Normally the hinge prop er ties for each of the six de grees of free dom are un cou pled from each other. How ever, you have the op tion to specify cou pled ax ialforce/bi -axial-mo ment be hav ior. This is called the P-M2-M3 or PMM hinge. See also the Fi ber P-M2-M3 hinge be low.
Ten sion is Always Pos i tive!
It is im por tant to note that SAP2000 uses the sign con ven tion where ten sion is al -ways pos i tive and com pres sion is al -ways neg a tive, re gard less of the ma te rial be ing used. This means that for some ma te ri als (e.g., con crete) the in ter ac tion sur face may ap pear to be up side down.
136 Hinge Properties
In ter ac tion (Yield) Sur face
For the PMM hinge, you spec ify an in ter ac tion (yield) sur face in three-di men sional PM2M3 space that rep re sents where yielding first oc curs for dif fer ent com bi na -tions of ax ial force P, mi nor mo ment M2, and ma jor mo ment M3.
The sur face is spec i fied as a set of PM2M3 curves, where P is the ax ial force (ten sion is pos i tive), and M2 and M3 are the mo ments. For a given curve, these mo -ments may have a fixed ra tio, but this is not nec es sary. The fol low ing rules ap ply:
• All curves must have the same num ber of points.
• For each curve, the points are or dered from most neg a tive (com pres sive) value of P to the most pos i tive (ten sile).
• The three val ues P, M2 and M3 for the first point of all curves must be iden ti cal, and the same is true for the last point of all curves
• When the M2-M3 plane is viewed from above (look ing to ward com pres sion), the curves should be de fined in a coun ter-clock wise di rec tion
• The sur face must be con vex. This means that the plane tan gent to the sur face at any point must be wholly out side the sur face. If you de fine a sur face that is not con vex, the pro gram will au to mat i cally in crease the ra dius of any points which are “pushed in” so that their tan gent planes are out side the sur face. A warn ing will be is sued dur ing anal y sis that this has been done.
You can ex plic itly de fine the in ter ac tion sur face, or let the pro gram cal cu late it us -ing one of the fol low -ing formulas:
• Steel, AISC-LRFD Equa tions H1-1a and H1-1b with phi = 1
• Steel, FEMA-356 Equa tion 5-4
• Con crete, ACI 318-02 with phi = 1
You may look at the hinge prop er ties for the gen er ated hinge to see the spe cific sur -face that was cal cu lated by the pro gram.
Mo ment-Ro ta tion Curves
For PMM hinges you spec ify one or more mo ment/plas ticro ta tion curves cor re spond ing to dif fer ent val ues of P and mo ment an gle q. The mo ment an gle is mea -sured in the M2-M3 plane, where 0° is the pos i tive M2 axis, and 90° is the pos i tive M3 axis.
Hinge Properties 137
You may spec ify one or more ax ial loads P and one or more mo ment an gles q. For each pair (P,q), the mo mentro ta tion curve should rep re sent the re sults of the fol -low ing ex per i ment:
• Ap ply the fixed ax ial load P.
• In crease the mo ments M2 and M3 in a fixed ra tio (cos q, sin q) cor re spond ing to the mo ment an gle q.
• Mea sure the plas tic ro ta tions Rp2 and Rp3 that oc cur af ter yield.
• Cal cu late the re sul tant mo ment M = M2*cos q + M3*sin q, and the pro jected plas tic ro ta tion Rp = Rp2*cos q + Rp3*sin q at each mea sure ment increment
• Plot M vs. Rp, and sup ply this data to SAP2000
Note that the mea sured di rec tion of plas tic strain may not be the same as the di rec -tion of mo ment, but the pro jected value is taken along the di rec -tion of the moment.
In ad di tion, there may be mea sured ax ial plas tic strain that is not part of the pro jec -tion. How ever, dur ing anal y sis the pro gram will re cal cu late the to tal plas tic strain based on the di rec tion of the nor mal to the in ter ac tion (yield) sur face.
Dur ing anal y sis, once the hinge yields for the first time, i.e., once the val ues of P, M2 and M3 first reach the in ter ac tion sur face, a net mo mentro ta tion curve is in ter -po lated to the yield -point from the given curves. This curve is used for the rest of the anal y sis for that hinge.
If the val ues of P, M2, and M3 change from the val ues used to in ter po late the curve, the curve is ad justed to pro vide an energy equiv a lent mo ment-ro ta tion curve. This means that the area un der the mo mentro ta tion curve is held fixed, so that if the re sul tant mo ment is smaller, the duc til ity is larger. This is con sis tent with the un der -ly ing stress strain curves of ax ial “fi bers” in the cross sec tion.
As plas tic de for ma tion oc curs, the yield sur face changes size ac cord ing to the shape of the M-Rp curve, de pend ing upon the amount of plastic work that is done. You have the op tion to spec ify whether the sur face should change in size equally in the P, M2, and M3 di rec tions, or only in the M2 and M3 di rec tions. In the lat ter case, ax ial de for ma tion be haves as if it is per fectly plastic with no hard en ing or col lapse.
Ax ial col lapse may be more re al is tic in some hinges, but it is computationally dif fi cult and may re quire non lin ear di rectin te gra tion timehis tory anal y sis if the struc -ture is not sta ble enough the re dis trib ute any dropped grav ity load.
138 Hinge Properties