Shear Lag of Steel Box Girders with Longitudinal...

SHEAR LAG OF STEEL BOX GIRDERS WITH

LONGITUDINAL STIFFENERS

BY

TANPITCHA PUMPHAKA

A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE

REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE

(ENGINEERING AND TECHNOLOGY)

SIRINDHORN INTERNATIONAL INSTITUTE OF TECHNOLOGY

THAMMASAT UNIVERSITY

ACADEMIC YEAR 2018

Ref. code: 25615922040208FTI

ii

Abstract

SHEAR LAG OF STEEL BOX GIRDERS WITH LONGITUDINAL STIFFENERS

by

TANPITCHA PUMPHAKA

Bachelor of Engineering (Civil Engineering), Sirindhorn International Institute of

Technology, Thammasat University, 2015

Master of Science (Engineering and Technology), Sirindhorn International Institute of

Technology, Thammasat University, 2018

This research studies about the shear lag effect and focuses on stress

concentration for steel box girders with longitudinal stiffeners. The research used the

finite element method for studying the shear lag effect and it is known that the finite

element mesh must be constructed with carefulness to evaluate stress concentration.

The study investigates the stress concentration in a flange due to the shear lag in a

simply supported box girders with longitudinal stiffeners by the three-dimensional

finite element method using shell elements under two loading conditions of

concentrated load at the mid span and uniformly distributed load along the beam length.

Definitely, parametric study with respect to the geometry of a box girders with

longitudinal stiffeners is carried out. The dependency of finite element mesh on the

shear lag is carefully treated. It is also reported that the stress distributions in the flange

are different from those of the elementary theory. Based on the results, empirical

formulas are proposed to compute stress concentration factors due to the shear lag

effect. The derived formulas could be used to improve the current design specification

of steel box girders with longitudinal stiffeners.

Keywords: Box girders, Longitudinal stiffeners, Shear lag, Stress concentration factor

Ref. code: 25615922040208FTI

iii

Acknowledgements

I would like to express my upmost appreciation and sincere gratitude to

Assoc. Prof. Dr. Taweep Chaisomphob, my research supervisor, for his guidance, and

valuable suggestions for my research study.

My sincere gratitude is also extended to Assoc. Prof. Dr. Winyu

Rattanapitikon, my chairperson of the examination committee and Col. Asst. Prof.

Dr. Nuthaporn Nuttayasakul, my examination committee; for their knowledge,

encouragements, and their advices which help my research from different perspectives.

I would like to thanks Sirindhorn International Institute of Technology,

SIIT for providing an opportunity to continue study in Master degree here. I also like

to express my gratitude to SIIT for providing me with financial support for both

academic and research.

Finally, my special thanks and gratitude are extended to my family

members for their supports and encouragements in moving forward and completing the

thesis.

Ref. code: 25615922040208FTI

iv

Table of Contents

Chapter Title Page

Signature Page i

Abstract ii

Acknowledgements iii

Table of Contents iv

List of Tables vi

List of Figures vii

1 Introduction 1

1.1 Statement of Problems 1

1.2 Objectives and Scopes 3

2 Literature Review 4

2.1 Shear Lag Problem 4

2.2 Effective Width Concept 4

2.3 Stress Concentration due to Shear Lag in Simply Supported 5

Box Girder

2.4 Finite Element Approach 7

3 Methodology 9

3.1 Introduction 9

3.2 Parametric Study 9

3.3 Finite Element Model 14

3.3.1 Boundary Conditions 15

3.4 Stress Concentration Factor 18

4 Results and Discussion 20

4.1 Comparison between the Results of Simply Supported Box Girder 20

and Fixed Ended Box Girder

Ref. code: 25615922040208FTI

v

4.2 Normal Stress Distribution in Upper Flange 22

4.3 Effect of Loading 25

4.4 Effect of Geometric Properties 27

4.4.1 Half Flange Width / Height of Web Ratio (B/H) 27

4.4.2 Height of Web / Span Length Ratio (H/L) 32

4.4.3 Thickness of Flange / Web Ratio (tf/tw) 37

4.4.4 Cross Sectional Area of Stiffeners / Area of Flange Ratio (As/Af) 45

4.5 Overall Effects of Stress Concentration Factor 57

4.6 Empirical Formulas 57

4.6.1 Proposed Formulas 57

4.6.2 Accuracy of the Proposed Formulas 59

5 Conclusion 62

References 63

Ref. code: 25615922040208FTI

vi

List of Tables

Tables Page

2.1 Kc evaluated from effective width approach and finite element analysis 6

4.2 The difference of some Kc value between simply supported box girder 20

and fixed ended box girder

4.2 Values of coefficients in equations 58

Ref. code: 25615922040208FTI

vii

List of Figures

Figures Page

1.1 Stress distribution due to shear lag 1

1.2 Structural geometry of box girder 2

2.1 Normal stress distribution in upper flange of box girder under Load C-2 5

3.1 Cross section and geometric properties 9

3.2 Spacing of stiffeners 11

3.3 Truck load 11

3.4 The maximum moment calculation 12

3.5 The deflection of box girder 13

3.6 Concentrated load (a) Load C-1, (b) Load C-2; Distributed load 14

(c) Load D-1, (d) Load D-2

3.7 A quarter of box girders 14

3.8 Finite element meshes 15

3.9 Boundary conditions at support 16

3.10 Symmetry boundary conditions at mid span 17

3.11 Symmetry boundary conditions along the length 17

3.12 Normal stress distributions in the upper flange 18

3.13 Variation of normal stress with respect to representative element size 19

4.1 Variation of Kc with respect to H/L: (a) Simply supported box girder, 21

(b) Fixed ended box girder

4.2 Geometric Properties of box girder with longitudinal stiffeners 22

4.3 Normal stress distribution in the upper flange (B/H = 1.0, H/L = 0.2, 23

tf/tw = 2.5)

4.4 Variation of Kc with respect to H/L (tf/tw = 1.3, 1.9, 2.5 and As/Af = 0.9) 26

4.5 Variation of Kc with respect to B/H (tf/tw = 1.3) 28



4.8 Variation of Kc with respect to H/L (tf/tw = 1.3) 33



Ref. code: 25615922040208FTI

viii

4.11 Variation of Kc with respect to tf/tw (B/H= 0.4 and As/Af = 0) 37




4.15 Variation of Kc with respect to tf/tw (B/H= 0.4 and As/Af = 0.3) 39












4.27 Variation of Kc with respect to As/Af (B/H = 0.4, tf/tw = 1.3) 45












4.39 Kc due to proposed formulas and finite element analysis 61

Ref. code: 25615922040208FTI

1

Chapter 1

Introduction

1.1 Statement of Problems

In the elementary beam theory, the normal stress in the longitudinal direction

produced by bending deformation is assumed to be proportional to the distance from

neutral axis and uniform across the flange width. If the flange gets wider, this

assumption becomes invalid and a phenomenon called shear lag will happen. The effect

of shear lag causes the longitudinal stress at flange/web connection to be higher than

the mean stress across the flange as shown in Figure 1.1.

Figure 1.1. Stress distribution due to shear lag

Shear lag effects are usually very large, especially near points of high

concentrated load or at reaction points in short span beams with thin-wide flanges. In

particular, shear lag effects may be significant in light-gauge, cold-formed sections and

in stiffened box girders. Shear lag has no serious consequences in a ductile structure, in

which any premature local yielding leads to a favorable redistribution of stress.

However, the increased stress due to shear lag may induce in a tension flange, which is

liable to brittle fracture or fatigue damage, or in a compression flange whose strength

is controlled by its resistance to local buckling (Trahair and Bradford, 1988). So the

formulas for solving the shear lag effect are important for structural design.

This study aims to investigate the shear lag effect on simply supported steel box

girders with longitudinal stiffener under two loading conditions. The simply supported

box girder is shown in Figure 1.2.

σmax

σmax

σ

Stress distribution due to elementary beam theory

Stress distribution due to

elementary shear lag

Web

Flange

2B

Ref. code: 25615922040208FTI

2

L/2

F

L/2

Figure 1.2. Structural geometry of box girders

Empirical formulas for evaluation of shear lag effect are given in term of the

effective width but this approach cannot give a precise value of stress concentration in

general. Moffatt and Dowling (1975) gave the definition of the effective width as

follows:

Be=1

2σmax∫ σydx

2B

0 (1.1)

where Be stands for the half effective width, and the numerator is the integration of the

normal stress in the flange, σy, while the denominator is the actual maximum normal

stress in the flange due to shear lag, σmax. Yamaguchi et al. (2008) are realized that the

evaluation of the maximum stress by the effective width approach invites error by itself.

So, in the present study, the empirical formulas are proposed for evaluating stress

concentration factors obtained by the present finite element analysis instead of the

effective width.

(a) Side view of concentrated

load at the mid span

(c) Cross section and

geometric properties

L/2 L/2

w

(b) Side view of distributed

load along the beam length

H

2B

tw

tf

ts ds

Ref. code: 25615922040208FTI

3

1.2 Objectives and Scopes

The main study of this research is to investigate the shear lag effect at the mid-

span of steel box girders with longitudinal stiffeners by using three-dimensional 4-node

shell finite element analysis. All the elements in each mesh are square. The element

meshes are used to study about the influence of finite element mesh on stress

concentration and used to eliminate discretization error by multimesh extrapolation

method for every girders under a specific loading condition. MARC (2016) program is

used in this finite element analysis. The main objectives is that:

To apply the finite element analysis to study the stress concentration due to

shear lag in steel box girders with longitudinal stiffeners under 4 ways of

loading. And sensitivity analysis of finite element mesh will be considered

To study the effects of the geometric properties of box girders. The following

geometric properties are half flange width (B), span length (L), height of web

(H), thickness of flange (tf), thickness of web (tw), cross sectional area of the

stiffeners (As) and cross sectional area of the flange (Af).

To propose the empirical formula to compute stress concentration due to shear

lag effect in steel box girder with longitudinal stiffeners

The AASHTO LRFD Bridge Design Specifications (2010) is used for the design

steel beams. By varying the proportions of geometric properties of the box girders, the

linear finite element analysis is performed.

Two loading conditions of concentrated load at the mid span and uniformly

distributed load along the girder length are used. Various ways to apply those loads are

considered. The shear lag effect on stress in simply supported box girder with

longitudinal stiffeners depends on the geometric properties of box girders in term of

H/L, B/H, tf/tw, and As/Af.

Ref. code: 25615922040208FTI

4

Chapter 2

Literature Review

2.1 Shear Lag Problem

This chapter describe about shear lag effect. The elementary beam theory, the

normal stress in the longitudinal direction produced by bending deformation is assumed

to be proportional to the distance from neutral axis and uniform across the flange width.

If the flange gets wider, this assumption becomes invalid and a phenomenon called

shear lag will happen. The effect of shear lag causes the longitudinal stress at

flange/web connection to be higher than the mean stress across the flange.

The three-dimensional finite element analysis of a box girder by shell element

is carried out to study the shear lag effect in the present study. Two loading conditions

of concentrated load at die mid-span and uniformly distributed load along the beam

length are employed. Multiple ways to apply those loads are considered. Much attention

is paid to finite element mesh as well, so as to minimize discretization error. Note that

not many researchers have explicitly addressed how the discretization error is

controlled in their shear lag study by the finite element method. To the best of the

authors’ knowledge, the work of Lee and Wu (2000) is one of the very few numerical

studies.

The normal stress in the longitudinal direction in the flange is of interest for

investigating the shear lag effect on stress. The stress in the mid-span cross section is

focused on in particular, since the largest stress is expected. The vertical displacement

at the mid-span is also computed to see the shear lag effect on deflection. An extensive

parametric study is conducted, based on empirical formulas are proposed. In all the

analyses, a finite element program, MARC (2016), is used.

2.2 Effective Width Concept

The concept of effective width was first proposed by Von Karman (1924) so as

to take care of the effect of shear lag in thin-walled structures. Because of its simplicity,

the effective width approach has been widely adopted for the evaluation of stress

concentration due to shear lag. To provide a simple approach for stress evaluations due

to shear lag, the effective width ratio has been provided in some current design codes.

The definition of the effective width is given as follows Moffatt and Dowling (1975):

Be=1

2σmax∫ σydx

2B

0 (1.1)

where Be stands for the half effective width, and the numerator is the integration of the

normal stress in the flange, σy, while the denominator is the actual maximum normal

stress in the flange due to shear lag, σmax. Eq. (1.1) simply implies that the analysis of

Ref. code: 25615922040208FTI

5

shear lag is associated with the analysis of σy. In general, the effective width ratio, λ

which is defined as the ratio of Be to the half actual width, B from Figure 1.1 is widely

used since it can roughly notify how much the influence of the shear lag is Eq. (2.1).

λ=Be

B (2.1)

If the distribution of σy over the flange is approximately uniform, λ will be close

to unity, which indicates that the degree of shear lag is very small. When the shear lag

effect is severe, the distribution of σy across the flange is completely non-uniform with

a very sharp gradient in the vicinity of web-flange intersections. In this way, σmax arises

considerably and results in significant reduction of Be. As a result, λ will be much

decreased and approaches zero.

2.3 Stress Concentration due to Shear Lag in Simply Supported Box Girder

Lertsima et al. (2004) and Yamaguchi et al. (2008) have recently studied about

the shear lag effect in simply supported box girder by using the three-dimensional finite

element analysis. The models were shell elements with two loading conditions of

concentrated load at the mid-span and uniformly distributed load along the beam length.

For the evaluation of the shear lag effect, empirical formulas are often given in

terms of the effective width. This is because only a simple modification of the width is

then required for the inclusion of the shear lag effect. However, this approach cannot

give a rigorous value of stress concentration in general. As an example problem,

consider the normal stress distribution in the flange at the mid-span of a simply

supported beam under Load C-2, which is shown in Figure 2.1. The definition of the

effective width b is given as Eq. (1.1).

Figure 2.1 Normal stress distribution in upper flange of box girder under Load C-2

For the normal stress distribution in Figure 2.1 where the maximum normal

stress σ/σbeam is 4.14, Be is computed as Be/B = 0.165. With this effective width, they

are supposed to be able to evaluate the maximum stress. However, the effective width

Ref. code: 25615922040208FTI

6

approach represented by Eq. (1.1) leads to the maximum stress of σ/σbeam = 3.52, which

includes a significant error of 15.0%.

Table 2.1 shows some more numerical results of stress concentration factor, Kc

evaluated from the effective width approach and the finite element analysis. Stress

concentration factor, Kc stands for the ratio of the maximum normal stress, σmax, which

is calculated by finite element analysis to the elementary beam theory stress, σbeam.

These are the results obtained from the box girder under Load C-2 shown in Figure 2.1

with different girder proportions, i.e. B/L = 0.1, 0.2, 0.3, 0.4 and 0.5. It is found that the

effective width approach gives less value of Kc than that of the finite element analysis.

In particular, the percentage discrepancy of Kc obtained by effective width approach

and by the finite element analysis increases when B/L becomes larger.

Table 2.1. Kc evaluated from effective width approach and finite element analysis

B/L

Kc

Discrepancy from FEA (%) Effective Width

Approach

Finite Element

Analysis

0.1 1.452 1.493 -2.72

0.2 1.963 2.107 -6.82

0.3 2.487 2.755 -9.73

0.4 3.038 3.469 -12.45

0.5 3.519 4.141 -15.03

Following the discussion in this section, it is realized that the evaluation of the

maximum stress from the effective width approach invites error by itself. Therefore, in

the present study, empirical formulas are proposed for directly evaluating stress

concentration factors obtained by the present finite element analysis instead of the

effective width.

Yamaguchi et al. (2008) proposed the empirical formulas for calculation the Kc

value in simply supported box girder in Eq. (2.2) to Eq. (2.4).

Concentrated load (Load C-1):

Kc= a1× (H

L) +1 (2.2)

where

a1=b1× (B

H)

c1 (2.2a)

b1=0.832×ln (tf

tw) +2.77 (2.2b)

Ref. code: 25615922040208FTI

7

c1=-0.034×ln (tf

tw) +1.744 (2.2c)


Kc= a2× (H

L) +1 (2.3)

where

a2=b2× (B

H)

c2 (2.3a)

b2=1.756×ln (tf

tw) +6.101 (2.3b)

c2=0.053×ln (tf

tw) +1.202 (2.3c)

Distributed load (Load D-1):

Kc= a3× (H

L)

2

+1 (2.4)

where

a3=b3× (B

H)

c3 (2.4a)

b3=1.225×ln (tf

tw) -0.494× (

tf

tw) +6.001 (2.4b)

c3=-0.041×ln (tf

tw) -0.006× (

tf

tw) +2.371 (2.4c)

Lertsima et al. (2004) and Yamaguchi et al. (2008) concluded that the real stress

can be much larger than those due to the beam theory. The proposed formulas would

be of some help to improve the current situation.

2.4 Finite Element Approach

Web behaved in accordance with the elementary theory had infinite in-plane

rigidity but no out-of-plane rigidity are assumed by Moffatt and Dowling (1975). They

employed the finite element method with rectangular third-order extensional-flexural

element. Tenchev (1996) analyzed the shear lag in orthotropic beam flanges and plates

with stiffeners by using two-dimensional plane stress finite element model. The

empirical formula of shear lag was obtained in term of ratios of half flange width to

half-length of beam, Young's modulus to shear modulus of flange, and thickness of

flange to thickness of web. Longitudinal stiffener has been accounted for by modifying

the ratio of Young's modulus to shear modulus. Lee and Wu (2000) reduced a shear lag

problem to a two-dimensional plane stress problem. The adaptive technique was used

Ref. code: 25615922040208FTI

8

to reduce discretization error. Yamaguchi et al. (2008) study about the shear lag effect

in simply supported box girder by using the three-dimensional finite element analysis

and propose the empirical formulas to compute the stress concentration factor to

account the shear lag effect.

Ref. code: 25615922040208FTI

9

Chapter 3

Methodology

3.1 Introduction

This Chapter aims to investigate the shear lag effect on stress concentration at

the mid-span of simply supported box girder with longitudinal stiffener. In the finite

element analysis, three-dimensional shell element is used. Various values of geometric

properties of a stiffened box girder are considered. MARC (2016) program is used in

this finite element analysis. Based on the numerical results, empirical formulas are

proposed to simplify the shear lag effect and to make the benefit for beam design.

3.2 Parametric Study

This study uses initial dimension of steel box girder from Steel Bridge Design

Handbook (Chavel and Carnahan, 2012) and checking with AASHTO LRFD Bridge

Design Specifications (2010) as shown in Figure 3.1 and following list of calculation

below.

Figure 3.1. Cross section and geometric properties

1. Dimensions of box girder

1.1 The cross-section proportion limits for flange width, 2B of box girder are specified

as:

2B ≥ H

6 (3.1)

1,800 ≥ 2,027

6

H = 2,027mm

2B = 1,800mm

tw = 9mm

tf = 12mm

ts = 12mm

ds = 120mm

Span length, L = 40,000 mm

Young’s modulus, E = 20,6000 MPa

Poisson’s ratio = 0.3

Ref. code: 25615922040208FTI

10

1,800 ≥ 337.83 O.K.

and

2B ≥ L

85 (3.2)

1,800 ≥ 40,000

85

1,800 ≥ 470.588 O.K.

1.2 The thickness of flange is specified as;

tf ≥ 1.1tw (3.3)

12 ≥ 1.1(9)

12 ≥ 9.9 O.K.

2. Stiffeners dimension

2.1 This box girder uses 5 stiffeners. The height, ds and the thickness, ts of one stiffener

shall satisfy:

ds

ts ≤ 0.48√

E

Fy (3.4)

120

12 ≤ 0.48√

20,6000

365

10 ≤ 11.403 O.K.

2.2 Checking the number of longitudinal stiffeners.

k = 5.34+2.84(

Is

(B)tf3)

13

(n+1)2 ≤ 5.34 (3.5)

Where k is plate buckling coefficient

Is is moment of inertia of a single longitudinal stiffeners (mm4)

n is number of longitudinal stiffeners

k = 5.34+2.84(

112

(12)(120)3

(900)(12)3

)

13

(5+1)2 = 0.23 ≤ 5.34 O.K.

Therefore, this box girder uses 6 stiffeners.

Ref. code: 25615922040208FTI

11

2.3 Compute spacing of each stiffener

Spacing = 2B / (no. of stiffeners+1) = 1,800 / 6 = 300 mm

Figure 3.2. Spacing of stiffeners

3. Design for truck load

Figure 3.3. Truck load

For this design, apply 1 HL-93 AASHTO truck or moving load to the box girder

as shown in Figure 3.3. The truck consists of three axles, front and two rear axles with

front axle weighing 35 kN and two rear axles weighing 145 kN. The distance between

front and rear axle is 4.27 m and that of two rear axles can be 9.14 m. The value of

maximum moment is calculated by SAP2000 program (2016).

L =40 m

35 kN 145 kN 145 kN

4.27 m 9.14 m

HL-93 AASHTO truck (moving load)

Ref. code: 25615922040208FTI

12

Figure 3.4. The maximum moment calculation

3.1 Moment design

Mu≤∅fMn (3.6)

Where ∅f is resistance factor for flexure = 1.0

Mn is nominal flexural resistance of the section

Mu is bending moment

Mu = 2.53 x 109 N-mm

If Dp ≤ Dt, then Mn = Mp (3.7)

Otherwise Mn = Mp (1.07 – 0.7(Dp / Dt)) (3.8)

Where Dp is the distance of the top of the composite section (this is steel box girder and

has no concrete deck on the top of beam. So that Dp is equal to 0)

Dt is the total depth of the section

0.1Dt = 0.1(2027) = 202.7mm > Dp = 0

So, Mn = Mp = ZxFy (3.9) Where Zx is plastic section modulus

Mn= {2{2[(9)(1007.5)(503.75)] + [(1809)(12)(1013.5)] + 5[(12)(120)(947.5)]}}(365)

Mn = 2.77 x 1010 N-mm

From Mu≤∅fMn (3.10)

So, 2.53 x 109 ≤ (1.0)(2.77 x 1010) O.K.

Mmax

= 2,530,260,355 N-mm

Ref. code: 25615922040208FTI

13

3.2 Deflection limitation

The deflection limit for bridges under vehicular load is L / 800. Beam deflection

from SAP2000 program (2016) is 28 mm:

Figure 3.5. The deflection of box girder

So, the maximum deflection is 28 mm < L/800 = 50 mm O.K.

The initial proportions are H/L = 0.05, B/H=0.4, tf/tw = 1.3, and As/Af = 0.3.

From the initial proportions can vary the proportion, and the following values are

considered: H/L=0.05, 0.1, 0.15, 0.2; B/H = 0.4, 0.6, 0.8, 1.0; tf/tw = 1.3, 1.9, 2.5. The

longitudinal stiffeners in a beam are considered: As/Af = 0, 0.3, 0.6, 0.9 in which As is

total area of stiffeners on each flange and Af is area of the flange. And the values of H,

tf, ts, and ds are fixed equal to 2027 mm, 12 mm, 12 mm, and 120 mm respectively. The

combination of all these values results in 192 models different from each other in

geometry.

Ref. code: 25615922040208FTI

14

3.3 Finite Element Model

Apply the load in various ways. The loading in the plane of the web are

considered. Herein the load applications that may cause local effects on the stress

distribution in the flange are avoided (Lertsima et al., 2004). To evaluate the local effect

on loading condition, two loading models are used for concentrated load: Load C-l is a

concentrated load at the middle of the web and Load C-2 is a uniformly distributed load

along the height of the web and two loading models are also used for uniformly

distributed load: Load D-l is a uniformly distributed load along the centerline of the

web and Load D-2 is a uniformly distributed load not only along the beam axis but also

along the web height of every cross section as shown in Figure 3.6.

Figure 3.6. Concentrated load (a) Load C-1, (b) Load C-2; Distributed load (c) Load

D-1, (d) Load D-2

For the FEM model, the structural is modeled by using three-dimensional 4-

node shell elements. It is noted that due to symmetry only a quarter of the box girder is

analyzed.

Figure 3.7. A quarter of box girders

Concentrated Load C-1 Concentrated Load C-2

Distributed Load D-1 Distributed Load D-2

Ref. code: 25615922040208FTI

15

All the elements in each mesh are square. Figure 3.8. Shows the number of

elements in the model are 4288, 17152, 68608, and 274432 for Mesh A to D,

respectively. The four meshes are used to study about the influence of finite element

mesh on stress concentration and used to eliminate discretization error by multimesh

extrapolation method for every girder under a specific loading condition.

Figure 3.8. Finite element meshes

3.3.1 Boundary Conditions

In finite element model of a quarter of the box girder, the general rule for a

symmetry displacement condition is that “the displacement vector component

perpendicular to the plane is zero and the rotational vector components parallel to the

plane are zero. For an anti-symmetry condition the reverse conditions apply

(displacements in the plane are zero; the rotation normal to the plane is zero)” (Tech

Tips, 2006).

Mesh A (4,288 elements) Mesh B (17,152 elements)

Mesh C (68,608 elements) Mesh D (27,4432 elements)

Ref. code: 25615922040208FTI

16

1. Boundary conditions at support (Ux = 0, Uy = 0, Uz = 0, θx = 0, θy = 0)

Figure 3.9. Boundary conditions at support

2. Symmetry boundary conditions at mid span (Ux = 0, θy = 0, θz = 0)

Ref. code: 25615922040208FTI

17

Figure 3.10. Symmetry boundary conditions at mid span

3. Symmetry boundary conditions along the length (Uz = 0, θx = 0, θy = 0)

Figure 3.11. Symmetry boundary conditions along the length

Ref. code: 25615922040208FTI

18

3.4 Stress Concentration Factor

The structural model is analyzed by finite element method, using shell elements.

Although the finite element method is very powerful, the results may depend on finite

element mesh employed in the analysis. The first study is the influence of finite element

mesh on the stress concentration. Figure 3.12 shows the normal stress distributions in

the upper flange at the mid-span. In this figure, σ is the normal stress obtained by the

present finite element analysis while σbeam is the normal stress due to the beam theory

and constant across the flange width. This is the result for a box girder (H/L = 0.05,

B/H = 0.4, Tf/Tw = 1.3, As/Af = 0.3) under Load C-2 by four finite element meshes,

Mesh A to D. All the elements in each mesh are square.

Figure 3.12. Normal stress distributions in the upper flange

Figure 3.13 shows the variation of the normal stress in the flange with respect

to a representative element size Δ. It is observed that the four lines in the graph become

almost straight for small Δ. The linear extrapolation shown by the dotted lines in the

graph can be used to estimate the converged stress. This extrapolation method is called

the multimesh extrapolation method by (Cook et al., 1989). Importantly, the four lines

in Figure 10 are almost straight, which is in accordance with the description of (Cook

et al., 1989). The arrow in the Figure 3.13 represents the stress concentration factor, Kc. Stress concentration factor, Kc stands for the stress concentration factor defined by

the ratio of the maximum normal stress in the flange to that of the elementary beam

theory, σFEM,max/σbeam.

Ref. code: 25615922040208FTI

19

Figure 3.13. Variation of normal stress with respect to representative element size

Ref. code: 25615922040208FTI

20

Chapter 4

Results and Discussion

4.1 Comparison between the Results of Simply Supported Box Girder and Fixed

Ended Box Girder

In real steel beams, how to know about the support of each beam is difficult.

This section will show the stress concentration factor, Kc in different support of the

beams to study the effect of Kc. value when changing beam support. Table 4.1 shows

the difference of some Kc value between simply supported box girder and fixed ended

box girder. The variation of Kc with respect to H/L for the cross section of As/Af = 0.9,

tf/tw = 2.5 under Load C-1, Load C-2, Load D-1 and Load D-2 in two types of boundary

condition is shown in Figure 4.1.

From the comparison, it is found that the Kc value of each model in simply

supported box girders close to that of fixed ended box girders. Therefore, this study

only uses simply supported box girders for stress concentration analysis.

Table 4.1. The difference of some Kc value between simply supported box girder and

fixed ended box girder

Case Proportion of Geometric Properties Load type

Kc, simply

supported

box girder

Kc, fixed

ended box

girder

1 H/L=0.2, B/H=0.4, As/Af =0.9, tf/tw =2.5 Load C-1 1.5608 1.5342

H/L=0.2, B/H=0.4, As/Af =0.9, tf/tw =2.5 Load C-2 2.2696 2.2397

H/L=0.2, B/H=0.4, As/Af =0.9, tf/tw =2.5 Load D-1 1.0750 1.0752














Ref. code: 25615922040208FTI

21

(a) Simply supported box girder


Figure 4.1. Variation of Kc with respect to H/L: (a) Simply supported box girder,


Simply supported box girder

Fixed ended box girder

Ref. code: 25615922040208FTI

22

4.2 Normal Stress Distribution in Upper Flange

The difference between the maximum normal stress (σmax) due to shear lag effect

and the normal stress obtained by the elementary beam theory (σbeam). The ratio of these

stresses is defined as the stress concentration factor (Kc) for simple measure of the shear

lag effect. The normal stress distribution in the upper flange at the mid span of simply

supported beam with B/H = 1.0, H/L = 0.2 and tf/tw = 2.5 is presented in Figure 4.3.

Stress concentration factor, Kc increases with the increase of As/Af. Cross sectional area

of stiffener, As is equal to area of one stiffener multiplied by no. of stiffeners (ts x ds x

no. of stiffeners) and cross sectional area of flange, Af is equal to two multiplied by area

of flange (2 x (2B + tw) x tf).

Figure 4.2. Geometric Properties of box girder with longitudinal stiffeners

(a) Normal stress distribution in the upper flange under Load C-1

H

2B

tw

tf

ts ds

Ref. code: 25615922040208FTI

23

(b) Normal stress distribution in the upper flange under Load C-2

(c) Normal stress distribution in the upper flange under Load D-1

(d) Normal stress distribution in the upper flange under Load D-2

Figure 4.3. Normal stress distribution in the upper flange

(B/H = 1.0, H/L = 0.2, tf/tw = 2.5)

Ref. code: 25615922040208FTI

24

The shear lag effect on simply supported box girder with longitudinal stiffeners

depends on five factors:

1. Type of loading

2. Half flange width/height of web ratio of the girder (B/H)

3. Height of web/span length ratio of the girder (H/L)

4. Thickness of flange/web ratio of the girder (tf/tw)

5. Cross sectional area of the stiffeners/area of the flange ratio of the girder

(As/Af)

Ref. code: 25615922040208FTI

25

4.3 Effect of Loading

Stress concentration factor, Kc for cross section of tf/tw = 1.3, 1.9, 2.5 and As/Af

= 0.9 under concentrated load and uniformly distributed load. The difference of Kc

between Load C-1 and Load C-2 is larger in case of wide flange and Kc for Load C-2 is

larger than Load C-1 in every case. While the difference of Kc between Load D-1 and

Load D-2 is insignificant as shown in Figure 4.4 and hence this study will consider only

case of Load D-1.

(a) As/Af = 0.9, tf/tw = 1.3

(b) As/Af = 0.9, tf/tw = 1.9

Ref. code: 25615922040208FTI

26

(c) As/Af = 0.9, tf/tw = 2.5

Figure 4.4. Variation of Kc with respect to H/L (tf/tw = 1.3, 1.9, 2.5 and As/Af = 0.9)

Ref. code: 25615922040208FTI

27

4.4 Effect of Geometric Properties

This section aims to analyze the stress concentration factor of simply supported

box girders with longitudinal stiffeners with the difference of the proportion of

geometric properties. Because the value of web height (H), thickness of flange (tf),

girder length (L), thickness of stiffener (ts) and depth of stiffener (ds) is fixed, the

geometric properties that affect to the stress concentration factor are half flange width

(B), thickness of web (tw), cross sectional area of flange (Af) and cross sectional area

of stiffeners (As). The dependence of shear lag on the B/H, H/L, tf/tw and As/Af is

recognized.

4.4.1 Half Flange Width / Height of Web Ratio (B/H)

Stress concentration factor, Kc becomes larger as B/H increases, as the flange

get wider, follow the shear lag phenomenon. Kc tends to grow with the increase of B/H

for large H/L. The influence of B/H on Kc is very small for H/L equal to 0.05 in the

both case of concentrated load (Load C-1 and Load C-2) and uniformly distributed load

(Load D-1) as shown in Figure 4.5, 4.6 and 4.7. And Kc for Load C-2 is larger than

Load C-1 when B/H increases.

(a) Variation of Kc with respect to B/H (tf/tw = 1.3) under Load C-1

Ref. code: 25615922040208FTI

28

(b) Variation of Kc with respect to B/H (tf/tw = 1.3) under Load C-2

(c) Variation of Kc with respect to B/H (tf/tw = 1.3) under Load D-1

Figure 4.5. Variation of Kc with respect to B/H (tf/tw = 1.3)

Ref. code: 25615922040208FTI

29



Ref. code: 25615922040208FTI

30




Ref. code: 25615922040208FTI

31




Ref. code: 25615922040208FTI

32

4.4.2 Height of Web / Span Length Ratio (H/L)

H/L has considerable influence on stress concentration factor, Kc: as H/L

becomes larger, Kc increases in general. The relationship between Kc and H/L is rather

linear in case of the concentrated load (Load C-1 and Load C-2) and nonlinear in case

of distributed load (Load D-1) for larger B/H as shown in Figure 4.8, 4.9 and 4.10. Kc

for Load C-2 is larger than Load C-1 in every case of increasing H/L. H/L on Kc is very

small for B/H equal to 0.4 under Load C-1, Load C-2 and Load D-1.

(a) Variation of Kc with respect to H/L (tf/tw = 1.3) under Load C-1

(b) Variation of Kc with respect to H/L (tf/tw = 1.3) under Load C-2

Ref. code: 25615922040208FTI

33

(c) Variation of Kc with respect to H/L (tf/tw = 1.3) under Load D-1

Figure 4.8. Variation of Kc with respect to H/L (tf/tw = 1.3)


Ref. code: 25615922040208FTI

34




Ref. code: 25615922040208FTI

35



Ref. code: 25615922040208FTI

36



Ref. code: 25615922040208FTI

37

4.4.3 Thickness of Flange / Web Ratio (tf/tw)

Stress concentration factor, Kc tends to grow with increase of tf/tw as shown in

Figure 4.11 to Figure 4.26. The graph shows variation of Kc with respect to tf/tw. Kc for

Load C-2 is larger than Load C-1 when increasing tf/tw.

Figure 4.11. Variation of Kc with respect to tf/tw (B/H = 0.4 and As/Af = 0)

Figure 4.12. Variation of Kc with respect to tf/tw (B/H = 0.6, As/Af = 0)

Ref. code: 25615922040208FTI

38



Ref. code: 25615922040208FTI

39

Figure 4.15. Variation of Kc with respect to tf/tw (B/H = 0.4, As/Af = 0.3)


Ref. code: 25615922040208FTI

40



Ref. code: 25615922040208FTI

41



Ref. code: 25615922040208FTI

42



Ref. code: 25615922040208FTI

43



Ref. code: 25615922040208FTI

44



Ref. code: 25615922040208FTI

45

4.4.4 Cross Sectional Area of Stiffeners/Area of Flange Ratio (As/Af)

Stress concentration factor, Kc tend to increase significally with the increase

As/Af as shown in Figure 4.27 to Figure 4.38. The effect of the shear lag is increase in

case of concentration load (Load C-1 and Load C-2) and Kc for Load C-2 is larger than

Load C-1 in every case.

(a) Variation of Kc with respect to As/Af (B/H = 0.4, tf/tw = 1.3) under Load C-1

(b) Variation of Kc with respect to As/Af (B/H = 0.4, tf/tw = 1.3) under Load C-2

(c) Variation of Kc with respect to As/Af (B/H = 0.4, tf/tw = 1.3) under Load D-1

Figure 4.27. Variation of Kc with respect to As/Af (B/H = 0.4, tf/tw = 1.3)

Ref. code: 25615922040208FTI

46





Ref. code: 25615922040208FTI

47





Ref. code: 25615922040208FTI

48





Ref. code: 25615922040208FTI

49





Ref. code: 25615922040208FTI

50





Ref. code: 25615922040208FTI

51





Ref. code: 25615922040208FTI

52





Ref. code: 25615922040208FTI

53





Ref. code: 25615922040208FTI

54





Ref. code: 25615922040208FTI

55





Ref. code: 25615922040208FTI

56





Ref. code: 25615922040208FTI

57

4.5 Overall Effects of Stress Concentration Factor

Loading conditions and the proportions of geometric properties of simply

supported steel box girders with longitudinal stiffeners from parametric study influence

on the value of the stress concentration factor, Kc. The following results have been made

in this study.

1. The difference of stress concentration factor, Kc between Load C-1 and Load C-2 is

larger in case of wide flange and Kc for Load C-2 is larger than Load C-1 in every

case. While the difference of Kc between Load D-1 and Load D-2 is insignificant.

So this study considers only Load D-1.

2. Stress concentration factor, Kc tends to grow with the increase of B/H for large H/L.

The influence of B/H on Kc is very small for H/L equal to 0.05 in the both case of

concentrated load (Load C-1 and Load C-2) and uniformly distributed load (Load

D-1).

3. H/L becomes larger, stress concentration factor, Kc increases in general. H/L on Kc

is very small for B/H equal to 0.4 under Load C-1 and Load D-1. The relationship

between Kc and H/L is rather linear in case of the concentrated load (Load C-1 and

Load C-2) and nonlinear in case of distributed load (Load D-1) for larger B/H.

4. Stress concentration factor, Kc tends to grow with increase of tf/tw.

5. Stress concentration factor, Kc tends to grow with the increase of As/Af for large

H/L.

4.6 Empirical Formulas

4.6.1 Proposed formulas

The empirical formulas for stress concentration of box girders including shear

lag effect are proposed by Yamaguchi et al. (2008) and a multiplier factor, ∅ is put in

the empirical formula to consider the effect of longitudinal stiffeners. The empirical

formulas are presented in the following:

Kc= ∅×aload type× (H

L)

c

+1 (4.1)

where

∅= (1+As

Af)

d

(4.1a)

Ref. code: 25615922040208FTI

58

aload type=bload type× (B

H)

cload type (4.1b)

bload type=e×ln (tf

tw) +f× (

tf

tw) +g (4.1c)

cload type=h×ln (tf

tw) +i× (

tf

tw) +j (4.1d)

The unknown coefficients are shown in Table 4.2 and determined from Statica

of Statsoft, Inc. The d value in Table 4.2 is made from the analysis of the present FEA

results by data fitting curves solving in Microsoft Excel (2013).

Table 4.2. Values of coefficients in equations

Load type c d e f g h i j

C-1 1 1.1 0.832 0 2.77 -0.034 0 1.744

C-2 1 1.2 1.756 0 6.101 0.053 0 1.202

D-1 2 1.15 1.225 -0.494 6.001 -0.041 -0.006 2.371

For the analysis of the present FEA results, the present numerical results give

the following formulas:


Kc=∅×a1× (H

L) +1 (4.2)

where

∅= (1+As

Af)

1.1

(4.2a)

a1=b1× (B

H)

c1 (4.2b)

b1=0.832×ln (tf

tw) +2.77 (4.2c)

c1=-0.034×ln (tf

tw) +1.744 (4.2d)


Kc=∅×a2× (H

L) +1 (4.3)

where

Ref. code: 25615922040208FTI

59

∅= (1+As

Af)

1.2

(4.3a)

a2=b2× (B

H)

c2 (4.3b)

b2=1.756×ln (tf

tw) +6.101 (4.3c)

c2=0.053×ln (tf

tw) +1.202 (4.3d)

Distributed load (Load D-1):

Kc=∅×a3× (H

L)

2

+1 (4.4)

where

∅= (1+As

Af)

1.15

(4.4a)

a3=b3× (B

H)

c3 (4.4b)

b3=1.225×ln (tf

tw) -0.494 (

tf

tw) +6.001 (4.4c)

c3=-0.041×ln (tf

tw) -0.006 (

tf

tw) +2.371 (4.4d)

It is noted that the above formulas are applicable for

0.05≤H

L≤0.2, 0.4≤

B

H≤1.0, 1.3≤

tf

tw ≤2.5 and 0≤

As

Af≤1.0.

4.6.2 Accuracy of the proposed formulas

Figure 4.39 shows Kc obtained by the proposed formulas of Eq. (4.2), Eq. (4.3)

and Eq. (4.4) together with the present finite element results. To quantify the accuracy,

the percentage error of the stress concentration factor calculated from this formula:

εi=KcEmp-KcFEA

KcFEA×100(%) (4.5)

where KcEmp and KcFEA are the Kc values obtained by the proposed formulas and the

present finite element analysis, respectively. The accuracy of the proposed formula for

each loading condition is calculated as mean square error by the following equation

(Tenchev, 1996):

ε ̅= √ 1

N∑ εi2

Ni=1 (4.6)

where εi is the error computed by Eq. (4.5) for a present finite element result and N is

the number of the present finite element results for a loading condition. N in Eq. (4.6)

Ref. code: 25615922040208FTI

60

is equal 192. Using Eq. (4.6), the mean square error is found to be 1.92% for Load C-

1, 2.84% for Load C-2 and 1.41% for Load D-1.

Figure 4.39 shows Kc due to proposed formulas and finite element analysis for

the cross section of tf/tw = 1.3 and As/Af = 0, 0.9 under Load C-1, Load C-2 and Load

D-1.

(a) tf/tw = 1.3 and As/Af = 0

Ref. code: 25615922040208FTI

61

(b) tf/tw = 1.3 and As/Af = 0.9

Figure 4.39. Kc due to proposed formulas and finite element analysis

Ref. code: 25615922040208FTI

62

Chapter 5

Conclusion

Three-dimensional finite element analysis of simply supported box girder with

longitudinal stiffeners has been performed to reveal the shear lag effects on the normal

stresses of the box girders. The stress concentration factor, Kc represents the shear lag

effect on stress concentration is defined as the ratio between the maximum normal stress

and the normal stress based on theory. Shell elements have been used for all analysis

models. Concentrated load and distributed load are applied to various cross-sections of

box girders. The results show that the shear lag effect increases with the increase of

H/L, B/H, tf/tw and As/Af.

The stress concentration factor, Kc values in four cases can be summarized as

follows;

1. Kc tend to grow with increase of H/L.

2. H/L on Kc is very small for B/H equal to 0.4 under Load C-1 and Load D-1.

3. B/H has considerable influence on Kc: as B/H become larger, Kc increase in

general.

4. B/H on Kc is very small for H/L equal to 0.05 under Load C-1, Load C-2 and

Load D-1.

5. Kc tends to grow with the increase of tf/tw for large H/L.

6. Kc tends to grow with the increase of As/Af for large H/L.

Based on the present numerical results, the empirical formulas have been

proposed to compute the stress concentration factors. And it is confirmed that the results

of proposed formulas are similar to the present finite element results. The formulas

proposed by this study could be adopted as the fundamental data for improvement of

the present design codes of steel box girders considering shear lag effects.

For future study, it is recommended that the investigation of the shear lag effect

on stress concentration in continuous box girders with longitudinal stiffeners by using

the three-dimensional finite element method should be studied. In the same way, the

study about shear lag effect on deflection in both simply supported box girders with

longitudinal stiffeners and continuous box girders with longitudinal stiffeners is an

interesting issue to improve the current formulas.

Ref. code: 25615922040208FTI

63

References

Moffatt, K.R., and Dowling, P.J., 1975. Shear Lag in Steel Box Girder Bridges. Struct.

Engng., 53, 10: 439-448.

Yamaguchi, E., Chaisomphob, T. and Sa-nguanmanasak, J., 2008. Stress concentration

and deflection of simply supported box girder including shear lag effect,

Structural Engineering and Mechanics, Vol. 28, No. 2, 207-220.

AASHTO, 2010. LRFD Bridge Design Specifications, American Association of State

Highway and Transportation officials, Washington, DC.

MARC Analysis Research Corporation, 2016. MARC Manuals Vol. A-D, Rev. K.6.

Palo Alto, Calif, USA.

Lee, C.K., and Wu, GJ., 2000. Shear Lag Analysis by The Adaptive Finite Element

Method, Part 1: Analysis of Simple Plated Structures. Thin-Walled Struct., 38:

285-309.

Von Karman, T., 1924. Beitrage zur Techischen Mechanic and Technischen Physik,

Agust Foppl Festschrift. Berlin, Germany.

Tenchev R.T., 1996. Shear lag in orthotropic beam flanges and plates with stiffeners.

Int J Solids Struct., 33, 9: 1317-1334.

Cook R.D., 1995. Finite Element Modeling for Stress Analysis. New York: John Wiley

& Sons.

Tech Tips, 2006. SolidWorks Express. How to use symmetry and anti-symmetry

boundary conditions. p. 1-3.

Chavel, B.; Carnahan, J. Steel Bridge Design Handbook: Steel Tub Girder Bridge.

2012. p. 15-19.

CSI SAP2000, 2016. v19.2.0. Computer & Structures, Inc. California, USA.

Lertsima, C., Chaisomphob, T., and Yamaguchi, E., 2004. Three-dimensional finite

element modeling of a long-span cable bridge for local stress analysis,

Structural Engineering and Mechanics, 18(1): 113-124.

Trahair, N.S., and Bradford, M.A., 1988. The Behavior and Design of Steel Structures

London: E & FN Spon.

Ref. code: 25615922040208FTI

Date post:	26-Jan-2021
Category:	Documents
Upload:	others
View:	2 times
Download:	0 times

Shear Lag of Steel Box Girders with Longitudinal...

Documents