Analysis and Design of a Continuous R. C. Raker Beam using Eurocode 2

Analysis and Design of Stadium Stadium Raker Beam Using EC2

Ubani Obinna U. (2016)

Analysis and Design of a Continuous Reinforced Concrete Raker Beam for Stadium Using Eurocode 2 Ubani Obinna Uzodimma Works/Engineering Services Department Ritman University, PMB 1321, Ikot Ekpene, Akwa Ibom State [email protected]

Abstract A continuous intermediate raker beam in the first tier of a football stadium was analysed using elastic method and designed using Eurocode 2. The raker beam was analysed for permanent and variable actions due to crowd load and permanent loads only. Due to its inclination, it was subjected to significant bending, axial, and shear forces. However, design results show that the effect of axial force was not very significant si gnificant in i n the quantity of shear reinforcement required. Asv/Sv ratio of 1.175 (3Y10mm @ 200 c/c) was found to satisfy shear requirements. The greatest quantity of longitudinal reinforcement was provided at the intermediate support with a reinforcement ratio of 1.3404%. The provided reinforcement was found found adequate to satisfy ultimate and serviceability requirements. requirements.

1.0 Introduction The most common construction concept of sports stadiums today is a composite type where usually precast concrete terrace units (seating decks) span between inclined (raker) steel or reinforced concrete beams and rest on each other, thereby forming a grandstand (Karadelis, 2012). The raker beams are usually formed in-situ with the columns of the structure, or sometimes may be preferably precast depending on site/construction constraints. This arrangement usually forms the skeletal frame of a stadium structure. In this paper, a raker beam isolated from a double tiered reinforced concrete grandstand that wraps around a football pitch has been presented for the purpose of structural analysis and design. A repetitive pattern has been adopted in the design which utilizes a construction joint of 25mm gap between different frame units. By estimate, each frame unit is expected to carry a maximum of 3600 spectators, under full working conditions. With ten different frames units, the stadium capacity is about 35000 after all other reservations have been taken into account. Each grandstand frame has precast L-shaped seating terrace units that span in between reinforced concrete raker beams inclined at angles between 20° - 22° with the horizontal. Crowd load and other loads are transferred from the seating units to the raker beams, which then transfers them to the columns and then to the foundations. Load from the service areas and concourse areas are also transferred using the same method.

Figure 1.1: 3D skeletal structure of each grandstand frame units (slabs and sitting areas removed)

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The three dimensional view of the skeleton of t he grandstand is shown in Figure 1.1, while a section through the grandstand is shown if Figure 1.2. Section through the L-shaped seating unit is shown in Figure 1.3.

Figure 1.2: Section through the grandstand

Figure 1.3: Section of the precast seating unit

In simple horizontal beams, vertical forces will produce vertical reactions only. However, once a beam is statically indeterminate and inclined, vertical forces will produce both vertical and horizontal reactions and as a result, axial forces which may be compressive or tensile in nature will be induced in the beam. In the design of horizontal floor beams in normal framed structures, the effect of axial force in the shear force capacity of the section is usually neglected. This is largely due to the fact that these forces are usually compressive, and in effect tends to increase the concrete resistance shear stress (Vc) and (V Rd,c) of the section in accordance with BS 8110-1:1997 and EC2 respectively. It is a well known phenomenon that compressive axial force increases the concrete resistance shear stress of a section, while tensile axial force will reduce the concrete resistance shear stress. So this neglect can be justified in terms of it being a conservative design which can only err in economy. However for inclined beams members in a frame (as in the case of a grandstand), axial force behaviour can vary greatly especially when the load is applied in the global direction (which is the prevalent scenario). In other words, based on the structural configuration of the structure and the loading, it is even common to see the

nature of axial forces moving moving from positive (tensile) to negative (compressive) (compressive) in the same span of an inclined member. If the load is however resolved and applied in the local direction of the inclined member, the axial forces will be absent. A good design will therefore require the use of less shear reinforcement in the axial compression zone, and more shear reinforcement in the axial tension zone. While the effect of axial forces may be neglected in horizontal floor beams under axial compression, it may be unsafe to neglect it in inclined beams because more often than not, some sections are usually usually under axial tension.





1.2 Materials, Model, and Loading Structural design of stadiums is critical and this becomes more obvious when EN 1990 (Eurocode – Basis Basis for Structural Design) classified it under ‘Consequence Class 3’ which by description means high consequence for loss in terms of human life, economy, environmental considerations, and otherwise if failure should occur. Several codes of practice across different countries and bodies have provisions made in them for the design of structures subjected to crowd loading (for example stadiums), but the level of expertise o ften associated with the processes in terms of analysis, des ign and construction is often perceived to be something left to a limited few. The application of static crowd imposed loads according to both BS 6399-1:1997( Loading on buildings - Code of practice for dead and imposed loads ) and EN 1991-1: ( Action on structures: General actions - Densities, self-weight, imposed loads imposed loads for buildings ) are given in Table 1 below:

Table 1.1: Values of variable actions on grandstands from BS 6399 and EN 1991

CODE

BS 6399-1:1996

EN 1991-1-1

CATEGORY(DESCRIPTION)

C5 (Areas susceptible to overcrowding e.g. grandstands) C5( Areas susceptible to large crowds, e.g sports halls including stands)

IMPOSED LOAD/VARIABLE ACTION (KN/m2) 5

CONCENTRATED LOAD (KN)

5.0 – 5.0 – 7.5 7.5 *

3.5 – 3.5 – 4.5* 4.5*

3.6

*Exact range of value to be set by various national annex Raker beams in stadiums usually support precast seating terrace units which may be L-shaped, or extended into a more complex shape (see Figure 1.5). These seating terrace units are designed as simply supported elements spanning between the raker beams (Karadelis, 2012, Salyards et al 2005). The crowd loading is supported directly by these terrace units, which then transfer the load to the raker beams through the bearings. This construction concept has been adopted in the design of Cape Town Stadium (South Africa) for the 2010 FIFA world Cup (Plate 1.5). The picture in (Figure 1.4) below shows the formwork and construction of in-situ raker beams at San Diego State University Student Activity Activity Center (Steele and Larson 1996).

Figure 1.4: Typical formwork and reinforcement for in-situ raker beam (Steele

Plate 1.5: Precast seating units being installed on raker beams at Cape Town Stadium (2010)

and Larson 1996).

In this design, each L-shaped seating unit is 7m long, which means that the raker beams are spaced at 7m centre to centre. The crowd loading is supported by the terrace seating units, which is then transferred to the raker beams through the end shears. The T he raker beams can be analysed as sub-frames or as full 3D structures in order to get the most realistic behaviour of the structure.





1.2.1 Partial Factor for load The partial factor for all permanent actions (dead load) Gk is 1.35 while the partial factor for all variable actions (imposed load) Qk is 1.5. No reduction factor was applied in the analysis, and the effect of wind was neglected. 1.2.2 Material Properties for the design Design compressive of concrete f ck = 35 N/mm 2 ck = Yield strength of steel f yk = 460 N/mm 2 yk =

Table 1.2: Values used in the computation of loading

Load

Value 25 KN/m3 5 KN/m2 2 KN/m2

Density of concrete Imposed load/variable action Weight of finishes, rails, seats, stair units 1.2.3 Concrete cover Exposure class = XC1 A concrete cover of 40mm is adopted for the section 1.2.4 Design equations according EC2 From EC2 singly reinforced concrete stress block; MRd = = FCz ------------ (1)

FC =

.  0.8 ; z = d – 0.4x 0.4x -------------- (2) .

Clause 5.6.3 of EC2 limits the depth of the neutral axis to 0.45d for for concrete class less than or equal to C50/60. Therefore for an under reinforced section (ductile); ----------------- (3) x = 0.45d ----------------Combining equation (1), (2) and (3), we obtain the ultimate moment of resistance (M Rd ) MRd = = 0.167 ---------------------- (4)



Also from the reinforced concrete stress block; MEd = = FSz ------------------ (5) FS =

  ------------------ (6) .

Substituting equ (6) into (5) and making

 . ------------------- =



the subject of the formular;

(7)

The lever arm z in EC2 is given ;

[0.5 [0.5  (0.(0.25 25  0.882) 882) ]

z = d

where K =

---------------------- (8)

  ---------------- (9)  

1.2.4.1 For doubly reinforced sections;

−  ----------------------- (10) .(− )  + A -------------------- (11) Area of tension reinforcement  = . Area of compression reinforcement A S2 =

S2




[0.5 [0.5  (0.(0.250.882 ′) ]

Where z = d


where K’ = 0.167

1.2.4.2 Check for deflection (Clause 7.4.2)

The limiting basic span/ effective depth ratio is given by;

⁄     L/d = K 111.5   3.2    1  if  ≤  --------------------------- (12)       ⁄ if  >  ---------------------- (13) L/d = K 111.5     −    Where; L/d is the limiting span/depth ratio K = Factor to take into account different structural systems

 ′

10− 

= reference reinforcement ratio = = Tension reinforcement ratio to resist moment due to design load = Compression reinforcement ratio

1.2.4.3 Shear design In EC2, the concrete resistance shear stress without shear reinforcement is given b y;

(100 ) .     ≥   ) 

VRd,c = [CRd,c.k.

CRd,c = 0.18/ ; k = 1+

+ k 1

]bw.d

≥

(Vmin + k 1

. 

) ----------------------- (14)

 < 0.02 (In which  is the area of tensile reinforcement    beyond the section considered; Vmin = 0.035     < 0.02 (d in mm);



=

which extends ( K 1 = 0.15; = NEd /Ac /Ac < 0.2fcd (Where N Ed is is the axial force at the section, Ac = cross sectional area of the concrete), fcd = design compressive strength of the concrete.

1.2.5 Load Analysis 1.2.5.1 Loading on precast seating unit Permanent Actions

Self weight of the 7m precast seating deck (see Figure 1.4) (GK1) = (25 0.25 0.15 7) + (25 0.95 0.15 7) = 31.5 KN

× × ×

× × × × × × ×

Weight of finishes, rails, seats (G K2) = (2

0.95

Variable Actions Imposed load for structural class C5 (Q K ) = (5

7) = 13.3 KN

0.95

7) = 33.25 KN

Total action on L-shaped seating terrace unit at ultimate limit state by Eurocode 2 (FE) = 1.35∑(GKi) + 1.5Q K = 1.35(44.8) + 1.5(33.25) = 110.355 KN

1.2.5.2 Loading on the raker beams

Height of beam = 1200mm Width of beam = 400mm





Self weight of raker beam

Concrete own weight (waist area) = 1.2m the local direction)

×

0.4m

×

25 KN/m 3 = 12.00 KN/m (normal to the inclination i.e. in

Height of riser in the raker beam = 0.4m; Width of tread in the raker beam = 0.8m; Angle of inclination ( 20.556° Stepped area (risers) =

1⁄2 × 0.4 × 25 = ×

)

=

5 KN/m (in the global direction)

For purely vertical load in the global y-direction, we convert the load fro m the waist of the beam by; UDL from waist of the beam = (12.00 cos 20.556°) = 11.236 KN/m  Total self weight (Gk) = 11.236 + 5 = 16.235 KN/m Self weight of raker beam at ultimate limit state; n = 1.35∑(GKi) = 1.35 16.235 = 21.917 KN/m

×

Load from precast seating units

End shear from precast seating unit = F E/2 = 110.355 = 55.1775 KN Total number of the precast seating units on the beam = 24/0.8 = 30 units For an intermediate beam supporting seating units on both sides; Total number of precast seating units = 2 30 = 60 units Therefore, total shear force transferred from the seating units to the raker beam = 55.1775

×

× 60 = 3310.65 KN .   Equivalent uniformly distributed load in the global direction at ultimate limit state =  = 137.94 KN/m Total load on intermediate raker beams at ultimate limit state in the global direction = 137.94 + 21.917 = 159.857 KN/m

1.3 Structural Analysis A full 3D elastic analysis of the whole stadium was performed using Staad Pro with all elements loaded at ultimate limit state. Also, the raker beam is isolated as a subframe and also analysed. The results from the two models are very comparable.

Figure 1.6: 3D Modelling of the grandstand on Staad Pro





The internal stresses on the intermediate raker beams from the analysis of the frame at ultimate limit state are shown in Figures 1.7 to 1.9.

Figure 1.7: Bending Moment Diagram

Figure 1.8: Shear Force Diagram

Figure 1.9: Axial Force Diagram





1.3.1 Summary of Analysis Results

The summary of the analysis result of the raker beams is shown in Table 1.3. Table 1.3: Analysis Results of the Raker Beam

Section

Moment (KN.m) 1967.54 948.078 2283.18 1249.787 1565.63

MA MABspan MB MBCspan MC

Section

Shear Force (KN) 934.62 983.88 999.52 918.98

QAB QBA QBC QCB

Section

NAB NBA NBC NCB

Axial Force (KN) 380.061(C) 339.376(T) 510.767(C) 208.670(T)

1.4 Structural Design The structural design of the of the raker beam using EN 1992-1-1has been carried out and all the parameters parameters used in the, and and steps followed followed are shown below below in the subsequent subsequent sections sections.. Design compressive of concrete f ck = 35 N/mm 2 ck = Yield strength of steel f yk = 460 N/mm 2 yk =

bw = 400mm; 400mm; h = 1200mm; Cc Cc = 40mm 1.4.1 Flexural Design of span AB (MABspan)

MABspan = 948.078 KNm d = h – Cc – Cc – ϕ /2 – ϕ ϕ/2 – ϕlink d = 1200 – 1200 – 40 – 40 – 16 – 16 – 10 10 = 1134mm

  .   ×   k=   =  ×  ×  = 0.0527

Since k < 0.167 No compression reinforcement required

[0.5 [0.5  (0.(0.25 25  0.882) 882) ]

z = d

[0.5 [0.5  (0.(0.25  0.882(0. 82(0.0527) 527) ]

= z = d

= 0.95d

  . = . ×.× .× × = 2199 mm

2

=

Provide 5Y25mm BOT (ASprov = 2450 mm 2)

To calculate the minimum area of steel required; (TABLE 3.1 EC2) fctm =

0.3 × m⁄

=

ASmin = 0.26

0.3

× 35⁄

= 3.2099 N/mm 2

× Fyk × × × 3.2099 ×400 ×1134 × × bw

Check if ASmin < 0.0013

= 822.962 mm2

d = 0.26

bw

d (589.68 mm2)

Therefore, ASmin = 822.962 mm2 Check for deflection; K = 1.5 for beam fixed at both ends L/d = K

111.5 111.5    3.2    1⁄  ≤  if





 ⁄     L/d = K 111.5 111.5  −        if  >   =  =  35 ×  = 0.00540 < 10−√ 35 L/d = 1.5 111.5√ 111.5√ 35× 35 × ..  3.2√ .2√ 35 35 ..  1⁄ = 1.5(20.695 + 0.5333) = 31.842

Modification factor      ×  ×       ×  . =

=

= 278.241 N/mm2

=

=1.11

=

Since the span is greater than 7m, allowable span/depth ratio =

 × 

= 19.374

Actual deflection L/d =

 ×

31.842

× 

= 1.11

×

31.842

 = 11.301 

Since 11.301 < 19.374, deflection is ok.

1.4.2 Flexural Design of support A (MA); MA = 1967.54 KNm k = 0.1093; la = 0.8919; AS1 = 4861 mm2; ASmin = 822.9785 mm2 Provide 4Y32mm + 4Y25mm TOP (A Sprov = 5180 mm2) 1.4.3 Flexural Design of support B (MB); MA = 2283.18KNm k = 0.1268; la = 0.8717; AS1 = 5772 mm2; ASmin = 822.9785 mm2 Provide 6Y32mm + 4Y20mm TOP (A Sprov = 6080 mm2)

1.4.3 Flexural Design of span BC (MBCspan) MBCSpan = 1249.787 kNm k = 0.0694; la = 0.9345; AS1 = 2947mm2; ASmin = 822.9785 mm2 Provide 5Y25mm + 2Y20mm BOT (ASprov = 3083 mm2)

Check for deflection

  35  =  ×  = 0.00679 > 10−√ 35  ⁄     L/d = K 111.5  −        if  >  111.5√ 3535 × ..−   0 = 28.066 L/d = 1.5 111.5√



=

Modification factor      ×  ×       × =

=

=

= 299.039 N/mm2






=

 = 1.0366 .

Since the span is greater than 7m, allowable span/depth ratio = 28.066

 × 

 ×


28.066

= 15.89

Actual deflection L/d =

× 

= 1.0366

×

 = 11.301 

Since 11.301 < 15.89, deflection is ok.

1.4.4 Flexural Design of support C (MC); MC = 1565.63 KNm k = 0.0870; la = 0.9163; AS1 = 3765 mm2; ASmin = 822.9785 mm2 Provide 5Y32mm TOP (ASprov = 4020 mm2)

Provide Y16 @ 200mm c/c on both faces as longitudinal side bars 1.4.5 Shear Design 1.4.5.1 Support A VEd = = 934.62 KN; N = 380.061 KN (Compression) (Compression) Taking shear at the centreline of support; V Ed = 934.62 KN VRd,c = [CRd,c.k.



(100 ) .  + k 1

≥

]bw.d (Vmin + k 1

. 

) bw.d

CRd,c = 0.18/ = 0.18/1.5 = 0.12 k = 1+

       = 1+

= 1.4199 < 2.0

Vmin = 0.035

   =

=

= Vmin = 0.035

×(1.4199) ×(35)

= 0.3504 N/mm 2

 = 0.011419 < 0.02; K = 0.15  ×  1

= NEd /Ac /Ac < 0.2fcd (Where N Ed is is the axial force at the section, Ac = cross sectional area of the concrete), fcd = design compressive strength of the concrete.)



=

. ×   = 0.7917 N/mm  × 

VRd,c = [0.12

×

1.4199

2

(100×0.011419 ×35 )

×0.7917

+ 0.15

]400

×1134

= 318111.948 N = 318.11 KN

Since VRd,c < VEd , shear reinforcem r einforcement ent is required.



Assume strut angle = 21.8° Let us now investigate the compression capacity of the strut;

1   1     . ×    .   × . ×  × . × .  10− (. + .) ( +   )

v1 = 0.6

f cd = cd =

= 0.6

Taking

VRd,max =

= 0.516

= 0.85; fcd =

= 19.833 N/mm 2; z = 0.9d

=

= 1440.64 KN > VEd

Since VEd < < VRd,max VEd,s =

   cot = 934620 N 





 =   (. ×  ×. ×  × .  .) = 0.9153 Trying 3Y10mm @ 200mm c/c (235/200 = 1.175) 1.175 > 0.9153 Hence shear reinforcement is ok. Following the steps described above; 1.4.5.2 Support B; Shear at VBA

VEd = = 983.88 KN; N = KN 339.376 (Tension) Note that due to the tensile axial force in the section, the second term of V Rd equation equation assumes a negative value.



= 0.0134;

  =

0.7070 N/mm 2; vmin = 0.3504 N/mm 2; VRd = 230.6532 KN

Since VRd,c < VEd , shear reinforcem r einforcement ent is required



Assuming that the strut angle = 21.8° = 19.8450 N/mm 2; z = 0.9d = 1020.6 mm; V RDmax =1440.64 KN v1 = 0.5160; f cd cd = Since VRDmax > VEd

 = 0.9635 

Trying 3Y10mm @ 200mm c/c (235/200 = 1.175) 1.175 > 0.9153 Hence shear reinforcement is ok.

1.4.5.3 Support B; Shear at VBC

VEd = = 999.52 KN; N = KN 510.767 (Compression) Note that due to the tensile axial force in the section, the second term of V Rd equation equation assumes a negative value.



= 0.0134;

 1.0641 =

N/mm2; vmin = 0.3504 N/mm2; VRd = = 351.161 KN





 = 0.9789 


1.4.5.4 Support C; Shear at VCB

VEd = = 918.98 KN; N = KN 208.670 (Tension) Note that due to the tensile axial force in the section, the second term of V Rd equation equation assumes a negative value.



= 0.0089;

 0.4347 =

N/mm2; vmin = 0.3504 N/mm 2; VRd = = 213.2707 KN









 = 0.9000 


1.5 Discussion and Conclusion It is very easy to see that the influence of axial force was not very pronounced in the results produced. It would have been very significant using BS 8110. The maximum reinforcement was seen at support B due the high magnitude of moment at that section. This phenomenon is consistent with horizontal continuous beams. See detailing sketches in Figure 2.0.

Figure 2.0: Reinforcement detailing sketches (not to scale)





References [1] BS 6399 part 1: 1996: Loading for Building code of practice for dead and imposed loads. British Standards Institution. [2] BS 8110 – 1:1997: Structural use of concrete Part1: Code of practice for design and construction. British Standard Institutions. [3] EN 1991-1-1 (2002): General Actions- Densities, self weight, imposed loads for buildings [4] EN 1992-1-1 (2004): Design of concrete structures: General Rules and rules for building [5] Jeff Steele, Mark Larsen (1996): Raker-Beam Construction Requires Rugged Steel Forms. Publication #C960738 The Aberdeen Group [6] Karadelis J (2009): Concrete Grandstands. Part 1: Experimental investigations. Proceedings to the Institution of Civil Engineers – Engineering and Computational mechanics. Volume 162,Issue 1 ISSN 1755-0777 [7] Salyards K.A., Honagan L.M (2005): Evaluation of a finite element model for dynamic characteristic characteristic prediction of stadium facility.



Analysis and Design of a Continuous R. C. Raker Beam using Eurocode 2

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