Reinforces Concrete Structures 1.

Tamás Juhász

Pécs 2019

The Reinforces Concrete Structures 1. course material was developed under the project EFOP 3.4.3-16-2016-00005 "Innovative university in a modern city: openminded, value-driven and inclusive approach in a 21st century higher education model".

EFOP 3.4.3-16-2016-00005 "Innovative university in a modern city: open-minded, value-driven and inclusive approach in a 21st century higher education model".

Tamás Juhász

Reinforces Concrete Structures 1.

Pécs 2019

A Reinforces Concrete Structures 1. tananyag az EFOP-3.4.3-16-2016-00005 azonosító számú, „Korszerű egyetem a modern városban: Értékközpontúság, nyitottság és befogadó szemlélet egy 21. századi felsőoktatási modellben” című projekt keretében valósul meg.

EFOP-3.4.3-16-2016-00005 Korszerű egyetem a modern városban: Értékközpontúság, nyitottság és befogadó szemlélet egy 21. századi felsőoktatási modellben

REINFORCED CONCRETE I JUHÁSZ TAMÁS ------------------------------------------------------------------------------------------------------------------------------------------------------------

SECTION GEOMETRY: FREE SPAN:

l'  4000

WIDTH OF SUPPORTS:

a1  200

WIDTH OF SECTION:

b  300

HEIGHT OF SECTION:

h  500

CONCRETE COVER:

cc  20

ACCIDENTAL REBAR REPLACEMENT:

δ  10

EFFECTIVE DEPTHS:

a2  200

VALUES ARE GIVEN IN mm, N AND THEIR COMBINATIONS

TENSILE BARS:

COMPRESS BARS:

d  min( 0.9 h h  50)  450

d'  50

p g

MATERIAL CLASSES: fck  20

C20/25 3

Ecm  33 10 ϕt  2.55 S500B

Eceff 

fyk  500 ξ c0 

γc  1.5

fck fcd   13.333 γc

1.05 Ecm 1  ϕt

 9760.563

fctm  2.9

ε cu  0.35%

fyk γs  1.15 fyd   434.783 γs 560

700  fyd

 0.493

ξ'c0 

α  1 fbd  2.3

Es  200  10 560 700  fyd

 2.111

LOADS: PERMANENT LOADS: g k  25

γg.sup  1.35

g d  γg.sup g k  33.75

LIVE LOADS, IMPOSED LOADS: q k  27

γq  1.5

ψ2  0.3

q d  γq  q k  40.5

LOAD COMBINATION FOR ULS ANALYSIS: p Ed  g d  q d  74.25 LOAD COMBINATION FOR SLS ANALYSIS (QUASI PARMANENT COMBINATION): p qp

p qp  g k  ψ2  q k  33.1

p Ed

 44.579 %

DESIGN VALUES OF EFFECTS:  a1  a2  h  4200  2 

LENGTH OF STATIC MODEL: leff  l'  min LOAD FUNCTION:

L( x )  p Ed

REACTIONS:

leff 1  RA    L( x ) dx 2  0

SHEAR FUNCTION:

RA  1.559  10

 V( x )  RA   p Ed dx  0

2 10 V ( x)

2100

4200

SHEAR (ULS):

 2 10

VEd.max  V( 0 )  1.559  10 x

BENDING MOMENT FUNCTION:

 M ( x )   V( x ) dx  0

2100

 M ( x)  1 10

 2 10

4200

BENDING MOMENT (ULS):  leff  M Ed  M    1.637  108 2





BENDING MOMENT (SLS): p qp 7 M qp   M Ed  7.299  10 p Ed

BENDING DESIGN OPTIMAL BENDING MOMENT: ξc0   2 M 0  ξ c0 d  b  α fcd  1    3.011  108 2





ΔM  M Ed  M 0  1.374  10

COMPRESS REINFORCEMENT: ΔM

A's.req 

if ΔM  0

fyd ( d  d')

0

0 otherwise

TENSILE REINFORCEMENT:

σ's 

ξc0 d

fyd if

 ξ'c0

0

0 if A's.req = 0 ε cu Es 1.25 ξc0 d  d' otherwise 1.25 ξ c0 d





Given xc  797.33179382256389671   x c  M Ed = x c b  α fcd  d      2    102.66820617743614829 * ORIGIN  1

As.req 

x c  102.668 2

ξc0 d  b  α fcd  A's.req σ's fyd

if ΔM  0

 944.547

x c  b  α fcd 2

otherwise

fyd

APPLIED REINFORCEMENT: TENSILE: φ  16 COMPRESS: φ'  10

 As.req  n  round  4   0.5 0  5  φ2 π    n' 

STIRRUP: ζ  max( 20 φ)

φk  8

 A's.req   0.5 0 if A's.req  0  0  φ'2 π   

round  4 

0 otherwise 2

A's.prov  n'

φ'  π 4

0

φ π As.prov  n   1005.31 4

As.prov As.req

 106.433  %

ALLOWABLE NUMBER OF REBARS IN EACH ROWS n 1 





b  2 cc  φk  ζ  δ φζ

 7.056

APPLIED NUMBER OF REBARS: 1ST ROW: n a 

2ND ROW:

 

trunc n 1

if n 1  n

5

 n  n a

n f 

n otherwise

if n 1  n

0

0 otherwise

EFFECTIVE DEPTH: nf φ d prov  h  cc  φk    ( φ  ζ)  δ  454 2 n

 c  φ  φ'  δ if A'  c  k s.prov  0 2  

d'prov 

0 otherwise VERIFICATION:

x c.prov  x c.prov d prov

As.prov  A's.prov fyd  109.273 b  α fcd

x c.prov

 0.241 PLASTIC

d'prov



PLASTIC

EC2 REGULATIONS FOR REINFORCEMENT RATIO:

ρsl 

As.prov  A's.prov b  d prov

 0.738  %

fctm   ρsl.min  max 0.13% 0.26   0.151  % fyk   ρsl.max  4%

ρsl.max  ρsl  ρsl.min

BENDING CAPACITY: x c.prov   M Rd  x c.prov b  α fcd  d    A's.prov fyd d prov  d'prov  1.728  108 2



EFFICIENCY:

M Ed M Rd







 94.741 %

SHEAR DESIGN: VEd ( x ) 



V d prov



if x  d

Vdprov



if x  leff  d



V( x ) otherwise VEd.red  V( d )

VEd.red  1.225  10

SHEAR RESISTANCE WITHOUT STIRRUPS: As.prov

ρl  b  d prov



200



d prov

k  min 2 

  

 

νmin  0.035  k  fck

1    0.18  3 v Rdc  max  k   100  ρl fck νmin  0.195 SHEAR STRENGTH:  γc 

SHEAR RESISTANCE:

VRdc( x )  v Rdc b  d prov

VRdc( 0 )  26611.111

SINCE THE EFFECTIVE DEPTH COULD CHANGE ALONG THE BEAM DUE TO THE CHANGE OF LONGITUDINAL REINFORCEMENT, VRd.c IS NOT CONSTANT.

155925

2650.903

V Ed( x) V ( x) 0

V Rdc( x)

1000

2000

3000

4000

 155925 x





t  root V( x )  VRdc( x ) x  DUE TO THE LINEAR SYMMETRIC SHEAR FUNCTION VRd.c CUTS THE BEAM TO TWO PARTS. AT THE SUPPORTS VEd > VRd.c WHILE ALONG THE MIDSECTION VEd < VRd.c. THEREFORE AT THE SUPPORTS THE SHEAR REINFORCEMENT IS DESIGNED WHILE THE MIDSECTION IS REINFORCED WITH THE MINIMUM STIRRUP DENSITY ACCORDING TO THE EC2 REGULATIONS. tn  leff  2  t  STRUT RESISTANCE:

fck   1 VRd.max  b  0.9 d prov 0.6  1    fcd  4.511  105 2 250   VEd.max REINFORCEABLE SECTION  34.566 % VRd.max REQUIRED STRIRRUP DENSITY: 2

Asw  2 

φk  π 4

Asw fyd sreq  0.9d   144.493 VEd.red

 100.531

REEQUIRED REINFORCEMENT RATIO: Asw ρw   0.232  % sreq  b EC2 REGULATIONS: UPPER BOUND OF SHEAR REINFORCEMENT: fck   1  0.6  1   f 2 250  cd  ρw.max   0.585  % ( 1  cos( 90) )  fyd THEREFORE: Asw smin   57.332 ρw.max b

ρw.min 

0.08 fck fyk

 0.072  %

Asw smax1   468.321 ρw.min b

MAXIMUM ALLOWABLE DISTANCE BETWEEN STIRRUPS: Vagyis:

LOWER BOUND OF SHEAR REINFORCEMENT :



smax2  0.75 d prov  340.5



smax  min smax1 smax2  340.5 LOWER BOUND OF STIRRUP DENSITY ALONG THE BEAM (IT IS TO BE APPLIED WHERE VRdc>VEd ) :

 smax  sprov.tn  10 round   0.5 0  340  10  WHILE AT SUPPORTS sprov.A 

sprov.tn if sreq  sprov.tn

 140

  sreq   10 round   0.5 0  if smin  sreq  sprov.tn   10  "HIBA" otherwise STIRRUP DENSITY CAN BE REDUCED IN ACCORDANCE WITH THE SHEAR CURVE

LONGITUDINAL REBAR REDUCEMENET, CURTAILMENT: ANCHORAGE LENGTH (BASIC VALUE): φ fyd lb.d   756.144 4  fbd ANCHORAGE LENGTH (NETTO VALUE): SMOOTH END:

HOOKED END:

lb.net  lb.d

lb.eq.k  0.7 lb.net  529.301

MINIMAL ANCHORING:





lb.min  max 10 φ 100 0.3 lb.d  226.843 nyomatéki ábra eltolásának mértéke a többlet húzóerő figyelmbevételére: al 

1 2

 0.9 d prov  204.3

A húzott mennyiség felének elhagyásának legkorábbi helye: n half  round 

2

 0.4  3



φ π As.half  n half   603.186 4 d half 

Given

 454

d prov if n f = 0 nf   φ h  cc  φk  2  na  

n half n half



 ( φ  ζ)  δ if n half  n a



x c.half  x c.half  b  α fcd = As.half  fyd solve x c.half  65.5636727705695948 x c.half  65.564

 

M Rd.half  x c.half  b  α fcd  d half  NUMBER OF CUT BARS: n l  n  n half  2

M Rd( x )  M Rd

x c.half  2

  1.105  108 

  xal    M 2 ( x )   V( x ) dx    0  M Rd.half ( x )  M Rd.half

M Ed  1.637  10

1 10

 M ( x)  M Rd( x)

1 10

2 10

 M Rd.half ( x)  M 2( x)

 1 10

 2 10

SERVICEABILITY LIMIT STATE DESIGN: DEFLECTION: CROSS-SECTIONAL PROPERTIES (INTACT SECTION):

Es αE   20.491 Eceff

M qp  7.299  10

IDEAL SECTION AREA:







AiI  h  b  αE  1  As.prov  A's.prov  1.696  10 NEUTRAL AXIS POSITION: 1 x iI 





 h  b  αE  1  As.prov d prov  A's.prov d'prov 2



AiI

 273.569

INERTIA:

IxiI 

b  x iI 3





b  h  x iI



 αE  1  As.prov d  x iI 











 A's.prov x iI  d'prov

  2

IxiI  3.818  10

CRACKING MOMENT: M cr 

fctm IxiI h  x iI

 4.89  10

M cr M qp

 67.002 %

CRACKED SECTION

CROSS-SECTIONAL PROPERTIES (CRACKED SECTION):

Given 1 x iII  x iII =





 x  b  αE As.prov d prov  αE  1  A's.prov d'prov 2 iII



ORIGIN  1

IxiII 



x iII b  αE As.prov  αE  1  A's.prov



 327.62885369493123555     190.29939705316096593 *

x iII  190.299 2

b   x iII

 2 2  2  α  A E s.prov  d  x iII2   αE  1  A's.prov  x iII2  d'prov 3    

IxiII  2.078  10

STRESS CONTROL: M qp M qp M qp σc   x iII  6.682 σs  αE d prov  x iII   189.741 σ's  αE   x  d'prov  136.926 2 2 IxiII IxiII  IxiII  iII2  REINFORCEMENT REMAINS ELASTIC

DEFORMATIONS ACCORDING TO SS1 AND SS2: INTACT SECTION (SS1): CURVATURE:

κI 

M qp Eceff  IxiI

 1.958  10

6

DEFLECTION:

eI 

p qp leff

  3.599 384 Eceff  IxiI

CRACKED SECTION (SS2): CURVATURE:

κII 

M qp Eceff  IxiII

 3.598  10

6

DEFLECTION:

eII 

  6.611 384 Eceff  IxiII

COMBINATION FACTOR FOR LONG TERM LOADING: 2

 Mcr  ζ'  1  0.5    77.553 % M qp   CALCULATED DEFORMATIONS: eEC 

eI if M cr  M qp

 5.935

ζ' eII  ( 1  ζ')eI otherwise κEC 

κI if M cr  M qp

 3.23  10

6

ζ' κII  ( 1  ζ')  κI otherwise VERIFICATION: emax 

leff 200

 21

emax  eEC OK CRACK WIDTH CONTROL: REBAR STARIN:

ε s 

CONCRETE STRAIN BESIDE THE REBARS :

ε c 

σs Es

 0.095  %

fctm Eceff

 0.03 %

TENSION STIFFENING: BETWEEN "A" AND "B" THEORETICAL CRACKS THE TENSILED CONCRETE STIFFENS THE REBARS EFFECTIVE CONCRETE AREA FOR TENSION STIFFENING: x iII h  h c.eff  min2.5 h  d prov h    3 2 





ε sc 

fctm Aceff Es As.prov

Aceff  b  h c.eff  34500

 0.05 %

DURABILITY OF LOADS: LONGTERM LOADING

k t  0.4

STRAIN DIFFERENCE BETWEEN THE CONCRETE AND THE REBARS: Δε  ε s  k t ε sc  k t ε c  0.063  % MAXIMUM RELATIVE DISTANCE BETWEEN TWO CRACKS: φ Aceff sr.max  3.4 cc  0.425  0.8 0.5  161.344 As.prov

COMPUTED CRACK WIDTH: wk  sr.max Δε  0.102 wk.max  0.3

wk.max  wk

CHECKED

Beton feszültség ellenőrzése tartós terhelésre: CHARACTERISTIC LOAD COMBINATION:

p car  g k  q k  52 p car leff

az ehhez tartozó nyomaték:

M car 

COMPRESSION STRESS IN CONCRETE:

M car σc.car  x  10.498 IxiII iII2

 1.147  10

σc.car.max  0.6 fck  12

ACCEPTABLE