109 Aunque los costos totales son mayores a los beneficios, existen otros que aunque no se pueden
4.5 Determinar el tiempo de recuperación de la inversión
This chap ter de scribes in de tail the vari ous as pects of the con crete de sign pro ce dure that is used by the pro gram when the user se lects the New Zea land code, NZS 3101- 95 (NZS 1995). Vari ous no ta tions used in this chap ter are listed in Table X-1.
The de sign is based on user- specified load ing com bi na tions. The pro gram pro vides a set of de fault load com bi na tions that should sat isfy re quire ments for the de sign of most build ing type struc tures.
The pro gram pro vides op tions to de sign or check all types of mo ment re sist ing frames as re quired for regu lar and seis mic de sign. For regu lar de sign, the frame should be iden ti fied as Or di nary. For Seis mic de sign, the frame must be iden ti fied as Duc tile, Lim ited, or Elas tic to rep re sent Duc tile mo ment re sist ing frames, frames with Lim ited duc til ity, and Elas ti cally re spond ing frames, re spec tively. The de tails of the de sign cri te ria used for the dif fer ent fram ing sys tems are de scribed in the fol -low ing sec tions.
Eng lish as well as SI and MKS met ric units can be used for in put. The code is based on Newton- Millimeter- Second units. For sim plic ity, all equa tions and de scrip tions pre sented in this chap ter cor re spond to Newton- Millimeter- Second units un less oth er wise noted.
Acv Area of con crete used to de ter mine shear stress, sq- mm Ag Gross area of con crete, sq- mm
As Area of ten sion re in force ment, sq- mm As¢ Area of compression re in force ment, sq- mm
As required( ) Area of steel re quired for ten sion re in force ment, sq- mm Ast To tal area of col umn lon gi tu di nal re in force ment, sq- mm Av Area of shear re in force ment, sq- mm
a Depth of com pres sion block, mm
ab Depth of com pres sion block at bal anced condition, mm amax Maxi mum depth of com pres sion block, mm
b Width of mem ber, mm
bf Ef fec tive width of flange (T Beam sec tion), mm bw Width of web (T Beam sec tion), mm
Cm Co ef fi cient, de pend ent upon col umn cur va ture, used to cal cu late mo ment mag ni fi ca tion fac tor
c Depth to neu tral axis, mm
cb Depth to neu tral axis at bal anced con di tions, mm
d Dis tance from com pres sion face to ten sion re in force ment, mm d¢ Con crete cover to cen ter of re in forc ing, mm
ds Thick ness of slab (T Beam sec tion), mm Ec Modu lus of elas tic ity of con crete, MPa
Es Modu lus of elas tic ity of re in force ment, as sumed as 200,000 MPa (NZS 3.8.3.3)
fc¢ Spec ified com pres sive strength of con crete, MPa fy Spec ified yield strength of flex ural re in force ment, MPa fyt Spec ified yield strength of shear re in force ment, MPa h Di men sion of beam or col umn, mm
Ig Mo ment of in er tia of gross con crete sec tion about cen troi dal axis, ne glect ing re in force ment, mm4
k Ef fec tive length factor
L Clear un sup ported length, mm
Table X-1
List of Symbols Used in the New Zealand code
M1 Smaller fac tored end mo ment in a col umn, N- mm M2 Larger fac tored end mo ment in a col umn, N- mm Mc Factored mo ment to be used in design, N- mm
Mns Non sway com po nent of fac tored end mo ment, N- mm Ms Sway com po nent of fac tored end mo ment, N- mm M* Fac tored mo ment at section, N- mm
Mx* Fac tored mo ment at sec tion about X-axis, N- mm M*y Fac tored mo ment at sec tion about Y-axis, N- mm Nb Ax ial load ca pac ity at bal anced strain con di tions, N Nc Criti cal buck ling strength of col umn, N
Nmax Maxi mum ax ial load strength al lowed, N N0 Ax ial load ca pac ity at zero ec cen tric ity, N N* Fac tored ax ial load at sec tion, N
vb Ba sic shear stress re sisted by con crete, MPa vc Shear stress re sisted by con crete, MPa Vc Shear force re sisted by con crete, N VD+L Shear force from span load ing, N
Vp Shear force com puted from prob able mo ment ca pac ity, N V* Fac tored shear force at a sec tion, N
Vs Shear force at a sec tion re sisted by steel, N a Re in forc ing steel over strength fac tor
a1 Av er age stress fac tor in equiva lent stress block
b1 Fac tor for ob tain ing depth of com pres sion block in con crete bd Ab so lute value of the ra tio of the maxi mum fac tored ax ial
dead load mo ment to the maxi mum fac tored to tal load moment db Mo ment mag ni fi ca tion fac tor for nonsway moments
ds Mo ment mag ni fi ca tion fac tor for sway mo ments ec Strain in con crete
es Strain in re in forc ing steel
jb Strength re duc tion fac tor for bending
Table X-1
List of Symbols Used in the New Zealand code (continued)
Design Load Combinations
The de sign load com bi na tions are the vari ous com bi na tions of the pre scribed load cases for which the struc ture is to be checked. For this code, if a struc ture is sub -jected to dead load (DL), live load (LL), wind (WL), and earth quake (EL) loads, and con sid er ing that wind and earth quake forces are re vers ible, the fol low ing load com bi na tions should to be con sid ered for de sign of con crete frames (NZS 4203- 92 2.4.3):
1.4 DL
1.2 DL + 1.6 LL (NZS 4203- 92 2.4.3.3)
1.2 DL ± 1.0 WL 0.9 DL ± 1.0 WL
1.2 DL + 0.4 LL ± 1.0 WL (NZS 4203- 92 2.4.3.3)
1.0 DL ± 1.0 EL
1.0 DL + 0.4 LL ± 1.0 EL (NZS 4203- 92 2.4.3.3)
These are also the de fault de sign load com bi na tions in the pro gram when ever the NZS 3101 95 code is used. The user should use other ap pro pri ate load ing com bi na tions if roof live load is sepa rately treated, other types of loads are pres ent, or pat -tern live loads are to be con sid ered.
Live load re duc tion fac tors can be ap plied to the mem ber forces of the live load case on an element- by- element ba sis to re duce the con tri bu tion of the live load to the fac tored load ing.
Strength Reduction Factors
The de fault strength re duc tion fac tor, j, is taken as
jb = 0.85 for bend ing and (NZS 3.4.2.2)
jc = 0.85 for com pres sion and (NZS 3.4.2.2)
js = 0.75 for shear. (NZS 3.4.2.2)
The user can, how ever, over write them.
Column Design
The user may de fine the ge ome try of the re in forc ing bar con figu ra tion of each con -crete col umn sec tion. If the area of re in forc ing is pro vided by the user, the pro gram checks the col umn ca pac ity. How ever, if the area of re in forc ing is not pro vided by the user, the pro gram cal cu lates the amount of re in forc ing re quired for the col umn.
The de sign pro ce dure for the re in forced con crete col umns of the struc ture in volves the fol low ing steps:
• Gen er ate ax ial force/bi axial mo ment in ter ac tion sur faces for all of the dif fer ent con crete sec tion types of the model. A typi cal bi ax ial in ter ac tion sur face is shown in Fig ure II-1. When the steel is un de fined, the pro gram gen er ates the in ter ac tion sur faces for the range of al low able re in force ment ra tios ¾ 0.008 to 0.08 for Or di nary mo ment re sist ing frames (NZS 8.4.6.1) and 0.008 to 18 fy for Seis mic (Duc tile, Lim ited, and Elas tic) mo ment re sist ing frames (NZS 8.5.4.2).
• Cal cu late the ca pac ity ra tio or the re quired re in forc ing area for the fac tored ax -ial force and bi ax -ial (or uni ax -ial) bend ing mo ments ob tained from each load ing com bi na tion at each sta tion of the col umn. The tar get ca pac ity ra tio is taken as one when cal cu lat ing the re quired re in forc ing area.
• De sign the col umn shear re in force ment.
The fol low ing three sub sec tions de scribe in de tail the al go rithms as so ci ated with these steps.
Generation of Biaxial Interaction Surfaces
The col umn ca pac ity in ter ac tion vol ume is nu meri cally de scribed by a se ries of dis -crete points that are gen er ated on the three- dimensional in ter ac tion fail ure sur face.
In ad di tion to ax ial com pres sion and bi ax ial bend ing, the for mu la tion al lows for ax -ial ten sion and bi ax -ial bend ing con sid era tions. A typi cal in ter ac tion dia gram is shown in Fig ure II-1.
The co or di nates of these points of the in ter ac tion dia gram are de ter mined by ro tat -ing a plane of lin ear strain in three di men sions on the sec tion of the col umn. See Fig ure II-2. The lin ear strain dia gram lim its the maxi mum con crete strain, ec, at the ex trem ity of the sec tion, to 0.003 (NZS 8.3.1.3).
The for mu la tion is based con sis tently upon the gen eral prin ci ples of ul ti mate strength de sign (NZS 8.3), and al lows for any dou bly sym met ric rec tan gu lar, square, or cir cu lar col umn sec tion.
The stress in the steel is given by the prod uct of the steel strain, es, and the steel modu lus of elas tic ity, Es, and is lim ited to the yield stress of the steel, fy (NZS 8.3.1.4). The area as so ci ated with each re in forc ing bar is as sumed to be placed at the ac tual lo ca tion of the cen ter of the bar and the al go rithm does not as sume any fur ther sim pli fi ca tions in the man ner in which the area of steel is dis trib uted over the cross-sec tion of the col umn (such as an equiva lent steel tube or cyl in der). See Figure X-1.
The con crete com pres sion stress block is as sumed to be rec tan gu lar, with a stress value of a1fc¢ (NZS 8.3.1.7) and a depth of the stress block of b1c, where
a1 =0.85 0.004- (fc¢ -55), (NZS 8.3.1.7)
b1 =0.85 0.008- (fc¢ -30), (NZS 8.3.1.7)
0.75£a1 £0.85, and (NZS 8.3.1.7)
0.65£b1 £0.85, and (NZS 8.3.1.7)
In de sign ing the col umn longitudinal re in force ment, the fol low ing lim its are im -posed on the steel ten sile strength and the con crete com pres sive strength:
Figure X-1
Idealization of Stress and Strain Distribution in a Column Section
fy £ 500 MPA (NZS 3.8.2.1) fc¢ £ 100 MPA (Or di nary and Elas tic) (NZS 3.8.1.1) fc¢ £ 70 MPa (Duc tile and Limited) (NZS 3.8.4.4) The in ter ac tion al go rithm pro vides cor rec tion to ac count for the con crete area that is dis placed by the re in force ment in the com pres sion zone.
The ef fects of the strength re duc tion fac tor, j, are in cluded in the gen era tion of the in ter ac tion sur faces. The maxi mum com pres sive ax ial load is lim ited to Nmax, where the maxi mum fac tored ax ial load re sis tance is given by:
Nmax= 0.85jc[a1fc¢(Ag -Ast) +f Ay st] (Or di nary, Elas tic), (NZS 8.4.1.5) Nmax= 0.70jc[a1fc¢(Ag -Ast) + f Ay st] (Duc tile, Limited). (NZS 8.5.1.4)
Check Column Capacity
The col umn ca pac ity is checked for each load ing com bi na tion at each check sta tion of each col umn. In check ing a par ticu lar col umn for a par ticu lar load ing com bi na -tion at a par ticu lar sta -tion, the pro gram uses the fol low ing steps:
• De ter mine the fac tored mo ments and forces from the analy sis load cases and the speci fied load com bi na tion fac tors to give N*, Mx*, and M*y.
• De ter mine the mo ment mag ni fi ca tion fac tors for sta bil ity.
• De ter mine the dy namic mo ment mag ni fi ca tion ef fect.
• Ap ply the mo ment mag ni fi ca tion fac tors to the fac tored loads ob tained in the first step. De ter mine whether the point, de fined by the re sult ing ax ial load and bi ax ial mo ment set, lies within the in ter ac tion vol ume.
The fol low ing three sec tions de scribe in de tail the al go rithms as so ci ated with these steps.
Determine Factored Moments and Forces
The fac tored loads for a par ticu lar load com bi na tion are ob tained by ap ply ing the cor re spond ing load fac tors to all of the load con di tions, giv ing N*, Mx*, and M*y. The fac tored mo ments are fur ther in creased, if re quired, to ob tain mini mum ec cen -trici ties of (15 + 0.03 h mm, where h is the di men sion of the col umn in the cor re -) spond ing di rec tion (NZS 8.4.11.5). The com puted mo ments are fur ther am pli fied
by us ing “Mo ment Mag ni fi ca tion Fac tors” to al low for “Lat eral Drift Ef fect” and
“Mem ber Sta bil ity Ef fect.”
Determine Moment Magnification Factors
The mo ment mag ni fi ca tion fac tors are ap plied in two stages. First the mo ments are sepa rated into their “sway” and “non sway” com po nents. The non sway com po -nents are am pli fied for lat eral drift ef fect. Al though this am pli fi ca tion may be avoided for “braced” frames ac cord ing to the code, the pro gram treats all frames uni formly to am plify non sway com po nents of mo ments. These am pli fied mo -ments are fur ther am pli fied for in di vid ual mem ber sta bil ity ef fect.
Lateral Drift Effect
For all frames, the mo ment mag ni fi ca tion fac tor for lat eral drift ef fect is ap plied only to the “sway” mo ment in the pro gram.
M =Mns + dsMs
The mo ment mag ni fi ca tion fac tors for mo ments caus ing sidesway in the ma jor and mi nor di rec tions, dsxanddsy, can be dif fer ent. The mo ment mag ni fi ca tion fac tors, dsxanddsy, can be taken as 1.0 if a P-D analy sis is car ried out. The pro gram as sumes that the pro gram analy sis mod els P-D ef fects; there fore, dsxanddsy are taken as 1.0.
It is sug gested that the P-D analy sis be performed at the fac tored load level (White and Hajjar 1991). The nec es sary fac tors for a P-D analy sis for the NZS 3101- 95 code should be (1.0 DL + 0.4 LL)/jc with the load ing stan dard NZS 4203, where jc is the strength re duc tion fac tor for com pres sion and is equal to 0.85.
The user is re minded of the spe cial analy sis re quire ments, es pe cially those re lated to the value of EI used in analy sis (NZS 8.4.11.5). In the pro gram, the EI val ues are com puted based on gross cross- section ar eas. The user has the op tion to re duce the EI val ues for analy sis pur poses us ing a scale fac tor on a section- by- section ba sis. If the pro gram as sump tions are not sat is fac tory for a par ticu lar mem ber, the user can ex plic itly spec ify val ues of dsx and dsy.
Member Stability Effects
All com pres sion mem bers are de signed us ing the fac tored ax ial load, N*, ob tained from the analy sis and a mag ni fied fac tored mo ment, Mc. The mag ni fied mo ment is com puted as,
Mc = dbM2 , (NZS 8.4.11.5)