Analysis of Combined Axial and Bending Loads on Columns Beam Column

Transcription

1 CE 537, Sprig 2006 Aalysis of Combied Axial ad Bedig 1 / 6 Loads o Colms Axial loads ad bedig momets both case ormal stresses o the colm crosssectio. We aalyze the ormal stresses from these combied loads i the same way that we aalyze the ormal stresses de to bedig oly i a beam, with two exceptios. 1. The sm of the ormal stresses is ow eqal to the axial load ( ), istead of eqal to zero, ad 2. We sm momets abot the cetroid of the colm cross-sectio, istead of the cetroid of the compressive stress o the cocrete. Beam Colm C s C c T d a/2 C T α 2 α 1 α 3 Σ F 0, -C + T 0 Σ, T x (d a/2) Σ F, -C s -C c + T Σ, C s α 1 + C c α 2 + T α 3 We calclate the loads o a colm at ltimate stregth jst as we do for a beam: 1. Assme a strai profile for the colm cross-sectio. Ultimate stregth of a colm occrs whe the compressive strai i the cocrete reaches 0.003, jst as for a beam 2. Calclate the stresses i the cocrete ad steel. 3. Calclate the stress resltats. 4. The sm of the stress resltats is eqal to the axial capacity of the colm ( ) 5. The sm of the momets cased by each stress resltat abot the cetroid of the colm is eqal to the momet capacity of the colm ( ).

2 CE 537, Sprig 2006 Aalysis of Combied Axial ad Bedig 2 / 6 Loads o Colms Whereas a beam has oly oe momet capacity, a colm has differet axial ad momet capacities for each ratio of /. This ratio is called the load eccetricity for the reaso demostrated i the figre below. e e

3 CE 537, Sprig 2006 Aalysis of Combied Axial ad Bedig 3 / 6 Loads o Colms Colm Iteractio Diagram. The plot of axial capacity ( ) vs. momet capacity ( ) is called a iteractio diagram. Each poit o the iteractio diagram is associated with a iqe strai profile for the colm cross-sectio. A iteractio diagram has three key poits, as show i the figre below. Each poit ad each regio betwee the poits is discssed below. re Compressio ε s Compressio-Cotrolled Failre Balaced Failre φ _max, ε s ε y φ, φ 2 Tesio-Cotrolled Failre 3 re Bedig ε s >> ε y oit 1: The colm is i pre compressio. The maximm axial capacity of the colm occrs i this state. oit 1 to oit 2 (compressio-cotrolled failre): The cocrete crshes before the tesio steel (layer frthest from the compressio face) yields. omet capacity decreases becase the steel does ot reach its fll stregth.

4 CE 537, Sprig 2006 Aalysis of Combied Axial ad Bedig 4 / 6 Loads o Colms oit 2 (Balaced failre): A so-called balaced failre occrs whe the cocrete crshes (ε c ) at the same the tesio steel yields (ε s 0.002). oit 2 to oit 3 (tesio-cotrolled failre): As compressio force is applied to the sectio, the compressio area ca icrease beyod the area balaced by the tesio steel. Larger compressio force leads to larger momet. oit 3: The colm behaves as a beam. The compressio area is limited by the area balaced by the tesio steel. Stregth Redctio Factor. The redced omial axial capacity (φ ) ad the redced omial momet capacity (φ ) are obtaied by calclatig the stregth redctio factor (φ) based o the strai i the tesio steel (the layer frthest from the compressio face). ax. Axial Capacity. ACI limits the axial force i a colm (sectio , pg 123) to ' φ 0.85φ [0.85 f ( A A ) + f A ] (flat portio at top of φ, φ crve), max c g s y s accots for accidetal eccetricity Varios methods exist for checkig the combied ormal stresses de to axial ad bedig i a colm. Two methods are discssed here: 1) Sigle oit sefl whe checkig colm for oly oe set of loads 2) lti-poit (fll iteractio diagram) sefl whe checkig colm for mltiple sets of loads Capacity Check for Oe Set of Loads φ Every poit o the iteractio diagram has a iqe ratio of φ φ φ p φ e. Therefore, if e ad φ > ad φ >, the the colm is adeqate. φ, max φ φ e φ, φ e 1, φ

5 CE 537, Sprig 2006 Aalysis of Combied Axial ad Bedig 5 / 6 Loads o Colms A example colm desig sig this procedre is provided separately. Capacity Check for ltiple Load Sets The capacity of a colm with several sets of loads (e.g. from differet load combiatios) ca most easily be checked by geeratig a colm iteractio diagram. φ φ, max 1 ε s , LC II LC II 3 φ, φ, LC III LC III, LC I LC I 4 ε s ε s φ A poit o the colm iteractio diagram ca be calclated by assmig a strai profile i the colm ad calclatig the resltig φ, φ. The strai profiles are kow for oit 1 (ε s ) ad oit 4 (ε s ε y ). oit 6 ca typically be calclated sig ε s 5 ε y Ideally, oit 2 shold be jst slightly greater tha φ _max, ad oit 3 ad ot 5 midway betwee adjacet poits.

6 CE 537, Sprig 2006 Aalysis of Combied Axial ad Bedig 6 / 6 Loads o Colms Example: t. 5 Let ε s f'c 3 ksi, 5 #9 bars i each face tesio +'ve d'2.5" ε s ' f'c C s y t " a d13.5" h16" C c b16" ε s T ( ), y 13.5 t y t " ε s' 0.003, ε s' " 2.5" " a b 1 y t 0.85 (5.0625") 4.303" C c f'c a b -0.85(3 ksi ) 4.303"(16") -176 k f s' 29,000ksi ( ) ksi, > -60 ksi, OK C s A s ' [f s' (-.85f'c)] (5)1.00i 2 [-44.1 ksi +.85(3 ksi )] -208 k T A s f s 5.00i 2 (60,000 ksi ) 300 k ΣF -208 k k +300 k -84 k Σ sice ε s > ε y k 16" k 16".4.303" k 16" ( 2.5") + ( 176 )( ) ( 13.5") 319 i φ 0.90 sice ε s φ φ 0.90( 84 ) 76 k 0.90( 319 k ft k ) 287 k ft ft k ft