ADDENDUM TO HR-273

PILE DESIGN FOR SKEWED INTEGRAL ABUTMENT BRIDGES

DECEMBER 1987

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ADDENDUM TO:

PILE DESIGN AND TESTS FOR INTEGRAL ABUTMENT BRIDGES

by L. F. Greimann, R. E~ Abehdroth, D. E. Johnson, P. B. Ebner

Final Report, HR-273, Iowa Department of Transportation, December, 1987

·TOPIC: PILE DESIGN FOR SKEW INTEGRAL ABUTMENT BRIDGES

1. 0 Biaxial Bending - Stress Criteria

Piles in skewed integral abutment bridges may be bent about both

.the strong ano weak axis as the bridge expands and contracts. The design

criteria presented in the report can be generalized to biaxial bending

for Case A (capacity of the pile as a structural member). Case Band

Case C criteria require no generali~ation.

When the pile undergoes biaxial bending, two separate equivalent

cantilevers must be developed. Each can be defined as described in the

report, Sec. 5.2.1. For bending about the strong or x axis, the length

of the equivalent cantiiever wiii be denoted by Lx and the pile head dis-

placement as Ax· For y axis bending, Ly is the length of the equivalent

cantilever and Ay is the displacement. The corresponding soil stiffnesses

are khx and khy· The generalized equations for Service Load Design are:

+ +

+ - + 0.472 Fy Fbx

. fa (1 - -

1) Fby

Fey ( 1)

(2)

in which

· fa = applied axial stress

fbx' fby =applied stre·ss for bending ab'Out the x and y axes, respectively

f y = yield stress

Fa = allowable ax.ial stress

Fbx, Fby = a·llowable stress for bending about .the .x and y axes, respectively

Fex'; Fey' = Euler buckling stress divided by a factor of safety

for buckling about the x and y axes, respectively

Cmx' Cmy = equivalent moment factor for x and y bending, respectively.

1.1. A 1ternati v~ Qne

As described in Sec. 5.2.3.1. of the report, Alternative One accounts

for first order thermal stresses. The moments for biaxial bending of

the fixed-head pile are

Mx =

My =

6Eixtix:

L2 x

6Eiytiy

L2 Y.

where Mx and My are the bending moments ·and Ix and Iy are the moments

of inertia about the strong and weak axis, respectively.

1. 2. A Her11a ti ve Two ·

(3)

(4)

Iri Alternative fwo, the stresses in the pile caused by thermal dis­

placement of the bridge are neglected but the P~ effect is included.

As in Sec. 5.2.3.2. of the report, the moments for the fixed-head pile

for thi~ alternative are:

Mx = ptix/2 (5)

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1.3. Comparison with Finite Element

In previous work conducted for the Iowa Department of Transportation

[l], the finite el~ment programs IAB2D and the three-dimensional version,

IAB3D, were used to analyze end bearing piles bent about the strong axis

(Ay equal zero) and about a 45 degree axis (Ax equal Ay). Fig. 6.20 (a

and b) and 6.22 (a and b) in [1] present the results as plots of P/P0

versus Ax, in which P is the pile capacity corresponding to Ax and P0

is the pile capacity for Ax equal zero. Six soil types are presented.

These results are compared to Eq. (1) and (2) in the following Fig. 1

and 2, respectively. (For the comparison purposes, the factor of safety.

has been removed from Eq. (1) & (2)). The conclusions from this comparison

are the same as in Sec. 5.2.3.3. of the report:

(1) Both Alternative One and Two are conservative and

(2) Alternative One is very conservative and would dictate

relatively short-integral abutment bridges.

Alternative Two is recommended if the pile has sufficient inelastic rotation

capacity.

2.0. Inelastic Rotation Capacity

Alternative Two requires that the pile have sufficient inelastic

rotation capacity to permit some redistribution of the pile forces during

the thermal expansion .. The following section generalizes the inelastic

rotation capaci~j developments in Sec. 6.1.1.2. of the r~port to the biaxial

case.

-4 ...

2.1. Uniaxial Bending

In this section, the conservative nature of the inelastic rotation

capacity in Sec. 6.1.1.2. of the report is demonstrated and a more appropri­

ate value is suggested. The inelastic rotation capacity of a plastic

hinge is presented in Eq. (6.51) -Of th~ report as

(7)

fo which Ci is an inelastic rotation capacity reduction factor given in

Eq. (6.52) of the report and

0p = Mptp (8) EI

is the elastic rotation corresponding to Mp. In the report, R.p was taken

as the length of the plastic hinge [2].

Work by Lukey and Adams [3] indicated that Eq. (7) and (8) are quite

conservative. They tested several simply-supported beams with a center

concentrated load and plotted ductility (3 Ci) versus bf/tf .. Eq. (6.52)

of the report fits thi~ plot well. Lukey and Adams used the length .e.p

equal to one-h~lf the span length of the simply supported bea~s or, in

general, the distance from the maximum moment to the inflection point.

The resulting rotation On is the rotation between the tangents to the ,..

elastic curve on each side of the plastic hinge. For a fixed head equiva­

lent cantilever, the length R.p is L/2. :.The inelastic rotation capacity

at a plastic hinge at a fixed support ~ill be one-half of the inelastic

rotation capacity at the plastic hinge :fri a simply supported beam, since

a fixed beam corresponds to one-half 'of a symmetrically loaded simple

beam. Hence, applying the results in Ref. [3] for a fixed head equivalent·

cantilever of length L, the inelastic rotation capacity becomes

Mpl eic = 3 Ci --

4EI (9)

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Other authors [4,5,6] have developed a different expression for inelas­

ti~ rotation capacity, but the study in Ref. [3] has demonstrated that

this expression was non-cons~rvative for large bf/tf values such as HP

shapes.

As a numerical example, consider an HP10x42 as a fixed head equivalent

cantilever with a 12 ft. length and Fy equal to 36 ksi. Eq. (7) and

(8), which are from the report, predict an inelastic rotation capacity

for strong axis bending of 0.0062 radians. Eq. (9) predicts 0.024 radians,

whereas the work in Ref. [4] yields 0.039 radians. Eq. (9) is recommended.

The inelastic rotation demand given by Eq. (6.48) of the report does

not account for removing the girder load after the pile has been displaced

to Point D' in Fig. 6.6. This unloading will cause an additional rotational

demand from Point D' to Point D by an amount ew· Therefore, the total

rotational demand can be expressed as

= 2 ( t, L (10)

Eq. (6.50) in the report is inappropriate. Also, ew should be interpreted

as the live load rotation only.

The inelastic rotation demand, Eq. {10), must be less than the inelas­

tic rotatioh capacity~ Eq. (9); therefore,

9 3M .. w. ·) t, ~ t, p ( 1 + - Ci -

4 4Mp ( 11)

in which

(12)

-6-

and Mw is the live load moment corresponding to Gw· Now, if the live

load stress is conservatively assumed to be equal to the allowable bending

stress of 0.55 Fy, Eq. ·(11) can be simplified by letting

3Mw

4Mp =

3(0.55 Fy)S

. 4 F Z y = 0·.4 (13)

where S is the section modulus. The sh~pe factor (Z/S) has been conser­

vatively taken as one. With this simplification, Eq. (11) can be rewritten

with a factor of safety as

(14)

in which, for a fixed-head pile,

(15)

With appropriate subscripts, Eq. (14) and {15) apply to both the x and

y axes.

2.2. Biaxial Bending

No published work could be found which described the inelastic rotation

capacity for biaxial bending. In fact, all of the available publications

related to strong axis bending only. T~e development in the previous . ·""·.

section implicidly assumed that the p~b)ished work (3] was a conservative

bound of the weak axis case. For biaxial b~nding, the strain at the extreme

fiber of the flange will be assumed to ~ontrol flange buckling and, thereby,

limit the inelastic rotation capacity. The inelastic strain demand at

the extreme fiber can be written as

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d e: iD = <Pi xD - +

2

_7._

b <PiyD -

2 (16)

in which <PixD and <PiyD are the inelastic curvature demands for the x and

y axes, respectively. They are proportional to the inelastic rotation

demands eixD and eiyD• which are discussed in Sec. 2.1. The criteria

that inelastic strain demand,e:iD• be less than the inelastic strain capacity,

~C· gives

<P;xo (d/2)

or,

<PixD -.- + <P;xc

<PiyD (b/2) + < 1

e: i c

<P;yo -- < 1 <PiyC

in which the inelastic curvature capacities are <PixC and <P;yc for the

(17)

(18)

x and y axes, respectively. Since curvatures are proportional to rotations,

Eq. (18) can a 1 so be written as

8 ixD 0ixC

GiyD + < 1

0iyC

Substituting Eqs. (9) and (10), specialized for x and y axis bending,

(19)

into Eq. (19) and applying the conditions expressed in Eqs. (11) and (13),

gives

t,x ,1 /).y \

\ - - 0.6 /- - 0.6 t,px i t,PY

+

~ :~~ < 1

/).ix (20)

- - 0.6 - 0.6 Apx

-8-

in which Ax and Ay represent the displacement for bending about the x

and y ~xes, respectively. The quantities Apx• Apy• Aix and Aiy are obtained

from Eqs. (12) and (14), spetialized to the x and y axes. Note that a

factor of safety has been incorporated into Eq. (14). For design purposes,

Eq. (20) can be conservatively bounded by the simple interaction equation.

(21)

which is appropriate for design.

3.0 Summary

In summary, the design criteria for stress in integral abutment piles

under biaxial bending are summarized in -Eq. (1) and (2). For Alternative

One which includes thermal stresses, the bending moments are given by

Eq. (3) and (4). Alternative Two which neglects thermal stresses has

moments as given in Eq. (5) and (6). The ductility requirements of Alterna­

tive Two are satisfied by Eq. (21) where Aix and Aiy are given by Eq.

(14).

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References

(1) L. F. Greimann, P. S. Yang, S. K. Edmunds, /l.. M. Wo.lde-Tinsae, 11 Design of Piles for Integral Abutment Bridges, 11 Final Report, HR-252, Iowa Department of Transportation, August 1984.

(2) Beedle: C. s. ·, Plastic Design of ·steel Frames, ··John Wiley, 1959, pg. 2,0,3 •. .

... ' .. :• .•: ~·.,. '~- , .

(3) Lukey, A. F. and Adams, P. F., 11 Rotation Capacity of BeAms Under Moment Gradient, 11 Journal of the Structural.Division~ ASCE, Vol. 95, ST6, June 1969, pp. 1173-1188.

(4) Lay, M. G. and Galambos, T. V., "Inelastic Beams Under Moment Gradi­ent," Journal of the Structural Division, ASCE, Vol. 93, STl, Feb. 1967, pp. 381-399.

(5) Galambos, T. V. and Lay, M. G., "Studies of the Ductility of Steel Structures," Journal of the Structural Division, ASCE, Vol. 91, ST4, Au~. 1965, pp. 125-151.

(6) Lay, M. G., "Flange Local Buckling in Wide-,Flange Shapes, 11 Journal of the St~uctural Division, ASCE, Vol. 91, ST6, Dec. 1965, pp. 95-116.

0 c.. ...... c..

0 c.. ...... c..

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l. 2 _:,. __ ..;..__....._ ___ .......... _______ __,

A,J ternati ve 2

1 -a- SOFT CLAY ~ STI.FF- CLAY .

-A.- VERY STIFF CLAY .. · ----FINITE .. ELEMENT

........ _... ___ DESIGN METHOD

4.0 1.0 . 2 .. 0 ' ~ ' , ,.~. .

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Fig. 1.

---<>-LOOSE SAND -6-MEDIUM SNAD

, -o- DENSE SAND . ·~ . FINITE ELEMENT

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Ultimate vertical lo~d ratio (end-bearing pile about strong axis).

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1.0

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-0-SOFT CLAY -o-STI FF CLAY --A-VERY STIFF CLAY

2.0 t.h, IN.

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4.0

O.OL--L-L--lt-.1------.~-'---------~------~ 0.0 1.0 2.0 3.0 4.0

(b) t.h, IN.

Fig. 2. Ultimate vertical load ratio (end-bearing piles about 45° axis) in Iowa soils.

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EXISTING GROUND LINE

STIFF CLAY

VERY STIFF CLAY

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SECTION A-A COMPOSITE SECTION I = 352932 1n 4

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SECTION B-B

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Figure.6.1. Section through abutment and soil profile.

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