Non-prismatic Sections

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 95) for ex am ples of non- prismatic Sec tions.

Section Properties

95

Steel Beam with Cover Plates at Ends

Concrete Column with Flare at Top

Section B

Section B Section A

Section A

Figure 20

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, L_c, is first cal cu lated as the to tal length mi nus the end off sets:

L_c =L-(ioff+joff)

See Topic “End Off sets” (page 99) 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.

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 93): 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

If a shear area is zero at ei ther end, it is taken to be zero along the full seg ment, thus elimi nat ing all shear de for ma tion in the cor re spond ing bend ing plane for that seg - ment.

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 99) in this Chap ter for more in for ma tion.