Design of AAC floor slabs aording to EN 160 Example 1: Floor slab with uniform load 1.1 Issue Design of a floor slab under a living room Materials Component with a ompressive strength lass AAC 4,5, densit lass 550, welded steel reinforement with tensile ield strength 500 MPa and ultimate tensile strength 550 MPa. EN 160, table 1 and EN 10080 1. Material properties Dr Densit Table 1: Densit lasses, dr densities in kg/m³ Densit 400 450 500 550 600 650 700 lass Mean dr densit ρ m > 350 400 > 400 450 > 450 500 > 500 550 > 550 600 > 600 650 > 650 700 Compressive strength Table : Compressive strength lasses for AAC in MPa Strength lass AAC AAC,5 AAC 3 AAC 3,5 AAC 4 AAC 4,5 AAC 5 f k,0,5 3,0 3,5 4,0 4,5 5,0 EN 160, 4...3 EN 160, 4..4 1.3 Tpe of element Profile
Sstem and dimensions.1 Sstem Longitudinal setion 4747 Minimum value for support length AAC omponent beams floor elements roof elements minimum requirement 60 mm 40 mm 35 mm EN 160, A.11 Reommended values AAC omponent support material minimum requirement beams masonr 100 mm floor elements roof elements wall elements masonr steel onrete masonr steel onrete wood steel onrete 70 mm 50 mm 50 mm 70 mm 50 mm 50 mm 50 mm 50 mm 50 mm
L eff l w + 3 1 a1,min + 3 1 a,min 4,70 + 3 0,07 4,747 m The omponent has to be designed for all load ases also for impats resulting from transport. The relevant load ase for that is the transport with a fork lifter and for the weak axis. Figure 1: Transport situation fork lift truk Assumption for distane forks: b s 1,00 m ( L bs ) L antilever (5,00 1,00),0 m. Cross setion h 50 mm b 65 mm.3 Conrete over and effetive depth 1 0 mm 0 mm Assumption for fire resistane lass: REI 60 With a granted diameter of 8 mm the effetive depth is: d 50 mm 0 8 6 mm 3
3 Loads Self-weight of AAC element: with 35 kg/m³ steel and 6 M-% moisture ontent of AAC Load Permanent loads,5 erami tiles (inl. glue) 6 ement sreed AAC (g 6, kn/m 3 ) permanent load, g k Variable loads, q K Transport weight of AAC element: ρ trans 7,75 kn/m³ 0,55 kn/m² 1,3 kn/m² 1,55 kn/m² 3,4 kn/m²,00 kn/m² EN 160, 4...4 (1) Thikness of slab 0,5 m EN 160, 4...4 (3) 4 Internal fores Internal fores are determined for a single omponent with a width of 65 mm. 4.1 Internal fores for harateristi ombinations Load ombinations a. to EN 1990 G d1 γ G b g k 1,35 0,65 3,4,89 kn/m Q d1 γ Q b q k 1,50 0,65,00 1,88 kn/m ( Gd + Qd1) l V Sd1 ( Gd + Qd 1) l M Sd1 8 1 eff 1 eff,89 + 1,88 4,747 11,3 kn 4,77 4,747 8 13,44 knm 4. Internal fores for frequent ombinations G d b g k 0,65 3,4,14 kn/m Q d ψ 1 b q k 0,5 0,65,00 0,63 kn/m Categor A, ψ 1 0,5 V Sd ( G d + Qd ) l eff,14 + 0,63 4,747 6,57 kn 4
( G M Sd d + Qd ) leff 8,77 4,747 7,80 knm 8 4.3 Internal fores for quasi-permanent ombinations G d3 b g k 0,65 3,4,14 kn/m Q d3 ψ b q k 0,3 0,65,00 0,38 kn/m Categor A, ψ 0,3 ( G d 3 + Qd V Sd3 3),14 + 0,38 l eff 4,747 5,98 kn ( Gd 3 + Qd 3) leff,5 4,747 M Sd3 7,10 knm 8 8 4.4 Internal fores for transport situations G T γ G ρ trans b h 1,35 0,65 7,75 0,5 1,64 kn/m V T γ T G L 1,3 1,64,00 4,6 kn T γt GT L M T antilever 1,3 1,6 64,00 4,6 knm where T 1,3 (assumption for dnami oeffiient due to manipulationn of omponents, when indiatedd onsideration of national regulations) 5
5 Design 5.1 Material properties Charateristi ompressive strength, f k 4,5 MPa 4500 kn/m² Basi Shear Strength, 0,5 0,063 f τ Rd k 0,063 4,5 0,5 / 1,73 0,0773 MPa γ Mean modulus of elastiit of AAC slab, E 5 (ρ m 150) 000 N/mm² Charateristi strength of steel, f k 500 MPa 500 N/mm² EN 160, 4..4 EN160, A.4.1. (A.6) EN 160, 4..7 EN 160, 4.3.1 5. Design for bending Finding equilibrium of stress / strain: 1000 MSd1 γ 13,44 1,44 1000 m d 158,5 α f A d 0,85 4,5 0,65 0,6 0,6 k Reading from design table (see Annex A): ε 3,00 ε s 8,41 k x 0,63 1000 ϖ 175, A s A α f γ ϖ k S γ f k 0,65 0,6 175, 0,85 4,5 1,15 1,51 ² 1000 1,44 500 hosen: 7 Ø 8,0 mm (A s1 3,5 ²) Upper reinforement: 1000 MT γ 1000 m d α f A d k 4,6 1,44 0,85 4,5 0,65 0,7 0,7 49,80 6
Reading from design table (see Annex A): ε 1,55 ε s 10,00 k x 0,134 1000 ϖ 5,14 A s A α f γ ϖ k S γ f k 0,65 0,7 5,14 0,85 4,5 1,15 0,45 ² 1000 1,44 500 hosen: 3 Ø 6,0 mm (A s 0,85 ²) 5.3 Minimum reinforement f flm 0,7 4,5 1,15 MPa h A t b 6,5 1,5 781² EN 160, A.3.4 (A.3) A s,min k A t f flm / f k 0,4 781 1,15 / 500 0,76 ² A s,min 0,76 ² < 3,5 ² 5.4 Design for shear fore EN 160, A.4 Determination of reinforement ratio: A ρ l s, exis 3,5 0,0049 < 0,005 b d 6,5,6 Minimum design value of shear fore: tk;0,05 V Rd1 0,5 f b w d 0,5 0,10 4500 / 1,73 0,65 0,6 γ 18,37 kn EN 160, (A.6) Design value of shear fore: V Rd1 τ Rd (1 0,83 d) (1 + 40 ρ l ) b w d 77,3 (1 0,83 0,6) (1 + 40 0,0049) 0,65 0,6 14,17 kn Higher value is determinant (ritial) : V Rd1,04 kn V Rd1 18,37 kn > 11,3 kn V Sd1 Therefore, no shear reinforement is required. 7
5.5 Spaing of Longitudinal Bars EN 160, 5..7.. Centre distane between bars : 50 mm s l1 d Therefore, we onsider the longitudinal bars at a distane of 70 mm entre to entre as per the limits. And the distane of longitudinal bars from the panel surfae is supposed to be 0 mm. 8
6 Anhoring of longitudinal reinforement EN 160, A.10.3 Cross-Setional View Desription reinforement laout: diameter ross bars: Ø t 5,5 mm distane between longitudinal bars: s t 70 mm distane to bottom side of panel: e + Ø sl + Ø t / 0 + 8,0 + 5,5 / 30,75 mm Effetive length of transverse bars: t t 3 t 4 t 5 t 6 s t / 70 mm < 14 Ø t 77 mm > t t 3 t 4 t 5 t 6 70 mm t 1 t 7 t 1 +t 1 t 7 +t 7 35 + 15 mm 50 mm < 14 Ø t 77 mm > t 1 t 7 50 mm t i < 8 Ø t 44 mm t t t 1 + t + t 3 + t 4 + t 5 + t 6 + t 7 450 mm Maximum tensile fore : F ld,max M d1,max / z 13,44 / (0,9 0,6) 66,08 kn F ld,support M d1,support / z,44 / (0,9 0,6) 1,01 kn 9
Assume 9 transverse ross bars with diameter 5,5 mm for half of the panel and the arrangement is shown below: Figure 1: Reinforement laout Design value for bearing strength at support (m 1,3; n p, transverse ompression at support): f ld,support 1,35 m (e / Øt) α fk γ 1/3 1,35 1,3 (30,75 / 5,5) 1,44 1/3, f k γ 0,85 4,5, 4,5 1,44 Bond Class B1 8,7 MPa 6,88 MPa therefore, f ld,support 6,88 MPa Design value for bearing strength at middle of span (m 1,067; n p, transverse ompression at support): 1/3 1,35 m (e / Øt) α fk f f ld,field, k γ γ 10
1,35 1,067 (30,75 / 5,5) 1,73 1/3 0,85 4,5, 4,5 1,73 5,65 MPa < 5,8 MPa therefore, f ld,field 5,65 MPa where, α is a redution oeffiient for long term effet on ompressive strength of AAC (α 0,85) Anhorage fore apait (F RA ): F RA,support 0,83 n t Ø t t t f ld,support 0,6 n l n t F wg / γ s 0,83 5,5 450 6,88 0,6 7 0,5 A sl f k / γ s 8,7 kn < 45,89 kn EN 160, A.3. Welding Strength Class S1 0,60 nl nt F F RA,max min 0,83 φtot tt fld ( nt ); γ S 110,33 kn < 06,53 kn wg As, F RA,support F ld,support and F RA,max F ld,max Figure : Anhorage of tensile fores As we an see from fig. that the anhorage apait fore does exeed the design tensile fore at eah setion of the panel. Therefore, the assumption is satisfied for the required onditions. So, we an use 18 ross bars with Ø 5,5 mm (for whole panel). The suffiient anhorage has to be proven also for the upper reinforement (aording to the method above) whih is not shown here. 11
7 Servieabilit Limit States EN 160, A.9.4 Craking moment, M r (b h / 6) f flm (0,65 0,5 / 6) (0,7 0,8 4,5) 6,33 knm EN 160, A.9.4.3 and 4..5 where, f flm is the flexural strength of AAC ( 0,7 0,8 f k ) As, M f > M r, therefore, the slab is onsidered to behave in a manner intermediate between unraked and raked ondition. 7.1 Defletion under unraked ondition 7.1.1 Short-term defletion Ratio of the modulus of elastiit of reinforing steel and AAC: E 00000 N / mm s n 100 E 000 N / mm EN 160, (A.4) Moment of area of AAC and reinforement: 3 b h l, brutto + n (7 π (ø 1 /) 4 / 4 + 3 π (ø /) 4 / 4) 81396,19 4 1 The upper longitudinal reinforement an be full taken into aount to determine the moment of inertia. The position of the entre of gravit of the reinforement laer is supposed to be,4 from the panel surfae. Parts of moment of inertia from onsideration of the reinforement: A s1 3,5 A s 0,85 Centre of gravit, b h h / + n ( As1 s1 + As s) S b h + n ( A + A ) s1 s 97,05 11,15 1999,5 where, s1 and s are the distanes from the entre of the reinforement steel to the bottom surfae of the slab. h I ST b h ( s ) + n ( As 1 ( s1 s ) + As ( s s ) ) 1563 (1,5 11,15) + 100 (3,5 (,4 11,15) + 0,85 (,7 11,15) ) 4 4094,8 E I E I + I ) i,447mnm ( C; BRUTTO st 000 (81396,19 + 4094,8) 10 1 8
Defletion due to load ombination (frequent ation ombinations): 5 MSd Leff 5 0,00780 4,747 el 0, 00748m 48 E I 48,447 i Leff el 0,00748 m 0,75 < 1,90 50 General note: The limit value for the maximum defletion ma be found in a national appliation doument. The reommended value for the alulated defletion of roof and floor omponents subjeted to quasi-permanent loads is (aording to EN 160) span/50. EN 160, 9.4.1, Note 1 7.1. Long-term defletion For long term defletion an effetive modulus of elastiit, E,eff E / (1 + φ) EN 160, (A.43) is used. Therefore, E,eff 1000 N/mm and n E s 00000 N / mm E, eff 1000 N / mm 00 Moment of area of AAC and reinforement, 3 b h l, brutto + n (7 π (ø 1 /) 4 / 4 + 3 π (ø /) 4 / 4 ) 8141,17 4 1 Centre of gravit, S b h h / + n ( As1 s1 + As s) b h + n ( A + A ) s1 s 5079,85 10,9 436,5 Moment of inertia for reinforement, h I ST b h ( s ) + n ( As1 ( s1 s ) + As ( s s ) ) 1563 (1,5 10,9) + 00 (3,5 (,4 10,9) + 0,85 (,7 10,9) ) 77640,70 4 E I E I + I ) 1000 (8141,17 + 77640,70) 10, eff i, eff ( C; BRUTTO st 1,591 MNm 13 8
Defletion due to load ombination 3 (quasi-permanent ombinations): 5 MSd 3 Leff 5 0,00710 4,747 48 E, eff Ii 48 1,591 Leff 0,01048 m 1,05 < 1,90 50 0,01048 m 7. Defletion under raked ondition 7..1 Short-term defletion The ratio of the modulus of elastiit of reinforing steel and AAC: E 00000 N / mm s n 100 E 000 N / mm EN 160, (A.43) In this ase, we onsider onl ompression zone of AAC and reinforement for the alulation of moment of inertia. Therefore, first we will find the x- equilibrium x 1 + 4 d A 1 A 11,9 where, x is height of ompression zone from top surfae of panel d is effetive height, A b E / ( A s1 E S ) The upper longitudinal reinforement an be full taken into aount to determine the moment of inertia. Moment of area of ompression zone AAC and reinforements, 3 b x l, brutto + n (7 π (ø 1 /) 4 / 4 + 3 π (ø /) 4 / 4) 7511,14 4 1 The position of the entre of gravit of the reinforement laer is supposed to be,4 from the panel surfae. Parts of moment of inertia from onsideration of the reinforement: A s1 3,5 A s 0,85 14
Centre of gravit is: S b x ( h x / ) + n ( As1 s1 + As s) b x + n ( A + A ) s1 s 16431,7 14,38 114,6 where, s1 and s are the distanes from the entre of the reinforement steel to the bottom surfae of the slab I ST b x ( h x / ) + n ( A ( ) + A ( ) ) 73867,74 4 s s1 s1 s s s s E I E I + I ) i I i E 1,63MNm ( C; BRUTTO st 000 (7511,14 + 73867,74 ) 10 8 Defletion due to load ombination (frequent ation ombinations): 5 MSd Leff 5 0,00780 4,747 el 0, 0113m 48 E I 48 1,63 i Leff el 0,0113 m 1,1 < 1,90 50 7.. Long term defletion For long term defletion an effetive modulus of elastiit is used, E,eff E / (1 + φ ) therefore, E,eff 1000 N / mm and n E s 00000 N / mm E, eff 1000 N / mm 00 Moment of area of AAC and reinforement, 3 b x l, brutto + n (7 π (ø 1 /) 4 / 4 + 3 π (ø /) 4 / 4 ) 757,1 4 1 Centre of gravit, S b x ( h x / ) + n ( As1 s1 + As s) b x + n ( A + A ) s1 s 1906,0 1,16 1579,6 15
Moment of inertia for reinforement, x I ST b x 4 1475,74 ( h s ) + n As1 ( s1 s ) + As ( s s E I E I + I ) 1000 (757,1 + 1475,7) 10, eff i, eff ( C; BRUTTO st 1,300 MNm ) 8 Defletion due to load ombination 3 (quasi-permanent ombinations): 5 48 M E Sd 3, eff L eff I i 5 48 0,00710 4,747 1,300 Leff 0,018 m 1,8 < 1,90 50 0,018 m 7.3 Combination of defletion unraked / raked 7.3.1 Short-term defletion The short term defletion for the intermediate situation (raked/unraked) due to frequent loads is: ( 1 k) p 0,473 1,1 + ( 1 0,473) 0,75 0, k p + 93 ΙΙ Ι EN 160 (A.44) sd where k 1 0,8 ( M r / M ) 1 0,8 ( 6,33 / 7,80 ) 0, 473 M r : raking moment M sd : bending moment for frequent ombination of loading p Ⅱ : short-term defletion for raked ondition p Ⅰ : short-term defletion for unraked ondition Leff el 0,93 < 1,90 50 16
7.3. Long-term defletion B onsidering an effetive modulus of elastiit (E,eff ) and quasi-permanent ombination of loading is: ( 1 k ) p 0,473 1,8 + ( 1 0,473) 1,05 1, k p + 16 ΙΙ Ι sd where k 1 0,8 ( M r / M ) 1 0,8 ( 6,33 / 7,80 ) 0, 473 M r : raking moment M sd : bending moment for frequent ombination of loading p Ⅱ : short-term defletion for raked ondition p Ⅰ : short-term defletion for unraked ondition Leff 1,16 < 1,90 50 17
Annex A 1000 M 1000 m d Sd1 γ α fk A d α f γ A s A ϖ k S γ fk M sd1 d A A s f k f k γ,dutile γ S bending moment under harateristi ombination of loading (respeting transport load situations) effetive depth of omponent ross setion of AAC, A b d ross setional area of reinforement harateristi ompressive strength of AAC harateristi ield strength of reinforing steel partial safet fator of AAC for dutile failure partial safet fator for reinforing steel stainless steel, f k 35 MPa steel, f k 500 MPa 18