 3 (σ 11 + σ 22 + σ 33 )= 1 3 σ ii (6.8) σ 11 p σ 12 σ 13 s ij = σ 21 σ 22 p σ 23 (6.9) σ 31 σ 32 σ 33 p where p denotes the hydrostatic pressure and s ij represents the deviatoric stress tensor. The invariants of deviatoric stress tensor J 1, J 2 and J 3 can be obtained the same way as for σ ij and can be written as: J 1 = s ii (6.10) J 2 = 1 2 s ijs ij = 1 ( s s s 2 3) (6.11) J 3 = s ij s jk s kl = s 1 s 2 s 3 (6.12) where s 1, s 2 and s 3 are the eigenvalues of s ij. The invariant J 2 is of great importance for the formulation of many constitutive models and can be expressed in principal stress coordinates as follows: J 2 = 1 [(σ 1 σ 2 ) 2 +(σ 2 σ 3 ) 2 +(σ 1 σ 3 ) 2] (6.13) 6 }{{} only related to shear 202

93 If σ ij is transformed into principal stress coordinates it reads: σ σ ij = 0 σ 2 0 (6.14) 0 0 σ 3 As will be seen later in this chapter some material models are expressed in the octahedral stress space. An octahedral stress plane is defined in such a way that its outward normal makes equal angles with each of the principal stress directions. The normal stress action on an octahedral plane can be expressed as: σ oct = 1 3 (σ 1 + σ 2 + σ 3 ) (6.15) and the shear stress acting in the octahedral plane can be written as: (2 ) τ oct = 3 J 2 (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 = 1 3 (σ 1 + σ 2 + σ 3 )=σ oct (6.17) J = 1 (σ 1 σ 2 ) 2 +(σ 2 σ 3 ) 2 +(σ 1 σ 3 ) 2 (6.18) 6 [ ( 1 θ = tan (σ )] 2 σ 3 ) (σ 1 σ 3 ) 1 (6.19) Fig. 6.1 shows that p is a measure for the distance along the hydrostatic axis (= space diagonal) between the origin and the deviatoric plane. The latter is defined as any plane perpendicular to the hydrostatic axis. Any stress state within the deviatoric plane can be expressed in polar coordinates. The deviatoric stress invariant J is a measure for the radius between the current stress state and the hydrostatic axis. Lode s angle θ defines the radial orientation. It can, depending on the convention, vary between +30 (triaxial extension σ 1 = σ 2 σ 3 ) and 30 (triaxial compression σ 1 σ 2 = σ 3 ) or between 0 and

94 Figure 6.1: Geometrical significance of p, J and θ (from Potts & Zdravković, 1999) The gradient of a relative displacement vector can be split into a symmetric second order tensor ɛ ij and into an anti symmetric second order tensor ω ij : ɛ ij = ɛ ij + ω ij (6.20) The latter represents the rigid body rotation where ɛ ij represents the strain tensor and can be written as: ɛ 11 ɛ 12 ɛ 13 ɛ ij = ɛ 21 ɛ 22 ɛ 23 (6.21) ɛ 31 ɛ 32 ɛ 33 The entities of the strain tensor can be expressed as: ɛ 11 = u 1 x 1 }{{} 1st order [ + 1 ( u1 ) 2 ( ) 2 ( ) ] 2 u2 u x 1 x 2 x 3 }{{} 2nd order ɛ 12 = 1 ( u1 + u ) ( u1 u 1 + u 2 u 2 + u ) 3 u 3 2 x 2 x 1 2 x 1 x 2 x 1 x 2 x 1 x 2 }{{}}{{} 1st order 2nd order (6.22) (6.23) and consequently as ɛ ij = ɛ ji = 1 2 (u i,j + u j,i + u k,i u k,j ) (6.24) The principal strains ɛ 1, ɛ 2 and ɛ 3 can be obtained by solving the eigenvalue problem: det (ɛ ij ɛδ ij ) = 0 (6.25) 204

95 As a result of the characteristic cubic polynomial equation the strain invariants I 1, I 2 and I 3 are: I 1 = ɛ 1 + ɛ 2 + ɛ 3 (6.26) I 2 = ɛ 1 ɛ 2 + ɛ 2 ɛ 3 + ɛ 3 ɛ 1 (6.27) I 3 = ɛ 1 ɛ 2 ɛ 3 (6.28) The strain tensor can be split into a volumetric and deviatoric component: ɛ ij = e v δ ij + e ij }{{}}{{} V D (6.29) e v = 1 3 (ɛ 11 + ɛ 22 + ɛ 33 )= 1 3 ɛ ii = 1 3 ɛ vol (6.30) ɛ 11 e v ɛ 12 ɛ 13 e ij = ɛ 21 ɛ 22 e v ɛ 23 ɛ 31 ɛ 32 ɛ 33 e v (6.31) where e ij is the deviatoric strain tensor and the invariants of which can be written as: J 1 = e ii (6.32) J 2 = 1 2 e ije ij = (e 1 e 2 + e 2 e 3 + e 3 e 1 ) (6.33) J 3 = s ij s jk s kl = e 1 e 2 e 3 (6.34) where e 1, e 2 and e 3 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 as limit equilibrium, stress field and limit analysis, the FEM is capable of satisfying all four fundamental requirements for a complete theoretical solution in continuum mechanics. That is satisfaction of equilibrium, compatibility, constitutive laws and boundary conditions. The FEM therefore provides extensive information which allows detailed insight into every level of structural behaviour. According to Potts & Zdravković (1999) the following steps are required for solving a problem with the FEM: 1. Element discretisation 2. Primary variable approximation 205

96 3. Formulation of element equations 4. Definition of boundary conditions 5. Solution of global equations 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 are then assigned to each sub-domain which in total forms the finite element mesh. This process is also called discretisation. Generally, such a mesh can involve one-, two- and three dimensional elements which are commonly called beam-, shell- and solid elements respectively. Every element consists of nodes which in the most basic element formulations are located at both ends in case of a beam element or at the corners in case of 2D and 3D elements. Advanced elements comprise additional 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 quality standards. These depend on the element type and concern the aspect ratio, minimum and maximum angles between the edges (distortion) amongst other criteria. Furthermore, the mixed 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 of rapid changes of displacements in the mesh (usually the region of interest) must be refined by the use of smaller elements. Both measures, however, increase the computational costs, which requires a mesh optimisation process. This optimisation process involves a mesh-sensitivity study which aims to investigate whether the results converge to the exact solution when the mesh density is increased. Numerical analyses of punching shear problems conducted by the author showed that the adopted shell elements with embedded reinforcement in their current formulation were incabable 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 strain distributions 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). Structural steel elements were discretised by 8 noded (serendipity class) isoparametric, Mindlin type shell elements (see Fig 6.3). One node in this element has 5 degrees of freedom (3 displacements and 2 rotations). Reinforcement was modelled discretely with 3 noded cable elements (see 206

97 Fig 6.4). One node in this element has 3 degrees of freedom (3 displacements). More details of 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) Primary variable approximation In the displacement-based FEM, the primary variable is the displacement which varies over the domain giving a displacement field. Other quantities are treated as secondary variables and can be computed from the primary variable. The key aspect of the FEM is that the displacements 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 influence of temperature and time dependency are ignored. Considering a 3D analysis, in case of a solid element, the displacement field is characterised by 3 displacements u 1, u 2 and u 3 in the respective directions of a Cartesian coordinate system. The displacement field within an element can thus be described by equation [N] represents the quadratic matrix of shape functions which interlinks internal- with nodal displacements, where the latter are treated as unknown degrees of freedom. The index n in equation 6.35 denotes nodal. u 1 u 2 u 3 =[N] 207 u 1 u 2 u 3 n (6.35)

98 The adopted element types used for NLFEA as presented in this thesis are higher order elements. 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 in DIANA (2008). Usually the element as it appears in the finite element mesh is derived from a parent element which 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 geometry from natural to global coordinates, such an element is called isoparametric. The global coordinates of an arbitrary point within a solid element can be obtained as follows: x 1 = n N i x 1i and x 2 = i=1 n N i x 2i and x 3 = i=1 n N i x 3i (6.36) i=1 where n denotes the number of nodes, x 1i, x 2i and x 3i are the nodal coordinates in the global coordinate system and N i are the interpolation functions. These are expressed in terms of natural coordinates. For each node in the element such an interpolation function exists which takes the value of +1 at the corresponding node and zero for the remaining nodes Formulation of element equations The incremental displacements within an element are expressed as follows: {Δd} =[N] {Δd} n (6.37) where Δu 1 {Δd} = Δu 2 Δu 3 Δu 1 and {Δd} n = Δu 2 Δu 3 n (6.38) The six first order strain-displacement relations can be written according to equation 6.24 as: Δɛ 11 = Δu 1 Δɛ 12 = 1 [ Δu1 + Δu ] 2 (6.39) x 1 2 x 2 x 1 Δɛ 22 = Δu 2 Δɛ 23 = 1 [ Δu2 + Δu ] 3 (6.40) x 2 2 x 3 x 2 Δɛ 33 = Δu 3 Δɛ 13 = 1 [ Δu1 + Δu ] 3 (6.41) x 3 2 x 3 x 1 208

99 Consideration of the above relations in equation 6.37 gives equation 6.42 which gives the incremental strains within an element as a function of the incremental nodal displacements. Δu 1 {Δɛ} =[B] Δu 2 Δu 3 =[B] {Δd} n (6.42) n The matrix [B] consists of the derivatives of the shape functions N i with respect to the natural coordinates. The Jacobian matrix is used to convert the derivatives from naturalinto global coordinates such that: { Ni ξ N i η } N T { i Ni =[J] ζ x 1 N i x 2 } N T i (6.43) x 3 where [J] can be written as: [J] = x 1 ξ x 1 η x 1 ζ x 2 ξ x 2 η x 2 ζ x 3 ξ x 3 η x 3 ζ (6.44) It is however necessary to express the global- in terms of natural coordinates such that: N i x 1 N i x 2 N i x 3 =[J] 1 N i ξ N i η N i ζ (6.45) The constitutive laws of a material relate stresses and strains, and therefore interlink equilibrium with compatibility. The constitutive behaviour of a material can therefore be written according to equation {Δσ} =[D] {Δɛ} (6.46) where {Δσ} represents the incremental Chauchy stress tensor, {Δɛ} represents the incremental strain tensor and [D] represents the constitutive matrix (also known as generalised Hooke s law) which is a fourth order tensor. The entities in such a constitutive matrix are called elasticity constants. A triclinic material which is the most general form requires 21 independent elasticity parameters. A monoclinic 209

100 material requires 13 independent parameters. Orthotropic material behaviour is defined by 9 parameters, whereas a transverse isotropic material requires 5 constants. The most basic form is isotropic material behaviour which requires 2 parameters which are commonly defined by the modulus of elasticity and Poisson s ratio. The element equations can now be obtained applying the principle of minimum potential energy. It asserts that a body deforms into a position in such a way that it minimises its potential energy. That is a body subjected to static loading conditions seeks an equilibrium condition 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 system maximises at equilibrium. The incremental total potential energy of a body ΔE can be written as: ΔE =ΔW ΔL (6.47) where ΔW is the incremental strain energy of the deformed body and ΔL is the work done by the body forces and traction forces. ΔW is obtained by integrating the strain energy density over the volume V of the element according to equation ΔW = 1 2 V {Δε} T {Δσ} dv (6.48) ΔL is the work done by the loads which consists of the work done by the body forces and surface or traction forces (see equation 6.49). The first part is obtained by integration over the volume of the element and the second part is obtained by integration over the surface. ΔL = V {Δd} T {ΔF } dv + A {Δd} T {ΔT } da (6.49) In equation 6.49 {Δd} T = {Δu 1, Δu 2, Δu 3 } represents the incremental displacement vector, {ΔF } T = {ΔF x1, ΔF x2, ΔF x3 } denotes the body force vector and {ΔT } T = {ΔT x1, ΔT x2, ΔT x3 } represents the surface traction vector. Equation 6.50 represents the principle of minimum potential energy. Substitution of equations 6.48 and 6.49 in equation 6.50 and making use of equations 6.37, 6.42 and 6.46 yields equation δδe = δδw δδl = 0 (6.50) [K E ] {Δd} n,e = {ΔR E } (6.51) 210

101 Equation 6.51 interlinks the nodal displacements {Δd} n,e with the nodal forces {ΔR E } of one single finite element. The element stiffness matrix K E is given in equation 6.52 and the right hand side load vector is given in equation [K E ]= [B] T [D][B]dV (6.52) V {ΔR E } = V [N] T {ΔF } dv + A [N] T {ΔT } da (6.53) It needs to be stressed that the integrals in equations 6.52 and 6.53 are evaluated for the natural 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 procedure for integration can be used. However, a coordinate transformation of the surface and the volume using the determinant of the Jacobian J according to equation 6.54 is required. dv = dx 1 dx 2 dx 3 = J dξ dη dζ (6.54) The integrals in equations 6.52 and 6.53 can usually not be obtained exactly and therefore need to be evaluated numerically. In the adopted software code DIANA, four integration procedures, namely the Simpson-, Gaussian, Newton-Cotes and Lobatto integration scheme are available. However, the Gaussian integration scheme was used for the analysis presented in this thesis because of its high numerical efficiency. Equation 6.55 represents the Gaussian quadrature for a two-dimensional integration. It shows that the integral is replaced by the weighted sum of function values which are evaluated at specific integration points. These integration points are also called Gauss-points where the locations of which are fixed within the geometry of the parent element f(ξ,η) dξ dη = ng ng i=1 j=1 w i w j f(ξ i,η j ) (6.55) In linear elastic analysis a full Gauss integration scheme gives exact values. The full integration scheme for the adopted solid and shell elements consists of 3x3x3 Gauss points in the ξ,η and ζ directions. Especially if material nonlinearity is introduced, and bearing in mind that the stresses within an element are evaluated at the Gauss points, the number of integration points needs to be increased to sustain the accuracy of the integration. On the other hand the numerical integration procedure is computationally expensive; that is a decrease of integration points significantly decreases the running time of a problem. It is therefore useful, for large numerical problems, to use reduced integration schemes for elements in less important areas in the domain. 211

102 After the element equations are formulated separately, the assembly of these into a set of global equations is required. The form of these is similar to equation 6.51 and can be written as: [K G ] {Δd} n,g = {ΔR G } (6.56) where [K G ] represents the global stiffness matrix, {Δd} n,g denotes the incremental displacement vector of all the nodes in the mesh and {ΔR G } n,g is the global right hand side load vector. The process of assembling the single element stiffness matrices into a global stiffness matrix is called direct stiffness method. The principle of this method is based on the summation of 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 this procedure can be found in Potts & Zdravković (1999) Definition of boundary conditions The choice of appropriate boundary conditions is of great importance in finite element analysis. 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 {ΔR G } n,g. The essential boundary conditions are displacement conditions (e.g. prescribed- or fixed displacements) which affect the global nodal displacement vector {Δd} n,g. Sufficent displacement conditions need to be assigned to the mesh in order to prevent ill conditioning in form of rigid body motions. In case of an ill-conditioned mesh the global stiffness matrix becomes singular ( ΔK G = 0) and equation 6.56 cannot be solved Solution of global equations In the final step of the FEM procedure the global set of equations is solved for the unknown nodal displacements {Δd} n,g. Hereby usually a huge set of simultaneous equations needs to be solved. This task has been the focus of research for the past decades and many different mathematical techniques to tackle this issue are available. One of the most prominent solution strategies is the Gaussian elimination method which is a direct solution method. If the size of the stiffness matrix increases, the direct methods become increasingly inefficient. In this case sparse solver techniques can be more efficient where the stiffness matrix is partitioned and these partitions are then solved separately. A detailed description of additional solution strategies can be found in Potts & Zdravković (1999). However, once the nodal displacements are obtained, the secondary variables such as stresses and strains can be computed from equations 6.42 and

103 6.4 Modelling concrete compressive behaviour The advances in experimental research on concrete as well as research on modelling concrete materials numerically made great progress during the last three decades. As a matter of fact the available literature on these subjects is vast and still increasing rapidly. For this reason it is hardly possible to cover the entire field of mechanical behaviour of concrete and concrete material modelling in this thesis. This section, therefore, gives a brief description of the basic mechanical behaviour of concrete subjected to different loading conditions. Based on that, some basic constitutive modelling approaches are thereafter presented and discussed. This section concludes with a more detailed presentation of the Total Strain model which has been adopted for the analysis in this 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 considered to be insignificant for the present problems Continuum considerations Especially when modelling concrete material behaviour the theoretical assumptions made in treating it as a continuum must be brought in accord with the observed nature of the material. Continuum mechanics is a phenomenological theory. Its purpose is not to describe continuum behaviour on the basis of the structure of a material such as, for instance, the atomic lattice or the pore structure. In this context it is noteworthy that the elasticity parameters mentioned in Subsection which describe material behaviour need to be obtained experimentally. In continuum mechanics it is assumed that the matter which forms a material is homogeniously distributed in such a way that it fills the entire space. In other words, a continuum features no voids or cracks which means that no part of a volume can be deformed to zero or to infinity. Furthermore, the continuum can be sub-divided into regions of infinitesimal size where 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 still the focus of extensive research activity. Concrete material consists of a skeleton of solid particles (aggregates) which form a pore structure. These pores are filled with cement paste which conglutinates the particles, whilst the cement paste itself consists of solid and liquid phases. Concrete naturally comprises microcracks and voids, all of which are violating the assumptions of a continuum as the pore structure and the exact location, shape and size of the aggregates are disregarded. Fig. 6.5 shows the tacit assumptions being made when treating concrete as a continuum. 213

104 Figure 6.5: Phenomenological conversion of concrete into a continuum However, continuum mechanics provides a robust and reliable theoretical concept which is essential for structural analyses to be comprehensible and efficient. To get continuum mechanics and the natural material behaviour into reasonable agreement, the length scales of the investigated domain need to be much greater than the size of its constituents. Unlike in steel or other similar materials, in structural concrete analysis the expression much is not clearly defined. It is therefore important to consider the accuracy of local effects (e.g. crack strains) in the scale of the discretisation Uniaxial compressive behaviour The unconfined uniaxial behaviour can be divided into four phases where f c represents the unconfined uniaxial peak strength. In the first part, up to f c /3, concrete behaves almost linear elastic and existing micro cracks in the pore structure do not progress. According to Kotsovos & Newman (1978) the stress level at f c /3 has been termed onset of localised cracking. In the second phase between f c /3 and f c /2 bond cracks start to propagate but the fracture process is stable; under constant load the released energy is still smaller than the required energy to progress fracture. In the third phase between f c /2 and 3f c /4 mortar cracks are initiated and bond cracks are propagating. The stress level of 3f c /4 has been termed onset of unstable fracture propagation where it coincides with the minimum volumetric strain. Progressive damage occurs between 3f c /4 and f c where with increasing compressive strain the stress-strain curve descends and undergoes compression softening. This phase is governed by the appearance of macroscopic cracks until crushing occurs at the ultimate compressive strain ɛ u. According to Chen (2007) the descending branch of the stress-strain curve is difficult to measure and strongly depends on the measuring method. Fig. 6.6 depicts a typical unconfined uniaxial compression response. The left hand figure (a) shows the stress-strain response whereas (b) shows the development of volumetric strain ɛ v = I 1 with increasing uniaxial pressure σ. The volumetric strain decreases rather linearly up to 3f c /4 where upon loading ɛ v sharply increases which indicates an expansion of the sample. 214

105 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. Up to 4f c /5 of axial stress the Poisson s ratio ranges between 0.15 ν At higher stress levels ν increases which shows that the value of a constant (elastic) Poisson s ratio becomes irrelevant. Figure 6.7: Relation between normalised stress and Poisson s ratio (from Chen, 2007) Biaxial behaviour Fig. 6.8 shows a typical bi-axial strength envelope which indicates that the uniaxial compressive strength increases under biaxial compressive loading. According to Nelissen (1972) 215

106 the strength increase under biaxial loading conditions is widely stress-path dependent. At a biaxial compression stress ratio of σ 2 /σ 1 =0.5 the concrete strength is enhanced by approx. 25%. At an equal biaxial compression state σ 2 /σ 1 = 1 the concrete strength is enhanced by about 16%. A nearly linear relation between compression-tension loading can be seen where no 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 is termed dilatancy. According to De Borst & Vermeer (1984) dilation is caused by frictional sliding, either along particles or along micro cracks. This phenomenon can be mechanically explained by two layers of loose granular material which have to undergo both a vertical and a horizontal displacement if these layers slide along each other. The vertical uplift causes the volume to increase under plastic deformation Triaxial behaviour Fig. 6.9 shows the axial stress-strain relationship for concrete subjected to different levels of compressive confinement pressure. It shows that both the concrete strength and the ductility increases with increasing levels of confinement (Mirmiran & Shahawy, 1997). With increasing levels of hydrostatic pressure the failure of the pore structure shifts from cleavage to crushing of cement paste (Kotsovos & Newman, 1978). 216

107 Figure 6.9: Triaxial stress-strain relationship for concrete (from Chen, 2007) Fig depicts the basic failure principles of concrete on a material level provided that the strength of the cement paste is lower than that of the aggregates. Triaxial test data showed that concrete failure can be described as a surface in the three dimensional principal stress space. The fact that the shapes of these failure surfaces for different kinds of concrete are similar, gave rise to the development of mathematically defined failure criteria. Experiments revealed 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 depicts such surfaces in the 3D stress space. The elastic limit surface denotes the onset of stable micro crack propagation and the failure surface represents the onset of unstable 217

108 crack propagation (Chen, 2007). Fig also shows that the meridians of the failure surfaces are curved. A meridian in this respect is defined as the intersection of a plane which contains the hydrostatic axis with any of the afore mentioned surfaces. Generally these meridians are smooth and convex where the curvature of which depend on I 1. (Schütz, 2010). Figure 6.11: Triaxial failure surface for concrete (from Chen, 2007) 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 by means of elasto-plasticity is briefly discussed. The implementation of elasto-plastic constitutive models in the framework of the finite element method allowed a close simulation of the complex three dimensional concrete behaviour. The yield function determines a three-dimensional stress state at which a material exhibits not 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 neither change its location nor its size. Such yield functions generally used for concrete are discussed in the next subsection. The values of the yield function F (σ) have the following physical meanings: 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 stress point when once touched the surface can only move on the surface. Fig gives a graphical interpretation of the yield function and the two conditions. The enclosed area of the yield function is called the elastic domain. 218