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General analysis of asymmetric thin-walled members
Thin-Walled Structures 7 (1989) 159-186
General Analysis of Asymmetric Thin-Walled Members
P. Chang & H. Hijazi Department of Civil Engineering,Universityof Maryland,CollegePark. MD 20742, USA (Received7 June 1987;accepted7 June 1988)
A BSTRA CT This paper presents the closed-form sohaion for the deflection and stresses of an asymmetric thin-walled member. The method is based on the assumption that the cross-section can deform out of plane when warping and shear-lag effects are significant. The out-of-plane deformation is represented by a linear warping function plus a truncated series of complete eigenfunctions. The differential equations are derived by using the principle of Minimum Potential Energy and solved by a symbolic manipulator. An example is used to illustrate the application of the method and the use of the closed-form solutions obtained. Results of the example are compared to the results of a finite element analysis and other approximate models. The comparison indicates that the maximum error of the proposed method is within O.1% of the value obtained by the finite element method in lateral displacement and 0.2% in axial displacement.
NOTATION Ayy, etc. bc
Property of the area defined in the Appendix Denotes solution without eigenwarping correction d, Constant consisting of eigenwarp constants and load E Modulus of elasticity Eigenwarp intensity function F,(x) Shear modulus G Constant consisting of eigenwarp function and linear warp, function defined in eqn (13) 159 Thin-Walled Structures 0263-8231/89/$03.50 (~ 1989 Elsevier Science Publishers Ltd, England. Printed in GreatBritain
160 [p
!." etc. J~,
g,
Keg k l , k~
k mx El
P.v, P: Fk S
S t U, V, w
U,(s) g,(s) Ux(x) Vi. Wi, Zi ot
t3 Ex
4, 77"
Y ys F~
o (x) o (x)
P. Chan¢ H. Hi/azi
Property of area defined in the Appendix Moment of inertia defined in the Appendix Warping constant Constant consisting of derivative of the linear warp function. defined in eqn (14) Constant consisting of derivative of the linear warp function. defined in eqn (15) Constants defined in eqn (24) Denotes shear center Moment about the longitudinal axis Denotes centroidal quantities Load in the y and z directions Normal distance from a point on the member to the centroid Denotes arc length Cross-sectional area of thin-walled member Thickness of thin-walled member Displacements in the x, y and z directions Eigenwarp function Warping function (linear) The axial deformation at the centroid of the section Unknown constant multiplier to eigenwarp intensity function Constant consisting of Art, etc. Constant consisting of lyy, etc. Axial strain Constant consisting of eigenwarp function defined in property 5 on page 5 Sectional area of the cross-section Potential energy Constant defined in eqn (24) Shear strain Combination of unknown constant defined in eqn (7) Rotation with respect to tile y axis Rotation with respect to the z axis Warp intensity function
1 INTRODUCTION In the past 40 years, the concept of tube structures has been widely used in tall buildings. The analysis of these buildings has been made possible with the advent of high-speed computers. However, the analysis of these
General analysis of asymmetric thin-walled members"
16 l
structures is time consuming and costly; an alternative method of analysis is desired, especially for the preliminary stage of the design. A large class of these alternative methods consists of homogenization of the actual structure so that analysis can be performed on the homogenized continuum. The analysis of this continuum must be as accurate as possible; otherwise important behaviors of the structure may be overlooked. The tube buildings resemble thin-walled members. Unfortunately, an accurate analysis of asymmetric thin-walled members is not presently available. Thin-walled members are defined as members in which the thickness is small compared to the other cross-sectional dimensions. The cross-sectional dimensions are in turn often small compared to the length of the member. The analysis of thin-walled members requires special attention because the in-plane shear stress and strain of these members can be relatively large compared to those in beams with solid cross-sections. The in-plane shear stress and strain induce two types of longitudinal deformation called shear lag and warping. Shear lag deformation refers to the differential longitudinal displacement resulting from shear deformation of the wall of the member, and warping deformation refers to the out-of-plane displacement due to torsion. When these deformations are significant, conventional beam analysis, which assumes no out-of-plane deformation, is inadequate. Thin-walled members have been studied extensively in the past 40 years. In the first attempts to account for shear-lag deformation in thin-walled members, a second-order differential equation was used to describe the intensity of the shear-lag effect. ~'2 These methods did not give accurate results. The use of a parabolic function to describe the longitudinal deformation of the cross-section and the use of the Minimum Potential Energy principle to obtain the magnitude of the differential displacement had some success. 3-5 However, this approach is limited to beams with specific crosssections. The Bar-Simulation method, which discretizes the beam into axial-load carrying bars and shear-force carrying plates, 6"7 also had some success. This method assumes that the longitudinal deformation varies cubically. Such an assumption is, similar to the parabolic assumption, only accurate for specific situations. Another disadvantage of the abovementioned methods is the difficulty in their use because they involve the solution of a set of complicated differential equations. To avoid such difficulty, designers have adopted an effective width method where the beam is replaced by one without shear lag deformation, but with a smaller width. 8-~° To account for the warping deformation, researchers have traditionally used an approximate warping function."-" The main drawback with this approach is that a specific warping function can not describe the longitudinal deformation accurately. In order to correct this problem, a series of ortho-
162
P. (Tmng. H. Hijazi
gonal functions is used to represent the warping deformation, z>-'-' However. a complete series requires a prohibitively large number of terms. The truncated series used by these researchers, among other problems in their formulation, yielded inaccurate results. And further improvement in the analysis of thin-walled members is necessary in the study of tall buildings through the m e t h o d of homogenized continuum. This paper presents a method that includes the out-of-plane deformation due to shear lag as well as warping. The out-of-plane deformation is assumed to be made up of a linear warping function and a truncated complete series. The resulting sixteen-order differential equation is solved by using the symbolic manipulator, MA CSYMA. Closed-form solutions for deflection, rotation and stresses are obtained. 2 MATHEMATICAL FORMULATION The mathematical formulation presented herein is developed for a structure with a cross-sectional shape that remains unchanged after deformation, This assumption has been used by many researchers because thin-walled members with closed cross-sections are relatively rigid in their own plane. In addition, diaphragms are often added to increase the in-plane stiffness in many applications such as tall buildings and bridge girders. The present study is motivated by the representation of a tube structure by a continuum. The floors of these structures serve as a diaphragm which makes the stiffness in the plane of the cross-section very large as assumed. Because the walls are relatively thin, it is reasonable to assume that the member is in a state of plane stress. Hence, the only significant strains are the axial strains, e,, and the shear strain, ys. To get e, and Ys, the assumptions were made that the deflection in the axial direction is made up of rigid-body rotation, rigid-body deflection, and a series of shape functions, with each multiplied by a weighing parameter. Thus, N
u(x.s) = U . ( x ) - ( y - y . ) O : ( x ) + ( z - z . ) O y ( x ) + U,~(x)+ ~ UiFi i
in which Ux(x) = O:(x) = 0y(x) = U,(s) = ~(x) = Ui(s) = F~(x) =
the axial deformation at the centroid of the section. rotation with respect to the z axis, rotation with respect to the y axis, warping function (linear). warp intensity function. eigenwarp function eigenwarp intensity function.
(l)
General analysiso( asymmetric thin-walledmembers ,,t = c e n t r o i d a l
163
quantities.
.s = a r c l e n g t h , u . v, w = d i s p l a c e m e n t s
in t h e x , v a n d - d i r e c t i o n s .
The term Y~ U~ Fi is an infinite series that corrects the longitudinal displacement of the cross-section to include the shear-lag effect and effects of warping not accounted for by the linear warping function. If {U," i = 1. . . . . x } is a complete set. then the term U, ~(x) is not necessary. However, only a finite number of terms in U, can be used: the addition of the term U,/;(x), then, improves the approximation significantly. The lateral deflections and the twist are approximated similarly. But since the cross-sectional shape remains unchanged, the correction terms I,;, W, and Z, are constants. Hence,
,.,(x) = v-(:-:,)o,,+ ~ v, f F~a, = ,,',(.~) = . . + o , - : , , ~ o ~ , +
~'u-(:-:~)0~,
(2)
~• w, f F,a,- = ,,.~+ 0.-.v~)0~,
(3)
O@r)=O,+ ~ Z; f F,ctr
(4)
For small deformation, the axial strain and shear strain are given by:
au Ox "" and y~
• -
~'
=
au as
dv, dv ch" ds
dw, dz dx ds
+ _ _ _ _ + w _ _ + r ~
dO,. dx
(5)
in which rk is the normal distance from a point on the member to the centroid of the cross-section, and 0~, is the rotation about the longitudinal axis. Substituting eqns (1)-(4) into (5) gives:
~*-
dU~ d.r
(y
v, = -0~+
dO. -Y")--~r + ( z - z " )
ds
0,+
+U'-'~r + ~UidFi ct-T
(6)
ds +--ag +---~-r,+ Y_ r,F;
(7)
t
in which F; = dUi/ds + Vidy/ds + W;dz/ds+ Z, rk, and the subscript k denotes the shear center. The total potential energy of the system is:
I IL f [ E t ~ +Gty~]dscK-IL
7"g -~" 5
(p,v+ .
P.w+rn.,Ox)cK
(8)
164
P. Chart,, H. Hiiazi
Definition. Let the subscript "bc" denote the part of the solution without the eigenwarping correction terms, then the axial strain. E~h~and the shear strain. )%~ are defined as % -
d U~ dO: dO, ctr - (y-y°)-8-(v +(:-:')-8-(~ + u,
(9)
and
( dr)d:+( ds Ov+-~V ~-~r+
"Y~c----- -0:+'-~-r
rk+-'~
(ll))
Let Vv. V: and Alk be the shear in the y direction, shear in the z direction, and the sum of torque, respectively:
d V,.
d V.
d_v
P"" ctr
dMk P., and 4r
-mr
(I 1)
Substituting eqns (6), (7) and (9)-(11) into eqn (8) yields:
1 ILl l
[Ete!¢~ + G t %" j d s d . r -
IL[
dF~Ui
dr+ V. ---~+ d dG1 ctr V~, d.r Mk --~.j dG\ 2"1 (12)
- ~L ~ (V~V"+ V:W~+MkZ,)F~dx T h e eigenwarping functions have the following properties: 25
(1) ~ EtUi Uj ds = 80, the eigenfunctions are orthonormal; (2) ~ EtUids = 0, the contour integral of the eigenwarping function is
zero;
(3) ~ E t U i ( y - y n ) d s = O, the contour integral of the eigenwarping functions and (y - y.) is zero; (4) ~ E t G ( z - zn) ds = O, the contour integral of the eigenwarping functions and (z - z.) is zero; (5) l~ ~ EtUiU, ds = ~ GtdU,/dsF, ds
General analysis of cts'ymmetricthin-walled members
165
in which/x, p.j =- :~GtF, Fi ds/~ EtU~ Ui ds, and the approximate functions F, have the following properties: (1) the contour integral of I-'itimes dy/ds is zero: ]; GtFi d.v/ch'ch = O: (2) the contour integral of F, times dz/ds is zero: ~;GtFidz/ch'ds = O: (3) the contour integral of l-'gtimes r~ is zero: J; GtF~ r~ ch" = O.
Definition. Let the section properties and the load terms be defined as shown below:
!,~=- f U,U~Etds
(13)
Krf = f trk dUt ds ds
(14)
Kff=f t ( d--~----L)2ds
(15)
di =- Vi Vy + Wi V: + Zirnk
(16)
Using the properties defined above and the definitions from eqns (13)-(16), the total potential energy of eqn (12) can be written in the form: 7r=Tr~+
.
t fL[EFs('tux L
= S Jo l
2
+ G Ayy -0~+-~(
+ lp(~)"+
+5_ttrF?+l"dx d.r ~-ttTl,,gF,-d,F, dx (17)
\dx /
+
I
I
, ovdo=+j {deTI
dx J + =\ dx J - 2l'= d-T d--T
+A= ¢ + dx/ +2A":
\ clx J J
°y+-~r]\-°=
2K,tr,-~+ktt~2]}clx- yL(p~.v+ P:w+m,O,)d.r
dx I (18)
in which S, J,,, Iyy, It.., Iy:, Ayy, A,, and Ay: are geometric properties of the cross-section defined in the Appendix. Minimizing the total potential energy in eqn (18) with respect to v, w, Oy, 0~, 0,,, U,,. r~ and Fi, the following differential equations are obtained:
166
P. Cha/t~ H. Hil/:t
dx
A,: - 0 : + d):
d_r A>:,. - O : + ~
dr
~-
da"
-'~'v - E l : : dx
+,4::
0,+ dr ]
= -P:(x)
( 19a}
+A,.:
0,.+ dr ]
= - P'(x)
(19b}
":-~7rj + GA,.,. - o . + "
"
+ GA,.. "
dr
0>.+
d,. j = o
- EJ~
+ GKa
~
= 0
(19c)
-0:+--~7~. + G A :: O,. + --.~-r ] = 0 (19d)
+Gay:
(19e)
dx ]
cu-
dv ]
(19f) + GKff ~ -
~
d2Fi
l,i ~
,
+ S-" ~; I,i Fi = 0
(19g)
i
d"- g dr:
, ~, ~; ~ = -d, - l,i + ~ I,~r~
(19h)
with the boundary conditions:
[.( ( d.)]j,=L [(d,,) (
8v A,..
-0.
8W Ayy
--0: + - ' ~
80 Y x
:--~
Oy +--.-~r
.,=o = 0
(20a)
+ Ay: 0>.+---~
,=,) = 0
(2Oh)
dx ] + A::
= 0
(20c)
= 0
(20d)
o
= o
80, tp-~-
Ka(~
=0
(20e)
(20f)
General amdvsis of asymmetric thin-walled members
~
(
EJ~
+
BE` \ d_r
I,~ ~
'~ ~ r ]
=o = 0
x ,, = 0
167
(20g)
(2tlh)
E q u a t i o n s (19) are t o o c o m p l e x to be solved in closed f o r m . T h e s y m b o l i c m a n i p u l a t o r , M A C S Y M A , will b e u s e d instead. T h e v a r i a b l e s are r e w r i t t e n to a f o r m m o r e c o n v e n i e n t for use in M A C S Y M A as follows:
Let X l = V
be the deflection in the v direction
X 2 ~- 14'
be the deflection in the z direction
X 3 =0 v
be the rotation about the v axis due to flexure
X4=O:
be the rotation about the z axis due to flexure
d0y x5 -
X6
be the derivative of 0~ with respect to x
~r d0= d.r
be the derivative of 0= with respect to x
d I,~ x7 = ~ -
dw
.r8 = ~
0:
-0y
be the rotation about the z axis due to shear
be the rotation about the v axis due to shear be the linear warping intensity function
X9 =
d~ " r i ° - dx XII = #x
d0x x t 2 - dx XI3 ~ f i
dE, Xl 4
be the derivative of x9 be the rotation about the longitudinal axis the derivative o f x , with respect to x be the eigenwarping intensity function be the derivative o f x u with respect to x
dx XI5 -~- U x
dUx xl6-- dx
be the rigid-body linear displacement be the derivative ofxt5 with respect to x.
168
P. Chanv, It. Hi/a=i
Then the differential equations in eqns (19) become: d'r7 drs G.4,:--~r + G A = dx G4
dx7
• ..--~+GA~.:
4rs 4r
P.(x)
(2 la)
(21b)
Py(x)
dr. E L , ~ - EI~. - " + GA,.,. X 7 + G A ~...rs = 0 " d.r " " • d.r6
dx~ +
El,: ~
- El:: dx
GA.v: x7 + G A
::
xs = 0
d.~ t X7 + X ,
(21e)
--
X8-
(21f)
-
x5
(21g)
-
x6
(21h)
-
Xl0
dx, X 3
dx 3
4r d.r,
d_r
(21d)
-
dx
dx
(21c)
d_r9
dx
(21i)
-,4 a,m d.r + B ~dx9 + GK~.rlz + ~ l,id i = 0
(21j)
i
d-lf l I
4r
(21k)
- - Xt 2
G l _ dxl2 ) lp---~r + Kaxlo + m , = 0
~
dx
tx~ xl3 = - d i -
dXl5
dx d.~'16
dx
--
-0
Xl 6
(211)
{ a ,o _ ,
,Cai ~ dx
I ~ ; X9
)
(21m)
(21n) (21o)
General analysis of asymmetric thin-walledmembers
169
T h e b o u n d a r y conditions in eqns (20) for a cantilever b e a m with the end x = 0 fixed written in terms of the new variables are: xt(0) = 0
(22a)
x_,(0) = 0
(22b)
x3(0) = 0
(22c)
x4(0) = 0
(22d)
xs(L) = 0
(22e)
x6(L) = 0
(22f)
XT(L) = 0
(22g)
xs(L) = 0
(22h)
xg(0) = 0
(22i)
xt0(L) = 0
(22j)
xlt(0) = 0
(22k)
Ipx12(L) + Krfxg(L) = 0
(221)
xl3(0) = 0
(22m)
xt4(L) = 0
(22n)
x15(0) = 0
(220)
xl6(L ) = 0
(22p)
Using these b o u n d a r y conditions, and the differential e q u a t i o n s in eqns (21), the following general closed-form solution is obtained: 1
v(x) - x t ( x ) - 24EGetfl {txG(I= Py + lrz Pz)x 4 -[4otG2 x7 (0)(Ay~ it. + A . !~ ) + 4aG~ xs(O)( Azz ly~ + At. l=)]x 3 + [12a~SEGx6(O) + 12/3E(Ayz Pz - A.. Py)]x z + [24aBEGxT(O)]x }
(23a)
P. Chart,#. H. Hijazi
170
-I
w(x) --x_,(x) -
24EGo~B {eG(!,: P, + / , , P:)x 4
- [ c~G- xT(0)(A,../,.,. + A,.,. 1,..) + 4e~G'-xs(O)(A:: I,., + A,.: /.,.:)]x 3 + [12oLBEGx5(O) + 12BE(As,, P: - A,: P,)Jx 2 - [24a/3EGxs(0)]xl
(23b)
1
O:.(x) =--x3(x) = 6--~ {[I~.~.P: + It: Py]x 3 - [3Gxd0)(A.,.: t,': + A:: l,'s) + 3Gx7(O)(A.,.,, ly: + A.~: l~r)]x 2 + 6BExd0)x} O:(x) --x.,(x) = ~
1
(23c)
{[Is: P: + I= Pr]x s - [3G.rs(0)(A:: t,': + Ay: I:: )
+ 3GxT(O)(A,,: ly: + Ayy L:)]x 2 + 6/3Ex6(0)x }
-alp {(
=--xg(x) = ~
(23d)
d~(0)
alp K~fmx -
CK/)
dx
(23e)
G-=x,t(x) = ~ Glp
[
~(~:)dsC+m~
GKtt
F/(x)~x13(x)=-i'i~(x)
kl
-~cosh(/zix)+
2/zi
k,x ", /x,
+
Ux(x)---xls(x) = Ux(L)x = 0
( 4)]
l'i+
kt
, /z7
Lx-
4r
(23f)
"
(23g) (23h)
w h e r e the initial c o n d i t i o n s are 1
x d 0 ) = 2--~(ly~ P: -/yy P~) L z
(24a)
Generul analysis of asymmetric t/fin-walled member~
1 x6(O) = .---~-7~ (6:
P,-/,,:
171
P:) L-"
(24b)
XT(0) = ~---~(A:: P,. - A,.: P:) L
(24c)
xs(O) = a-~(Ayy P: - Ay: Py)L
(24d)
-/,.1 L
1
1 .qo(0)
=
,,¢ cosh { ~
\aC/
~ { cosh[ ~
( Kffm., - Ipi~ik,_)
-
(24e)
/zi
-
m x t Xtz(0)
-
- --'~--r
i
(24f)
(24g)
Glp
in w h i c h
=- alp(blp - G K ~ ) a = - ( E J ~ - ~1,,)
(24h) (24i)
i
b =---(GKtf
-
~ la.i2 ,17i )
(24j)
i
k, m (V, fy + W,~: + Z,m~) L
(24k)
k2 = - ( V i i , + W i L + Z i m , )
(241)
a = Ayy A= - A"yz
13- t.
t:~
-
t~z
(24m) (24n)
P. Chang,H. Hiiazi
172
3 EXAMPLES The proposed method will be illustrated using numerical examples. In the first and second examples, a thin-walled member with a rectangular crosssection is used (Fig. 1). The load on the structure in the first example is symmetric in the vertical direction, and the load for the second example is a distributed torque. These loads are used because it can be seen from eqns (2l) that the effects due to bending and torsion are uncoupled, and a structure subjected to both bending and torsion can be modeled by two
40"
t
t f : 0-05625"
J I ~0"
tw= 0"3275"
Fig. !. Cross-sectional dimensions of beam used in examples 1 and 2. loadings, one representing pure bending and the other representing pure torsion. The effect of the combined load is the superposition of the two separate loads. Example 3 illustrates the decomposition of a general load into the pure bending and pure torsion components for a thin-walled beam with asymmetric cross-section.
3.1 Example h Rectangular beam under pure bending load The geometric properties of the cross-section are calculated using the formulas listed in the Appendix, and in Table 1. The eigenwarping functions are the eigenvectors of the cross-section. These eigenvectors can be obtained easily by modeling the cross-section by finite elements. And the weight parameters /a.~, V,, Wi and Zi are calculated by the following formulas:
f (ix]EtUi+Gt-~)
ds = 0
(25a)
fGtF~ d-~ds=O
(25b)
GtFi ~dz ds = 0
(25c)
f GtFir, ds = 0
(25d)
General anah'~i.~ of asymmetric thin-walled members"
173
TABLE 1 G e o m e t r i c Properties of the Structure in E x a m p l e s 1 and 2
L 1,~
100in 5 880 in ~ St~ in 4
I:: /y: E v A,,. A::
0
30
0 ~ ) ksi
0.3 4"5 in 2 13"5 in 2
A.,.:
(}
/p Jt
5 860 in ~ 1 733 in 4 5-153 x 1()~ K - in 4 I)'337 5 in 0-056 25 in
EJ,,. &. tf kff
krf zk Yk
4 187 in 4
- 4 187 in 4 0-0 in 0 . 0 in
The eigenwarping functions and their associated weight parameters are shown in Fig. 2. The linear warping function, U,, is given by the following equation: zk(Y -- Y,,) + yk(z -- Z.) + At
U, = ~ t ( s ) -
(26)
in which ~b, is the sectorial area of the cross-section, z~ and Yk are the shear center coordinates, z, and y, are the centroidal coordinates, and A , = - ~ t4~, ch'/~, t ch'. Using eqn 126), the linear warping function is calculated and shown in Fig. 2. Substituting the geometric properties listed in Table I. and the calculated shape functions and weighting parameters in Fig. 2 into eqn (23), and simplifying, the deflections become: 1 zP: El_. ( - L - ' x + x 2 L - x3/3) + E Ui F/(x) u- 2 --
(27a)
v=0
(27b)
4--
~42
z"-
1 2
-
-
Ox(x) = 0
1 P.
i
(27c) (27d)
-(-L2x+LxZ-f~) o , ( x ) - 2 -Et-=
127e)
0.(x) = 0
127f)
P. Chain,'. H. tttla=t
174
=
.-_-s.
t
~o
.~
~o > t< io
> .,,¢
:<
0
0 t~
0 ii N
>
-.2
~bi ii
ixl
General analysis of asymmetric thin-walledmembers
175
in which L-x
Fi(x) = C, cosh(~/x) + C:sinh(ta.ix) +----r-(P.. Wi) it;
(27g)
L C, - -~(PzW~)
(27h)
C2 = -Ct
1 +/a. i L sinh(/xi L) ~i L cosh(/zi L)
(27i)
The beam was also analysed using the finite element method. The finite element mesh used is shown in Fig. 3. The elements used are twodimensional plane stress elements. The model is constrained such that the cross-sectional shape remains unchanged throughout the analysis as
Fig. 3. Finite element representation of model in Examples 1 and 2.
assumed in the proposed method. The displacements u and w are plotted against the finite element solutions in Figs 4 and 5. Also included in these figures are deflections predicted by other theories. It can be observed that the proposed method gave highly accurate solutions compared to the results of a more rigorous and time-consuming finite element analysis. It can also be observed that the proposed method is superior to the other approximate methods. 3.2 Example 2: Rectangular thin-walled beam under pure torsional load
The beam used in this example has the same cross-section as the beam used in Example 1. However, the load applied is a uniformly distributed torque along the longitudinal axis, rnx. Because the beam's cross-section has not changed, the geometric properties of the cross-section in Table 1 remain
P. Chang, H. Hiiazi
17e~
0.4 i"E
0.3
Z
E
0.2 .to
t~
0.1
0.0 "r 0.(
/,
Finite Element a x o
Proposed Method Bauchau Timoshenko i |
i'
0.2
0.4
| "
0.6 0.8 Length, x/L
i
1.0
.2
Fig. 4. Axial displacement of rectangular thin-walled beam.
°~
I
C
I
i
i
E i
I I
J
t
o
f
0
~
Finite Element ,", ProposedMethod x Bauchau o Timoshenko
,
0.0
0.2
,
0.4
,
0.6 Length,
i
0.8
,
Z
1.0
x/L
Fig. 5. Lateral displacement of rectangular thin-walled beam.
.2
General analysis of asymmetric thin-w,dled members
177
valid. Substituting these properties and the loading rn, into eqn (23). the following deflections are obtained: u = U,(s)ij(.r)+ ~ Ui Fi(x)
(28a)
i
r
=
(28b)
-zO,
w = y0x
(28c)
l[I/
0~(.r) = C - ~
GK,
/~(~) d~: + m.,(Lx - x:/2)
]
(28d)
in which
F,(.r) = CI cosh0zix) + 6". sinh(/l,r) - l,i ~(x) + ¢(.r) = C3 cosh(kx) + C~ sinh(kx) - C3(L - x )
L-x
, m., Z~ tz;
(28e) (28f)
and C1 =
-L ,_ mx Zi L /.ti sinh(/.ti L) /zi b cosh(p.i L)
(28h)
C 3 : L [Kam,,/ip + Ei(Py Vi +m,,Zi)i,i] " ,c , , -GKtt + GK'a/lp + _~ It, 17~
(28i)
C4
(28j)
C2 = - C1
=
I
(28g)
+ p.~ L
- C 3 ( k s i n h ( k L ) - 1/L)
" " - Yi I~ p.i E J . - z, a:,
k cosh (k L )
(28k)
T h e d i s p l a c e m e n t s u and 8, o b t a i n e d from eqns (28) are c o m p a r e d to the d i s p l a c e m e n t s o b t a i n e d using other theories as well as displacements o b t a i n e d f r o m a finite element analysis in Figs 6 and 7. T h e results clearly s h o w that the closed-form solution given in eqns (23) is accurate and superior to the o t h e r a p p r o x i m a t e m e t h o d s used w h e n c o m p a r e d to the results of the finite e l e m e n t analysis.
P. LTnmg, H. [ti]aci
178
5
'
i
I
4 0
E U
--¢ ¢x
2
"
i5
1 x
Finite Element ProposedMethod Bauchau
[] x 0-¢~
0.0
,
,
0.2
1
,
0.4
I
0.6
,
0.8
J
,
1.0
.2
Length, x ' L
Fig. 6. Axial displacement of rectangular thin-walled beam.
0.03
0.02
J
o
0.01 ,,
m
=
0.0
0.2
0.4
0.6
Proposed Method
0.8
1.0
Length, x/L
Fig. 7. Axial rotation of rectangular thin-walled beam.
1.2
General analysis of asymmetric thin-walled member~
179
3.3 Example 3: Trapezoidal thin-walled beam under axisymmetric loading To illustrate the use of the proposed method when the member and the load are asymmetric, a thin-walled member with a trapezoidal cross-section is used. The beam is subjected to two uniformly distributed loads in the v direction, Py~ and P~:" These loads are located at distances eyj and e,,_ from the shear center. The beam dimension and load configuration are shown in Figs 8 and 9. 40" o-
_1
1 14
_1
20" O"
1
Fig. 8. Cross-sectional dimensions of the example structure.
z
l
m
Fig. 9. Actual and equivalent loads on example structure.
The applied load was replaced by an equivalent set of loads with a transverse load and a distributed torsional moment applied along the shear center, as shown in Fig. 9. The geometric properties of the cross-section are calculated using the formulas given in the Appendix, and listed in Table 2. The eigenwarping functions are the eigenvectors of the cross-section modeled as a plane frame, and the weight parameters ~, Vi, W, and Zi are calculated by eqns (25). The eigenwarping functions and their associated weight parameters are shown in Fig. 10. The linear warping function, U,, which is given by eqn (26), is also shown in Fig. 10. Substituting the geometric properties listed in Table 1 and the calculated shape functions and weighing parameters in Fig. 10 into eqns (23), and simplifying, the deflections become:
P. C/tang ft. ktijazl
IN()
1 vP,,
(-L2x+?(2L-'[~3)
It- 2 ~ ,
l"--
1 P~.
L: .r2
2 EI-~.
2
.r 3 L
+Ut(S)~{X)+
r4
3 + -
P'
X-'U'F'(r)~I
(29a)
Lx -
+ GA~, (29b)
w=yO~+ ~ W, f F,(x)ctr
t [p G K , ' O.,(x) = --G-%-
(29c)
~ ¢(~)d~+m,(L.r- x-'/2)]
(29d)
in which ~(.r) = Cl cosh(/z~x) + C,_sinh(/~x) -
I,i ~(x) +
L--X
, ....(m.~Z~ + P, V, ) /2? (29e)
~(.r) = (73 cosh(kx) + C~ sinh(kx) - C
(L - x ) ~ L
(29f)
and -L C, = ----T-_(m~ Z, + Py V~) /z, C: = - Cl
(29g)
1 +/ii L sinh(/zi L) /zi L cosh(/zi L)
(29h)
C3 = L [Katnfllp + vi (Py Vi + m~Zi)Li] -
OK. aK'-./¢ +
+
C~ = -C3(k sinh(kL) - l/L) k k2 w G k , - GkZatlp- ~i ~¢ 12ti ~i2 2
EJ~¢ - - i It i
g, 1
cosh(kL)
(29i)
(29j)
(29k)
T h e b e a m was also analysed using the finite e l e m e n t m e t h o d . T h e finite e l e m e n t m e s h used is shown in Fig. 11. The two-dimensional plane stress e l e m e n t s are used in the finite e l e m e n t model. T h e m o d e l is constrained
General ttnalysis of asymmetric thin-walled members
TABLE 2
Geometric Properties of the Example Structure L E/v,,
El:: El.,:
Itll in 3-87 x 10" K-in 2 4-87 x 10~ K-in-" 0
E v
GAyy GA= GA~: GIp GJ, Gktf J~ z~ )'k
Z : 0
W: 321x10 -4 V=O It1:0 ~2:~.10x10-3
30 000 ksi 0.3 85 548 K 16 317 K
0 5 568 156 K-in-' 4 705 660 K-in-" 518 377 K-in-" 2 571 in 6 - 1-07 in 0.0 in
W:O V = 9"34x10 "5 Z = 9 " 4 2 x 1 0 "(5 It1 : 2 7 2 0 ~ 2 : 1"~3 x l O "2
W: 0 Z :-3.64xi0 "7
Z=O
: 9 . 1 7 x i 0 "6 I t 1 : 5 8 5 5 / 4 2 : 7 3 7 x 1 0 "3
V : 7.80XI0 "5 It I = o5510 ~ 4 2 = 3 " 0 8 x 1 0 " 2
Z : 0
W : -2-g2x 10 "2 V : O Itl=O t2 = 1 ~ 2 ~ x I 0 - 2
W : - 4 . 3 X 1 0 "4 V : O It 1 : 0 ~2 : 2 . 7 7 5 x 1 0 " 2
Ut
Fig. 10. Eigenfunctions and associated weight parameters of cross-section in Fig. 8.
181
P. Chan£, H. Hiiazz
Fig. 11. Finite element representation of model.
0.8
S
A
"o
s.
0.6
p
,
]
I
o X
r0 0 ee
0.4
F
0.2 i
,<
f
0.0 0.0
0.2
0.4
o
0.6 Length,
Finite Element I Proposed Method !
0.8
1.0
1.2
x/L
Fig. 12. Comparison of axial rotation.
such that the cross-sectional shape remains unchanged throughout the analysis as assumed in the proposed method. The displacements u, v, w and x are plotted against the finite element solutions in Figs 12, 13 and 14. Also included in these figures are deflections predicted by other theories. It can be observed that the proposed method gave highly accurate solutions compared to the results of a more rigorous and time-consuming finite element analysis.
4 CONCLUSIONS A closed-form solution of an asymmetric thin-walled member under arbitrary loading was obtained. The method is based on the assumption that
183
General analysis of asymmetric thin-walled members
A r'°B
I
E
!
!
,
Ii
'
,
I
,,
Proposed I .°,~°°°~°,
;
J1
l,)
i5
_.o
J
--I
~ - -
.
i
0 0.0
0.2
0.4
F,n,te E,omen, i
a
0.6 Length,
0.8
.
Method I
1.0
1.2
x/L
Fig. 13. Comparison of lateral displacement.
1
A e" °m
o. O X
E O E¢ )
oB
/
X
/
/ ,l°
Finite Element Proposed Method
i
0 0.0
0.2
0.4
0.6 Length,
I
0.8 x/L
Fig. 14. Comparison of axial displacement.
1 .0
1.2
18,4
P. Chang, H. Htiuzi
both the shear-lag deformation and warping deformation of a thin-walled m e m b e r are significant. These assumptions are approximated using a finite set of eigenwarping functions and a linear warping function. The differential equations are obtained through the principle of Minimum Potential Energy and solved using the symbolic manipulator. M A C S Y M A . T h e m e t h o d contains far less constraints than the existing methods listed in Section 2. Because closed-form solutions are presented, users of the proposed m e t h o d only need to calculate the required geometric properties of the beam being analysed. Numerical examples of a beam with trapezoidal cross-section are presented. The axial displacement, axial rotation and lateral displacement obtained were then compared to the results of a finite element analysis with a relatively fine mesh. The results from the closed-form equations match the finite element results exactly. It should be pointed out that the methods that neglect the warping deformation gave no axial rotation for the example beam. The methods that neglect the shear-lag deformation resulted in smaller deformation in all directions compared to the finite element solution, especially in the axial direction. Methods that assumed symmetrical cross-section or methods that assumed no warping or shear-lag d e f o r m a t i o n could not be used.
REFERENCES 1. Kuhn, P., Some Notes on the Numerical Solution of Shear Lag and Mathematically Related Problems, NACA TN No. 704, 1939. 2. Newton, R. E., Shear lag and torsion bending of four-element box beams, J. Aeronaut. Sci., 12 (October, 1945) 461-7. 3. Hildebrand, F. B. 8: Reissner, E., Least Work Analysis of the Problem of Shear Lag in Box Beams, National Adv. Comm. Aeron., TN No. 893. 1943. 4, Chang, P. C. 8: Foutch, A. M., Static and dynamic modeling and analysis of tube frames, J. Struct. Engng, 110(12) (1984) 2955-75. 5. Coull, A. & Bose. B., Simplified analysis of frame tube structures. ]. Struct. Div. A S C E , STII, 101 (1975)2223-40. 6. Kuhn, P., Stresses in Aircraft and Shell Structures, McGraw-Hill, New York, 1956. 7. Evans, H. & Taharian, A., The prediction of the shear lag effect in box girders, Proc. Inst. Civil Engng, Part 2 (1977) 69-92. 8. Abdel-Syed, G., Effective width of steel deck plate in bridges, J. Struct. Div. A S C E , 95(St7) (July 1969) 1459-74. 9. Moffat, K. R. & Dowling, P. J., Steel Box Girders, Parametric Study on the Shear Lag Phenomenon in Steel Box Girder Bridges, CESLIC Report BG 17. September, 1972, pp. 1123-30. 10. Malcom, D. J. & Redwood, R. G., Shear lag in stiffened box-girder, J. Struct. Div. A S C E , 96(ST7) (1970) 1403-19.
General analysis o( asymmetric thin-walledmembers
185
I1. Goodier, N. J., Torsional and flexural buckling of bars of thin-walled open section under compressive and bending loads. J. Appl. Mech., 913) (Sept. 1942) 103-7. 12. Von Karman. T. & Christensen, N. B.. Methods of analvsis for torsion with variable twist, J. Aeronaut. Sci. (11), 2 (1944) 110--24. 13. Benscoter, S. U., A theory of torsion bending for multicell beams, J. Appl. Mech.. 2 !, Tr, ASME (1) (i954) 25-34. 14. Vlasov. V. Z., Thin-walled Ehtstic Beams. 2nd edn. Israel Program for Scientific Translations, Jerusalem, 1961. 15. Dabrowski, R.. Curved Thin-walled Girders Theory and Analvsi,s. SpringcrVerlag, Berlin/Heidelberg/New York, 1968. 16. Bazant. Z. P.. Non Uniform Torsion of Thin-walled Bar~ of Variable Section. Publications, International Association tk~r Bridges and Structural Engineering. Zurich, 1967. 17. Kollbrunner, C. F. & Basler, K., Torsion in Structures, Springer-Verlag, Berlin-Heidelberg-New York, 1967. 18. Wrighte, R. N., AbdeI-Samad, S. R. & Robinson, A. R., BEF analogy for analysis of box girders, J. Struct. Div. ASCE, 94(ST7) (July 1968) 1719-43. 19. Dalton, D. C. & Richmond, B., Twisting of thin-walled box girders of trapezoidal cross-section, Proc. ICE. 39 (1968) 61-73. 20. Gent, A. R. & Shebini, V. K., Parametric Studv: Report on Torsional Warping, Report to DOE (UK), Imperial College, London, 1972. 2 I. Billington, D. J., Ghavami, K. & Dowling. P. J., Steel Box Girders Parametric Stud.v of Cross-sectional Distortion Due to Eccentric Loading. Engineering Structure Laboratory Report, CE SLIC Report No. BG 16, Imperial College. London, 1972. 22. Khan, A. H. & Tottenham. H., The method of bimoment distribution fi~r the analysis of continuous thin-walled structures subjected to torsion. Pr~c. ICE (2), 63 (1977) 843--63. 23. Von Karman, T. & Chien, W. Z,, Torsion with variable twist. J. ,4erottattt. Sci., 13(10) (Oct. 1946) 503--10. 24. Argyris. J. H. & Dunne, P, C., The general theory of cylindrical and conical tubes under torsion and bending loads. J. Royal Aeronaut. Sot., 51 (1947) 199-269; 757-84; 844-930. 25. Bauchau, O. A., A beam theory for anisotropic materials, J. Appl..~lech., Tr. ASME, 52(2) (1985) 416-22.
BIBLIOGRAPHY Moffat, K. R. & Dowling, P. J., Shear lag in steel girder bridges, Struct, Engineer, 53(10) (1975) 439-48. Moffat, K. R. & Dowling, P. J., British shear lag rules for composite girders, J. Struct. Div. ASCE, Ig4(ST7)July 1978. Reissner, E., Least work solution of shear lag problems, J. Aeronaut. Sci., 8(7) (1941) 284-91. Reissner, E., Analysis of shear lag in box beams by the principle of minimum potential energy, Q. Appl. Math., 4(3) (1946) 268-78.
18¢~
p. Chang. H. Hijazi
Report, CESLIC Rept. No. GB 16, Imperial College, London, 1972. Richard. B.. Twisting of thin-walled box girders, lCE Proc., 33 (April 1966)
659-75.
A P P E N D I X : SECTION P R O P E R T I E S
Ayv = f t ---:---" ds
(AI) (A2) (A3) (A4)
Ivy = f t(y - y,)2 cls
(A5)
/:: = ~ t(z - z.)2 ds
(A6)
Iv: = f t ( y - Y , ) ( z - z , ) d s
(A7)
4A 2 -7--~ds
(A8)
J~ = f tU~t(s)ds
(A9)
I, = f tU,(s) Ui(s)ds
(AIO)
S = ~tds
(All)
General Analysis of Asymmetric Thin-Walled Members
P. Chang & H. Hijazi Department of Civil Engineering,Universityof Maryland,CollegePark. MD 20742, USA (Received7 June 1987;accepted7 June 1988)
A BSTRA CT This paper presents the closed-form sohaion for the deflection and stresses of an asymmetric thin-walled member. The method is based on the assumption that the cross-section can deform out of plane when warping and shear-lag effects are significant. The out-of-plane deformation is represented by a linear warping function plus a truncated series of complete eigenfunctions. The differential equations are derived by using the principle of Minimum Potential Energy and solved by a symbolic manipulator. An example is used to illustrate the application of the method and the use of the closed-form solutions obtained. Results of the example are compared to the results of a finite element analysis and other approximate models. The comparison indicates that the maximum error of the proposed method is within O.1% of the value obtained by the finite element method in lateral displacement and 0.2% in axial displacement.
NOTATION Ayy, etc. bc
Property of the area defined in the Appendix Denotes solution without eigenwarping correction d, Constant consisting of eigenwarp constants and load E Modulus of elasticity Eigenwarp intensity function F,(x) Shear modulus G Constant consisting of eigenwarp function and linear warp, function defined in eqn (13) 159 Thin-Walled Structures 0263-8231/89/$03.50 (~ 1989 Elsevier Science Publishers Ltd, England. Printed in GreatBritain
160 [p
!." etc. J~,
g,
Keg k l , k~
k mx El
P.v, P: Fk S
S t U, V, w
U,(s) g,(s) Ux(x) Vi. Wi, Zi ot
t3 Ex
4, 77"
Y ys F~
o (x) o (x)
P. Chan¢ H. Hi/azi
Property of area defined in the Appendix Moment of inertia defined in the Appendix Warping constant Constant consisting of derivative of the linear warp function. defined in eqn (14) Constant consisting of derivative of the linear warp function. defined in eqn (15) Constants defined in eqn (24) Denotes shear center Moment about the longitudinal axis Denotes centroidal quantities Load in the y and z directions Normal distance from a point on the member to the centroid Denotes arc length Cross-sectional area of thin-walled member Thickness of thin-walled member Displacements in the x, y and z directions Eigenwarp function Warping function (linear) The axial deformation at the centroid of the section Unknown constant multiplier to eigenwarp intensity function Constant consisting of Art, etc. Constant consisting of lyy, etc. Axial strain Constant consisting of eigenwarp function defined in property 5 on page 5 Sectional area of the cross-section Potential energy Constant defined in eqn (24) Shear strain Combination of unknown constant defined in eqn (7) Rotation with respect to tile y axis Rotation with respect to the z axis Warp intensity function
1 INTRODUCTION In the past 40 years, the concept of tube structures has been widely used in tall buildings. The analysis of these buildings has been made possible with the advent of high-speed computers. However, the analysis of these
General analysis of asymmetric thin-walled members"
16 l
structures is time consuming and costly; an alternative method of analysis is desired, especially for the preliminary stage of the design. A large class of these alternative methods consists of homogenization of the actual structure so that analysis can be performed on the homogenized continuum. The analysis of this continuum must be as accurate as possible; otherwise important behaviors of the structure may be overlooked. The tube buildings resemble thin-walled members. Unfortunately, an accurate analysis of asymmetric thin-walled members is not presently available. Thin-walled members are defined as members in which the thickness is small compared to the other cross-sectional dimensions. The cross-sectional dimensions are in turn often small compared to the length of the member. The analysis of thin-walled members requires special attention because the in-plane shear stress and strain of these members can be relatively large compared to those in beams with solid cross-sections. The in-plane shear stress and strain induce two types of longitudinal deformation called shear lag and warping. Shear lag deformation refers to the differential longitudinal displacement resulting from shear deformation of the wall of the member, and warping deformation refers to the out-of-plane displacement due to torsion. When these deformations are significant, conventional beam analysis, which assumes no out-of-plane deformation, is inadequate. Thin-walled members have been studied extensively in the past 40 years. In the first attempts to account for shear-lag deformation in thin-walled members, a second-order differential equation was used to describe the intensity of the shear-lag effect. ~'2 These methods did not give accurate results. The use of a parabolic function to describe the longitudinal deformation of the cross-section and the use of the Minimum Potential Energy principle to obtain the magnitude of the differential displacement had some success. 3-5 However, this approach is limited to beams with specific crosssections. The Bar-Simulation method, which discretizes the beam into axial-load carrying bars and shear-force carrying plates, 6"7 also had some success. This method assumes that the longitudinal deformation varies cubically. Such an assumption is, similar to the parabolic assumption, only accurate for specific situations. Another disadvantage of the abovementioned methods is the difficulty in their use because they involve the solution of a set of complicated differential equations. To avoid such difficulty, designers have adopted an effective width method where the beam is replaced by one without shear lag deformation, but with a smaller width. 8-~° To account for the warping deformation, researchers have traditionally used an approximate warping function."-" The main drawback with this approach is that a specific warping function can not describe the longitudinal deformation accurately. In order to correct this problem, a series of ortho-
162
P. (Tmng. H. Hijazi
gonal functions is used to represent the warping deformation, z>-'-' However. a complete series requires a prohibitively large number of terms. The truncated series used by these researchers, among other problems in their formulation, yielded inaccurate results. And further improvement in the analysis of thin-walled members is necessary in the study of tall buildings through the m e t h o d of homogenized continuum. This paper presents a method that includes the out-of-plane deformation due to shear lag as well as warping. The out-of-plane deformation is assumed to be made up of a linear warping function and a truncated complete series. The resulting sixteen-order differential equation is solved by using the symbolic manipulator, MA CSYMA. Closed-form solutions for deflection, rotation and stresses are obtained. 2 MATHEMATICAL FORMULATION The mathematical formulation presented herein is developed for a structure with a cross-sectional shape that remains unchanged after deformation, This assumption has been used by many researchers because thin-walled members with closed cross-sections are relatively rigid in their own plane. In addition, diaphragms are often added to increase the in-plane stiffness in many applications such as tall buildings and bridge girders. The present study is motivated by the representation of a tube structure by a continuum. The floors of these structures serve as a diaphragm which makes the stiffness in the plane of the cross-section very large as assumed. Because the walls are relatively thin, it is reasonable to assume that the member is in a state of plane stress. Hence, the only significant strains are the axial strains, e,, and the shear strain, ys. To get e, and Ys, the assumptions were made that the deflection in the axial direction is made up of rigid-body rotation, rigid-body deflection, and a series of shape functions, with each multiplied by a weighing parameter. Thus, N
u(x.s) = U . ( x ) - ( y - y . ) O : ( x ) + ( z - z . ) O y ( x ) + U,~(x)+ ~ UiFi i
in which Ux(x) = O:(x) = 0y(x) = U,(s) = ~(x) = Ui(s) = F~(x) =
the axial deformation at the centroid of the section. rotation with respect to the z axis, rotation with respect to the y axis, warping function (linear). warp intensity function. eigenwarp function eigenwarp intensity function.
(l)
General analysiso( asymmetric thin-walledmembers ,,t = c e n t r o i d a l
163
quantities.
.s = a r c l e n g t h , u . v, w = d i s p l a c e m e n t s
in t h e x , v a n d - d i r e c t i o n s .
The term Y~ U~ Fi is an infinite series that corrects the longitudinal displacement of the cross-section to include the shear-lag effect and effects of warping not accounted for by the linear warping function. If {U," i = 1. . . . . x } is a complete set. then the term U, ~(x) is not necessary. However, only a finite number of terms in U, can be used: the addition of the term U,/;(x), then, improves the approximation significantly. The lateral deflections and the twist are approximated similarly. But since the cross-sectional shape remains unchanged, the correction terms I,;, W, and Z, are constants. Hence,
,.,(x) = v-(:-:,)o,,+ ~ v, f F~a, = ,,',(.~) = . . + o , - : , , ~ o ~ , +
~'u-(:-:~)0~,
(2)
~• w, f F,a,- = ,,.~+ 0.-.v~)0~,
(3)
O@r)=O,+ ~ Z; f F,ctr
(4)
For small deformation, the axial strain and shear strain are given by:
au Ox "" and y~
• -
~'
=
au as
dv, dv ch" ds
dw, dz dx ds
+ _ _ _ _ + w _ _ + r ~
dO,. dx
(5)
in which rk is the normal distance from a point on the member to the centroid of the cross-section, and 0~, is the rotation about the longitudinal axis. Substituting eqns (1)-(4) into (5) gives:
~*-
dU~ d.r
(y
v, = -0~+
dO. -Y")--~r + ( z - z " )
ds
0,+
+U'-'~r + ~UidFi ct-T
(6)
ds +--ag +---~-r,+ Y_ r,F;
(7)
t
in which F; = dUi/ds + Vidy/ds + W;dz/ds+ Z, rk, and the subscript k denotes the shear center. The total potential energy of the system is:
I IL f [ E t ~ +Gty~]dscK-IL
7"g -~" 5
(p,v+ .
P.w+rn.,Ox)cK
(8)
164
P. Chart,, H. Hiiazi
Definition. Let the subscript "bc" denote the part of the solution without the eigenwarping correction terms, then the axial strain. E~h~and the shear strain. )%~ are defined as % -
d U~ dO: dO, ctr - (y-y°)-8-(v +(:-:')-8-(~ + u,
(9)
and
( dr)d:+( ds Ov+-~V ~-~r+
"Y~c----- -0:+'-~-r
rk+-'~
(ll))
Let Vv. V: and Alk be the shear in the y direction, shear in the z direction, and the sum of torque, respectively:
d V,.
d V.
d_v
P"" ctr
dMk P., and 4r
-mr
(I 1)
Substituting eqns (6), (7) and (9)-(11) into eqn (8) yields:
1 ILl l
[Ete!¢~ + G t %" j d s d . r -
IL[
dF~Ui
dr+ V. ---~+ d dG1 ctr V~, d.r Mk --~.j dG\ 2"1 (12)
- ~L ~ (V~V"+ V:W~+MkZ,)F~dx T h e eigenwarping functions have the following properties: 25
(1) ~ EtUi Uj ds = 80, the eigenfunctions are orthonormal; (2) ~ EtUids = 0, the contour integral of the eigenwarping function is
zero;
(3) ~ E t U i ( y - y n ) d s = O, the contour integral of the eigenwarping functions and (y - y.) is zero; (4) ~ E t G ( z - zn) ds = O, the contour integral of the eigenwarping functions and (z - z.) is zero; (5) l~ ~ EtUiU, ds = ~ GtdU,/dsF, ds
General analysis of cts'ymmetricthin-walled members
165
in which/x, p.j =- :~GtF, Fi ds/~ EtU~ Ui ds, and the approximate functions F, have the following properties: (1) the contour integral of I-'itimes dy/ds is zero: ]; GtFi d.v/ch'ch = O: (2) the contour integral of F, times dz/ds is zero: ~;GtFidz/ch'ds = O: (3) the contour integral of l-'gtimes r~ is zero: J; GtF~ r~ ch" = O.
Definition. Let the section properties and the load terms be defined as shown below:
!,~=- f U,U~Etds
(13)
Krf = f trk dUt ds ds
(14)
Kff=f t ( d--~----L)2ds
(15)
di =- Vi Vy + Wi V: + Zirnk
(16)
Using the properties defined above and the definitions from eqns (13)-(16), the total potential energy of eqn (12) can be written in the form: 7r=Tr~+
.
t fL[EFs('tux L
= S Jo l
2
+ G Ayy -0~+-~(
+ lp(~)"+
+5_ttrF?+l"dx d.r ~-ttTl,,gF,-d,F, dx (17)
\dx /
+
I
I
, ovdo=+j {deTI
dx J + =\ dx J - 2l'= d-T d--T
+A= ¢ + dx/ +2A":
\ clx J J
°y+-~r]\-°=
2K,tr,-~+ktt~2]}clx- yL(p~.v+ P:w+m,O,)d.r
dx I (18)
in which S, J,,, Iyy, It.., Iy:, Ayy, A,, and Ay: are geometric properties of the cross-section defined in the Appendix. Minimizing the total potential energy in eqn (18) with respect to v, w, Oy, 0~, 0,,, U,,. r~ and Fi, the following differential equations are obtained:
166
P. Cha/t~ H. Hil/:t
dx
A,: - 0 : + d):
d_r A>:,. - O : + ~
dr
~-
da"
-'~'v - E l : : dx
+,4::
0,+ dr ]
= -P:(x)
( 19a}
+A,.:
0,.+ dr ]
= - P'(x)
(19b}
":-~7rj + GA,.,. - o . + "
"
+ GA,.. "
dr
0>.+
d,. j = o
- EJ~
+ GKa
~
= 0
(19c)
-0:+--~7~. + G A :: O,. + --.~-r ] = 0 (19d)
+Gay:
(19e)
dx ]
cu-
dv ]
(19f) + GKff ~ -
~
d2Fi
l,i ~
,
+ S-" ~; I,i Fi = 0
(19g)
i
d"- g dr:
, ~, ~; ~ = -d, - l,i + ~ I,~r~
(19h)
with the boundary conditions:
[.( ( d.)]j,=L [(d,,) (
8v A,..
-0.
8W Ayy
--0: + - ' ~
80 Y x
:--~
Oy +--.-~r
.,=o = 0
(20a)
+ Ay: 0>.+---~
,=,) = 0
(2Oh)
dx ] + A::
= 0
(20c)
= 0
(20d)
o
= o
80, tp-~-
Ka(~
=0
(20e)
(20f)
General amdvsis of asymmetric thin-walled members
~
(
EJ~
+
BE` \ d_r
I,~ ~
'~ ~ r ]
=o = 0
x ,, = 0
167
(20g)
(2tlh)
E q u a t i o n s (19) are t o o c o m p l e x to be solved in closed f o r m . T h e s y m b o l i c m a n i p u l a t o r , M A C S Y M A , will b e u s e d instead. T h e v a r i a b l e s are r e w r i t t e n to a f o r m m o r e c o n v e n i e n t for use in M A C S Y M A as follows:
Let X l = V
be the deflection in the v direction
X 2 ~- 14'
be the deflection in the z direction
X 3 =0 v
be the rotation about the v axis due to flexure
X4=O:
be the rotation about the z axis due to flexure
d0y x5 -
X6
be the derivative of 0~ with respect to x
~r d0= d.r
be the derivative of 0= with respect to x
d I,~ x7 = ~ -
dw
.r8 = ~
0:
-0y
be the rotation about the z axis due to shear
be the rotation about the v axis due to shear be the linear warping intensity function
X9 =
d~ " r i ° - dx XII = #x
d0x x t 2 - dx XI3 ~ f i
dE, Xl 4
be the derivative of x9 be the rotation about the longitudinal axis the derivative o f x , with respect to x be the eigenwarping intensity function be the derivative o f x u with respect to x
dx XI5 -~- U x
dUx xl6-- dx
be the rigid-body linear displacement be the derivative ofxt5 with respect to x.
168
P. Chanv, It. Hi/a=i
Then the differential equations in eqns (19) become: d'r7 drs G.4,:--~r + G A = dx G4
dx7
• ..--~+GA~.:
4rs 4r
P.(x)
(2 la)
(21b)
Py(x)
dr. E L , ~ - EI~. - " + GA,.,. X 7 + G A ~...rs = 0 " d.r " " • d.r6
dx~ +
El,: ~
- El:: dx
GA.v: x7 + G A
::
xs = 0
d.~ t X7 + X ,
(21e)
--
X8-
(21f)
-
x5
(21g)
-
x6
(21h)
-
Xl0
dx, X 3
dx 3
4r d.r,
d_r
(21d)
-
dx
dx
(21c)
d_r9
dx
(21i)
-,4 a,m d.r + B ~dx9 + GK~.rlz + ~ l,id i = 0
(21j)
i
d-lf l I
4r
(21k)
- - Xt 2
G l _ dxl2 ) lp---~r + Kaxlo + m , = 0
~
dx
tx~ xl3 = - d i -
dXl5
dx d.~'16
dx
--
-0
Xl 6
(211)
{ a ,o _ ,
,Cai ~ dx
I ~ ; X9
)
(21m)
(21n) (21o)
General analysis of asymmetric thin-walledmembers
169
T h e b o u n d a r y conditions in eqns (20) for a cantilever b e a m with the end x = 0 fixed written in terms of the new variables are: xt(0) = 0
(22a)
x_,(0) = 0
(22b)
x3(0) = 0
(22c)
x4(0) = 0
(22d)
xs(L) = 0
(22e)
x6(L) = 0
(22f)
XT(L) = 0
(22g)
xs(L) = 0
(22h)
xg(0) = 0
(22i)
xt0(L) = 0
(22j)
xlt(0) = 0
(22k)
Ipx12(L) + Krfxg(L) = 0
(221)
xl3(0) = 0
(22m)
xt4(L) = 0
(22n)
x15(0) = 0
(220)
xl6(L ) = 0
(22p)
Using these b o u n d a r y conditions, and the differential e q u a t i o n s in eqns (21), the following general closed-form solution is obtained: 1
v(x) - x t ( x ) - 24EGetfl {txG(I= Py + lrz Pz)x 4 -[4otG2 x7 (0)(Ay~ it. + A . !~ ) + 4aG~ xs(O)( Azz ly~ + At. l=)]x 3 + [12a~SEGx6(O) + 12/3E(Ayz Pz - A.. Py)]x z + [24aBEGxT(O)]x }
(23a)
P. Chart,#. H. Hijazi
170
-I
w(x) --x_,(x) -
24EGo~B {eG(!,: P, + / , , P:)x 4
- [ c~G- xT(0)(A,../,.,. + A,.,. 1,..) + 4e~G'-xs(O)(A:: I,., + A,.: /.,.:)]x 3 + [12oLBEGx5(O) + 12BE(As,, P: - A,: P,)Jx 2 - [24a/3EGxs(0)]xl
(23b)
1
O:.(x) =--x3(x) = 6--~ {[I~.~.P: + It: Py]x 3 - [3Gxd0)(A.,.: t,': + A:: l,'s) + 3Gx7(O)(A.,.,, ly: + A.~: l~r)]x 2 + 6BExd0)x} O:(x) --x.,(x) = ~
1
(23c)
{[Is: P: + I= Pr]x s - [3G.rs(0)(A:: t,': + Ay: I:: )
+ 3GxT(O)(A,,: ly: + Ayy L:)]x 2 + 6/3Ex6(0)x }
-alp {(
=--xg(x) = ~
(23d)
d~(0)
alp K~fmx -
CK/)
dx
(23e)
G-=x,t(x) = ~ Glp
[
~(~:)dsC+m~
GKtt
F/(x)~x13(x)=-i'i~(x)
kl
-~cosh(/zix)+
2/zi
k,x ", /x,
+
Ux(x)---xls(x) = Ux(L)x = 0
( 4)]
l'i+
kt
, /z7
Lx-
4r
(23f)
"
(23g) (23h)
w h e r e the initial c o n d i t i o n s are 1
x d 0 ) = 2--~(ly~ P: -/yy P~) L z
(24a)
Generul analysis of asymmetric t/fin-walled member~
1 x6(O) = .---~-7~ (6:
P,-/,,:
171
P:) L-"
(24b)
XT(0) = ~---~(A:: P,. - A,.: P:) L
(24c)
xs(O) = a-~(Ayy P: - Ay: Py)L
(24d)
-/,.1 L
1
1 .qo(0)
=
,,¢ cosh { ~
\aC/
~ { cosh[ ~
( Kffm., - Ipi~ik,_)
-
(24e)
/zi
-
m x t Xtz(0)
-
- --'~--r
i
(24f)
(24g)
Glp
in w h i c h
=- alp(blp - G K ~ ) a = - ( E J ~ - ~1,,)
(24h) (24i)
i
b =---(GKtf
-
~ la.i2 ,17i )
(24j)
i
k, m (V, fy + W,~: + Z,m~) L
(24k)
k2 = - ( V i i , + W i L + Z i m , )
(241)
a = Ayy A= - A"yz
13- t.
t:~
-
t~z
(24m) (24n)
P. Chang,H. Hiiazi
172
3 EXAMPLES The proposed method will be illustrated using numerical examples. In the first and second examples, a thin-walled member with a rectangular crosssection is used (Fig. 1). The load on the structure in the first example is symmetric in the vertical direction, and the load for the second example is a distributed torque. These loads are used because it can be seen from eqns (2l) that the effects due to bending and torsion are uncoupled, and a structure subjected to both bending and torsion can be modeled by two
40"
t
t f : 0-05625"
J I ~0"
tw= 0"3275"
Fig. !. Cross-sectional dimensions of beam used in examples 1 and 2. loadings, one representing pure bending and the other representing pure torsion. The effect of the combined load is the superposition of the two separate loads. Example 3 illustrates the decomposition of a general load into the pure bending and pure torsion components for a thin-walled beam with asymmetric cross-section.
3.1 Example h Rectangular beam under pure bending load The geometric properties of the cross-section are calculated using the formulas listed in the Appendix, and in Table 1. The eigenwarping functions are the eigenvectors of the cross-section. These eigenvectors can be obtained easily by modeling the cross-section by finite elements. And the weight parameters /a.~, V,, Wi and Zi are calculated by the following formulas:
f (ix]EtUi+Gt-~)
ds = 0
(25a)
fGtF~ d-~ds=O
(25b)
GtFi ~dz ds = 0
(25c)
f GtFir, ds = 0
(25d)
General anah'~i.~ of asymmetric thin-walled members"
173
TABLE 1 G e o m e t r i c Properties of the Structure in E x a m p l e s 1 and 2
L 1,~
100in 5 880 in ~ St~ in 4
I:: /y: E v A,,. A::
0
30
0 ~ ) ksi
0.3 4"5 in 2 13"5 in 2
A.,.:
(}
/p Jt
5 860 in ~ 1 733 in 4 5-153 x 1()~ K - in 4 I)'337 5 in 0-056 25 in
EJ,,. &. tf kff
krf zk Yk
4 187 in 4
- 4 187 in 4 0-0 in 0 . 0 in
The eigenwarping functions and their associated weight parameters are shown in Fig. 2. The linear warping function, U,, is given by the following equation: zk(Y -- Y,,) + yk(z -- Z.) + At
U, = ~ t ( s ) -
(26)
in which ~b, is the sectorial area of the cross-section, z~ and Yk are the shear center coordinates, z, and y, are the centroidal coordinates, and A , = - ~ t4~, ch'/~, t ch'. Using eqn 126), the linear warping function is calculated and shown in Fig. 2. Substituting the geometric properties listed in Table I. and the calculated shape functions and weighting parameters in Fig. 2 into eqn (23), and simplifying, the deflections become: 1 zP: El_. ( - L - ' x + x 2 L - x3/3) + E Ui F/(x) u- 2 --
(27a)
v=0
(27b)
4--
~42
z"-
1 2
-
-
Ox(x) = 0
1 P.
i
(27c) (27d)
-(-L2x+LxZ-f~) o , ( x ) - 2 -Et-=
127e)
0.(x) = 0
127f)
P. Chain,'. H. tttla=t
174
=
.-_-s.
t
~o
.~
~o > t< io
> .,,¢
:<
0
0 t~
0 ii N
>
-.2
~bi ii
ixl
General analysis of asymmetric thin-walledmembers
175
in which L-x
Fi(x) = C, cosh(~/x) + C:sinh(ta.ix) +----r-(P.. Wi) it;
(27g)
L C, - -~(PzW~)
(27h)
C2 = -Ct
1 +/a. i L sinh(/xi L) ~i L cosh(/zi L)
(27i)
The beam was also analysed using the finite element method. The finite element mesh used is shown in Fig. 3. The elements used are twodimensional plane stress elements. The model is constrained such that the cross-sectional shape remains unchanged throughout the analysis as
Fig. 3. Finite element representation of model in Examples 1 and 2.
assumed in the proposed method. The displacements u and w are plotted against the finite element solutions in Figs 4 and 5. Also included in these figures are deflections predicted by other theories. It can be observed that the proposed method gave highly accurate solutions compared to the results of a more rigorous and time-consuming finite element analysis. It can also be observed that the proposed method is superior to the other approximate methods. 3.2 Example 2: Rectangular thin-walled beam under pure torsional load
The beam used in this example has the same cross-section as the beam used in Example 1. However, the load applied is a uniformly distributed torque along the longitudinal axis, rnx. Because the beam's cross-section has not changed, the geometric properties of the cross-section in Table 1 remain
P. Chang, H. Hiiazi
17e~
0.4 i"E
0.3
Z
E
0.2 .to
t~
0.1
0.0 "r 0.(
/,
Finite Element a x o
Proposed Method Bauchau Timoshenko i |
i'
0.2
0.4
| "
0.6 0.8 Length, x/L
i
1.0
.2
Fig. 4. Axial displacement of rectangular thin-walled beam.
°~
I
C
I
i
i
E i
I I
J
t
o
f
0
~
Finite Element ,", ProposedMethod x Bauchau o Timoshenko
,
0.0
0.2
,
0.4
,
0.6 Length,
i
0.8
,
Z
1.0
x/L
Fig. 5. Lateral displacement of rectangular thin-walled beam.
.2
General analysis of asymmetric thin-w,dled members
177
valid. Substituting these properties and the loading rn, into eqn (23). the following deflections are obtained: u = U,(s)ij(.r)+ ~ Ui Fi(x)
(28a)
i
r
=
(28b)
-zO,
w = y0x
(28c)
l[I/
0~(.r) = C - ~
GK,
/~(~) d~: + m.,(Lx - x:/2)
]
(28d)
in which
F,(.r) = CI cosh0zix) + 6". sinh(/l,r) - l,i ~(x) + ¢(.r) = C3 cosh(kx) + C~ sinh(kx) - C3(L - x )
L-x
, m., Z~ tz;
(28e) (28f)
and C1 =
-L ,_ mx Zi L /.ti sinh(/.ti L) /zi b cosh(p.i L)
(28h)
C 3 : L [Kam,,/ip + Ei(Py Vi +m,,Zi)i,i] " ,c , , -GKtt + GK'a/lp + _~ It, 17~
(28i)
C4
(28j)
C2 = - C1
=
I
(28g)
+ p.~ L
- C 3 ( k s i n h ( k L ) - 1/L)
" " - Yi I~ p.i E J . - z, a:,
k cosh (k L )
(28k)
T h e d i s p l a c e m e n t s u and 8, o b t a i n e d from eqns (28) are c o m p a r e d to the d i s p l a c e m e n t s o b t a i n e d using other theories as well as displacements o b t a i n e d f r o m a finite element analysis in Figs 6 and 7. T h e results clearly s h o w that the closed-form solution given in eqns (23) is accurate and superior to the o t h e r a p p r o x i m a t e m e t h o d s used w h e n c o m p a r e d to the results of the finite e l e m e n t analysis.
P. LTnmg, H. [ti]aci
178
5
'
i
I
4 0
E U
--¢ ¢x
2
"
i5
1 x
Finite Element ProposedMethod Bauchau
[] x 0-¢~
0.0
,
,
0.2
1
,
0.4
I
0.6
,
0.8
J
,
1.0
.2
Length, x ' L
Fig. 6. Axial displacement of rectangular thin-walled beam.
0.03
0.02
J
o
0.01 ,,
m
=
0.0
0.2
0.4
0.6
Proposed Method
0.8
1.0
Length, x/L
Fig. 7. Axial rotation of rectangular thin-walled beam.
1.2
General analysis of asymmetric thin-walled member~
179
3.3 Example 3: Trapezoidal thin-walled beam under axisymmetric loading To illustrate the use of the proposed method when the member and the load are asymmetric, a thin-walled member with a trapezoidal cross-section is used. The beam is subjected to two uniformly distributed loads in the v direction, Py~ and P~:" These loads are located at distances eyj and e,,_ from the shear center. The beam dimension and load configuration are shown in Figs 8 and 9. 40" o-
_1
1 14
_1
20" O"
1
Fig. 8. Cross-sectional dimensions of the example structure.
z
l
m
Fig. 9. Actual and equivalent loads on example structure.
The applied load was replaced by an equivalent set of loads with a transverse load and a distributed torsional moment applied along the shear center, as shown in Fig. 9. The geometric properties of the cross-section are calculated using the formulas given in the Appendix, and listed in Table 2. The eigenwarping functions are the eigenvectors of the cross-section modeled as a plane frame, and the weight parameters ~, Vi, W, and Zi are calculated by eqns (25). The eigenwarping functions and their associated weight parameters are shown in Fig. 10. The linear warping function, U,, which is given by eqn (26), is also shown in Fig. 10. Substituting the geometric properties listed in Table 1 and the calculated shape functions and weighing parameters in Fig. 10 into eqns (23), and simplifying, the deflections become:
P. C/tang ft. ktijazl
IN()
1 vP,,
(-L2x+?(2L-'[~3)
It- 2 ~ ,
l"--
1 P~.
L: .r2
2 EI-~.
2
.r 3 L
+Ut(S)~{X)+
r4
3 + -
P'
X-'U'F'(r)~I
(29a)
Lx -
+ GA~, (29b)
w=yO~+ ~ W, f F,(x)ctr
t [p G K , ' O.,(x) = --G-%-
(29c)
~ ¢(~)d~+m,(L.r- x-'/2)]
(29d)
in which ~(.r) = Cl cosh(/z~x) + C,_sinh(/~x) -
I,i ~(x) +
L--X
, ....(m.~Z~ + P, V, ) /2? (29e)
~(.r) = (73 cosh(kx) + C~ sinh(kx) - C
(L - x ) ~ L
(29f)
and -L C, = ----T-_(m~ Z, + Py V~) /z, C: = - Cl
(29g)
1 +/ii L sinh(/zi L) /zi L cosh(/zi L)
(29h)
C3 = L [Katnfllp + vi (Py Vi + m~Zi)Li] -
OK. aK'-./¢ +
+
C~ = -C3(k sinh(kL) - l/L) k k2 w G k , - GkZatlp- ~i ~¢ 12ti ~i2 2
EJ~¢ - - i It i
g, 1
cosh(kL)
(29i)
(29j)
(29k)
T h e b e a m was also analysed using the finite e l e m e n t m e t h o d . T h e finite e l e m e n t m e s h used is shown in Fig. 11. The two-dimensional plane stress e l e m e n t s are used in the finite e l e m e n t model. T h e m o d e l is constrained
General ttnalysis of asymmetric thin-walled members
TABLE 2
Geometric Properties of the Example Structure L E/v,,
El:: El.,:
Itll in 3-87 x 10" K-in 2 4-87 x 10~ K-in-" 0
E v
GAyy GA= GA~: GIp GJ, Gktf J~ z~ )'k
Z : 0
W: 321x10 -4 V=O It1:0 ~2:~.10x10-3
30 000 ksi 0.3 85 548 K 16 317 K
0 5 568 156 K-in-' 4 705 660 K-in-" 518 377 K-in-" 2 571 in 6 - 1-07 in 0.0 in
W:O V = 9"34x10 "5 Z = 9 " 4 2 x 1 0 "(5 It1 : 2 7 2 0 ~ 2 : 1"~3 x l O "2
W: 0 Z :-3.64xi0 "7
Z=O
: 9 . 1 7 x i 0 "6 I t 1 : 5 8 5 5 / 4 2 : 7 3 7 x 1 0 "3
V : 7.80XI0 "5 It I = o5510 ~ 4 2 = 3 " 0 8 x 1 0 " 2
Z : 0
W : -2-g2x 10 "2 V : O Itl=O t2 = 1 ~ 2 ~ x I 0 - 2
W : - 4 . 3 X 1 0 "4 V : O It 1 : 0 ~2 : 2 . 7 7 5 x 1 0 " 2
Ut
Fig. 10. Eigenfunctions and associated weight parameters of cross-section in Fig. 8.
181
P. Chan£, H. Hiiazz
Fig. 11. Finite element representation of model.
0.8
S
A
"o
s.
0.6
p
,
]
I
o X
r0 0 ee
0.4
F
0.2 i
,<
f
0.0 0.0
0.2
0.4
o
0.6 Length,
Finite Element I Proposed Method !
0.8
1.0
1.2
x/L
Fig. 12. Comparison of axial rotation.
such that the cross-sectional shape remains unchanged throughout the analysis as assumed in the proposed method. The displacements u, v, w and x are plotted against the finite element solutions in Figs 12, 13 and 14. Also included in these figures are deflections predicted by other theories. It can be observed that the proposed method gave highly accurate solutions compared to the results of a more rigorous and time-consuming finite element analysis.
4 CONCLUSIONS A closed-form solution of an asymmetric thin-walled member under arbitrary loading was obtained. The method is based on the assumption that
183
General analysis of asymmetric thin-walled members
A r'°B
I
E
!
!
,
Ii
'
,
I
,,
Proposed I .°,~°°°~°,
;
J1
l,)
i5
_.o
J
--I
~ - -
.
i
0 0.0
0.2
0.4
F,n,te E,omen, i
a
0.6 Length,
0.8
.
Method I
1.0
1.2
x/L
Fig. 13. Comparison of lateral displacement.
1
A e" °m
o. O X
E O E¢ )
oB
/
X
/
/ ,l°
Finite Element Proposed Method
i
0 0.0
0.2
0.4
0.6 Length,
I
0.8 x/L
Fig. 14. Comparison of axial displacement.
1 .0
1.2
18,4
P. Chang, H. Htiuzi
both the shear-lag deformation and warping deformation of a thin-walled m e m b e r are significant. These assumptions are approximated using a finite set of eigenwarping functions and a linear warping function. The differential equations are obtained through the principle of Minimum Potential Energy and solved using the symbolic manipulator. M A C S Y M A . T h e m e t h o d contains far less constraints than the existing methods listed in Section 2. Because closed-form solutions are presented, users of the proposed m e t h o d only need to calculate the required geometric properties of the beam being analysed. Numerical examples of a beam with trapezoidal cross-section are presented. The axial displacement, axial rotation and lateral displacement obtained were then compared to the results of a finite element analysis with a relatively fine mesh. The results from the closed-form equations match the finite element results exactly. It should be pointed out that the methods that neglect the warping deformation gave no axial rotation for the example beam. The methods that neglect the shear-lag deformation resulted in smaller deformation in all directions compared to the finite element solution, especially in the axial direction. Methods that assumed symmetrical cross-section or methods that assumed no warping or shear-lag d e f o r m a t i o n could not be used.
REFERENCES 1. Kuhn, P., Some Notes on the Numerical Solution of Shear Lag and Mathematically Related Problems, NACA TN No. 704, 1939. 2. Newton, R. E., Shear lag and torsion bending of four-element box beams, J. Aeronaut. Sci., 12 (October, 1945) 461-7. 3. Hildebrand, F. B. 8: Reissner, E., Least Work Analysis of the Problem of Shear Lag in Box Beams, National Adv. Comm. Aeron., TN No. 893. 1943. 4, Chang, P. C. 8: Foutch, A. M., Static and dynamic modeling and analysis of tube frames, J. Struct. Engng, 110(12) (1984) 2955-75. 5. Coull, A. & Bose. B., Simplified analysis of frame tube structures. ]. Struct. Div. A S C E , STII, 101 (1975)2223-40. 6. Kuhn, P., Stresses in Aircraft and Shell Structures, McGraw-Hill, New York, 1956. 7. Evans, H. & Taharian, A., The prediction of the shear lag effect in box girders, Proc. Inst. Civil Engng, Part 2 (1977) 69-92. 8. Abdel-Syed, G., Effective width of steel deck plate in bridges, J. Struct. Div. A S C E , 95(St7) (July 1969) 1459-74. 9. Moffat, K. R. & Dowling, P. J., Steel Box Girders, Parametric Study on the Shear Lag Phenomenon in Steel Box Girder Bridges, CESLIC Report BG 17. September, 1972, pp. 1123-30. 10. Malcom, D. J. & Redwood, R. G., Shear lag in stiffened box-girder, J. Struct. Div. A S C E , 96(ST7) (1970) 1403-19.
General analysis o( asymmetric thin-walledmembers
185
I1. Goodier, N. J., Torsional and flexural buckling of bars of thin-walled open section under compressive and bending loads. J. Appl. Mech., 913) (Sept. 1942) 103-7. 12. Von Karman. T. & Christensen, N. B.. Methods of analvsis for torsion with variable twist, J. Aeronaut. Sci. (11), 2 (1944) 110--24. 13. Benscoter, S. U., A theory of torsion bending for multicell beams, J. Appl. Mech.. 2 !, Tr, ASME (1) (i954) 25-34. 14. Vlasov. V. Z., Thin-walled Ehtstic Beams. 2nd edn. Israel Program for Scientific Translations, Jerusalem, 1961. 15. Dabrowski, R.. Curved Thin-walled Girders Theory and Analvsi,s. SpringcrVerlag, Berlin/Heidelberg/New York, 1968. 16. Bazant. Z. P.. Non Uniform Torsion of Thin-walled Bar~ of Variable Section. Publications, International Association tk~r Bridges and Structural Engineering. Zurich, 1967. 17. Kollbrunner, C. F. & Basler, K., Torsion in Structures, Springer-Verlag, Berlin-Heidelberg-New York, 1967. 18. Wrighte, R. N., AbdeI-Samad, S. R. & Robinson, A. R., BEF analogy for analysis of box girders, J. Struct. Div. ASCE, 94(ST7) (July 1968) 1719-43. 19. Dalton, D. C. & Richmond, B., Twisting of thin-walled box girders of trapezoidal cross-section, Proc. ICE. 39 (1968) 61-73. 20. Gent, A. R. & Shebini, V. K., Parametric Studv: Report on Torsional Warping, Report to DOE (UK), Imperial College, London, 1972. 2 I. Billington, D. J., Ghavami, K. & Dowling. P. J., Steel Box Girders Parametric Stud.v of Cross-sectional Distortion Due to Eccentric Loading. Engineering Structure Laboratory Report, CE SLIC Report No. BG 16, Imperial College. London, 1972. 22. Khan, A. H. & Tottenham. H., The method of bimoment distribution fi~r the analysis of continuous thin-walled structures subjected to torsion. Pr~c. ICE (2), 63 (1977) 843--63. 23. Von Karman, T. & Chien, W. Z,, Torsion with variable twist. J. ,4erottattt. Sci., 13(10) (Oct. 1946) 503--10. 24. Argyris. J. H. & Dunne, P, C., The general theory of cylindrical and conical tubes under torsion and bending loads. J. Royal Aeronaut. Sot., 51 (1947) 199-269; 757-84; 844-930. 25. Bauchau, O. A., A beam theory for anisotropic materials, J. Appl..~lech., Tr. ASME, 52(2) (1985) 416-22.
BIBLIOGRAPHY Moffat, K. R. & Dowling, P. J., Shear lag in steel girder bridges, Struct, Engineer, 53(10) (1975) 439-48. Moffat, K. R. & Dowling, P. J., British shear lag rules for composite girders, J. Struct. Div. ASCE, Ig4(ST7)July 1978. Reissner, E., Least work solution of shear lag problems, J. Aeronaut. Sci., 8(7) (1941) 284-91. Reissner, E., Analysis of shear lag in box beams by the principle of minimum potential energy, Q. Appl. Math., 4(3) (1946) 268-78.
18¢~
p. Chang. H. Hijazi
Report, CESLIC Rept. No. GB 16, Imperial College, London, 1972. Richard. B.. Twisting of thin-walled box girders, lCE Proc., 33 (April 1966)
659-75.
A P P E N D I X : SECTION P R O P E R T I E S
Ayv = f t ---:---" ds
(AI) (A2) (A3) (A4)
Ivy = f t(y - y,)2 cls
(A5)
/:: = ~ t(z - z.)2 ds
(A6)
Iv: = f t ( y - Y , ) ( z - z , ) d s
(A7)
4A 2 -7--~ds
(A8)
J~ = f tU~t(s)ds
(A9)
I, = f tU,(s) Ui(s)ds
(AIO)
S = ~tds
(All)