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DR. CELAL N. KOSTEM Fritz Engineering Lab., 13

Lehigh University Bethlehem, PA 18015 USA

----- -- --

THE INTERACTION OF PRIMARY-AND SECONDARY MEMBERS

. IN MULTIGIRDER COMPOSITE BRIDGES USING FINITE ELEMENTS

FRITZ ENGINEERING· · LABORATORY LIBRARY·

Bv . . . . . .

THOMAS A. FISHER . . . . . .

CELAL N. KOSTEM

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

fRITZ ENGINEERING LABORATORY REPORT No. 432.5

I I I I I I I I I I I I I I I I I I I

THE INTERACTION OF PRIMARY AND SECONDARY MEMBERS

IN

MULTIGIRDER COMPOSITE BRIDGES

USING FINITE ELEMENTS

FRITZ ENGINEERING LABORATORY UBRARY

by

Thomas A. Fisher

Celal N. Kostem

Fritz Engineering Laboratory

Department of Civil Engineering Lehigh University

Bethlehem, Pennsylvania

June 1979

Fritz Engineering Laboratory Report No. 432.5


1.

2.

3.

4.

TABLE OF CONTENTS

ABSTRACT

INTRODUCTION

1.1 Problem Statement

1.2 State of the Art

1.3 Fatigue of Welded Structures

~4 Objectives and Scope of Research

FINITE ELEMENT MODELING OF THE BRIDGE SUPERSTRUCTURE

2.1 Description of the Prototype Structure

2.2 Behavior of the Model under Preliminary Loading

Cases

2.3 Critical Load Location

2.4 Parametric Study

2.5 Summary of Observations

FINITE ELEMENT MODELING OF THE WEB GAP REGION

3.1 Description of the Substructure Model


3.2.1 Bracing Members with Moment Connections

3.2.2 Bracing Members with Pinned Connections


DISCUSSION

4.1 Discussion of Observations with Respect to the

Interaction of Primary and Secondary Members

ii

Page

1

2

2

3

9

13

14

14

17

19

20

26

28

28

38

39

44

48

52

53


5.

6.

7.

8.

9.

10.

TABLE OF CONTENTS (continued)

4.1.1 Variable Load Location

4.1. 2 Variable Bracing Member End Restraint

4.2 Discussion of Observations with Respect to

the Secondary Stresses Developed in the Web

·Gap Region

4.2.1 Variable Flange Thickness

4.2.2 Variable Web Thickness

4. 2 .• 3 Variable Gap Length

4.2.4 Variable Bracing Member End Restraint

4 • .3 Interpretation of Observations with Respect

to Structural Fatigue

SUMMARY AND CONCLUSIONS

TABLES

FIGURES

REFERENCES

APPENDIX A - DESCRIPTION OF THE SIMULATED PIN

CONNECTION

ACKNOWLEDGMENTS

iii

Page

53

54

56

56

58

60

65

67

72

77

112

221

223

231


ABSTRACT

An analytical investigation of a 24.38 m (80 ft) simple span multi-

girder (4 girders) composite highway bridge with cross framing is pre

sented. The research employed the finite element method .. HS20-44 truck

loading, as defined in the present AASHTO Specification was used.

The interaction between the primary members, i.e. girders, and

secondary members, i.e. cross framing, is examined. A parametric study

encompassing the variation of the girder dimensions and the secondary

member connection details is conducted. An examination of the secon

dary stresses developed within the web gap region of cut short trans-

verse connection plates is also presented.

The stiffness of the web in the vicinity of the bracing member

connections, and the method of connecting the cross framing members to

the transverse connection plates were found to significantly affect

the interaction of the primary and secondary members. It was also

determined that the effectiveness of cross framing in distributing the

live load is dependent upon load location.

Good agreement was obtained between analytically predicted

stresses, and experimental and field data. Web stresses for tight

fit stiffeners were found to range from 92.7 MPa (13.4 ksi) to

292.3 MPa (42.4 ksi). It was concluded that these stresses could be

safely eliminated by welding the stiffener to the tension flange.

In lieu of welding the transverse connection plate to the tension

flange, it is recommended that a gap length of 8 t to 10 t be used. w w

1


1. INTRODUCTION

1.1 Problem Statement

It has been traditionally assumed that the floor systems of

multigirder composite bridges prevent· twisting of the main girders

(Ref. 1). It is current design practice, therefore, to consider

only the in-plane displacement of these girders (Ref. 2). The out

of-plane deformations which cause secondary stresses, and are induced

by slab behavior, or the interaction of primary (main girders) and

secondary members (cross frames, diaphragms, lateral bracing, etc.),

are not accounted for in the design. Recent cracking of several

bridges, however, as shown in Figs. 1, 2, and 3, has stressed the

importance of considering this out-of-plane movement (Ref. 3).

A study at Lehigh University was conducted which examined out•·

of-plane web displacement caused by the interaction of primary and

secondary members (Ref. 4). Figures 4, 5, and 6 show a schematic of

the out-of-plane displacement tests conducted, and the actual test

setup, respectively. Cracking similar to that observed in Figs. 1

and 2 occurred during these tests and is shown in Fig. 7. The

results of these tests are plotted in Fig. 8.

The data in Fig. 8 indicates that the gap length (g), and the

magnitude of the out-of-plane displacement are the controlling

factors in determining the secondary stresses, thus the fatigue life

of the detail (Ref. 4). It has been reported by J. W. Fisher that

secondary stresses are also developed where the connections have

2


been assumed to be "simple" (Ref. 3). ·Reference 3 states that "No

practical riveted, bolted or welded connection can be completely

flexible." Thus, the end restraint of connected members is also an

area in which the bridge designer and engineer should pay strict

attention if fatigue cracking is to be reduced. Figures 9 and 10

show examples in which "simple" connections have developed moment

resisting capabilities and caused fatigue cracking.

This study will focus on the out-of-plane web displacement

induced by cross framing in a simple span multigirder composite

bridge. It will examine the effects of secondary member end re

straint, and transverse connection plate gap length (g) on the over

all behavior of the bridge. It will examine also their effects on

the magnitude of secondary stresses developed within this gap length.

This study is an attempt to predict analytically the secondary

stresses that actually exist in a real bridge superstructure and to

assist in developing guidelines for typical welded girder details.

1.2 State of the Art

A significant volume of research regarding the distribution of

loads on bridge decks has been conducted since 1948 (Ref. 5). At

that time N. M. Newmark conducted tests on quarter-scale models of

typical I-beam bridges and reported that the distribution of loads

to the beams was determined by the "average stiffness of the slab"

(Ref. 5). A value of relative stiffness (H) was calculated which

compared the stiffness of the beam to a width of slab equal to the

3


span of the beams. This expression for H appears below with the

variables defined:

Eb = modulus of elasticity of beam OMPa)

~ =moment of inertia of beam (mm4)

a = span length (mm)

E = modulus of elasticity of slab OMPa)

(Eq. 1)

I = moment of inertia of unit width of slab--can be taken

as h3 I 12 (mm4 )

h = slab thickness (mm).

This report further stated that "Transverse bridging is not

particularly effective except for loads at or close to the section

where the transverse frames are located" (Ref. 5). This "bridging"

is also not desirable unless the slab is thin, thus flexible. Con-

stants for the computation of girder live-load moments were formu

lated in this report. They are presently used in the 1977 AASHTO

Specification in a revised version (Ref. 2).

A different viewpoint regarding the effectiveness of diaphragms

appeared in the literature in 1957. In two independent papers by

A. M. Lount (Ref. 6), and A. White and W. B. Purnell (Ref. 7), it

was determined that the diaphragms contributed significantly in

distributing the loads. The former paper by Lount was a theoretical

elastic grid analysis of a six girder, s~ple span bridge, 36.58 m

(120 ft) long, and 15.24 m (50 ft) wide with cross frames spaced at

4


7.32 m (24ft). In this paper Lount compared the elastic analysis

to the existing AASHO Specification. Large discrepancies were

revealed between the maximum live load bending moments computed by

the two methods. The latter paper, however, was an experimental

study of a three span continuous structure. Strain gages were placed

on the cross framing as well as the main girders and data accumulated

for various locations of live load. The concluding remarks from

this paper stated that "Effective lateral distribution of test loads

was obtained, with about 80% of lateral transfer of ioad being

through the roadway slab and only 20% being carried by the diaphragm."

Although no specific percentages were stated in Lount's paper the

agreement in results between these two reports strongly indicates

that diaphragms do assist in distributing the live load. The

concensus of most of the literature listed in Reference 8 agrees

with this idea. This theory has been reinforced recently by a

finite element analysis conducted by Mertz and Rhimbos (Ref. 9).

It can be stated from this literature that the effectiveness of

diaphragms and cross framing, in distributing live loads, ranges

from 0 to 20 percent, and depends upon the location of the measure

ments. This same literature, however, does not examine the local

effects that this distribution has on the main girders. The forces

developed in the diaphragms or cross framing, due to this load dis

tribution, are perpendicular to the longitudinal direction of the

girders (i.e. out-of-plane). These out-of-plane forces cause out

of-plane displacements of the girders. It is these displacements

that are presently causing the problems in many welded structures.

5


The magnitude of the displacements or secondary stresses alone

is not the overall concern. It is the fact that these unaccounted

for stresses occur in members that undergo cyclic loading. The

combination of high stress and cyclic loading creates an undesirable

condition (fatigue cracking) that could lead to premature failure of

the member. Section 1.3 briefly explains fatigue and its effects

on the integrity of a structure.

The present AASHTO Specification (1977) considers the lateral

distribution of live load through diaphragms and· cross framing;

however, it does not consider the out-of-plane displacements men-

tioned above. J. W. Fisher explained in a series of reports the

problems associated with out-of-plane web displacement at lateral

connection details (Refs. 3,10,11). Initially, in 1974, he presented

a simplified model which examined the rotational and/or lateral

displacements in the webs of stringers with lateral support. Figure

11 shows a schematic of this deformation in the stringer web (Ref.

11). The out-of-plane bending moment caused by these deformations

is expressed by the following equation and the terms are as defined

below and shown in Fig. 11:

4 EI 9 + 6 EI 6. M = L La (E"q. 2)

e = rotation of the flange relative to the web

L = web length between connection plate and flange (gap length)

6. = out-of-plane displacement

I = moment of inertia of web alone over a finite width.

6

I I I I I I I I I I I I I I I I I I ·I

Fisher suggested two methods to alleviate this problem:

1. Increase the gap length (L) to 100 mm (4 in) or more, which

will reduce the out-of-plane bending moment, and thus reduce

the cyclic stress range.

2. Extend the connection to the top flange and attach it to

this flange so that the web flexure is minimized.

In the same paper Fisher also cited an example of out-of-plane

displacement that past experience in bolted and riveted connections

did not expose. It was associated with girders that had transverse

stiffeners welded to the web and cut short of the flange. During

shipment of these girders fatigue cracks developed at the ends of

the stiffeners as shown in Fig. 12. It was determined that swaying

of the girders during shipment caused a rotation of the flange

relative to the web. Fisher's simplified out-of-plane bending model

(Eq. 2) was used, and revealed that the swaying motion created large

cyclic stress in the gap length which resulted in cracking of the

web. The fatigue cracking was not attributed solely to this high

cyclic stress but was also due to the presence of a weld termination

which is a source of high stress concentration.

To prevent this fatigue cracking it was recommended that stiff

eners be terminated four to six times the web thickness away from the

flange as shown in Fig. 13. Thought should be given also to temporary

loadings that may cause web rotation. A special note in this paper

was placed on the fact "That 'tight fit' stiffeners, which allow a

7


1.59 mm (1/16 in) gap according to AWS (American Welding Society),

will permit rotation to occur which is about the same as cut short

stiffeners" (Ref. 11). 'Tight fit' stiffeners should therefore be

chamfered so that "The web-to-stiffener welds are four to six times

the web thickness above the web-to-flange weld" as shown in Fig. 14.

In 1977 the American Institute of Steel Construction published

a second report entitled "Bridge Fatigue Guide- Design and Details",

by J. W. Fisher. This report reiterated the material presented in

the previous paper and examined two additional examples. It included

the out-of-plane displacements at floor beam connection plates and

lateral gusset plates. Recommendations were made as to how the

problems could be solved. The floor beam connection plate problem

could be alleviated by welding the plate to the tension flange as

shown in Fig. 15. The stress range in this flange, however, must

satisfy Category C restrictions (Ref. 2). The suggested lateral

gusset connections, as detailed in Fisher's report, are shown in

Fig. 16.

Fisher further emphasized the importance of out-of-plane dis

placement by citing a simple example. He showed that if only the

out-of-plane movement, ~' in Eq. 2 was used for a transverse

stiffener with a web gap and web thickness of 12.7 mm (0.5 in), an

out-of-plane movement of only .00254 mm (.0001 in) would create a

web bending stress of 124.11 MPa (18 ksi) (Ref. 3).

8


1.3 Fatigue of Welded Structures

The local problems and failures presented in previous sections

have-been related to structural fatigue. This section was included,

therefore, to give the reader a better understanding of fatigue and

its relationship to bridges.

Structures that undergo cyclic loading, such as bridges,

experience a phenomenon known as structural fatigue. It is a

problem that must be considered by the engineer, since it occurs

in members whose maximum nominal stress is less than the yield stress

of the material. Welded structures are particularly susceptible to

structural fatigue because of the small (possibly large) fabrication

discontinuities that are inherent to the process.

Fatigue life, as defined in fracture-mechanics, is the time

required to initiate a crack and to propagate the crack to a critical

size. Upon reaching this critical dimension, brittle fracture of

the member occurs •

.. Fatigue life is divided, therefore, into two stages, a) initia

tion stage, and b) propagation stage. These stages are shown in

Fig. 17 (Ref. 12). It is obvious from this figure that the initia

tion stage represents a major portion of the total fatigue life of

a member. It can be seen, therefore, that the presence of any

initial flaws or cracks, which act as initiated fatigue cracks, will

shorten the total fatigue life considerably.

9


Fabrication of welded girders creates many initial discontinui-

ties, as previously stated. Typical defects that may occur are

undercutting, incomplete fusion, weld cracking, porosity, slag inclu-

sions, weld overfill, and arc strikes. These defects may vary in

size from .025 mm to .762 mm (.001 in to .03 in), and all act as

initiated fatigue cracks. Since we are examining welded structures,

it is understood that- their fatigue life is essentially composed of

the stage of crack propagation or ~ubcritical-crack-growth. In this

stage a propagating fatigue crack follows one of the existing crack-

n growth "laws", e.g., da/dN ::a A•AK , where

a = crack length

N • number of cycles

~K = stress-intensity-factor fluctuation.

A and n are constants (Ref. 12). The expression above for da/dN

represents the rate of fatigue crack growth and is influenced by

several factors. First, the magnitude of stress range (cr -cr i ) max m n

greatly affects fatigue life because as stress range increases,

fatigue crack growth rate also increases. Initial flaw size is a

second factor that strongly influences fatigue life. This is due to

the fact that fatigue crack growth rate is very low for small cracks

and very high for large cracks. The effects of applied stress level

and crack size on crack growth rate are shown in Fig. 18 (Ref. 13).

The third factor that greatly effects fatigue life is material

toughness. Its influence on life depends, however, upon the pro-

gression of the state of stress, e.g., plane strain to elastic-

10


plastic behavior, or elastic-plastic to plastic behavior. Large

effects on fatigue life result when moving from plane strain to

elastic-plastic behavior. Smaller effects on fatigue life occur

when progressing from elastic-plastic to plastic behavior. These

effects on fatigue life, along with the effects due to stress range

and crack size are shown in Fig. 19 (Ref. 12).

A study conducted at Lehigh University examined the fatigue

life of cut short transverse stiffeners that were displaced out-of

plane (Ref. 4). An attempt was made to correlate experimental data

with fatigue crack growth theory. Five girders were tested with

approximately ten details per girder. Gap lengths equal to 1.25,

2.5, 5, 10, and 20 times the web thickness (tw) were examined for

out-of-plane displacements between 0.013 mm and 2.5 mm (.0005 in

to 0.1 in). A schematic of the test, the actual test setup, typical

fatigue cracking, and a plot of test results are shown in Figs. 4

through 8, as previously mentioned.

The moment at the weld toe of the stiffener was computed using

A, the out-of-plane displacement, only, from Eq. 2, and is expressed

as

6 M = 6 EI -:a gap g~

(Eq. 3)

g is the gap length and I is the moment of inertia of a unit web

strip. The stress range at this location is

A sr = 3 E tw ~ gap

11

(Eq. 4)


The predicted fatigue life (N) expected from the crack growth

relationship

(Eq. 5)

can be estimated as

(Eq. 6)

af and ai are the final and initial crack length, respectively. C is

a constant. (Note: English units shown in parentheses in Eqs. 5 and

6.)

Using Eq. 6, a comparison of fatigue lives for gap lengths of 5

and 10 times the web thickness was made and the results were Ng5/Ngl0

equals 0.015. A similar comparison of test data showed a ratio for

Ng5/Ngl0 of 0.15. The difference between the observed and predicted

values was "Due to rotation of the beam flange and other variances

of the assumed model" (Ref. 4). It was recommended that short web

gaps be avoided if any out-of-plane displacement was expected during

the life of the structure. It was also stated that relatively large

out-of-plane displacements [0.25 mm (0.01 in)] could be tolerated in

large web gaps (10 tw to 20 tw); however, cracks would form in very

small gaps (1.25 tw and 2.5 tw) for displacements of only 0.025 mm

(0.001 in).

12


1.4 Objectives and Scope of Research

There are two specific objectives of the reported research.

The first objective is to determine how primary and secondary bridge

members interact under various locations of loading, and various

bracing member connection details. This will be accomplished by

examining the forces and deformations in the primary and secondary

members of a simple span, multigirder, composite highway bridge with

cross framing.

The second objective of the research study is to determine the

secondary stresses developed in the \veb gap at cut short transverse

stiffeners. The study will examine the effects of the following

variables on the stress within this gap:

1. web thickness

2. bottom flange thickness

3. gap length

4-. end restraint of bracing members •

A global analysis of the subject bridge will be conducted

followed by a refined analysis (substructure model), focusing on

the area of concern. It is expected that the three dimensional

analysis will assist the bridge designer in better understanding the

local problem of cut short stiffeners, and will yield data that may

be correlated to real observations or other research studies. It is

also expected that the parametric study will indicate the best

connection detail for the specific conditions examined.

13

I I I I I I I I I I

, I , I

I I

I I

I I I I

2. FINITE ELEMENT MODELING OF THE BRIDGE SUPERSTRUCTURE

2.1 Description of the Prototype Structure

The subject bridge shown in Figs. 20, 21 and 22 represents a

"typical" superstructure for a multigirder composite steel bridge.

The structure was taken from Reference 14 and is a simple span bridge

24.38 m (80 ft) long, and 10.16 m (33.33 ft) wide. It consists of

a 177.8 mm (7 in) thick composite deck slab supported by four main

girders, 1.22 m (4 ft) deep and 2.54 m (8.33 ft) apart. The top

flange of the main girders is 304.8 mm (12 in) wide and 14.29 mm

(9/16 in) thick while the bottom flange is 304.8 mm (12 in) wide

but varies in thickness from 25.4 mm (1 in) to 44.54 mm (1-3/4 in).

The web of the main girders has a constant thickness of 7.94 mm

(5/16 in). The cross framing is spaced at 6.10 m (20 ft) and

consists of angles at interior locations, and angles and channels

at end bearings. The roadway is composed of two 4.27 m (14 ft)

wide lanes with two outside sidewalks 0.81 m (2.66 ft) wide.

An isometric view of the finite element discretization of this

bridge appears in Fig. 23. SAP IV - A Structural Analysis Program

for Static and Dynamic Response of Linear Systems was used to conduct

the analysis (Ref. 15). Symmetry about midspan was used for three

reasons - a) to reduce the computation time, b) to produce a model

of manageable size, and c) to provide a model that could be un

symmetrically loaded about its centerline. Only half of the actual

structure is shown therefore, in Fig. 23. In using symmetry,

extreme care must be taken to prevent inaccuracies from occurring.

14


This care was exercised during model development. It required that

all section properties of the cross framing and transverse stiffeners

at midspan be reduced by a factor of 0.5.

The deck slab was modeled with 240 plate bending elements.

Figure 24 shows the reference plane of these elements in the finite

element model as well as the centroidal plane of the actual deck

slab. The slight eccentricity (e) of the actual location from the

model location, discussed in Section 2.2, created minor differences

in the overall structural behavior and, therefore, was neglected.

Figure 24 also shows the constant main girder depth of 1250 mm

(49.16 in) used in the analysis, even though the bottom flange varied

in thickness. The small differences in depth that actually existed

were considered negligible in the overall bridge behavior, thus the

depth was assumed to be constant.

The typical cross-sectional discretization at the interior cross

frames, and the end bearing cross frames is also shown in Fig. 24.

The girder web was divided into four plate bending elements through

its depth. This element type was chosen since the study was to

focus on the out-of-plane displacement of the web, and could not

be simulated using a plane stress element. The specific dimensions

of the elements were selected so that the finite element model

resembled the actual structure as closely as possible. The heavy

line in Fig. 24 reveals the close agreement between the discretized

and actual structures.

15


The top flange of the main girders was modeled with truss

elements. This was done because the bridge was a composite system

and bending of the embedded top flange would be negligible. The

bottom flange was simulated, however, with beam elements, since its

out-of-plane stiffness was considered an important factor in exam

ining the problem. The transverse connection plates and stiffeners

were also modeled with beam elements. Milled transverse connection

plates were modeled by permitting axial and shear transfer at the

milled end, and removing all moment transfer at this location. Small

gaps at the end of transverse stiffeners that were not connection

plates were simulated by permitting axial transfer only at the bottom

node and removing all shear and moment transfer at this node. Larger

gaps (225.4 mm (8.875 in)] at the end of transverse connection plates

were represented by removal of the beam element between the bottom

two nodes of the girder.

The cross framing was modeled with two types of elements.

Initially, beam elements were used as the cross framing members in

order to simulate the moment and shear end restraint conditions.

Truss elements were then used to represent the simple, pin-ended

connection.

Preliminary loading of the finite element model was conducted

to verify that the model simulated the real structure. HS20-44 truck

loading was distributed to each girder according to AASHTO Specifi

cations and the impact factor for a 24.38 m (80 ft) span was computed.

A load of 88.52 kN--22.13 kN per girder (19.9 kips--4.975 kips per

16


girder) was placed at midspan and represented factored wheel loads

plus impact. This loading corresponded to the "live-load" deflection

analysis conducted in Reference 14, and thus it permitted a direct

comparison between the planar analysis of the AASHTO Specifications

and a three-dimensional bridge analysis. These live-load deflection

responses and the response under other loadings are tabulated in

Table 1, and discussed further in Section 2.2.

2.2 Behavior of the Model under Preliminary Loading Cases

Prior to beginning the parametric study it was deemed necessary

to determine that the finite element model responded similar to

known theoretical response. The classical methods of deflection

computation were used to obtain the vertical bridge deflection at

midspan and at quarter span (Ref. 16). These values are tabulated

in Table 1 for dead load, and live load plus impact cases. Also

included are the deflection values for a "modified" classical method

in which the deck slab was placed at the centroid of the girder's

top flange. This was done to better simulate the finite element

model. Figure 25 shows the actual cross section and the transformed

cross sections used in the classical computations.

Table 1 shows also the midspan and quarterspan deflections from

two finite element analyses. FEM 1 represents the bridge structure

with a 177.8 mm (7 in) thick deck slab and 203.2 mm (8 in) thick

curbs and sidewalks. FEM 2 represents the structure with a 333.5 mm

(13.13 in) thick slab and 322.8 mm (12.7 in) thick curbs and

17


sidewalks. The latter model has an effective stiffness equal to the

actual bridge, where the centroidal plane of the slab is a distance e

above the centroid of the top flanges, as shown in Fig. 24.

The equations used in the classical method to compute deflec

tions and stresses are shown in Tables 1 and 2, respectively. These

equations assume that the member's material is elastic, homogeneous,

and isotropic. The member is assumed to be prismatic, loaded in

. the plane of the web, and supported at points along its centroidal

axis. It is further assumed that the Bernoulli-Navier assumption,

that plane sections remain plane after bending, also applies.

Very close agreement exists between the "modified" classical

method and FEM 1, as well as between the classical method and FEM 2.

Although this close agreement existed for the vertical deflections,

it was decided that stresses at various locations should be checked

also.

The girder stresses at midspan and quarter span are tabulated

in Table 2. Close agreement exists between the two classical methods

and the two finite element models. Since better agreement was

observed between the classical methods and FEM ~ it was decided that

this model would be used for the parametric study.

Additional examination of FEM 1 was conducted to ensure local

continuity. This model was modified by removing the stiffener (beam

element) between the lowest two nodal points on the girder at the

midspan and quarter span cross framing connections. This simulated

18


a 225.4 mm (8.875 in) gap length (g), as shown in Fig. 4. Comparisons

of horizontal and vertical deflections, as well as girder stresses,

were made between this modified model and FEM 1. No significant

differences were observed in either deflections or stresses. Another

modification of FEM 1, separate from the first, was conducted in

which the cross framing members were changed from beam to truss

elements. No significant differences of deflections or stresses

existed when a comparison to FEM 1 was made.

The insignificant differences that occurred when FEM 1 was

modified, indicated that a finer analysis of the problem area would

have to be conducted. This refined analysis is described in detail

in Chapter 3, and from this point on will be referred to as the

substructure model.

2.3 Critical Load Location

Live loading of bridges may vary greatly due to the variety of

truck weights, sizes, and locations on the bridge. It was therefore,

necessary to establish a typical test loading vehicle and locate a

critical position of this vehicle on the bridge. The typical test

vehicles are shown in Fig. 26. The critical position was defined as

the position which produced the greatest relative horizontal dis

placement between the two lowest nodes of the girders at the interior

cross frames. Equivalent wheel loads were distributed to the deck

slab nodes through the use of simple statics.

19


FEM 1 was loaded with the test vehicles shown in Fig. 26 at

nine various transverse positions. These nine positions are shown

in Fig. 27. While at these locations the test vehicles were also

moved longitudinally to three other positions. A typical loading

sequence at each transverse position is shown in Figs. 28(a) through

28(i).

The maximum relative horizontal displacements for each of the

nine transverse positions are shown in Table 3. The values for

exterior and interior girders are given at both the midspan cross

frames and the quarter span cross frames. These values were obtained

at all transverse positions with vehicle 'A' closest to midspan, as

shown in Fig. 28(a). Due to symmetry, this position simulates two

trucks at midspan, back to back.

The tabulated values in Table 3 indicated that the "critical

position" was the one in which vehicle 'A' was closest to the curb.

This vehicle and position were used therefore in the parametric

study. The values in Table 3 also indicated that the interior girder

at midspan (Girder No. 3 in Fig. 27) was the critical area that should

be examined in the substructure model.


The parametric study that was conducted, examined the bridge

response with test vehicle 'A' in the critical location, previously

determined in Section 2.3, and shown in Fig. 28(a). Web thickness,

bottom flange thickness, bracing member end restraint, and stiffener

20

I I end conditions were varied, and produced thirty-six case studies.

I These case studies are shown in Table 4.

The maximum relative horizontal displacement, the item of

I interest, occurred at an interior girder, as previously shown in

I Table 3. It was decided, therefore, that the end forces of the

bracing members that framed into this girder should be examined.

I Figure 29 reveals these forces with all values nondimensionalized

with respect to Case No. 1. This was accomplished by dividing the

I end forces in each case study by the corresponding end force in Case

I No. 1. The actual forces for Case No. 1 are shown in parentheses.

The diagonal member end forces are the member forces at the point

I where the diagonals cross. The horizontal member end forces are the

forces where these members frame into the adjacent girders.

I •

The two nodal points, designated 3 and 7 in Fig. 29, represent

I the points between which the maximum relative horizontal displacement

I was measured. Table 5 lists this relative displacement for all

thirty-six case studies. The vertical displacement of nodal point 3,

I and the total stress in the bottom flange at midspan, and 2.44 m

(8 ft) from midspan are listed also, in Table 5. All values in this

I table are nondimensionalized with respect to Case No. 1 by dividing

I each case value by the corresponding value of Case No. 1. The actual

values for Case No. 1 appear in parentheses at the top of each column.

I The total stress in the bottom flange of the' interior girder,

I shown in Table 5, represents the combination of stresses caused by

I 21

I

I I I I I I I I I I I I I I

I I

I I I I

bending about the major and minor axes of the composite girder. The

girder bottom flange in the finite element model was simulated with

beam elements. The axial stress in these beam elements, therefore,

corresponds to the major axis bending of the composite girder, while

the strong axis bending in these beam elements corresponds to the

minor axis bending of the composite girder. The shear stresses,

torsional stresses, and weak axis bending stresses of these beam

elements were not included in the total stress computation because

they were generally one-hundredth of the axial stress. This was

considered negligible. The strong axis bending stress, however, was

one-tenth of the axial stress; thus, it was included.

Each grouping of three loading conditions (i.e. Case No. 1,

Case No. 2, and Case No. 3) examined the bridge response while only

the stiffener end condition was varied. This is shown in Table 4.

Examining the data in Fig. 29 and Table 5, it becomes apparent that

within each grouping the 225.4 mm (8.875 in) gapped stiffener

represents the most drastic change from Case No. 1. Figure 29 shows

large variations in member forces, from a factor of 0.08 to 32.20,

while Table 5 indicates increases in relative deflection as high as

60%.

No values of stress in the web gap were included in Table 5

since only one data point existed between nodes 3 and 7. Stresses

from the finite element analysis were computed for this point, which

lies 304.8 mm (12 in) from the actual gap location. The stresses

observed at this point were considered to be insignificant to the

22


study. However, if the values for relative deflection and gap length

from each case study are inserted into Eq. 4, we see that the computed

stress range varies from 97.7 MPa to 149.0 MPa (14.2 ksi to 21.6 ksi).

These stress ranges are prohibitive in bridge structures, thus the

gapped stiffener should be closely examined in the substructure model

of Chapter 3.

Comparison within each group also revealed that the milled

stiffener condition varied very slightly from the welded stiffener

condition. Observed stresses and deflections varied within 2%. Since

these variations were small they may be considered negligible in the

global analysis. Most values of force in the bracing members were

within 10% of the comparable welded stiffener values. A large varia

tion, and change in direction of the end moment did occur in the

right horizontal member. This large variation, however, may be

neglected since the absolute value of the end moment in Case No. 1

was extremely small when compared to the end moments of the other

members. Although the observations above indicated only small dis

crepancies from the welded stiffener condition, it has been

previously shown (Fig. 1) that fatigue cracking may develop at the

end of "tight fit" or milled stiffeners. This condition is examined,

therefore, in the substructure model.

The effectiveness of an increase in web thickness can be deter

mined if Cases 1, 2, and 3 are compared to Cases 4, 5, and 6, re

spectively. The only change between the corresponding cases is a

change in web thickness from 7.94 mm to 9.53 mm (5/16 in. to 3/8 in.)-

23

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a 20% increase in thickness. Table 4 reveals similar comparisons

that can be conducted to determine the effectiveness of this para-

meter. An overall view of these comparisons showed that the bracing

member forces, vertical deflections, and bottom flange stresses,

generally varied up to 6%. This would indicate that an increase in

web thickness of 20% has a very small effect on the overall response

of the structure. The relative horizontal displacement, however,

between nodes 3 and 7 varied as much as 15% when comparable gapped

stiffener cases were examined. This is not a large variation, but

in the local problem examined in the study, it may become significant

when a more refined analysis is conducted. Large differences (42%)

did exist in the right horizontal end moment, but this can be

neglected as before because of the extremely small magnitude of the

end moment in Case No. 1. Also noted was a 26% change in the end

moment and end shear of the right diagonal. This was noted and is

examined further in Chapter 3.

Comparison of member forces, flange stresses, vertical deflec-

tions, and relative horizontal displacements in Cases 1 and 19, 2

and 20, 3 and 21, etc., revealed that an increase in bottom flange

thickness of 6 mm (1/4 in) caused a reduction of up to 10% in these

values. This might be of significant importance in the substructure

model, thus it is included in Chapter 3.

The changes that occur when the bracing member end restraints

are varied must be determined by examining deflections and bottom

flange stresses only. The change in end restraint will cause the

24


member forces to change and, therefore, would provide very little

useful information. Comparison of deflections and stresses between

Cases 1, 2, and 3 and Cases 7, 8, and 9, respectively, showed minute

differences, except for the gapped stiffener condition. A variation

of 9% existed between the relative horizontal displacement of nodes

3 and 7 when Case No. 3 and Case No. 9 were compared. A variation of

7% was observed when the same values were compared for Cases 21 and 27.

This indicated that a shear connection at the end of a bracing member

would increase the relative horizontal displacement and thus create

a larger secondary stress than a moment connection.

Comparison of Cases 1, 2, and 3 with Cases 13, 14, and 15,

respectively, revealed minute variations in most of the deflections

and stresses. However, large differences of relative horizontal

displacement were observed in the cases involving the gapped stiffener

condition. This difference was a 35% increase over the moment end

condition values. A similar variation of 24% was observed when Case

No. 21 was compared with Case No. 33. Thus it is observed that an

ideal "pin-ended" member would produce a worse condition than the

shear connection previously mentioned. (The only difference between

the shear connection and the "pin-ended" condition is that in the

former condition the diagonal member is capable of transferring

shear and moment as well as axial load. This can be seen in Fig. 29~

The stiffness of an actual connection lies between the moment

connection and pin-ended connection. The above comparisons have

shown that the moment end condition provides the most resistance to

25

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out-of-plane web movement and the pin-ended condition provides the

least. It would be advantageous if these two conditions could be

simulated in the substructure model so that an upper and lower

bound of out-of-plane displacement could be obtained.


A close examination of the data presented in Section 2.4 reveals

various trends. These observed trends are listed below:

1) The 225.4 mm (8.875 in.) gapped stiffener had a large

influence on the out-of-plane displacement pattern of

the girder web.

2) The milled stiffener appeared to have little effect on the

out-of-plane displacement of the girder web; however, since

there have been known instances of cracking it should be

examined further.

3) A web thickness increase of 20% did not affect the overall

bridge response more than 6%. However, relative horizontal

displacements between nodes 3 and 7 varied as much as 15% for

the gapped stiffener condition. A 26% change in the end

moment and end shear of the right diagonal also occurred.

4) An increase in flange thickness of 6 mm (1/4 in) resulted in

a reduction in forces, stresses, and deflections up to 10%.

5) The moment end restraint provided the most resistance to

relative horizontal displacement, while the pin-ended member

provided the least.

26

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It is expected that the observations listed above will become

clearer if the variables that caused them are included in a more

refined analysis of the region near the web gap. This refined analy

sis is presented in Chapter 1.

27


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II I

3. FINITE ELEMENT MODELING OF THE WEB GAP REGION

3.1 Description of the Substructure Model

Chapter 2 examined the overall structural response of the

bridge superstructure. The discretization employed in the proto-

type structure, as described in Chapter 2, did not provide a

detailed description of the stress and deformation patterns within

the web gap region. This indicated that a refined analysis of the

web gap region would have to be conducted in order to determine the

stresses and displacements in this region. The area to be examined

in this analysis was determined to be Girder No. 3 (as specified

in Fig. 27) at midspan, and this area is shown in Fig. 30. This

area was designated the "critical" area because the maximum relative

displacement between the end of the cut short stiffener and the

bottom flange occurred at this location for the critical loading

condition, as shown in Table 3. Since the study was examining the

stress within the web gap region, and it was known that the

relative displacement, ~' created the dominant web stress in this

region, it was decided that the area of maximum relative deflection

in the prototype model should be examined in the substructure model.

Thus, the "critical" area represents the location of maximum web

stress for the out-of-plane displacement problem only.

The critical area consisted of a section of girder at midspan

with the bracing members framed into transverse connection plates

which were welded to the girder web. The transverse connection

plates had a variable gap length, and the section of girder was

28


I I

I I I

914.40 mm (36 in) long and 809.63 mm (31.875 in) deep. The diagonal

bracing members extended to a point 1270 mm (50 in) on either side

of the web. This was the location in the prototype model where the

diagonals crossed. The horizontal bracing members extended 2540 mm

(100 in) on either side of the girder web. This represented the

end at which the horizontal members framed into the adjacent girders.

(The ends of the diagonal and horizontal bracing members are the same

as those shown in Fig. 29.) Figure 31 shows the location of the

critical area with respect to the bridge superstructure. The section

of Girder No. 3 within this area is shown shaded in Fig. 32.

A more detailed examination of the membrane stresses and bending

moments in the web of Girder No. 3 near midspan had to be conducted

in order to determine the "optimal" dimensions of the substructure

model. These dimensions were "optimal" in that they provided a model

that produced a detailed and fully acceptable representation of the

stresses and deformations in the web gap region. The membrane stresses

and bending moments from the parametric study, detailed in Chapter 2,

were plotted for Case Studies No. 1 and No. 3. They are shown in

Figs. 33 through 44, and the local coordinate system of the elements

is indicated. These figures show stresses and moments at the centroid

of elements that are as far as three elements off the midspan line.

Similar curves were drawn for several other case studies, but were not

included in the text due to their close agreement with the two cases

presented.

29.

I I I I I I I I I I

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It was essential to the validity of the substructure model that

St. Venant's principle be followed, as applied to possible imper-

fections or stress concentrations at simulated loading points and

boundary conditions. Therefore, dissipation of the effects of these

imperfections or stress concentrations was sought, while proceeding

away from them. The only method available to examine the dissipation

of these local effects in the prototype model was to examine the

membrane stresses and bending moments at various girder cross

sections while proceeding away from the area of prime interest

(i.e., the web gap region which would be most susceptible to changes

in the bracing member connection details). This area of interest

is labeled Q-Q in Figs. 33 through 44. The extent of the dissipa-

tion of local effects was determined by comparing the bending

moments and membrane stresses of Case Study No. 1 to the correspond-

ing values of Case Study No. 3. The extent of dissipation in a

given case study was also examined.

The plots of membrane stress and bending moment in Figs. 33

through 44 showed smooth variations while proceeding away from

midspan. The only abrupt changes occurred in Case Study No. 3 for

membrane stress S (Fig. 40), and bending moment M (Fig. 44). yy xy

Both of these were attributed to the presence of the gapped stiffen-

ers and bracing members, and indicated the local effects caused by

them. Gradual changes were also observed when moving up the web

from the bottom flange. An abrupt change was seen again when S yy

(Fig. 40) and M (Fig. 44) were examined for Case Study No. 3. As xy

30


previously stated, this was due to the presence of the bracing

members and gapped stiffeners.

A comparison between corresponding stresses and moments of

Case Study No. 1 and Case Study No. 3 generally disclosed poor

agreement (greater than 75% difference in values) for all elements

within one element of midspan. This was expected since these

elements were closest to the area of interest and were most suscep

tible to the local effects caused by the bracing members and gapped

stiffeners at midspan. An exception to the general trend was

membrane stress S ; in which case, excellent agreement (±2%) was XX

observed.

Upon proceeding in the X-direction to the second element group

from midspan, it was observed that better correlation existed

between the values from Case No. 1 and Case No. 3. This was anti

cipated since the local effects dissipate while moving away from

the region most susceptible to the local effects. The best corre-

lation,as expected, between stresses and moments from Case No. 1 and

Case No. 3 was observed at the third group of elements off the

midspan line. However, the improvement in correlation of this

group over the second element group could not justify an increase

in the size of the substructure model.

Similar comparisons of membrane stresses and bending moments

were made between Cases 1 and 3 while proceeding away from the

bottom flange in the Y-direction. An improvement in correlation was

observed upon each successive movement away from the region most

31


susceptible to local effects (line Q-Q). It was determined that

the improvement in correlation between the third and fourth element

groups above the bottom flange was not large enough to warrant an

increase in model size. The girder section that is shown with

dashed lines in Fig. 45 was then selected. The dimensions of this

girder section are specified in Fig. 45. These dimensions were

also chosen because they provided data points from which boundary

toads could be determined.

A discretization of the finite element substructure model

appears in Fig. 46. Since the area of main concern was the web

gap region at the end of the cut short stiffeners, the size of the

finite elements changed from a coarse mesh to a fine mesh as this

region was approached. The web in the substructure model was

simulated with 570 plate bending elements, and the bottom flange

and transverse connection plates were modeled with beam elements,

as done in the prototype structure. The diagonal and horizontal

bracing members, however, were not modeled in the same manner as

in the prototype structure. The actual bracing members were struc

tural angles with specific dimensions. Since the mesh size was

fine where these bracing members framed into the transverse connec

tion plates, it was decided that the use of beam elements framing

into single nodal points, as done in the prototype model (see Fig.

24(a)), would not accurately represent the connection detail. Any

eccentricities in the actual detail would not appear, and the

transfer of load through the depth of the connection would not be

32


represented. Thus, 54 plane stress elements were used to simulate

each diagonal member, and 88 plane stress elements were used to

represent each horizontal member.

It was expected that a moment connection at the end of a

bracing member could be simulated if the elastic properties of all

the plane stress elements corresponded to those of the actual

structure. It was further anticipated that the pinned connection

could be simulated by the reduction of the elastic properties in

several specific plane stress elements. A thorough explanation of

this appears in Section 3.2 and Appendix-A.

Figure 47 shows a cross section of the actual horizontal and

diagonal bracing members used in the bridge superstructure. The

"equivalent" plane stress element cross sections used in the sub

structure model appear adjacent to these. Equivalency was deter

mined by comparing the section properties of the actual members

with the properties of the plane stress cross section. Since it

was known from examination of the prototype model that bending of

the bracing members about the Y-axis (Fig. 47) was negligible, the

only properties compared were those in Fig. 47. The use of plane

stress elements would not have been permitted if bending about the

Y-axis was significant. The properties shown in Fig. 47 were the

only bracing member properties pertinent to the interaction of the

girders and the bracing members.

It was necessary that the response of the substructure model

closely resemble the response of the prototype model. To accomplish

33


this, various degrees-of-freedom of boundary nodal points were fixed.

These boundary nodal points were along edges of the substructure model

that were continuous in the prototype model. Figure 46 shows these

edges as lines AB, AC, BD, EF, GH, IJ, and KL. Also shown in Fig. 46

are the directions in which the boundary nodal points along these edges

were fixed.

The web plate of the substructure model, defined by points A-B-

D-C in Fig. 46, is shown in Fig. 48. Also shown are the plots of mem-

brane stresses S and S for Case No. 1. These stresses are plotted XX yy

on the substructure boundaries to which they apply, and were plotted

by merely connecting the adjacent centroidal data points (i.e. A-1, A-2,

2-3) with a straight line. Since no data points existed below point 3

or to the right of point 1, the straight line between the two preceding

data points was extended to the boundary of the substructure model.

Figure 48 will be used to assist in describing the procedure used to

load the substructure model.

Loading of the substructure model for each case study was

accomplished using the following procedure:

1. Girder section boundary nodes

a) Obtain the membrane stresses and bending moments at the centroid of the plate bending elements that lie on the boundary of the substructure model, from the prototype case study data (i.e. points A, 1, 2, and 3 in Fig. 48). (Stresses and moments applied to the top boundary (line A-1-B) were S , S , M , M . Stresses and moments applied to th~ylef£Ysid~ybou6Xary (line A-2-3-C) were S , S , M , M .)

XX xy XX xy

b) Linearly interpolate between, and linearly extrapolate beyond these values to obtain the stresses and moments at all the substructure's boundary nodal points.

34


I I I I

c)

d)

e)

f)

g)

Determine the forces and moments to be applied at these boundary nodal points by multiplying the nodal stresses by the appropriate area (S x (X1/2 + x2/2) t ), and by yy w multiplying the nodal moments by the appropriate length (Myy x (X1/2 + x2/2)). This is typical for both the top

and left boundaries.

Obtain the beam element end forces for the bottom flange beam element @-G) from the prototype case ~tudy data.

Obtain the beam element end forces for the transverse connection plate beam element @)- CZ) from the prototype case study data.

Using the end forces from d and e above, compute the forces in the bottom flange and transverse connection plate at points C and B, respectively.

Add these forces to the forces computed from c above.

2. ·Bracing member free end nodes

a) Diagonals

1) Obtain the bracing member free end forces from the prototype case study data (see Fig. 29).

2) Apply the end moments to nodes M and N, as shown in Fig. 46 where applicable.

3) Determine the normal stresses and shear stresses at the free ends due to the end forces.

4) Compute the nodal forces by multiplying the stresses by the appropriate area - similar to the computations made for the plate bending elements that represented the web (see Fig. 47).

5) Transform the normal and shear nodal forces into global axes forces by using force transformations.

b) Horizontals

1) Obtain the bracing member free end forces from the prototype case study data (see Fig. 29).

2) Apply the end moment at the neutral axis of the equivalent horizontal, as shown in Fig. 47, and compute the stresses at the nodal points in the cross section.

35


3) Compute the normal stress and shear stress at the free ends due to the end forces.

4) Add the normal stress due to bending to the normal stress due to axial load, and compute the axial nodal forces by multiplying the nodal stress by the appropriate area - similar to the computations made for the plate bending elements that represented the web (see Fig. 47).

5) Compute the shear forces by multiplying the nodal stresses by the appropriate area.

The substructure model was deformed (loaded) using boundary

forces (hereby designated as boundary loads) instead of boundary

displacements because better insight existed regarding the stress

pattern, than existed regarding the displacement pattern of the

prototype structure. This method of applying boundary loads also

provided a means of correlating the substructure model response

to the prototype structure's response. This means of correlation

was accomplished by comparing the relative displacements between

nodal points in the prototype model with the relative displacements

of the same nodal points in the substructure model. Relative

deflections were also used since the study focused on the relative

deflection of the bottom flange and the end of a cut short connec-

tion plate. The comparison of these relative displacements was

the only method used to determine if the substructure model and the

prototype model responded in a similar manner.

The nodal points, whose relative displacements were compared,

are shown in Fig. 48 as solid circles. The prototype nodal point

number appears in parentheses adjacent to each point, and the

36


substructure nodal point number is shown in brackets. Table 6

defines and lists the relative deflections that were compared for

Case No. 1 and Case No. 3. Excellent correlation (±1%) was observed

for both case studies when relative deflections in the global X

direction were compared. Very good agreement (±10%) was observed

when most values of relative displacement in the Z-direction were

correlated. Exceptions to this occurred in Case No. 1 when

~~z(3_ 7 ) and ~oz(7_ 13 ) were compared to ~oz(620_ 603 ) and ~oz(603_ 592 ),

respectively. A similar exception occurred in Case No. 3 when

~oz( 7_ 13 ) was compared to ~5z(603_ 592 ). Since the magnitude of

these differences was very small (.0033 mm (.00013 in)), they were

deemed negligible to the response of the substructure model. Poor

agreement (over 100% difference) was observed upon comparing

My(3-39)' My(7-43)' and My(l3-47) to My(620-93)' 65y(603-76)'

and ~5y(S92_ 6S)' respectively. This was expected since the nodal

points along line A-C in Fig. 48 were fixed in the Y-direction,

and any out-of-plane force (Y-direction) applied away from this

line would displace the free end (line B-D in Fig. 48) considerably.

Since the relative deflection of main importance was not between

points in different cross sections along the length of the bridge,

but was between points within the same cross section, this large

discrepancy was neglected.

The relative deflection of most importance was in the Y

direction between nodal points 3 and 7 of the prototype structure.

A comparison of this relative deflection with that from the

37


substructure model showed fair correlation (±38%) for Case Study

No. 3, and good correlation (±18%) for Case Study No. 1. Although

this agreement was not as close as expected, it was decided to

accept the substructure model since the magnitudes of the relative

displacement were within the range of test values examined in

Reference 4 (i.e. between 0.013 mm (.0005 in) and 2.5 mm (0.1 in)).


The parametric study described in Chapter 2 examined the sub

ject bridge under fixed loads while varying the flange thickness,

web thickness, gap length, and bracing member end restraint.

Various trends were observed, one of these being that a bracing

member with a moment connection provided the most resistance to

relative horizontal displacement. The pin-ended bracing member

provided the least. It was decided, therefore, to attempt to

simulate both of these end conditions in the substructure model.

The remainder of this section has been divided into two parts in

order to examine both end restraint conditions, while varying

other parameters. Section 3.2.1 will examine the substructure model

under various web thicknesses, flange thicknesses, and gap lengths

with a moment connection at the end of the bracing members. These

case studies with the appropriate parameters are shown in Table 7.

Section 3.2.2 will examine the substructure model under the same

varied parameters but with a "pinned" connection at the end of the

bracing members. The case studies examined in this part of Section

3.2 are listed in Table 8.

38

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3.2.1 Bracing Members with Moment Connections

The moment connections at the end of the bracing members were

simulated by framing the equivalent horizontal and diagonal

members into the web of the girder, as shown in Fig. 49. A constant

modulus of elasticity (200000 MPa (29000 ksi)) was used for all

the plane stress elements in these members. The transverse conn

ection plates shown in Fig. 49 were modeled with beam elements

instead of plane stress or plate bending elements for several

reasons: a) the beam elements provided the appropriate stiffness,

b) the choice of plane stress or plate bending elements would have

increased the model size significantly, and c) the increased model

size would not have improved the accuracy to any great extent.

Also specified in Fig. 49 is the location of the equivalent bracing

members with respect to the centroid of the bottom flange. This

closely simulated the cross framing connections that were used in

the prototype model, as shown in Fig. 24(a). This orientation

also permitted large flexibility in selecting gap lengths for the

transverse connection plates.

As previously stated, all case studies examined in this part

of Section 3.2 are listed in Table 7. All boundary loads for these

cases were computed as specified in Section 3.1. The welded

stiffeners (transverse connection plates) in Cases 1, 4, 19, and

22 were modeled by extending the beam elements that represented

these stiffeners to the bottom flange. Axial load, shear, and moment

transfer were permitted in all of these beam elements. The milled

39


(tight fit) stiffeners in Cases 2, 5, 20 and 23 were simulated in

the same manner as the welded stiffeners, except that only axial

and shear load were permitted to be transferred at the milled end.

In Figures 48 and 49, the welded and milled stiffeners extend from

point B to point D. Point D represents the welded or milled end of

the stiffener.

The 225.4 mm (8.875 in) gapped stiffener in Cases 3, 6, 21, and

24.could not be simulated precisely in the substructure model. This

was due to the existence of only 203.2 mm (8 in) between the bottom

of the equivalent bracing members and the bottom flange of the girder,

as shown in Fig. 49. Therefore, the largest gap length examined in

the substructure model was 203.2 mm (8 in). Other gap lengths

between 0.0 mm and 203.2 mm (8 in) were examined for each case

study above, even though only one set of boundary loads existed for

each of th~se cases. The gap lengths examined were 12.7 mm, 25.4 mm,

50.8 mm, 101.6 mm, 152.4 mm, and 203.2 mm (.5 in, 1 in, 2 in, 4 in,

6 in, and 8 in). These lengths corresponded to removal of the

stiffener beam elements between point D and point 14, point 13,

point 12, point 11, point 10, and point 9, respectively, in Fig. 48.

The use of boundary loads from Cases 3, 6, 21, and 24 instead

of boundary loads from Cases 1, 2, 5, 6, etc., was done in order

to obtain conservative results for the gapped condition. These

case loadings were deemed conservative by examining the resultant

forces at nodal point 7 for the member end forces shown in Fig. 29.

These resultant forces are tabulated in Table 9 for each case study

40

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that is examined in Section 3.2. The resultant forces of main

importance were the moment and the out-of-plane force, F • These y

were of main concern since the rotation, e, and the displacement,

~' seen in Eq. 2, were caused by this moment and force, respectivel~

Examination of these resultants showed a reduction in both values

of 70% to 85% when proceeding from the welded stiffener condition

to the 225.4 mm (8.875 in) gapped stiffener condition (i.e. Case

No. 1 to Case No. 3). It is evident that the resultant forces

for gap lengths less than 225.4 mm (8.875 in) should lie between

the two extreme values shown in Table 9 (i.e., between Case No. 1

and Case No. 3). It is also expected that these resultant forces

should gradually decrease as the 225.4 mm (8.875 in) gapped condi-

tion is approached. If the resultant forces from this gapped

condition were used in a substructure model with gap lengths less

than this value, the stresses and deflections obtained would be

less than the "actual" values. Thus, if prohibitive stresses were

obtained within the web gap region for this "minimal" loading,

larger stresses would be expected under the "actual" loading.

The area of prime interest in the substructure model was the

web section within the gap of the cut short transverse connection

plates, as previously stated. This web section is shown in Fig. 50,

and the force creating the dominant stress is described. This

force was the bending moment M , and as shown in Fig. 50, created yy

a maximum fiber stress in the web. This stress was the only web

stress considered throughout the remainder of this study since it

41


was noted that the membrane stress in this direction was only one

tenth of the stress created by M yy

The distortion of the cross section between point 9 and point

D in Fig. 49 for Cases 1, 2, and 3a through 3f are shown in Figs.

51 through 58. Similar deflection patterns were observed for the

remaining cases listed in Table 7. These deflection patterns are

not presented in the text; however, relative displacements and

relative rotations between the flange and the end of the stiffener

for these cases are tabulated in Table 10. These relative hori-

zontal displacements and relative rotations are plotted as a func

tion of gap length in Figs. 59 and 60, respectively. Figure 61

shows a plot of maximum web gap stress versus gap length. The

web stress was obtained from M as detailed in Fig. 50. These yy

stresses are also listed in Table 11.

Comparison of Figs. 51 and 52 showed a reduction in flange

rotation of approximately 50% when the stiffener was tight fit

(milled) instead of welded to the bottom flange. This created a

relative rotation between the flange and the web of 0.00139 radians,

which did not exist in the welded condition. This was typical for

all welded-milled comparisons as shown in Table 10. A comparison

of maximum web stress between these two conditions (see Table 11)

showed an extremely large increase when proceeding from the welded

to the milled condition. Since the relative horizontal displace

ments for the welded and milled conditions were essentially equal

(see Table 10), the increased stress could only be attributed to the

42


relative rotation between the flange and the web. The magnitude of

web stress in the milled condition was about 102 MPa (15 ksi) which

agreed with field observations (Ref. 17). This high web stress

was located at the milled end of the stiffener.

Figure 59 reveals that an increase in bottom flange thickness

of 14% had no influence on the relative horizontal displacement

between the bottom flange and the end of the cut short stiffener.

Figures 60 and 61 show that this same increase in bottom flange

thickness did not affect relative rotation or web gap stress either.

The same three figures (59, 60 and 61) do show, however, that

a 20% increase in web thickness did affect the displacement,

rotation, and stresses within the web gap region. Figure 59 reveals

that relative horizontal displacement was reduced When the web

thickness was increased. This should be expected since an increase

in web thickness increases the stiffness of the section, thus

reducing the relative displacements. The percent reduction in

relative displacements increased as the gap length increased. This

reduction was 16% for a 12.7 mm (.5 in) gap length and 30% for a

203.2 mm (8 in) gap length.

Figure 60 shows that relative rotation in the web gap region

was reduced about 10% as the web thickness was increased, except

for the 203.2 mm (8 in) gap length. The general reduction was

expected since the loading was constant and the stiffness had

been increased.

43


Figure 61 reveals a reduction in the maximum web gap stress

when the web thickness was increased, except for the 12.7 mm ~5 in)

gap length and the milled stiffener. In these cases web stress

increased with an increase in web thickness. As the gap length was

enlarged, however, the increased thickness became more significant.

In fact, the reduction in stress increased from 16% for a 50.8 mm

(2 in) gap length to 31% for a 203.2 mm (8 in) gap length.

Further examination of Figure 59 shows that for a fixed

loading condition (i.e. boundary loads from prototype Case No. 3·

were used for substructure cases 3a through 3f), an increase in

gap length caused an increase in relative horizontal displacement.

Figure 60 also shows a similar increase in relative rotation as

gap length increased, except when proceeding from the 152.4 mm

(6 in) gap length to the 203.2 mm (8 in) gap length. These increases

in relative displacement and relative rotation caused the web stress

to increase also as gap length increased. This is the general

trend shown in Fig. 61. This figure shows that maximum web stress

increased as gap length increased from 50.8 mm (2 in) to 203.2 mm

(8 in). It was also observed that maximum web stress increased

while proceeding from the 12.7 mm (.5 in) gap length to the milled

condition.

3.2.2 Bracing Members with Pinned Connections

The pinned connections at the end of the bracing members were

simulated by framing the equivalent horizontal and diagonal bracing

44


members into the girder web as shown in Fig. 62. Several plane

stress elements through the depth of these equivalent members were

assigned a reduced modulus of elasticity, thereby minimizing the

moment transfer capability of the equivalent bracing member. These

elements are shown shaded in Fig. 62. Various reduced modulus of

elasticity values were examined. The value selected was that value

which most closely approximated the pinned connection. Simple

statics was used to determine this, and these computations and

comments are given in Appendix A.

The closest approximation was achieved by using a reduced

modulus of elasticity of 20.0 MPa (2.9 ksi). This change in the

substructure model was the only difference between the two finite

element models used to analyze the moment and pinned connections.

The boundary loads were changed, but these corresponded to the

appropriate prototype model "pinned" case studies, as shown in

Table 8.

The gapped stiffener case studies shown in Table 8 (i.e. Cases

15a through 15f, etc.) were loaded with boundary loads obtained

from the corresponding prototype model, which had a 225.4 mm (8.875

in) gapped stiffener. This procedure was the same as that detailed

in Section 3.2.1.

Figures 63 through 70 show the distortion of the web gap region

for Cases 13, 14, and 15a through 15f. Similar deflections were

observed for the remaining case studies in Table 8 but are not shown

in the text. The relative horizontal displacement and relative

45


rotation between the flange and the end of the stiffener for each

of these cases are tabulated in Table 12. Figures 71 and 72 show

respective plots of these relative displacements and relative

rotations versus gap length. Figure 73 is a plot of maximum web

gap stress versus gap length. The web stress was obtained from M YY

as previously detailed in Fig. 50. These stresses are also listed

in Table 13.

Comparison of Figures 63 and 64 showed a reduction in flange

rotation of about 50% when the stiffener was tight fit (milled)

instead of welded to the bottom flange. This created a relative

rotation between the flange and the web of 0.00396 radians, which

did not exist in the welded condition. This was typical for all

welded-milled comparisons as shown in Table 12. This observation

was also consistent with the results from Section 3.2.1 for the

bracing members with moment connections.

A comparison of maximum web stress between the welded and

milled conditions, as shown in Table 13, disclosed an extremely

large increase in stress when proceeding from the welded to the

milled condition. This increase was also observed in Section 3.2.1

and, as stated there, must be attributed to the relative rotation

between the flange and the web. The magnitude of the observed web

stress was about 275 MPa (40 ksi) and this agreed with observed

field data (Ref. 17). This high web stress also existed at the

milled end of the stiffener.

46


Figures 71 and 73, respectively, show that an increase in

bottom flange thickness of 14% did not influence the relative

horizontal displacement, or web gap stress. Figure 72 reveals

that this increase in flange thickness also affected relative

rotation very little. A small increase was observed when Case No.

15 was compared to Case.No. 33 but this increase was less than 8%.

These results were consistent again with the observations for the

bracing members with moment connections.

Figures 71, 72 and 73 show that a 20% increase in web thickness

did influence the relative displacement, relative rotation, and

stresses within the web gap region. Figure 71 shows that relative

horizontal displacement was reduced when the web thickness was

increased. The percentage of reduction in relative displacement

increased from 7% for a 12.7 mm (.5 in) gap length to 20% for a

203.2 mm (8 in) gap length.

The relative rotation was also affected by the increase in web

thickness and Figure 72 shows that the percent reduction was

dependent upon the flange thickness. Comparison of Cases 15 and 18

(44.5 mm [1-3/4 in] thick bottom flange) disclosed a reduction in

relative rotation that was less than 10%. Comparison of Cases 33

and 36 (50.8 mm [2 in] thick bottom flange), however, showed

reductions up to 20%.

Various trends were observed in Fig. 73 when the web thickness

was increased. It was observed for gap lengths less than 50.8 mm

47


(2 in) that the maximum web gap stress increased by about 30% when

the web thickness was increased. Thus, an increase in web thickness

for small gap lengths had an undesirable effect on the stress within

the gapped region. For gap lengths greater than 50.8 mm (2 in),

however, the maximum web gap stress was reduced by about 15% as

the web thickness was increased.

Further examination of Figures 71 and 72 revealed that both

relative horizontal displacement and relative rotation increased

as gap length increased. An exception to this trend for relative

rotation occurred when the gap length increased from 152.4 mm (6 in)

to 203.2 mm (8 in). This observation was consistent with that

observed in Fig. 60 for the bracing members with moment connections.

It is expected that this reduction in relative rotation occurred

because the stiffness of the bracing members that frame into the

girder became much more influential for the larger gap length.

Further examination of Figure 73 however, discloses a trend

that is opposite to that observed in Section 3.2.1. It shows that

the maximum web gap stress decreased as the gap length increased.

Upon reaching a gap length of 101.6 mm (4 in), it was observed that

the stress remained almost constant. The observed constant stress

was about 66 MPa (9.5 ksi).


A close examination of the data presented in Section 3.2

discloses the following trends:

48

I I I I I I I I I I I

I I I I I I I

A) Bracing members with moment connections

1) A reduction in flange rotation of 50% occurred when the milled condition was compared to the welded condition. This produced a relative rotation between the web and the flange of about 0.00139 radians, which did not exist in the welded condition. The maximum web gap stress increased from 1.0 MPa (.15 ksi) for the welded case to 102 MPa (15 ksi) for the milled case. This high web stress existed at the milled end of the stiffener and was attributed to the relative rotation previously mentioned.

2) A 14% increase in bottom flange thickness did not affect the relative horizontal displacement, relative rotation, or maximum stress within the web gap region.

3) a) A 20% increase in web thickness reduced the relative horizontal displacement at an increasing rate as gap length increased from 12.7 mm ( .5 in). A 16% reduction was observed for a gap length of 12.7 mm (.5 in) and a 30% reduction was observed for a 203.2 mm (8 in) gap length.

b) This increase in web thickness also consistently reduced relative rotation by about 10% except for the 203.2 mm (8 in) gap length.

c) The 20% increase in web thickness reduced the maximum web gap stress at an increasing rate (16% to 31%) while proceeding from a 50.8 mm (2 in) gap length to a 203.2 mm (8 in) gap length. The magnitude of stress for these gap lengths ranged from 54.4 MPa (7.9 ksi) to 101.2 MPa (14.7 ksi). However, for gap lengths less than 25.4 mm (1 in) it was observed that an increase in web thickness caused a 10% to 20% increase in stress. A comparison of relative horizontal displacements and relative rotations for these small gap lengths revealed that stresses increased while the displacements and . rotations decreased. Stresses for these small gaps ranged from 58.1 MPa (8.4 ksi) to 112.1 MPa (16.3 ksi).

4) Under a fixed loading condition an increase in gap length from 12.7 mm (.5 in) to 203.2 mm (8 in) caused a continuous increase in relative horizontal displacement and relative rotation in the web gap region. An exception to this occurred in the relative rotation while proceeding from a 152.4 mm (6 in) gap length to a 203.2 mm (8 in) gap length. This occurred because of the influence of the stiffness of the bracing members at the larger gap length. It was observed that under a fixed loading condition the maximum web stress increased as gap length increased from 50.8 mm

49


(2 in) to 203.2 mm (8 in). It was also observed that stress increased while proceeding from the 12.7 mrn (.5 in) gap length to the milled condition. All observed stresses were greater than 55.2 MPa (8.0 ksi).

B) Bracing members with pinned connections

1) A reduction in flange rotation of 50% occurred when the milled condition was compared to the welded condition. This produced a relative rotation between the web and the flange of about 0.00380 radians, which did not exist in the welded condition. The maximum web gap stress increased from 2.2 MPa (.32 ksi) for the welded case to 275 MPa (40 ksi) for the milled case. This high web stress existed at the milled end of the stiffener and was attributed to the relative rotation previously mentioned.

2) A 14% increase in the thickness of the bottom flange did not affect the relative horizontal displacement, relative rotation, or maximum stress in the web gap region.

3) a) A 20% increase in web thickness reduced the relative horizontal displacement 7% for a 12.7 mrn (.5 in) gap length. This reduction increased as gap length increased until a 20% reduction occurred for a 203.2 mm (8 in) gap length.

b) The increase in web thickness reduced relative rotation; however, the percent reduction depended upon bottom flange thickness. A less than 10% reduction occurred for cases with a 44.5 mrn (1-3/4 in) bottom flange, while a reduction up to 20% was observed for cases with a 50.8 mm (2 in) bottom flange.

c) The 20% increase in web thickness reduced the maximum stress by about 15% for gap lengths greater than 50.8 m (2 in). Web stresses for these gap lengths ranged from 58.1 MPa (8.4 ksi) to 77.5 MPa (11.2 ksi). However, for gap lengths less than 50.8 mm (2 in) the maximum web stress increased about 30%. Web stresses for these gap lengths ranged from 73.3 MPa (10.6 ksi) to 288.9 MPa (41.9 ksi).

4) Under a fixed loading condition an increase in gap length from 12.7 mm (.5 in) to 203.2 mm (8 in) revealed a continuous increase in relative horizontal displacement and relative rotation. An exception to this trend occurred when the relative rotation for a 52.4 mm (6 in) gap length was compared to the relative rotation for a 203.4 mm (8 in) gap length. A reduction in rotation occurred when this comparison was made. This reduction was attributed to

50


the influence of the stiffness of the bracing members. It was also observed that maximum web stress decreased as gap length increased to a length of 101.6 mm (4 in). A constant stress of about 66 MPa (9.5 ksi) was observed for gap lengths larger than 101.6 mm (4 in).

A comparison of the moment connection data and the pinned

connection data, along with a discussion of observations, is

included in Chapter 4.

51


4. DISCUSSION

It was stated in Chapter 1 that it is traditionally assumed that

the floor systems of multigirder composite bridges prevent twisting

of the main girders. Results presented in Chapter 2 (Table 5) and

Chapter 3 (Tables 10, 12, and Figs. 51 through 58, 63 through 70)

indicated this to be a gross assumption with respect to the bridge

superstructure examined in this study. Specifically, relative dis

placements of the girder web, comparable to those obtained experi

mentally in Ref. 4, were obtained when the subject bridge was loaded

with a typical vehicle. These displacements occurred at the section

of the girder where the cross framing was located and are shown in

Figs. 51 through 58 and Figs. 63 through 70. Web stress in this region

ranged from 58.1 MPa (8.4 ksi) to 292.3 MPa (42.4 ksi). The fact that

typical connection details caused localized stresses of this magnitude

which are not considered in design, implied that current design practice

should be reassessed, maybe even modified.

The magnitude of the localized stresses has been shown to vary

with changes in stiffness of the girder, as well as with changes in

the rigidity of the bracing member end connection. The remainder of

Chapter 4 discusses these variations of stress in light of current

design practice, and the present methods available for considering

these localized stresses. The implications of the observations made

in Chapters 2 and 3, as they apply to the objectives stated in

Section 1.4, are also discussed.

52


It should be noted that these implications are applicable only

to the bridge superstructure examined in this study and to similar

superstructures. Because of the limited nature of the bridge

geometry, loading, and parametric study reported herein, the findings

should not be directly applied to all steel bridge superstructures of

this type.. However, it is expected that the findings will be appli

cable to many existing bridges because the bridge superstructure

examined in this study was a typical multigirder composite structure.

4.1 Discussion of Observations with Respect to the Interaction

of Primary and Secondary Members

4.1.1 Variable Load Location

The loading scheme of the prototype structure disclosed that the

"critical" loading position was the position shown in Fig. 28(a).

This loading produced the maximum relative horizontal displacement

between nodal points 3 and 7 of Girder No. 3. The data in Table 3

shows that this relative displacement was reduced as the vehicle was

moved from Position 1 to Position 9. It was observed that this re

lative deflection was also reduced as the vehicle was moved away

from midspan. These observations indicated that the interaction of

primary and secondary members depended upon the position of the

vehicle with respect to the cross framing, as well as the vehicle's

position with respect to the longitudinal centerline of the bridge.

It may be concluded, therefore, that the effectiveness of the cross

framing in distributing the live load was dependent upon load location.

53

I I This is consistent with the literature. However, the

I local effects caused by connecting the secondary and primary members

together should be of concern also, since these connections transfer

I up to 20% of the live load. The data in Table 3 disclosed that

local effects occurred near these connections even for the doubly

I symmetric loading of Position 9·. These relative horizontal dis-

I placements (local effects) would not cause problems if the transverse

connection plate was welded to the bottom flange. However, a web

I stress of 84.7 MPa (12.3 ksi), would be developed according to Eq.

I 4, if the maximum relative displacement of Position 1 occurred with

a gap length of 225.4 mm (8.875.in). A stress of this magnitude

I could cause fatigue cracking, and should be considered in design.

I 4.1.2 Variable Bracing Member End Restraint

The prototype structure was examined with the bracing members

I framed into the girders in three ways: a) moment connections, b)

I shear connections, c) pi~ connections. .The change in bracing member

forces that occurred when the "connection" was varied is shown in

I Fig. 29. From a comparison of· member forces for Cases 1, 7, and.l3, '

it was evident that the bracing.member end restraint greatly in-

I fluenced the interaction of the primary and secondary members.

I Examination of data in Table 5 disclosed minor differences in the

bottom flange stress and the vertical deflection when the bracing

I member end restraint was changed. Minor differences for all

I conditions were also observed in the relative horizontal displacement

between nodal points 3 and 7, except for the gapped stiffener

I 54

I

I I I I I I I I I I I I I I I I I

condition. For this condition the relative displacement increased by

9% when the moment connection was changed to a shear connection, and

increased by 23% when the moment connection was changed to a pin

connection. This indicated that the interaction of the primary and

secondary members was also influenced by the web stiffness at the

location of the cross framing connection. This dependence on "web

stiffness" was reflected in the change in bracing member forces when

Case No. 1 and Case No. 3 in Fig. 29 were compared. The dependence

on web stiffness was revealed again when the resultant force, F , and y

resultant moment in Table 9 were also compared for Cases 1 and 3.

Examination of the substructure model with various "web

stiffnesses" (i.e. a change in transverse stiffener gap length changes

the web stiffness at the connection) reinforced the idea that the

interaction of primary and secondary members was dependent upon web

stiffness. This was disclosed by the variations in relative dis-

placement, relative rotation, and maximum web stress which occurred

when the gap length was varied. This is shown in Figs. 59 through

61, and_ Figs. 71 through 73.

Figures 74 through 77 indicate the differences of web distortion

caused by the moment connection and the pin connection. It is evident

from these figures that assuming a simple connection in design, and

providing a "fixed" connection in the actual structure, and vice versa,

would change the distortion of the web gap region considerably.

This could lead to fatigue cracking, as shown in Figs. 9 and 10.

This cracking occurred in connections of members that were much

55


stiffer than the cross framing members examined in this study. This

fact is not of great importance since the amount of distortion and

the presence of stress concentrations have more influence on fatigue

cracking than does individual member stiffness.

It can be concluded from these observations that the method of

connecting the bracing member to the connection plate will influence

the forces in the bracing members, and the distortion of the web gap

region. The more flexible the connection is (i.e. closer approxi

mation to a pin connection), ·the more the web gap region will distort.

It can also be concluded that the stiffness of the web in the vicinity

of the connection greatly influences the interaction of the primary

and secondary members. An increase in gap length increases the web

flexibility which increases the web gap distortion. It can also be

stated that a potential fatigue crack location may be created if a

connection is designed with a certain assumed flexibility, and the

actual connection has a different flexibility.

4.2 Discussion of Observations with Respect to the Secondary

Stresses Developed in the Web Gap Region

4.2.1 Variable Flange Thickness

The observations in Chapters 2 and 3 indicated that a 14% increase

in the thickness of the bottom flange at midspan reduced the overall

stresses and deflections of the bridge by about 10%. However, the

56


stress and deformation patterns within the web gap region were not

affected.

It is postulated that increases in thickness of "thin" bottom

flanges (i.e. 19.05 mm (.75 in))would show more influence on the

stresses and distortions within the web gap region, than were observed

in this study. This is due to the fact that the out-of-plane stiff-

ness of thin flanges approaches the out-of-plane stiffness of the web

at the transverse connection. Any increase in this flange stiffness

will cause the difference between the web and flange stiffnesses to

increase. This will continue until a point is reached beyond which

any further increase in the flange stiffness remains insignificant

with respect to the web stiffness. It is further postulated that

this "limiting" value of flange stiffness was equaled or exceeded in

the original bottom flange examined in this study. The 14% increase

in thickness did not increase the stiffness significantly with respect

to the out-of-plane stiffness of the web and, therefore, no apprec-

iable changes in stresses or distortions within the web gap region

were observed.

It can be concluded that increasing the thickness of flanges that

are "stiff" in the out-of-plane direction, does not reduce the

secondary stresses in the web gap region. It is expected, however,

that increasing the thickness of "thin" flanges will reduce the web

stresses.

57


4.2.2 Variable Web Thickness

It has been shown that a 20% increase in web thickness reduced the

relative horizontal displacement within the web gap region of the pro-

totype structure by 15%. Examination of the substructure model dis-

closed that this reduction actually varied from 7% to 30%, and was

dependent upon the gap length, as well as the bracing member end

restraint (connection detail). This range of values suggested that

the increase in web thickness might be significant in reducing the

stresses and distortions within the web gap region.

Further examination of distortions in this region disclosed that

the web thickness increase had also reduced the relative rotation by

10 to 20 percent (see Figs. 60 and 72). This corresponded to a 75%

reduction in resultant moment about nodal point 7, and a 23% increase

in the resultant out-of-plane force, F (Table 9 -- compare Case No. y

3 to No. 6, etc.). These observations indicated that the change in

web stiffness, resulting from an increase in web thickness, caused

the redistribution of the bracing member forces. This redistribution

reduced the resultant moment, and increased the resultant force, F . . y

The decrease in moment thereby reduced the relative rotation. The

increase in the resultant out-of-plane force, F , however, was offset y

by the increase in web stiffness; thus, reduction of the relative

horizontal displacement occurred.

A reduction of distortion in a structure usually corresponds to

a reduction of stress. Since the discretization of the prototype

structure was coarse, the corresponding reduction in the web gap

58


stress was not observed. Such a reduction, however, was observed in

the stresses within the web gap region of the substructure model. This

reduction ranged from 15% to 30% for gap lengths greater than 50.8 mm

(2 in). Such reductions in stress are significant, and could cause a

design to be accepted instead of being rejected when structural fatigue

is considered. Examination of gap lengths less than 50.8 mm (2 in)

revealed that although the distortion within the web gap region was

reduced, the stresses increased by 10 to 30 percent.

It can be concluded from the above observations that the increased

web thickness significantly reduced the secondary stresses and dis

tortions within the web gap region when the gap length exceeded 50.8 mm

( 2 in). Since the change in web thickness produced a comparable

change in web stress, the procedure of increasing the web thickness

to meet fatigue considerations would be economical and helpful to the

designer. For small gaps (less than 50.8 mm (2 in)) however, the

increased thickness adversely affected the web stress; thus, this

procedure should not be used for gap lengths less than 50.8 mm (2 in).

It was noted that the relative rotation within the web gap

region increased when web thickness was increased for the 203.2 mm

(8 in) gap length. This change in the general trend was caused by the

bracing members which framed into the girder just above the end of

the stiffener, as shown in Figs. 49 and 62. These members added a

large stiffness to the girder web at this location. This· large

stiffness was similar to the "limiting" value of stiffness mentioned

in Section 4.2.1. Additions of small amounts of stiffness, such as

59


II

I I

increasing the web thickness, would be insignificant to the total

stiffness at the connection, and thus would cause relatively no change

in rotation within the web gap region. Since significant reductions

in relative horizontal displacement did occur, an overall reduction

of stress within the web gap region was observed.

4.2.3 Variable Gap Length

Observations from Chapter 3 disclosed that the milled condition

generally yielded the highest web gap stress (292.3 MPa (42.4 ksi)),

and the welded stiffener produced the lowest (1.0 MPa (.15 ksi)).

This increase was attributed to a relative rotation between the flange

and the web which occurred in the milled condition. The magnitude of

stresses for the milled condition ranged from 92.7 MPa (13.4 ksi) to

292.3 MPa (42.4 ksi). These stresses agreed with observed field data

and accentuated the presence of stress concentrations in this detail.

The stress buildup indicated above was due to the geometry of the

region being considered. Besides this stress buildup, additional stress

concentrations should be expected due to the presence of two welds

adjacent to one another, as shown in Fig. l(b). This further reduces

the fatigue life of the detail. The small distortion that occurs due

to the relative rotation is focused into a small gap (approximately

1.59 mm (1/16 in)), and results in a large web stress. The stress con

centrations and corresponding web stresses can be reduced by coping

the stiffener as shown in Fig. 14. The effects of these stress con

centrations can be reduced further, and almost eliminated if the

60


transverse connection plate is welded to the bottom (tension) flange

as shown in Fig. 15. This slight change in the detail should satisfy

the fatigue restrictions of Ref. 2, Category "C", with only minor ad

justments to the original girder design,.because the end of the milled

stiffener is very close to the bottom flange and is also classified

as a Category "C" detail.

It can be concluded that prohibitive stresses occurred in the web

gap region when the transverse connection plate was tight fit to the

tension flange. These stresses can almost be eliminated by the

present recommended procedures, which are specified in Ref. 3.

The gapped condition examined in the prototype structure greatly

influenced the out-of-plane displacement pattern of the girder web.

Significant stresses were not directly obtained from ~his finite

element model; however, substitution of observed displacement values

into Eq. 4 produced stresses ranging from 97.7 MPa (14.2 ksi) to

149.0 MPa (21.6 ksi).

The various gap lengths in the substructure model were examined

under a constant loading condition. It was observed that the distortion

of the web gap region increased as the gap length was increased. How

ever, upon reaching the 203.2 mm (8 in) gap length, the relative

rotation in this region was observed to decrease (see Figs. 60 and 72).

This occurred because the bracing members framed into the girder

immediately above this point, and provided a large additional stiffness

that resisted the rotation. These observations revealed the large

61

I I changes in web gap distortion that result from changes in "local

stiffness" of the girder web.

I The observed web stresses for the various gap lengths ranged

I from 55.2 MPa (8.0 ksi) to 167.6 MPa (24.3 ksi). This range en-

I compassed the stresses determined from Eq. 4 for the prototype

structure. It can be concluded therefore, that Eq. 4 with data from

an overall three dimensional structural analysis can be used to obtain

a rough approximation of the stress in the web gap region.

As previously stated in Section 3.1, the boundary loads applied

I to the gapped stiffener cases of the substructure model were less than

the "actual loads". The result:f.ng web stresses shown in Figs. 61 and

I 73 should, therefore, be increased. Since the constant loading

I condition used was formulated from the prototype structure with a

225.4 mm (8.875 in) gap length, a larger increase in stress should be

I applied to the smaller gap lengths than to the larger gap lengths.

An approximation of this increase in stress for each gap length

was formulated by applying the boundary loads of Case No. 2 to Case

I No. 3c. The maximum stress in the web gap region was found to increase

I from 65.3 MPa (9.5 ksi) to 73.5 MPa (10.7 ksi). This represented an

increase of 13%. It was then assumed that the percent increase in

I stress varied linearly and was zero for the 203.2 mm (8 in) gap length.

The percent increase in stress for each gap length was then determined

I and is shown in Fig. 78. These increases raise the magnitudes of

I stress but do not alter the general trends observed in Figs. 61 and 73.

I 62

I

I I I I I I I I I I . I I I I I I I I I

It was previously stated in Section 4.2.2 that a reduction of

distortion in a structure usually corresponds to a reduction of stress.

The opposite of this statement (i.e. an increase of distortion in

a structure usually corresponds to an increase in stress) is also

valid. This latter trend is what was observed in Figs. 59 through 61

for bracing members with moment connections. However, Figs. 71

through 73 revealed that stress decreased as distortion of the web

gap region increased for bracing members with pinned connections.

This variation of stresses can be visualized if the data in Table 14

is examined.

In this table the total web stress, crt; has been divided into

stress caused by rotation, cre, and stress caused by displacement, cr~.

These values of stress were computed by transforming Eq. 2 into Eq. 7,

as shown below, and substituting the appropriate relative rotations

and displacements from Tables 10 and 12 into Eq. 7 .

4Et 8 3Et ~ = - ___ Lw~- + --~w __ _

L2 (Eq. 7)

The sign of the rotation term is negative because it was observed that

the rotation tended to relieve the stress caused by the relative

displacement.

Examination of cr8 and cr~ for Cases 3c through 3f., and 6b through

6f disclosed that cre decreased at a higher rate than cr~. as gap length

was increased. Because of this difference in rate of change in cr8 and

cr~, an overall increase in crt was observed. The absolute values of


crt are plotted versus gap length in Figs. 79 and 80, and appear as

dashed lines. The curves f.or Cases 3 and 6 lie below the curves ob-

tained from the substructure model, however, all curves indicate the

same trend. This trend, as previously stated, showed that for bracing

members with moment connections, an increase in gap length resulted

in an increase in web stress. From these observations it can be

concluded that the stress in the web gap region is not only dependent

upon the gap length, and the magnitude of the relative out-of-plane

displacement, as indicated in Chapter 1, but also depends upon the

relative rotation within the web gap region. It can also be concluded

that Eq. 7 provides an unconservative estimate of the stress in the

web gap region for bracing members with moment connections.

Examination of cr8 and cr~ in Cases 15a through 15e, and 18a

through 18e revealed that cre decreased at a slower rate than cr~ while

gap length was increased. Thus, an overall reduction in cr resulted. t

These values of cr are also plotted in Figs. 79 and 80 and appear as t

dashed lines. The curve for Case 15 lies below, as well as above the

curve which shows the substructure model values. The curve for Case

18 generally lies above the corresponding substructure model curve.

All of these curves, however, show the same basic trend. This trend,

as previously mentioned, shows a reduction in stress as gap length

was increased. These observations reinforce the conclusion drawn

above, that stress within the web gap region is dependent upon gap

length, relative rotation, and relative displacement. It can also be

concluded that Eq. 7 generally provides a conservative estimate of the

stress in the web gap region for bracing members with pinned connections.

64


4.2.4 Variable Bracing Member End Restraint

Analysis of the prototype structure in Chapter 2 disclosed that

the moment end restraint condition provided the most resistance to

relative horizontal displacement, and the pin connection provided the

least. Chapter 3 presented data regarding the deformations and stresses

within the web gap region for both of these end restraint conditions.

Figures 74 through 77 compare the distortions in this region caused by

the two bracing member connection details, and verify the observations

mentioned above from Chapter 2. As previously stated in Section

4.1.2, significant changes in the distortions of the web gap region

occurred when the connection detail was changed. The most significant

variations occurred in the relative rotation. Figures 76 and 77 show

that the relative rotation doubled when the connection was changed

from a moment to a pin connection. Since the deformations within the

web gap region determine the local stress pattern, the observations

above indicated that the connection detail should greatly affect

this stress pattern. This has been observed and is shown in Figs.

81 and 82·. Examination of these figures disclosed a large influence

for gap lengths less than 50.8 mm (2 in) and a smaller influence

for gaps larger than this length. In fact, for gap lengths larger

than 50.8 mm (2 in) approximate constant stresses of 86.2 MPa

(12.5 ksi) and 66.0 MPa (9.5 ksi) were observed for the 7.94 rnm

(5/16 in) and 9.53 rnm (3/8 in) web thicknesses, respectively. Thus it

can be stated that a change in the bracing member connection detail

greatly affects the stress in the web gap region for small gap lengths

(less than 50.8 rnm (2 in)). For gap lengths larger than 50.8 mm (2 in),

65

I I however, a change in connection detail has a relatively small influence

I on the stress within the gap region.

It should be noted that the 50.8 mm (2 in) gap length discussed

I above represents a ratio between gap length and web thickness of about

I 6. The present recommended gap length for cut short transverse con-

nection plates in the positive moment region is 4 t to 6 t (Ref. 3). w w

I The observations above and those previously mentioned in Section

4.2.2, indicate that this recommendation should be reassessed, maybe

I changed to 8 t to 10 t • w w

I Included in Figs. 81 and 82 are plots of maximum web gap stress

I versus gap length in which the stress was computed by Eq. 4, and

represents only the displacement term, cr~, in Eq. 7. These curves

I appear as dashed lines and considerably overestimate the web stress.

Therefore, the use of Eq. 4 to estimate the stress within the web

I gap region is at most a gross approximation, and Eq. 7 should be

I used in lieu of Eq. 4 whenever possible.

Further examination of Fig. 79 disclosed that the theoretical

I curves (Eq. 7) provided an upper and lower bound of stress for certain

I gap lengths, but basically these curves yielded good approximations

of the stress. The theoretical curves in Fig. 80, however, provided

I well defined upper and lower bounds of stress for relatively all gap

lengths. It is postulated that the more well defined bounds occurred

I because the stiffer 9.53 mm (3/8 in) web (Fig. 80) provided a more

I uniform resistance for all gap lengths than the thinner web did

(Fig. 79).

I 66

I


As previously stated in Section 2.4, the stiffness of an actual

connection lies between the moment connection and pin-ended connection.

The deformations and stresses determined in Chapter 3 for these

conditions, and mentioned above, generally defined upper and lower

limits. Equation 7 also provided upper and lower bounds. It can

be concluded, therefore, that the distortions and stresses within the

web gap region for an actual bracing member connection should lie

within the values determined in this investigation. It can also be

concluded that Eq. 7 can be used to obtain the upper and lower bounds

of stress within the gap region if appropriate data is available.

4.3 Interpretation of Observations with Respect to Structural Fatigue

Chapter 1 showed instances in which the secondary stresses

developed by out-of-plane web displacements created problems in

cyclically loaded members. Fatigue cracking occurred and occasionally

lead to premature failure of the member.

The findings presented in Chapter 3 indicated that the magnitude

of stress for tight fit stiffeners was prohibitive when considering

fatigue. This stress ranged from 91.3 MPa (13.2 ksi) to 292.3 MPa

(42.4 ksi), and approached, and even exceeded, the yield strength of

many structural steels used in bridge construction. This stress agreed

with field observations of similar conditions in which fatigue cracking

developed rapidly (Ref. 17). It can then be concluded that tight fit

stiffeners should be avoided.

67


'I IJ

The range of stress for the other gap lengths examined in this

study was determined to be 55.2 MPa (8.0 ksi) to 167.6 MPa (24.3 ksi).

Fatigue life estimates using Eq. 6 with a modified material constant

were determined for these various gap lengths in order to compare the

theoretical data of this study with the experimental data from Ref. 4.

The material constant of

1.21 X 10-13 CNe:::~:3 cyclJ r X 10-10~(~::~~2 cycles) l was changed to 2.178 x l0-13 (3.6 x l0-10) and an initial crack size

ai, of .762 mm (.03 in) was selected. This change in material

constant, and the selection of the largest weld defect mentioned in

Section 1.3 were done in order to minimize the fatigue life estimates.

A final crack size of 28.575 mm (1.125 in) was also selected but this

value had very little influence on the fatigue life estimates. This

final crack size was selected because it represented the length of a

typical crack for the experimental data (Fig. 8) from Ref. 4.

Table 15 lists the fatigue life estimates for Cases 3, 6, 15, and

18, a through f. Cases 21, 24, 33, and 36, a through f were not pre-

sented in this table because the stresses in these cases have been

shown to be the same as Cases 3, 6, 15, and 18 a through f, respec-

tively (see Figs. 61 and 73). Included in Table 15 are the fatigue

life estimates that were obtained when the web stress was computed

by Eq. 7. The stress values used in these fatigue life estimates

were the absolute value of the stresses shown in Table 14. Comparison

of the two fatigue life estimates in Table 15 for each gap length

disclosed large discrepancies. These occurred because the stress

68


was cubed in the denominator of Eq. 6. Any differences in stress,

therefore, would have been magnified considerably when fatigue life

estimates were computed.

A better visualization of the fatigue aspects was obtained when

the substructure model findings and experimental data from Fig. 8

were plotted together. This is shown in Fig. 83. The actual ratios

of gap length to web thickness for the substructure model ·are shown

in Table 15. The symbol that represents the gap length for each case

in Fig. 83 is also shown in Table 15, adjacent to the ratio.

Figure 83 indicated that the theoretical data of Chapter 3 closely

resembled the experimental data. It was noted that the theoretical

fatigue life estimates generally overestimated the actual fatigue lives

of the details. Since the overestimate was not excessive, it can be

concluded that the theoretical data of this study provided good fatigue

life estimates for cut short transverse stiffeners.

Better agreement should exist between theoretical and experimental

stresses, because of the relationship between fatigue life and stress,

previously mentioned and shown in Eq. 6. This improved agreement was

verified by using the experimental data in Fig. 8 and Eq. 8 to compute

the experimental stress. Equation 8, which is shown below, is Eq. 6

evaluated for the initial and final crack lengths, and material con-

stant previously stated.

N = 1.58 X 1012

S (MPa) 3 r

69

= ( 4.82 X 10~ )

S (ksi) 3 r

(Eq. 8)


Rearranging Eq. 8 and solving for the stress (S ) results in Eq. 9. r

S (MPa) = r

12 1/3 1.58 X 10

N (Eq. 9)

Table 16 lists the experimental data from Fig. 8 and the corresponding

stress computed from Eq. 9. The stresses marked with an (x) resulted

from relative deflections that were similar to those observed in this

investigation. These stresses are plotted in Figs. 81 and 82 as x's

and show the improved correlation. From this agreement between ex-

perimental and theoretical stresses, it can be concluded that the

procedure used in this study is valid for predicting stress within

the web gap region of cut short transverse connection plates.

Since the predicted stresses were accurate, they were compared to

the acceptable stress levels for fatigue in Ref. 2. This was done in

order to determine which gap lengths were permissible when considering

structural fatigue. Table 1.7.2Al of Ref. 2 is reproduced in Table

17. In comparing the data from Chapter 3 with Table 17, it must be

understood that the "range of stress" determined in the substructure

model resulted from live load only. This "range of stress" actually

represented the lowest and highest values of stress in the web gap

region due to this live load. Each value of stress previously mentioned

throughout this report, therefore, actually represented a value of

stress range that must be compared to F , as defined in Table 17. sr

The bridge superstructure examined in this study was a "redundant

load path structure", and the detail under examination was a Category

"C", as defined in Table 17. For the sake of comparison, it was

70

I I assumed that the bridge had to sustain up to two million cycles of

I stress. The maximum stress range permitted for these conditions was

89.63 MPa (13.0 ksi). Comparison of this value with the substructure

I stress range values, labeled cr in Table 15, disclosed that all gap max

lengths with moment connected bracing members were acceptable, except

I for Cases 3e and 3f. These cases would have been permitted if the

I bridge was required to sustain 500,000 stress cycles. This comparison

further disclosed that Cases 15c through 15f and Cases 18c through 18f

I were also acceptable. The remaining cases would not be permitted

for the conditions above, but would have been acceptable if only

I 500,000 cycles of stress were required.

I From these observations it can be concluded that acceptability of

I stress range with respect to the restrictions of Table 17 was closely

dependent upon the bracing member connection detail. These observa-

I tions also indicated that gap lengths equal to or less than 25.4 mm

(1 in) should be avoided when the bracing member has a pin connection.

I The stresses within the web gap region for bracing members with

I moment connections and gap lengths less than 50.8 mm (2 in) were sig-

nificantly smaller than those for the bracing members with pinned

I connections. Since the stiffness of an actual connection is not

finite and could approach the pin connection, these smaller gap lengths

I should always be avoided.

'I It has been shown that the interaction of primary and secondary

members, and the stress in the web gap region were dependent upon the

I flexibility of the secondary member connection detail and the

I 71

I

I

I

I I I I I I I I I I I I I

flexibility of the web in the vicinity of this connection. Various

conclusions have been drawn regarding the effects of the parameters

examined, and these conclusions are summarized in Chapter 5.

72


5. SUMMARY AND CONCLUSIONS

This investigation was conducted to identify overall and local

effects in a typical multigirder composite highway bridge caused by

the interaction of primary girders and secondary cross framing members.

Current specifications do not take into account the interaction be

tween primary and secondary members; consequently, the stresses induced

by this interaction are not considered (Ref. 2). Present recommenda

tions from the literature, which may be used to reduce the fatigue

cracking caused by these secondary stresses, have been presented

(Refs. 3 and 11).

A finite element analysis of a simple span multigirder composite

bridge with cross framing was conducted. This was followed by a

refined analysis of the primary-to-secondary member connection. A

parametric study was carried out in which the variables were bottom

flange thickness, web thickness, transverse connection plate gap

length, and secondary member end restraint. Observations and con

clusions were made regarding the effects of these variables on the

bridge response, and the secondary stresses developed. A summary of

the conclusions with respect to the bridge response follows:

1. The effectiveness of cross framing in distributing

the live load is dependent upon load location.

2. Local effects occur in the web gap region for all

loading locations and can cause prohibitive fatigue stresses.

73


3. The method of connecting the secondary cross framing

members to the transverse connection plates influences the

forces in the secondary members and the out-of-plane deforma

tion pattern of the web gap region.

4. The stiffness of the web in the vicinity of the

secondary member connection influences the interaction of primary

and secondary members.

5. A potential fatigue crack location may be created if

a connection is designed with a certain assumed flexibility,

and the actual connection has a different flexibility.

A summary of conclusions regarding the secondary stresses

developed in the web gap region follows:

1. The procedure used in this investigation was valid

for predicting the secondary stresses within the web gap

region of cut short transverse connection plates, and also

provided good fatigue life estimates of the detail.

2. Gap lengths less than 50.8 mm (2 in-- 6t ), w

including "tight fit" stiffeners should be avoided if fatigue

cracking is to be reduced. Present recommended design pro-

cedures in Ref. 3 may be used to eliminate the large secondary

stresses developed when the tight fit stiffener is used.

3. The secondary stress developed in the web gap region

is dependent upon the gap length, relative out-of-plane dis-

placement and out-of-plane rotation between the tension flange

74


and the end of the cut short transverse connection plate.

Equation 7 provides a good estimate of this secondary stress and

should be used in lieu of Eq. 4 whenever possible.

4. Increasing the thickness of flanges that are "stiff"

in the out-of-plane direction, does not reduce the secondary

~tress in the web gap region. However, increasing the thickness

of "thin" flanges should reduce this stress.

5. Increasing the web thickness when the gap length

exceeds 50.8 mm (2 in·-- 6t ) reduces the distortions and w

secondary stresses in the web. This is an economical design

procedure that can be used to create an acceptable girder

design when fatigue is considered.

Several recommendations for design and further study appear below:

1. The secondary stresses created by the interaction of

primary and secondary members should be considered in the design

specification since fatigue cracking may result and cause

failure of the primary members.

2. The transverse connection plate should be welded to

the tension flange whenever possible. In lieu of this, a gap

length of 8 t to 10 t should be used to minimize the fatigue w w

cracking at the end of cut short connection plates.

3. Establishing an average value of 9 in Eq. 7 for

typical connection details so that this equation could be used

75

I I I I I I I I I I·

I I I I I I I I I

as a design tool for evaluating the secondary stresses in the

web gap region.

4. Verifying the theoretical results of this investi

gation through testing of a full scale cross framing connection

detail.

5. Evaluating the effect on economics of girder design

when the transverse connection plate is welded to the tension

flange.

76


TABLE 1

VERTICAL DEFLECTION - mm (in)

Dead Load

Interior Exterior Girder Girder

Quarter 11.18 -Classical Span (.436)

Method 15.49 Midspan

(. 611) -Quarter 12.80

!Modified (.504) -Span Classical Method

Midspan 17.96 (. 707) -

Quarter 12.29 13.11 Span ( .484) (. 516)

FEM 1

Midspan 16.94 17.98 (. 667) (.708)

Quarter 10.62 10.97 Span ( .418) (.432)

FEM 2

Midspan 14.66 15.06 (.577) (.593)

Classical formulas used:

A. Dead Load

a) Quarter. Span

19 w ~} 6. = 2048 EI

77

Live Load

Interior Girder

0.99 (.039)

1.45 ( .057)

1.14 (. 045)

1.68 (.066)

1.21 (.048)

1. 79 (.070)

1.04 ( .041)

1.52 (.060)

Plus Impact

Exterior Girder

- -

-

-

-

1.21 (.048)

1. 79 (.070)

1.04 ( .041)

1.50 (. 059)


TABLE 1 (continued)

b) Midspan

5 w t 4

!::. = 384 EI

B. Live Load Plus Impact

a) Quarter Span

11 p Q,3

b. = 768 EI

b) Midspan

p Q,3

!::. = 48 EI

!::. = vertical deflection (mm)

t = span length (mm)

E = modulus of elasticity (MPa)

I =moment of inertia of composite section (mm4 )

w = dead load of slab and girder (16.0 N/mm)

P = live load plus impact (22130.0 N)

Note: 1. FEM 1 consists of a 177.8 mm

(7 in) deck slab with 203.2 mm

(8 in) curbs and sidewalks

2. FEM 2 consists of a 333.5 mm

(13.13 in) deck slab with 322.8 mm

(12.71 in) curbs and sidewalks

78

-------------------TABLE 2

DEAD LOAD STRESSES

•,

MIDSPAN

Exterior Interior Modified Girder Girder

Classical Classical Method Method FEM 1 FEM 2 FEM 1 FEM 2

- -y (J y (J (J (J (J (J

Element (mm) (MPa) (mm) ~Pa) (MPa) (MPa) (MPa) (MPa) No. [in) [ksi] [in] ksi] [ksi] [ksi] [ksil [l<si]

1 928.1 46.54 863.5 50.06 46.75 42.61 43.71 41.58 [ 36.54] [ 6.75] [34.00] [ 7.26] r. 6.78] [ 6.18] [ 6.34] [ 6. 03]

16 618.5 31.03 554.0 32.13 30.20 28.89 27.86 27.99 [24.35] [ 4.50] [21. 81] [ 4. 66] [ 4.38] [ 4.19] [ 4.04] [ 4.06]

31 231.1 11.58 166.6 9.65 9.93 12.00 8.21 11.24 [ 9.10] [ 1. 68] [ 6.56] [ 1.40] [ 1.44] [ 1. 74] [ 1.19] [ 1. 63]

46 -83.6 -4.21 -148.1 -8.62 -6.41 -1.65 -7.58 -2'.14 [-3.29] [-0.61] [-5.83] [ -1.25] [-0.93] [ -0.24] [ -1.10] [-0.31]

Bot. 1040.8 52.20 976.2 56.61 52.20 - 48.89 -Flg. [40.98] [ 7.57] [38.44] [ 8.21] [ 7. 57] [ 7.09]

~- -------------------------

00 0

QUARTER SPAN

!Element No.

7

8

22

23

Classical Method

-y (J

(mm) (MPa) [in] [ksi]

928.1 34.68 [36.54] [5.03)

928.1 34.68 [36.54] [ 5. 03]

618.5 23.10 [24.35] [3.35]

618.5 23.10 [24.35] [3. 35]

TABLE 2 (continued)

Exterior Interior Modified Girder Girder Classical Method FEM 1 FEM 2 FEM 1 FEM 2

-y (J (J 0' (J (J

(mm) (MPa) (MPa) (MPa) (MPa) (MPa) [in] [ksi] [ksi] [ksi] [ksi] [ksi]

863.5 37.30 36.54 33.10 33.51 31.85 [34.00) [5.41] [5.30] [4.80] [4.86] [4.62]

863.5 37.30 34.61 31.23 31.65 29.99 [34.00] [5.41] [5.02] [4.53] [4.59] [4.35]

554.0 23.93 23.86 22.55 21.44 21.51 [21.81] [3.47] [3.46] [3.27] [3.11] [3.12]

554.0 23.93 23.17 21.86 20.82 20.75 [21.81] [3.47] [3.36] [3.17] [3.02] [3.01]

-------------------TABLE 2 (continued)

QUARTER SPAN

Exterior Interior Modified Girder Girder

Classical Classical Method Method FEM 1 FEM 2 FEM 1 FEM 2

- -y cr y cr cr cr cr cr

Element (mm) (MPa) (mm) (MPa) (MPa) (MPa) (MPa) (MPa) No. [in) [ksi) [in) [ksi) [ksi) [ksi) [ksi) [ksi)

231.1 8.62 166.6 7.17 8.27 9.58 6.62 8.83 38 [ 9.10) [1.25) [ 6.56) [ 1.04 J [ 1.20) [1.39] [ . 96] [1.28)

39 231.1 8.62 166.6 7.17 8.96 10.00 7.38 9.31 [ 9.10) [1.25] [ 6.56) [ 1.04] [ 1.30) [1.45] [1.07] [1.35)

-83.6 -3.10 -148.1 -6.41 -4.48 -1.10 -5.52 -1.52 53 [-3.29) [- .45) [-5.83] [ -.93] [ -.65] [-.16] [-.80] [-.22)

54 -83.6 -3.10 -148.1 -6.41 -4.07 -.76 -4.90 -1.17 [-3.29] [-.45) [-5.83] [ -.93] [ -.59] [-.11] [-.71] [- .17]

Bot. 1040.8 38.64 976.2 42.20 39.37 - 36.06 -Flg. [40.98] [5.64] [38.44] [ 6.12] [ 5.71] [5.23]


I I

I I I

TABLE 2 (continued)

Note: 1) Minus (-) indicates compression

2) For classical computations

a) ~ = ~L + ~L ) from Ref. 14 1 2

i) Midspan

~ = 1155.1 kN-m (10224 in-kips)

ii) Quarterspan

MD = 860.9 kN-m c162o in-kips)

b) Classical method

I = 2.304xlol 0 mm4 (55364 in4)

c) Modified classical method

I= 1.992xl& 0 mm4 (47859 in4)

d) a = (~ x y)/I

Element Number

L _j Element Centroid

L ,

4.§ ~

Is 3_._ ~ .~~ - neutral axis (classical]

31

§ -(:l -" ...-l Ll'l . \0 00 ....... 38, 39 ...;t . N 0'1 ....... ...;t 16 .....,

-22, 23

1 -7, 8 _...._

. I nentl='a.l

§ § -!:l\0

"l"'lC"'! C"') . \0 00-.;t . 00 N

~~ -0'1 ........ !:lO ..............

...-lOO C"') ....., 00 C"')

N \0 . . N \0 N ....... ...;t .....,

112.71 nnn

(4.44 in) 82

axis (modified class. )j •

§ - ~ -- (:l (:l (:l ...-l ...-l

...-l Ll'l " N ...;t " 00

" . ...;t . 0'1 N \0 . 0 . . " 00 ...;t 0 ...;t 0'1 C"') 0 ...;t ...;t ....., ....... ....., .....,


Transverse iPosition No.

1

2

3

4

5

6

7

8

9

TABLE 3

MAX]MUM RELATIVE HORIZONTAL DISPLACEMENTS

mm (in)

Midspan Quarter

Exterior G. Interior G. Exterior G.

.67361 .90399 .45034 (.02652) (.03559) (. 01773)

.57277 .86360 .34696 (.02255) (.03400) (.01366)

.50470 .73558 .30937 (.01987) (. 02896) ( .01218)

.47396 .63195 .30277 (.01866) (.02488) (.01192)

.42342 .56617 .23851 (.01667) (.02229) (.00939)

.38862 .51511 .28448 (.01530) (.02028) (.01120)

.35357 .46050 .27762 (.01392) (.01813) (.01093)

.29312 .33960 .26492 (.01154) (. 01337) (.01043)

.24867 .22530 .25425 (.00979) (. 00887) (.01001)

83

Span

Interior G.

.59665 (.02349)

.57429 (.02261)

.49251 (.01939)

.42088 (.01657)

.40818 (.01607)

.38710 ( .01524)

.36779 (.01448)

.29235 (. 01151)

.21184 (.00834)


. 0 z Q) Ul til u

1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36

-c:: Ul Ul o,.l Ul 1-1 1-1 1-1 Q) Q) 11'\Q.l c:: c:: I"C Q) Q) co Q)

4-l 4-l • 4-l 4-l 4-l C04-l o,.l o,.l ....... •.-1 .u .u .u Cl) Cl) eoo

E 't:l 't:l 't:l Q) Q) ._:f"Q)

't:l ...... • c. ...... ...... II'\ C.

Q) o,.l N til ::: ~ Nc.!l

X X

X X

X X

X X

X X

X X

X X

X X

X X

X X

X X

X X

X X

X X

X X

X X

X X

X X

TABLE 4

LOADING CASE PARAMETERS

- -Ul Ul c:: c:: - c:: Ul c:: o,.l Eo..l c: - 0 c:: 0 ~..:t E o,.l c:: o,.l 0 o,.l N

o,.l +J o,.l .u ........ co \0 u .u u 11'\~Q) •'t:l Q) ...... co ..c: Q) ..c: u ..c: Q) • I 00 0 c: 00 ........ ........ .u c:: .u Q) .u c:: .,:f".-IC:: II'\ til c:: II'\ ~ o,.l c:: o,.l c:: o,.l c:: ..:t til til ....... ....... ~ 0 ~ c:: ~ 0 't:l.-1 't:l c:: ......

u 0 u 't:IC:J:x. C::o..lJ:x.

~ ~ 00 oou 00 c:: til til C::+J c:: C:'t:l til E ..:t E

o,.l c:: o,.l 1-1 o,.l Q) c:: 0 11'\-......o ..:t ~ u Q) u til u c:: .,:f"o..l.U 1".-I.U 0"1,.0 11'\,.0 til E til Q) til c:: . .u • I .U

• Q) • Q) 1-1 0 !-I..C: 1-!o..l II'\ ...... 0 .-4.-10 "'::: 0'\::: I:Q~ I:QCI) I:QP., N'-'I:Q ~'-'I:Q

X X X X X X X X X

X X X X X X X X X

X X X X X X X X X

X X X X X X X X X

X X X X X X X X X

X X X X X X X X X

X X X X X X X X X

X X X X X X X X X

X X X X X X X X X

X X X X X X X X X

X X X X X X X X X

X X X X X X X X X

84


Case No.

1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36

Vertical Displacement

0z(3) (14.20 mm)

1 1.000 1.000 0.959 0.959 0.959 1.000 1.000 1.000 0.959 0.959 0.959 1.000 1.000 1.000 0.959 0.959 0.959 0.922 0.922 0.922 0.886 0.886 0.886 0.922 0.922 0.922 0.886 0.886 0.887 0.922 0.922 0.922 0.886 0.886 0.887

TABLE 5

DEFLECTIONS AND STRESSES

Relative Total Stress in Bottom Flg. Horizontal Displacement

2.44 m from b. 5y(7-3) Midspan Midspan (0.73 mm) (44.13 MPa) (40 .54 MPa)

1 1 1 1.004 0.998 0.995 1.294 0.995 0.991 0.963 0.969 0.966 0.966 0.966 0.961 1.167 0.966 0.959 0.992 1.001 1.000 0.996 0.998 0.995 1.373 0.995 0.990 0.955 0.969 0.966 0.958 0.966 0.961 1.178 0.966 0.959 0.992 1.001 1.000 0.998 0.998 0.995 1.593 0.995 0.985 0.956 0.969 0.966 0.960 0.967 0.963 1.353 0.966 0.954 0.926 0.898 0.903 0.932 0.895 0.898 1.193 0.895 0.895 0.894 0.875 0.872 0.898 0.872 0.869 1.078 0.872 0.867 0.918 0.900 0.903 0.924 0.895 0.898 1.268 0.894 0.893 0.886 0.875 0.872 0.890 0.872 0.869 1.086 0.872 0.867 0.918 0.900 0.903 0.926 0.897 0.898 1.484 0.894 0.886 0.886 0.875 0.874 0.893 0.872 0.869 1.259 0.872 0.862

85

-------------------

Nodal Points

Case No. 1

J.:..39

[620-93]

7-43

[603-76]

13-47

[592-65]

3-7

[620-603]

7-13

(603-592]

TABLE 6

RELATIVE DEFLECTIONS (rom (in))

66x(a-b) 66y(a-b)

Prototype Substructure Prototype

.13157 .02337 (.00518) (.00092)

.13183 (.00519)

.10033 .03048 (.00395) (.00120)

.10084 (.00397)

.04775 .05563 (.00188) ( .00219)

.04750 ( .00187)

- .72593 - (.02858)

--

- 1.2591 - (.04957)

--

65 m 5 -x(a-b) x(PT.a) 65 m a -y(a-b) y(PT.a) 65z(a-b) = 5z(PT.a) -

Substructure

1.4305 (.05632)

1.2057 (.04747)

.5080 (.02000)

.59461 ( .02341)

1.0498 (. 04133)

5x(PT.b) 5 y(PT.b) 5z(PT.b)

66 z(a-b)

Prototype Substructure

.05131 (.00202)

.05588 (.00220)

.04420 (.00174)

.04648 (.00183)

.03759 (.00148)

.03835 (.00151)

.00711 (.00028)

.00381 ( .00015)

.00914 (.00036)

.00737 (.00029)

-------------------

. -

Nodal Points

Case No. 3

3-39

[620-93]

7-43

[603-76]

13-47

[592-65]

3-7

[620-603]

7-13

[603-592]

M x(a-b)

TABLE 6 (continued)

RELATIVE DEFLECTIONS (mm (in))

M y(a-b) 65 z(a-b)

Prototype Substructure Prototype Substructure Prototype Substructure

.13183 .05258 .04521 ( .00519) (.00207) (.00178)

.13183 1.8456 ( .00519) (.07266)

.10109 .16256 .04521 (.00398) (.00640) (. 00178)

.10084 .85827 (.00397) (.03379)

.04801 .11963 .03810 (.00189) ( .00471) (.00150)

.04775 .27432 (.00188) (.01080)

- .93929 .01321 - (.03698) (. 00052)

- 1.2931 - ( .05091)

- 1.0036 .00889 - (.03951) (.00035)

- .98857 - (.03892)

Note: The substructure model data used above is for Case No. 3 with a 200 mm (8 in) gap at the end of the transverse connection plate (Case No. 3f of Table 7).

.04724 (.00186)

.04877 (.00192)

.03912 (.00154)

.01448 (. 00057)

.00610 (.00024)


TABLE 7

LOADING CASE PARAMETERS FOR BRACING MEMBERS WITH MOMENT CONNECTIONS

,0 Q) ,0 ~ Q)

~ -fll fll Cll - r:::: 1-1 1-1 ~-~- r:::: - ..-I Q) Q) Q) r:::: - - - ..-I r:::: r:::: r:::: C::..-1 - - r:::: r:::: r:::: ..-I .;t -Q) Q) Q) r:::: r:::: ..-I ..-I ..-I 1.0 -cu r:::: Q)

4-l 4-l 4-l II'\ ..-I ..-I ...-1 00 C""lOO ..-100 4-l 4-l 4-l . .;t 1.0 00 - - I r:::: r:::: ..-I ..-I ..-I 0 ...-1 "'

.._, '-" '-" II'\ C""l ...-1 ttl "'ttl .u .u .u .._, .._, .._, .._, .._, '-"...-I '-"...-I . Cl) Cl) Cl) E ~ E """ """ 0 E E ~ E E ~ E E E z '"C) '"C) '"C) E E E E E E E

Q) Q)

~ 1.0 .;t "' 0 0

Q) '"C) ...-1 ..... .;t 00 . . . .;t C""l 11'\.U oo.u Ul ...-1 ...-1 ~ . . . ...-1 "' C""l 0\ II'\ • .u •.U ttl Q) ..-I "' II'\ 0 0 II'\ 0 . . .;t 0 0 0 u ~ ~ ~...-I "' II'\ ...-1 ...-1 "' ..... 0\ .;ti:Q 11'\I:Q

1 X X X 2 X X X 3a X X X 3b X X X 3c X X X 3d X X X 3e X X X 3f X X X 4 X X X 5 X X X 6a X X X 6b X X X 6c X X X 6d X X X 6e X X X 6f X X X

19 X X X 20 X X X 21a X X X 21b X X X 21c X X X 21d X X X 21e X X X 21£ X X X 22 X X X 23 X X X 24a X X X 24b X X X 24c X X X 24d X X X 24e X X X 24£ X X X

88


I

TABLE 8

LOADING CASE PARAMETERS FOR BRACING MEMBERS WITH PINNED CONNECTIONS

,..0 QJ ,..0 ~ QJ

~ -fiJ fiJ fiJ - c: J.l J.l J-1- c: - ..... QJ QJ QJ c: - - - ..... c: c: c: c: ..... - - c: c: c: ..... -:t -QJ QJ QJ Q c •.-4 ..... ..... \0 -QJ c QJ

4-1 4-1 4-1 Lf'\ •r1 ..... ...... 00 t"100 ..... 00 4-1 4-1 4-1 . -:t \0 00 - - I c c ..... •.-4 ..... 0 r-i co-l ..._., ..._., ..._.,

Lf'\ t"1 ...... tU co-l tU .u .u .u ..._., - ..._., ..._., ..._., ..._., ...... ,_..,..... . Ul Ul Ul § 8 8 """ """ 0 8 8 8 8 8 8 8 8 8 z "lj "lj "lj 8 8 8 e e e e 8 e QJ QJ

~ \0 -:t co-l 0 0 QJ "lj ...... " -:t 00 . . -:t t"1 Lf'\.U oo.u

fiJ ...... ...... • . . ...... co-l t"1 0'1 Lf'\ • .u • .u tU QJ ..... tU co-l Lf'\ 0 0 Lf'\ 0 . . -:to 0 0 u ~ ::E: C,!) ...... co-l Lf'\ ...... ...... co-l " 0'1 -:t~ Lf'\~

13 X X X 14 X! X X 15a X X X 15b X X X 15c X X X 15d X X X 15e X X X 15f X X X 16 X X X 17 X X X 18a X X X 18b X X X 18c X X X 18d X X X 18e X X X 18f X X X 31 X X X 32 X X X 33a X X X 33b X X X 33c X X X 33d X X X 33e X X X 33f X X X 34 X X X 35 X X X 36a X X X 36b X X X 36c X X X 36d X X X 36e X X X 36f X X X

89

I I I I I I I I I I I I I I I I I I

II

TABLE 9

RESULTANT FORCES AT NODAL POINT 7

Resultant Resultant Resultant Moment Force - F Force - F y z

Case No. N-m (k-in) kN (kips) kN (kips)

1 -185.40 3.914 .805 ( -1. 641) (.880) (.181)

2 -176.02 3.936 .805 (-1.558) (.885) (.181)

3 30.17 1.090 .947 (~267) ( .245) (.213)

4 -181.67 3.919 .734 ( -1. 608) (.881) (.165)

5 -173.76 3.945 .734 (-1.538) ( .887) ( .165)

6 7.57 1..339 1.067 (.067) (.301) ( .240)

19 -185.63 3.852 .681 ( -1. 643) (.866) ( .153)

20 -172.52 3.888 .681 ( -1.527) (.874) ( .153)

21 29.71 1.099 .814 (.263) (.247) (.183)

22 -182.01 3.861 .623 (-1.611) (.868) (.140)

23 -170.49 3.905 .623 (-1.509) (.878) (. 140)

24 7.23 1.348 .743 (.064) (.303) (.167)

Minus (-) indicates clockwise rotation; positive forces as indicated

in Fig. 29.

90


Case No.

13

14

15

16

17

18

31

32

33

34

35

36

TABLE 9 (continued)

RESULTANT FORCES AT NODAL POINT 7

Resultant Resultant Moment Force - F

y N-m (k-in) kN (kips)

0 3.719 (.836)

0 3.736 ( .840)

0 .898 (.202)

0 3.727 (.838)

0 3.759 (. 845)

0 1.165 (. 262)

0 3.661 (.823)

0 3.705 (.833)

0 .898 (.202)

0 3.674 (.826)

0 3. 723 (.837)

0 1.165 (.262)

91

Resultant Force - F z

kN (kips)

.761 (.171)

.761 (.171)

.925 (.208)

.694 ( .156)

.694 ( .156)

.836 (.188)

.641 (.144)

.641 (.144)

.801 (.180)

.583 (.131)

.587 (.132)

.721 (.162)


TABLE 10

RELATIVE HORIZONTAL DISPLACEMENT AND RELATIVE ROTATION

Relative Horizontal Relative Point D

Displacement Rotation Relative to My M Point below

X

Case No. (mm) [in] (radians) (see Fig. 48)

1 .03353 .00000 14 (.00132)

2 .03404 .00139 14 (.00134)

3a .02362 .00101 14 (.00093)

3b .05639 .00147 13 (.00222)

3c .14376 .00195 12 (.00566)

3d .40411 .00238 11 ( .01591)

3e .77927 .00256 10 (.03068)

3£ 1.2426 .00144 9 (.04892)

4 .03150 .00000 14 (.00124)

5 • 03277 .00140 14 (.00129)

6a .01981 .00090 14 (.00078)

6b .04572 .00132 13 (.00180)

92

I I

I I

I I I I I I I I I I ·I·· I I I I I

TABLE 10 (continued)



Displacement Rotation Relative to M M Point below y X


6c .11176 .00176 12 (.00440)

6d .29845 .00216 11 (.01175)

6e .55728 .00233 10 (.02194)

6£ • 86512 .00160 9 (.03406)

19 .03454 .00000 14 (.00136)

20 .03378 .00137 14 (.00133)

21a .02388 .00099 14 (.00094)

21b .05690 .00144 13 (.00224)

21c .14478 .00191 12 (.00570)

21d .40665 .00235 11 (.01601)

2le .78410 .00253 10 (. 03087)

21£ 1.2502 .00141 9 (.04922)

93






X


22 .03150 .00000 14 (.00124) .

23 • 03251 .00141 14 (.00128)

24a .01981 .00088 14 (.00078)

24b .04572 .00129 13 (.00180)

24c .11100 .00173 12 (.00437)

24d .29337 .00212 11 ( .01155)

24e .54534 .00231 10 (.02147)

24£ .85039 .00160 9 (.03348)

94


TABLE 11

MAXIMUM STRESS WITHIN WEB GAP DUE TO M yy

Stress Location of (MPa) Maximum Stress

Case No. [ksi] · (Points Refer to Fig.

1 .97 9 (.14)

2 92.74 14 (13.45)

3a 59.23 14 (8.59)

3b 60.40 14 (8.76)

3c 65.30 14 (9 .47)

3d 73.16 11 (10.61)

3e 98.87 10 (14.34)

3f 100.18 9 (14.53)

4 1.03 9 ( .15)

5 111.91 14 (16.23)

6a 65.23 14 (9.46)

6b 57.44 14 (8.33)

6c 55.16 14 (8.00)

6d 56.81 14 (8. 24)

95

48)



MAX1MUM STRESS WITHIN WEB GAP DUE TO M yy

Stress Location of (MPa) Maximum Stress

Case No. [ksi] (Points Refer to Fig.

6e 63.71 10 (9.24)

6f 69.29 9 (10.05)

19 1.03 9 ( .15)

20 91.29 14 (13.24)

21a 58.06 14 (8.42)

21b 59.71 14 (8.66)

21c 65.02 14 (9.43)

21d 74.33 11 (10.78)

21e 99.98 10 (14.50)

21£ 101.22 9 (14. 68)

22 1.03 9 ( .15)

23 112.11 14 (16.26)

24a 64.12 14 (9.30)

24b 56.40 14 (8.18)

96

48)


Case No.

24c

24d

24e

24f


MAXIMUM STRESS WITHIN WEB GAP DUE TO ~y

Stress Location of (MPa) Maximum Stress [ksi] (Points Refer to Fig. 48)

54.40 14 (7.89)

54.81 14 (7.95)

62.81 10 (9 .11)

69.36 9 (10.06)

97


TABLE 12



Displacement Rotation Relative to My 6.9 Point below

X


13 .07671 .00000 14 (.00302)

14 .08433 .00396 14 (.00332)

15a .03988 . 00205 14 (.00157)

15b .08941 .00296 13 (.00352)

15c .20803 .00390 12 (.00819)

15d .51511 .00477 11 (.02028)

15e . 91491 .00518 10 (.03602)

15£ 1.3612 .00356 9 ( .05359)

16 .. 06528

.00000 14 (.00257)

17 .07747 .00363 14 (.00305)

18a .03708 .00189 14 (.00146)

18b .08179 .00276 13 (.00322)

18c .18390 .00371 12 ( .00724)

98






X


18d .43739 .00463 11 (.01722)

18e .75514 • 00510 10 (.02973)

18£ 1.0955 .00384 9 (.04313)

31 .07620 .00000 14 (.00300)

32 .08331 .00397 14 (.00328)

33a .04267 .00220 14 (.00168)

33b .09474 .00317 13 (.00373)

33c .21793 .00418 12 (.00858)

33d .53188 .00513 11 (.02094)

33e .93624 .00561 10 (.03686)

33£ 1.3879 .00391 9 (. 05464)

34 .06274 .00000 14 (.00247)

35 .07645 .00367 14 (.00301)

99


Case No.

36a

36b

36c

36d

36e

36£




Displacement Rotation Relative to My ~e Point below

X

(mm) [in] (radians) (see Fig. 48)

.03632 .00189 14 (.00143)

.08026 .00276 13 (.00316)

.18186 .00371 12 (.00716)

.43383 .00462 11 (.01708)

• 75159 .00509 10 (.02959)

1.0930 • 00383 9 ( .04303)

100


I I , I

Case No.

13

14

15a

15b

15c

15d

15e·

15f

16

17

18a

18b

18c

18d

TABLE 13

MAXIMUM STRESS WITHrn WEB GAP DUE TO Myy

Stress Location of (MPa) Maximum Stress (ksi] (Points Refer to Fig. 48)

2.55 9 (. 37)

264.42 14 (38.35)

125.14 13 (18.15)

95.70 14 (13.88)

78.19 14 (11. 34)

70.74 14 (10.26)

69.16 14 (10.03)

78.60 9 (11.40)

2.41 9 ( .35)

288.90 14 (41. 90)

167.55 13 (24.30)

122.39 12 (17.75)

73.29 14 (10.63)

61.57 14 (8.93)

101


Case No.

18e

18f

31

32

33a

33b

33c

33d

33e

33f

34

35

36a

36b


MAXIMUM STRESS WITHIN WEB GAP DUE TO Mvv


58.26 14 (8.45)

58.12 14 (8.43)

2.48 9 (.36)

265.25 14 (38.47)

141.35 13 (20.5)

100.25 14 (14.54)

79.43 14 (11.52)

69.98 14 (10.15)

67.23 14 (9. 75)

77.50 9 (11. 24)

2.21 9 (. 32)

292.28 14 (42 .·39)

167.41 13 (24.28)

121.90 12 (17.68)

102


Case No.

36c

36d

36e

36f


MAXIMUM STRESS WITHIN WEB GAP DUE TO ~y


73.57 14 (10.67)

61.78 14 (8.96)

58.75 14 (8 .52)

58.61 14 (8.50)

103

I I I I I I I I I I I I I I I I .I I I

TABLE 14

STRESS DUE TO ROTATION AND DISPLACEMENT

Case No. cre crt::. crt (MPa) (MPa) (MPa) [ksi] [ksi] [ksi]

3a - 504.92 697.36 192.44 (-73.23) (101.14) (27.91)

3b - 367.43 416.18 48.75 (-53.29) (60.36) (7.07)

3c - 243.67 265.25 21.58 ( -35-. 34) (38. 47) (3.13)

3d - 148.73 186.17 37.44 (-21.57) (27.00) (5. 43)

3e - 106.67 159.76 53.09 (-15.47) (23.17) (7. 70)

3£ - 45.02 143.28 98.26 (-6. 53) (20.78) (14.25)

6a - 539.88 701.84 161.96 (-78.30) (101.79) (23.49)

6b - 395.91 404.94 9.03 (-57.42) (58. 73) (1. 31)

6c - 263.94 247.46 - 16.48 (-38.28) (35.89) (-2.39)

6d - 161.96 165.20 3.24 (-23.49) (23.96) (0.47)

6e - 116.46 137.07 20.61 (-16.89) (19.88) (2.99)

6£ - 59.99 119.70 59.71 (-8.70) (17.36) (8.66)

15a - 1024.80 1177.25 152.45 (-148.63) (170.74) (22.11)

15b - 739.83 659.85 - 79.98 (-107.30) (95.70) (~11. 60)

15c - 487.41 383.78 - 103.63 (-70.69) (55.66) (-15.03)

15d - 298.07 237.60 - 60.47 (-43.23) (34.46) (-8. 77)

15e - 215.81 187.54 - 28.27 ( -31. 30) (27.20) (-4.10)

15£ - 111.22 156.93 45.71 (-16.13) (22.76) (6.63)

104



STRESS DUE TO ROTATION AND DISPLACEMENT

Case No. cre cr6. crt (MPa) (MPa) (MPa) [ksi] [ksi] [ksi]

18a - 1133.74 1313.70 179.96 (-164.43) (190. 53) (26.10)

18b - 827.81 724.32 - 103.49 (.;.120.06) (105.05) (-15.01)

18c - 556.36 407.1,5 - 149.21 (-80.69) (59.05) ( -21. 64)

18d - 347.16 242.08 - 105.08 (-50.35) (35 .11) (-15.24)

18e - 254.98 185.75 - 69.23 (-36.98) (26.94) (-10.04)

18f - 143.97 15.17 - 128.80 (-20.88) (2.20) ( -18·. 68)

Note: The stresses created by the distortions of the web are

fiber stresses. Corresponding to these fiber stresses

and each value of crt will be a tensile stress (+) on one

side of the web and a compressive stress (-) on the other

side. The negative sign in crt above does not indicate

compression, but merely indicates that the stress due to

rotation, cre, was dominant. The lack of a symbol for

crt indicates that cr6. was dominant.

105


I I

Case No.

3a

3b

3c

3d

3e

3£

6a

6b

6c

6d

6e

6£

GaE Length Web Thickness

(g/tw)

1.6 t)

3.2 ~

6.4 £

12.8 II

19.2 C)

25.6 .C)

1.33 ®

2.66 ~

5.33 A

10.66 ~

16.0 ®

21.33 ®

TABLE 15

FATIGUE LIFE ESTIMATES

Substructure

(J Fatigue Life max (MPa) (N)

6 [ksi] [cycles x 10 ]

59.23 7.60 (8.59)

60.40 7.17 (8.76)

65.30 5.67 (9.47)

73.16 4.04 (10.61)

98.87 1.63 (14.34)

100.18 1.57 (14.53)

65.23 5.69 (9.46)

57.44 8.34 (8.33)

55.16 9.41 (8.00)

56.81 8.62 (8.24)

63.71 6.11 (9.24)

69.29 4.75 (10.05)

106

Eq. 7

I crt! Fatigue Life

(MPa) (N) 6 [ksi] [cycles x 10 ]

192.44 0.22 (27.91)

48.75 13.64 (7. 07)

21.58 100 (3 .13)

37.44 30.10 (5.43)

53.09 10.56 (7. 70)

98.26 1.67 (14.25)

161.96 0. 37 (23.49)

9.03 100 (1. 31)

16.48 i 100 (2.39)

3.24 100 (0.47)

20.61 100 (2.99)

59.71 7.42 (8.66)


~I

Case No.

15a

15b

15c

15d

15e

15f

18a

18b

18c

18d

18e

18f

GaE Length . Web Thickness

(g/tw)

1.6 0

3.2 ~

6.4 A

12.8 [!]

19.2 0

25.6 0

1.33 0

2.66 <>

5.33 ~

10.66 0

16.0 0

21.33 0


FATIGUE LIFE ESTIMATES

Substructure Eq. 7

cr Fatigue Life crt Fatigue Life max (N) (N) (MPa) 6 (MPa) 6 [ksi] [cycles x 10 ] [ksi] [cycles x 10 ]

125.14 0.81 152.45 0.45 (18.15) (22.11)

95.70 1.80 79.98 3.09 (13.88) (11. 60)

78.19 3.31 103.63 1.42 (11. 34) (15 .03)

70.74 4.46 60.47 7.15 (10.26) (8. 77)

69.16 4.78 28.27 69.94 (10. 03) (4.10)

78.60 3.25 45.71 16.54 (11. 40) (6.63)

167.55 0.34 179.96 0.27 (24.30) (26.10)

122.39 0.86 103.49 1.43 (17.75) (15.01)

73.29 4.01 149.21 0.48 (10.63) (21. 64)

61.57 6.77 105.08 1.36 (8.93) (15.24)

58.26 7.99 69.23 4.76 (8.45) (10.04)

58.12 8.05 128.80 0.74 (8. 43) (18.68

107


TABLE 16

STRESS RANGE FROM EXPERIMENTAL DATA

Fatigue Life Gap Length Relative Stress Range (N) (g) Deflection (o) s

6 mm (in) mm (in) MPar (ksi) (cycles x 10 )

.77 12.7 .254 126.87 (.5) (.01) (18.40)

1.00 12.7 1.27 116.46 (.5) (.05) (16.89)

3.67 12.7 .0254 75.50 (. 5) (. 001) (10.95)

5.99 12.7 .0254 64.12 (.5) (.001) (9.30)

9.59 12.7 .0254 54.82 (.5) (.001) (7.95)

.75 25.4 .127 128.04 (1.0) (. 005) (18.57)

.77 25.4 .0254 127.28 (1.0) (. 001) (18.46)

.76 25.4 .0635 127.56 (1.0) (.0025) (18.50)

2.89 25.4 .0254 81.71 (1.0) (. 001) (11. 85)

.85 25.4 .127 123.21 (1.0) (.005) (17.87)

.87 50.8 .254 121.97 (2.0) ( .01) (17.69)

2.73 50.8 .0508 83.29 (2.0) (. 002) (12.08)

3.69 50.8 .0508 75.36 (2.0) (.002) (10.93)

.47 50.8 .127 150.24 (2.0) (. 005) (21.79)

108

(x)

(x)

(x)

(x)

(x)

(x)




Fatigue Life Gap Length Relative Stress Range (N) (g) Deflection Co) s

6 r (cycles x 10 ) mm (in) n1m (in) MPa (ksi)

1 .. 42 50.8 .127 103.63 (2.0) (. 005) (15.03)

.44 50.8 1.27 153.14 (2.0) (.05) (22.21)

.47 50.8 .127 150.24 (2.0) (.005) (21. 79)

2.73 50.8 .0508 83.29 (2.0) (. 002) (12.08)

.42 50.8 .254 155.00 (2.0) ( .01) (22.48)

3.92 50.8 .254 73.84 (2.0) (.01) (10. 71)

6.61 50.8 .0508 62.06 (2.0) (.002) (9.00)

.20 50.8 .127 198.51 (2.0) (. 005) (28.79)

6.61 101.6 .0584 62.06 (4.0) (. 0023) (9. 00)

.31 101.6 1.27 172.65 (4.0) (.05) (25.04)

1.57 101.6 .254 100.25 (4.0) (. 01) (14.54)

10.0 101.6 .254 54.06 (4.0) (.01) (7. 84)

3.66 101.6 .127 75.57 (4.0) (. 005) (10.96)

5.9 101.6 .127 64.47 (4.0) (. 005) (9.35)

109

(x)

(x)

(x)

(x)

(x)

(x)

(x)


, I I



Fatigue Life Gap Length Relative Stress Range (N) (g) Deflection (a) s

X 106

) r

(cycles nun (in) mm (in) MPa (ksi)

1.38 101.6 .254 104.67 (4.0) ( .01) (15.18)

2.82 101.6 .127 82.40 (4.0) (.005) (11. 95)

.44 203.2 1.27 153.14 (8.0) (.05) (22.21)

.43 203.2 1. 27 154.65 (8.0) (.05) (22.43)

Note: All data presented in this table is for a web thickness of 9.53 rmn (3/8 in).

110

(x)

(x)

(x)

- -

I-' I-' I-'

- - - - - - - - - - - - - -

TABLE 17 ------ALLOI.fABLE RANGE OF STRESS - REDUNDANT LOAD PATH STRUCTURES} - (Allowable Range of Stress, Fsr)

Category (see Table 1.7.2A2) For For For For over 100,000 cycles 500,000 cycles 2,000,000 cycles 2,000,000 cycles

··-f---ksi MPa ksl HPa ksi HPa ksi HPa

A 60 413.69 36 248.21 24 165.47 24 165.47

B 45 310.26 27.5 189.60 18 124.10 16 110.31

c 32 220.63 19 131.00 13 89.63 10 68.95 12* 82.74*

D 27 186.16 16 110.31 10 68.95 7 48.26

E 21 144.79 12.5 86.18 8 55.15 5 34.47

F 15 103.42 12 82.74 9 62.05 8 55.15

NONREDUNDANT LOAD PATII STRUCTURES2

A 36 248.21 24 165.47 24 165.47 24 165.47

B 27.5 189.60 18 124.10 16 110.31 16 110.31

c 19 131.00 13 89.63 10 68.95 9 62.05 12* 82.74* 11* 75.84*

D 16 110.31 10 68.95 7 48.26 5 34.47

E 12.5 86.18 8 55.15 5 34.47 2.5 17.24

F 12 82.74 9 62.05 8 55.15 7 48.26

* For transverse stiffener welds on girder webs or flanges. 1structure types with multi-load paths where a single fracture in a member cannot lead to the collapse.

2

For example, a simply supported single span multi-beam bridge or a multi-elenient eye bar truss member have redundant load paths.

Structure types '~ith a single load path where a single fracture can lead to a catastrophic coJlnpse. For example, flange and web plates in one or two girder bri.dges, main one-element truRs members, hanger plntes, caps at single or two column hents have nnnredundant lnad paths.

- - -

I I I I I I I I I I I I I I I I Fig. 1

I I I

Cracking plate in were not

A --,

Section A-A

(a) See Fig. I (b) below

(b)

in web gap at floor beam-to-girder connection negative moment region where connection plates welded to tension flange. (Taken from Ref. 3)

112


r

0

0

0 0

0 0

0 0

0 : 0 I 0 1 I ol I 0 I o: I 0 I I

'lr-~--------------------~1~0~111 I I

J

J:~ 1 ~~: I I I --.--r----:1:!'---,

'--\ ~ t \_See Fig. 2(b) below for

crocking at transverse diophro~m web connect1on plate

(a) Relative deformation

(b) Cracking at transverse diaphragm web connection plate (skewed bridge)

Fig o 2 Deformation in Multi-b:eam Bridge (Taken from Ref o 3)

113


, I

I

----------------

(a)

(b) Fig. 3 Cross-frame detail which has developed fatigue

crack growth (Taken from Ref. 3)

114

•

I I I I I I I I I· I I I I I.

II , I

I I I

Jack

--------

Deflection Gage

Wooden Stiffener

~ Up And Dow.n f Movement

Fig. 4 Schematic of Out-of-Plane Displacement Test

115

I

I I I I I I I I I I I

I I

I I I I I I I

Fig. 5 Actual Test Setup

116


Fig. 6 Actual Test Setup

I I

117

I

I I I I I I .I

I I I I I I I I I I I I

Web

Fatigue Crack Developed In N Stress Cycles

Flange

Fig. 7 Typical Cracking at End of Stiffener

118

---- -·--------------

0 fTI .,

0.1 ............ .._

r fTI 0 --i 0 0.01 z

::J .

...................... ...... '"'"'- . v~ • ......_

OA

........ .....................

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

0

Gop Length g /

• g = 20 · fw o g= IO·fw

• g = 5 · fw ... g = 2.5· fw 6 g = 1.25 fw

............. ........... ....... .

...._~ . . ------- -- -~

105 106

CYCLES TO FAILURE

Fig. 8 Test Results of Out-of-Plane Displacement

0 J11 ., r J11 0 -i 0 z .. 3 3


0 0

Floorbeom

Fig. 9

Stringer

Crack in web connection angle

See Fig.9(b} below

(a)

(b) Fatigue crack in standard connection angle

(Taken from Ref. 3)

120


'I I

(a)

(b)

See Fig. 10 (b) below

Fig. 10 Crack in stiffener plate of stringer-to-floor-beam connection (Taken from Ref. 3)

121

I I I I I I I

'I II I I I I I I I I I I

} e ~ I

~----L_

~ "'\ :r- ---- - ---I

L ~

\

l:l. \ t_ - I I I I I I I I I I

Floor Beam or Floor Beam Truss

Fig. 11 Schematic of Deformation in Stringer Web

122


;I II

Stiff

Fig. 12

Web A Stiff

Flange Flange

Cross Section Elevation

Schematic of Crack Formation at the End of Transverse Stiffener

123


Fig. 13 Transverse Stiffener with Reconnnended Gap

124


II

I

Fig. 14 Transverse "Tight Fit" Stiffener

125

I I 1-


.·.··~~·:·~/. J 1· •• I.' ,· I I • • • • • J.l ·.. . . . . ... Composite slab

-<.D 0 -3 --c

E t . w E E o;l{) C\J-

r- -I I I I I I I I I I I I 1: - -.+z=~:::::::l

_j L- 59,mm or 4tw . ( 2 ) gap

(a) Floorbeam acting composite with slab

~~~ .. . :1 .. ' .... , .• . . ~ ., .. !·,.· _.:·, ... · . .A • .. s, '•I' •J! • • ·-·- • ~ .. : .·.·,r•., ~.:~ . . . ·l·" . •••• . . . . , '"" . ..

L J V"'"''J

r--:..-_ ,. I I I I -> I r 1-

I I I -J:-:.~-

~l"N l j

(b) Floorbeam without composite action.

Fig. 15 Recommended Floor Beam Connection Plates-in Negative Moment Regions and Near End Supports

126


I I

I I

IOOmm (411)

(4tw to 6tw) Or 50 mm (2 11)

(whichever is larger)

(a) Gusset Plate Welded To Web And Vertical

r£175 mm(3")

(b) Gusset Plate Welded To Web; Transverse Member

Bolted To Vertical

Fig. 16 Suggested Lateral Gusset Connections at Transverse Stiffeners

127

-------------------

...... N 00

--"v- -

I t I

(c) Stiffener End For Right Angle Bridges In Positive Moment Regions. Weld Stiffener To Flange At Supports And In Negative Moment Regions.

Cope(4tw to 6twl Or 50mm Min.

-~uaJ..-_--.:(=2~~~.:...-.) . ...-I~~L---.

(d) Attach Stiffener To Flange In Curved Girder Bridges And Where Out-Of-Plane Movement May Be Large

12.7mm x 100 mm Plate ( ~211 X 411)

(e) Welded Along Web-Flange Only.- No Seal Welds.

Fig. 16 Suggested Lateral Gusset Connections at Transverse Stiffeners (c~ntinued)


w _J <( (.) (J)

(!)

0 _J .. b <l .. z 0

ti ::> 1-(.) ::> _J u.. (J) (J) w 0:: 1-(J)

--INITIATION LIFE --PROPAGATION LIFE --TOTAL LIFE

NUMBER OF CYCLES TO FAILURE, N, LOG SCALE

Fig. 17 Schematic S-N Curve Divided into Initiation and Propagation Stages

(Taken from Ref. 12)

129


c I

w N (J)

3: a· J

c::r _.J lJ..

a· I

NUMBER OF CYCLES-N

Fig. 18 Crack Propagation Data Showing Effect of Applied Stress Level. Fatigue Crack Propagation Rate Increases with Stress and Crack Length. (Taken from Ref. 13)

130


FLAW SIZE-a

0 cr--+ ( Plastic Behavior)

0 cr---+------------ --------(Elastic -Plastic)

Behavior

(Plane Strain) acr----4-+-Behavior 0 o

,_ !I .. ,

I

NUMBER OF CYCLES OF FATIGUE LOADING, N

I - Improvement In Life Due To Lower Stress Level li- Improvement In Life Due To Smaller Initial Flaw Size m- Improvement In Life Due To Moderate Improvement

In Notch Toughness N- Improvement In Life Due To Large Improvement

In Notch Toughness

Fig. 19 Schematic Showing Effect of Notch Toughness, Stress, and Flaw Size on Improvement of Life of a Structure Subject to Fatigue Loading

131

-------------------

1: 24.38 m C To C Bearings

:I (80 1)

1-Cross Framing .. , -I .

~· Roadwa_y - Stiffener Plates 2.54m ~ i\This Side (8.33 1) 8.53 m (28')

~ I \ I 2.54m r - -

{8.33 I) ~ ~ 1-' \ w N

) ~ <t Girders 2.54m

~ {8.33 I)

Fig. 20 Framing Plan of Subject Bridge

-------------------<t Bearing

Sym. About <t_ Span

12.20m (40') Top Flange~--------------------------------~~--~.~-------1

PLATE-304.8mm x 14.29mm (1211

X 9/16') ~Om ~Om

Diaphrag~ Spacing ~--5~0~8~m--m--(~2~0~.~)--------~~--------~(2~0~'~)--------~~ ( 1.66 )

5 @ 1.016 m = 5.08 m 5@ 1.22 ~ = 6.10m Stiffener Spacing 1-J.~-+----_:_------"7-------;-:---t-----:-:---=--:o--:::-:-i~-------1

(5@ 3.331 = 16.66

1) (5@) 4'=20')

Note: All Material ASTM A36

End Diaphragm

Bearing Stiffener Plate

Stiffener Plate ,-t-r<Typ. {One Side Only)

100 mm x 7. 94 mm (4

11 )( 5/16

11

I

Web Plate 1220 mm x 7.94 mm

{ 4811

X 5/1611

)

Diaphragm ConnectiJ Plate 152.4 mm x 9.53 mm

4.88m { 16') Bottom Flange PLATE-304.8 mm x 25.4mm

{1211

x 111

)

7.32m{24') (G"x 3/ 8")

PLATE-304.8 mm x 44.54m~ { 12"x 1-3/4

11)

Fig. 21 Elevation of Subject Bridge Girder

- - - - - -· - - - - - - - - - - - - -

0.81m

(2.661

)

I. 27m {4.16')

10.16m(33.331

) 0 ToO

8.53 m ( 281

) Clear Roadway

4.265m 4.265 m 0.81m

( 141

) ( 14 ) (2.661

)

L 100 mm x 75 mm x 8 mm ( L 4 x 3 x 5/16- Typ.)

Sym. About <t_ Roadway L100mmx75mmx8mm

50mm Min. (2")

LIOOmm x 89mm x 8mm {L4x 31/~ x 5/16-Typ.)

2.54 m ( 8.33

1)

Half Section At End Bearing

I. 27m (4.16

1)

177.8 mm (L4x3x5/16-Typ.) (711)

1.27 m (4.16'.)

19mm {3/4") Typ. For Tension Flange

2.54m (8.33'')

Half Section At Intermediate Cross Framing

Fig. 22 Typical Cross Section

Typ.

I. 27m (4.16 I)

-------------------z y

SUPPORT MIDSPAN

Fig. 23 Finite Element Discretization of Bridge

-------------------

1250mm 49.16

11

Centroidal Plane of Actual Deck Slab

Reference Plane of Deck Slab Elements and Centroid of Girder Top Flange

380mm 15.0

11

390mm 15.5

11

G35mm • _ 635mm •j• 635 mm_

1

_ 635 mm_

1 1- 2511 I 25

11 25

11 25

11

. .. •' .. . b. · .. ~ .. .. ..

Discretization

1- 2.54m

8.33'

(a) Interior Cross Frame

Fig. 24 Typical Cross Framing

e = 121.41 mm 4.78

11

_[19.05mm 0.75

11

c::t::5-r 152.4mm

1

T 6" -

_________ , _________ _

1250mm 49.16

11

Centroidal Plane of Actual Deck Slab

Reference Plane of Deck Slab Elements and Centroid of Girder Top Flange

380mm 15.0

11

390mm 15.5

11

230mm 8.88"

_635mm • - 635mm" "635 mm. I "635 .. mm.l I 25" I 25" I 2511

25 • . . .. . ..

4 A c.·

Discretization 152.4mm

1

611 ----------~2~.5~47m~------------~-~

8.331

(b) End Bearing Cross Frame

Fig. 24 (cont'd) Typical Cross Framing

e = 121.41 mm 4.78

11

-------------------

2.54m

100"

1.22m x 7.94mm 48" X 5/16 II

304.8mm x 44.54mm 1211

X 13/411

(a) Typical Composite Girder Section

Fig. 25 Composite Girder Section at Midspan

----- -·-------------

WEFF = ~=0.3175m(l2.5") ,_ _,

Neutral

1.06m

41.8511

I Axis

(b) Classical Transformed Section

WEFF ,_ ., I .I I I

Neutral Axis

lOOm 39.31"

-L-- •~.--_ ___.

(c) Modified Classical Transformed Section

Fig. 25 (cont'd) Composite Girder Section at Midspan

-------------------<t_ Axle Cl Axle 't Axle

'jj)

1- 4.27m -1- 4.27 m -I -Q)

141

141 Q)

.c. 3: 17.79 kN 71.16 kN 71.16kN ~

4k 16 k 16k

'jj) 1.83m - 61 Q) Q)

.c. 3: 17.79kN 71.16kN 71.16kN ~

4k 16 k 16k

Test Vehicle A: Truck Loading HS 20-44

~ .p. 0

't Axle 't Axle Q)

,_ 4.27m -I Q)

141 .c.

3: 17.79kN 71.16kN ~

4k 16k

1.83m Q) 61 Q) .c. 3: 17.79kN 71.16 kN ~

l6k 4k

Test Vehicle 8: Truck Loading H20-44

Fig. 2~ Test Loading Vehicles

-------------------Position No. I

0.46m 18

11 L Girder No. CD

Position No. 2

I \ I

-r-

0.46m ,_ II

18

Girder No. Q)

p 1.83m

p

7211

1.68m 66

11

10011

_ ,_ 2.54m _, _ 2.54m 100" 100

11

2.54m

® ® @

p p

r f I I .._ ..... r-" -

.71m 1.83m

2811

72 11

c:::::.

-I- _, 2.54m -1- 2.54m 2.54m 0.46m II II II II

100 100 100 18

®

Fig. 27 Transverse Loading Positions

-------------------Position No. 3

I \ r

--.;;;;;;~ ..... ....-

""' !::::.. c:::!:::a

0.46m L 2.54m -1-1811

10011

Girder No. CD ® ~ .p. N

Position No. 4

0.46m 1- 2.54m . I . 18

11 100

11

Girder No. CD ®

Fig. 27 (cont'd)

p

..

~ 0.355m 1.83m

1411

72.11

c:b

2.54m -1- 2.54m 100

11 100

11

®

1.83m 72

11

2.54m -I- 2.54m

10011

10011

®

Transverse Loading Positions

p

-I c:b

-1

p

-I 0

I I l

0.355m

1411

I ~

0.46m

1811

- - - - - -· - - - - - - - - - - - - -Position No. 5 p

1.83m p

7211

1.98m .56 I. 27m I. 27m 78

11 22

11 50

11 50

11

0.46m I~ 2.54m ~I~ 2.54m _,_

· 2.54m -1 ~ 1811

10011

10011 100

11 18

Girder No . CD ® 0 @

.....

.p. w Position No. 6 1.83m p

7211

1.63m .91m 1.63m 64 11 36

11 64 11

0.46m 1- 2.54m -1~ 2.54m -I- 2.54m -I 0.46m

1811

10011

10011

10011

1811

Girder No. CD ® ® @

Fig. 27 (cont'd) Transverse Loading Positions

-------------------Position No. 7 p

. 1.83 m p

7211

I. 27m I. 27m 1.98m 5011

5011

7811

0.46m L 2.54m . 1. 2.54m .I . 2.54m -1 ~ 1811

10011

10011

10011

18

Girder No. Q) ® ® @ I-' +:'-+:'-

Position No. 8 p p

1.83m

7211

0.46m 1- 2.54m -I· 2.54m -I- 2.54m -I 0.46m

1811

10011

10011

10011

1811

Girder No. CD. ® ® @


-------------- -·----

Position No. 9 p p

0.355m 1- 1.83m -1 0.355m

1411

7211

1411

.....

I .

_ ,_ _, _ 2.54m _, ~ ~ 0.46m 2.54m 2.54m V1

1811

10011

10011

.10011

18

Girder No. CD ® ® @



t Cross Framing Midspan

l I I

1220mm

71.16KN 4811

1"'-

16k

Quarter Span

71.16KN-... 16k

-'-

610mm -H ~7.79KN-... 2411

4k

- I T r. Bearing --=--'t.. ~

j Cf. Girders -I

Fig. 28(a) Discretized Plan of Deck Slab Showing Longitudinal Loading Positions

146

I I I I I I I I II I I I I I I I I I

~ Cross Framing Midspan

I I 1220mm

i.-71.J6KN-.. 4811

1-.·

16 k

Quarter Span

L~ 17. 79KN-.. 4k

l-et Girders • j • I -1 Cl Bearing

Fig. 28(b) Discretized Plan of Deck Slab Showing Longitudinal Loading Positions

147

l

-L...

T

I I 1-

I I I I I 1-.I I I I I I I I I I

Cl Cross Framing Midspan

l d2o~m

..-17.79KN ..... 48

11

4k

Quarter Span

.-71.16 KN, 4k

.1_

-1 .I T <l Bearing --=------=-------'

I Cf. Girders • I

Fig. 28(c) Discretized Plan of Deck Slab Showing Longitudinal Loading Positions

148

I I .-I_-__ _

-~-

'1 I -I-



I~

I~

I~

15l25~m 60

11

~71.16 KN..:-......

16k

Quarter Span

305mm ~~oo-71.16KN' 12

11

l 16k I

910mm ~ ~o-17. 79KN' 36

11

4k

I ... Cf. Girders • I -1 -1 Cf_ Bearing

Fig. 28 (d) Discretized Plan of Deck Slab Showing Longitudinal Loading Positions

149

l

1-L-

T

I I I· I I I I I I I I I I I I I I I I

<k_ Cross Framing Midspan

I I 1525mm 60"

I~ ~1.16 KN ...... -

16k

Quarter Span

305mm

I~ ~ 17.79KN-.. 1211

l 4k I

1-J- Girders -I -I .1 ct_ Bearing

Fig. 28(e) Discretized Plan of Deck Slab Showing Longitudinal Loading Positions

150

l

-~-

J

I I I -1- -~ . ~ -·· ----

I I I I I I I I I I I I I I I·


l I 1525mm 60

11

·-_ ....

I~ ~7.79KN-4k

Quarter Span

305mm 12

11

I~ ~71.16KN' 16k ·t

~-~Girders .j • I -~ Bearing

Fig. 28(f) Discretized Plan of Deck Slab Showing Longitudinal Loading Positions

151

l

1-'-

r


(( Cross Framing Midspan

l I I

1830mm 72

11

H~ 71.16KN-....

16k

Quarter Span

71.16KN, ,_,_ 16k "

t200n ~m 17.79KN, 8 II

H~ 4k

t

1-t:f.. Girders • I -1 _, T <t Bearing ~-------..1

Fig. 28(g) Discretized Plan of Deck Slab Showing Longitudinal Loading Positions

152


'k_ Cross Framing Midspan

l I I I

1830mm 7211

H ~ 71.16KN' 16k

Quarter Span

17.79KN' 4k

_..._

J-~ Girders • j _, _, T ~~~B_e_a_r_in~g--------~

Fig. 28 (h) Discretized Plan of Deck Slab Showing Longitudinal Loading Positions

153


<k_ Cross Framing Midspan

l I I I

1830mm 72

11

I~ ~H7.79KN' 4k

Quarter Span

f.\ 71.16KN, !-1...

16k

j .. i Girders -I _, _, T <t Bearing ~-

Fig. 28(i) Discretized Plan of Deck Slab Showing Longitudinal Loading Positions

154


Interior Girder 1(.08 kN)

(4.91 kN)1 J (10.96 N-m)1 1(2.79 kN)

1(.05 kN) ~.f • t {- 1(43.1 N-m)

( ~ 1(67.95 N-m)

))\~6 N-m)l I' ). I. 1(.57 kN) 1 .03 kN.~_J , ... ·----

(11.68 kN)1 1(.01 kN)

LOADrnG CASE NO. 1

.98

.• 96 1.~ 7 .95)) ~

· i.or"t \ (~ · 94 ~ +~... 1.o2

.97 5.51 .77

LOADING CASE NO. 2

Fig. 29 End Forces of Bracing Members that Frame into Interior Girder

155

2.70


.98 1. 02

LOADING CASE NO. 4

.96

.• ;!6 ~ .92 ~ ~1.00

.99--t \ (~ .93 ~ ~---~ 1.10

.95 3.52 .82

LOADING CASE NO. 5

1.13 4.36 -~

~~~ ~~\~~0 .95 --f~ ~0 ···~ t .. 2.60

.66 32.20

LOADING CASE NO. 6

Fig. 29 (continued)

156


LOADING CASE NO. 7

LOADING CASE NO. 8

LOADING CASE NO. 9

Fig._ 29 (continued)

157

2.87


LOADING CASE NO. 10

LOADING CASE NO. 11

2. 77

LOADING CASE NO. 12

Fig. 29 (continued)

158

I I I 1.01 .99

I ~~~ ·10~ I 1.00

· ~ . o1 oj• 1.13

I LOADING CASE NO. 13

I I 1.02 ~(-2. 0~ ,~.98

I 1.oo•rr ~ o)Oj.,. 1.16


I I 1.20 ~ 1.13

I . I(~~ •1°( : ~ 1 ... 3.02

I .95


I Fig. 29 (continued)

I I 159

I

I I I

.98 ~(0 o~y 1.00

I • 9~~~o ~ o)oj ... 1.21

I I

LOADING CASE NO. 16

I I .98 ~(..0 . ~\..----.99

I .99 ... ff- ~ 0

)01 ... 1.22


I I

1.15 "J~ ~y 1.13

I· ... ,oc ~ ,1 ... 2.90

I .95

LOADING CASE NO. 18

! I I Fig. 29 (continued)

I I 160

I

I I I I I I I I I I I I I I I I I II

I

1.20

LOADING CASE NO. 19

.96

.93 .94 ~ .90 ~ '\1.01

.99 .. t ~ (~ .9~ • ....,__.~ 1.23

.93 2.02 .87

2.86

LOADING CASE NO. 21

Fig. 29 (continued)

161

I I

I I


.97

LOADING CASE NO. 22

.94

.9~91 ~ . .88~ ~1.02

.98 .. t ~ (~ ·91

-) +~~ 1.29

.92 .24 .91

LOADING CASE NO. 23

-~5 72

.94 .. t~ ~ ~t .. .64 28.69

2.75

LOADING CASE NO. 24

Fig. 29 (continued)

162


.16 . 09

•lo(g • 98 \.

.9~ . ~~1.02

. (~ 0~~~--1.32 LOADING CASE NO. 25

.16 • 05

.95~ \ ............. 1.00

•laC (~1o o)OJ~•- 1 • 35 .98

LOADING CASE NO. 26

3.03

LOADING CASE NO. 27

Fig. 29 (continued)

163


'I I I

.16 • 05

.93~ ~1.02

.. ,0~ (~) 0~ 01~- 1.38

.98

LOADING CASE NO. 28

LOADING CASE NO. 29

2.91

LOADING CASE NO. 30

Fig. 29 (continued)

164


.99

LOADING CASE NO. 31

.95 ~1.00

. ~~0 0~\ .,.lo(o ~ o)Oj...-.~-1. 36

LOADING CASE NO. 32

3.17

LOADING CASE NO. 33

Fig. 29 (continued)

165


.93

LOADING CASE NO. 34.

LOADING CASE NO. 35

1.09 1.15

~~~ ~y ..,lae ~. 0)1 ... 3.04

LOADING CASE NO. 36

Fig. 29 (continued)

166

-------------------

809.63 mm

(31.875 in)

BOTTOM FLANGE

1270 mm

(50 in)

TRANSVERSE CONNECTION s, WEB PLATE PLATES (STIFFENERS)

DIAGONAL BRACING

MEMBER

2540 mm

(100 in) (100 in)

Fig. 30 Critical Area Examined in the Refined Analysis

-------------------

SUPPORT MIDSPAN

SUBSTRUCTURE

MODEL

Fig. 31 Relationship of Critical Area to Prototype Structure

-------------------

SUPPORT MIDSPAN

.508 m .508 m .610 m .610 m r ,

(1.67 ft) (1. 67 ft) (2.0 ft) (2 .• 0 ft)

4 @ 1.016 m ~ 4.064 m 3 @ 1.219 m = 3.657 m -- - ~ .

(4 @ 3.33 ft = 13.33 ft) (3 @ 4.0 ft = 12.0 ft) CRITICAL AREA

(SUBSTRUCTURE) ~ 1\..

' -.--- ~ ~~ / ~ ~

/ ~ ~

809.63 mm / 7

., (31.875 in)

914.40 mm (36 in)

Fig. 32 Relationship of Critical Area to Girder No. 3

-------------------TOP FLANGE

1--" "--1 0

Lx BOTTOM FLANGE

CENTROID OF ELEMENT

-5.568

(-.8075)

..... 7. 550

1

. (1. 095)

I +23.581 I (3.420)

+ 36.337

I (5.270)

Fig. 33 Membrane Stress S [MPa XX

MIDSPAN

I - + -6.341 I_

- + 5.974 (- .9196) I <-. 8664)

+ 7.833 + 7.488 I (1.136) I (1.086)

I _,._24.801

I _.- + 24.608 I (3 .597) I (3.569)

I I Q ..... 38.260 + 38.702

1 (5.549) 1 (5.613)

Q

(ksi)] Case No. 1

-------------~-----

TOP FLANGE

-2.643

(-.3836)

-1.521 -+-1---

(-.2206) I

-.196 --(-. 0284)

L (. 85~7...:..)__._ X (.1243)

BOTTOM FLANGE

CENTROID OF ELEMENT

-3 086 I ·-+--(-.4475) I

-1.~! _ _,_ (-.2672) I

.132

(.0191)

Fig. 34 Membrane Stress S [MPa (ksi)] Case No. 1 yy

MIDSPAN

-.159

(-.0231)

I _......_ + 1.054 I (.1529)

I Q ....__+ 2.147

1 <. 3114 >

Q

-------------------TOP FLANGE

BOTTOM FLANGE

CENTROID OF ELEMENT \

--~-4.604 I <-.6677)

-4.516 -+--

1·(-.6550)

L4.087 --+.-~~ I(- .5928) .

L3.561 J --;- ' I<-. 5164 >

- I .7~+...---(- .1038) I

I -1.~+-+--(- .1544) I

-1.037 I -+-+--

(- .1504) I

-.4~1J. __ <-. 0644 > 1

Fig. 35. Membrane Stress S [MPa (ksi)] Case No. 1 xy

MIDSPAN

I . 4ij1 --+--

1 (.0697)

I ---1+_.448 I (.0650)

-.043

(-.0063)

...

__ ! \ -.682 Q

1 <-. 0989 >

Q

-----------·----------TOP FLANGE

Lx BOTTOM FLANGE

CENTROID OF ELEMENT \

~2.476 l + 3.584

I (. 5566xl0- 3 ) I ( .8058xl0 - )

I - ~ 1. 732 + 4.202

,. (.3895xl0- 3 ) ,(.9446xl0-3

.658 ! 4. 759 f-

(.1480xl0- 3 ) 1(1.070xl0-3

/14.438 I 3. 277 T -3 ..... I (. 9977xl0 ) I<· 7367xl0-

3

Fig. 36 Bending Moment M [N-m/m (kip-in/in)] Case No. 1 XX

MIDSPAN

I -- + 2.863

I (.6437xl0- 3

I r4- 1. 845

,(.4149xl0-3

)

·' "" • 788 -(.1772xl0- 3)

.543 Q

l<-.1220x10- 3

Q

- - - - - - - - - - - - - - - - -· - -CENTROID OF ELEMENT MIDSPAN ... , TOP FLANGE \

x8.189 110.119 I 9.363 -3 -,-+ -3 ___,_ + -3 I (1.841x10 ) · 1(2. 275x10 I (2. 105x10 )

5.693 7.740 I 6.356 - .. .... 3 + -3 + -3

r1.280x10- ) rl· 740x10 r1.429xl0 )

12.577 15.898 I 3.850 + -3 j(l· 326x10-

3 + -3 ) r·5793x10. ) ,(.8656x10 )

LX / 114.167 I 4. 742 l Q -1.187 i: -3 i(l· 066x10-

3 ~(-.2669x10- 3 l(3.185x10 )

BOTTOM FLANGE Q

Fig. 37 Bending Moment M [N-m/m (kip-in/in)] Case No. 1 yy '

- - - - - - - - - .. - - - - - - -:- -TOP F LA E NG

Lx BOTTOM FLANGE

CENTROID OF ELEMENT ~

\t1.267 -3 I _. 316 - f. -I(. 2848x 10 ) (- .0710x10 )

.254 .036 - f. (.0570x10- 3) - (.0080x10- 3

I

-.622 I -2.043 1- --I

3) (- .1398x10- 3) (- .4593x10

j .583 L 18.669

... -3 ~ -3 I ( .1310x10 ) 1(1. 949x10 ,

Fig. 38 Bending Moment M [N-m/m (kip-in/in)] Case No. 1 xy

MIDSPAN ... r

_J-1.419 ~~- .3190x10-

3

-1.182 -1 -3 (- .2658x10

·' .036 - 1-(.0080x10- 3)

J I 3. 633 Q

+ -3 1(.8168x10 )

Q

-------------------CENTROID OF ELEMENT

TOP FLANGE

(-. 8084)

_..7 .543

r1. 094)

,.....

13.588 ....... 0\

1(3.421)

Lx 136.350 T 1(5.272)

BOTTOM FLANGE

Fig. 39· Membrane Stress s XX

1-6.343 -+-

(-.9199)

+7 .847 I (1.138)

I _,}4. 725 I (3.586).

138.329 .... I (5.559)

· (MPa-ksi) Case No. 3

MIDSPAN

1-5.974 -+..---1 (-.8664)

+7 .516 I (1.090)

I ..::.....__ + 24.739 I (3.588)

138.143 + I (5 .532)

Q

Q

-------- -,----------TOP FLANGE

BOTTOM FLANGE

CENTROID OF ELEMENT

I -3.!3..!-.__ + --(-.4526) I

-1.529 -1.886 -+-+-- -+--~-

(-.2218)

1

.

-.181

(-.0262)

.827

(.1200)

Fig. 40

(-.2736) I

-.651 I _..._._ __ (- .0944) I

.534 ---(. 0775)

Membrane Stress S (MPa-ksi) Case No. 3 yy

MIDSPAN

--l-1.013 I .1469)

I -.161

(-.0234)

11.053 --1-+---

1 ( .1527)

-.185

(-.0268)

Q

Q

~-· ~~ .• , .... ~ ... ~- .~---. ---··. -~ . ., ... ,. -·····~·~---.---~·--- . ---------------- -·--TOP FLANGE

LX BOTTOM FLANGE

CENTROID OF ELEMENT \.

~-4.598 -. 712 I -+ I c-.6669) c- .1033 > I ,

_l-4.511 I

-1.~+ I ( _ .6542) ( _ .1513) I

1-4.099 -.954 I + --1-I (-.5945) (-.1384) I

1-3.565 -. 527 I j J T . -+ I c-.5171) c-. 0765 > I

Fig. 41 Membrane Stress S (MPa-ksi) Case No. 3 xy

MIDSPAN

-l-476

I c.o690)

I r+ _.423 I ( .0613)

-.077 -· I (- .0112)

11-.454 Q

1 c-.0658)

Q

-------------------CENTROID OF ELEMENT MIDSPAN .. I TOP FLANGE \

\! 1.982 I _ I .153 - +3.008

j ( .4455x10- 3 ) I (.6762x10 - l -3 ) \ ( .0345x10

11.419 I I 4.750 + 4.020 -- j (.3191x10-

3) -3 +

,(.9037x10 I (-1.068 X

10-3)

.631 14.501 I "-3 .672 ... -3 i(l. 012x10-

3 ~+ I (.1419xl0 ) (-.8256 X

j1o-3) J

LX /16.201 _j I 1.157 /+ 4. 777 Q -r -3 .... I (1. 394x10 ) 1(1.609x10-

3 I (1. 074x10-3

)

BOTTOM FLANGE Q

Fig. 42 Bending Moment M [N-m/m (kip-in/in)] Case No. 3 XX .

-------------------

..... CXl 0

TOP FLANGE

Lx BOTTOM FLANGE

CENTROID OF ELEMENT . MIDSPAN

" ~ 6.561 I 8.304

-r-+ 3 i (1.475x10- 3 ) I (1. 867x10-

14.608 I + 7.339 + -3 -3 r (1.036x10 ) I (1.650x10

!2.174 13.697 · I -3 .... -) I ( .4888xl0 ) I (.8312x10 )

) -z 119.980 L 116.382 T (4.492x10-3

) -I (3.683x10 )

Fig. 43 Bending Moment M [N-m/m (kip-in/in)] Case No. 3 YY

• I + 3.962 -

1(.8907x10- 3 )

12.246 -3 I ( .5050xl0 )

I .. + 3.068 -3

~ 1(.6898x10 )

7 I 16.462 Q

+ -3 I (3. 701x10 )

Q

I

- - - - - - -·- - - - - - - - - - - -CENTROID OF ELEMENT MIDSPAN

TOP FLANGE \ • \J 1.394 I .Q91 I -. 004 - ...

----- t(-.OOOBxl0- 3 I ( .3134xl0-3

) (.0205x10 - )

I I .527 . 798 ' 2.213 - ... (.1185xl0- 3 )

- ~ -~ t-

(.1795x10 ) ·'6xl0-

3)

1-2.365 . I " -.210 -20.941 + 1- -3 --~ 3) (- .0471xl0 ) (-.5318xl0 (-4.708 I

/ I X 10- 3 )

Lx / 19.154 __..4.943

Q .384

-3 ..... '1

) 1<5 .158xl0- 3 ) I (.0864x10 ) I (2 .058x10-

BOTTOM FLANGE Q

Fig. 44 Bending Moment M [N-m/m (kip-in/in)] Case No. 3 xy .

-------- ------------

r--' CX> N

TOP FLANGE

BOTTOM FLANGE

I + I

I -+

I I +

I I ·.

;-I

l -+ .

I

I -r-- -·-----

I I

809.63 mm I (31.875 in) -I-

I -r

I~ Fig. 45 Girder Section Dimensions

MIDSPAN ... I

I --+

I

I ---- ~-- ,_, -.,

I I

-+

I I + I

914.40 mm (36 in)

-------------------z

y

X

BOTTOM FLANGE (CD)

G

H

E M F

o FIXED ALONG LINE AC y

o FIXED ALONG LINE BD X

D

HORIZONTAL BRACING MEMBER

TRANSVERSE CONNECTION PLATES (a long line BD)

&z FIXED AT PTS. C, AND E THROUGH L

& FIXED AT POINT B y

Fig. 46 Discretization of the Finite Element Substructure Model

K

L

I I I I I I .I


ly 19.279 nnn 19.30 nun

(.76 in) (. 759 in) \,_ J ~--~~~~+~==--=-=-=-~~~~N~.A~1~- N.A.

25.9lnnn [email protected]=71.63nun

(3@. 94in=2. 82in) (1. 02 in)

y 75nun

(3 in)

L 100 mm x 7 5 nnn x 8 m (L4 X 3 X 5/16)

A/2 = 674.24 nnn2 (1.045 in2

)

I /2 = 343390.92 nnn4 (.825 in4

) X

ACTUAL DIAGONAL

89mm

1(3.5 in)

23.673 mm

(.932 in) N.A.

.. 11... 6 .lOnun (.24 in)

A = 752.00 mm2 (1.166 in

2) I = 306013.33 mm4 (.735 in

4)

X

EQUIVALENT DIAGONAL

[email protected]=76.2mm

(3@1in=3in) N.A. 98.425 nun ----'--14-+ ---...,...-

X I X 32.639

~4--(3.875 in)

- (1. 285

19.05 mm 22.225 mm

(.75 in) ~I (.875 in)

L 100 mm x 89 mm x 8 mm (L4 X 3-1/2 X 5/16)

2 2 A/2 = 725.81 mm (1.125 in )

4 . 4 I /2 = 530695.05 mm (1.275 in )

X

ACTUAL HORIZONTAL

A = 752.26 mm2 (1.166 in2)

I = 518624.34 mm2 (1.246 in4

) X

EQUIVALENT HORIZONTAL

Note: Half of actual properties compared due to the use of symmetry.

Fig. 47 Cross Section of Actual and Equivalent Cross Framing Members

184

-------------------- 1.842 MP

a

7.833 MP (-. 2 6 72 ks i) - .159 MP a a

s (Case No. 1) (-. 0231 ksi) (D (1.136 ksi) yy

~I 7 '-r--

190.5 mm s (Case No. 1)

A ~ 2(2 CD (7.5 in) XX

B _....__ 114.3 rom -,.---- y (4.5 in) - 4Y Local - -

I L. 76.2 mm X (47) [ 6~ l ® (13)[592] (3.0 in) - ·-j__

[email protected]=203.2mm 1 (4 @ 2 in - 8 in)

24.801L MP a G) _§) 38.1 mm - -o. 597/ ksil

(1. 5 in) I

[email protected]=l52.4mm J

(

43) r1n (6@ 1 in = 6 in) __ (7)[603] 22.225 mm,............, I

/_;}8.260 MP~ -@ 9 (.875 in) 16@12. 7~~203.2mm lL"I{5 • 549 ksii ~@ 16@ . 5 1n = 8 in) ~

~ cp (39) [93] . 4 c

Lx 304.8mm [email protected]=38lm\!l (3)[620]

(12 in)

\@ - f \-5@3 in- 15 in) [email protected]=76.2mm

152.4mm=304.8mm . (6@ .5 in = 3 in (2@ 6 in = 12 in) [email protected]=l52.4mm

)

GLOBAL (6@ 1 in = 6 in)

Fig. 48 " Discretized Section of Girder used in Substructure Model


Plates

Equivalent Me ers

B

D

Web

203.2 nnn

(8 in)

Centroid of Bottom Flange

Fig. 49 Simulated Bracing Member with Moment Connection

186


( ....-

I y

X .I ~

~ I (

(_ Local Coord~nat es

AJ Partial Elevation of

Girder Web

Section A-A

End of

Stiffener

Gap

~M 'fi"' yy

I

1 ..

a

+

.. .. M

yy

b

M [N- m/m (kip-in/in)] yy

cr = 6 M

yy

t w

2

Web Stress

cr

Fig. 50 Web Gap Region and Dominant Stress in this Region

187


Bottom of

Horizontal

-

,._--WEB

1.613 mm ( .0635 in)

\ \ \

.00264 RAD

\ l \

- -

Fig. 51 Web Gap Distortion for Case No. 1

188

--

I I I I I I I I I I I I I I.

I I I I I

Bottom of

Horizontal

WEB

1.658 mm

\ \ \ \

2.209 mm

( .0870 in)


189


WEB Bottom of

Horizontal 1.567 mm .0617 in)

(jf \ \

+ .00255 RAD

2.053 mm \ (.0808 in) \

End of

~ Stiffener

9x E/S =

12.7 mm GAP

(.5 in) 2.077 mm

(. 0818 in)

= 9 at end of stiffener X

.00231 RAD

Fig. 53 Web Gap Distortion for Case No. 3a

190


I I I I I

Bottom of

Horizontal

2.024 mm (.0797 in)

End of Stiffener

25.4 mm GAP

(1 in)

...._ __ WEB

1.567 rmn (.0617 in)

.00255 RAD

= .00278 RAD

( .0819 in)

Fig. 54 Web Gap Distortion for Case No. 3b

191


Bottom of

Horizontal

50.88 mm GAP (2 in)

-

WEB

1.563 mm

(. 0615 in)

• 00136 R<\1)

2.102 mm

(. 0827 in)

.00330 RAD

Fig. 55 Web Gap Distortion for Case No. 3c

192


I I I I I

Bottom of

Horizontal

End of Stiffener

101.6 mm GAP

(4 in)

WEB

1.552 mm ( .0~11 in) .

+ .00252 RAD 1. 818 nnn

~ 9x E/S a .00389 RAD

2.222 mm

(. 0875 in)

\ \

Fig. 56 Web Gap Distortion f or Case No. 3d

193


Bottom of

Horizontal

End of Stiffener

152.4 mm GAP (6 in)

-

1-4-- WEB

1. 536 nun

(. 0605 in)

.00169 RAD

2.455 nun

(.0967 in)

.00249 RAD

.00425 RAD = 9x E/S

1.676 mm (.0660 in)

Fig. 57 Web Gap Distortion for Case No. 3e

194


Bottom of Stiffener

and Bottom of Horiz.

WEB

1.528 mm

<.o6o2 in)

= .00331 RAD

\ \ \ 'l"\' . 00692 RAD

.00187 RAD

2. 771 mm

(.1091 in)

\ \ \

Fig. 58 Web Gap Distortion for Case No. 3f

195

I I GAP LENGTH mm

0.056° 50 100 150 200

I 1.40

I 0.048

1.20

I - E c: - E

I >. -

(() Case No. 3 a 21 >.

<l 0.040 (()

1.00 <l

I 1-

1-'-z w z ~ w l.LJ :E

I u l.LJ <t 0.032 u .....1 0.80 <t a. .....1 (J) a.

I 0 (J)

0 .....1 <t .....1

I 1- 0.024

<t z 0.60 1-0 z N 0 0::

N

I 0 0:: ::c 0

::c l.LJ 0.016 0 40 l.LJ

I > 1-

. > <t 1-.....1 <t l.LJ .....1

I 0:: l.LJ 0::

0.008 0.20

I Case No. 6 8 24

I 0 2 4 6 8

I GAP LENGTH , in

I Fig. 59 Relative Horizontal Displacement versus Gap Length

I 196

I


I I

I I

GAP LENGTH , mm o.oo28 o....-___ s,...o ____ to,....o ____ l5-r-o ____ 2...,.-oo

0.0024

-~ 0.0020 0 -o 0 "--)(

Cb

<: 0.0016 z 0 I-<(

I-~ 0.0012

LLJ > I<( _J

~ 0.0008

0.0004

Case No.3

Case No. 6 8 24

Note: Data Points Plotted For Zero Gap Length Are From Milled Condition {i.e. Case No. 2 Plotted With Case No. 3)

0 2 4 6 GAP LENGTH in

Fig. 60 Relative Rotation versus Gap Length

197

8

I I I I I I I.


0

30

In = 25 b

GAP LENGTH , mm 50 100 150

Note= Data Points Plotted For Zero Gap Length Are From Milled Condition (i.e. Case No.2 Plotted With Case No.3)

200

200

0 a.. :E

b

150 en en IJJ 0:: ten

a.. <t C!)

-------100 ~

Case No. 6 B 24

5

0 2 4 6 GAP LENGTH , in

Fig. 61 Maximum Web Gap Stress versus Gap Length

198

8

3:

::! => :::E X <t ::!


Transverse Connection

Plates

Equivalent Bracing Members

z

..... ---~y

F"ig. 62

B Girder Web

0

100

(8 in)

D

Centroid of.

Bottom Flange

Simulated Pinned Connection

199


Bottom of 3.693 mm Horizontal ( .1454 in)

(9j \ \ ~ .00604 RAD

\ \ \

D ----- -.00601 RAD

4.920 mm (.1937 in)


200


Bottom of

Horizontal

--D

-

WEB

4.100 mm

(.1614 in)

\ \ \ \ \

~ .00673 RAD

.00277 RAD

5.466 mm

(. 2152 in)


201


Bottom of

Horizontal

4.902 mm ( .1930 in)

End of Stiffener

12.7 mm GAP (. 5 in)

,_ __ WEB

3.744 mm

(.1474 in)

= .00418 RAD

----.00213 RAD

Fig. 65 Web Gap Distortion for Case No. 15a

202


Bottom of

Horizontal

4.856 mm (.1912 in)

End of Stiffener

25.4 mm GAP

(1 in)

~--WEB

3.764 mm

(.1482 in)

~ ~ .00618 RAD

\ \

~g = X E/S

.00203 RAD

4.945 mm

(.1947 in)

Fig. 66 Web Gap Distortion for Case No. 15b

203

.00499 RAD


Bottom of Horizontal

4. 729 mm

(.1862 in)

End of Stiffener

50.8 mm GAP

(2 in) D

---

WEB

3.785 mm

(.1490 iri)

+ .00621 RAD

\ \ \ ~~ E/S = .00584 RAD

-----.00195 RAD

4.938 mm

( .1944 in)

Fig. 67 Web Gap Distortion for Case No. lSc

204


f l

" I""

WEB

Bottom of 3.792 mm

Horizontal \ I (.1493 in)

.I l ..

(if I I

i .00623 RAD

End of Stiffener

\ ~9 X E/S = .00669 j 1

RAD

101.6 mm GAP \ \ 4.430 mm

(4 in) \ ( .1744 in)

\ D I I - --\

.00192 RAD

4.945 mm

(.1947 J.n)

Fig. 68 Web Gap Distortion for Case No. 15d

205


WEB

Bottom of 3.785 mm

Horizontal ( .1490 in)

.00621 RAD

End of Stiffener

~ Qx E/S= .00715 RAD

\

152.4 mm GAP

(6 in)

\ 4.107 mm

\ (.1617 in)

\ \

.00197 RAD

5.022 mm

( .1977 in)

Fig. 69 Web Gap Distortion for Case No. 15e

206

I I I I I I .I


End of Stiffener and Bottom of

Horizontal

203.2 mm GAP

(8 in)

WEB

\ \ \

3.942 mm

(.1552 in)

Qx E/S = .00560 RAD

+ .00785 RAD

\ \

.00204 RAD

5.304 mm

(.2088 in)

Fig. 70 Web Gap Distortion for Case No. 15f

207


GAP LENGTH , mm

0_056

0 ~---...;;.5-r-O ____ IO-r-O ____ I5..,..0 ____ 2"'MOO 1.40

-c:

>. (()

0.048

<J - 0.040

1-z lJ.J ::2 lJ.J u :5 0.032 C... en 0

....J <t 1-z 0.024 0 N a:: 0 :t:

lJ.J > 0.016 1-<t ....J lJ.J a::

0.008

0

Case No. 15 8 33

Case No. 1.~ 8 36

2 4 6 GAP LENGTH in

-E E ->.

(()

1.00 <J

1-z lJ.J ::2 lJ.J

0.80 ~ ....J c.. en 0

....J <t

0.60 1-z 0 t:::! a:: 0 :t:

8

0.40 lJ.J > 1-<t ....J lJ.J a::

0.20

Fig. 71 Relative Horizontal Displacement versus Gap Length

208

I I I I I I I I I I I I I I I I I I I·

0 0.0056

0.0048

(/)

g 0.0040 -c 0 ~

>< Q)

<l - 0.0032

z 0 1-c::r 1-0 a::

LLJ > 1-c::r _J

~ 0.0016

0.0008

0

GAP LENGTH , mm 50 100 150 200

Case No. 18 a 36

Note : Data Points Plotted For Zero Gop . Length Are From Milled Condit ion '•

2 4 6 8 GAP LENGTH in


_209

I I GAP LENGTH , mm

I 0 50 100 150 200

I 60 Note= Data Points Plotted For Zero Gap

I Length Are From Milled Condition

I - 50 -(/)

.X 0 a.. I b ::E -b Cf)

I Cf)

w 40 Cf) 0: Cf)

.f- w

I Cf) 0:

1-a.. Cf) <{ (!)

I £Il w 3: £Il

w

I ::E 3: :::> ::E 150 ::E

:::> X ::E I <{

::E Case No. 15 8 33 X <{

::E

I I 50

Case No. 18 8 36

I 0 2 4 6 8

I GAP LENGTH , in

I Fig. 73 Maximum Web Gap Stress versus Gap Length

I 210

I

I I GAP LENGTH , mm

0 50 100 150 200

I 0.056 lA-O

I Case No. 15 S 33

0.048 1.20 I -- Web Thickness = 7.94 mm E c:::

I 5/1611 E >.

co >. co <J 0.040 1.00 <J

I ~ ~ z z w w ~ ~ I w w u

0.032 0.80 u <( <( _J

...J c..

I (f) c.. (f) o· 0

...J. _J

I <(

<( ~ 0.024 0.60 .~ z 0 z N 0

N I 0: 0: 0

~ 0 ~

I w 0.016 0.40 w > > ~ <( ~

<(

I _J

_J w w 0: 0:

0.008 Case No. 3 a No. 21 0.20

I I

0 2 4 6 8

I GAP LENGTH , in


I 211

I

I I GAP LENGTH , mm

0.056° 50 100 150 200

I 1.40

I Web Thickness = 9.53 mm 0.048 3ta .. 1.20 I -E c:

.S -I >. >.

~ ~

<l 0.040 Case No. 18 <l 1.00

I .._ .._ z z w w ~ :E w w

I u u <t 0.032 0.80

c::x: _J _J a. a.

I en en Q Q

_J _J c::x: <t

I .._ .._ z 0.024 0.60 z 0 0

·N N

I 0:: 0:: 0 0 :::c :::c w

0.016 w

I > 0.40 > .._ .._ c::x: c::x: _J _J

I w w 0:: 0::

0.008 0.20

I Case No. 6 a 24

I 0 2 4 6 8

I GAP LENGTH , in


I 212

I


GAP LENGTH 1 mm O.OOS6 or--__ ___;:5:.,.;:0;.__ __ ....;.1..;;.,.00~------::15•0----2-r-OO

0.0048

1/)

c: 0.0040 0

"'0 0 ... -)(

<l)

<l ~ 0.0032

z 0 ~ <X: 1-0 0:::

w > 1-c::x: ...J w 0::: 0.0016

0

Case No.3 S 21

Web Thickness = 7.94 mm 5/tGn

2 4 6 GAP LENGTH 1 in


213

8

I I GAP LENGTH , mm

I 0.0056° 50 100 150 200

I I

0.0048

I -VI c: 0

I -o 0 ... -)(

I Q:) Case No. 18 B 36 ~

I z 0 1-<{

I 1-0 a:: w

I > 1-<{

...J

I w a::

Case No. 6 B 24

I I Web Thickness = 9.53 mm

·I 3/a ..

I 0 ·2 4 6 8

GAP LENGTH , in

I Fig. 77 Relative Rotation versus Gap Length

I 214

I


I I GAP LENGTH , mm

I 0 50 100 150

Web Thickness = 7.94mm

I 5/1611

I 60

-I c - a..

(/) :E ..:.:: -50

I b b

en en en

I en w w 0:: 0:: 40 1-1- en en

I a.. a.. <l: <l:

(!) (!)

I lD CD 30 w w

3: 3:

I ~ \ Case No.3 8 21 ~ :::> :::> \ :E ~

'\--Case No. 33 (eq. 7) 150 X

I X

20 ,, <l: <l: :E. :E \1 ·')(Case No. 15 (eq. 7) \ /" '•

I ~ / ' 100 / / ..... ' •'' .

\ / ..... ' ..... ..... '""' ..........

I 10 \ ........................... / /

\.Case No. 15 S 33 ' .... :::IOC'..._--' 50 ' _,..,.. ' .....__- -_.,..:::_ ' ---- ..... ......._.---I ' ...... --'-.case No.3 S 21 (eq.7)

I 0 2 4 6 8 GAP LENGTH , in

I Fig. 79 Maximum Web Gap Stress versus Gap Length

I 216

I


0

60

-If)

.:::. 50

b

(/) (/)

LLJ a:: 40 1-C/)

a.: <( (!)

£Il 30 LLJ 3:

:2 ::> :2 X 20 <(

:2

0


Web Thickness = 9.53mm 3/s ..

Case No. 18 a 36

J.. Case No. 18 S. 36 (eq. 7)

/ .............. / ........

' ' .................. , ............... ,

200

(/) (/)

LLJ

250' ~ (/)

a.. <(

200 (!) £Il LLJ 3:

150 :2 ::> :2 X <(

100 :2

Case No.6 a 24 ,.,/~ 50 /

\--Case No.6 a 24 (eq. 7) .-./ /

\ -- --......... ...... ----------- ---.,.,.,. 2 4 6 8

GAP LENGTH , in


217


(/)

.X:

b

C/) C/)

0

60

50

w 40 a::

'1-C/)

0


Stress For These Gap Lengths Was Greater Than 413.7 MPa (60 ksi)

-,-'r---Case No. 15 & 33(eq.4)

\ \ \ \ \ \ 'ACose NO. 3 8 21 (eq.4)

\ \ \ \ \. \ ' \ ' \ X

' ' ' ' ' ', '-... . ......... ......... ...... ......

.............. ........

Cas~ No. 15 8 33

X

X

Web Thickness = 7. 94 mm 5/ts"

2 4 GAP LENGTH , in

......._ __ ---........

6


218

8

0 a.. ~

b

C/) CJ)

w 250 ~

CJ)

a.. <t

200 (!)

CD w ~

150 ~ :E X <t :E

50


GAP LENGTH , mm 0 50 100 150 200

60

~ 50

b

0

Stress For These Gap Lengths Was Greater Than 413.7 MPa (60 ksi)

-,-~Case No.l8 a 36 (eq. 4}

\ \

\--~Case No. 6 8 24 (eq. 4)

'\ \ \ \ \ \ \ \

\ \ \ ' \

X

' ' ' ' ' ', ' ' ' ' ...... Case No. 18 a 36 '-.... ' .......... ...... ......

............

Case No.6 a 24

X

Web Thickness = 9.53 mm 3/a ..

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

2 4 6 GAP LENGTH , in

Fig. 82 Ma~imum Web Gap Stress versus Gap Length

219

8

400

350 -c a.. :.-2 -

300 b

Cf) Cf)

I.JJ

250 ~ Cf)

a.. <:t:

200 (!)

co I.JJ 3:

150 :.-2 ::::> :.-2 X <:t:

100 :.-2

50

-------------------

N N 0

0.1 ....... ............ ....... ...... ·-·· .............. • """'

0 fT1

..._ """' ....... ••

II r fT1 ()

~ 0.01 0 z

:J

0.001

. Gop Length g Experimental 18

• g = 20tw 0

• g = 10 tw 0

• g = 5 tw fl.

• g = 2.5 tw 0

• g = 1.25 tw 0 0

"""' """'-.

Case No . 15

0

(!]

£

~

0

0

0

3 ()

ll

A

~ f)

6 ®

181

£

~

@

• ...... ..._ ll 0 ..............

£ --~ ----~1:.--ll.

105 106

CYCLES TO FAILURE

Fig. 83 Experimental and Theoretical Results of Out-of-Plane Displacement

1.0

0 fT1 II r rn ()

~

0 z


8. REFERENCES

1. Pippard, A. J. S. and de Waele, J. P. A., "The Loading of Interconnected Bridge Girders," Journal of the Institution of Civil Engineers, London, England, Paper No. 5176, 1938, pp. 97-114.

2. 1977 Standard Specifications for Highway Bridges, American Association of State Highway and Transportation Officials, Washington, D. C.

3. Fisher, J. W., "Bridge Fatigue Guide- Design and Details," American Institute of Steel Construction, New York, N. Y., 1977.

4. Fisher, J. W., Hausammann, H. and Pense, A. W., "Retrofitting Procedures for Fatigue Damaged Full Scale Welded Bridge Beams," Fritz Engineering Laboratory Report No. 417-3, Lehigh University, Bethlehem, Pa., 1978.

5. Newmark, N. M., "Design of I-Beam Bridges," ASCE Proceedings, Structural Division, March 1948, pp. 305-331.

6. Lount, A. M. , "Distribution of Loads on Bridge Decks," Journa 1 of Structural Division, Proceedings of ASCE, Paper No. 1303, July 1957.

7. White, A. and Purnell, W. B., "Lateral Distribution on I-Beam Bridge," Journal of Structural Division, Proceedings of ASCE, Vol. 83, Paper No. 1255, May 1957,

8. Zellin, M. A., Kostem, C. N. and VanHorn, D. A., "Structural Behavior of Beam-Slab Highway Bridges - A Summary of Completed Research and Bibliography," Fritz Engineering Laboratory Report No. 387.1, Lehigh University, Bethlehem, Pa., May 1973.

9. Mertz, D. and Rimbos, P., "Effects of X-Bracing on the Behavior of Plate ·Girder Bridges," Unpublished C.E. 409 Report, Civil Engineering Dept., Lehigh University, Bethlehem, Pa., May 1976.

10. Fisher, J. W., "Fatigue Cracking in Bridges from Out-of-Plane Displacements," Canadian Structural Engineering Conference, 1978.

11. Fisher, J. W., "Guide to 1974 AASHTO Fatigue Specifications," American Institute of Steel Construction, 1974.

12. Rolfe, S. T. and Barsom, J. M., "Fracture and Fatigue Control in Structures - Applications of Fracture Mechanics," PrenticeHall, Inc., Englewood Cliffs, N. J., 1977.

221


13. Hertzberg, R. W., "Deformation and Fracture Mechanics of Engineering Materials," John Wiley and Sons, New York, N.Y., 1976.

14. United States Steel Corporation, "Composite: Welded Plate Girder," Highway Structures Design Handbook, Vol. 2, Sec. 4, 1969.

15. Bathe, K. J., Wilson, E. L. and Peterson, F. E., "SAP IV- A Structural Analysis Program for Static and Dynamic Response of Linear Systems," Earthquake Engineering Research Center Report No. EERC 73-11, University of California, Berkeley, Ca., June 1973 (revised April 1974).

16. Manual of Steel Construction, American Institute of Steel Construction, 7th Ed., 1970.

17. Fisher, J. W., Fisher, T. A. and Kostem, C. N., "Displacement Induced Fatigue Cracks," presented at the symposium honoring Professor Theodore V. Galambos on the occasion of his 50th Birthday, April 17, 1979 (to be published in "Engineering

. Structures", IPC Science and Technology Press Limited, England).

222


APPENDIX A

DESCRIPTION OF THE SIMULATED PIN CONNECTION

Several methods of representing the pin connection in the sub

structure model were available. This appendix has been included to

describe the method used and the reasons it was selected.

The cross framing members in the substructure model were sim

ulated by plane stress elements. During the early stages of model

development, it was anticipated that a pin connection at the end of

these cross framing members could be modeled by reducing the elastic

properties in several specific plane stress elements. Other methods

of simulating this connection, possibly more exact, were considered.

These methods, however, entailed rediscretization of the basic sub

structure model which had a moment connection at the end of these

members. Since this would have required almost double the effort, it

was decided that accuracy of results would be slightly sacrificed for

convenience.

The pin connection was simulated, therefore, by reducing the

modulus of elasticity of certain plane stress elements. These plane

stress elements are shown shaded in Fig. A-1. It was expected that

reduction of this property would cause redistribution of the stress

pattern through the depth of the equivalent bracing member. This re

distribution would place most of the force in the stiffer plane stress

element. By accomplishing this, a majority of the member force would

223


have been moved closer to the neutral axis (designated aR, bR, a1

,

b1

in Fig. A-1) of the equivalent member. This would reduce the moment

transferred at this location, and thus approach the pin condition

which produces a resultant moment of zero about the pin. By selectively

reducing the modulus of elasticity, the best "pin" connection could be

determined.

Figure A-1 shows the cross section of the substructure model where

the equivalent bracing members framed into the girder web. Also shown

are the boundary loads that were applied to the free ends of the bracing

members for Cases 15a through 15f. These loads were computed by the

procedure described in Section 3.1. The resultant moments of these

forces for each separate bracing member about points aR, bR, a1

, and

b1 are shown in Fig. A-1 as ~MaR, ~MbR, ~Ma1 , and ~Mb1 , respectively.

Figures A-2 through A-5 show the changes in these resultant moments

when the modulus of elasticity was reduced by different amounts.

The reduced modulus of elasticity values of 2000 MPa (290.0 ksi)

and 20 MPa (2.9 ksi) were examined. Case study No. 15d was the specific

substructure model used with these reduced values to validate the "pin"

connection. Figures A-2 and A-3 show the forces in the plane stress

elements at sections R-R and L-L from Fig. A-1, respectively. The

reduced modulus of elasticity for this condition was 2000 MPa

(290.0 ksi). The neutral axes of the diagonal members at sections

R-R and L-L are located at points aR and a1 , respectively. The cor

responding resultant moments are shown as ~MaR and r.Ma1

. Similarly;

the resultant moments about the neutral axes of the horizontal members

224


are shown as ~MbR and ~MbL. A comparison of these values with the cor

responding values in Fig. A-1 shows a large reduction in all the result

ant moments. A further reduction was attempted by reducing the modulus

of elasticity to 20 MPa (2.9 ksi), as previously stated. Figures A-4

and A-5 show the forces in the plane stress elements at sections R-R

and L-L, respectively, for this condition. Comparison of the resultant

moments in these figures with the resultant moments from Figs. A-2 and

A-3 reveals changes in the resultant moment. It was evident, there

fore, that reduction of the elastic properties had an effect on the

distribution of stress in the equivalent bracing members. Further

examination of different elastic properties between the two values

already presented was not conducted. Since both reduced modulus of

elasticity values produced resultant moments close to zero, it was

decided that a value of 20 MPa (2.9 ksi) would be used. All case

studies listed in Table 8 use this reduced modulus of elasticity for

the shaded plane stress elements shown in Figs. A-1 and 62. It must

be emphasized that a true pin connection was not simulated. The modeled

connection was actually "closer" to a shear connection.

225


~ • 2477.0 Nm (21.92 in-kips~

MbL = .139 Nm (.0012 in-kips))

Ll = 1. 698 kN (. 382 kips)

L2 = 2.233 kN (.502 kips)

L3 = 1.072 kN (.241 kips)

L4 = .538 kN (.121 kips)

L5 = .810 kN (.182 kips)

L6 = 1. 624 kN (. 365 kips)

L7 = 1. 624 kN (.365 kips)

L8 = 3.945 kN (.887 kips)

L9 = 3.131 kN (. 704 kips)

B

D

Ma = 1342.0 Nm (11.88 in-kips~ . R 1)

MbR = .185 Nm (.0016 in-kips))

7tR5R:!7 R9

2540 mm (100 in)

Rl = • 921 kN (.207 kips)

R2 = 1.210 kN ( .272 kips)

R3 = .583 kN ( .131 kips)

R4 = .289 kN (. 065 kips)

R5 = .125 kN (.028 kips)

R6 = .249 kN (.056 kips)

R7 = .249 kN (.056 kips)

R8 = .609 kN (.137 kips)

R9 = .485 kN (.109 kips)

Fig. A-1 Cross Section of Substructure Model where Bracing Members Frame into Girder Web

226

I I I I I I

·I I I I I I I I I I I I I

Fig. A-2

GIRDER WEB

38.1 mm (1. 5 in)

._ 480 kN (.108 kips)

a~= 67.90 Nm (.601 in-kips)

._ 1.756 kN (.170 kips)

-----1 ... ~ 1. 828 kN (. 411 kips)

---11 ... ~ 1.165 kN (. 262 kips)

LMbR = 242.12 Nm (2.143 in-kips)

.. 5.186 kN (1.166 kips)

Free Body of Right Bracing Members (Section R-R) for E1 = E0/100 = 2000 MPa (290.0 ksi)

. 227

I I I I I I

·I I I I I I I I I I I I I

GIRDER WEB

2.820 kN ( .634 kips) .... ,._ ___ _

1. 904 kN (. 428 kips) .... .,..._ __

2.384 kN (.536 kips)

1.566 kN (.352 kips)

~MbL = 31.41 Nm (.278 in-kips)

.676 kN (.152 kips) ------t-

6.507 kN (1.463 kips) ---.. •~

38.1 mm (1. 5 in)

Fig. A-3 Free Body of Left Bracing Members (Section L-L) for E1 = E0/100 = 2000 MPa (290.0 ksi)

228

I I I I I I I

I I I I I I I I I

GIRDER WEB

38.1 mm (1. 5 in)

~ 2.962 kN (.666 kips)

yr~ = 41.46 Nm (.367 in-kips)

~ .116 kN (. 026 kips)

.... ~--- .027 kN (.006 kips)

---• ... ~ .031 kN (.007 kips)

---• ... ~ .022 kN ( .005 kips)

L: MbR = 40. 45 nM (. 358 in-kips)

.... 1.779 kN (.400 kips)

Fig. A-4 Free Body of Right Bracing Members (Section R-R) for E1 = E

0/10000 = 20 MPa (2.9 ksi)

229

I I I I I I ·I I I I I I I I I I I I I

GIRDER WEB

5.498 kN (1.236 kips) .... ,..._, __

= 79.99 nM (.708 in-ki~ . -.--~MaL~

.013 kN (.003 kips) ~

.027 kN ( .006 kips) ..... .._ __

.040 kN (.009 kips) ...

.027 kN (.006 kips)

• 009 kN (. 002 kips)

L:MbL = 235.00 Nm (2.080 in-kips1;Mb L

11.06 kN (2.487 kips) ._

38.1 nnn (1. 5 in)

Fig. A-5 Free Body of Left Bracing Members (Section L-L) for E1 = E0 /10000 = 20 MPa (2.9 ksi)

. 230


10. ACKNOWLEDGMENTS

The authors extend their appreciation to Dr. John W. Fisher

for his technical contributions in the conduct of the research, Dr.

Stephen C. Tumminelli, Messrs. John A. Grant and Hans Hausammann for

their suggestions, and Ms. Rebecca A. Villari for her editorial

assistance in the preparation of the report. Thanks are also due to Ms.

Shirley Matlock and Mrs. Dorothy Fielding for the fast and accurate

typing of the report.

The assistance of the staff of the Lehigh University Computing

Center deserves special recognition. The reported research required

extensive computer resources; without the Center staff's understanding

and cooperation, the successful completion of the research would not have

been possible.

231

Documents

c;, f --of tdigital.lib.lehigh.edu/fritz/pdf/432_5.pdf · 2012-08-01 · 1 t --- c;, t f ~ of \ DR. CELAL N. KOSTEM Fritz Engineering Lab., 13 Lehigh University Bethlehem, PA 18015