The Hashemite University Department of Civil Engineering ( ) Dr. Hazim Dwairi 1

Size: px

Start display at page:

Download "The Hashemite University Department of Civil Engineering ( ) Dr. Hazim Dwairi 1"

Camilla Black
5 years ago
Views:

1 Department of Civil Engineering Letre 8 Slender Colmns Definition of Slender Colmn When the eentri loads P are applied, the olmn deflets laterally by amont δ,, however the internal moment at midheight: M ( ) P e + δ The defletion δ inreases the moments for whih the olmn mst be designed. 1

2 Definition of Slender Colmn Failre ors when the load- moment rve O-B for the point of maximm moment intersets the interation diagram of the ross setion. Definition of Slender Colmn A slender olmn is defined as the olmn that has a signifiant redtion in its axial load apaity de to moments reslting from lateral defletions of the olmn. In the derivation of the ACI ode. a signifiant redtion was arbitrarily taken anything greater than 5%. Less than 10 % of olmns in braed or non- sway frames and less than half of olmns in nbraed or sway frames wold be lassified as slender following ACI Code Proedre.

3 Bkling The differential eqation for olmn in state of netral eqilibrim is: Leonhard Eler soltion: n: nmber of half-sine waves in length of olmn " EIy Py P n π EI l Bkling The lowest vale for P will or with n 1.0 This gives the Eler Bkling Load: Effetive length onept P π EI 1 ( l) n π EI ( kl) Effetive Length Fator k P π EI l P π EI l 3

4 Effetive Length Fator Effetive Length Fator 4

5 Effetive Length Fator Effetive Length Fator 5

6 Effetive Length Fator ψ EI EI / l b / l of olmns of beams Ψ A and Ψ B are top and bottom fators of olmns. For a hinged end Ψ is infinite or 10 and for a fixed end Ψ is zero or 1. Assmptions for nomographs: E 4700 f 1. Symmetrial retanglar frames. Eqal load applied at top of olmns 3. Unloaded beams. 4. All olmns bkle at the same moment I I b 0.35I 0.70I g g ' Nomographs for k 6

Nomographs for k As a reslt of these very idealized assmptions, nomographs tend to nderestimate the vales of the effetive length fator k for elasti frames of pratial dimensions p to 15%.

7 Nomographs for k As a reslt of these very idealized assmptions, nomographs tend to nderestimate the vales of the effetive length fator k for elasti frames of pratial dimensions p to 15%. This leads to an nderestimate of the magnified moment, M. The lowest pratial vales for k in a sway frame is abot 1. de to frition in the hinges. When smaller vales obtained from nomographs, it is good pratie to se k 1.. 7

8 Slenderness Effet For olmns in nonsway frames, ACI Se allows the slenderness effets to be negleted if: k : effetive length fator l : olmn nspported length r : radis of gyration r 0.3h (Retanglar) r 0.5D kl r (Cirlar) M < 34 1 M 1 Slenderness Effet For olmns in nbraed frames, ACI Se allows the slenderness effets to be negleted if: If kl r >100 kl r < seond-order order analysis. design shall be based on a 8

9 Moment Magnifier Design Proedre 1) Length of Colmns.. The nspported length l is the lear height between slabs or beams apable of giving lateral spport to the olmn. ) Effetive Length Fator.. an be estimated from the nomographs. 3) Braed or Unbraed Frames.. Inspet braing elements, sh as walls, whether stiffer than olmns (braed) or not (nbraed). 4) Consideration of Slenderness Effets.. Chek slenderness ratio: kl? r Moment Magnifier Design Proedre 5) Minimm Moment. ACI Eqn. (10-14) 14) states that for olmns in braed frames, minimm moment M : M P ( ),min h 6) Moment Magnifier. ACI Se states that olmns on nonsway frames shall be designed for P and M : M C m δ M ns M M ; δ 1 ns Cm P P

10 Moment Magnifier Design Proedre Where M is the larger end moment M 1 /M is positive for single rvatre and negative for doble rvatre. Bkling load, P is: and: EI 0.EI g + EsI se 0.40EI EI 1+ β 1+ β d d g β d max. fatored axial dead load in the olmn total fatored axial load in the olmn The figre shows a typial frame in an indstrial bilding. The frames are spaed 6.0 m apart. The olmns rest on a 1. m-sqare footings. The soil bearing apaity is 190 kn/m. Design olmns C- D and D-E.. Use f 0 MPa and f y 40 MPa for beams and olmns. Use lower ombination and strength-redtion redtion fators from ACI setions 9. and 9.3 (Example 1- : Magregor and Wight 4 th edition in SI nits) 10

11 5mm mm 400mm 7300mm mm mm 100mm 7600mm (kn.m) 1) Callate the olmn loads from frame analysis a first-order-elasti elasti analysis of the frame gave the following fores and moments Colmn CD Colmn DE Servie load, P Servie moment at top of olmns Servie moments at bottom of olmns Dead 350 kn Live 105 kn Dead -80 kn.m Live -19 kn.m Dead -8 kn.m Live -11 kn.m Dead 0 kn Live 60 kn Dead 57.5 kn.m Live 15.0 kn.m Dead -43 kn.m Live -11 kn.m 11

12 ) Determine the fatored loads a) Colmn CD P 1. x x kn Moment at top 1. x x kn.m Moment at bottom 1. x x kn.m ACI se. 10.0, M is always +ve, and M 1 is +ve if the olmn bent in single rvatre. Sine CD is bent in doble rvatre, M +16.4kN.m and M kn.m b) Colmn DE P 360 kn, M +93 kn.m, M kn.m 3) Make a preliminary seletion of the olmn size (assme ρ t 0.015) P Ag ( trial ) ' 0.40 f + f ρ Bease of slenderness and large moments se 350mm x 350mm olmns throghot 0.40 ( ) ,894mm ( ) y t 3 1

13 4) Are the olmns slender? l a) Colmn CD: mm From Table1 -, k r mm kl r M 34 1 M (ACI ) (ACI10.11.) 39.5>38.9 Colmn CD jst slender ) Are the olmns slender? l b) Colmn DE: mm From Table1 -, k r mm kl r M 34 1 M (ACI ) (ACI10.11.) 54.8 > 5.1 Colmn CD is slender

14 5) Chek whether the moments are less than the minimm ACI Se reqires that braed slender olmns be designed for minimm eentriity of ( h).. For 350-mm olmn, this is 5.5 mm. Colmn CD : P e Colmn DE : P e min min kN. m 9.kN. m Sine atal moments exeed these vales, the olmns shall be designed for atal moments 3 6) Compte EI Use a onservative estimate by 0.40EI EI 1+ β E ,019 MPa I g mm E I 10, N. mm g d g 4 14

15 6) Compte EI a) Colmn CD βd , EI b) Colmn DE 1. 0 βd , EI N. mm N. mm ψ 7) Compte the effetive-length fators We will se the nomograph this time jst for demonstration, one shold se the same method throghot all allations. 75 mm E I b E I b b 0.47 / l / l 6 E ψ E 9 E b b / 7300 / E ( / ψ D 9 E / 9100 / 7300) 400mm 5mm 385mm I g x 10 9 mm 4 I b 0.35 x Ig 5.7 x 10 9 mm 4 I 0.70 x / x 10 6 mm 4 15

16 Colmn CD is restrained at C by the rotational resistane of the soil nder the footing, ths: 4E I / l ψ I f ks Where I f is the moment of inertia of the ontat area between the footing and the soil and k s is the sbgrade reation I f mm , / 7300 ψ C Ψ D Ψ C Ψ D kde 0.65< 0.86 USE 0.86 k 0.710< 0.77 CD USE 0.77 Ψ E 16

17 Sbgrade Modls Allowable bearing apaity (kn/m ) K s (kn/m 3 ) ) Compte magnified moments a) Colmn CD C m π EI P δ ns USE π ( kl ) ( ) < /( ) δ ns 1.0 (i.e.setion of maximm moment remains at the end of the olmn) 9 M 1.0 x kn.m kN 17

18 8) Compte magnified moments a) Colmn DE C m ns π EI P δ π ( kl ) ( ) /( ) This olmn is affeted by slenderness 9 M 1.5 x kn.m 1809kN 9) Selet the reinforement a) Design olmn CD for P 588 kn and M 16.4 kn.m USE 350mm x 350mm with 4φ5 b) Design olmn DE for P 360 kn and M kn.m USE 350mm x 350mm with 4φ5 18

Case Study in Reinforced Concrete adapted from Simplified Design of Concrete Structures, James Ambrose, 7 th ed.

Case Study in Reinforced Concrete adapted from Simplified Design of Concrete Structures, James Ambrose, 7 th ed. ARCH 631 Note Set 11 F013abn Case Stdy Refored Conrete adapted from Simplified Design of Conrete Strtres, James Ambrose, 7 th ed. Bildg desription The bildg is a three-story offie bildg tended for spelative