TAPERED BEAM ELEMENT
27
where aðzÞ ¼ E IðzÞ gðzÞ; A0w ðzÞ ; bðzÞ ¼ E IðzÞ G A2w ðzÞ N : gðzÞ ¼ 1 þ G Aw ðzÞ Equation (3.10) is the governing equation for the equilibrium of tapered beam elements. It should be noted significantly that Equation (3.10) has a general form for any solid or latticed nonprismatic members, other than the forenamed I-shaped sectional tapered members, as long as an appropriate expression is replaced for shear factor in Equation (3.5).
3.1.2 Stiffness Equation Let ¼ Lz ; Equation (3.10) is converted to nondimensional form by að Þ y00 bð Þ L N y0 L2 N y ¼ bð Þ L2 Q1 L2 ðM1 Q1 L Þ:
ð3:11Þ
By using the Chebyshev polynomial, the functions yð Þ; að Þ and bð Þ can be approached by yð Þ ¼
M X
yn n ;
ð3:12aÞ
an n ;
ð3:12bÞ
bn n :
ð3:12cÞ
n¼0
að Þ ¼
M X n¼0
bð Þ ¼
M X n¼0
Substituting Equation. (3.12) into Equation (3.11) leads to " M n X X n¼0
# ai ðn þ 2 iÞðn þ 1 iÞynþ2 i n
i¼0
L N
" M n X X n¼0
¼ L2 Q1
# bi ðn þ 1 iÞynþ1 i n L2 N
i¼0
M X
M X
yn n
ð3:13Þ
n¼0
bn n L2 M1 þ L3 Q1 :
n¼0
According to the principle that the factors at the two sides of Equation (3.13) for the same exponent of should be equal (Eisenberger, 1995), we have for n ¼ 0: 2a0 y2 L N b0 y1 L2 N y0 ¼ L2 Q1 b0 L2 M1 ;
ð3:14Þ
for n ¼ 1: 6a0 y3 þ 2a1 y2 L Nð2b0 y2 þ b1 y1 Þ L2 N y1 ¼ L2 Q1 b1 þ L3 Q1 ;
ð3:15Þ