Elastic-plastic analysis of tubular transmission structures

004s7949/a s3.00+0.00 PergamonPESS plc

ELASTIC-PLASTIC ANALYSIS OF TUBULAR TRA~SMrSSION STRUCTURES ROBERT

LEMASTER, VICHIEN NOPRATVARAKORN and TIMOTHY THQSS

Sverdrup Technology, inc., P.O. Box 884, Tullahoma, TN 37388, U.S.A. (Received 6 April 1987)

Abstract-The potential application of elastic-plastic analysis and design methods to tubular steel H-frame structures is discussed. The ma~cmati~l theory is su~uently presented for a large deflection beam element with elastic-plastic material characteristics. The numerically integrated beam element and associated computer program are based on an incremental Newton-Raphson solution methodology. The beam element was used to study the failure characteristics of a 500 kV tubular steel H-frame structure and a substation take-off structure. Computed results were found to agree well with full-scale structure test results.

NOTATION A B : H s T w i

t I4 u x

I

surface area volume Green’s strain shear modulus instantan#us slope of stress vs plastic strain integral Ziegler’s effective stress 2nd Piola stress work modulus of elasticity body force

length surface traction displacement vefocity spatial coordinate

f: P u

center of yield surface hardening parameter equivalent plastic strain integral torsional angle of rotation plastic strain evolution factor density yield surface radius

SY~l~ !?rscored overscored 1 overscored * overscored ’ underscored Operators 6

two-dimensional matrix column matrix quantity evaluated at node point base vector time derivative deviatoric component tensor

d

variation differential

Subscriprs 0 c Lj, k, m, n

centroid covariant tensor component indices

reference configuration

Lefr superscripts j

i

Right superscripts indices for contravariant components of a i,.i,k,m,n (P) Deriviatives nli

i, i ?I 5.II

tensor plastic component covariant partial aflax’ a%iaW

Electrical transmission structures are designed to withstand climatic loads such as wind and ice, erection and conductor stinging loads, and unbalan~d longitudinal loads resulting from a broken conductor or insulator and differential ice or wind. For each of these load conditions, there are different probabilities of occurrence and associated design objectives. There is a high probability that an electric trans~ssion structure will be exposed to varying climatic loads, and the design objective is to provide a reliable structure which will remain in service and will not require excessive maintenance every time a storm passes through. Conversely, there is a relatively low probability that a structure will be exposed to broken conductor or insulator loads, and the design objective is to contain the failures and minimize damage to the line. As stated in [I], ‘While the industry recognizes that failures occur occasionally, the extent of the failure should be limited by providing adequate strength in each structure to resist the lon~tudina1

Gresdc F tl

2

referenced to material con~guration at beginning of load increment referenced to material configuration at end of load increment

referenced to material configuration used

in evaluating the components of the tangent stiffness matrix ~uilib~um iteration index

loads without losing more than three or four structures.’ Tubular steel structures are being used on an increasing basis in overhead electric transmission Iines and are designed to remain elastic under all

603

ROBERTLEMASTERet af.

604

design conditions [2,3]. In general, the size of tubular transmission structures is controlled by large unbalanced longitudinal loads. Following a failure, tubular steel structures behave quite differently from lattice transmission structures. Self-supporting lattice structures generally behave in a brittle manner following a failure and will not resist a signi~cant load in a post-failure state. Conversely, tubular steel transmission structures fail in a ductile manner and can sustain a significant load following a failure. In the light of the inherent ductility manifested by tubular electric transmission structures, it may be beneficial to take advantage of this feature in the design process. This is particularly true for low probability of occurrence loads in which the main objective is to contain the failure within three or four structures. A prerequisite for taking advantage of the ductile failure characteristics of tubular electric transmission structures is to be able to compute the pre- and post-yield response of not only a single structure, but also of several structures representing a segment of an electric transmission line. The required computational capabilities encompass static and dynamic response with associated elastic and elastic-plastic material behavior, small strains with large rotations, and elastic/elastic-plastic stability phenomena. The purpose of this paper is to describe a nonlinear beam element which has been developed and applied to the post-yield analysis of tubular steel electric transmission structures. The element is based on a numerical integration of the governing equations and is applicabIe to compact, cio~d-polygonal sections. Comparisons between computed results and experimental data obtained from two full-scale structure tests conducted at the Transmission Line Mechanical Research Facility (TLMRF) [4] are presented. GOVERNING EQUATIONS

In this section the governing equations are presented and cast in a form compatible with an incremental-Newton-type solution scheme. The Lagrangian statement of the rate of virtual work is [5]

and differential surface area; and A is the current body force vector defined relative to the reference configuration. This equation is an integral statement of equiiib~um, and states that the virtual work performed by the internal stresses must equal the rate of virtual work performed by the externally applied loads. S~~Ii~lly, eqn {I) may be written as SkV,= SlPr,

(2)

or alternatively, srit;-

W,=O,

(3)

where SW*=

P&W

j4

d4

-I-

8% s B0

p,$

y d& +

Y&i&dB,

(5) is the rate of external virtual work. If equilibrium between the surface tractions and internal stresses is not satisfied, eqn (3) will not be equal to zero. Consider a Taylor’s series expansion of eqn (3) about a state which is not in equilibrium, then “(s~,-&s~)=‘(s~,-~s~) + -$ ‘(6 F& - 6 l&) dy -I- O(d&‘,

L

@EdA,+

(6)

where the left superscripted number 1 or 2 denotes the configuration to which the kinetic and kinematic variables and integrals are referenced (note that configuration 1 is not in ~~~b~urn). If co~guration 2 in eqn (6) is chosen to be a configuration which is in equilibrium, then (7)

and eqn (6) becomes

j _j

(4)

is the rate of internal virtual work, and

‘(SlP,--SW,>=0 j&

d4,

j

&Wd&, 8,

0)

where p, is the material density in the reference configuration; $ is the current acceleration vector at a material point; & is a virtual velocity vector at a material point; B, is the material volume in the reference configuration; T is the 2nd Viola stress tensor; S& is the virtual rate of the Green strain tensor &; A, is the surface of the reference ~nfiguration; t, is the current surface traction vector at a point on the surface, defmed relative to the reference configuration unit outward normal vector

The right side of eqn (8) represents the amount by which configuration 1 is out of equilibrium and can be considered an unbalanced force. The left side of eqn (8) represents a system of equations with unknown displacement increments &. The displacement increments dp are estimates of the dispiacements required to bring the unbalanced forces into equilibrium. Equation (8) can be generalized to yield a Newton-Raphson-type solution scheme in which it-

605

Elastioplastic analysis of tubular transmission structures erations are performed to ensure ~~lib~urn;

that is,

where the superscriptj denotes the configuration used to compute the derivatives. The superscript i denotes the number of the equilibrium iteration. The improved displacement after i ~~lib~~ iterations is obtained as

After performing reduces to

the partial differentiation,

eqn (9)

is the effective load vector. In eqn (15), (F,,) is the vector of externally applied node point loads, and {Tf is a vector whose components are the 2nd Piola stress tensor components. The [B], [G] and [D] matrices are defined sym~lically as W1

=

WI{@I

(16)

(Sy:) = [Gl(SP]

(17)

(dT) = W]{dE),

(18)

where (&)r = (68,, Sz&,. . . , Si&) are the virtual node point velocities. The [D] matrix in eqn (18) is a material coefficient matrix which relates the stress increments to strain increments. For an elasticplastic material description, eqns (14) and (15) are integrated numerically. In the following sections, the components of [of, [B], [G] and [S] matrices are presented. ELASTIC-PLASTICCONSTITUTMT EQUATIONS: IDI MATRIX

If the internal body force J, and external surface traction t, are not functions of the dispIacements (i.e. they are conservative), then eqn (11) may be reduced to

(12)

The elastic-plastic constitutive equations used in this investigation are based on the equations presented by Tanaka [6]. The constitutive equations can be used to describe both combined isotropic and Ziegler’s modified kinematic hardening. In the following paragraphs, the general constitutive equations are presented first. Subsequently, the equations are specialized to those used in conjunction with the beam element. The governing thr~-dimensional elastic-plastic constitutive equations [7] are

da = it1 - 0% dTb,trl)’

(20)

da,f?I= HB dvl,

WI

In matrix form, eqn (12) can be written as where dT is the differential of the 2nd Piola stress tensor T; dE is the differential of the Green’s strain tensor &; 6 is the elastic material coefficient tensor; H is the instantaneous slope of the stress versus plastic strain integral curve; v is the equivalent plastic strain integral defined by

where

+

[~lr[Sl~Gl d(%)

1

(14)

is the tangent stiffness matrix, {da) is the vector of incremental node point displacements and rotations, and

where d&(p) is the plastic ~nt~butio~ of the strain differential d&; $ is deviatoric effective stress tensor defined by s=r-g,

(23)

where the overscored prime denotes ‘the deviatoric

606

ROBERT

LEMASTERer al.

components of’, and g is the deviator of g which represents the instantaneous center of the yield surface; a, is the instantaneous radius of the yield surface defined by (24)

?ss)=&Si.

/I is a parameter ranging from zero to one which controls the amount of isotropic and kinematic hardening used to describe the material, for p = 1.0 the equations reduce to isotropic hardening, for /I = 0.0 the equations reduce to Ziegler’s modified kinematic hardening, for 0.0 < B < 1.0 the equations represent combined kinematic and isotropic hardening. The only non-zero components of the stress and strain tensors, T and g, for a beam element are T”, T’*, T” and E,,, E,*, E,,. Therefore, for the special case of a beam element, eqns (19) through (2 1) may be written in matrix form as

w

=

C

PAN

W[Cl

[C]-~~~,(r1)2+{~}T[C]{~}

1 WI

(25)

(26)

da,(ttI= HB h-3

[Cl=

0

G

[ OOG

0

1

(32)

d’=$Hq,(r~)~+{SJT[C]{S}'

(33)

where

The following steps are used to integrate the constitutive equations at each element integration point: (1) compute the new estimate for the radius of the yield surface using eqns (32) and (33) in combination with eqn (27); (2) compute the new estimate for the stress components using eqn (25); (3) compute the new estimate for the stress coordinates defining the center of the yield surface using eqn (26); (4) compute a new estimate a(q) for the radius of the yield surface using the results from steps (2) and (3) in conjunction with eqn (24); (5) determine the ratio of a,(q) computed in step (1) and a(q) computed in step (4); that is, ratio = g

;

(6) scale the stress components computed in steps (2) and (3) by the ratio in step (5). (28)

Y = modulus of elasticity G = shear modulus

{dZ-}r= {dZ-“, dT12, dT’?}

(29)

{dEjr = {dE,,, 2dE,,, 2dE,,}

(30)

(31)

The components

{dE”“} = dA{S},

(27)

where YOO

computed using the equation

of the plastic strain increments are

In step (l), dn is computed using a constant value for H. The value at the beginning of the interval or at the point of transition from elastic to elasticplastic response during the interval is used. However, the value of H is updated during the integration of eqn (27). This technique provides consistent and close approximations to the yield surface radius for curved stress-strain curves. The stress and stress coordinates for the center of the yield surface computed in steps (2) and (3) are tangent approximations. To ensure that the stresses lie on the yield surface, they are scaled radially back to the yield surface according to steps (4) through (6). This technique is essentially a tangent approximation with radial correction (predictor-corrector) as discussed in [8].

TROID OF CROSS-SECTION

Fig. 1. Typical cross-section.

Elastic-plastic analysis of tubular transmission structures

The components of the virtual rate of Green’s strain may be found from the expression

INCREMENTAL STRAIN-DISPLACEMENT EQUATIONS: lBl MATRIX

An arbitrary cross-section at any point along the length of an element is shown in Fig. 1 along with a reference coordinate system with base vectors &, & and &. The kinematic equations used to define the deformation at any point on the cross-section are 2

I1 -UC-x

u

u2=

uf-

2 u,,,-x

3 u,,,

6$=;{6V”]ix”]j+

Wb)

u’=u;+x%,.

(34c)

The subscript c indicates that the subscripted variable is on the axis of the element which passes through the centroid of the cross-section. These equations do not account for warping of the cross-section in torsion, or for lateral shear deformation. These two phenomena are generally unimportant in tubular electric transmission structures. The deformation of the centroidal axis at any point along the length of the beam is related to the node point displacement through the interpolation equations [9].

Wm]jxm]i}.

(37)

For a Cartesian coordinate system, the deformation gradient is xyj=

(344

x’tl,

607

ST +

UI;,

(38)

where Sy is the Kronecker delta function, Substituting eqn (38) into eqn (37) yields the following relationship for the virtual rate of Green’s strain:

(39)

s~~==fIsv,,i+6~i,j+6~“,jU:j+6~=,juI”i}.

-initial displacement contribution

small deformation contribution

For an updated Lagrangian analysis technique [5], the initial displacement contribution is zero, and eqn (39) reduces to

Pa) For a beam element, the only non-zero virtual rate of 8, =fsn” +f6c’O

and Sl?,, . Wb) Green’s strain components are 68,, , CL??,,

24;=f,ti2 +f2ziS +f3c6 +f4ii12

(35c)

Therefore W-4

u;=f,E’

+f2ii9 +f31i5+f41s”,

(354 (41b)

where 26&, = SV,,, + SV,,,. f+l+*($3(;)

By substituting eqns (34), (35) and (36) into eqns (4l), a matrix expression relating the virtual rate of Green’s strain to the virtual node point velocities is obtained:

h=3($-2($ f

(xY ’ - (I)2

(4lc)

W4

2W 7+x’

{Sk} = [B](6P}.

(36d)

(42)

The following expression relating the incremental Green’s strain components to the incremental node point displacements is obtained in a manner similar to eqn (42):

(36e) {dE} = [B]{dC}. (36f)

f 5. I [El=

[ 00

-2fi 0

I

-x%. II

0 0

0 X2h.I -x3&.,

x%.,,

0 0

-x%.,,

0 0

(43)

The [B] matrix in eqns (42) and (43) is

&I

0 0

-di,,

0 0

-X%.1,

0 0

0

al.11

-X2%.11

a.., -x3&1

0 0

0 0.

I WI

ROBERT LEMASTER er al.

608 INITIAL STRRSS STIFFNESS MATRIX: ICI AND 1st MATRICES

INTEGRATION OF THE ELEMENT EQUATIONS

The initial stress contribution to the tangential stiffness matrix comes from the term

The integrals contained in eqns (14) and (15) are evaluated numerically using a Newton-Cotes algorithm, which can be written as F(x], x2, x3) dB,

in eqn (12). The components of the increment in the virtual rate of Green’s strain can be obtained from the expression d(6~~)=f(6V”Jidu”lj+6VmI,du”l,).

(46) where Wi, Wj, W, are the weights applied to values of the function F evaluated at the sample points x’, x2, x3. This integration method requires the evaluation of the [B], [D], [G] and [S] matrices at each sample point in the element. Typical sample point locations are shown in Fig. 2. The actual number of sample points and their locations depends on the number of sides making up the polygonal crosssection.

Using eqn (46), eqn (45) may be written as

=

~(6V”IiT”du”lj+6V”ljT”dumIi)d(~B,) I

= {6V}T

[GIT[sl[Gld(kB,){d~},

(47) 500 kV H-FRAME STRUCIWRE

where the [G] matrix is defined by {6Y’iI =

[Gl{6VII

(48)

and [S] is a matrix of stress terms required to satisfy (49)

{Sp}T[G]T[S][G]{dri} = 6V”],T”du”lj. The [G] and [S] matrices obtained (39, (36), (48) and (49) are: f 5. I

[G] =

from eqns (34),

The element was used to analyze the full-scale 500-kV transmission structure shown in Fig. 3. The beam element representation of the structure is shown in Fig. 4. The purpose of the analysis was to determine whether a numerically integrated elasticplastic beam element could compute the location and model of a failure which occurred while the structure was being tested at the TLMRF. The subject failure occurred at the 90% load level

-Xtr,,,,

0

fl,,

0

0

0

0

-f,. 0

0

0

0

0

0

0

0

0

I

0

0

0

0

0

0

T”

0

0

T12

0

0

T13

0

0

0

T”

0

0

T’2

0

0

T13

0

0

0

T”

0

0

T’*

0

0

T13

T’*

0

0

0

0

0

0

0

0

0

T12

0

0

0

0

0

0

0

0

0

T12

0

0

0

0

0

0

T'-'

0

0

0

0

0

0

0

0

0

T13

0

0

0

0

0

0

0

0

0

T13

0

0

0

0

0

0

(51)

Elastic-plastic

\

analysis of tubular transmission structures

0

MATERIAL

*

ELASTIC/ELASTIC-PLASTIC

609

NONLINEARITIES

NUMERICAL INTEGRA~ON

BUCKLING

POINTS

Fig. 2. Compact-ciosed-section-tapered beam element.

while the structure was exposed to a high transverse wind and unbalanced outside conductor load combination. The faiiure shown in Fig. 5 occurred in an x-brace member loaded in combined bending and compression. The stress-strain curve of the material obtained from a sample taken from the failed x-brace member is shown in Fig. 6. Note that although the 0.2% off-set yield stress value for the material is approximately 411 MPa (60 ksi), plastic deformation actually begins at approximately 240 MPa (35 ksi). The element did compute an elastic-plastic instability in the x-brace at the same load level and at the location where the x-brace failed during the test. However, the program also indicated that the main compression shaft-which was loaded in combined bending, compression, and torsion-would buckle at a 70% load level. This phenomena was not observed during the test, and is currently under investigation.

SU~TA~ON

TAKEOFF STRUCTURE

The test structure was a 115-kv substation take-off structure as shown in Fig. 7. The member shapes were hollow rectangular sections assembled from ASTM A36 plates welded together. The structure was exposed to a complex combination of transverse, longitudinal, and vertical loads. The 100% level associated with these loads was dictated by current design practice and was sufficient to cause the extreme fibers in the main cross member to just reach the material yield stress. The loads were increased simultaneously from the 0 to 100% levels with hold points at 50, 75, 90 and 95%. From the 100% level, all loads except the ground wire attachment points were increased simultaneously from the 100 to 200% load levels in 10% increments. The ground wire attachment point C.&S. *8,$---D

loads were held constant at the 100% load level. The structure remained intact and did not experience a catastropic failure during the loading sequence. Eighty-eight electric resistance strain gages were placed at strategic locations on the structure. Each gage was arranged to measure axial and flexure strains and utilized a separate Wheatstone bridge circuit. Strain versus percent load plots were obtained for each measurement location. The response of the structure was computed using the geometric and material nonlinear beam element. Figure 8 shows the geometry and location of the numerical integration points associated with the dement. During the analysis, the loads were applied to the structure in 10% increments, with an average of four equilibrium iterations performed following each increment. Execution time on a VAX 11/780 computer was approximately one CPU hour. An evaluation of the elastic-plastic analysis results indicated that the most significant yielding of the material occurred at the mid-span of the cross-beam. This was confirmed by the test results which indicate the development of a plastic hinge at the mid-span of the cross-beam. In Figs 9 through 14, six pairs of analysis and test results are presented for the mid-span of the crossbeam. The first plot in each pair shows the computed stress versus strain curve at the integration point closest to a strain gage installation. This plot shows the degree of yielding occurring at that point in the beam. In general, the yielding is very moderate and does not exceed the 0.2% offset strain for the material. The second plot in each pair shows both the computed and measured strains as a function of percent load. These figures show that the computed elastic-plastic strain growth followed the test

610

ROBERT

LEMASTER et al.

Fig. 3. 500 kV H-frame structure. data fairly well-the two exceptions are shown in Figs 12(b) and 14(b). Figure 1l(b) demonstrates a significant amount of strain growth at each of the test hold points. This strain growth is attributed to creep in the test structure. The slope of the test strain versus percent load curve between the hold points agrees with the computed slope, which indicates that the elastic-plastic response between the hold points was being computed accurately. No creep mechanism was accounted for in the analysis. The structural analysis terminated while attempting to achieve an equilibrium configuration at a load level of 190%. The software termination

indicated that the structure, as modeled, was unstable in the 190% load range. However, the test structure was capable of carrying the 200% level loads. The results indicated that the structure had developed a plastic hinge at the mid-span of the beam. After the plastic hinge was fully developed, the beam began to carry the loads in tension instead of bending. The transition between the bending and tension load carrying mechanisms is the point at which the structural software fails to converge. An adaptive numerical algorithm which determines the load application increment based on solution characteristics would have facilitated this type of computation and is currently being implemented.

Elastic-plastic analysis of tubular transmission structures

611

Fig. 4. H-frame structural model.

SUMMARYAND CONCLUSIONS

In general, the use of a numerically integrated beam element to investigate the failure mechanisms of full-scale tubular structures under complex loading was quite successful. The analyses performed using the element described in this paper provided an improved understanding of the post-yield strength of the respective structures. The computational resource requirements were relatively insignificant when compared to the cost of constructing and testing a prototype structure. The analysis of a substation take-off structure indicated that the structure could react without significant damage, approximately twice as much load as current design practice permits. Comparisons between computed elastic-plastic strain increments

agreed well with measured values. However, creep during test hold periods caused the total strain (elastic-plastic creep) to exceed the computed values. The incremental-Newton-type solution method could not compute the response of the structure during the transition between a bending load reaction mechanism and axial force reaction mechanism. Solution methods exist which should remove this limitation [lo-121. The failure load level and failure mode of an x-brace member loaded in combined bending and compression was accurately computed for a 500 kv H-frame structure. The element did compute a combined compression, torsion, and bending instability in one of the H-frame vertical shafts. This failure mode was not observed during test and is currently under investigation.

612

ROEIERT LEMASTER et al.

Fig. 5. Buckled x-brace member.

5oo r-r-i--

----I-0 0.000

I

0.002 6

0.004

0.006

I 0.006

(m/m) Fig. 6. Stress-strain curve for x-brace material.

!

I

0.010

0.012

Elastic-plastic analysis of’tubular transmission structures

-.I

it-4

1.22 m jr.0”)

1.22 n

(4‘~S=

LONG~TUO~NAL

I/

TRANSVLRSE

Fig. I. Sub-station take-off structure.

Fig. 8. Geometry and numerical integration points associated with nonlinear beam element.

613

ROBERTLEMASTERer

614

al.

0

-50

-100

1 -150

I

E

-200

-250

-0.175

-0.150

-0.125

-0.100

-0.075

-0.025

-0.000

(10-2)

STRAIN

Fig. 9(a). Computed stress-strain

Xl

-0.050

response for location Bl.

o-2

0.00

-

TEST

+

ANALYSIS

-0.05

-0.16

-0.20

0

50

100

150

PERCENT LOAD

Fig. 9(b). Comparison of computed and measured strain for location BI.

200

Elastic-plastic

analysis of tubular transmission structures

615

0

-250 -0.20

-0.15

-o.,o

-0.05

STRAIN

Fig. 10(a). Computed stress-strain

response for location B2.

x10-2 0.00

-0.05

-0.10

2 a z *

-0.15

-0.20

-0.25 0

100

PERCENT

-0.00

(10-2)

LOAD

Fig. 10(b). Comparison of measured and computed strain for location B2.

ROBERTLEMASTERet al.

616

(103

STRAIN

Fig. I l(a). Computed stress-strain

response for location B4.

-0.05

0

50

150

100

PERCENT

LOAD

Fig. II(b). Comparison of measured and computed strain for location B4.

1

200

Elastic-plastic

analysis of tubular transmission structures

617

260

200

f

160

scn s E u)

100

50

0

I,,,

0.025

0.000

,,I,

0.050

,I,,

0.075

),I,

0.100

0.125

,,I(

0.150

0.175

(103

STRAIN

Fig. 12(a). Computed stress-strain

I

0

I

I

I

I

1

I

response for location BS.

I

50

I

100

PERCENT

I

I

-

TEST

+

ANALVSIS

I

I

,

I

150

LOAD

Fig. 12(b). Comparison of measured and computed strain for location BS.

I

618

ROBERTLEMAWERet al.

200

K

150

3 v) E a

100

50

0

I 0.00

I

I

I

I

I

,

I

0.05

I 0.10

1

I

1

I

1

1

0.15

. 0.20

(lo-q

STRAIN

Fig. 13(a). Computed stress-strain

I

response for location B6.

__

TEST

+

ANALYSIS

-0.05 0

50

100

PERCENT

150

LOAD

Fig. 13(b). Comparison of measured and computed strain for location B6.

200

Elastic-plastic

-0.150

analysis of tubular transmission structures

-0.125

-0.100

-0.075

-0.050

619

-0.025

0.000

(103

STRAIN

Fig. 14(a). Computed stress-strain

response for location B8.

x10-2 0.00

-

TEST

+

ANALYSIS

-0.05 E

-0.10

l-0

50

100

150

PERCENT LOAD

Fig. 14(b). Comparison of measured and computed strain for location B8.

200

ROBERTLEMASTER er al.

620

Acknowledgements-The work reported in this paper was supported by the Electric Power Research Institute under contract RP2016-03. The data from the 500-kv H-frame structure were obtained during a co-sponsored research test between Valmont Industries, Inc., Valley, Nebraska, and EPRI. The data from the sub-station take-off structure were obtained during a co-sponsored research test between the Western Area Power Authority, Denver, Colorado, and EPRI. In addition to the financial support provided by these organizations, the efforts of the TLMRF operations staff played an important role in the accomplishment of this work. REFERENCES

Guidelines for transmission line structural loading, p. 117. Prepared by the Committee on Electrical Transmission Structures of the Committee on Analysis and Design of Structures of the Structural Division of the American Society of Civil Engineers. American Society of Civil Engineers, New York (1984). Design of steel transmission pole structures, p. 10. Prepared by Task Committee on Steel Transmission Poles of the Committee on Analysis and Design of Structures of the Structural Division of the American Society of Civil Engineers. American Society of Civil Engineers, New York (1978). Tapered Tubular Steel Structures-ANSI/NEMA ITI-1983, p. 6. American National Standards Institute. Inc., New York (1984)

4. TLMRF the Transmission Line Mechanical Research Facility. Electric Power Research Institute, Palo Alto, CA (1983). 5. K. I. Bathe, Finite Element Procedures in Engineering Analysis. Prentice-Hall, Englewood Cliffs, NJ (1982). 6. M. Tanaka, Elastoplastic constitutive laws based on combined isotropic and kinematic hardening. Osaka (Japan) University Faculty of Engineering Technology Reports, No. 25, 101-115 (1975). 7. R. A. LeMaster, Finite deformation-finite element formulations for elastic-visccplastic materials. Thesis presented to the University of Tennessee, at Knoxville, TN, in partial fulfillment of the requirements for the degree of Doctor of Philosophy (1983). 8. R. D. Kriea and D. B. Kriea. Accuracies of numerical sohrtion methods for thi elastic-perfectly plastic model. J. Pressure Vessel Technol. 99, 510-515 (1977). 9. R. S. Barsoum and R. H. Gallagher, Finite element analysis of torsional and torsional-flexural stability problems. Int. J. Numer. Meth. Engng 2, 335-352 (1970). 10. M. A. Crisfield, A fast incremental/iterative procedure that handles snap-through. Comput. Strucr. 13, 55-62 (1981). 11. J. Podovan and S. Tovichakchaikul, Self-adaptive predictor-corrector algorithms for static nonlinear structural analysis. Compnr. Struct. 15, 365 (1982). 12. E. Riks, An incremental approach to the solution of snapping and buckling problems. Inr. J. Solids Struct. IS, 529-551 (1979).