Non prismatic Sec tions may be de fined for which the prop er ties vary along the ele -ment length. You may spec ify that the ele -ment length be di vided into any number of seg ments; these do not need to be of equal length. Most com mon situa tions can be mod eled us ing from one to five seg ments.
The varia tion of the bend ing stiff nesses may be lin ear, para bolic, or cu bic over each seg ment of length. The ax ial, shear, tor sional, mass, and weight prop er ties all vary line arly over each seg ment. Sec tion prop er ties may change dis con tinu ously from one seg ment to the next.
See Figure 20 (page 106) for ex am ples of non- prismatic Sec tions.
Segment Lengths
The length of a non- prismatic seg ment may be speci fied as ei ther a vari able length, vl, or an ab so lute length, l. The de fault is vl = 1.
When a non- prismatic Sec tion is as signed to an ele ment, the ac tual lengths of each seg ment for that ele ment are de ter mined as fol lows:
• The clear length of the ele ment, Lc, is first cal cu lated as the to tal length mi nus the end off sets:
Lc =L-(ioff+joff)
See Topic “End Off sets” (page 111) in this Chap ter for more in for ma tion.
• If the sum of the ab so lute lengths of the seg ments ex ceeds the clear length, they are scaled down pro por tion ately so that the sum equals the clear length. Oth er -wise the ab so lute lengths are used as speci fied.
• The re main ing length (the clear length mi nus the sum of the ab so lute lengths) is di vided among the seg ments hav ing vari able lengths in the same pro por tion as the speci fied lengths. For ex am ple, for two seg ments with vl = 1 and vl = 2, one third of the re main ing length would go to the first seg ment, and two thirds to the sec ond seg ment.
Section Properties 105
106 Section Properties
Axis 2
Axis 2 End I
End I
End J
End J l=24
seci=B secj=B
vl=1
seci=A secj=A
l=30
seci=B secj=B
vl=1 l=50
seci=A secj=A seci=A secj=B eivar33=3
Steel Beam with Cover Plates at Ends
Concrete Column with Flare at Top
Section B
Section B Section A
Section A
Figure 20
Examples of Non-prismatic Sections
Starting and Ending Sections
The prop er ties for a seg ment are de fined by speci fy ing:
• The la bel, seci, of a pre vi ously de fined pris matic Sec tion that de fines the prop -er ties at the start of the seg ment, i.e., at the end clos est to joint I.
• The la bel, secj, of a pre vi ously de fined pris matic Sec tion that de fines the prop -er ties at the end of the seg ment, i.e., at the end clos est to joint j. The start ing and end ing Sec tions may be the same if the prop er ties are con stant over the length of the seg ment.
The Ma te rial would nor mally be the same for both the start ing and end ing Sec tions and only the geo met ric prop er ties would dif fer, but this is not re quired.
Variation of Properties
Non- prismatic Sec tion prop er ties are in ter po lated along the length of each seg ment from the val ues at the two ends.
The varia tion of the bend ing stiff nesses, i33×e1 and i22×e1, are de fined by speci fy -ing the pa rame ters eivar33 and eivar22, re spec tively. As sign val ues of 1, 2, or 3 to these pa rame ters to in di cate varia tion along the length that is lin ear, para bolic, or cu bic, re spec tively.
Spe cifi cally, the eivar33-th root of the bend ing stiff ness in the 1-2 plane:
eivar33
i33 e1×
var ies line arly along the length. This usu ally cor re sponds to a lin ear varia tion in one of the Sec tion di men sions. For ex am ple, re fer ring to Figure 19 (page 103): a lin ear varia tion in t2 for the rec tan gu lar shape would re quire eivar33=1, a lin ear varia tion in t3 for the rec tan gu lar shape would re quire eivar33=3, and a lin ear varia tion in t3 for the I- shape would re quire eivar33=2.
The in ter po la tion of the bend ing stiff ness in the 1-2 plane, i22 e1× , is de fined in the same man ner by the pa rame ter eivar22.
The re main ing prop er ties are as sumed to vary line arly be tween the ends of each seg ment:
• Stiff nesses: a e1× , j g12× , as2 g12× , and as3 g12×
• Mass: a×m + mpl
• Weight: a×w + wpl
Section Properties 107
If a shear area is zero at ei ther end, it is taken to be zero along the full seg ment, thus elim i nat ing all shear de for ma tion in the cor re spond ing bend ing plane for that seg -ment.
Advanced Location Parameters
Normally the full vari a tion of a non-pris matic sec tion oc curs over the length of a sin gle frame ob ject. When the ob ject is auto-meshed into mul ti ple frame el e ments, each el e ment will rep re sent a por tion of the full non-pris matic length.
Sim i larly, if you ex plic itly di vide a non-prismatic frame ob ject into mul ti ple frame ob jects, it is nec es sary to spec ify for each frame ob ject what por tion of the to tal nonprismatic vari a tion ap plies to each ob ject. This is done by as sign ing to each ob -ject:
• The length of the to tal nonpris matic sec tion. This will be the same for each ob ject de rived from a sin gle par ent frame ob ject, and will be lon ger than each de -rived ob ject.
• The rel a tive lo ca tion
For ex am ple, con sider a sin gle nonpris matic frame ob ject of length 8 me ters, di -vided into four equal-length ob jects. For each de rived ob ject, the as signed non-pris matic length should be 8 me ters and the rel a tive start ing lo ca tions should be 0.0, 0.25, 0.5, and 0.75, re spec tively.
Effect upon End Offsets
Prop er ties vary only along the clear length of the ele ment. Sec tion prop er ties within end off set ioff are con stant us ing the start ing Sec tion of the first seg ment. Sec tion prop er ties within end off set joff are con stant us ing the end ing Sec tion of the last seg ment.
See Topic “End Off sets” (page 111) in this Chap ter for more in for ma tion.