DERECHO A LA HUELGA COMPARADO EN VENEZUELA Y ESPAÑA

The statically indeterminate integral abutment region can be idealized as shown in Figure 5-1, where the simplified model is superimposed on an outline of the bridge components. The abutment is idealized as a rigid body from its soffit to the neutral axis of the superstructure. The superstructure is idealized as a frame element with a flexural stiffness of Ks and axial stiffness of ks connect to the top of the rigid body. The axial, lateral, and

rotational stiffnesses of the soil-pile system are represented as a stiffness matrix [K], as described in Section 4.3.2, at the abutment soffit. The matrix coefficients can be calculated following the procedure described in Section 4.3.2 and illustrated in Figure 4-8.

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The response of this idealized model can be determined analytically using mechanics-based equations of equilibrium and compatibility. Figures 5-2(a) and (b) show the deflections and the free body diagram of the idealized integral abutment, respectively, as the bridge expands. The sign conventions for the force effects and deflections are as shown in Figure 4-13. Since the abutment is rigid, the rotations at the pile head (Ѳp) and at the end of the superstructure (Ѳs) must equal the rotation of the abutment (Ѳa) given by:

Ѳa =

∆_s− ∆_p

h ( 5-1 )

where ∆_s and ∆_p are the horizontal displacements at the end of the superstructure and at the pile head, respectively, induced by the temperature variation, temperature gradient, and earth pressure, and h the vertical distance between the neutral axis of the superstructure and the soffit of the abutment.

For a linear elastic response, the moment at the end of the superstructure (Ms ) is:

Ms = Ks(Ѳs− Ѳst) ( 5-2 )

where K_s is the flexural stiffness of the superstructure, Ѳ_s is the rotation at the end of the superstructure due to temperature variation, temperature gradient, and earth pressure, and Ѳ_st is the rotation at the end of the unrestrained superstructure due to temperature variation and temperature gradient only. Similarly, the axial force at the end of the superstructure (Ns) satisfies:

Ns = ks(Δst−Δs) ( 5-3 )

where k_s is the axial stiffness of the superstructure, Δ_s is the lateral displacement at the end of the superstructure due to temperature variation, temperature gradient, and earth pressure, and Δ_st is the lateral displacement at the end of the unrestrained superstructure due to temperature variation and temperature gradient only.

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(a) Deformations (b) Free Body Diagram Figure 5-2: Deformations and Free Body Diagram of the Simplified Model

Vertical force equilibrium requires that:

Np = Vs+ Wa ( 5-4 )

where Np is the axial force applied to the head of the pile, Vs is the shear force at the end

of the superstructure, and Wa is the weight of the abutment. Similarly, horizontal force

equilibrium, requires that:

N_s = P’_p+ ΣV_p ( 5-5 )

where P’p is the resultant of the earth pressures, located a distance (ep) from the neutral axis

of the superstructure, and Vp, is the total shear force at the heads of all piles beneath one

abutment. Finally, moment equilibrium about the point where the neutral axis of the superstructure intersects the vertical axis of the abutment requires:

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where Mp is the total moment at the heads of all piles beneath one abutment. As described

in Section 4.3.2, Mp and ΣVpcan be determined from ∆p, Ѳa, and the coefficients in the

stiffness matrix (K_yy, K_yѲ, K_Ѳy, and K_ѲѲ) representing the soil-pile system as:

ΣM_p = K_Ѳy∆_p+ K_ѲѲѲa ( 5-7 )

ΣV_p = K_yy∆_p+ K_yѲѲa ( 5-8 )

Substitution of Equations 5-2, 5-7 and 5-8 into Equation 5-6, yields:

∆p[−Kyyh + 2KyѲ− K_ѲѲ h − K_s h] + ∆s[−KyѲ+ K_ѲѲ h + K_s h] = P’pep + KsѲst ( 5-9 ) If ∆_s, Ѳ_st, Kyy, KyѲ, KѲѲ, Ks, P’p, ep, and h are known, the lateral pile head deflection (∆p)

can be computed by rearranging Equation 5-9 as:

Then Ѳ_a can be determined using Equation 5-1, the force effects in the pile can be determined from Equations 5-7 and 5-8 and the force effects in the superstructure can be determined from Equation 5-5 and 5-6.

5.2.1

Validation of Equations

The bridge deformations and load effects obtained from the SAP-B analyses can be used to verify Equations 5-1 to 5-9a. Tables 5-1 (a) and (b) compare the deformations and load effects, respectively, at the pile head and at the end of the superstructure obtained from these independent procedures. For consistency with the assumption used to derive Equations 5-1 to 5-9a, the abutment is idealized in the SAP-B analysis as rigid from its soffit to the neutral axis of the superstructure. The results are essentially identical, confirming the validity of Equations 5-1 to 5-9a.

∆p = −∆_s[−KyѲ+K_hѲѲ+K_hs] + P’pep+ KsѲst −K_yyh + 2K_yѲ−KѲѲ h − Ks h (5-9a)

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Table 5-1(a): Deformations at the Pile Head and the End of the Superstructure for Case 1 (Fully Rigid Abutment)

SAP-B Eq. 5-9(a) SAP-B Eq. 5-1

Δp (mm) Δp (mm) Ѳp (rad) Ѳs (rad) Ѳa (rad)

B- 4.8 4.9 0.0029 0.0029 0.0029

K- 5.5 5.4 0.0027 0.0027 0.0027

L- 5.6 5.6 0.0026 0.0026 0.0026

Table 5-1(b): Load Effects at the Pile Head and the End of the Superstructure for Case 1 (Fully Rigid Abutment)

Mp (kN.m) Vp (kN) Ns (kN) Ms (kN.m)

SAP-B Eq. 5-7 SAP-B Eq. 5-8 SAP-B Eq. 5-5 SAP-B Eq. 5-6

B- -482 -483 2067 2068 3140 3140 -7727 -7729

K- -493 -494 1805 1806 2878 2878 -7045 -7048

L- -494 -495 1750 1752 2823 2824 -6902 -6906

Tables 5-2 (a) and (b) repeat the comparison for the case where the abutment is idealized in the SAP-B analysis as rigid only from the neutral axis of the superstructure to the soffit of the girder. In this case Equations 5-1 to 5-9a yield comparable results to those obtained from the SAP-B analysis, with slightly higher load effects at the pile head and superstructure end, and a lower displacement at the pile head. However, the maximum difference for the pile deflection is 2.1%, and the maximum force effects are only slightly overestimated. The SAP-B results in Tables 5-1 and 5-2 are almost identical, suggesting that the abutment beneath the girder soffit acts essentially as a rigid body. Equations 5-1 to 5-9a can therefore be used to estimate the responses of integral abutment bridges, and to check the results of numerical models.

Table 5-2(a): Deformations at the Pile Head and the End of the Superstructure for Case 1 (Partially Rigid Abutment)

SAP-B Eq. 5-9(a) SAP-B Eq. 5-1

Δp (mm) Δs (mm) Δp (mm) Ѳp (rad) Ѳs (rad) Ѳa (rad)

B- 4.8 12.5 4.9 0.0030 0.0028 0.0029

K- 5.5 12.6 5.4 0.0028 0.0026 0.0027

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Table 5-2(b): Load Effects at the Pile Head and the End of the Superstructure for Case 1 (Partially Rigid Abutment)

Mp (kN.m) Vp (kN) Ns (kN) Ms (kN.m)

SAP-B Eq. 5-7 SAP-B Eq. 5-8 SAP-B Eq. 5-5 SAP-B Eq. 5-6

B- -453 -469 2013 2036 3086 3108 -7554 -7630

K- -467 -519 1760 1853 2833 2925 -6901 -7199

L- -468 -501 1708 1746 2781 2818 -6763 -6898