finite element analyses of irregular thick plates for signal pole design

0045-7949(95)00134-4

Compuhrs & Swucrures Vol. 58, No. I, pp. Z-232, 1996 Copyright 0 1995 Elswier Science Ltd

Printed in Great Britain. All rights reserved 0045.7949196 $9.50 + 0.00

TECHNICAL NOTE

FINITE ELEMENT ANALYSES OF IRREGULAR THICK PLATES FOR SIGNAL POLE DESIGN

Gongkang Fu and S. J. Boulos Engineering Research and Development Bureau, New York State Department of Transportation,

Albany, NY 12232, U.S.A.

(Received 28 June 1994)

Abstract-The current specification of the American Association of State Highway and Transportation Officials does not specify any analysis method for the base plate design of span-wire-traffic-signal-poles. A simplified analysis method is developed here using finite element analysis for 28 signal poles, with calibration of the models by full-scale load testing. This method decomposes the base plate into three elementary components corresponding to respective critical regions of stress concentration. It is consistent with the allowable stress design concept of the current design code.

INTRODUCTION

Thick plates are widely used as components of engineering structures. The base plate of span-wire-mounted traffic-signal pole (referred to here simply as “signal pole”) is an example of such application. The pertinent U.S. design specifications do not provide any analysis method for design of the base plate, except general requirements for strength using the concept of allowable stress design [l]. This paper presents a study of applying the finite element analysis (FEA) to examine structural adequacy of the base plates of signal poles supplied to the New York State Department of Transportation (NYSDOT), U.S.A. and to develop a simplified analysis method for base plate design, because the current procedures were found unreliable. This design method is intended to be consistent with the concept of allowable stress design adopted by the current code [1] and it may be included in the specifications for design of signal poles.

FINITE ELEMENT MODELING

A typical signal pole supplied to NYSDOT is composed of a round or polygonal steel post with continuous changing or differential diameters inserted in and welded to a square steel base plate. The base plate is anchored to a concrete footing by four steel bolts. A reinforced hand-hole is provided in the post for access to electrical connections. Typical details are shown in Fig. 1. Dead, wind and ice loads are required to be covered in the design of the pole [1,2]. Their combinations and corresponding strength requirements are included in the code [l]. In this paper, signal poles are indentified by a letter indicating the manufacturer, design load in kins and height in feet. For examule. C530 is a pole supplied by a man;facturer with its name starting with C, with a desinn load of 22.2 kN (5 kiosk and a leneth of 9.14m (30 ft). T\o critical loading ‘cases ire conside;ed here: parallel and diagonal loading where the wire support- ing traffic signals runs parallel and diagonally to the square base plate, respectively.

FEA was used to analyze 28 poles in this study, covering a reasonably wide range of geometry variation for the poles

supplied to NYSDOT and redesign for some of these poles. Their dimensions are listed in Table 1, where an additional alphabetic suffix is used to identify those poles otherwise identical. The analysis was performed using graphics inter- active finite element total system (GIFTS) software [3] on an IBM Model 80 personal computer and was limited to the elastic range. Two quarter models were developed to respectively analyze the poles under the diagonal and parallel loading, taking advantage of their symmetric and antisym- metric behaviors. For the analysis of the base plate, the bottom 457 mm (18 in) of the post was also included in the models to include stress concentration in the joint area between the post and the base plate. Eight-node linear and isoparametric solid elements [3,6] were used to model the structure, as shown in Fig. 2, with five layers of elements for the base plate. The anchor bolts were modeled by beam elements.

VERIFICATION OF ANALYSIS MODELS BY

FULL-SCALE LOAD TESTING

Full scale load test was conducted for four signal poles under either parallel or diagonal loading, to verify the models of FEA. The load test was performed by mounting the poles horizontally and applying loads vertically by an upward hydraulic jack at 457 mm (18 in) from its tin (where the span wire is mounted in s&ice cbndition). Figure 3 shows the structural support for load testing, with a hand- hold seen on the top surface of the post in the bottom figure. Applied load levels were measured by a pressure gage connected to the loading jack, with a resolution of 0.995 kN (233.6 lb). Only four load tests involving two signal poles are discussed here, since the rest of load tests either had no response measurements for the base plate or involved a specially built pole not from the standard stock of the manufacturers. The interested reader is referred to Ref. [4] for details of these load tests.

Figures 4 and 5 show the strain gages for the posts and base plates of two tested poles C530 and C832. Design of the instrumentation was based on a preliminary FEA and was indented to measure the maximum responses to the two critical loading cases. Readings for these strain gages during

221

222 Technical Note

101.6 Hsndh

Round or Multi-Sided Steel Post u

x 165.1 zmn Frame 6 cover

(See Detail B)

I \

Steel stamp on Top of Base Plate (12.7 mm numerals)

1

Ease Plate (See Detail C)

-See Detail A

-Diameter at Top of Post, DT

76.2 mm Full Coupling with Chase Nipple Inside Located 270' Clockwise From Hsndhole

WMTRERtlEAD OPTION

DETAiL A

Diameter at Bottom of Post, DB

l(4) "H" Dia. lloles ior Anchor Bolts (per chart) with (2) Sex. Nuts per Bali.

B.C.

L: Base Plate Side Length EC: Bolt Circle Diameter T: Base Place Thickness

BASE PLAN

Wall Thickne

read

DETAIL C

Lgth 203.2 mm -1

ilsndhole Deta

DETAIL B

Fig. 1. Typical signal pole supplied to NYSDOT.

9.5x57.2 m Bar

6.4x25.4 m Steel Locking Bat

II-gage Cover (3.0 mm)

6.4x50.8 mm Stainless Steel Mschlne Screw

lil

load testing were taken by a computer driven data acquisition system. After the load tests, samples were taken from the post, base plate and anchor bolts for material test, with the results for these two poles given in Table 2. A second sample from C832 base plate was tested after the first showed an unexpectedly low strength, which was thus con- firmed. For simplicity of the discussion below, the structural response obtained in strain has been converted to stress, according to the elastic stress-strain constitutive relation, although it is obviously not valid for inelastic ranges.

A C530 pole was subjected to diagonal loading with strain gages on the post observing tensile strains (Fig. 4). It was loaded successively up to 10.9 and 12.9 kN (2460 and 2907 lb) in two cycles. Among the strain gages on the base plate, R4 showed the highest stress level. Its dominant component S, (bending stress in the J’ direction defined in Fig. 4) is shown in Fig. 6. The FEA and test results are in good agreement within the elastic range. Inelastic behavior of the pole was caused by partial yielding of an anchor bolt [4]. By linear extrapolation of the base plate’s elastic load-stress relation, S, would be 3 I1 MPa (45.1 ksi) at its design load 22.2 kN (5000 lb), which is much higher than its nominal yield stress 248 MPa (36 ksi). Thus the base plate was concluded to be deficient. The same pole was diagonally loaded again in another test, after being turned 180” about

its central axis. Similar results were obtained and assumed symmetry was verified [4].

In another load test, a C832 pole was first loaded diagonally, with the strain gages on the post under tension (Fig. 5). The pole was loaded up to 25.9 and 31.8 kN (5814 and 7155 lb), successively, in two cycles. Among the strain gages on the base plate, R8 and RlO showed highest stress levels symmetrically. Their locations and definitions of stress directions are shown in Fig. 3. Figure 7 shows the load- stress curves for RIO, with only the dominant components included. They again demonstrate validity of the FEA model within the elastic range. Residual strains were observed at the ends of the both cycles. Yielding was initiated in an anchor bolt at a load between 20.9 and 2.5.9 kN (4696 and 5814 lb) [4]. By linear extrapolation, and base plate was found also deficient for its nominal yielding stress 248 MPa (36 ksi).

The same C832 pole was reset for parallel loading (Fig. 5), and loaded through five cycles. Among the base plate strain gages, R8 and RlO showed highest stress levels. Figure 8 shows the load-stress curves of their respective dominant components. FEA predicted these stresses fairly accurately. By linear extrapolation of the elastic part of the stress-load relation, the maximum principal stress at RlO would be 331 MPa (48.1 ksi) at its design load 35.6 kN

Pole

Technical Note

Table 1. Dimension details of sample poles

Post dimensions (mm) Base plate dimensions (mm) DT DB w L T BC

Anchor dimensions

(mm) AD

223

Existing poles : C326 219 213 6.35 432 38.1 432 31.8 C328 219 213 6.35 451 38.1 451 31.8 c430 213 324 7.11 451 44.5 457 38.1 c530 273 324 7.94 584 44.5 584 38.1 C530b 273 324 7.94 451 50.8 457 44.5 Cl32 324 406 1.94 635 63.5 635 38.1 C832 324 406 9.53 559 57.2 559 50.8 S324 178 254 4.16 533 38.1 533 38.1 S328 191 279 4.76 533 38.1 533 38.1 s334 216 330 4.76 686 38.1 635 38.1 s434 241 356 4.76 686 44.5 635 44.5 s530 261 368 4.16 686 44.5 635 44.5 S632 305 419 4.76 686 50.8 131 50.8 S832 381 483 4.76 813 57.2 813 50.8 s934 419 533 4.76 838 57.2 851 51.2 S1036 445 572 4.76 889 57.2 851 51.2 U226 114 261 4.55 358 38.1 356 31.8 u530 249 356 6.35 521 50.8 508 44.5 U636 297 425 6.35 660 50.8 597 44.5 U832 305 419 7.94 622 63.5 597 50.8 U840 279 470 7.94 660 63.5 635 57.2 u1040 391 533 7.94 699 69.9 699 57.2 u1044 401 546 1.94 737 69.9 711 57.2 Redesign cases :

632 305 419 4.16 686 57.2 137 50.8

934a 419 533 4.16 838 69.9 851 57.2 934b 419 533 4.16 838 76.2 851 57.2 226 174 267 4.55 358 44.5 356 31.8 1040 401 546 7.94 131 82.6 Ill 51.2

Dimension definitions (see Fig. 1): DT = diameter at top of post; DB = diameter at bottom of post; w = wall thickness of post at its base; L = side length of square base plate; T = thickness of base plate; BC = bolt circle diameter; AD = anchor bolt diameter.

(8008 lb), which is much higher than its nominal yielding stress 248 MPa (36 ksi). Inelastic behavior under higher

diagonal load test above [4]. This shows, as expected, diag-

loads shown in Fig. 8 was initiated in an anchor bolt, onal load being the governing loading case for the anchor

although the maximum stress level was much lower than the bolts. This appeared to have been ignored in the design of this pole.

PARALLEL LOADING

\

DIAGONAL LOADING

Fig. 2. Finite element models for diagonal and parallel loadings.

c&s 5a,1-0

224 Technical Note

Table 3 permits numerical comparison of test and FEA results of the diagonal load test for C832, as a typical case including nondominant components of stress. These results are consistent with one another, especially for the pro- nounced stresses indicating critical responses to the load. Relatively larger differences between FEA and test results (for example, in R7) are attributed to inevitable discrepan- cies between locations of the strain gage and its corresponding element in FEA and/or higher noise to signal ratios in test data acquisition when the strain signal was low. Further note that good agreement between FEA and test results was also observed for tip deflections and stresses on the post and in the anchor bolts [4]. This also verified the beam assump- tion for the post adopted by the current AASHTO design code.

BASE PLATE BEHAVIOR AND SIMPLIFIED DESIGN METHOD

Figure 9 shows typical top surface stress contours of a base plate under diagonal and parallel loadings, as obtained by FEA for the design load. Stress is expressed as a percentage of the pole’s nominal yield stress F,. = 248 MPa (36 ksi) using the von Mises criterion. The shaded areas are critical regions with stresses of 110-140% of the nominal yielding stress, apparently associated with the deflection shown there. For the parallel loading, critical regions B and C represent maximum stresses under the given load con- tributed by dominant bending component (S, in Region B) and shear component (T,,, in Region C), respectively. Figures 10 and 1 I show simtlar stress concentration patterns

Fig. 3. Structural support for full scale load testing (top: distance view; bottom: close view).

Technical Note 225

Pole i.d.

Table 2. Material coupon test results

Steel type 0.2% Yield Ultimate (nominal yield) strength strength

Sample (MPa) (MPa) (MPa)

c530 post A252 (345) 315 490 plate A36 (248) 256 436 bolt A36M55 (379) 417 611

C832 post A53 (345) 326 489 plate* A36 (248) 194 310

200 304 bolt A36M55 (379) 405 595

*A second sample was tested for verification.

for poles S334 and S324. These patterns were observed for all 28 poles analyzed using the finite element models discussed above. These critical regions are thus addressed in the following proposed analysis method for the design of base plate.

For each potential failure mode (i.e. critical region), a part of the base plate is isolated and modeled by an elementary component (e.g. a beam or bar). A critical cross section is then identified, as well as the corresponding load. The subsequent analysis becomes straightforward, with the assistance of an empirical coefficient, modifying the section’s elastic capacity to reach an equivalent section modulus with respect to the maximum stress. These equivalent coefficients CL, /I and y were determined empirically by considering 23 representative signal poles supplied to NYSDOT and five redesign cases of inadequate poles, with the dimensions shown in Table 1.

This analysis method is intended to be consistent with the current allowable stress design concept adopted by the

t

Load

PLAN

I--- -u 101.6 mm

76.2 mm

50.8 mm

A-A

Fig. 4. Instrumentation for load test of C530 under diagonal loading (0: three-arm strain rosetts; 1: single-arm strain

gages).

AASHTO code [l] with respect to strength requirements, to obtain critical stresses within the elastic range. The resulting stresses are to be used to meet the AASHTO strength requirements:

f<kF (1)

where f is computed stress, F is allowable stress and k is given by the current AASHTO code [Il_for example, 1.4 for Group II load including dead- and wind-load- effects. f and F can have subscripts b and v as shown below to indicate bending and shear stresses, respectively. All three critical stresses must be considered for proportioning.

I

PLAN

I I

A-A Fig. 5. Instrumentation for load test of C832 under diag-

onal and parallel loadings.

226 Technical Note

Fig. 6. Comparison of test and FEA results: C530 under diagonal loading.

(a) Bending slress due to diagonal load (for critical region A)

fb = Anchor force x moment arm/equivalent flexural elastic section modulus

M EC-DE

=Bc &* ___ /{a(1.414L - DB)T2/6}

2 (2a)

where& is the computed bending stress, M is the moment at the post base due to the design load and a is an empirical coefficient for an equivalent section modulus, in terms of the critical stress:

a = (4.304 - O.O2021BC/T - 4.30408/L

+4.503(DB/L)2 - 0.97SO(L - 0.707BC)/

(L - DE) - 1.686BC/L}/C, (2b)

cm = 1.097 (2c)

Fig. 7. Comparison of test and FEA results: C832 under diagonal loading (top: bending stress; bottom: shear stress).

Fig. 8. Comparison of test and FEA results: C832 under parallel loading (top: bending stress; bottom: shear stress).

Equations 2a-2c were obtained by simplifying the problem as a cantilever beam under a concentrated load at its free end, applied by an anchor bolt, as shown in Fig. 12a. This assumes that the critical point being checked is on Section S,

(b ) Bending stress due to parallel loading qor critical region B)

F,, = midspan (maximum) moment/equivalent elastic

flexural section modulus

= & {l/4 - (1 - DE/L’)/3

+ (1 - DB/L’)4/12)/{j(L - DB)T*/12}, (3a)

where fb iS the computed bending stress; L’ = max(0.707BC.DBJ meaning the maximum of the two values; /I is an empirically determined coefficient for an equivalent section modulus, with respect to the critical stress:

fl = jl57.6 - 21.85L/DB - 0.33OOBC/T

-259.3DBIL -48.13(L*T/DB/(L - DB))“2

+ 194.6(DB/L)’ + 127.4T/BC

- 21.65(DB/BC)}/CB (W

ca = 1.080. (3c)

This case is treated as a beam with both ends built-in and a span of 0.707BC under a triangularly distributed load applied by the post, as shown in Fig. 12b. Formula (3a) checks a critical point on Section S,.

Technical Note 221

Table 3. Numerical comparison of testing and FEA results in base plate stress (pole C832 under diagonal 15.9 kN load)

Strain

yi; sx

Stress components (MPa)

Test FEA SY TXY sx SY TXV

Difference Dominant in dominant

stress stress component comtwnent (%)

R7 f3.65 -3.65 i-21.7 +1.86 -9.31 +32.6 TXY + 17.7% R8 +1.79 -91.0 +41.4 +2.21 -90.9 +43.2 SY +o.l% R9 - 13.7 -34.5 -54.1 -15.4 -53.6 -56.5 TXY +4.5% RlO +1.59 -91.2 -41.4 +2.21 -90.9 -43.2 SV +0.2%

Note: see Fig. 5 for loading and gage locations.

JOB: dd530 1.000g+00 , qEFL. ApI STRCSSES @NV.) , , i.OQOE+Ol 17-WA-92 13 06

“&?t;f: ii

0.000E+00 1.000E+Ol

s 2.000E+Ol 3.000E+Ol

F' 4.000E+Ol 5.000E+Ol

8 6.000E+Ol 7.000E+Ol

I e.OOOE+Ol

;: 9.000E+Ol 1.000E+02

k l.lOOE+02 1.200Ec02

; 1.300E+02 1.400E+02

4tm 9TFw3E9 @NV.) , LX] Jo& C~PlYO

. , 1 . OpOE*Ol 22-APR-92 8: 41

Fig. 9. Base plate stress concentration of C530 pole.

Technical Note

HooEC

IY DEFLS .

IY

JOB: rta-d334 5-WAY-92 9: 24

L----x JOB: x54334 a * oOo~+oo QEFL. A/a sm &NV.) . , 2kKO:I . 3-MAY-92 8: 23 .

Fig. 10. Base plate stress concentration of S334 pole.

(c) Shear stress due to parallel loading lfor criticai region C) 4 based on the elasticity solution, depending on ratio b/T [5]

f, = Torque by anchors/equivalent elastic torsional and y is an empirical coefficient, to reach an equivalent cross

section modulus section for critical stress prediction:

= ;,{yC’bT2).

y = (2tO.O - 66.9BC/DB - O.l719(BC - DB){T

(4a) -714.8DBIL + 358.3(DBjL)2 -48.16

wheref, is computed shear stress. M, ’ IS the torque induced x (L - 0.707BC)/(L - DB) - 288.2(BC - DB)/ by the anchor forces on the post base, which in turn are due to the design load. b = min{0.707BC,DB) meaning the

(1.414L - DB)+ 381.OBC/LjC, (4b)

minimum of the two values. C’ is a coefficient given in Table CY = 1.094. (4c)

Technical Note 229

Equations (4a)-(4c) are derived, based on simplification of the problem by considering a rectangular bar under torque M/2 applied by a pair of anchor bolts, as shown in Fig. 12~. The maximum shear stress occurs on section S,.

Figure 13 shows comparison of the calculated stress f by the suggested method andf,,, by FEA, for the three critical stress cases. The conservatism (overestimating) infobserved there is introduced by an amplification factor C, = WI, + oi (i = a, B, y), where m,(m, + a,) and u,(M, + 6,) are, respectively, mean and standard deviation of f/fFEA for respective

cases of critical stresses. M, is around 1.0 and 0, is about 0.09 for each case.

ILLUSTRATIVE EXAMPLE

The suggested analysis method is applied here to pole S632 for its evaluation and modification as an example for illustration. F” = 343 MPa (50 ksi) is used for proportioning and kF, = I .4*0.66*345 = 0.92 x 345 = 319 MPa (46.2 ksi)

3 x 1. ooor$ coo , LJEFL. Aplo SIR~SSES (ENV. I , 9 -

OEii s lY 1

OQOE+O

L--x 2&tD$pDo , 9En.WawEssEsEN V.) .

L-XI . l.Opo2+01

JOB: xm*224 ia-dt4-s m sa

-_- VON MlSCS CRII'EHIA

0 c%tw~

B

l:OOOE+Ol l.SOOE+Ol 2.OOOE+Oi

E

2.5002+01 po~.to~

I 4:0002+01

i! 4.5ooE+oi E.OOOE+Oi

B S.EOOE+O1 6.000E+Ot

0 J*%KG

ii 7:5OOE*Ol 0.000E+Oi

JOB:- 19. JUtI- 92 8: 18

Fig. 11. Base plate stress concentration of S324 pole.

230 Technical Note

(4

Saction s,

EC 1 bK. r -.. . . .../... *. _ _&A... z-

-... . . . . . DESIGN LOAD L-. ./ . . : -

i

\ 1.4 ML-D8

_

FORCES

ANCHOR FORCES

Fig. 12 (a and b)--Continued opposite.

Technical Note 231

-... -..* . ..i . . . . .._..._ .-..

‘2.. . . . . ..y

. . . . . . . . ..,..,_ I

. . . ..- . . . . . .

/...-

ANCHOR FORCES 1 v

Fig. 12. Simplified models for design analysis: (a) bending stres under diagonal loading; (b) bending stress under parallel loading; (c) shear stress under parallel loading.

(a)

E Q % up

Bending under parallel load

Bending under diagonal load

4@3-

350 -

300-

200 250 300 350 400

GEA

(cl Shear under parallel load n 250 -

2 200 -

3 B C&Q ISO-

Fig. 13. Comparison of critical stress (MPa) by FEA and proposed design analysis method; (a) bending stress under diagonal loading; (b) bending stress under parallel loading; Cc) shear stress under parallel

loading.

232 Technical Note

Table 4. Coefficient C’ for torsion 151

blT C’

1.0 1.2 1.5 2.0

2.5 3.0 4.0 5.0

10.0 cx

0.208 0.219 0.231 0.246 0.258 0.267 0.282 0.291 0.312 0.333

and kF, = 1.4 x 4 x 345 = 0.56 x 345 = 193 MPa (28 ksi) are assumed for load group II including dead and wind load effects and load group III including dead, ice. and half wind load effects. Note that they are usually the governing load groups for the state of New York.

Step I

From Table 1. L = 686 mm (27 in), T = 50.8 mm (2 in). BC = 737 min (29 in) and DB = 419 min (16.50 in). By defi- nition, the pole is 9.75m (32 ft) in height and its design load is 26.7 kN (6 kips).

For maximum bending stress under diagonal loading, anchor force = 6 x (32 - 1.5) x 12/29(4448) = 337 kN (75.72 kips), moment arm = 0.5 x (29 - 16.5)(25.4) = 159 mm (6.25 in), equivalent coefficient a = {4.304 - 0.02021(29/2) - 4.304(16.5/27) + 4.503(16.5/27)’ - 0.9750 (27 - 0.707(29))/(27 - 16.5) - 1.686(29/27))/1.097 = 0.5909. \ ,,,~ equivalent section ’ modulus = 0.5909(1.414 x 27 - 16.5)2’/6(25.4)’ = 140 x lo3 mm3 (8.541 ins), maximum bending stress = 75.7 x 6.25/8.541(6.90) = 382 MPa (55.4 ksi) r 319 MPa (46.2 ksi) NG.

Step I?

Increase the thickness T by 6 mm (0.25 in): L = 686 mm (27in), T = 57mm (2,25in), BC = 737mm (29in) and DB = 419 mm (16.50 in).

For maximum bending stress under diagonal loading, anchor force = 337 kN (75.72 kips), moment arm = 159 mm (6.25 in), equivalent coefficient a =

i 4.304 - 0.02021

(2912.251 - 4.304(16.5127) + 4.503(16.5127) - 0.9750(27 - 0:7b7(29))/(27 - 16.5)-1.686(29/27);1.097 = 0.6206, equivalent section modulus = 0.6206(1.414*27 - 16.5)2.2j2/6 (25.4)3 = 186 x lo3 mm3 (11.35 in’), maximum bending stress = 75.72(6.25)/l 1.35(6.90) = 288 MPa (41.7 ksi) < 319 MPa (46.2 ksi) OK.

For maximum bending stress under parallel loading, M = 6* (32 - 1.5) *12 (4.45*25.4) = 248 x lo3 kN-mm (2196 kip-in), L’ = max{0.707*29,16.5)(25.4) = 521 mm (20.50 in), DB/L’ = 16.5/20.50 = 0.8049, 1 -DB/L’= 0.1951, midspan moment = 2196/4/0.80492{0.25 - 0.1951/3 + 0.19514/ 12](4.45*25.4) = 17.7 x lo3 kN-mm (156.8 kip-in), p = { 157.6 - 21.85(27/16.5) - 0.3300 (29/2.25) - 259.3(16.5/27) - 48.13[(27) (2.25)/ 16.5 / (27 - 16.5)1”* + 194.6(16.5/27)* + 127.4(2.25/29) - 21.65(16.5/ 29)jfi .080 = 0.8070, equivalent section modulus = 0.8070 (29 - 16.5)*2.252/12(25.4)3 = 69.7 x 10’ mm3 (4.26 in3). \-~ maximum’ bending stress = 156.8/4.256(6.9) = 254 MPa (36.84 ksi) c 319 MPa (46.2 ksi)OK.

For maximum shear stress under parallel loading, M/2 = 6*(32 - 1.5)*12/2(4.45*25.4) = 124 x 103kN - mm (1098 kip-in), b = min{0.707*29,16.5}(25.4) = 419 mm

(16.5 in), b/T = 7.333. C’ = 0.301. y = (210.0 - 66.9 (29/16.5) - 0.1719(29 - 16.5)/2.25 - 714.8(16.5/27) + 358.3 (16.5/27)* - 48.16(27 - 0.707(29))/(27 - 16.5) - 288.2(29 - 16.5)/(1.414(27) - 16.5) + 381.0(29/27)}/1.094 = 1.545 equivalent torsional section modulus = 1.545(0.3d;j 16.5(2.25*)(25.4)’ = 637 x lo3 mm3 (38.85 in-‘). maximaam “1..

shear stress =‘1098/38.85(6.90) = 195 MPa ‘(28.28 ksi) = 193 MPa (28 ksi) OK.

Based on the experience of several such redesign examples for deficient existing poles, it is found that increasing the base plate thickness is most effective to reduce the stress level in the base plate.

CONCLUSIONS

Both the diagonal and parallel loadings may be critical and should be considered in designing the signal poles. Critical regions A, B and C are stress concentration areas, respectively, under the diagonal and parallel loadings. They must be checked in the base plate design. The suggested semi-empirical analysis method simplifies the analysis by decomposing the base plate into three simple components for respective critical stresses. This method presents clear mechanical origins of stress concentration in a simple manner. Hand calculation is sufficient for its design application

Acknowledgernears-The Federal Highway Administration, U.S.A. provided partial support for this study. Cooperation of G. Scaife and staff of Carlan Manufacturing Companv. Long Island, New York, is gratefully appreciated for the load testing. Manv New York State Department of Trans- portation personnel make the completion of this study possible. Special thanks are due to L. N. Johanson for his valuable comments in the course of this study, S. Alampalli, D. M. Berkley, W. J. Deschamps, E. Dillon, M. J. Gray, G. L. Howard and J. La11 for their assistance in the load testing and FEA, D. Sandhu for her assistance in the statistical analysis and I. A. Aziz, A. D. Emerich and D. L. Noonan for their assistance in editing and preparing some of the figures.

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6.

REFERENCES

Standard specifications for structural support for highway signs, luminaries and traffic signals. American Association of State Highway and Transportation Officials, Washington, DC (1985). Method for calculating the loads applied to span wire traffic signal poles. Engineering Instruction E183-38, Facilities Design Division, New York State Department of Transportation, 14 September (1983). CASA /GIFTS User’s Reference Manual. Software Package Version 6.4.3. CASA/GIFTS, Tucson, AZ. October (1989). S. J. Boulos, G. Fu and S. Alampalli, Load testing, finite element analysis and design of steel traffic signal poles. Final report to FHWA on R1229, FHWA/NY/RR- 93/159. New York State Department of Transportation, July (1993). F. P. Beer and E. R. Johnston, Mechanics of Materials. McGraw-Hill, New York (1981). 0. C. Zienkiewicz and R. L. Taylor, The Finite Element Method, 4th edn, Vol. 1, Basic Formulation and Linear Problems. McGraw-Hill, New York (1989).

finite element analyses of irregular thick plates for signal pole design

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