concentration factor of in-plane stresses beyond elastic limit in ...

June 2021 | 56

CONCENTRATION FACTOR OF IN-PLANE STRESSES BEYOND ELASTIC LIMIT IN

GUSSET PLATES

Mohammed Taher,

Teaching Assistant, Faculty of Engineering and Technology, Future University in Egypt, Egypt.

E-mail: [email protected]

Mohamed M. Korashy,

Assistant Professor, Faculty of Engineering, Ain Shams University, Egypt.

E-mail: [email protected]

Ahmed M. Ebid,

Associate Professor, Faculty of Engineering and Technology, Future University in Egypt,

Egypt. E-mail: [email protected]

S. Y. Aboul Haggag Associate Professor, Faculty of Engineering, Ain Shams University, Egypt. E-mail: [email protected]

Abstract— Gusset plates are main elements in steel connections but their behavior under

loads is still not completely understood. Due to lack of researches, design codes don’t specify

clear procedures to design gusset plates. This research aims to improve the understanding of

stress distribution in gusset plates. It investigates the factors affecting the inelastic stress

concentration factor (Kt) within the gusset plate of lower chord truss bolted connection. FEM

model was developed and verified using literature experimental test results. A parametric study

of 55 different truss connections was conducted based on the developed model to examine the

stress distribution within gusset plates. The considered factors were gusset plate thickness,

dimensions and bolt diameter/spacing. The effect of each parameter on (Kt) factor was studied.

The results showed that (Kt) value is proportional with plate thickness and effective width and

inversely proportional with bolt diameter/spacing.

Index Terms— FEM, Gusset Plate, Stress distribution, Stress concentration factor, Bearing

type connection, Bolted Connection, Inelastic Behaviour

1 INTRODUCTION

Gusset plate connections are commonly used in trusses to transfer the forces between the

connected members. Connection members could be connected to gusset plate either by

welding or bolting. Each method has its advantages and disadvantage. One of the main

disadvantages of the bolted connections is the stress concentration around bolt holes. Previous

researches that discussed the gusset plate connections didn’t suggest a specific design

procedure for gusset plate to achieve the connection safety requirements. Also, very few

researches had got deep into detailed modeling of the connection. Whitmore [1] made an

experimental study to determine the distribution of stresses in aluminum gusset plates. The

reported that the maximum tensile and compressive stresses can be determined by assuming

that the force in each member is distributed and transferred to the gusset plate in a zone

Tianjin Daxue Xuebao (Ziran Kexue yu Gongcheng Jishu Ban)/

Journal of Tianjin University Science and Technology

ISSN (Online): 0493-2137

E-Publication: Online Open Access

Vol:54 Issue:06:2021

DOI 10.17605/OSF.IO/N5DWR

June 2021 | 57

obtained by an effective width that is perpendicular to the axis of the steel member in the

gusset plate. Kirsch [2] presented a study for stress distribution in infinite isotropic plate with a

circular hole under uniaxial tensile stresses. He proved that within elastic stress range, the

maximum stress around the hole is between 2 to 3 times the average stresses in the plate.

Walbridge et al. [3] prepared an analytical study for the tensile behavior of steel gusset plates

considering the effect of inelasticity of material, stiffness of member, large displacement and

the initial imperfections. He concluded that gusset plate’s behavior does not influence by

shortening in tensile capacity. C. Chou and P. Chen [4] investigated the behavior of the central

gusset plates in buckling-restrained braced frames. Different variables were considered such

as gusset plate thickness, geometry and utilizing of free-edge stiffeners. They concluded that

the ultimate capacity of the gusset plate increased with using stiffeners along gusset free

edges. Y. Cui et al. [5] presented a study on four welded gusset plate connections, using FE

analysis in order to investigate the ultimate strength of the welded connections subjected to

tensile stresses. The considered Parameters were the size of the gusset plate, brace

eccentricity, and the angle of brace. Results showed that welds in Whitmore zone transferred

forces from the connected members to the gusset plate. The effective length of weld was

governed by size of the gusset plate, angle of brace and the brace eccentricity. Fang et al. [6]

presented an experimental and numerical study on the gusset plate behavior under

eccentrically compressive loads applied. Studied parameters included size and thickness of

gusset plate and connection layout. It was observed that the mode of failure was due to the

splice member yielding, while stiffened splice member exhibited better behavior. Y. Zeynali et

al. [7] prepared a study of the influence for slip-critical connections in gusset bolted plates

using 28 samples of double gusset plates. The considered variables were material properties,

number of bolts, type of connections, and friction coefficient value. It was demonstrated that the

ratio of the ultimate stress of slip-critical connections to ultimate stress of bearing connections

increases with increasing of friction coefficient and decreasing of both thickness and material

resistance. K. S. Vivek et al. [8] presented a study on the inelastic performance of the mild-

steel gusset plates connected to a diagonal brace subjected to a tensile force. Non-linear finite

element analysis was performed considering inclination angle of the brace, hardening type and

bolt diameter. It was found that gusset plates with under estimated thicknesses will fail by

excessive yielding due to stress concentration at holes. Sh. Maleki and M. Ghaderi-Garekani

[9] studied the geometry effect of gusset plate on the block shear strength of welded

connections by using a nonlinear FEM under combined loading. Obtained results indicated that

the inclined planes of failure are expected for the combined loading conditions, and the state of

stress of these planes was a combination of shear and tensile stresses. They used these

results to describe the behavior of the planes of block shear rupture in such cases and

provided a better estimation for block shear strength in welded connections subjected to

combined loads. They produced a new equation to consider the effect of the ruptured block on

gusset plate.







June 2021 | 58

2 OBJECTIVE OF RESEARCH

The main objective of this research is to investigate the effect of considered parameters

(gusset plate thickness, dimensions, bolt diameter/spacing) on the stress concentration factor

of truss lower cord connection’s bolted gusset plate. The stress concentration factor (Kt) is

defined as the ratio between the maximum Von Mises stresses within gusset plate calculated

from FEM to the average normal stresses in the gusset plate calculated using Whitmore’s

effective width method. Non-pretension high strength bearing type bolts were considered in this

study. Cross sections of connected members and diameter of connecting bolt were selected to

prevent any failure outside the gusset plate.

3 RESEARCH METHODOLGY

In order to achieve the descripted objective, the research is divided into two phases. The first

phase aimed to select the geometrical and loading configurations for the tested connections,

while the second phase aimed to study the stress distribution within the selected connections

under the selected loading conditions. The first phase was carried out considering linear

analysis using SAP2000 V20 software, while the second phase was carried out considering

nonlinear and large deformations analysis using ANSYS V15.0 software.

3.1 1st Phase – Truss models using SAP2000 A parametric study was carried out to select the most critical geometrical and loading

configurations for the tested connections. The study consisted of nine symmetrical simply

supported N-trusses with different selected geometrical configurations to satisfy the following

conditions:

• Inclination angle of diagonal members is ranged between 30o and 60o.

• Truss span is ranged between 8 and 16 times its maximum height.

• The slope of the upper cord is ganged between 0 and 1:10.

• Subdivides were used when panel width exceeds 2m.

Each model was loaded by equal concentrated loads at the top chord nodes, the loads were

calculated based on spacing between trusses of 6.0m, super imposed load and live load is

considered 60 kg/m2 while the own weight of truss was automatically calculated using pre-

designed cross sections. Fig. 1 shows schematic for the models and the loads applied. Fig. 2

shows the detailed dimensions for each model, location of the selected connection and force in

each member in the connection.

The critical connection is defined as the connection that has the maximum Normalized

Summation of Forces (NSF) which is the summation of absolute member forces divided by the

maximum absolute member force.

Fig. 1 - schematic for geometrical and loading configurations of studied models







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Fig. 2 - details of truss models and selected critical joints

Truss 1

T1-J3 (NSF=2.27)

Truss 2

T2-J4 (NSF=2.29)

Truss 3

T3-J5 (NSF=2.10)

Truss 4

T4-J4 (NSF=2.59)

Truss 5

T5-J4 (NSF=2.36)

Truss 6

T6-J4 (NSF=2.30)

Truss 7

T7-J7 (NSF=2.18)

Truss 8

T8-J4 (NSF=2.13)

Truss 9

T9-J2 (NSF=3.16)







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3.2 2nd Phase – Joint models using ANSYS V15 The aim of this phase is to study the each one of the selected truss joints considering different

gusset plate thickness and different bolts spacing. To achieve this goal, a reliable FEM model

should be developed to figure out the optimum technique to simulate each element in the joint

considering its material properties and the interface between elements. This preliminary model

must be verified by experimental test results. Accordingly, this phase is divided into two stages,

the first stage is to develop and verify the preliminary model and the second stage is to

generate the parametric study models based on the verified preliminary model.

3.2.1 1st Stage - Develop and verify the preliminary FE model

Zeynali Y et al. [7] investigated a bearing bolted gusset plate specimen subjected to tensile

loading. The specimen was a double gusset plate connected to a thick steel plate which was

used to load the specimen. Full detailed geometries and material properties were presented in

the research besides the results of the experimental test. The preliminary FE model was built

based on these data. Since the specimen was symmetric, only one half of the specimen was

modeled. Fig. 3 shows the half model and bolt meshing. Boundary conditions of model were

defined as per the experimental study to allow the load transfer plate to move in the direction of

the loading. The load was set as displacement in the direction of the loading.

Fig. 3 - Details preliminary FE model, (a) Geometry & boundary conditions, (b) Meshing of bolt hole and bolt shank (invisible bolt head)

Gusset plate was modeled using SHELL181 elements; while loading plate, bolts and nuts were

modeled using SOLID185 elements. The gap interface between gusset plate and both bolts

and members were simulated using CONTA175 and CONTA174 elements respectively.

TARGET170 element was used to identify the considered interfaced surfaces for each

element. The stress-strain curves for the used materials are shown in Fig. 4.







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Fig. 4 - Stress – Strain curves for used materials

Fig. 5 shows the load - displacement curves for both experimental and preliminary FE model,

while Fig. 6 shows the distribution of both normal and shear stresses in the gusset plate. It

could be noted that maximum tensile stress occurred between bolts of the last row and

maximum shear stress appears along the bolt gauge line. These maximum stresses occurred

at the locations of the fractures in the experimental study.

Fig. 5 - Load - displacement curves for both experimental and preliminary FE model







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Fig. 6 - Stresses distribution in gusset plate – Preliminary FE model

3.2.2 2nd Stage – FE models of parametric study

Based on the verified preliminary FE model, 55 FE models were generated to simulate the nine

selected truss joints with different gusset plate thicknesses and different bolt spacing. Detailed

description of the parametric study models is summarized in Table 1. Fig. 7 shows a typical

model for the considered truss joints.

Boundary conditions were chosen to allow members to move along their axes only. In addition,

the exact point of intersection of angels’ centerlines at the gusset plate is not allowed to

translate in any direction. All vertical and diagonal members have a single angel cross section.

While the chords have double angles cross sections. All members and bolts sizes were

selected to ensure that the failure will occur in the gusset plate. Member loads are distributed

stress on their cross section. Loads were applied step by step till the connection failed or the

total load reached.

Fig. 7 - Typical model for the considered truss joints

(a) Normal stresses

(b) Shear stresses







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Table 1 Detailed description of the parametric study models

ID Joint

Gusset thick. (mm)

Bolt space ----------- Bolt dia.

Output load -------------- Target load

ID Joint

Gusset thick. (mm)

Bolt space ----------- Bolt dia.

Output load -------------- Target load

1

Truss 1 Joint 3

4 3.13 Not Failed

28

Truss 5 Joint 4

8 3.13 0.46

2 6 3.13 Not Failed

29 10 3.13 0.61

3 8 3.13 Not Failed

30 12 3.13 0.7

4 10 3.13 Not Failed

31 14 3.13 Not

Failed


32 14 3.44 Not

Failed

6

Truss 2 Joint 4

4 3.13 0.34

33 14 3.75 Not

Failed

7 6 3.13 0.70

34 14 4.06 Not

Failed

8 8 3.13 Not Failed

35 16 3.13 Not

Failed


36

Truss 6 Joint 4

4 3.13 0.25

10 12 3.13 Not

Failed 37 6 3.13 0.52

11

Truss 3 Joint 5

4 3.13 0.21

38 8 3.13 0.67

12 10 3.13 0.60

39 10 3.13 0.85

13 12 3.13 0.81

40 12 3.13 Not

Failed 14 14 3.13 0.86

41 14 3.13 Not

Failed

15 16 3.13 0.97

42

Truss 7 Joint 7

4 3.13 0.35

16 18 3.13 Not

Failed 43 6 3.13 0.77

17 18 3.44 Not

Failed 44 8 3.13 Not





Failed

20 Truss 4

Joint 4 4 3.13 0.39

47 14 3.13 Not

Failed

2 6 3.13 0.50 4 Trus 16 3.13 Not







June 2021 | 64

1 8 s 8 Joint 4






Failed 51 16 4.06 0.96

25 12 3.44 Not

Failed 52

Truss 9 Joint 2

14 3.13 Not Failed

26 12 3.75 Not




Failed

Note: 16mm diameter bolts are used in all models

55 14 4.06 Not

Failed

4 RESULTS

Models results showed that the distribution of maximum stresses tends to locate around the

holes. The size of these spots of stress concentration increased by increasing the applied load

on the connection until they merged in one zone includes all holes. This zone is called

Whitmore zone. Fig. 8 presents the theoretical boundaries of Whitmore zone.

Fig. 8 - Whitmore effective zone [1]

For each model of the parametric study, the connection was loaded gradually till the target

loads from SAP2000 results achieved or the connection failed before that. The average stress

in the gusset was calculated using Whitmore width and considering bolt holes diameter of

18mm, besides that, the maximum Von Mises stress in gusset plate was recorded. Stress

concentration factor (Kt) was calculated by dividing the maximum Von Mises stress by the

average stress. Table 2 summarized the outputs of the parametric study. Fig. 9 presents

selected samples of Von Mises stress distribution in gusset plate. Fig. 9-a, b, c illustrates that

failure occurred in the 12mm thick gusset plate and the Whitmore zone is completely

developed, for thicker plate (18mm) the connection did not fail and although the stresses are

concentrated around bolt holes but the Whitmore zone is not fully developed yet. Fig. 9-d, e, f

indicates that – within the studied range- bolt spacing has no significant effect in the stress







June 2021 | 65

concentration factor (Kt). It also shows that increasing bolt spacing increases the stress

concentration around the bolt holes especially the edge ones.

Table 2 Detailed outputs of the parametric study models

ID

Whitmore width (mm)

Bolt space. --------- Bolt dia.

Avg. Stress (MPa)

Max. Stress (MPa)

Kt

ID

Whitmore width (mm)

Bolt space. --------- Bolt dia.

Avg. Stress (MPa)

Max. Stress (MPa)

Kt

1

80

3.13 117 275 2.35

28

148

3.13 87 360 4.15

2 3.13 78 273 3.50

29 3.13 93 360 3.88

3 3.13 58 272 4.65

30 3.13 90 360 4.01

4 3.13 47 271 5.79

31 3.13 110 358 3.26

5 3.13 39 266 6.82

32 3.44 110 358 3.24

6

120

3.13 119 360 3.03

33 3.75 110 359 3.25

7 3.13 167 360 2.16

34 4.06 110 360 3.26

8 3.13 179 315 1.76

35 3.13 97 315 3.26

9 3.13 143 305 2.13

36

120

3.13 104 360 3.45

10 3.13 119 305 2.56

37 3.13 148 360 2.43

11

148

3.13 89 360 4.06

38 3.13 143 360 2.52

12 3.13 103 360 3.51

39 3.13 146 360 2.46

13 3.13 116 360 3.10

40 3.13 144 320 2.23

14 3.13 105 360 3.43

41 3.13 123 310 2.52

15 3.13 104 360 3.46

42

120

3.13 104 360 3.45

16 3.13 96 306 3.18

43 3.13 152 360 2.36

17 3.44 96 311 3.23

44 3.13 150 330 2.19

18 3.75 96 313 3.26

45 3.13 120 325 2.70

19 4.06 96 315 3.28

46 3.13 100 320 3.19

20 80 3.13 133 360 2.71

47 3.13 86 310 3.61







June 2021 | 66

21

3.13 115 360 3.12

48

150

3.13 93 303 3.28

22 3.13 173 292 1.69

49 3.44 93 311 3.36

23 3.13 139 280 2.02

50 3.75 93 309 3.34

24 3.13 115 275 2.38

51 4.06 89 360 4.06

25 3.44 115 277 2.40

52

80

3.13 54 241 4.45

26 3.75 115 279 2.42

53 3.44 54 241 4.45

27 4.06 115 281 2.43

54 3.75 54 243 4.49

Note: 16mm diameter bolts are used in all models

55 4.06 54 243 4.49

5 DISCUSSION

Kirsch [2] reported that the stress concentration factor around circular hole in isotropic plate

subjected to uniaxial tensile stress could be calculated using the following equation:

(

)

(

)

……………………………………………………………..(1)

Where φ is hole diameter and w is the plate width.

This equation is valid within the elastic stress range only where the maximum stress around the

hole is less than the yield stress of the plate material. Accordingly, this study is concerned in

studying the inelastic stress concentration factor beyond yielding stress. Table 2 shows that the

values of maximum Von Mises stress exceed the yield stress of the gusset plates (y =240

MPa). In order to study the relation between stress concentration factor (Kt) and the average

stress in gusset plate, the results listed in Table 2 was divided into two groups; the first group

includes the inelastic zone models which achieved the target load and the maximum stress in

the plate is between yield stress and ultimate stress. The second group includes the beyond

inelastic zone models where the maximum stress in the plate exceeded the ultimate stress

before the connection achieved the target load. The relation between stress concentration

factor (Kt) and the average stress for both groups are presented in Fig. 10. From the best fitting

curve in both charts, Eq. 2 could be extracted considering the ultimate stress of gusset plate

(u) =360 MPa.







June 2021 | 67

………………..………………………………..(2)

Where avg, max, y and max, u are average, maximum, yield and ultimate stress in (MPa)

respectively, φ is hole diameter and w is the Whitmore width.

Kt =max

avg=

3.0 3.14

𝑊+ 3.66

𝑊

2

1.53

𝑊

3

, max ≤ y

140

avg 0.84 , y < max ≤ u

u

avg , max > u







June 2021 | 68

ts: Gusset plate thickness, P: Bolt spacing, Po/Pt: Output load / Target load, Kt: stress concentration factor

Fig. 9 - Von Mises stress distribution in selected gusset plates

a) Model-13 Truss 3, Joint 5

ts=12mm, P=50mm, Po/Pt=0.81, Kt=3.10

d) Model-31 Truss 5, Joint 4

ts=14mm, P=50mm, Po/Pt=1.00, Kt=3.26

b) Model-16 Truss 3, Joint 5

ts=18mm, P=50mm, Po/Pt=1.00, Kt=3.18

e) Model-32 Truss 5, Joint 4

ts=14mm, P=55mm, Po/Pt=1.00, Kt=3.25

c) Model-19 Truss 3, Joint 5

ts=18mm, P=65mm, Po/Pt=1.00, Kt=3.28

f) Model-33 Truss 5, Joint 4

ts=14mm, P=60mm, Po/Pt=1.00, Kt=3.25







June 2021 | 69

Fig. 10 - Relation between average stress in gusset plate and Stress concentration factor (Kt)

6 CONCLUSIONS

This research presents a numerical study for the inelastic stress concentration factor (Kt) in

bolted gusset plates of lower cord truss connection (which is mainly subjected to tensile

stresses), the study results could be concluded in the following points:

• The value of stress concentration factor equals 3.0 within the elastic stress range, after

Kirsch [2]

• Within the inelastic zone where the stress around the hole exceeds the yielding stress

and below the ultimate stress, the stress concentration factor is in inverse exponential

proportional with the average stress value.

• Previous point means that (Kt) value is proportional with Whitmore width and gusset

plate thickness and inversely proportional with hole diameter/spacing.

• Within the studied range of (bolt spacing / bolt diameter) parameter, the results

illustrated that it has a minor effect (Kt) values.

• The proposed equation of the (Kt) within the inelastic zone is valid for the tested material

only (mild steel) with yield and ultimate stress of 240 MPa and 360 MPa respectively.

• Beyond the inelastic zone where the stress around the hole reaches the ultimate stress,

the stress concentration factor equals the ultimate stress divided by the average stress.

(a) Inelastic zone models (b) Beyond inelastic zone models







June 2021 | 70

REFERENCES

[1] R. E. Whitmore, Experimental investigation of stresses in gusset plates, Bulletin No.16, University of Tennessee, USA, 1952. [2] Kirsch, 1898, Die Theorie der Elastizität und die Bedürfnisse der Festigkeitslehre. Zeitschrift des Vereines deutscher Ingenieure, 42, 797–807. [3] Walbridge S., Grondin G. and Cheng J., “Gusset Plate Connections under Monotonic and Cyclic Loading.” Canadian Journal of Civil Engineering, NRC Research Press 32(5) (2005), pp. 981–995. [4] Chou Ch. and Chen P., “Compressive Behavior of Central Gusset Plate Connections for a Buckling-Restrained Braced Frame”, Journal of Constructional Steel, Research 65 (2009), pp. 1138–1148. [5] Cui Y., Asada H., Kishiki Sh. and Yamada S., “Ultimate Strength of Gusset Plate Connections with Fillet Welds”, Journal of Constructional Steel, Research 75 (2012), pp. 104–115. [6] Fang Ch., Yam M., Cheng J. and Zhang Y., “Compressive Strength and Behaviour of Gusset Plate Connections with Single-Sided Splice Members”, Journal of Constructional Steel, Research 106 (2015), pp. 166–183. [7] Zeynali Y., Samimi M., Mazroei A., Marnani J. and Rohanimanesh M., “Experimental and Numerical Study of Frictional Effects on Block Shear Fracture of Steel Gusset Plates with Bolted Connections”, Thin-Walled Structures 121 (2017) Elsevier Journal, pp. 8–24. [8] Vivek K., Priyanka U., Reddy C. and Ram K., “Non – Linear Finite Element Analysis of Mild Steel Gusset Plates”, Materials Today: Proceedings 18 (2019) Elsevier Journal, pp. 3726–3732.







[9] Maleki Sh. and Ghaderi-Garekani M., “Block shear failure in welded gusset plates under combined loading”, Journal of Constructional Steel Research 170 (2020), 106079.

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