5. Analysis

To achieve and check out the aseismic design concept according to Figures 9 and 10 several analytical approaches have been used: modal analysis, push-over analysis and design capacity method. Seismic actions parallel to both the façades and to the floor diagonals have been considered.

The building weight has been either increased by 10% [3] or decreased by 20% [4] depending on which case

was non-beneficial.

5.1. Modal Analysis

An elastic 3D-model of the entire structure from the foundation slab up to the very top of the building has been used. It has been assumed that the model is built-in at the base. Additional to each seismic action a simultaneous 30%—component in the perpendicular direction has also been considered [3].

The modal analyses have been used 1) to check out the compliance with the limit imposed by [3] for the elastic relative story displacement (1% of the story height), and 2) to proportion all structural members which could undergo plastic deformation during the design earthquake. To account for the plastic seismic response of the structure the design elastic seismic forces have been reduced with a “behavior” factor q equal to 3.6. The proportioning refers primarily to 1) the steel sections of the dissipative parts of the facade bracings, and 2) the longitudinal reinforcement of both ends of the coupling floor beams as well as of the cross-sections at the core bottoms.

5.2. Push-Over Analysis

The non-linear 3-D model used within the push-over analyses assumes that the tower is built-in at the ground level and made only of bars. The model has considered the geometrical effect of the box-type cores. The vertical distribution of the seismic loads has been assumed to be either rectangular or triangular [3].

The plastic behavior of the bars is modeled by means of bilinear relationships force—plastic deformation as depicted in Figure 11.

The bilinear relationships bending moments—end rotations for the RC bars (i.e. the floor coupling beams and the bottom of cores) were deducted by means of the constitutive relationships of concrete and reinforcement recommended in [5] with the mean values of both strengths and strains recommended within the Annex A. Both the confining effect of the transverse reinforcement and the influence of the axial forces have been taken into consideration. The reinforcement bars in the potential yielding zones have been designed with steel grade S500C in order to ensure the necessary ductility [5].

The bilinear relationships axial force—axial displacement for the steel bars of the façade bracing correspond to the values given in Table 1. Special requirements were imposed on the production of S235 to make sure that the actual yield stress complies with the assumed value. The façade bracings are designed with welded sturdy H-profiles. Their slenderness is less than 25 and the cross-sections correspond to class 1 according to [6].

On this account both tensile and compressive diagonals have been considered active. However, to account for the second order effects the assumed axial forces and ultimate deformations of compressive diagonals where reduced to 80% and, respectively, 75 % of the corresponding values of tensile diagonals.

For the calculation of the ultimate axial deformations of the façade bracings only the “dissipative” parts of the diagonals which are designed with S235 steel grade have been considered (see detail in Figure 10). Their length is 4000 mm at the 1^st bracing and 3400 mm above.

The results of the push-over analyses are depicted in Figure 12.

The red line corresponds to the displacement demand in Figure 9 and according to [3] is called “target displacement” (TD). TD has been evaluated according to the Annex B to [3] for the design spectrum from Figure 2 and resulted equal to 1.44 × 75 cm = 1.08 m, where 75 cm is the maximum displacement of the nonlinear equivalent system with a single degree-of-freedom (approximately the same for both seismic directions) and 1.44 is a factor accounting for the effect of multiple degreesof-freedom. The push-over analyses have been carried out up to a displacement 1.5 times higher than TD.

Figure 12 shows that the structure is expected to respond to the design earthquake without any significant damage and that the maximum base shear forces induced by the seism will range between 100 MN and 170 MN, depending on the direction of the seismic action and on the seismic load distribution over the building height. As expected the constant distribution yields higher seismic loads.

For comparison the corresponding values according to the modal analyses are 64 MN for the base shear force, practically independent of the earthquake direction, and 48 cm and 34 cm for the elastic top displacement when the seismic action is parallel to the façade and, respectively, to the floor diagonal.

A rough idea on the influence of the plastic behavior on the seismic response can be obtained by comparing the results of the two analyses. The ratio between the maximum seismic responses according to the push-over analysis and according to the modal analysis varies between 1.5 and 2.2 in terms of both forces and displacements. These values give a hint of the magnitude of the actual behavior factor q.

The push-over analyses yield some important detailed information on the plastic seismic response of the HDand LD-members from Figure 10, which are crucial for the energy dissipation. They are to be discussed hereafter only for the triangular vertical distribution of the seismic loads as this case is more plausible for the minaret. As expected for a mast-like tower with braced corner cores, the magnitudes of the plastic deformations of the facade bracings and of the coupling floor beams become larger when the seismic action is parallel to the façade, whereas the core dimensioning is associated with the seismic action along the floor diagonal.

The behavior of the façade bracings is depicted in Figure 13, which shows all 11 façade bracings over the minaret height as well as the steel profiles within the walls coupling the cores at their top. The given steps of the top displacement correspond to the last 5 cross— marks on the blue line in Figure 12. The points depicted “pink” denote the situations when the yielding is initialized; the points depicted “blue” the situations when the plastic deformations are smaller than 50% of the ultimate limit. As expected the compressive diagonals yield first and more than the tensile ones. At TD the largest plastic deformation occurs in the 4th bracing and reaches some 6% of the ultimate deformation. At 1.5 times TD the largest plastic deformations occur in the 6^th bracing from bottom and amount to some 30% of the ultimate value in the compressive diagonal and, respectively, 20% in the tensile one. The 1^st bracing and the steel bracing within the coupling walls at the top of the cores remain elastic even at 1.5 times TD.

Concerning the plastic behavior of the coupling floor beams the results of the push-over analyses indicate that:

• At TD all beams of the floors situated between the 1^st and the 9^th bracing in Figure 13 yield, whereas the maximum plastic hinge rotation at the upper side amounts to 22% of the ultimate value and that at the lower side to 3% of its ultimate value.

• At 1.5 times TD the beam yielding extends to all floors up to the coupling walls at the top of the cores. The maximum plastic deformations occur within the floors between the tops of 3^rd and the 8^th bracings in Figure 13 and amount to 43% (at upper side) and 12% (at the lower side) of their respective ultimate values.

Concerning the plastic behavior of the cores at minaret base the push-over analysis yields the following results

• At TD no plastic deformation has occurred.

• The first plastic hinge occurs within the tensile core when the top displacement amounts to a value 1.4 times higher as TD.

• At 1.5 times TD the maximum plastic rotation at the bottom of the tensile core amounts to 1% of its ultimate value while all other cores behave still elastically.

6. Structural Members

The main structural members of the minaret are either RC elements (basement walls, all floor elements and the summit tube) or steel elements (façade bracings, substructure of the summit envelope) or composite elements (the cores and the coupling walls at the core tops). Some fundamental details concerning the proportioning of the bracings, of the cores and of the coupling floor beams will be presented hereafter.

When using the Euro Codes the proportioning of the structural elements must fulfill the requirement

(1)

E_d denotes the design internal forces and R_d the corresponding resistances. For the aseismic design the R_d values are determined by means of the material design strengths, which result from dividing the characteristic strengths by the factors γ_c = 1.2 for concrete and γ_s = 1 for steel.

In the case of a HD-member the E_d value results on the basis of the modal analysis. After their proportioning and detailing the actual resistance R_eff of the element can be calculated by using all existing cross-section components and more realistic material strengths as the design values, e.g. the mean values.

In the case of LDor E-members the E_d value results on the basis of push-over analyses by considering a top displacement equal to TD or more. The higher the top displacement considered, the safer is the proportioning of these elements.

In the special case of possible fragile collapses (e.g. bolted connections of steel members or shear resistance of RC elements required to behave ductile) the E_d values results by means of the “capacity design” method.

6.1. Steel Bracing

To the steel bracing belong not only the facade diagonals but also the steel profiles embedded within the core external walls (see Figure 4). The axial forces in the façade diagonals are given by their real resistance R_eff. On the basis of these forces the associated axial forces within the other steel bracing members can be determined by simply using equilibrium conditions. For the vertical steel members situated in the three external corners of the cores these axial forces represent only a local modification ΔN of the global axial forces N_glob resulting from the seismic response of the structure (Figure 14). Indeed the vertical steel members reinforce the core cross section and the N_glob forces are associated with the sectional forces N and M in Figure 6. Due to the stud connectors the local modifications ΔN are transferred to the other components of the core wall, i.e. to concrete and longitudinal reinforcement bars, so that the effect of ΔN disappears at a certain distance beneath the corresponding node and is already included within the global force N_glob. It has been assumed by the proportioning of the strut connectors on the vertical steel profiles that the transfer of the ΔN takes place over a length of a story height. When the vertical steel member is elastically stressed, ΔN is transferred both above and beneath the corresponding truss node. When the vertical steel member

yields, ΔN is transferred only above the corresponding truss node, i.e. there where an unloading takes place.

Typical details of the steel bracing in elevation and in plan are depicted in Figures 15 and 16, respectively. The exterior vertical member is embedded at the junction of both external core walls. Its cross-section is made of two identical parts placed within each of the two external walls. The two parts are welded together at every node of the spatial truss (Figure 17). The interior vertical member is embedded in the external core wall at its junction with the internal core wall. To reduce the slenderness of the façade diagonals the central node of the X-cross joint is hold horizontally in the plane of the corresponding RC floor.

All connections to be executed on the construction site are bolted. The positions of all these connections were dictated by geometrical, transportation and erection criteria.

Basically the design of the cross section areas and of the connections in Figure 15 follows the concept already described in conjunction with Figure 10.

• The parts of the façade diagonals with the cross section area A_s,2 made of highly ductile steel grade S235 act as “fuses” for the entire steel bracing.

• The other parts of the facade diagonals and the embedded diagonals have the same cross-section areas but are made of steel grade S355. The ratio between the yield strengths of the two steel grades is equal to 355/235 ≈ 1.5, i.e. large enough to ensure that the S355 parts remain elastic.

• The bolted and fillet weld non dissipative connections are also designed with a sufficiently safe overstrength relative to the steel profiles. The resistance R_bd of the bolted shear connections fulfills the relation [3]

(2)

R_sd denotes the resistance of the connected dissipative part and is calculated with the cross section area A_s,2 and with the design yield stress f_sy,d of S235. The factor 1.1 accounts for the possibility that the real value of f_sy,d may be larger than considered, the factor 1.25 accounts for the necessary overstrength of a non dissipative connection and the factor 1.2 accounts for the necessary overstrength of shear resistance relative to the design bearing resistance. The resistance R_wd of fillet weld connections fulfills the relation [3]

(3)

The acting force E_wd is associated to R_sd and equal to the local force ΔN in Figure 14.

• The non dissipative connections made by means of full penetration butt welds are deemed to satisfy the overstrength criterion [3].

The two parts of the exterior vertical member in Figure 17 can receive different local forces ΔN depending on the earthquake direction (Figures 18 and 19). In the

case “c” the two parts are similarly loaded, so that their connection is not stressed. On the contrary in the case “b” the connection is mostly stressed. The existing eccentricity between the forces ΔN yields a local rotational moment within the core external wall, which additionally stresses the strut connectors and which has to be considered by the proportioning of the horizontal reinforcement of the core external walls.

6.2. Coupling Floor RC Beams

During the design earthquake plastic hinges occur in both directions at both ends of the main beams of the RC floors (Figure 3). Additionally to the flexure produced by seismic action and gravity loads the outer main beams are also subjected to permanent tensile forces. Indeed the steel diagonals are compressed by the structure weight, push the adjacent cores apart of each other and therefore tension the floor beams (Figure 20(a)). The imposed compressive force D and subsequently the imposed tensile force N_B increase with the relative vertical shortening Δv of the cores and decrease with the elongation u of the RC beams. The creep of concrete core acts negatively while the cracking of the floor beams is beneficial. The final N_B corresponds to (3) in Figure 20(b), where ε_v = Δv/H denotes the vertical strain associated with the relative vertical shortening Δv and the factor A is equal with 2/3∙E_s∙A_sD∙sin²α∙cosα. Following notations are used: E_{s =} steel modulus of elasticity, A_s = total sectional area of the beam’s longitudinal reinforcement, A_sD = the sectional

area of the steel diagonal, ψ = parameter accounting for the beneficial effect of the tensile concrete on the actual beam elongation.

The straight line (1) in Figure 20(b) describes the effect of Δv on N_B and the straight line (2) corresponds to the beam elongation.

There is a correspondence between the final N_B value and the calculated crack opening. It has been attested that the crack openings comply with the limit of 0.3 mm imposed by [5] for indoor RC elements.

6.3. Composite Cores

The cores have a composite cross-section (Figure 16). The concrete has the class C50/60, the steel used for the reinforcement bars has the grade S500 and the embedded steel members are made of steel grade S355.

The necessary vertical reinforcement arose from the bending of the tower in a diagonal direction, whereas the necessary horizontal reinforcement arose from the bending of the tower in a direction parallel to façade. The results of the push-over analyses for the top displacement equal to the TD—value have been used. However the proportioning has been performed by means of bending moment envelopes of the results of both modal and pushover analyses, as the push-over analysis did not capture the influence of higher vibration modes. A typical example is given in Figure 21.

The vertical reinforcement at the bottom of the minaret has been finally decided by the internal forces of the tensile core. The safety margins of the both most stressed core cross-sections are outlined in Figure 22 by means of the difference between the acting bending moments M_Ed and the resistant moments M_Rd. The corresponding axial forces N are 265 MN (tension) and, respectively, 609 MN (compression).

The proportioning of the horizontal reinforcement is based on the envelope of the tower shear forces depicted in Figure 23. It has the form prescribed by [3]. The assumed value V_Ed,base corresponds to the rectangular distribution of the seismic forces over the tower height and to a top displacement 1.5 times higher than TD (Figure 12). This aimed at ensuring a higher resistance to shear than to bending. The ratio between the total seismic forces considered for the two situations is 140 MN/100 MN = 1.4.

The required horizontal reinforcement of the core walls arises from the design shear forces of the walls. To obtain these, the floor shear force from Figure 23 had to be distributed first to the cores and afterwards to the core walls resisting shear. The distribution factors were chosen by means of the elastic model used for the modal analyses. Both the earthquake direction parallel to façade as well the direction along the floor diagonal were analyzed.

Concerning the distribution of V_Ed between the four cores it has been found that 1) it is practically equal, i.e. V_Ed,core ≈ 0.25 × V_Ed, when “parallel” earthquake and that (ii) each of the two cores situated on the diagonal along which the “diagonal” earthquake acts overtakes only some 20% of the floor shear force, so that the relevant design value became V_Ed,core ≈ 0.3 × V_Ed.

Concerning the distribution of V_Ed,core between the resisting walls it has been found that it can be considered proportional to the wall thickness if a corrective factor is used for the external wall. The value of 1.15 has been found conservative in the case of the “parallel” earthquake, which has proved to be decisive for the required horizontal reinforcement. The required horizontal reinforcement A_sw/s (cm²/m) has been determined by means of Equation (1) with E_d taken equal to the value of V_Ed,wall as yielded by the described distribution of V_Ed  and R_d taken equal to the value

(4)

h_w denotes the length of the wall cross section.

The required horizontal reinforcement has been provided over the entire story height neglecting therefore the major beneficial contribution of the steel bracing. Indeed over the height of the stories with braced façade the steel diagonals overtake the most part of V_Ed whereas over the height of the other stories the steel diagonals embedded in the external walls have an important contribution to R_wd,wall. The additionally provided shear resistance accounts for the negative effects of local horizontal tensile stresses (e.g. see the discussion made in conjunction with the case “b” in Figure 18) and prevents the cracking caused by the composite action.

7. Foundation System

To this system belongs the rigid box-type enlargement of the minaret foot over the height of the two underground levels, the foundation slab and the “barrettes”. All structural members are of cast-in RC with concrete C50/60 (the basement walls) and C30/37 (floors, the foundation slab and the “barrettes”) and with reinforcement bars of steel grade S500. In accordance with the general design philosophy of the minaret, the foundation system has a higher seismic resistance as the tower itself. Moreover the resistance is increased gradually from the tower base to the foundation soil.

The design of the underground levels and of the foundation slab has been performed by means of a 3D-elastic model loaded with forces at the core bottoms and supported by elastic springs at the centers of “barrettes” and concomitantly by the subgrade soil.

The walls were modeled with membrane finite elements and the slabs (floors and foundation) with slab— membrane elements.

The forces at the core bottoms correspond to the results of the push-over analyses for a top displacement equal to 1.5 times TD. The axial forces and the biaxial bending moments at the base of each core have been considered by means of equivalent vertical forces applied at the corners of each core. The base shear forces have been given at the geometric center of each core. Different directions of the seismic action have been considered.

Each supporting point resists 3D displacements, i.e. two horizontal and one vertical. The corresponding spring stiffness values have resulted from the analyses of the soil-structure interaction reported below. The RC walls have to resist huge in-plane forces caused both by the cantilever action associated with the enlargement of the tower foot over the basement height and by the large seismic forces. These forces have required wall thicknesses up to 1.5 m and up to 8 layers of reinforcement grids.

A special attention has to be paid to two “hanging” effects. The first corresponds to the anchorage of the tensile forces within the steel members at tower foot and the second to the anchorage of the tensile “barrettes”, i.e. of those withstanding the foundation uplifting.

The first effect can occur either at the junction of the tensile façade diagonal with the interior vertical steel member or at the minaret corner when the vertical steel member is subjected to tension. The corresponding anchorage has been realized by embedding steel members over the entire basement height (Figure 24) and by providing sufficient vertical and horizontal reinforcement to spread the tensile forces and lead them into the foundation slab.

The second effect occurs at the border of the foundation slab at the junction with the “barrettes” which prevent the building’s uplifting. Due to the magnitude of the “barrette” tensile forces (up to 37 MN) and to the cantilever caused by the eccentric peripheral wall (Figure 8) the foundation slab must resist very large shear forces.

To realistically capture the soil-structure interaction an elastic 3D model with volumetric finite elements has been used. It consists of a 9 m thick foundation slab, of the “barrettes” and of the foundation soil. With the chosen thickness the modeled foundation slab has the same flexural stiffness as the entire box-type basement. The soil characteristics have been chosen according to the results of in situ soil tests carried out by the National Geotechnical Laboratory at Algiers (LCTP). A vertical section through the foundation soil is given in Figure 25. The barrettes traverse a thick sand layer and penetrate into the marl layer.

The forces acting at the core bottoms have been taken larger than the ones used to design the basement and foundation slab. Thus the vertical forces at the core corners associated with the seismic action have been amplified by the factor 1.05 whereas the horizontal forces by the factor 1.15. The higher amplification of the horizontal forces should cover the possibility that the total seismic force could act lower as assumed. Eight directions of

the seismic action have been considered, i.e. for both directions of each of the following seismic actions: parallel to the two adjacent façades and to the two floor diagonals. That was necessary in order to capture the influence of the different depths of the soil layers.

The modeling has shown that the soil is capable of carrying the forces induced and, as same time, has provided the internal forces and the 3D settlements at the “barrette” tops. These results have been used 1) to proportion the required barrette reinforcement, and 2) to evaluate the springs stiffness needed for the modeling of the basement and of the foundation slab.

8. Concluding Remarks

The design of the lateral stiffness and resistance of the minaret has been decisively affected by the expected seismic action. The wind action has been investigated by means of wind-tunnel tests carried out with 1/400 models. Both rigid and deformable models have been investigated. They have shown that the expected base shear force induced by the local wind conditions will reach about 25 MN, which is several times smaller than the design base shear force induced by the design earthquake. Push-over analyses indicated values of 100 MN - 140 MN for the latter forces, depending on the earthquake direction and on the distribution of the seismic forces over the tower height. According to these analyses the expected maximum displacement at the tower’s top amounts to 1.08 m and is associated with a non-linear seismic response.

The aseismic design has been performed on the basis of the performance criteria recommended by [3], which aims at ensuring optimal energy dissipation during the design earthquake and at protecting simultaneously the structural members which are to respond quasi elastically. The requirement for elastic response arises either for those structural members and connections whose collapse may be fragile or for components of the foundation system whose behavior depends on the soil response.

Consequently the member’s strength has been gradually increased from the highly ductile members over the less ductile ones up to the members which should respond elastically. The first category includes the façade steel bracing and the RC main floor beams. The second includes the bottom region of the tower cores and the coupling structural walls at the top of the cores. The latter category includes the rest of the core height and the members of the foundation system, i.e. the box-type basement, the foundation slab and the “barrettes”.

The push-over analyses confirmed the design philosophy. Plastic deformations were registered only within the zones with ductile behavior and their maximal values have had a large safety margin to the ultimate ones, even for top displacements larger than 1.5 times TD.

Due to the strong seismic conditions, to the tower slenderness and to the discontinuous façade bracing, the structural design of the minaret has faced some challenging aspects. The paper has outlined the most important of them.

9. Acknowledgements

We do thank all the colleagues who participated at the design and detailing of the minaret tower. The contributions of K. Golonka, M. Friedrich and M. Neacsu with his team deserve a special acknowledgement.

REFERENCES