1、Section 4 ANALYSIS REQUIREMENTS 4.1 GENERAL The requirements of this section shall control the se- lection and method of seismic analysis of bridges. Four analysis procedures are presented. Procedure 1. Uniform Load Method Procedure 2. Single-Mode Spectral Method Procedure 3. Multimode Spectral Meth
2、od Procedure 4. Time History Method In each method, ali fixed column, pier, or abutment supports are assumed to have the same ground motion at the same instant in time. At movable supports, displace- ments determined from the analysis prescribed in this chapter, which exceed the minimum seat width r
3、equire- ments as specified in Article 6.3 or 7.3, shall be used in design without reduction by the Response Modification Factor (Article 3.7). 4.2 SELECTION OF ANALYSIS METHOD Minimum requirements for the selection of an analysis method for a particular bridge type are given in Table 4.2A. Applicabi
4、lity is determined by the “regularity” of a bridge which is a function of the number of spans and the distribution of weight and stiffness. Regular bridges have less than seven spans, no abrupt or unusual changes in weight, stiffness, or geometry and no large changes in these parameters from span-to
5、-span or support-to-support (abutments excluded). They are defined in Table 4.2B. Any bridge not satisfying the requirements of Table 4.2B TABLE 4.2A Minimum Analysis Requirements Seismic Regular Bridges Not Regular Performance with Bridges with Category 2 Through 6 Spans 2 or More Spans A Not requi
6、red Not required B, C, D Use Procedure Use Procedure 1 or 2 3 is considered to be “not regular.” A more rigorous, gener- ally accepted analysis procedure may be used in lieu of the recommended minimum such as the Time History Method (Procedure 4). Curved bridges comprised of multiple simple spans sh
7、all be considered to be “not regular” bridges if the sub- tended angle in plan is greater than 20”; such bridges shall be analyzed by either Procedure 3 or 4. 4.2.1 Special Requirements for Single-Span Bridges and Bridges in SPC A Notwithstanding the above requirements, detailed seis- mic analysis i
8、s not required for a single-span bridge or for bridges classified as SPC A. 4.2.2 Special Requirements for Curved Bridges A curved continuous-girder bridge may be analyzed as if it were straight provided all of the following require- ments are satisfied: (a) the bridge is regular as defined in Table
9、 4.2B ex- cept that for a two-span bridge the maximum span length ratio from span-to-span must not exceed 2; (b) the subtended angle in plan is not greater than 30”; and TABLE 4.2B Regular Bridge Requirements Parameter Value Number of Spans 23456 Maximum subtended 90” 90” 90” 90” 90” Maximumspanleng
10、th 3 2 2 1.5 1.5 angle (curved bridge) ratio from span-to-span stiffness ratio from span-to-span (excluding abutments) Maximum bentlpier - 4432 Note: All ratios expressed in terms of the smaller value. 453 454 HIGHWAY BRIDGES 4.2.2 (c) the span lengths of the equivalent straight bridge are equal to
11、the arc lengths of the curved bridge. If these requirements are not satisfied, then curved con- tinuous-girder bridges must be analyzed using the actual curved geometry. 4.2.3 Special Requirements for Critical Bridges More rigorous methods of analysis are required for cer- tain classes of important
12、bridges which are considered to be critical structures (e.g., those that are major structures in size and cost or perform a criticai function), andor for those that are geometrically complex and close to active earthquake faults. Time history methods of analysis are recommended for this purpose, pro
13、vided care is taken with both the modeling of the structure and the selection of the input time histories of ground acceleration. Time history methods of analysis are described in Article 4.6. 4.3 UNIFORM LOAD METHOD- PROCEDURE 1 The uniform load method, described in the following steps, may be used
14、 for both transverse and longitudinal earthquake motions. It is essentially an equivalent static method of analysis which uses a uniform lateral load to approximate the effect of seismic loads. The method is suitable for regular bridges that respond principally in their fundamental mode of vibration
15、. Whereas ail dis- placements and most member forces are calculated with good accuracy, the method is known to overestimate the transverse shears at the abutments by up to 100%. If such conservatism is undesirable then the single mode spectral analysis method (Procedure 2) is recommended. Step 1. Ca
16、lculate the static displacements v,(x) due to an assumed uniform load po as shown in Figure 4.4A and Figure 4.4B. The uniform loading po is applied over the length of the bridge; it has units of forcelunit length and may be arbitrarily set equal to 1.0. The static displacement v,(x) has units of len
17、gth. Step 2. Calculate the bridge lateral stiffness, K, and total weight, W, from the following expressions: (4 - 1) (4 - 2) K=- POL VS,MAX W = I w(x)dx where L = total length of the bridge vs, MAX = maximum value of v,(x) and w(x) = weight per unit length of the dead load of the bridge superstructu
18、re and tributary substructure The weight should take into account structural ele- ments and other relevant loads including, but not limited to, pier caps, abutments, columns and footings. Other loads such as live loads may be included. (Generally, the inertia effects of live loads are not included i
19、n the analy- sis; however, the probability of a large live load being on the bridge during an earthquake should be considered when designing bridges with high live-to-dead load ratios which are located in metropolitan areas where traffic con- gestion is likely to occur.) Step 3. Calculate the period
20、 of the bridge, T, using the expression: T=2z - (4 - 3) where g = acceleration of gravity (length/time2) Step 4. Calculate the equivalent static earthquake loading pe from the expression: (4 - 4) where C, = the dimensionless elastic seismic response coefficient given by Equation (3-1) pe = equivalen
21、t uniform static seismic loading per unit length of bridge applied to repre- sent the primary mode of vibration. Step 5. Calculate the displacements and member forces for use in design either by applying pe to the struc- ture and performing a second static analysis or by scaling the results of Step
22、1 by the ratio pJpo. 4.4 SINGLE MODE SPECTRAL ANALYSIS METHOD-PROCEDURE 2 The single mode spectral analysis method described in the following steps may be used for both transverse and longitudinal earthquake motions. Examples illustrating its application are given in the Commentary. Step 1. Calculat
23、e the static displacements v,(x) due to an assumed uniform loading p, as shown in Figure 4.4A. 4.2.2 DIVISION IA-SEISMIC DESIGN 455 FIGURE 4.4A Bridge Deck Subjected to Assumed Transverse and Longitudinal Loading The uniform loading po is applied over the length of the bridge; it has units of forceh
24、nit length and is arbitrarily set equal to 1. The static displacement v,(x) has units of length. Step 2. Calculate factors a, , and y: a = jv,(x)dx = j w(x)v, (x)dx Y = jw(x)v,(x)2dx (4 - 5) (4 - 6) (4 - 7) where w(x) is the weight of the dead load of the bridge su- perstructure and tributary substr
25、ucture (forcehnit length). The computed factors, a, , y, have units of (length2), (force X length), and (force X length2), respectively. The weight should take into account structural ele- ments and other relevant loads including, but not limited to, pier caps, abutments, columns and footings. Other
26、 loads such as live loads may be included. (Generally, the inertia effects of live loads are not included in the analy- sis; however, the probability of a large live load being on the bridge during an earthquake should be considered when designing bridges with high live-to-dead load ratios which are
27、 located in metropolitan areas where traffic con- gestion is likely to occur.) Step 3. Calculate the period of the bridge, T, using the expression: where g = acceleration of gravity (length/time2). Step 4. calculate the equivalent static earthquake loading p,(x) from the expression: (4 - 9) (a) Plan
28、 Transverse Loading (b) Elevation Longitudinal Loading FIGURE 4.4B Bridge Deck Subjected to Equivalent Transverse and Longitudinal Seismic Loading where, C, = the dimensionless elastic seismic response co- efficient given by Equation (3-l), p,(x) = the intensity of the equivalent static seismic load
29、ing applied to represent the primary mode of vibration (forcehit length). Step 5. Apply loading p,(x) to the structure as shown in Figure 4.4B and determine the resulting member forces and displacements for design. 4.5 MULTIMODE SPECTRAL ANALYSIS METHOD-PROCEDURE 3 The multimode response spectrum an
30、alysis should be performed with a suitable space frame linear dynamic analysis computer program. 4.5.1 General The multimode spectral analysis method applies to bridges with irregular geometry which induces coupling in the three coordinate directions within each mode of vibration. These coupling eff
31、ects make it difficult to cat- egorize the modes into simple longitudinal or transverse modes of vibration and, in addition, several modes of vi- bration will in general contribute to the total response of the structure. A computer program with space frame dy- namic analysis capabilities should be u
32、sed to determine coupling effects and multimodal contributions to the final response. Motions applied at the supports in any one of the two horizontal directions will produce forces along both principal axes of the individual members be- cause of the coupling effects. For curved structures, the long
33、itudinal motion shall be directed along a chord con- necting the abutments and the transverse motion shall be applied normal to the chord. Forces due to longitudinal and transverse motions shall be combined as specified in Article 3.9. 456 HIGHWAY BRIDGES 4.2.3 4.5.2 Mathematical Model The bridge sh
34、ould be modeled as a three-dimensional space frame with joints and nodes selected to realistically model the stiffness and inertia effects of the structure. Each joint or node should have six degrees of freedom, three translational and three rotational. The structural mass should be lumped with a mi
35、nimum of three transla- tional inertia terms. The mass should take into account structural elements and other relevant loads including, but not limited to, pier caps, abutments, columns and footings. Other loads such as live loads may be included. (Generally, the inertia ef- fects of live loads are
36、not included in the analysis; how- ever, the probability of a large live load being on the bridge during an earthquake should be considered when designing bridges with high live-to-dead load ratios which are located in metropolitan areas where traffic congestion is likely to occur.) 4.5.2(A) Superst
37、ructure The superstructure should, as a minimum, be modeled as a series of space frame members with nodes at such points as the span quarter points in addition to joints at the ends of each span. Discontinuities should be included in the superstructure at the expansion joints and abutments. Care sho
38、uld be taken to distribute properly the lumped mass inertia effects at these locations. The effect of earth- quake restrainers at expansion joints may be approxi- mated by superimposing one or more linearly elastic members having the stiffness properties of the engaged re- strainer units. 4.5.2(B) S
39、ubstructure The intermediate columns or piers should also be mod- eled as space frame members. Generally, for short, stiff columns having lengths less than one-third of either of the adjacent span lengths, intermediate nodes are not neces- sary. Long, flexible columns should be modeled with in- term
40、ediate nodes at the third points in addition to the joints at the ends of the columns. The model should con- sider the eccentricity of the columns with respect to the superstructure. Foundation conditions at the base of the columns and at the abutments may be modeled using equivalent linear spring c
41、oefficients. 4.5.3 Mode Shapes and Periods The required periods and mode shapes of the bridge in the direction under consideration shall be calculated by established methods for the fixed base condition using the mass and elastic stiffness of the entire seismic resisting system. 4.5.4 Multimode Spec
42、tral Analysis The response should, as a minimum, include the effects of a number of modes equivalent to three times the num- ber of spans up to a maximum of 25 modes. 4.5.5 Combination of Mode Forces and Displacements The member forces and displacements can be esti- mated by combining the respective
43、 response quantities (e.g., force, displacement, or relative displacement) from the individual modes by the Complete Quadratic Combination (CQC) method. The member forces and displacements obtained using the CQC method of com- bining modes is generally adequate for most bridge systems. 4.6 TIME HIST
44、ORY METHOD-PROCEDURE 4 Any step-by-step, time history method of dynamic analysis, that has been validated by experiment and/ or comparative performance with similar methods, may be used provided the following requirements are also satisfied: (a) The time histories of input acceleration used to de- s
45、cribe the earthquake loads shall be selected in consul- tation with the Owner or Owners representative. Un- less otherwise directed, five spectrum-compatible time histories shall be used when site-specific time histories are not available. The spectrum used to generate these five time histories shal
46、l preferably be a site-specific spectrum. In the absence of such a spectrum, the re- sponse coefficient given by Equation (3-1), for the ap- propriate soil type, may be used to generate a spectrum. (b) The sensitivity of the numerical solution to the size of the time step used for the analysis shall
47、 be deter- mined. A sensitivity study shall also be carried out to investigate the effects of variations in assumed mate- rial properties. (c) If an in-elastic time history method of analysis is used, the R-factors permitted by Article 3.7 shall be taken as 1.0 for all substructures and connections.
