A RCHITECTURAL S TRUCTURES : F ORM, B EHAVIOR, AND D ESIGN A RCH 331 D R. A NNE N ICHOLS S UMMER 2018 lecture nineteen concrete construction: http:// nisee.berkeley.edu/godden materials & beams Concrete Beams 1 Architectural Structures Su2018abn Lecture 19 ARCH 331

Concrete Beam Design • composite of concrete and steel • American Concrete Institute (ACI) – design for maximum stresses – limit state design • service loads x load factors • concrete holds no tension • failure criteria is yield of reinforcement • failure capacity x reduction factor • factored loads < reduced capacity – concrete strength = f’ c Concrete Beams 2 Architectural Structures Su2018abn Lecture 19 ARCH 331

Concrete Construction • cast-in-place • tilt-up • prestressing • post-tensioning arch.mcgill.ca http:// nisee.berkeley.edu/godden Concrete Beams 3 Architectural Structures Su2018abn Lecture 19 ARCH 331

Concrete Beams • types – reinforced – precast – prestressed • shapes – rectangular, I – T, double T’s, bulb T’s – box – spandrel Concrete Beams 4 Architectural Structures Su2018abn Lecture 19 ARCH 331

Concrete Beams • shear – vertical – horizontal – combination: • tensile stresses at 45 • bearing http://urban.arch.virginia.edu – crushing Concrete Beams 5 Architectural Structures Su2018abn Lecture 19 ARCH 331

Concrete • low strength to weight ratio • relatively inexpensive – Portland cement • types I - V – aggregate • course & fine – water – admixtures • air entraining • superplasticizers Concrete Beams 6 Architectural Structures Su2018abn Lecture 19 ARCH 331

Concrete • hydration – chemical reaction – workability – water to cement ratio – mix design • fire resistant • cover for steel • creep & shrinkage jci-web.jp Concrete Beams 7 Architectural Structures Su2018abn Lecture 19 ARCH 331

Concrete • placement (not pouring!) • vibrating • screeding • floating • troweling • curing • finishing Concrete Beams 8 Architectural Structures Su2018abn Lecture 19 ARCH 331

Reinforcement • deformed steel bars (rebar) – Grade 40, F y = 40 ksi – Grade 60, F y = 60 ksi - most common – Grade 75, F y = 75 ksi – US customary in # of 1/8” (nominal) • longitudinally placed – bottom – top for compression reinforcement Concrete Beams 9 Architectural Structures Su2018abn Lecture 19 ARCH 331

Reinforcement • prestressing strand • post-tensioning • stirrups • detailing – development length – anchorage – splices http:// nisee.berkeley.edu/godden Concrete Beams 10 Architectural Structures Su2018abn Lecture 19 ARCH 331

Composite Beams • concrete – in compression • steel – in tension • shear studs Concrete Beams 11 Architectural Structures Su2018abn Lecture 19 ARCH 331

Behavior of Composite Members • plane sections remain plane • stress distribution changes E y E y 1 2 f E f E 1 1 2 2 Concrete Beams 12 Architectural Structures Su2018abn Lecture 19 ARCH 331

Transformation of Material • n is the ratio of E’s E n 2 E 1 • effectively widens a material to get same stress distribution Concrete Beams 13 Architectural Structures Su2018abn Lecture 19 ARCH 331

Stresses in Composite Section • with a section E E 2 steel transformed to one n E E material, new I 1 concrete – stresses in that My material are f c determined as usual I transformed – stresses in the other material need to be Myn f adjusted by n s I transformed Concrete Beams 14 Architectural Structures Su2018abn Lecture 19 ARCH 331

Reinforced Concrete - stress/strain Concrete Beams 15 Architectural Structures Su2018abn Lecture 19 ARCH 331

Reinforced Concrete Analysis • for stress calculations – steel is transformed to concrete – concrete is in compression above n.a. and represented by an equivalent stress block – concrete takes no tension – steel takes tension – force ductile failure Concrete Beams 16 Architectural Structures Su2018abn Lecture 19 ARCH 331

Location of n.a. • ignore concrete below n.a. • transform steel • same area moments, solve for x x bx nA ( d x ) 0 s 2 Concrete Beams 17 Architectural Structures Su2018abn Lecture 19 ARCH 331

T sections • n.a. equation is different if n.a. below flange f f h f h f b w b w x h h f b h x x h b nA ( d x ) 0 f 2 f f f w s 2 Concrete Beams 18 Architectural Structures Su2018abn Lecture 19 ARCH 331

ACI Load Combinations* • 1.4D • 1.2D + 1.6L + 0.5(L r or S or R) • 1.2D + 1.6(L r or S or R) + (1.0L or 0.5W) • 1.2D + 1.0W + 1.0L + 0.5(L r or S or R) • 1.2D + 1.0E + 1.0L + 0.2S • 0.9D + 1.0W • 0.9D + 1.0E *can also use old ACI factors Concrete Beams 19 Architectural Structures Su2018abn Lecture 19 ARCH 331

Reinforced Concrete Design • stress distribution in bending 0.85 f’ c b a/2 C C c a= 1 c d h NA A s T T actual stress Whitney stress block Wang & Salmon, Chapter 3 Concrete Beams 20 Architectural Structures Su2018abn Lecture 19 ARCH 331

Force Equations • C = 0.85 f ´ c ba 0.85 f’ c • T = A s f y a/2 a= 1 c C • where – f ´ c = concrete compressive T strength – a = height of stress block – 1 = factor based on f ´ c f 4000 c 0.85 (0.05) 0.65 1 1000 – c = location to the n.a. – b = width of stress block – f y = steel yield strength – A s = area of steel reinforcement Concrete Beams 21 Architectural Structures Su2018abn Lecture 19 ARCH 331

Equilibrium • T = C 0.85 f’ c a/2 • M n = T(d-a/2) C a= 1 c d – d = depth to the steel n.a. • with A s T A f s y – a = 0.25 * 0.65 ( ) 0.65 0 . 85 f b t y (0.005 ) c y – M u M n = 0.9 for flexure* – M n = T(d-a/2) = A s f y (d-a/2) Concrete Beams 22 Architectural Structures Su2018abn Lecture 19 ARCH 331

Over and Under-reinforcement • over-reinforced – steel won’t yield • under-reinforced – steel will yield • reinforcement ratio http://people.bath.ac.uk/abstji/concrete_video/virtual_lab.htm A ρ – s bd – use as a design estimate to find A s ,b,d – max is found with steel 0.004 (not bal ) – *with steel 0.005, = 0.9 Concrete Beams 23 Architectural Structures Su2018abn Lecture 19 ARCH 331

A s for a Given Section • several methods – guess a and iterate 1. guess a (less than n.a.) 0 85 . f ba 2. c A s f y 3. solve for a from M u = A s f y (d-a/2) M u 2 a d A f s y 4. repeat from 2. until a from 3. matches a in 2. Concrete Beams 24 Architectural Structures Su2018abn Lecture 19 ARCH 331

A s for a Given Section (cont) • chart method – Wang & Salmon Fig. 3.8.1 R n vs. M n n 1. calculate R 2 bd 2. find curve for f’ c and f y to get 3. calculate A s and a • simplify by setting h = 1.1d Concrete Beams 25 Architectural Structures Su2018abn Lecture 19 ARCH 331

Reinforcement • min for crack control • required 3 f c A ( bd ) s f y • not less than 200 A ( bd ) s f y • A s-max : 0 375 a ( . d ) 1 cover • typical cover – 1.5 in, 3 in with soil • bar spacing spacing Concrete Beams 26 Architectural Structures Su2018abn Lecture 19 ARCH 331

Shells http:// nisee.berkeley.edu/godden Architectural Structures Concrete Beams 27 Su2018abn ARCH 331 Lecture 19

Annunciation Greek Orthodox Church • Wright, 1956 http://www.bluffton.edu/~sullivanm/ Concrete Beams 28 Architectural Structures Su2018abn Lecture 19 ARCH 331

Annunciation Greek Orthodox Church • Wright, 1956 Concrete Beams 29 Architectural Structures Su2018abn Lecture 19 ARCH 331

Cylindrical Shells • can resist tension • shape adds “depth” • not vaults • barrel shells Concrete Beams 30 Architectural Structures Su2018abn Lecture 19 ARCH 331

Kimball Museum, Kahn 1972 Concrete Beams 31 Architectural Structures aasarchitecture.com Su2018abn Lecture 19 ARCH 331

Kimball Museum, Kahn 1972 • outer shell edges Concrete Beams 32 Architectural Structures Su2018abn Lecture 19 ARCH 331

Kimball Museum, Kahn 1972 • skylights at peak Concrete Beams 33 Architectural Structures Su2018abn Lecture 19 ARCH 331

Approximate Depths Concrete Beams 34 Architectural Structures Su2018abn Lecture 19 ARCH 331

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