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Analysis of flexible frames by energy search
Com~~crr & Strucrures Vol. 32. No. I. pp. 7546. Printed in Great Britain.
ANALYSIS
1989 0
OF FLEXIBLE
FRAMES
BY ENERGY
lY345-7949189 s3.00 + 0.00 1989 Pergamon Press plc
SEARCH
HANY A. EL-GHAZALY~ and GERARD R. MONFORTON~ TDepartment of Civil Engineering, Kuwait University, P.O. Box 5969, 13060 Safat, Kuwait IDepartment of Civil Engineering, University of Windsor, Windsor, Ontario, Canada N9B 3P4 (Received
15 March
1988)
Abstract-A geometrically nonlinear finite element formulation based on a moving Eulerian coordinate system is used to predict the structural response of flexible planar frames. The total potential energy of the system is expressed in terms of the strain inducing displacements and parameters that reflect the coupling between axial (tension or compression) and bending effects. A conjugate gradient energy search procedure is employed to minimize directly the total potential energy of the structural system. The method includes the effects of prestressing and prescribing displacements in addition to abrupt changes in geometric configuration due to buckling of compression members and slackening of tension elements. Numerical examples illustrate and verify the accuracy of the technique presented.
NOTATION
A,,B,,C,,D, A, I
‘1
E F I J k k, I M P? 4 s u, Cl
equations were formulated with respect to a system of fixed local coordinates (Lagrangian) which may be inaccurate for large rotations unless relatively small load increments are considered. Later, Baron and Venkatesan published a paper[2] dealing with the nonlinear analysis of beam-column elements where the undeformed geometry was used in formulating the stiffness equations. A valuable assessment of the solution techniques which are applicable to the geometrically nonlinear behaviour of structures was given by Haisler, Stricklin and Stebbins in [3]. The writers presented a complete review of the incremental stiffness procedure as first proposed by Turner, Dill, Martin and Melosh (41 and concluded with self-correcting initial value formulations [S]. Martin and Carey [6] presented a general theory for the problem of geometric nonlinearity based on an incremental load approach. The theory was applied to various elements, amongst which was the beamcolumn element. The nonlinear term $(du/dx)2 was included in addition to the nonlinear term f(du/dx)2. In 1973, Oran published two consecutive papers [7,8] for the development of consistent tangent stiffness matrices for both planar and space frame elements. The explicit definition of the element tangent stiffness matrix was given [7] with respect to a system of Eulerian (deformed) coordinates. The use of stability and bowing functions was demonstrated to couple the transverse and the axial stiffnesses. No method of analysis was given, but it is understood that the formulations are to be used within the framework of an incremental load procedure. In [8], Oran extended his previous work to account for the analysis of space frames. Yang [9] used an incremental procedure to predict the behaviour of geometrically nonlinear structures. He described the use of a linearized midpoint incremental approach to reduce the errors included in the conventional linear incremental method. Young [lo] recalled the geometric stiffness matrix previously developed by Martin in [l 11, but
integration constants (i = 0, 1,2) cross-sectional area and moment of inertia, respectively ith displacement component modulus of elasticity member force directed along chord line number of elements number of degrees of freedom tension-bending and compression-bending coupling constants, respectively initial member length end moment subscripts indicating member ends deforment chord length displacement components in x, y directions, respectively strain energy end force normal to chord line external work Eulerian coordinates local coordinates reference coordinates angle between reference and local axes strain prestrain potential energy angle of rotation
INTRODUCTION need for dependable geometric nonlinear methods of analysis is evident in the design of flexible lightweight structures. In this type of structure, situations involving geometric nonlinearity, but restricted to linear material behaviour, constitute a significant class of analysis problems. Moreover, the structural design of such structures is often governed by elastic stability requirements. An excellent review to the problem of geometric nonlinearity is given by Mallett and Marcal [l]. The stability critical situation was explained as the mutual degradation of the initial axial and flexural stiffnesses which finally leads to buckling. The equilibrium The
75
76
HANYA. EL-GHAZALY and GERARD R. MONFORTON
instead of using approximate coefficients to reflect the bending-axial interaction, the exact hyperbolic and trigonometric functions were used to obtain accurate expressions for the coefficients. Young showed that the customary assumption that the deflection curve is a cubic is erroneous in the cases where the value of the axial load is large. In [12] Bouchet and Biswas presented a method for the analysis of cantilever structures. The method was restricted to the analysis of cantilevers and stacks where the solution proceeds from the free end without the necessity of computing global elastic and geometric stiffness matrices. The energy search method is another approach to solve the structural analysis problem and was proposed in 1965 by Bogner, Mallett, Minich and Schmit in [13]. In this method, the structural analysis problem is viewed as a mathematical programming problem where the total potential energy of the structure is minimized using optimization techniques to yield the displacement vector which corresponds to the equilibrium position of the structure. The method is applicable to nonlinear as well as linear analyses. In [ 141,a comparison between the analytical solution, proposed in [ 131, and some experimental test results to some model skeletal structures was reported. Mallett [ 151discussed the basis of introducing mathematical programming techniques into the structural analysis problem. Mallett and Berke[l6] employed the energy search formulations, as originally reported in [13], to solve examples covering both trusses and frames. Bogner [17] refined the work for pin ended truss element by incorporating eigenvalues to detect the buckling of compression members. The approach in [17] was extended in [18] to include material nonlinearity. Holzer and Somers [ 191 dealt with the analysis of frame structures using the function minimization technique. In [20] the mathematical programming techniques were extended to the geometric and material analysis of structures. A comprehensive review by Noor in [21] surveyed the capabilities of existing computer programs dealing with solutions of nonlinear structural and solid mechanics problems. Although a large number of programs was cited, only two use a Eulerian reference frame; all others use either a total or updated Lagrangian reference system. In [22] is presented a general formulation for the elastic and inelastic nonlinear analyses of multistory frames. Kassimali [23] employed Eulerian coordinates for the large deformation elastic-plastic analysis of frames. Kam er al. [24] presented a general tangential stiffness matrix for plane frame member. The basic purpose of this study is to develop a consistent formulation and method for the geometric nonlinear analysis of flexible planar frame structures. The sources of geometric nonlinear behaviour include large deflections and rotations, coupling between axial and bending deformations and abrupt changes in geometry under increasing loads.
STRUCTURALELEMENTFORMULATION In the following, the formulation of the geometrically nonlinear analysis of a planar structural frame element is presented for incorporation within an energy search approach. Expressions for the potential energy of the element, as well as its analytic gradient, are obtained since both are required for implementation of the function minimization technique employed subsequently. A Eulerian local coordinate system attached to the deformed element is used in conjunction with a nonlinear strain-displacement relationship which reflects the coupling action between bending and axial deformations. Coordinates and displacements
A typical frame element in the undeformed and deformed positions is shown in Fig. 1. The y, 7 axes represent a reference coordinate system with displacements in the !?, 7 directions denoted by I, C, respectively. The undeformed length of the member is 1 = [(Xq - &)’ + (q - q)*]“2
(1)
where (XP, yP) and (Xq;,,X,J define the coordinates of the p and q ends of the member before deformation. The distance between the ends of the element in the deformed state defines the chord length s given by S = {[(‘& + lzq)- (TP - r.?,J* + [( r, + Lsp)- ( s, + CJ2) ‘I2 (2) where ri,,, $ and r&, Cqare the displacement components at the ends of the member measured in the 1 and P reference directions. The local coordinates & ? define the direction of the element in the underformed position and displacements in the & F directions are denoted by fi,,, I$ and Cqiq, fiqat ends p and q, respectively; correspondingly, rotations of the p and q ends, measured anticlockwise with respect to the 2 axis, are denoted by g,, and 8,. The angle 4 measured from the 8 axis to the chord line of the deformed element is given by
and the chord length may be expressed as s = [(I +
ziI - ziP )* +
(G4 - 0’P )*I”*.
(4)
Upon application of external loads, the element undergoes rigid body and strain inducing displacements. The strain inducing displacements are measured with respect to a local Eulerian coordinate system (x, y) which moves with the element during deformations. The origin of the Eulerian frame of reference is located at the displaced position of end p and the x axis is defined by the chord line joining
Analysis of flexible
+
%I4
71
frames by energy search
Rn
G -4
+
A.D.=After deformation B.D.=Before deformation
Fig. I. Frame discrete element.
the displaced position of the ends of the element. Displacements in the x and y directions are denoted by u and O, respectively, and O,,, 0, represent the rotation of the element ends measured from the x axis. The introduction of the Eulerian coordinate system incorporates several advantages, including the ability to express the strain energy with respect to the deformed configuration, which in turn allows satisfaction of equilibrium in terms of the deformed position of the structure. Since the Eulerian coordinates move with the element during deformation, rigid body movements are separated from strain inducing deformations. The angle 8, (3) defines a rigid body rotation and only three displacement variables, namely uq, t?,,and 8,, are associated with strain inducing deformations. Therefore, the chord length is s=I+u,
(5)
and the displacements u,,, v,,, t+ characterize rigid body motion which are imposed to be zero with respect to the x and y coordinates. In addition, the Eulerian coordinate system may justify the use of a simplified straindisplacement relationship. The strain in the frame element can be generally expressed as +GY)=$+Q
(6)
where 6(x, y) is the strain at any point in the element, e,, is the initial centric prestrain due to prestressing, e,, is the strain of deformation which is given as C/(X,Y) = 4 - Yt?v,v+ 4u;.
(7)
The introduction of the nonlinear term iv:, reflects the coupling between the transverse and the axial stiffnesses. Note that although the derivation of (7) is based on the assumption that rotations are small (sin 6 z 6; cos 0 ‘v l), this assumption is in general satisfied with respect to the Eulerian axes even though actual rotation (gP, & in Fig. 1) may be large. Increasing the number of elements is also beneficial in satisfying the assumptions embodied in (7). Potential energy The strain energy density of a general prestrained frame element, induced due to the application of external loads is given as dU,=
a dc.
(8)
Under the assumption of linear elastic material behaviour and upon integrating (8) to obtain US which is then integrated over the cross-sectional area A results in the following strain energy expression:
The work done by the element end forces is expressible as
w, = P~U,+ q_e, + ale,.
(10)
Under the assumption of small rotation with respect to the x axis, 0, and 6, may be replaced by vXPand v,~,
HANYA. EL-GHAZALY and GERARD R. MONFORTON
78
respectively; therefore (10) reads V, = P,a, + M,a, + M,v, .
(11)
Note that P, is the component of force along the chord at end q (Fig. 1) while M,,, M4 are end moments. The forces P,, V,, and Vq do not contribute to (IO, 11) since the corresponding displacements (up9 up and up) are zero with respect to the Eulerian system. The total potential energy of the frame element in Fig. 1 is thus given as np = u, - w .r. Substituting gives
the expressions
Note that the solution to (17) depends on whether the axial-bending coupling action produces tension (k* > 0), compression (k2 < 0) or zero (k 2 = 0) axial strain. Using (17) the strain energy of the element (9) reduces to
(19) Substitution of the solution to (17) into (19) allows the strain energy to be expressed as
(12)
in (9, 11) into
us_i{kJr[’
(12)
5:
hJ{;lLJ
(20)
where the expression for k,(i,j = 1,2, 3) in (20) are functions of k and s and depend upon the value of k2 being positive, negative or zero. The resulting expressions for v and for the k, of (20) are: -{P,u,
+ M,P, + %a,).
(13) (a) No axial strain (k2 = 0):
Application of the variational principle of stationary potential energy (6n, = 0) results in the following governing differential equations for the element:
v(x) = Aox’+ B,g2 + C,g + Do
(21)
where -$ (c, + u, + fvt) = 0
(14a)
A,=f (e,+ ep); B, = I d 1 t’.WX- ;i;; Kc, + a, -t
Kohl = 0.
- f (28, + e,);
(14b) co = epr; Do = 0
Equation (14a) states that the axial strain along the chord (x-direction) of the element is constant; therefore, the force component F directed along the chord is
k,,=O;
4EI k22=k33=T;
k2) = k32= 7.
(22) (23)
(b) Compressive axial strain (k2 = -k:O): F = EA(c P + u‘I + iv;) = k2EI
(13
v(x) = A, sin k,x + B, cos k,x + C,x + D, (24)
where k is the ‘coupling constant’ given by where k2=;(cp+u,T+jv;).
(16)
A’=&(lk,Q, Substitution
- cos k,s - k,s sin k,s)e,
of (16) into (14b) gives - (1 - cos k,s)e,] (17)
a,~.~,,- k 2a.~.~ = 0.
The transverse displacement, v, can be expressed in terms of k, s and the rotations of the element ends, 8, and O,,,by solving (17) while applying the following boundary conditions (Fig. 1). Endp(x
=0):
End q(x = s):
(25a)
v -0;
v.~=~ZI,
(lga)
v = 0;
u,~= e,,.
(18b)
B’-- -!-k,Q, [(sin k,s -
k,s cos k,s)e, + (k,s - sin k,s)e,]
‘I
= ._!_(1 Q,
cos k,s)(ep + e,)
D, = -B,
(25b)
(25c) (25d)
Analysis of flexible frames by energy search
(264 EI kzz= kX, = 2 k,{(k,s)’ 2Ql
- (k,s)* sin k,s
+ (1 - cos k,s) [2 sin k,s -2k,s
cos k, - (k,s)2 sin k,s]}
k,, = kj2 = $
k,{ -(k,s)‘cos
(26W
k,s + (k,s)2 sin k,s
I -2( 1 - cos k,s)(sin k,s - k,s cos k,s]}
79
Evaluation of coupling constant
When the element experiences axial strain (k* # 0), inspection of (20) in conjunction with (26) or (30) reveals that the strain energy, U,, is a nonlinear function of the coupling constant (k or k,) for particular values of e,, e4 and s = I+ uq. In the energy search approach adopted herein to generate solutions, the variables e,, 0, and ug are predetermined at every stage of the search process and the corresponding value for the coupling constant can be determined from (16). In the case of tensile axial strain (k2 > 0) for example, rearranging (16) gives
(26~)
I
n,=!krQ, = 2 - 2 cos k,s - k,s cos k,s.
(27)
2
Ep- jvr
A
and
Substitution of (28) into (32), integrating and imposing the boundary conditions
(32) both sides
(c) Tensile axial strain (kz > 0):
u(x=O)=O; v(x) = A, sinh kx + B, cash kx + c,x + D2 (28)
u(x=s)=r$=s--
(33)
where gives A2 = L
kQ
[( 1 - cash ks + ks sinh ks)fI,, k2 = + [d, + d2(e; + e:) + 2d3epeq]> 0
- (1 - cash B2 = -
1
kQ
ks)BJ
(34)
(294 where
[(sinh ks - ks cash ks)f?,, d, = 6P+ uq/s
+
”
= L (1 - cash Q D,=
(ks - sinh ks)0,]
(29b)
ks) (0, + 0,)
(294
-B,
k,, =$fk’s
(35a)
d2 = [ - (ks)’ + (ks)2(2 + cash ks) x sinh ks + 2(ks) (1 + cash ks - 2 cash* ks) - 2( 1 - cash ks)
(294
x sinh ks]/(4ksQ2)
(35b)
W-4 d3 = [(ks)3 cash ks - 3(ks)*
k,, = k,, = z2Qz k {(ks)3 - (ks)* sinh ks
x sinhks -6(ks)(l
- (1 - cash ks) [2 sinh ks - 2ks cash ks + (ks)2 sinh ks]} k23 = k32 = j$
+2(1 -cash
+ 2( 1 - cash ks)(sinh ks - ks cash ks)}
(30~)
ks)
x sinh ks]/(4ksQ2).
WI
k ( - (ks)3 cash ks + (ks)* sinh ks
-coshks)
(35c)
Since k appears on both sides of (34) an incremental iterative method is employed to determine its value. A similar procedure for determining k, associated with compressive axial strain gives
and Q =2-2coshks+kssinhks. CAS12;1--F
(31)
-k:
= ; [d, + d2(0f + 0:) + 2d33,8,8,]< 0
(36)
HANY A. EL-GHAZALY and GERARD R. MONFORTON
80
where
In addition, the following equilibrium valid for all cases: d, = cp + uJs
(37a)
V,=
-V,=+(M,+
relations are
M,).
(41)
dz = [(k,s)’ - (k,~)~(2 + cos k,s)sin k,s Analytic gradient of strain energy
+ 2(k,s)(l + cos k,s - 2 COGk,s) - 2( 1 - cos k,s)sin k,s]/(4k,sQ,)
(37b)
The gradient vector of U, with respect to displacements in the Eulerian coordinates of the element is defined as
d, = [ -(k,~)~ cos k,s + 3(k,~)~ sin k,s - 6(k,s)
x (1 - cos k,s) + 2( 1 - cos k,s)sin k,~]/(4k&). (37c) Force-displacement
equations
The relationships between the forces and displacements at the end of an element that is in equilibrium with respect to the deformed geometry can be established by application of the principle of minimum potential energy (8~~ = SU, - 6 I+‘,) to the elemental form of I[,, (13). Note the definitions of the forces shown in the positive directions (Fig. 1): F is directed along the chord line; V,, and V, are perpendicular to the chord line; M,, and M,, are the moments at the ends of the element. The resulting force-displacement equations are: (a) No axial strain (k2 = 0): F=O
(38a)
M, = 7
(20, + 0,)
The components of {VuS} are obtained by application of the principle of minimum potential energy (Vrr,,= VU,, - VW, = 0) to (12) which gives
where P,, = F, M,, and M, are defined in (38), (39) and (40) for k = 0, k2 < 0 and k? > 0, respectively. In order to use an energy search procedure to predict the response of structural systems composed of an assemblage of elements, the gradient of US for each element is required in terms of displacements measured with respect to the & P reference coordinate system. First, the gradient with respect to displacements measured in the local undeformed & I’ coordinates is defined as
(38b)
= -2au au, au5 au, au, aq r {
M, = 7
(0, + 20,).
Using (44) and the displacement (Fig. 1)
M, = -k,El[(sin
uy=s -I;
(39a)
M,, = -k, EZ[(k,s cos k,s - sin k,s)0, + (sin k,s - k,s)OJ/Q,
(39b)
(39c)
PII= (4Oa)
M, = kEI[(ks cash ks - sinh ks)B, + (sinh ks - ks)eJQ
e,=e;-&,
(40b)
-St;.
t:
;;
0
1
0
Sil
6
7;.
s;;,
-t”,
0
0
(45)
(46)
-;I 1
where
(474
M, = kEI[sinh ks - ks)8,
+ (ks cash ks - sinh ks)f?,]/Q.
transformations
-s;; -;; -6,‘
(c) Tensile axial strain (k2 > 0): F=k2EI
e,=&-&
w
while noting the definitions of 8, and s in eqns (3) and (4), respectively, the transformation matrix [T,] is
k,s - k,s)f?,
+ (k,s cos k,s - sin k,s)e,]/Q,.
I
@ bB,
(38~)
(b) Compressive axial strain (k’ = -kiO): F = -k;EI
ati,’ aiy aB,‘a17,’
(40~)
c, = (t&- Q/s*.
Wb)
Analysis of flexible frames by energy search
The gradient of V, with respect to displacements measured in the reference 1, P coordinates is defined as
that minimize follows:
81
the potential
energy function
7~~as
Given n,({D}) Find {D} such that n,({ D }) is minimum (48)
Using (48) and the displacement (Fig. 1) li = lil, -l- 61,;
transformations
I?= -er,.+fil.r
(494
where I, = cos /3 = (Xq - Q/r
(49b)
1,.= sin b = ( Yq- Y&/i gives
r21=
Substituting
1.V -!V
0
!V
L
0
0
0
0
0
00
0
0
10
0
0
o
o
o I I
-!v
0
0
0
0 !v
I.,
0
0
0
00
0
1,
(50)
where
PI = [TJ[T,l=
-St,
- t.l
-St)
t5
0
where I is the number of elements contributing to the total strain energy of the structure. If any element goes out of service during the search process, the contribution of that element to the strain energy is simply neglected. J in (52) is the total number of independent displacements. P, and 0, are thejth load and the corresponding displacement, respectively. The necessary condition for the occurence of a local minimum of z,, at {D} = {D}* is
aD:
(514
= P-IWJ
(52)
agD I*) =o;
(50) into (48) results in WJ
where
(51b)
St, St, 0
j=l,2
)...)
J.
(53)
Since z,, is a highly nonlinear function of the displacement {D}, (53) is a set of J nonlinear equations representing the first derivative of the potential energy function with respect to each of the generalized coordinates {D 1. The most obvious approach to finding the displacements is to solve (53). Unfortunately, the task of solving a large set of nonlinear equations may be very difficult. The function n,, may be so complex, such as that encountered in the present study, that it is virtually impossible to write the equations in closed form. The use of mathematical programming techniques in direct minimization of the potential energy function allows powerful numerical methods of unconstrained minimization to be used [25-281. The method of conjugate gradients known as the Fletcher and Reeves method[27] was used in this study because of its modest storage requirement. In addition, the incorporation of a scaling transformation proposed in [29] effectively improved convergence. Details of the procedure used herein are given in [27,29]. NUMERICAL EVALUATION
and
The following four examples demonstrate the capa-
(51c) bility of the formulation and method of solution.
(514 METHOD OF ANALYSIS
According to the principle of energy, the structural analysis an unconstrained minimization displacement degrees of freedom
stationary potential may be viewed as problem where the (D} assume values
Fixed-fixed
beam
This example investigates the behaviour of a fixedfixed beam under an increasing central vertical load and a uniform axial tensile prestrain. Figure 2 shows the beam dimensions and the necessary data for the analysis. In the experimental analysis, the edges of the beam are welded to two very rigid steel blocks and then a uniform prestrain of approximately
HANYA. EL-GHAZALYand GERARDR. MONFORTON
82
t’_60in.
4
Prestroining
Fig. 2. Schematic drawing of the experimental set-up. (For ail figures: 1 in. = 25.4 mm: 1lb. = 4.45 N; I ksi = 6.9 MN/m2.) 80 ~6 was applied. The ends were tightly bolted to the loading frame, then the external load was applied gradually. The compa~son (Fig. 3) between the experimental results and the analytical solution, using the present method and formulations, shows good agreement. Of special interest is the relative ease of the present formulations in considering the effect of the initial axial prestrains on the behaviour of the structure. The initial prestrain is simply one of the element properties in the input data to the computer program.
I
I
The analysis of the diamond square frame shown in Fig. 4(a) is considered using eight elements. The geometry as well as the section properties are shown in the same figure. Figure 4(b) shows the structure after deformations as the load is increased up to 60 lb. The problem represents a very large deflection case where the ratio of the deflections to the original dimension, in the direction of the displa~ment, reaches a value of 0.45 at a load of 60 lb. The frame was solved using an exact analysis by Jenkins and Seitz in (301. The solution (Fig. 5) demonstrates the reliability of the present method of analysis in handling problems of large deformations and rotations (a rotation of about 30” was reported at a load level of 601b). The same frame was analyzed by Mallett and Berke in [16] also using eight elements and an energy search approach. The solution by Mallett and Berke deviates from the solution obtained using the exact analysis and the deviation increases as the load increases. Shallow circular arch
-.-Llnror
aolutlon
---
E~porlmrntal rraulte
-
Prrent onolysit
/’
Fig. 3. Load~efl~tioo
Diamond square Pame
A,(ln3 curve of the fixed-fixed beam.
The shallow circular aluminium arch shown in Fig. 6(a) was tested experimentally [31] as well as analyzed using the nonlinear finite elements formulations by Marcal [32]. The method of analysis predicts the buckling load of the flexible shallow arch under vertical load. The behaviour of the arch in the unstable region after buckling was also predicted using the present method of analysis. The formulations were also used to follow the behaviour of the arch in the post-buckling zone. The circular arch was modeled as eight straight frame elements as shown in Fig. 6(b). In the portion A-B of the curve in Fig. 7, the load was gradually increased and the corresponding vertical displacement was calculated and plotted. Increasing the load beyond point B is characterized by a significant decrease in the arch stiffness which finally leads to
83
Analysis of flexible frames by energy search
E=lOOOO kri
Sec.m.m
*i
I
T
0.0625 in.
(a)
Horiz. scale = Vertmale
P=60lbs
-
-
P
=601bs
(b)
Fig. 4. (a) Diamond frame before deformation. (b) Diamond frame after deformation. buckling at a load of 33 lb. Increasing the load beyond point B was also characterized by convergence problems and solutions were then obtained by prescribing the displacement of the arch apex and calculating the corresponding load. The circular arch was tested experimentally by Gjelsvik and Bodner and the results are reported in[31]. The dash-dot curve in Fig. 7 represents the finite element solution by Marcal [32]. Marcal introduced an initial displacement matrix in addition to the initial stress matrix.
In [32] the arch was represented by sixteen straight beam-column elements and buckling was predicted at a vertical load of 27.2 lb; behaviour was not predicted in the post-buckling region since the procedure was a load-controlled type of analysis. Guyed tower
The nonlinear analysis of a planar three level guyed tower is presented to demonstrate the response of the tower under increasing wind loads (Fig. 8). The
----*--
0
1.0
2.0
3.0
Mallatt 6 Borks(l6) Exact solution (30) Prasrnt analyalr
4.0
A (III.) Fig. 5. Load-defection
curves for diamond frame.
6.0
6.0
HANY A. EL-GHAZALY and GERARD R. MONFORTON
84
Sec. L-L i
1 f OJ87Sin I T I-- I in.4
I
t-L I-L
I
I f
Rod.=133.11 in.
I-
i‘
4236in
I_
f
I-
,-
4.247in.
a_.
I_
I
4.257in.
i
4.26in.
4.26in.
II-
L ,
4257in.
1
t
-
4.247in.
4236in.
I (b)
Fig. 6. (a) Circufar arch. (b) Idealized arch. tower is
modeled using three beam-column elements, pinned at the base and supported at the three levels by prestressed elastic guys. The mathematical model for a guyed tower is essentially a flexible beamcolumn with elastic supports. The tower shaft is assumed to have infinite shear rigidity to justify neglecting the shear strains. The guys are considered straight elastic cable elements. The dimensions and material properties of the tower shaft are given in Table 1 for each of the three spans. The structural properties of the guys, as well as the initial pretension in each guy, are given in Table 2. The complete structure under the reference load is shown in Fig. 9 where the constant vertical loads represent the weight of the tower shaft as
well as any equipment that may be mounted on the tower. The horizontal loads represent increasing wind Ioads. A load parameter y is used as a multiplier to relate the current intensity to the reference wind intensity. The three curves in Fig. 9 represent the load-deflection curves for the three guyed levels. No solution was obtained at >’ equals 10 and the instability limit is estimated to lie between y equals 9.75 and 10. The load-deflection curves were also plotted in the range from y equals 0 to 1 which represents normal wind conditions. Figure 10 shows the load-deflection curves in this range. The sudden change in the direction of the load-deflection curve of level 1 is attributed to the slackening of the leeward guys at the
36.0
70 kips
32.0 -
-I-
Fmiteelrmwt
---
Expsrimsnta\
(32)
I
I
data 63)
260-
24.0 -
A
0
8
0.4
,
0.6
I
I.2
I
I.6
A(in.) Fig. 7. Load-deflection
curve of the shallow arch.
/-r-3600
Fig. 8. Guyed
In. +38OOin.
tower under reference 1 kip = 4.45 MN.
--/
loads (y = 1.0)
Analysis of flexible frames by energy search Table 1. Mast properties (E = 30,000 ksi)
Span no.
1 2 3
Area (in.?) 60 60 60
Table 2. Guy properties (E = 20,000 ksi)
Initial prestrain (in@.) -1.16 x 1O-4 -0.924 x 1O-4 -0.527 x 1O-4
Moment of inertia (in.‘) 300,000 300,000 300,000
85
Area Level no.
Initial prestrain
(in.‘)
1
in/in.
1.0 1.5 2.0
2 3
0.15 x 10-r 0.13 x 10-Z 0.125 x 1O-2
;pe;$f _w
0
200
400
600
600
(
COO
1200
1400
1600
1600
2000
A(in.) Fig. 9. Load+leflection curve for the guyed tower.
first and second levels which takes place at y = 0.37; the load-deflection curves of the second and the third
(2) The moving
Eulerian coordinates used in the study is effective when considering the effects of large deformations. (3) The method proposed is valid for the analysis of plane frames exhibiting large displacements and can be used to predict buckling loads as well as post-buckling behaviour (including snap-through buckling). When the axial force is relatively large, the exact (4) hyperbolic (k* > 0) and trigonometric (k* < 0) transverse displacement functions used herein give accurate results. (5) The energy search method used in the study proved to be an efficient, powerful and flexible solution procedure for the geometric nonlinear analysis of structures.
levels abruptly change their directions at y = 0.37 due to the slackening of the leeward guys at the first and the second levels and again change their directions at y = 0.51 due to the slackening of the leeward guys at the third level. CONCLUSIONS
The following conclusions present study:
are derived from the
(1) The method and formulations presented ensure that equilibrium is satisfied with respect to the deformed geometry.
0
20
Fig. 10. Load-deflection
40
60
60
-
Guys 2,4
-
Guys 2 84
100
120
66
slacken.d~y=O.SI~
slackrnrd{y=0.37)
140
160
curve for the guyed tower within the reference range.
86
HANY A. EL-GHAZALYand GERARDR. MONFORTON
REFERENCES I. R. H. Mallett and P. V. Marcal, Finite element analysis of nonlinear structures. J. Strucr. Dir. AXE ST9, 2081-2104 (1968). 2. F. Baron and M. S. Venkatesan, Nonlinear formulations of beam-column effects. J. Strucf. Div. ASCE ST4, 1305-1340 (1971). 3. W. E. Haisler, J. A. Stricklin and F. J. Stebbins, Development and evaluation of solution procedures for geometrically nonlinear structural analysis. AIAA J. 10, 264272 (1972). 4. M. J. Turner, E. H. Dill, H. C. Martin and R. J. Melosh, Large deflections of structures subjected to heating and external loads. J. Aerosp. Sci. 27, PP. 97-106. 5. J. A. Stricklin, W. E. Haisler and W. A. Riesemann, Self-correcting initial value formulations in nonlinear structural mechanics. AIAA J. 9, 2066-2067 (1971). 6. H. C. Martin and G. F. Carey. Introduction IO Finite Element Analysis Theory and Applications, pp. 15ll183. McGraw-Hill, New York (1973). 7. C. Oran. Tangent stiffness in plane frames. J. Struct. Div. ASCE ST6, 973-985 (1973). 8. C. Oran, Tangent stiffness in space frames. J. Struct. Dir. ASCE ST6, 98771001 (1973).
9. T. Y. Yang, Matrix displacement solution to elastica problems of beams and frames. Int. J. Solids Struct. 9, 8299842 (1973). IO. D. Young. Stiffness matrix for a beam with an axial force. AIAA J. 11 246247 (1973). I I. H. C. Martin. On the derivation of stiffness matrices for the analysis of large deflection and stability problems Matrix Methods in Structural Mechanics. AFFDL-TR-66-80, Air Force Flight Dynamics Laboratory, Wright-Patterson Air Force Base. Ohio, November, pp. 697-7 I5 (I 966). 12. A. V. Bouchet and M. Biswas, Nonlinear analysis of towers and stocks. J. Strucr. Dir). ASCE ST& 1631l1641 (1977). 13. F. K. Bogner, R. H. Mallett. M. D. Minich and L. A. Schmit. Development and evaluation of energy search methods of nonlinear structural analysis. AFFDLTR-65- I 13. Air Force Flight Dynamics Laboratory, Wright-Patterson Air Force Base, Ohio (1965). 14. F. K. Bogner, R. H. Mallett, M. D. Minich and L. A. Schmit, Development in discrete element finite deflection structural analysis by function minimization. AFFDL-TR-68-126, Air Force Flight Dynamics Laboratory, Wright-Patterson Air Force Base, Ohio, September (I 968). 15. R. H. Mallett and L. A. Schmit, Nonlinear structural analysis by energy search. J. Struct. Dia. ASCE ST3, 221-234 (1967).
16. R. H. Mallett and L. Berke, Automated methods for the finite displacement analysis of three-dimensional truss and frame assemblies. AFFDL-TR-66-102, Air Force Flight Dynamics Laboratory, Wright-Patterson Air Force Base, Ohio (1966). 17. F. K. Bogner, Analysis of tension structures. AFFDLTR-68-150, Air Force Flight Dynamics Laboratory, Wright-Patterson Air Force Base, Ohio (1968). 18. G. R. Monforton and N. M. El-Hakim, Analysis of truss-cable structures. Cornput. Struct. 11, 327-335 (1980). 19. S. M. Holzer and A. E. Somers, Nonlinear/solution process: energy approach. J. Engng Mech. Div. ASCE
EM4, 629647 (1977). 20. D. E. Grierson, A. Franchi, D. DeDonato and L. Corradi, Mathematical programming and nonlinear finite element analysis. Comput. Merh. appl. Mech. Engng 17/18, 497-518 (1979). 21. A. K. Noor, Survey of computer programs for solution
of nonlinear structural and solid mechanics problems. Comput. Struct. 13, 425465
(1981). 22. M. H. El-Zanaty and D. W. Murray, Nonlinear finite element analysis of steel frames. J. Struct. Engng ASCE
109, (1983). 23. A. Kassimali, Large deformation analysis of elasticplastic frames. J. Struct. Engng ASCE 109, ST8 (1983). 24. T. Y. Kam, E. C. Rossow and R. B. Corotis, Inelastic tangential stiffness for 2-D frames, J. Struct. Engng ASCE 109, ST1 I (1983). 25. M. J. Box, A comparison of several current optimization methods, and the use of transformations in constrained problems. Comput. J. 9, 67-77 (1966). 26. R. Fletcher and M. J. D. Powell, A rapidly convergent descent method for minimization. Comput. J. 6, 163-168 (1963). 27. R. Fletcher and C. M. Reeves, Function minimization by conjugate gradient. Comput. J. 7, 149-154 (1964). 28. W. I. Zangwill, Minimizing a function without calculating derivatives. Cornput. J. 10, 293-296 (1967).
29. R. L. Fox and E. L. Stanton, Development in structural analysis by direct energy minimization. AIAA J. 6, 10361042 (1968). 30. J. A. Jenkins and T. B. Seitz, Analysis
of large deflections in structures. Air Force Institute of Technology, Wright-Patterson AFB, Ohio, School of Engineering, Rept. No. GAM Mech., 65 B-8, June (1965). 31. A. Gjelsvik and S. R. Bodner, The energy criterion and snap buckling of arches. J. appl. Mech. Div. ASCE EMS, 87-134 (1962). 32. P. V. Marcal, Effects of initial displacement on problems of large deflection and stability. Tech. Report ARPA E54, Brown University, November (1967).
ANALYSIS
1989 0
OF FLEXIBLE
FRAMES
BY ENERGY
lY345-7949189 s3.00 + 0.00 1989 Pergamon Press plc
SEARCH
HANY A. EL-GHAZALY~ and GERARD R. MONFORTON~ TDepartment of Civil Engineering, Kuwait University, P.O. Box 5969, 13060 Safat, Kuwait IDepartment of Civil Engineering, University of Windsor, Windsor, Ontario, Canada N9B 3P4 (Received
15 March
1988)
Abstract-A geometrically nonlinear finite element formulation based on a moving Eulerian coordinate system is used to predict the structural response of flexible planar frames. The total potential energy of the system is expressed in terms of the strain inducing displacements and parameters that reflect the coupling between axial (tension or compression) and bending effects. A conjugate gradient energy search procedure is employed to minimize directly the total potential energy of the structural system. The method includes the effects of prestressing and prescribing displacements in addition to abrupt changes in geometric configuration due to buckling of compression members and slackening of tension elements. Numerical examples illustrate and verify the accuracy of the technique presented.
NOTATION
A,,B,,C,,D, A, I
‘1
E F I J k k, I M P? 4 s u, Cl
equations were formulated with respect to a system of fixed local coordinates (Lagrangian) which may be inaccurate for large rotations unless relatively small load increments are considered. Later, Baron and Venkatesan published a paper[2] dealing with the nonlinear analysis of beam-column elements where the undeformed geometry was used in formulating the stiffness equations. A valuable assessment of the solution techniques which are applicable to the geometrically nonlinear behaviour of structures was given by Haisler, Stricklin and Stebbins in [3]. The writers presented a complete review of the incremental stiffness procedure as first proposed by Turner, Dill, Martin and Melosh (41 and concluded with self-correcting initial value formulations [S]. Martin and Carey [6] presented a general theory for the problem of geometric nonlinearity based on an incremental load approach. The theory was applied to various elements, amongst which was the beamcolumn element. The nonlinear term $(du/dx)2 was included in addition to the nonlinear term f(du/dx)2. In 1973, Oran published two consecutive papers [7,8] for the development of consistent tangent stiffness matrices for both planar and space frame elements. The explicit definition of the element tangent stiffness matrix was given [7] with respect to a system of Eulerian (deformed) coordinates. The use of stability and bowing functions was demonstrated to couple the transverse and the axial stiffnesses. No method of analysis was given, but it is understood that the formulations are to be used within the framework of an incremental load procedure. In [8], Oran extended his previous work to account for the analysis of space frames. Yang [9] used an incremental procedure to predict the behaviour of geometrically nonlinear structures. He described the use of a linearized midpoint incremental approach to reduce the errors included in the conventional linear incremental method. Young [lo] recalled the geometric stiffness matrix previously developed by Martin in [l 11, but
integration constants (i = 0, 1,2) cross-sectional area and moment of inertia, respectively ith displacement component modulus of elasticity member force directed along chord line number of elements number of degrees of freedom tension-bending and compression-bending coupling constants, respectively initial member length end moment subscripts indicating member ends deforment chord length displacement components in x, y directions, respectively strain energy end force normal to chord line external work Eulerian coordinates local coordinates reference coordinates angle between reference and local axes strain prestrain potential energy angle of rotation
INTRODUCTION need for dependable geometric nonlinear methods of analysis is evident in the design of flexible lightweight structures. In this type of structure, situations involving geometric nonlinearity, but restricted to linear material behaviour, constitute a significant class of analysis problems. Moreover, the structural design of such structures is often governed by elastic stability requirements. An excellent review to the problem of geometric nonlinearity is given by Mallett and Marcal [l]. The stability critical situation was explained as the mutual degradation of the initial axial and flexural stiffnesses which finally leads to buckling. The equilibrium The
75
76
HANYA. EL-GHAZALY and GERARD R. MONFORTON
instead of using approximate coefficients to reflect the bending-axial interaction, the exact hyperbolic and trigonometric functions were used to obtain accurate expressions for the coefficients. Young showed that the customary assumption that the deflection curve is a cubic is erroneous in the cases where the value of the axial load is large. In [12] Bouchet and Biswas presented a method for the analysis of cantilever structures. The method was restricted to the analysis of cantilevers and stacks where the solution proceeds from the free end without the necessity of computing global elastic and geometric stiffness matrices. The energy search method is another approach to solve the structural analysis problem and was proposed in 1965 by Bogner, Mallett, Minich and Schmit in [13]. In this method, the structural analysis problem is viewed as a mathematical programming problem where the total potential energy of the structure is minimized using optimization techniques to yield the displacement vector which corresponds to the equilibrium position of the structure. The method is applicable to nonlinear as well as linear analyses. In [ 141,a comparison between the analytical solution, proposed in [ 131, and some experimental test results to some model skeletal structures was reported. Mallett [ 151discussed the basis of introducing mathematical programming techniques into the structural analysis problem. Mallett and Berke[l6] employed the energy search formulations, as originally reported in [13], to solve examples covering both trusses and frames. Bogner [17] refined the work for pin ended truss element by incorporating eigenvalues to detect the buckling of compression members. The approach in [17] was extended in [18] to include material nonlinearity. Holzer and Somers [ 191 dealt with the analysis of frame structures using the function minimization technique. In [20] the mathematical programming techniques were extended to the geometric and material analysis of structures. A comprehensive review by Noor in [21] surveyed the capabilities of existing computer programs dealing with solutions of nonlinear structural and solid mechanics problems. Although a large number of programs was cited, only two use a Eulerian reference frame; all others use either a total or updated Lagrangian reference system. In [22] is presented a general formulation for the elastic and inelastic nonlinear analyses of multistory frames. Kassimali [23] employed Eulerian coordinates for the large deformation elastic-plastic analysis of frames. Kam er al. [24] presented a general tangential stiffness matrix for plane frame member. The basic purpose of this study is to develop a consistent formulation and method for the geometric nonlinear analysis of flexible planar frame structures. The sources of geometric nonlinear behaviour include large deflections and rotations, coupling between axial and bending deformations and abrupt changes in geometry under increasing loads.
STRUCTURALELEMENTFORMULATION In the following, the formulation of the geometrically nonlinear analysis of a planar structural frame element is presented for incorporation within an energy search approach. Expressions for the potential energy of the element, as well as its analytic gradient, are obtained since both are required for implementation of the function minimization technique employed subsequently. A Eulerian local coordinate system attached to the deformed element is used in conjunction with a nonlinear strain-displacement relationship which reflects the coupling action between bending and axial deformations. Coordinates and displacements
A typical frame element in the undeformed and deformed positions is shown in Fig. 1. The y, 7 axes represent a reference coordinate system with displacements in the !?, 7 directions denoted by I, C, respectively. The undeformed length of the member is 1 = [(Xq - &)’ + (q - q)*]“2
(1)
where (XP, yP) and (Xq;,,X,J define the coordinates of the p and q ends of the member before deformation. The distance between the ends of the element in the deformed state defines the chord length s given by S = {[(‘& + lzq)- (TP - r.?,J* + [( r, + Lsp)- ( s, + CJ2) ‘I2 (2) where ri,,, $ and r&, Cqare the displacement components at the ends of the member measured in the 1 and P reference directions. The local coordinates & ? define the direction of the element in the underformed position and displacements in the & F directions are denoted by fi,,, I$ and Cqiq, fiqat ends p and q, respectively; correspondingly, rotations of the p and q ends, measured anticlockwise with respect to the 2 axis, are denoted by g,, and 8,. The angle 4 measured from the 8 axis to the chord line of the deformed element is given by
and the chord length may be expressed as s = [(I +
ziI - ziP )* +
(G4 - 0’P )*I”*.
(4)
Upon application of external loads, the element undergoes rigid body and strain inducing displacements. The strain inducing displacements are measured with respect to a local Eulerian coordinate system (x, y) which moves with the element during deformations. The origin of the Eulerian frame of reference is located at the displaced position of end p and the x axis is defined by the chord line joining
Analysis of flexible
+
%I4
71
frames by energy search
Rn
G -4
+
A.D.=After deformation B.D.=Before deformation
Fig. I. Frame discrete element.
the displaced position of the ends of the element. Displacements in the x and y directions are denoted by u and O, respectively, and O,,, 0, represent the rotation of the element ends measured from the x axis. The introduction of the Eulerian coordinate system incorporates several advantages, including the ability to express the strain energy with respect to the deformed configuration, which in turn allows satisfaction of equilibrium in terms of the deformed position of the structure. Since the Eulerian coordinates move with the element during deformation, rigid body movements are separated from strain inducing deformations. The angle 8, (3) defines a rigid body rotation and only three displacement variables, namely uq, t?,,and 8,, are associated with strain inducing deformations. Therefore, the chord length is s=I+u,
(5)
and the displacements u,,, v,,, t+ characterize rigid body motion which are imposed to be zero with respect to the x and y coordinates. In addition, the Eulerian coordinate system may justify the use of a simplified straindisplacement relationship. The strain in the frame element can be generally expressed as +GY)=$+Q
(6)
where 6(x, y) is the strain at any point in the element, e,, is the initial centric prestrain due to prestressing, e,, is the strain of deformation which is given as C/(X,Y) = 4 - Yt?v,v+ 4u;.
(7)
The introduction of the nonlinear term iv:, reflects the coupling between the transverse and the axial stiffnesses. Note that although the derivation of (7) is based on the assumption that rotations are small (sin 6 z 6; cos 0 ‘v l), this assumption is in general satisfied with respect to the Eulerian axes even though actual rotation (gP, & in Fig. 1) may be large. Increasing the number of elements is also beneficial in satisfying the assumptions embodied in (7). Potential energy The strain energy density of a general prestrained frame element, induced due to the application of external loads is given as dU,=
a dc.
(8)
Under the assumption of linear elastic material behaviour and upon integrating (8) to obtain US which is then integrated over the cross-sectional area A results in the following strain energy expression:
The work done by the element end forces is expressible as
w, = P~U,+ q_e, + ale,.
(10)
Under the assumption of small rotation with respect to the x axis, 0, and 6, may be replaced by vXPand v,~,
HANYA. EL-GHAZALY and GERARD R. MONFORTON
78
respectively; therefore (10) reads V, = P,a, + M,a, + M,v, .
(11)
Note that P, is the component of force along the chord at end q (Fig. 1) while M,,, M4 are end moments. The forces P,, V,, and Vq do not contribute to (IO, 11) since the corresponding displacements (up9 up and up) are zero with respect to the Eulerian system. The total potential energy of the frame element in Fig. 1 is thus given as np = u, - w .r. Substituting gives
the expressions
Note that the solution to (17) depends on whether the axial-bending coupling action produces tension (k* > 0), compression (k2 < 0) or zero (k 2 = 0) axial strain. Using (17) the strain energy of the element (9) reduces to
(19) Substitution of the solution to (17) into (19) allows the strain energy to be expressed as
(12)
in (9, 11) into
us_i{kJr[’
(12)
5:
hJ{;lLJ
(20)
where the expression for k,(i,j = 1,2, 3) in (20) are functions of k and s and depend upon the value of k2 being positive, negative or zero. The resulting expressions for v and for the k, of (20) are: -{P,u,
+ M,P, + %a,).
(13) (a) No axial strain (k2 = 0):
Application of the variational principle of stationary potential energy (6n, = 0) results in the following governing differential equations for the element:
v(x) = Aox’+ B,g2 + C,g + Do
(21)
where -$ (c, + u, + fvt) = 0
(14a)
A,=f (e,+ ep); B, = I d 1 t’.WX- ;i;; Kc, + a, -t
Kohl = 0.
- f (28, + e,);
(14b) co = epr; Do = 0
Equation (14a) states that the axial strain along the chord (x-direction) of the element is constant; therefore, the force component F directed along the chord is
k,,=O;
4EI k22=k33=T;
k2) = k32= 7.
(22) (23)
(b) Compressive axial strain (k2 = -k:O): F = EA(c P + u‘I + iv;) = k2EI
(13
v(x) = A, sin k,x + B, cos k,x + C,x + D, (24)
where k is the ‘coupling constant’ given by where k2=;(cp+u,T+jv;).
(16)
A’=&(lk,Q, Substitution
- cos k,s - k,s sin k,s)e,
of (16) into (14b) gives - (1 - cos k,s)e,] (17)
a,~.~,,- k 2a.~.~ = 0.
The transverse displacement, v, can be expressed in terms of k, s and the rotations of the element ends, 8, and O,,,by solving (17) while applying the following boundary conditions (Fig. 1). Endp(x
=0):
End q(x = s):
(25a)
v -0;
v.~=~ZI,
(lga)
v = 0;
u,~= e,,.
(18b)
B’-- -!-k,Q, [(sin k,s -
k,s cos k,s)e, + (k,s - sin k,s)e,]
‘I
= ._!_(1 Q,
cos k,s)(ep + e,)
D, = -B,
(25b)
(25c) (25d)
Analysis of flexible frames by energy search
(264 EI kzz= kX, = 2 k,{(k,s)’ 2Ql
- (k,s)* sin k,s
+ (1 - cos k,s) [2 sin k,s -2k,s
cos k, - (k,s)2 sin k,s]}
k,, = kj2 = $
k,{ -(k,s)‘cos
(26W
k,s + (k,s)2 sin k,s
I -2( 1 - cos k,s)(sin k,s - k,s cos k,s]}
79
Evaluation of coupling constant
When the element experiences axial strain (k* # 0), inspection of (20) in conjunction with (26) or (30) reveals that the strain energy, U,, is a nonlinear function of the coupling constant (k or k,) for particular values of e,, e4 and s = I+ uq. In the energy search approach adopted herein to generate solutions, the variables e,, 0, and ug are predetermined at every stage of the search process and the corresponding value for the coupling constant can be determined from (16). In the case of tensile axial strain (k2 > 0) for example, rearranging (16) gives
(26~)
I
n,=!krQ, = 2 - 2 cos k,s - k,s cos k,s.
(27)
2
Ep- jvr
A
and
Substitution of (28) into (32), integrating and imposing the boundary conditions
(32) both sides
(c) Tensile axial strain (kz > 0):
u(x=O)=O; v(x) = A, sinh kx + B, cash kx + c,x + D2 (28)
u(x=s)=r$=s--
(33)
where gives A2 = L
kQ
[( 1 - cash ks + ks sinh ks)fI,, k2 = + [d, + d2(e; + e:) + 2d3epeq]> 0
- (1 - cash B2 = -
1
kQ
ks)BJ
(34)
(294 where
[(sinh ks - ks cash ks)f?,, d, = 6P+ uq/s
+
”
= L (1 - cash Q D,=
(ks - sinh ks)0,]
(29b)
ks) (0, + 0,)
(294
-B,
k,, =$fk’s
(35a)
d2 = [ - (ks)’ + (ks)2(2 + cash ks) x sinh ks + 2(ks) (1 + cash ks - 2 cash* ks) - 2( 1 - cash ks)
(294
x sinh ks]/(4ksQ2)
(35b)
W-4 d3 = [(ks)3 cash ks - 3(ks)*
k,, = k,, = z2Qz k {(ks)3 - (ks)* sinh ks
x sinhks -6(ks)(l
- (1 - cash ks) [2 sinh ks - 2ks cash ks + (ks)2 sinh ks]} k23 = k32 = j$
+2(1 -cash
+ 2( 1 - cash ks)(sinh ks - ks cash ks)}
(30~)
ks)
x sinh ks]/(4ksQ2).
WI
k ( - (ks)3 cash ks + (ks)* sinh ks
-coshks)
(35c)
Since k appears on both sides of (34) an incremental iterative method is employed to determine its value. A similar procedure for determining k, associated with compressive axial strain gives
and Q =2-2coshks+kssinhks. CAS12;1--F
(31)
-k:
= ; [d, + d2(0f + 0:) + 2d33,8,8,]< 0
(36)
HANY A. EL-GHAZALY and GERARD R. MONFORTON
80
where
In addition, the following equilibrium valid for all cases: d, = cp + uJs
(37a)
V,=
-V,=+(M,+
relations are
M,).
(41)
dz = [(k,s)’ - (k,~)~(2 + cos k,s)sin k,s Analytic gradient of strain energy
+ 2(k,s)(l + cos k,s - 2 COGk,s) - 2( 1 - cos k,s)sin k,s]/(4k,sQ,)
(37b)
The gradient vector of U, with respect to displacements in the Eulerian coordinates of the element is defined as
d, = [ -(k,~)~ cos k,s + 3(k,~)~ sin k,s - 6(k,s)
x (1 - cos k,s) + 2( 1 - cos k,s)sin k,~]/(4k&). (37c) Force-displacement
equations
The relationships between the forces and displacements at the end of an element that is in equilibrium with respect to the deformed geometry can be established by application of the principle of minimum potential energy (8~~ = SU, - 6 I+‘,) to the elemental form of I[,, (13). Note the definitions of the forces shown in the positive directions (Fig. 1): F is directed along the chord line; V,, and V, are perpendicular to the chord line; M,, and M,, are the moments at the ends of the element. The resulting force-displacement equations are: (a) No axial strain (k2 = 0): F=O
(38a)
M, = 7
(20, + 0,)
The components of {VuS} are obtained by application of the principle of minimum potential energy (Vrr,,= VU,, - VW, = 0) to (12) which gives
where P,, = F, M,, and M, are defined in (38), (39) and (40) for k = 0, k2 < 0 and k? > 0, respectively. In order to use an energy search procedure to predict the response of structural systems composed of an assemblage of elements, the gradient of US for each element is required in terms of displacements measured with respect to the & P reference coordinate system. First, the gradient with respect to displacements measured in the local undeformed & I’ coordinates is defined as
(38b)
= -2au au, au5 au, au, aq r {
M, = 7
(0, + 20,).
Using (44) and the displacement (Fig. 1)
M, = -k,El[(sin
uy=s -I;
(39a)
M,, = -k, EZ[(k,s cos k,s - sin k,s)0, + (sin k,s - k,s)OJ/Q,
(39b)
(39c)
PII= (4Oa)
M, = kEI[(ks cash ks - sinh ks)B, + (sinh ks - ks)eJQ
e,=e;-&,
(40b)
-St;.
t:
;;
0
1
0
Sil
6
7;.
s;;,
-t”,
0
0
(45)
(46)
-;I 1
where
(474
M, = kEI[sinh ks - ks)8,
+ (ks cash ks - sinh ks)f?,]/Q.
transformations
-s;; -;; -6,‘
(c) Tensile axial strain (k2 > 0): F=k2EI
e,=&-&
w
while noting the definitions of 8, and s in eqns (3) and (4), respectively, the transformation matrix [T,] is
k,s - k,s)f?,
+ (k,s cos k,s - sin k,s)e,]/Q,.
I
@ bB,
(38~)
(b) Compressive axial strain (k’ = -kiO): F = -k;EI
ati,’ aiy aB,‘a17,’
(40~)
c, = (t&- Q/s*.
Wb)
Analysis of flexible frames by energy search
The gradient of V, with respect to displacements measured in the reference 1, P coordinates is defined as
that minimize follows:
81
the potential
energy function
7~~as
Given n,({D}) Find {D} such that n,({ D }) is minimum (48)
Using (48) and the displacement (Fig. 1) li = lil, -l- 61,;
transformations
I?= -er,.+fil.r
(494
where I, = cos /3 = (Xq - Q/r
(49b)
1,.= sin b = ( Yq- Y&/i gives
r21=
Substituting
1.V -!V
0
!V
L
0
0
0
0
0
00
0
0
10
0
0
o
o
o I I
-!v
0
0
0
0 !v
I.,
0
0
0
00
0
1,
(50)
where
PI = [TJ[T,l=
-St,
- t.l
-St)
t5
0
where I is the number of elements contributing to the total strain energy of the structure. If any element goes out of service during the search process, the contribution of that element to the strain energy is simply neglected. J in (52) is the total number of independent displacements. P, and 0, are thejth load and the corresponding displacement, respectively. The necessary condition for the occurence of a local minimum of z,, at {D} = {D}* is
aD:
(514
= P-IWJ
(52)
agD I*) =o;
(50) into (48) results in WJ
where
(51b)
St, St, 0
j=l,2
)...)
J.
(53)
Since z,, is a highly nonlinear function of the displacement {D}, (53) is a set of J nonlinear equations representing the first derivative of the potential energy function with respect to each of the generalized coordinates {D 1. The most obvious approach to finding the displacements is to solve (53). Unfortunately, the task of solving a large set of nonlinear equations may be very difficult. The function n,, may be so complex, such as that encountered in the present study, that it is virtually impossible to write the equations in closed form. The use of mathematical programming techniques in direct minimization of the potential energy function allows powerful numerical methods of unconstrained minimization to be used [25-281. The method of conjugate gradients known as the Fletcher and Reeves method[27] was used in this study because of its modest storage requirement. In addition, the incorporation of a scaling transformation proposed in [29] effectively improved convergence. Details of the procedure used herein are given in [27,29]. NUMERICAL EVALUATION
and
The following four examples demonstrate the capa-
(51c) bility of the formulation and method of solution.
(514 METHOD OF ANALYSIS
According to the principle of energy, the structural analysis an unconstrained minimization displacement degrees of freedom
stationary potential may be viewed as problem where the (D} assume values
Fixed-fixed
beam
This example investigates the behaviour of a fixedfixed beam under an increasing central vertical load and a uniform axial tensile prestrain. Figure 2 shows the beam dimensions and the necessary data for the analysis. In the experimental analysis, the edges of the beam are welded to two very rigid steel blocks and then a uniform prestrain of approximately
HANYA. EL-GHAZALYand GERARDR. MONFORTON
82
t’_60in.
4
Prestroining
Fig. 2. Schematic drawing of the experimental set-up. (For ail figures: 1 in. = 25.4 mm: 1lb. = 4.45 N; I ksi = 6.9 MN/m2.) 80 ~6 was applied. The ends were tightly bolted to the loading frame, then the external load was applied gradually. The compa~son (Fig. 3) between the experimental results and the analytical solution, using the present method and formulations, shows good agreement. Of special interest is the relative ease of the present formulations in considering the effect of the initial axial prestrains on the behaviour of the structure. The initial prestrain is simply one of the element properties in the input data to the computer program.
I
I
The analysis of the diamond square frame shown in Fig. 4(a) is considered using eight elements. The geometry as well as the section properties are shown in the same figure. Figure 4(b) shows the structure after deformations as the load is increased up to 60 lb. The problem represents a very large deflection case where the ratio of the deflections to the original dimension, in the direction of the displa~ment, reaches a value of 0.45 at a load of 60 lb. The frame was solved using an exact analysis by Jenkins and Seitz in (301. The solution (Fig. 5) demonstrates the reliability of the present method of analysis in handling problems of large deformations and rotations (a rotation of about 30” was reported at a load level of 601b). The same frame was analyzed by Mallett and Berke in [16] also using eight elements and an energy search approach. The solution by Mallett and Berke deviates from the solution obtained using the exact analysis and the deviation increases as the load increases. Shallow circular arch
-.-Llnror
aolutlon
---
E~porlmrntal rraulte
-
Prrent onolysit
/’
Fig. 3. Load~efl~tioo
Diamond square Pame
A,(ln3 curve of the fixed-fixed beam.
The shallow circular aluminium arch shown in Fig. 6(a) was tested experimentally [31] as well as analyzed using the nonlinear finite elements formulations by Marcal [32]. The method of analysis predicts the buckling load of the flexible shallow arch under vertical load. The behaviour of the arch in the unstable region after buckling was also predicted using the present method of analysis. The formulations were also used to follow the behaviour of the arch in the post-buckling zone. The circular arch was modeled as eight straight frame elements as shown in Fig. 6(b). In the portion A-B of the curve in Fig. 7, the load was gradually increased and the corresponding vertical displacement was calculated and plotted. Increasing the load beyond point B is characterized by a significant decrease in the arch stiffness which finally leads to
83
Analysis of flexible frames by energy search
E=lOOOO kri
Sec.m.m
*i
I
T
0.0625 in.
(a)
Horiz. scale = Vertmale
P=60lbs
-
-
P
=601bs
(b)
Fig. 4. (a) Diamond frame before deformation. (b) Diamond frame after deformation. buckling at a load of 33 lb. Increasing the load beyond point B was also characterized by convergence problems and solutions were then obtained by prescribing the displacement of the arch apex and calculating the corresponding load. The circular arch was tested experimentally by Gjelsvik and Bodner and the results are reported in[31]. The dash-dot curve in Fig. 7 represents the finite element solution by Marcal [32]. Marcal introduced an initial displacement matrix in addition to the initial stress matrix.
In [32] the arch was represented by sixteen straight beam-column elements and buckling was predicted at a vertical load of 27.2 lb; behaviour was not predicted in the post-buckling region since the procedure was a load-controlled type of analysis. Guyed tower
The nonlinear analysis of a planar three level guyed tower is presented to demonstrate the response of the tower under increasing wind loads (Fig. 8). The
----*--
0
1.0
2.0
3.0
Mallatt 6 Borks(l6) Exact solution (30) Prasrnt analyalr
4.0
A (III.) Fig. 5. Load-defection
curves for diamond frame.
6.0
6.0
HANY A. EL-GHAZALY and GERARD R. MONFORTON
84
Sec. L-L i
1 f OJ87Sin I T I-- I in.4
I
t-L I-L
I
I f
Rod.=133.11 in.
I-
i‘
4236in
I_
f
I-
,-
4.247in.
a_.
I_
I
4.257in.
i
4.26in.
4.26in.
II-
L ,
4257in.
1
t
-
4.247in.
4236in.
I (b)
Fig. 6. (a) Circufar arch. (b) Idealized arch. tower is
modeled using three beam-column elements, pinned at the base and supported at the three levels by prestressed elastic guys. The mathematical model for a guyed tower is essentially a flexible beamcolumn with elastic supports. The tower shaft is assumed to have infinite shear rigidity to justify neglecting the shear strains. The guys are considered straight elastic cable elements. The dimensions and material properties of the tower shaft are given in Table 1 for each of the three spans. The structural properties of the guys, as well as the initial pretension in each guy, are given in Table 2. The complete structure under the reference load is shown in Fig. 9 where the constant vertical loads represent the weight of the tower shaft as
well as any equipment that may be mounted on the tower. The horizontal loads represent increasing wind Ioads. A load parameter y is used as a multiplier to relate the current intensity to the reference wind intensity. The three curves in Fig. 9 represent the load-deflection curves for the three guyed levels. No solution was obtained at >’ equals 10 and the instability limit is estimated to lie between y equals 9.75 and 10. The load-deflection curves were also plotted in the range from y equals 0 to 1 which represents normal wind conditions. Figure 10 shows the load-deflection curves in this range. The sudden change in the direction of the load-deflection curve of level 1 is attributed to the slackening of the leeward guys at the
36.0
70 kips
32.0 -
-I-
Fmiteelrmwt
---
Expsrimsnta\
(32)
I
I
data 63)
260-
24.0 -
A
0
8
0.4
,
0.6
I
I.2
I
I.6
A(in.) Fig. 7. Load-deflection
curve of the shallow arch.
/-r-3600
Fig. 8. Guyed
In. +38OOin.
tower under reference 1 kip = 4.45 MN.
--/
loads (y = 1.0)
Analysis of flexible frames by energy search Table 1. Mast properties (E = 30,000 ksi)
Span no.
1 2 3
Area (in.?) 60 60 60
Table 2. Guy properties (E = 20,000 ksi)
Initial prestrain (in@.) -1.16 x 1O-4 -0.924 x 1O-4 -0.527 x 1O-4
Moment of inertia (in.‘) 300,000 300,000 300,000
85
Area Level no.
Initial prestrain
(in.‘)
1
in/in.
1.0 1.5 2.0
2 3
0.15 x 10-r 0.13 x 10-Z 0.125 x 1O-2
;pe;$f _w
0
200
400
600
600
(
COO
1200
1400
1600
1600
2000
A(in.) Fig. 9. Load+leflection curve for the guyed tower.
first and second levels which takes place at y = 0.37; the load-deflection curves of the second and the third
(2) The moving
Eulerian coordinates used in the study is effective when considering the effects of large deformations. (3) The method proposed is valid for the analysis of plane frames exhibiting large displacements and can be used to predict buckling loads as well as post-buckling behaviour (including snap-through buckling). When the axial force is relatively large, the exact (4) hyperbolic (k* > 0) and trigonometric (k* < 0) transverse displacement functions used herein give accurate results. (5) The energy search method used in the study proved to be an efficient, powerful and flexible solution procedure for the geometric nonlinear analysis of structures.
levels abruptly change their directions at y = 0.37 due to the slackening of the leeward guys at the first and the second levels and again change their directions at y = 0.51 due to the slackening of the leeward guys at the third level. CONCLUSIONS
The following conclusions present study:
are derived from the
(1) The method and formulations presented ensure that equilibrium is satisfied with respect to the deformed geometry.
0
20
Fig. 10. Load-deflection
40
60
60
-
Guys 2,4
-
Guys 2 84
100
120
66
slacken.d~y=O.SI~
slackrnrd{y=0.37)
140
160
curve for the guyed tower within the reference range.
86
HANY A. EL-GHAZALYand GERARDR. MONFORTON
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9. T. Y. Yang, Matrix displacement solution to elastica problems of beams and frames. Int. J. Solids Struct. 9, 8299842 (1973). IO. D. Young. Stiffness matrix for a beam with an axial force. AIAA J. 11 246247 (1973). I I. H. C. Martin. On the derivation of stiffness matrices for the analysis of large deflection and stability problems Matrix Methods in Structural Mechanics. AFFDL-TR-66-80, Air Force Flight Dynamics Laboratory, Wright-Patterson Air Force Base. Ohio, November, pp. 697-7 I5 (I 966). 12. A. V. Bouchet and M. Biswas, Nonlinear analysis of towers and stocks. J. Strucr. Dir). ASCE ST& 1631l1641 (1977). 13. F. K. Bogner, R. H. Mallett. M. D. Minich and L. A. Schmit. Development and evaluation of energy search methods of nonlinear structural analysis. AFFDLTR-65- I 13. Air Force Flight Dynamics Laboratory, Wright-Patterson Air Force Base, Ohio (1965). 14. F. K. Bogner, R. H. Mallett, M. D. Minich and L. A. Schmit, Development in discrete element finite deflection structural analysis by function minimization. AFFDL-TR-68-126, Air Force Flight Dynamics Laboratory, Wright-Patterson Air Force Base, Ohio, September (I 968). 15. R. H. Mallett and L. A. Schmit, Nonlinear structural analysis by energy search. J. Struct. Dia. ASCE ST3, 221-234 (1967).
16. R. H. Mallett and L. Berke, Automated methods for the finite displacement analysis of three-dimensional truss and frame assemblies. AFFDL-TR-66-102, Air Force Flight Dynamics Laboratory, Wright-Patterson Air Force Base, Ohio (1966). 17. F. K. Bogner, Analysis of tension structures. AFFDLTR-68-150, Air Force Flight Dynamics Laboratory, Wright-Patterson Air Force Base, Ohio (1968). 18. G. R. Monforton and N. M. El-Hakim, Analysis of truss-cable structures. Cornput. Struct. 11, 327-335 (1980). 19. S. M. Holzer and A. E. Somers, Nonlinear/solution process: energy approach. J. Engng Mech. Div. ASCE
EM4, 629647 (1977). 20. D. E. Grierson, A. Franchi, D. DeDonato and L. Corradi, Mathematical programming and nonlinear finite element analysis. Comput. Merh. appl. Mech. Engng 17/18, 497-518 (1979). 21. A. K. Noor, Survey of computer programs for solution
of nonlinear structural and solid mechanics problems. Comput. Struct. 13, 425465
(1981). 22. M. H. El-Zanaty and D. W. Murray, Nonlinear finite element analysis of steel frames. J. Struct. Engng ASCE
109, (1983). 23. A. Kassimali, Large deformation analysis of elasticplastic frames. J. Struct. Engng ASCE 109, ST8 (1983). 24. T. Y. Kam, E. C. Rossow and R. B. Corotis, Inelastic tangential stiffness for 2-D frames, J. Struct. Engng ASCE 109, ST1 I (1983). 25. M. J. Box, A comparison of several current optimization methods, and the use of transformations in constrained problems. Comput. J. 9, 67-77 (1966). 26. R. Fletcher and M. J. D. Powell, A rapidly convergent descent method for minimization. Comput. J. 6, 163-168 (1963). 27. R. Fletcher and C. M. Reeves, Function minimization by conjugate gradient. Comput. J. 7, 149-154 (1964). 28. W. I. Zangwill, Minimizing a function without calculating derivatives. Cornput. J. 10, 293-296 (1967).
29. R. L. Fox and E. L. Stanton, Development in structural analysis by direct energy minimization. AIAA J. 6, 10361042 (1968). 30. J. A. Jenkins and T. B. Seitz, Analysis
of large deflections in structures. Air Force Institute of Technology, Wright-Patterson AFB, Ohio, School of Engineering, Rept. No. GAM Mech., 65 B-8, June (1965). 31. A. Gjelsvik and S. R. Bodner, The energy criterion and snap buckling of arches. J. appl. Mech. Div. ASCE EMS, 87-134 (1962). 32. P. V. Marcal, Effects of initial displacement on problems of large deflection and stability. Tech. Report ARPA E54, Brown University, November (1967).