Southern Illinois University Carbondale OpenSIUC Publications Department of Civil and Environmental Enineerin Fall 9--04 FOOTING FIXITY EFFECT ON PIER DEFECTION Jen-kan Kent Hsiao Yunyi Jian Southern Illinois University Carbondale, alexjian@siu.edu Follow this and additional works at: http://opensiuc.lib.siu.edu/cee_pubs Recommended Citation Hsiao, Jen-kan K. and Jian, Yunyi. "FOOTING FIXITY EFFECT ON PIER DEFECTION." International Journal of Bride Enineerin (IJBE) Vol., No. No. (Fall 04): pp. 0-0. This Article is brouht to you for free and open access by the Department of Civil and Environmental Enineerin at OpenSIUC. It has been accepted for inclusion in Publications by an authorized administrator of OpenSIUC. For more information, please contact opensiuc@lib.siu.edu.
International Journal of Bride Enineerin (IJBE), Vol., No., (04), pp. 0-0 FOOTING FIXITY EFFECT ON PIER DEFECTION J. Kent Hsiao and Alexander Y. Jian, Southern Illinois University Carbondale, Dept. of Civil and Environmental Enineerin, USA e-mail: hsiao@enr.siu.edu, alexjian@siu.edu ABSTRACT: The rotational restraint coefficient at the top of a pier and the rotational restraint coefficient at the bottom of the pier (that is, the deree of fixity in the foundation of the pier) are used to determine the effective lenth factor of the pier. Moreover, the effective lenth factor of a pier is used to determine the slenderness ratio of the pier, while the deree of fixity in the foundation of a pier is used to perform the first-order elastic analysis in order to compute the pier deflection. Finally, the slenderness ratio of the pier is used to determine if the effect of slenderness shall be considered in the desin of the pier, while the manitude of the pier deflection resultin from the first-order analysis is used to determine if the second-order force effect (the p- effect) shall be considered in the desin of the pier. The computations of the slenderness ratio and the deflection of a pier, however, have conventionally been carried out by assumin that the base of the pier is riidly fixed to the footin, and the footin in turn, is riidly fixed to the round. Other derees of footin fixity have been nelected by the conventional approach. In this paper, two examples are demonstrated for the slenderness ratio computation and the first-order deflection analysis for bride piers with various derees of footin fixity (includin footins anchored on rock, footins not anchored on rock, footins on soil, and footins on multiple rows of end-bearin piles) recommended by the AASHTO RFD Bride Desin Specifications. The results from the examples indicate that the deree of footin fixity should not be nelected since it sinificantly affects the manitude of the slenderness ratio and the deflection of the pier. KEYWORDS: Bride desin; Deflection; Foundations; Piers; ateral loads. INTRODUCTION A sinle pier, as shown in Fi., and a bent with multiple piers, as shown in Fi., are the substructures of brides. The sinle pier and the bent with multiple piers are considered as a sway cantilever element and a sway frame, respectively, in the lonitudinal direction. For piers not braced aainst sidesway, the slenderness effects
Footin fixity effect on pier deflection must be considered where the slenderness ratio is or larer []. The slenderness ratio is calculated as Kl u /r where : K l u r is the effective lenth factor of a pier, is the unbraced lenth of a pier, and is the radius of yration of the cross section of a pier. Fiure. Sinle pier Fiure. Bent with multiple piers The effective lenth factor (K) of a cantilever pier in the lonitudinal direction, as shown in Fi., or a pier in a sway frame in the lonitudinal direction as shown in Fi. can be obtained by usin the Alinment Chart for Determinin Effective enth Factor, K, for Unbraced Frames, presented in the AASHTO RFD Bride Desin Specifications [] or by usin the followin two equations []: For K <, K 4 0. G a 0. G b 0. 0G G a b () For K, K 0. 9 πa 0. 8 4ab () Ga Gb in which: a G G a b
J. Kent Hsiao, Alexander Y. Jian b 6 G G a b 6 An alternate equation [] to compute the K value is K.6Ga Gb 4.0 Ga Gb 7.5 () G G 7.5 a b where: in which: G E c I c c E I EcIc G c E I (4) is the rotational restraint coefficient at the column (pier) end [the subscripts a and b refer to the two ends of the column (pier) under consideration], is the modulus of elasticity of the column, is the moment of inertia of the column, is the unbraced lenth of the column, is the modulus of elasticity of the beam, is the moment of inertia of the beam, and is the unsupported lenth of the beam. When computin the G value, I c should be replaced by 0.70 I c and I should be replaced by 0.5 I [4,5,6], in order to take into account the influence of axial loads, the presence of cracked reions in the member, and the effects of the duration of the loads. Note that I c is the moment of inertia of the ross concrete section of the column and I is the moment of inertia of the ross concrete section of the beam. The AASHTO RFD Bride Desin Specifications [] recommends that in the absence of a more refined analysis, the followin G values can be used for various footin conditions: G =.5 for footins anchored on rock, G =.0 for footins not anchored on rock, G = 5.0 for footins on soil, and G =.0 for footins on multiple rows of end-bearin piles. In addition, the Specifications specify that the moment on a compression member may be increased by multiplyin the moment by a moment manification factor if the sidesway ( ) of the compression member due to factored lateral or ravity loads calculated by conventional first-order elastic frame analysis is reater than l u /500. Similar to the computation of the G value, 0.70 I c
4 Footin fixity effect on pier deflection should be used for the moment of inertia of the column and 0.5 I should be used for the moment of inertia of the beam [6, 7] for the first-order elastic frame analysis. The aforementioned moment manification factor may be taken as: Pu P K e (5) where: P u K P e is the moment manification factor, is the factored axial load, is the stiffness reduction factor (0.75 for concrete members), and is the Euler bucklin load. The Euler bucklin load shall be determined as: EI P e (6) Kl where: E is the modulus of elasticity, I is the moment of inertia about the axis under consideration, K is the effective lenth factor in the plane of bendin, and is the unsupported lenth of a compression member. l u u In lieu of a more precise calculation, EI in Eq.(6) shall be taken as the reater of: or EcI EsIs EI 5 (7) E c I d d EI.5 (8) where: E c is the modulus of elasticity of concrete, I is the moment of inertia of the ross concrete section about the centroidal axis, is the modulus of elasticity of lonitudinal steel, E s
J. Kent Hsiao, Alexander Y. Jian 5 I s β d is the moment of inertia of lonitudinal steel about the centroidal axis, and is the ratio of the maximum factored permanent load moment to the maximum factored total load moment. The determination of the effective lenth factor and the computation of the deflection of a pier, however, have been conventionally carried out by assumin that the base of the pier is riidly fixed to the footin, and the footin in turn, is riidly fixed to the round. The deree of fixity in the foundation has been nelected by the conventional approximate approach. The accuracy of the approximate approach, therefore, is questionable. Two approaches for the determination of the effective lenth factor and the computation of the deflection of piers are presented in this paper. Approach I is an approximate approach which assumes the bases of the piers are riidly fixed to the round, while Approach II is a refined approach which considers the deree of fixity in the foundation of the piers. Results from both approaches are then compared with each other. APPROACH I AN APPROXIMATE APPROACH Considerin the sinle pier in the lonitudinal direction, as shown in Fi., Approach I (an approximate approach) treats the base of the pier as rotationally fixed and translationally fixed, and, additionally, treats the top of the pier as rotationally free and translationally free, as shown in Fi.. In addition, considerin the piers in the bent in the lonitudinal direction, as shown in Fi., Fiure. Ideal condition for a cantilever sinle pier in the lonitudinal direction Fiure 4. Ideal condition for piers in a bent in the lonitudinal direction
6 Footin fixity effect on pier deflection Approach I treats the bases of the piers as rotationally fixed and translationally fixed, and, additionally, treats the tops of the piers as rotationally fixed and translationally free as shown in Fi. 4. The desin value of the effective lenth factor, K, therefore, is. and. for the pier in Fis. and 4, respectively []. In Approach I, the idealized conditions of the pier(s), as shown in Fis. and 4, are also used to compute the deflection of a cantilever sinle pier and the piers in a bent, respectively, in the lonitudinal direction. APPROACH II A REFINED APPROACH In this approach, the deree of fixity in the pier s foundation is considered by usin an appropriate G value (correspondin to the deree of fixity in the pier s foundation) at the bottom of the pier. In addition, the rotational riidity at the top of the pier in the pier cap of a bent in the lonitudinal direction is also considered. In order to achieve the appropriate G value at the bottom of a cantilever sinle pier in the lonitudinal direction, a tie beam is introduced to the base of the pier, as shown in Fi. 5(a). Similarly, in order to achieve the appropriate G values at the foundations of the piers in a bent in the lonitudinal direction, a continuous tie beam is introduced to the bases of the piers, as shown in Fi. 6(a). The structure shown in Fis. 5(a) and 6(a) can be treated as the substructure shown in Fis. 5(b) and 6(b), respectively. The pinned ends of the tie beam shown in Fis. 5(a) and 6(a) are located at the inflection points of the tie beam in the swin structure shown in Fis. 5(b) and 6(b), respectively. The inflection point in the tie beam in each bay is assumed to be located at the midspan of the bay width. In Approach II, Fis. 5(a) and 6(a) are used as the structural systems to compute the deflection of a cantilever sinle pier and a bent with multiple piers, respectively, in the lonitudinal direction. / / (a) Fiure 5. Modified structural system for a cantilever sinle pier in the lonitudinal direction
J. Kent Hsiao, Alexander Y. Jian 7 / / (b) Fiure 5. Modified structural system for a cantilever sinle pier in the lonitudinal direction / / (a) / / (b) Fiure 6. Modified structural system for a bent with multiple piers in the lonitudinal direction
8 Footin fixity effect on pier deflection 4 PIER DEFECTION COMPUTATION EXAMPES USING APPROACHES I & II Two examples are demonstrated for the computation of pier and bent deflections in the lonitudinal direction due to lateral and ravity loads. Concrete with a modulus of elasticity, E = 5,00 MPa, is used for the pier and the bent. Example : Compute the deflection in the lonitudinal direction of the pier subjected to the factored lateral force (70 kn) and ravity load (6000 kn), as shown in Fi. 7, for the followin four different footin conditions: () the footin anchored on rock, () the footin not anchored on rock, () the footin on soil, and (4) the footin on multiple rows of end-bearin piles. 70 kn 6000 kn 0.6 m.8 m 0.6 m. m Elevation Fiure 7. oaded sinle pier illustrated for Example Side view Approach I: Referrin to the pier in the lonitudinal direction shown in Fi. 7, the base of the pier is treated as rotationally and translationally fixed for all four different footin conditions, while the top of the pier is treated as rotationally and translationally free, as shown in Fi.. Therefore, the desin value of the effective lenth factor of the pier is.. The pier heiht is 0.6 m. For a rectanular compression member, the radius of yration (r) may be taken as 0.0 times the overall dimension in the direction in which stability is bein considered [].
J. Kent Hsiao, Alexander Y. Jian 9 Therefore, the radius of yration of the pier about the transverse axis can be computed as.8 m 0. = 0.549 m. The transverse axis slenderness ratio of the pier, therefore, can be computed as: Kl r. 0.6m 0.549m 9.6 Since the slenderness ratio is larer than, the slenderness effects on the pier must be considered. From the first-order elastic analysis usin 0.70 I c as the moment of inertia for the pier bendin about the transverse axis, the deflection at the top of the pier can be computed as: Fh EI kn 5. mm 70kN 0.70 0,60mm 0mm 80mm 9.4mm where: F is the factored lateral load, h is the heiht of the pier, E is the modulus of elasticity of the pier, and I is the moment of inertia of the pier. Since the deflection, 9.4 mm, is larer than l u /500 (= 6.9 mm), the deflection (which corresponds to the bendin moment at the base of the pier) due to the factored lateral load must be multiplied by the moment manification factor, δ. From Eq. (5), the moment manification factor can be computed as: Pu P K e 6000kN 0.75. 0 5 kn.065 where P e Kl EI u 6.6. 0 kn 0,60mm mm. 0 5 kn Assumin that Eq. (8) controls over Eq. (7) and β d = 0 (there is no factored permanent load moment about the transverse axis of the pier), one has:
0 Footin fixity effect on pier deflection EI E c I.5 d kn 5. mm 0mm 80mm.5 0 6.6 0 kn mm Therefore, the final deflection of the pier in the lonitudinal direction due to the slenderness effects is: = (.065) (9.4 mm) = 9.7 mm Table summarizes the results obtained from Approach I. Note that this table includes all of the four footin conditions: () the footin anchored on rock, () the footin not anchored on rock, () the footin on soil, and (4) the footin on multiple rows of end-bearin piles. Table. Deflections of the pier in Example computed by Approach I footin condition effective lenth factor slenderness ratio pier base rotation (rad) deflection calculated by first-order analysis moment manification factor, δ final deflection of the pier all. 9.6 0 9.4 mm.065 9.7 mm Approach II: Referrin to the pier in the lonitudinal direction shown in Fi. 7, assin G a =.5 for footins anchored on rock, G a =.0 for footins not anchored on rock, G a = 5.0 for footins on soil, and G a =.0 for footins on multiple rows of end-bearin piles as per the G values recommended by the AASHTO RFD Bride Desin Specifications []. Since the top of the pier is rotationally free about the transverse axis of the pier and translationally free alon the lonitudinal direction of the pier, G b = at the top of the pier. From Eq. () or the Alinment Chart for Determinin Effective enth Factor, K, for Unbraced Frames, presented in the AASHTO RFD Bride Desin Specifications [], one has: K =.5 for the footin anchored on rock, K =.9 for the footin not anchored on rock, K =.4 for the footin on soil, and K =. for the footin on multiple rows of end-bearin piles. Since the transverse axis slenderness ratios (Kl u /r) of the pier with four different footin conditions are all larer than, the slenderness effects on the pier must be considered for all of the four different footin conditions. In order to use the 0.70 I c value as the moment of inertia for the pier bendin about its transverse axis, a reduced cross section of.68 m. m (as shown in Fi. 8)
J. Kent Hsiao, Alexander Y. Jian is used to replace the oriinal cross section of.8 m. m for the pier. Also, referrin to Fi. 5(a), a modified structural system with different lenths of added tie beams is made, as shown in Fi. 8, in order to achieve the appropriate G values at the base of the pier for the four different footin conditions. 70 kn.68 m Joint b. m 0.6 m. m.68 m Joint a / = 5.54 m / = 5.54 m.08 m.08 m 5.80 m 5.80 m 0.6 m 0.6 m for G a =.5 for G a =.0 for G a = 5.0 for G a =.0 Fiure 8. Modified structural system for Approach II in Example For example, as shown in Fi. 5(b) and Fi. 8, if / = 5.54 m for the tie beam, the span lenth of the tie beam will be =.08 m on each side of joint a. Therefore, the G a value at the base of the pier can be computed usin Eq. (4) as: G a EcI E c I c E c E.m.68m 0.6m.m.68m.08m.5
Footin fixity effect on pier deflection Considerin the pier with G a =.5 for the footin anchored on rock, the deflection at the top of the pier due to the factored lateral force (70 kn) alon the lonitudinal direction of the bent can be computed usin the followin steps: () Referrin to Fi. 9(a), the bendin moment at the base of the pier (at joint a ) can be computed as: M ab = 70 kn 0.6 m = 797 kn-m. () The end moments at joint a of the tie beam therefore can be computed as: M ad = M ac = M ab / = 797 kn-m / = 99 kn-m. () Referrin to Fi. 8, the moment of inertia about the transverse axis of the pier and the tie beam can be computed as: I = (0 mm)(680 mm) / = 4.46 0 mm 4. (4) Referrin to Fi. 9(b), the rotation at joint a of the tie beam can be computed as θ a = [(M ad )(l ad )] / [(E ad )(I ad )] = [(,99,000 kn-mm)(5,540 mm)] / [(5. kn/mm )(4.46 0 mm 4 )] = 0.00065 rad. (5) Referrin to Approach I, when the base of the pier is considered riidly fixed to the round, the deflection at the top of the pier was computed as 9.4 mm. Since the rotation at the base of the pier is θ a = 0.00065 rad for the condition in which the footin is anchored on rock (G a =.5), the total deflection (includin the deflection caused by the rotation at the base of the pier) can be computed as: = 9.4 mm + (0,60 mm)(0.00065) = 5.89 mm. Since the deflection, 5.89 mm, is larer than l u /500 (= 6.9 mm), the deflection due to the factored lateral load must be multiplied by the moment manification factor. Similar to the procedure shown in Approach I, the moment manification factor can be computed by usin Eq. (5) as: Pu P K e 6000kN 0.75 9. 0 4 kn.095 Note that K =.5 has been used for the computation of P e. Therefore, the final deflection of the pier in the lonitudinal direction due to the slenderness effects is: = (.095) (5.89 mm) = 7.40 mm
J. Kent Hsiao, Alexander Y. Jian 70 kn b θ a 0.6 m c θ a a d 5.54 m θ a 5.54 m (a) M ad a θ a d l ad = 5.54 m (b) Fiure 9. Joint rotation at the base of the pier with G a =.5 The deflections in the lonitudinal direction of the pier for four different footin conditions are computed usin Approach II and are summarized as shown in Table, where the footin conditions are () the footin anchored on rock, () the footin not anchored on rock, () the footin on soil, and (4) the footin on multiple rows of end-bearin piles. Table. Deflections of the pier in Example computed by Approach II footin condition () () () (4) effective lenth factor.5.9.4. slenderness ratio 47. 54.7 64. 4.4 pier base rotation (rad) 0.00065 0.0005 0.0074 0.00045 deflection calculated by first-order analysis 5.89 mm.66 mm.66 mm.65 mm moment manification factor, δ.095..9.079 final deflection of the pier 7.40 mm 5.66 mm 7.7 mm 4.7 mm
4 Footin fixity effect on pier deflection Example : Compute the deflection in the lonitudinal direction of the bent subjected to the factored lateral and ravity loads, as shown in Fi. 0, for the followin four different footin conditions: () footins anchored on rock, () footins not anchored on rock, () footins on soil, and (4) footins on multiple rows of end-bearin piles. Approach I: Referrin to the bent in the lonitudinal direction shown in Fi. 0, the bases of the piers are treated as rotationally and translationally fixed for all four different footin conditions, while the tops of the piers are treated as rotationally fixed and translationally free (since the cap of the piers is treated as a riid element), as shown in Fi. 4. Therefore, the desin value of the effective lenth factor of the piers is... m 850 kn 850 kn 850 kn 850 kn 60 kn. m. m 6.0 m.07 m Dia. (Typ.) 6.0 m.07 m Dia. spa @ 4.7 m Elevation Side view Fiure 0. oaded bent with multiple piers illustrated for Example The unbraced lenth of each of the piers is 6.0 m. For a circular compression member, the radius of yration (r) may be taken as 0.5 times the diameter []. Therefore, the radius of yration of each of the piers can be computed as.07 m 0.5 = 0.68 m. The transverse axis slenderness ratio of each of the piers, therefore, can be computed as: Kl u r. 6.0m 0.68m 7.
J. Kent Hsiao, Alexander Y. Jian 5 Since the slenderness ratio is larer than, the slenderness effects on the piers must be considered. Usin 0.70 I c as the moment of inertia for each of the piers bendin about the transverse axis, the deflection at the top of each of the piers in the lonitudinal direction of the bent can be computed by usin the first-order elastic analysis as: Fh EI 60kN / 4 kn 5. mm 0.70 600mm 55mm 4 4.5mm where: F is the factored lateral load applied to each of the piers, h is the heiht of each of the piers (measured from the top of the footin to the bottom of the pier cap), E is the modulus of elasticity of the piers, and I is the moment of inertia of each of the piers. Since the deflection due to the factored lateral load is.5 mm, which is less than l u /500 (= 4.07 mm), the deflection does not need to be increased by bein multiplied by the moment manification factor. Therefore, δ =.0. Table summarizes the results obtained from Approach I. Note that this table includes all of the four footin conditions: () footins anchored on rock, () footins not anchored on rock, () footins on soil, and (4) footins on multiple rows of endbearin piles. Table. Deflections of the bent in Example computed by Approach I footin condition K value of exterior piers K value of interior piers Kl u of r exterior piers Kl u of r interior piers bent deflection calculated by firstorder analysis moment manification factor, δ final deflection of the bent all.. 7. 7..5 mm.0.5 mm Approach II: In order to use the 0.70 I c value as the moment of inertia for the piers bendin about the transverse axis, a reduced cross section of 0.978 m dia. is used to replace the oriinal cross section of.07 m dia. for the piers. Also, in order to use the 0.5 I value as the moment of inertia for the pier cap bendin about the transverse axis, a reduced cross section of 0.940 m 0.940 m is used to replace the oriinal cross section of. m. m for the pier cap. In addition, referrin to
6 Footin fixity effect on pier deflection Fi. 6(a), a modified structural system is made for the bent in the lonitudinal direction, as shown in Fi.. In the modified structural system, a continuous tie beam is added to the bases of the piers in order to achieve the appropriate G values at the foundations of the piers in the bent in the lonitudinal direction. The cross section of the added tie beam is 0.579 m 0.579 m, 0.488 m 0.488 m, 0.49 m 0.49 m, and 0.64 m 0.64 m for G a =.5,.0, 5.0, and.0, respectively. For example, if the cross section of the tie beam is 0.579 m 0.579 m, the G a value at each of the pier base can be computed by usin Eq. (4) as: G a EcI E c I c E E c 0.978m 4 6.7m 0.579m 0.579m 4.7m 4.5 Note that G a =.5 is for footins anchored on rock, G a =.0 is for footins not anchored on rock, G a = 5.0 is for footins on soil, and G a =.0 is for footins on multiple rows of end-bearin piles. Also note that the pier heiht shown in Fi. is measured from the top of the footin to the mid-heiht of the pier cap. 60 kn 0.940 m 0.940 m (Pier cap) 6.7 m 0.978 m dia. (Typ. for piers). m spa @ 4.7 m. m Fiure. Modified structural system for Approach II in Example
J. Kent Hsiao, Alexander Y. Jian 7 Referrin to Fi., one has G b = 0. at each of the tops of the interior piers and G b = 0.44 at each of the tops of the exterior piers by usin Eq. (4). With the G a and G b values of each pier in the bent determined, the effective lenth factor, K, of each pier in the bent can be determined by usin Eq. () or the Alinment Chart for Determinin Effective enth Factor, K, for Unbraced Frames, presented in the AASHTO RFD Bride Desin Specifications []. The determined K values of the exterior and interior piers for the four different footin conditions are summarized in Table 4. Also, as shown in Table 4, since the slenderness ratios (Kl u /r) of all of the piers with four different footin conditions in the direction in which stability is bein considered are all larer than, the slenderness effects on all of the piers must be considered for all of the four different footin conditions. Table 4. Deflections of the bent in Example computed by Approach II footin condition K value of exterior piers K value of interior piers Kl u of r exterior piers Kl u of r interior piers bent deflection calculated by firstorder analysis moment manification factor, δ final deflection of the bent ()..7 9.8 8.9.94 mm.0.94 mm ().46.4.. 4.87 mm.048 5.0 mm ().59.54 6. 5. 5.75 mm.057 6.08 mm (4).5. 8.5 7.5.5 mm.0.5 mm Referrin to Fi., the deflection of the bent due to the factored lateral force (60 kn) alon the lonitudinal direction can be computed by usin the computer software SAP000 [8]. The results are shown in Table 4. Since both of the deflections (.94 mm and.5 mm) of the bent with footin conditions () and (4) are less than l u /500 (= 4.07 mm), these deflections do not need to be increased by bein multiplied by the moment manification factor. Therefore, δ =.0. However, since both of the deflections (4.87 mm and 5.75 mm) of the bent with footin conditions () and () are larer than l u /500, these deflections must be multiplied by the moment manification factor, δ. Considerin the bent with footin condition (), the moment manification factor can be computed by usin Eq. (5) as: Pu P K e 0.75 4 850kN 80,85kN 86,86kN.048
8 Footin fixity effect on pier deflection where and P EI 6.46 0 kn mm e 80,85kN (for exterior piers) Kl.46 600mm u P EI 6.46 0 kn mm e 86,86kN (for interior piers) Kl.4 600mm u Assumin that Eq. (8) controls over Eq. (7) and β d = 0 (there is no factored permanent load moment about the transverse axis of the pier), one has: EI E c I.5 d kn 5. mm.5 0 070mm 4 4 6.46 0 kn mm Therefore, the final deflection of the bent with footin condition () due to the slenderness effects is: = (.048) (4.87 mm) = 5.0 mm Usin the same approach, the moment manification factor, δ, and the final deflection of the bent with footin condition () can be computed and the results are shown in Table 4. 5 CONCUSIONS The computation of the deflection of a sinle cantilever pier has conventionally been carried out by assumin that the base of the pier is riidly fixed to the round. Similarly, the computation of the deflection in the lonitudinal direction of a bent with multiple piers has conventionally been carried out not only by assumin that the bases of the piers are riidly fixed to the round but also by assumin that the tops of the piers are rotationally fixed to the pier cap. The rotational restraint coefficient(s) at the base of a sinle cantilever pier, as well as at the bases and tops
J. Kent Hsiao, Alexander Y. Jian 9 of the piers in a bent in the lonitudinal direction, have completely been nelected by the conventional approach (which is classified as Approach I in this paper). In this paper, two examples, one for a sinle cantilever pier and the other for a bent with multiple piers, are demonstrated for the slenderness ratio computation and the first-order deflection analysis for the pier(s) by usin two approaches. Approach I is an approximate approach assumin the footins are riidly fixed to the round, while Approach II is a refined approach considerin various derees of footin fixity. The various derees of footin fixity include () footins anchored on rock, () footins not anchored on rock, () footins on soil, and (4) footins on multiple rows of end-bearin piles, as specified by the AASHTO RFD Bride Desin Specifications. Example indicates that Approach II results in the larest moment manification factor of δ =.9 (for the footin on soil condition), while the δ value resulted from Approach I is only.065. That is, the percentae of moment increase that resulted from Approach II is 9. %, which is about times of that (6.5 %) of the result from Approach I. Also, Approach II results in the larest deflection of 7.7 mm (for the footin on soil condition) which is about 4 times of that (9.7 mm) of the result from Approach I. Example indicates that Approach II results in the larest moment manification factor of δ =.057 (for the footins on soil condition), while the moment manification factor can be nelected in Approach I (that is δ =.0). Also, Approach II results in the larest deflection of 6.08 mm (for the footins on soil condition), which is about 4 times of that (.5 mm) of the result from Approach I. The results from the examples indicate that the deree of footin fixity should not be nelected since it sinificantly affects the manitude of the bendin moment(s) and the deflection(s) of the pier(s). REFERENCES [] American Association of State Hihway and Transportation Officials, AASHTO RFD Bride Desin Specifications, American Association of State Hihway and Transportation Officials, Washinton, DC, 0. [] Duan,, Kin, WS, Chen, WF, K-factor equation to alinment charts for column desin, ACI Structural Journal, Vol. 90, No., pp. 4-48, May-June, 99. [] Dumonteil, P, Simple equations for effective lenth factors, Enineerin Journal, AISC, rd quarter, pp. -5, 99. [4] Portland Cement Association, Notes on ACI 8-08 Buildin Code Requirements for Structural Concrete, Portland Cement Association, Skokie, I, 008. [5] Hassoun, MN, Al-Manaseer,A, Structural Concrete Theory and Desin, 4 th Ed., John Wiley & Sons, Inc., Hoboken, New Jersey, 008. [6] McCormac, JC, Brown, RH, Desin of Reinforced Concrete, 9 th Ed., John Wiley & Sons, Inc., Hoboken, New Jersey, 04.
0 Footin fixity effect on pier deflection [7] Wiht, JK, MacGreor, JG, Reinforced Concrete, Mechanics & Desin, 6 th Ed., Pearson Education, Inc., Upper Saddle River, New Jersey, 0. [8] SAP000 Version 4.., SAP000 Manuals, Computers and Structures, Inc., Berkeley, CA, 00.