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Lateral Buckling Analysis of a Steel Pony Truss

by Derek Matthies

A study submitted in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE

Major: Civil Engineering (Structural Engineering)

Committee Members: Fouad Fanous - Major Professor Robert Abendroth - Committee Member Vernon Schaefer - Committee Member

Iowa State University of Science and Technology Ames, IA 2012

ii

Contents

List of Symbols and Abbreviations.................................................. Abbreviations.................................................. .............................................. iv List of Figures ...................................................... .............................................................................................................. ........................................................................... ................... v 1.

2.

Introduction Introduction and Objective ....................................................... ...................................................................................................... ............................................... 1 1.1

Introduction Introduction ..................................................... ............................................................................................................ ................................................................. .......... 1

1.2

Objectives Objectives ........................................................ ............................................................................................................... ................................................................. .......... 2

Background Background .................................................. ........................................................ ........................................................................... ................... 3 2.1

Buckling Behavior......................................................................................... ................... 3

2.2

Euler Buckling............................................................................................... ................... 4

2.3

Buckling of Bars on Elastic Supports .................................................. ............................ 6

2.4

Buckling of Un-braced Top Chord Truss Members......................................................... 7

2.4.1

Analysis according to Engesser .................................................... ................................................................................ ............................ 7

2.4.2

Buckling Load using the Energy Method....................................................... ................................................................. .......... 9

2.4.3

Buckling Solution with Variable Axial Load ................................................. ........ 10

2.4.4

Buckling of a Pony Truss Top Chord with Elastic Ends ........................................ 12

2.4.5

Analysis of a Pony Truss Top Chord According to Holt ....................................... 14

2.4.6

Buckling Load with Initial Initial Out-of-plane Out-of-plane Deformations Deformations ......................................... 16

2.5 3.

4.

Pony Truss Design according to AASHTO Specifications Specifications............................................ 17 17

Finite Element Analysis .................................................. ...................................................... 18 3.1

Finite Element Model of the Compression Compression Chord ................................................. ........ 18

3.2

Analysis of Top Chords as a Bar on Elastic Supports.................................................... 19

3.3

Finite Element Model of the Pony Truss ...................................................... ....................................................................... ................. 20

Discussion and Results Results of the Analysis Analysis of of a Pony Pony Truss Truss Top Chord .................................... 23 4.1

Effective Buckling Length Factor – Lateral Support Stiffness Relationships ........... ...... ......... .... 23

4.2

Example Calculations Calculations for the Buckling Load of a Pony Truss ...................................... 24

4.2.1

Calculations Calculations following Engesser’s Engesser’s Procedure ................................................ ........ 25

4.2.2

Calculations Calculations following Bleich’s Procedure .................................................... ............................................................ ........ 25

4.2.3

Calculations Calculations following Timoshenko’s Timoshenko’s Procedure Procedure ................................................... 26

4.2.4

Calculations Calculations following Lutz and Fisher’s Fisher’s Procedure.............................................. 26

4.2.5

Calculations Calculations following Holt’s Procedure Procedure ............................................... ................. 26

4.2.6

Calculations Calculations using the Energy Method .................................................. ................. 27

4.3

Analysis of the Pony Truss using Finite Element .................................................. ........ 28

ii

Contents

List of Symbols and Abbreviations.................................................. Abbreviations.................................................. .............................................. iv List of Figures ...................................................... .............................................................................................................. ........................................................................... ................... v 1.

2.

Introduction Introduction and Objective ....................................................... ...................................................................................................... ............................................... 1 1.1

Introduction Introduction ..................................................... ............................................................................................................ ................................................................. .......... 1

1.2

Objectives Objectives ........................................................ ............................................................................................................... ................................................................. .......... 2

Background Background .................................................. ........................................................ ........................................................................... ................... 3 2.1

Buckling Behavior......................................................................................... ................... 3

2.2

Euler Buckling............................................................................................... ................... 4

2.3

Buckling of Bars on Elastic Supports .................................................. ............................ 6

2.4

Buckling of Un-braced Top Chord Truss Members......................................................... 7

2.4.1

Analysis according to Engesser .................................................... ................................................................................ ............................ 7

2.4.2

Buckling Load using the Energy Method....................................................... ................................................................. .......... 9

2.4.3

Buckling Solution with Variable Axial Load ................................................. ........ 10

2.4.4

Buckling of a Pony Truss Top Chord with Elastic Ends ........................................ 12

2.4.5

Analysis of a Pony Truss Top Chord According to Holt ....................................... 14

2.4.6

Buckling Load with Initial Initial Out-of-plane Out-of-plane Deformations Deformations ......................................... 16

2.5 3.

4.

Pony Truss Design according to AASHTO Specifications Specifications............................................ 17 17

Finite Element Analysis .................................................. ...................................................... 18 3.1

Finite Element Model of the Compression Compression Chord ................................................. ........ 18

3.2

Analysis of Top Chords as a Bar on Elastic Supports.................................................... 19

3.3

Finite Element Model of the Pony Truss ...................................................... ....................................................................... ................. 20

Discussion and Results Results of the Analysis Analysis of of a Pony Pony Truss Truss Top Chord .................................... 23 4.1

Effective Buckling Length Factor – Lateral Support Stiffness Relationships ........... ...... ......... .... 23

4.2

Example Calculations Calculations for the Buckling Load of a Pony Truss ...................................... 24

4.2.1

Calculations Calculations following Engesser’s Engesser’s Procedure ................................................ ........ 25

4.2.2

Calculations Calculations following Bleich’s Procedure .................................................... ............................................................ ........ 25

4.2.3

Calculations Calculations following Timoshenko’s Timoshenko’s Procedure Procedure ................................................... 26

4.2.4

Calculations Calculations following Lutz and Fisher’s Fisher’s Procedure.............................................. 26

4.2.5

Calculations Calculations following Holt’s Procedure Procedure ............................................... ................. 26

4.2.6

Calculations Calculations using the Energy Method .................................................. ................. 27

4.3

Analysis of the Pony Truss using Finite Element .................................................. ........ 28

iii

5.

4.3.1

Two Dimensional Dimensional Analysis ................................................. ................................... 28

4.3.2

Three Dimensional Analysis ........................................................ .................................................................................. .......................... 30

4.4

Effects of Compression Chord Moment of Inertia on the Stiffness of the Elastic Supports ................................................. ....................................................... ........................................................................ ................. 32

4.5

Analysis with Modified Elastic Stiffness ...................................................... ....................................................................... ................. 35

Summary, Conclusions, Conclusions, and Recommendations Recommendations .................................................. ................. 38 5.1

Summary ............................................... ........................................................ ......................................................................... ................. 38

5.2

Conclusions Conclusions ..................................................... ............................................................................................................ ............................................................... ........ 39

5.3

Recommendations Recommendations .................................................... .......................................................................................................... ...................................................... 39

Appendix A ................................................. ........................................................ .................................................................................. .......................... 40 Appendix B ................................................. ........................................................ .................................................................................. .......................... 43 Appendix C ................................................. ........................................................ .................................................................................. .......................... 45 Appendix D ................................................. ........................................................ .................................................................................. .......................... 47 References References ................................................... ........................................................ .................................................................................. .......................... 50

iv

���� �� ������� ��� ������������� A b C Ce CE C0 c d E Et h I Ib Ic Id K k k s l L Ld Le M m n P Pcr Pd Q q qo r U v V x y α ∆ δ πp θ ϕ ψ

area of compression chord length of floor beams spring stiffness of interior supports spring stiffness of end supports Engesser’s spring stiffness required spring stiffness with rigid end supports ratio of C to C0 length of diagonal end members modulus of elasticity tangent modulus of elasticity height of truss moment of inertia moment of inertia of floor beam moment of inertia of vertical web member moment of inertia of diagonal web member 2 column stiffness: K = P/EI effective length coefficient joint spring stiffness distance between panels total length of the truss length of diagonal web member effective length bending moment number of buckling modes number of bays compression load critical buckling load axial load on diagonal end members virtual load distributed compressive force maximum compressive load with a varying load distribution distributi on 2 radius of gyration: r = I/A internal energy factor of safety external work distance from end support displacement of chord at point x elastic foundation constant maximum displacement of compression chord relative displacement of vertical web members potential energy angle of rotation at joints stiffness of compression chord Schweda’s elastic end support coefficient

v

���� �� ������� Figure 1.1

Pony Truss Bridge………...…...….… Bridge………...…...….……….……………………… …….……………………………...……….1 ……...……….1

Figure 2.1

Equilibrium Equilibrium Path for Initially Initially Straight Column…..…......………..……… Column…..…......………..……………….4 ……….4

Figure 2.2

Equilibrium Equilibrium Path for Slightly Crooked Column…….…......……..…… Column…….…......……..……………….4 ………….4

Figure 2.3

Euler Buckling………………………………...… Buckling………………………………...………..……...………………….5 ……..……...………………….5

Figure 2.4

Buckling Modes for a Bar with Pin Ends.....………………..……… Ends..... ………………..…………………….6 …………….6

Figure 2.5

Elastically Supported Bar......………….……… Bar..... .………….…………………………………………. ………………………………….7 7

Figure 2.6

Column on elastic supports…………………… supports…………………………………………….. ………………………...………... .………...9 9

Figure 2.7

Varying Axial Load Distribution………...……. Distribution ………...…….………………………………. ………………………………...10 ..10

Figure 2.8

Compression Compression Chord with Elastic Ends…..…………………………………........12 Ends…..…………………………………........12

Figure 3.1

Finite Element Idealization of the Top Chord as a Bar on Elastic Supports…….19

Figure 3.2

Finite Element Idealization of the Pony Truss……………………… Truss…………………………………...21 …………...21

Figure 3.3

Stress-Strain Stress-Strain Curve…..……………………………………… Curve…..………………………………………………………......22 ………………......22

Figure 4.1

Compression Compression Chord Design Curve……..…………… Curve……..……………………………………......23 ………………………......23

Figure 4.2

Energy Method Design Curve ……....………………………………………. ……....…… …………………………………......24 .....24

Figure 4.3

2-D Compression Chord Elements.....…….…………………………… Elements.....…….…………………………………......29 ……......29

Figure 4.4

3-D Compression Chord Elements………………..……………… Elements………………..…………………………......30 …………......30

Figure 4.5

Nonlinear Load vs. Displacement Curve.……….. Curve.………. . ………………….………. ………………….………...... .....30 30

Figure 4.6

Pony Truss Top Chord Analysis……..……………………… Analysis……. .……………………………………….... ………………......31 ..31

Figure 4.7

Rigid Frame Boundary Conditions.…..……… Conditions.…..……………...…………………… ……...………………………......33 …......33

Figure 4.8

Rigid Frame Displacements…..……...………………… Displacements…..……...………………………………………......33 ……………………......33

Figure 4.9

Load Application of Pony Truss Frames…..……...…………… Frames…..……. ..………………………….... ……………......34 ..34

Figure 4.10

Pony Truss Lateral Displacement at Frame.…………………………………... Frame.…………… ……………………......35 ...35

Figure 4.11

Compression Compression Chord with New Stiffness………………………..………… Stiffness………………………..……………......37 …......37

Figure 4.12

Compression Chord with Elastic Ends…………………………… Ends……………………………………….... …………......37 ..37

1

1. ������������ ��� ��������� 1.1 ������������ Lateral stability of steel members under compression has been of interest to researchers for years. Among these members: columns under axial compression load, unbraced compression flange of steel girders, and the top chord of a pony truss for which vertical clearance requirements prohibit direct lateral bracing. The pony truss, while no longer used in constructing new bridges, may find applications in similar situations such as a walkway for a conveyer system between grain elevators. The structural behavior of the previously listed members has been studied by several researchers. In the following chapter, the behavior of an axially loaded bar and the top chord of a pony truss are briefly summarized. The calculation of the critical load for a pony truss top chord using published relations has been examined and compared to the results obtained using an analytical method. The compression chord of the pony truss structure, where vertical clearance prohibits lateral bracing, is elastically supported in the horizontal plane by the truss vertical and diagonal web members, which together with the floor beams form rigid frames as show in Fig. 1.1c.

n Panels

��

Fig. 1.1a Pony Truss Elevation View A

Fig. 1.1c Section A-A A Fig. 1.1b Pony Truss Plan View

To analyze the compression chord of a pony truss, the chord can be treated as a bar on elastic supports (Ballio, 1983). This member with intermediate elastic restraints will buckle in half-waves depending on the stiffness of the elastic restraints. The buckled shape of the bar will fall somewhere between the extreme limits of a half-wave length of unity and the number of

2

spans between the end restraints. From the buckled shape, the effective length of the compression chord can be used to determine the critical load. The method on how to determine the effective length has long been the focus of compression chord buckling. The failure of several pony truss bridges at the end of the nineteenth century prompted the research of compression chord buckling. Engesser (sited in Galambos, 1988) was one of the first researchers to investigate the problem and develop an approximate formula to determine the required stiffness for the elastic restraints that corresponds to a specified effective wave length, k.

Engesser’s approach for determining the stiffness of the elastic restraints and its effects on

the compression chord was based off the assumption that the connection between the web members and the floor beam is rigid. This theory used the frame consisting of the floor beam and vertical and diagonal members at each panel point location to provide stiffness for the compression chord. However, the theory in question is if the idealized structure is a conservative approach of the actual frame stiffness. In other words, one may argue that investigating the behavior of the bridge as a three dimensional system may result in a higher stiffness coefficient of these lateral supports. To the writer’s knowledge, all of the research for determining the critical buckling load on a compression chord with elastic supports is based on Engesser’s assumption. From this assumption, others, such as Timoshenko (1936), Bleich (1952) and Holt (1952), provided methods of solving for the effective buckling length factor, k.

1.2 ���������� The objective of the work presented herein was to verify the results of the published solutions for determining the effective length factor using the finite element method. These objectives were accomplished by performing the following tasks: 1. Conduct a literature search to review available information that is related to the stability of the top chords in truss structures. 2. Verify the results of analyzing a top chord of a pony truss using the approaches given in published literature and the results obtained using the finite element method. 3. Recommend the most applicable published analysis technique for determining the critical load of an unbraced top chord of a truss system.

3

2. ���������� 2.1 �������� �������� The failure of an axially loaded bar in compression is defined by limit states which are an identifying condition of design criteria. Limit states for a structural member include strength limit states, which may result in yielding or rupture, or serviceability limits states (i.e. deflection, vibration, slenderness or clearance). Although not a limit state, buckling presents a failure mode due to high compressive stresses which causes the member to no longer be in equilibrium. Usually buckling occurs before the column reaches the full material strength. The buckling strength of compression members has long been studied to relate the empirical methods of analysis to the actual results. The elastic buckling of an axially loaded column in compression occurs when a certain critical load is reached causing the member to suddenly bow out. The deviation of the member axis will result in additional bending that gives rise to large deformations, which in turn cause the member to collapse. The load at which collapse occurs is referred to as the buckling load and is thus a design criterion for compression members. In linear mechanics of deformable bodies, displacements are proportional to the applied loads. The essence of buckling, however, is a disproportionate increase in displacement resulting from a small increase in load. For example, Fig. 2.1, from Brush (1975), shows the loaddisplacement relation (referred to equilibrium path) for an axially loaded column. Each point of this path represents an equilibrium configuration of the structure. However, as the applied load reaches a critical value, i.e., Euler Load, the equilibrium path will follow the secondary path shown in Fig. 2.1. Points along the primary (vertical) equilibrium path represent the configuration of a compressed, perfectly straight column, but as the critical load is reached, a secondary path is formed representing the bent equilibrium configurations. The critical load is defined as the minimum load for which the structure remains in equilibrium before instability is reached and failure occurs. Of course, no real column can be perfectly straight, and hence the load displacement relation will not follow that shown in the Fig. 2.1 but rather a different load displacement will be obtained. The load displacement relationship of an imperfect axially loaded column is shown in Fig. 2.2. When comparing Fig 2.1 and 2.2 for the straight and crooked columns, the figures show that the equilibrium paths generally converge as the lateral

4

displacements increase. Analyses for both columns, straight and slightly crooked, lead to large lateral displacements at the critical load. �

=

��

=

�

��������� ���� ������� ���� �Δ

Δ �� � � �/�

+Δ

Fig. 2.1 Equilibrium Paths for Initially Straight Column

Δ �� � � �/�

Fig. 2.2 Equilibrium Paths for Slightly Crooked Column

2.2 ����� �������� Buckling of axially loaded bars in compression was investigated by Leonhard Euler in 1744. His work was based on a straight, prismatic, concentrically-loaded column with pin-ended connections. In his work, Euler stated that if the applied load, P, was less than the critical value, the bar remained straight and underwent only axial compression. By that definition all fibers would remain elastic until buckling occurred. According to Salmon (2009), Euler’s formula was not widely accepted initially since the test results on columns did not agree with his theory. The discrepancies, however, were due to the fact that the elastic limit was exceeded before the elastic buckling was attained. Euler’s formula was finally validated in 1889 when Considère and Engesser independently published works showing that one must use the tangent elastic modulus, Et, to account for the fibers beyond the proportional limit. For the reader’s interest, the following summarizes the derivation of the Euler buckling load. Figure 2.3 illustrates the deflected shape on an axially loaded bar. The bending moment, M, at a distance, x, can be related to the curvature as follows:

= =

(2.1)

5

�

� � �/�

Δ

�

�/�

� Fig. 2.3 Euler Buckling (Thandavamoorthy 2005)

The solution for the linear differential equation above can be written as

= s i n + c o s / 0 = sin 0 = sin = / =

where, K, is equal to

(2.3)

. The constants A and B in Eq. 2.3 can be calculated utilizing the

support conditions at both ends of the bar. For a pinned-end column, the boundary conditions can be set as y = 0 at x = 0 and y = 0 at x = L. These conditions will result in: B=0

and

(2.4)

Equation 2.4 can then be written as

By substituting K =

(2.5)

into equation 2.5 and solving for P, Euler buckling equation yields

the critical buckling load.

(2.6)

6

where m = 0, 1, 2…… is referred to as the number of buckling modes. The deformed shapes for the first three buckling modes are shown in Fig. 2.4.

���

����

���

����

���

���

Fig. 2.4 Buckling Modes for a Bar with Pin Ends

2.3 �������� �� ���� �� ������� �������� A bar supported by rigid supports at the ends with equally spaced elastic restraints between the ends can have several modes of buckling depending on the stiffness of the supports. If the stiffness of the elastic support is sufficiently large, the bar will buckle in half-waves of a length equal to the distance between supports as shown by Fig. 2.5a. The bar will then behave similar to a bar on rigid supports. However, if the elastic supports are very flexible, then bar will behave similar to a bar not supported by restraints and deflect in one-half wave as shown by Fig. 2.5b. As the elastic stiffness varies between the two extreme limits, the bar will buckle somewhere between one half wave and the number of spans between the rigidly supported ends such as Fig. 2.5c. Therefore, the stiffness of the elastic supports that are provided by the vertical and diagonal members of a pony truss is vital in controlling the buckling load and the buckling length of the top chord.

7

(a)

P

P

P

P

(c) P

P

(b)

Figure 2.5 Elastically Supported Bar (Bleich, 1952)

2.4 �������� �� ��������� ��� ����� ����� ������� As mentioned above, the compression chord from the unbraced top chord in a steel truss, such as a pony truss, can be idealized for buckling analysis as a continuous beam that is braced by elastic springs, which correspond to the stiffness of the transverse frames at each panel point. Therefore, design of the transverse frames formed by the web members and floor beams will have a direct effect of the critical buckling load of the chord members. AASHTO section 6.14.2.9 (2007) addresses these issues and gives recommendations on the design of the vertical web as well as the connection to the floor beam. However, unless one considers the effect of imperfections of the compression chord, the calculated critical load is an upper limit. The following sections summarize some of the published work that is related to the analysis of the unbraced top chord of steel trusses.

2.4.1 �������� ��������� �� ��������

The analysis proposed by Engesser in the late 1800’s can be applied with some reasonable accuracy to analyze a bar that is pinned at its ends and is supported on equally spaced intermediate elastic springs provided that the half-wavelength of the buckled shape is at least 1.8 times the spring spacing (See Galambos 1988). However, one must realize that Engesser’s solution can only be used as a preliminary design tool and more comprehensive analysis is needed. Engesser examined the top-chord buckling problem of pony trusses and summarized his findings in a paper that was published in 1884. In the following years, he used his work to explain the failures of pony truss bridges and provide a rational method of design for similar structures. Engesser developed a simple formula to calculate the required stiffness, C req, of the

8

elastic support to reach the desired critical load that is based on a specific buckling length. In his work, Engesser suggested that one needs to assume an effective length factor, k, of 1.3. The top chord, including the end posts, is straight and of uniform cross section. Engesser also provided the following assumptions: 1. Its ends are taken as pin-connected and rigidly supported. 2. The equally spaced elastic supports have the same stiffness and can be replaced by a continuous elastic medium. 3. The axial compressive force is constant through the chord length. Engesser’s solution for the required stiffness of a pony-truss transverse frame which is derived in Appendix C is

=

(2.7)

If Creq is met at each frame location, the chord with the length between panels, l, will achieve the specified design load, P cr. Several researchers suggest that one needs to assume a factor of safety, v, of two when calculating the design load. In other words, the load P cr, can be taken as vP, where P is the calculated top chord member load. In addition, once the calculations show that the stress induced in the member exceeds the limit specified in the design specifications, the flexural rigidity EI should be modified using the tangent modulus, E tI. By combining Euler’s buckling equation and Eq. 2.7, the required spring constant is

=

(2.8)

The use of Engesser’s original approach in design is summarized as follows: 1. Carry out a structural analysis to calculate the maximum load in the top chord members. 2. Introduce a factor of safety not less than 2.0 and calculate the design load, P = v*load from step one above. 3. Use an admissible structural analysis technique to calculate the elastic constant, C, of the provided lateral restraint.

9

4. Utilize the information calculated above to estimate the ratio, Cl/P. One approach that can be used to calculate the provided lateral supports stiffness, C, is detailed in Appendix A. 5. Use the above calculated ratio and the number of panels, n, to calculate the effective length factor, k, using Table 2.2 from Holt (1956) or Fig. 4.1 citing other authors. 6. Apply k, found in step five, to the equations found in Chapter E of the AISC (2011) manual to determine the nominal compression capacity of the compression chord.

2.4.2 �������� ���� ����� ��� ������ ������

Similar to the approach for a simply supported column, a column simply supported at both ends with equally spaced interior elastic supports, as shown in Fig. 2.6, can be defined by an equation which represents the buckled shape. The total length, L, is the number of bays, n, multiplied by each bay length, l. Using the energy method, the deflected shape can be defined by the equation

= sin

(2.9)

where m is the mode number, and the number of modes can be related to n-1. Using the sin curve, the boundary conditions are y = 0 at x = 0 and y = 0 at x = L.

(a)

P

A

B

P

L = nl (b)

P

P

Figure 2.6 Column on Elastic Supports

To solve for the external work and internal energy of the member, the first and second differential equations for the line can be solved as

′ = cos

(2.10)

10

′′ = sin

(2.11)

With P as the axial load, the potential energy of the system can be set equal to the external work, V, plus the internal energy, U to find the critical buckling load. A full derivation of a bar on two elastic springs using the energy method is presented in Appendix B. Since the energy method uses an assumed buckled shape for the chord, the solution is obtained by some degree of approximation. If the assumed shape is properly chosen to satisfy the boundary conditions, the energy method provides a satisfactory approximation. For the case presented in Fig. 2.6, the potential energy can be represented as

= + = ′ + ′′ + ∑

(2.12)

The final term in equation 2.12 represents the energy of the elastic supports as a function of the spring constant C. By solving for P, the critical buckling can be found as a function of the mode number.

2.4.3 �������� �������� ���� �������� ����� ����

Timoshenko (1936) extended the work of Engessers’ to include the effects of a varying the axial load along the top chord of a truss structure. He assumed the compression load varies parabolically along the length of the chord with the load equal to zero at the ends then reaching a maximum value at the center (see Fig. 2.7). In addition, Timoshenko’s solution assumed that the ends were pin connections.

� �

�

�

Figure 2.7 Varying Axial Load Distribution (Timoshenko, 1936)

11

Similar to Engesser, Timoshenko assumed an equivalent elastic foundation, α, is related to the lateral support spring constant, C, and the distance between each lateral support, l, as

= = ∑ + ∑

(2.13)

The strain for the chord energy can be represented as

(2.14)

In his solution, Timoshenko also stated that if the bridge is uniformly loaded, the compressive forces that are transmitted to the chord by the diagonals are proportional to the distance from the middle of the bridge span as

= 1−

(2.15)

where x is the distance from the left support in the figure and q o is the maximum force of the axial load represented by

= = − −

(2.16)

The external work, V, can then be calculated using the information given above as follows

(2.17)

Substituting the information given in Eqs. 2.14 and 2.17 into the total potential energy relationship, one can then obtain the following:

=

(2.18)

Finally, since the elastic supports are treated as a continuous elastic medium, Eq. 2.19 can be developed to relate the critical buckling load to the effective length factor by combining Eqs. 2.8, 2.16 and 2.18.

=

(2.19)

12

2.4.4 �������� �� � ���� ����� ��� ����� ���� ������� ���� 2.4.4.1 �������� ��������� �� ������ (1952)

Bleich obtained his solution by using finite difference as an exact approach to quantify the buckling load of the chord. His solution was based on the ends being pin connections and equally spaced intermediate supports of equal rigidity. Bleich also assumed the chord had a constant moment of inertia and constant axial compressive force over the entire length. Later, Schweda extended Bleich’s results to include chords with elastic ends.

���

� � ��

�

� �� �

� ��

� � ��

�

���

���

�� �

Figure 2.8 Compression Chord with Elastic Ends (Bleich, 1952)

The theoretical exact solution proposed by Bleich for a chord supported on rigid ends was

where vP = Pc and

=

= = ϕ =

(2.20)

The stiffness of the chord is represented by ϕ which is equal to

(2.21)

To eliminate the tangent modulus, E t, which varies per the axial load, Euler’s buckling equation is substituted into equation 2.21, simplifying the stiffness to

ϕ =

(2.22)

For a chord with n spans in equation 2.20, there are n -1 different half-wave buckling configurations. As n increases, the spring constant required for an infinite number of spans, C∞, increases to a limiting value. Bleich (1952) showed that C n can be replaced by C ∞ for any

13

span where n is greater than six which corresponds to an error less than 1%. So, for trusses with more than six spans, equation 2.16 simplifies to

= = Φ

(2.23)

where Φ is given in Table 2.1 from Bleich (1952) and is valid for the elastic and plastic range of buckling. Table 2.1 Values of Φ in Eq. 2.23 (Bleich, 1952)

1/k

Φ

1/k

Φ

1/k

Φ

1/k

Φ

0.3

0.111

0.5

0.309

0.70

0.614

0.90

1.102

0.32

0.126

0.52

0.335

0.72

0.652

0.91

1.138

0.34

0.142

0.54

0.361

0.74

0.692

0.92

1.177

0.36

0.160

0.56

0.388

0.76

0.734

0.93

1.219

0.38

0.179

0.58

0.417

0.78

0.777

0.94

1.264

0.40

0.198

0.60

0.447

0.80

0.822

0.95

1.316

0.42

0.218

0.62

0.478

0.82

0.870

0.96

1.375

0.44

0.239

0.64

0.510

0.84

0.921

0.97

1.444

0.46

0.261

0.66

0.544

0.86

0.976

0.98

1.530

0.48

0.285

0.68

0.578

0.88

1.036

0.99

1.652

1.00

2.000

Since Bleich’s theory assumes a constant axial force, which is rarely the case in practice, the center bay of the chord should be designed with the appropriate k value for the maximum load and then used for the remaining bays. Although the previous assumption of designing the bridge for the center span only would yield conservative results, the assumption of rigid ends can result in unsafe buckling loads when using Bleich’s theory. Thus, in order to continue on Bleich’s exact buckling theory, Schweda provides results to determine the required stiffness for a chord supported elastically on the ends. The spring constant of the end supports is denoted by C e and the intermediate supports by C. Schweda assumed the load was a constant axial force throughout the length of the

14

chord with equally spaced elastic supports similar to Bleich’s theory. Figure 2.8 shows the compression chord with the diagonals extended a length, d , subjected to the compressive force vPd and pinned at points –n and +n. The spring constant must be larger the spring constant of

the chord with rigid end, C 0. Thus, the value of C = cC0 where c > 1.1. The required spring constant with the axial load in the diagonal is

= +

(2.24)

where CE is Engesser’s equation (Eq. 2.8). Schweda calculated ψ as a function of the number of bays with respect to c and k . These values are listed in Appendix D. If all of the crossframes are identical, C e in equation 2.24 is equal to C and the equation becomes

=

(2.25)

2.4.5 �������� �� � ���� ����� ��� ����� ��������� �� ����

Holt’s research in the 1950s tested pony truss bridges in an attempt to compare the actual buckling load of the compression chord with the design equations. His research not only tested the primary constraints mentioned above but also the effects of secondary factors. The following secondary factors were considered in his research (Holt, 1956): 1. Torsional stiffness of the chord and web members. 2. Lateral support given to the chord by the diagonals. 3. Effect of web-member axial stresses on the restraint provided by them. 4. Effect of non-parallel-chord trusses. 5. Error introduced by considering the chord and end post to be a single straight member. The results of Holt’s analysis proved that the error in determining the critically buckling load by neglecting the above factors was relatively small. His conclusion stated that the load capacity of a pony truss bridge would be satisfactorily predicted by previous buckling analyses mentioned. As shown by Bleich (1952), if the truss has at least ten panels, then the effective length factor depends only on the stiffness of the transverse frames. Thus, the appropriate

15

effective length factor is a function of Cl/P c which is shown in the results section. Holt noted that Bleich’s analysis showed adequate results for the entire range of effective length values where Timoshenko’s results show adequate results for k > 2. A summary of Holt’s results can be seen in Table 2.2. Based on the results of his research, Holt (1957) also recommended the following on the design of the end posts The end post should be designed as a cantilever to carry, in addition to its axial load, a transverse force of 0.3% of its axial load at its upper end.

Table 2.2 (Galambos, 1988)

1/k 1.00 0.980 0.960 0.950 0.940 0.920 0.900 0.850 0.800 0.750 0.700 0.650 0.600 0.550 0.500 0.450 0.400 0.350 0.300 0.293 0.259 0.250 0.200 0.180 0.150 0.139 0.114 0.100 0.097 0.085

4 3.686

3.352 2.961 2.448 2.035 1.750 1.232 0.121 0

1/k for Various Values of Cl/ P c and n n 6 8 10 12 3.616 3.660 3.714 3.754 3.284 2.944 2.806 2.787 3.000 2.665 2.542 2.456 2.595 2.754 2.303 2.252 2.643 2.146 2.094 2.593 2.263 2.045 1.951 2.460 2.013 1.794 1.709 2.313 1.889 1.629 1.480 2.147 1.750 1.501 1.344 1.955 1.595 1.359 1.200 1.739 1.442 1.236 1.087 1.639 1.338 1.133 0.985 1.517 1.211 1.007 0.860 1.362 1.047 0.847 0.750 1.158 0.829 0.714 0.624 0.886 0.627 0.555 0.454 0.530 0.434 0.352 0.323 0.187 0.249 0.170 0.203

14 3.785 2.771 2.454

16 3.809 2.774 2.479

2.254 2.101 1.968 1.681 1.456 1.273 1.111 0.988 0.878 0.768 0.668 0.537 0.428 0.292 0.183

2.282 2.121 1.981 1.694 1.465 1.262 1.088 0.940 0.808 0.708 0.600 0.500 0.383 0.280 0.187

0 0.135 0.045 0

0.107 0.068

0.103 0.055

0.121 0.053

0.112 0.070

0.017 0

0.031

0.029

0.025

0.003 0

0.010

0

0

16

2.4.6 �������� ���� ���� ������� ������������ ������������

Initial out-of-plane deformations of the compression chord can reduce the critical buckling load determined by the previously mentioned methods. There are two primary causes of out-ofplane deformations that need to be taken into consideration. A vehicle load on the floor beams would cause a displacement of the chord at the location of the load creating initial lateral displacements in the chord. The chord could also have initial crookedness and unintentional eccentricities due to manufacturing. Such lateral deflections would reduce the maximum load capacity of the chord. Lutz and Fisher (1985) addressed this issue in their publication to the Structural Stability Research Council in 1985. Their work was similar to the stiffness criteria George Winter proposed in 1960. Winter (1960) proposed the ideal stiffness, C, needed to fully brace the compression member over the length, l, is equal to ideal stiffness

where the stiffness required, C req, is usually twice the

. Lutz and Fisher used Engesser’s formula for a perfectly straight

compression chord and developed a factor of safety to account for the out-of-plane stiffness. The stiffness equation by Engesser was

2

where Le = k l and π /4

≅

= 2.5

(2.26)

2.5. For trusses with a small l relative to L e equation 2.26 provides an

accurate bracing stiffness for the solution. However, as l increases relative to L e, then equation 2.26 will result in unsafe errors. Thus, Lutz and Fisher proposed the following empirical equation to determine the required stiffness

= 2.5 +1.5

(2.27)

Then they extended the applicability of k factors to less than 1.3 going as low as 1.0. The fully braced case where k = 1 corresponds to Winter’s stiffness of

. Equation 2.27 provides a

minimum value of stiffness for the compression chord to reach the required critical load however; there are currently no design procedures available to account for initial imperfections. Design recommendations by AASHTO only make a note of the design vertical truss members and the connection to the floor beam.

17

2.5 ���� ����� ������ ��������� �� ������ �������������� AASHTO Specifications (2007) for the LRFD design of half-through trusses recommends design loads for both the top chord of the truss and the web verticals. AASHTO states in section 6.14.2.9 that The top chord shall be considered as a column with elastic lateral supports at the panel points. The vertical truss members and the floor beams and their connections in half-through truss spans shall be proportioned to resist a lateral force of not less than 300 pounds per linear foot applied at the top-chord panel points of each truss considered as a permanent load for Strength 1 Load Combination and factored accordingly. By applying the appropriate vehicle or other live load cases to the truss, the floor beam can be designed. The floor beams, in addition to the vertical truss members designed with the 300 plf applied load, provide the elastic lateral supports at the panel points.

18

3. ������ ������� �������� The following investigation focused on validating the published results for calculating the effective buckling length factor, k, that can be determined by the methods mentioned in the previous chapter. The example used for the analysis was studied by Galambos 1988 (see Fig. 3.2) in his book. The top chord of the truss consisted of a 10”x10”x5/8” box section that was designed for a maximum compressive force of 360 kips. The vertical and web members were composed of a W10x33 sections in addition to the floor beams for the rigid frame, which were W27x84 sections. The Young’s modulus for the members was assumed to be 29,000 ksi. The investigation by Galambos was carried out assuming a factor of safety of 2.

3.1 ������ ������� ����� �� ��� ����������� ����� The buckling analysis that is presented in this paper was conducted using the ANSYS 12.1 general purpose finite element program. ANSYS is a commercial engineering software that is capable of analyzing different engineering properties on the structure very quickly with a host of different elements available. Using ANSYS, the nonlinear material properties of the compression chord can also be investigated. In the following paragraphs, the element type and why it was used will be explained in more detail. A beam3 element was used in the compressions chord model and is a uniaxial element with tension, compression, and bending capabilities. Each node has three degrees of freedom: translations in the x and y axes and rotation about the z axis. This element is a 2-D element, which provided the analysis for the basic compression chord case. These results were compared to the results for a 3-D element, beam4, to show the adequacy of model. The beam4 element is similar to the beam3 element except the beam4 element has six degrees of freedom: translations in the x, y and z axes and rotations about the x, y and z axes. Stress stiffening and large deflection capabilities are also included. When using the beam4 element, the translations in the z direction must be restrained to simulate the actual buckling properties of the compression chord since the chord has considerably more stiffness against buckling in the z direction. The combin14 element is an element with no mass and was used as the spring in all of the analysis. This combination element has longitudinal or torsional capabilities in 1-D, 2-D or 3-D applications. The longitudinal spring-damper option, which was used in this analysis, is a

19

uniaxial tension-compression element with up to three degrees of freedom at each node: translations in the nodal x, y and z directions. When analyzing the nonlinear properties of the compression chord, the beam23 element was used. The beam23 element is a uniaxial element with tension-compression and bending capabilities which also has plastic, creep, and swelling capabilities. This element has three degrees of freedom at each node: translations in the x and y directions and rotation in the z direction. The element is defined by the area, moment of inertia, and height. A more in-depth description of each element is available from ANSYS 12.1 (SAP Inc., 2009).

3.2 �������� �� ��� ������ �� � ��� �� ������� �������� Figure 3.1 shows the boundary conditions of the 2-D model, which is supported in the x and y directions at the base and only in the x direction at the top to simulate the pin and roller connections. A beam3 element was used in the compressions chord model and is a uniaxial element with tension, compression, and bending capabilities.

Fig. 3.1 Finite Element Idealization of the Top Chord as a Bar on Elastic Supports

20

3.3 ������ ������� ����� �� ��� ���� ����� The pony truss described above was also modeled, as shown in Fig. 3.2. The model was composed of beam4 (3-D beam elements) and 3-D link8 elements. Link8 is a 3-D truss element with three translation degrees of freedom at each node. The diagonal members were modeled using these 3-D truss elements. This was done in order to compare the loads with the analysis mentioned above in chapter 2 since those authors neglected the effects of the diagonal web members. The truss was restrained in the x, y and z directions on one end and only the y and z directions on the other end for pin and roller connections as seen in the Fig 3.2. 3-D beam elements were used to model the top and bottom chords of the truss structure. 4

The moment of inertia for the 10x10 box section was calculated to be 418.3 in . By rearranging Eq. 2.7, the tangent modulus of elasticity can be calculated as 7344 ksi for the chord. This tangent modulus is used in the nonlinear model. The ANSYS program allows the user to carry out a nonlinear buckling analysis. In the work presented herein, the analysis was carried out considering the effects of the nonlinear material behavior only. For this purpose, the user needs to provide the stress-strain relationship of the material. In the nonlinear model of the compression chord, the material nonlinearity was modeled using a multilinear isotropic hardening option (MISO) for the material with the stressstrain profile as shown in Fig. 3.3. The proportional limit was specified as per the AISC manual as 0.4Fy, and the yield value for this model was 36 ksi. The tangent modulus was found above using Engesser’s equation and varies as the critical load changes. Notice that in Fig. 3.3, the portion as the material reaches yield was defined using very small slope. This was necessary to avoid overshooting and any problems that may cause non-convergence to the solution.

21

Fig. 3.2 Finite Element Idealization of the Pony Truss

The nonlinear solution of the compression chord in ANSYS used the Newton-Raphson option to converge on the displacements of the solution. An initial load larger than the predicted buckling load was applied to the chord and ANSYS then uses load steps to continuously apply the load in small increments to iterate the solution. For each iteration, the program calculates a new element stiffness matrix based on the element strains in the stress-strain profile provided. For the nonlinear model in this research, it was assumed that a converged solution was reached when the difference in displacements between load steps was equal to or less than 0.1%. The results of analyzing the truss described above using the different available analyses techniques are summarized in the following chapter.

22

������������� ����� �0 �� �0 � � � � � � � � � � �

��

�� � ���� ��� �0 �� �0 �

� � ��000 ���

0 0

0.00� 0.00� 0.00� 0.00� 0.00� 0.00� 0.00� 0.00� 0.00� 0.0�

������ �������

Fig. 3.3 Stress-Strain Curve

23

4. ���������� ��� ������� �� ��� �������� �� � ���� ����� ��� ����� 4.1 ��������� �������� ������ ������ � ������� ������� ��������� ������������� For the comparison of the equations in chapter two with the FEM, the equations needed to be rearranged for the calculation of the effective length factor, k. This factor is required to check the capacity of the top chord member following the recommendation used in AISC compression calculations. The relationships of the suggested procedures that can be used in calculating the effective length factor are shown in the Figures 4.1 and 4.2 below. Figure 4.1 relates the inverse of the effective length factor to the stiffness of the lateral support, and Fig. 4.2 depicts the energy relation between the stiffness of the lateral elastic support and Euler’s load, P e, to the critical compressive load, P cr, of the member.

��������� ������ ������� �.�00

�.000

0.�00

�������� � � �

0.�00

������ ����������

0.�00

�����������

0.�00

0.000 0.000

0.�00

�.000

�.�00

�.000

�.�00

�.000

�.�00

�.000

����

Fig. 4.1 Compression Chord Design Curves

�.�00

24

�������� �������� ���� ��0

�00

�0 � � � � � �

�0

�0

�0

0 0

�00

�000

��00

�000

��00

�000

�������

Fig. 4.2 Energy Method Design Curve

4.2 ������� ������������ ��� ��� �������� ���� �� � ���� ����� Given below are the calculations that were used to develop the buckling load of the truss structure that was examined by Galambos (1988). The dimensions of the truss structure were previously listed in chapter three of this document. In this example, a structural analysis of the truss system illustrated that the top chord member being investigated was subjected to a compression force of 360 kips. A factor of safety of 2, as suggested in the literature, was used in the analysis. From the specified truss geometry, the provided lateral spring stiffness, C, was calculated using Eq. A1.1 as

C = = ∗ ∗ = 6.75 kip/in

25

Then, the ratio of C l /P was determined as

= .∗ = 1.50

Using the calculated ratio C l /P in conjunction with Fig 4.1, one can calculate the value of 1/k according to the different analysis techniques that were previously summarized. 4.2.1 ������������ ��������� ���������� ��������� 1/k = 0.78 (From Fig. 4.1) k = 1.28

= ..∗ = 50.3

Following the design steps that are given in the AISC (2011) manual, one can calculate the critical buckling stress, F e, and the nominal buckling load, P n, using the ASD and LRFD approaches, respectively as

F = = . = 113.2 ksi

F = 0.658 F = 0.658 . 36 = 31.5 ksi Hence, the critical buckling load is:

P = FA = 31.5∗25.0 =

This procedure is then repeated using the other methods mentioned in chapter two. 4.2.2 ������������ ��������� �������� ��������� 1/k = 0.768 (From Fig. 4.1) k = 1.30

= ..∗ = 51.1 F = = . = 109.8 ksi

F = 0.658 F = 0.658 . 36 = 31.4 ksi

26

P = FA = 31.4∗25.0 =

4.2.3 ������������ ��������� ������������ ��������� 1/k = 0.604 (From Fig. 4.1) k = 1.66

= ..∗ = 64.9 F = = . = 67.9 ksi

F = 0.658 F = 0.658 . 36 = 28.8 ksi

P = FA = 28.8∗25.0 =

4.2.4 ������������ ��������� ���� ��� �������� ��������� 1/k = 0.719 (From Fig. 4.1) k = 1.39

= ..∗ = 54.5 F = = . = 96.2 ksi

F = 0.658 F = 0.658 . 36 = 30.8 ksi

P = FA = 30.8∗25.0 =

4.2.5 ������������ ��������� ������ ��������� 1/k = 0.75 (From Table 2.2) k = 1.33

= ..∗ = 52.3 F = = . = 104.7 ksi

F = 0.658 F = 0.658 . 36 = 31.2 ksi

P = FA = 31.2∗25.0 =

27

4.2.6 ������������ ����� ��� ������ ������

P = = ∗∗. = 11.84 kips = ..∗ = 912.2 = 61.67 P = PR = 11.84∗61.67 = (From Fig. 4.2)

A summary of these results is shown in Table 4.1. As noted in Galambos (1988) the factor of safety for the compression chord on elastic supports was 2.0 when determining the C l/ P ratio. However, it was later addressed that the compression chord in Galambos example was designed for the maximum panel load using AASHTO’s 1983 formula which uses a factor of safety of 2.12 (Ziemian, 2010). In essence, the results should be compared to the design load of P=360*2.12 = 763 kips. The factor of safety listed in this table was calculated as the ratio between the estimated critical buckling load and the applied member load of 360 kips.

Table 4.1 Critical Load Results Method Engesser Bleich Timoshenko Lutz & Fisher Holt Energy Method

k 1.28 1.30 1.66 1.39 1.33 -

Pn 788 785 721 770 779 730

F.S 2.19 2.18 2.00 2.14 2.16 2.03

The results above show the methods of analysis in chapter two reasonably predict the critical load of the compression chord for the pony truss example. However, it may be noted that using the energy method underestimates the critical buckling load since it is based on an assumed deformed buckling shape.

28

4.3 �������� �� ��� ���� ����� ����� ������ ������� 4.3.1 ��� ����������� ��������

To verify the critical load calculations of the compression chord, the truss was analyzed in ANSYS to solve for the critical buckling load. As mentioned in chapter three, the validity of these tests was checked with both a 2-D and 3-D element model. Using a 2-D beam3 element with the tangent modulus, E t = 7344 ksi, the critical buckling load was calculated to be 719.213 kips as shown in Fig. 4.3. The 3-D element analysis of the compression yielded the exact same buckling load of 719.213 kips (Fig. 4.4) validating the model. The compression chord buckling load was also determined using a nonlinear approach. This buckling load was found by plotting the Load vs. Displacement in the vertical direction and then noting the load at which large displacements occur with only a small increase in load. Using Fig. 4.5 the buckling load in the nonlinear model was equal to 720 kips. As seen in the graph, there are two changes in slope which account for the change in modulus at the proportion limit and the yielding limit. All of the FEM solutions prove the reasonability of the calculations determined in the previous section with C = 6.75 and E t = 7344 since the three compression chords analysis all had approximately the same buckling load of 720 kips. The next step of the analysis was to check the critical load on the compression chord when the entire truss was modeled.

29

Fig. 4.3 2-D Compression Chord Elements

Fig. 4.4 3-D Compression Chord Elements

30

Fig. 4.5 Nonlinear Load vs. Displacement Curve

4.3.2 ����� ����������� ��������

The pony truss model was analyzed similar to the compression chords in that a compression load was applied to the top chord on each side of the truss. In order to only focus on the lateral displacements of the compression chord, the truss was restrained in a manner to create two symmetrical sides of the truss. To accomplish this symmetrical model, the truss was restrained against translation in the z direction and rotations in the x and y directions at the center of the floor beams. By applying these boundary conditions, the resultant load could be compared to the compression chord models. The critical buckling load, when modeling the entire pony truss, increased dramatically to a load of 1121 kips on the compression chord which can be seen in Fig. 4.6. It is the opinion of this author that the increase in critical buckling load is due to the effect of inertia from the compression chord. The spring stiffness, when analyzing the compression

31

chord by previous work, was composed only of the frame stiffness as calculated in Appendix A. However, the lateral stiffness of the chord itself provides an addition stiffness which explains the significant increase in load when modeling the entire truss. The goal in the preceding paragraphs is to analyze this theory using finite element models.

Fig. 4.6 Pony Truss – Top Chord Analysis

32

4.4 ������� �� ����������� ����� ������ �� ������� �� ��� ��������� �� ��� ������� �������� When calculating the elastic stiffness restraining the compression chord from buckling in Appendix A, the stiffness is a product of the rigid frame only and ignores the contributions of the stiffness from the top chord. This discrepancy was examined using the finite element method. To analyze this theory, several models of the truss structure considering different moments of inertia of the top chord were analyzed by applying two lateral loads at the panel points as seen in Fig 4.9. The results of this analysis were then used to calculate the stiffness of the lateral supports. In order to determine if the inertia of the compression chord has an effect on the stiffness of the compression chord, the frame itself was analyzed as a control procedure before examining the entire truss. The frame was analyzed as a simply supported structure with a load of 1 kip placed on each vertical web as seen in Fig. 4.7. The load causes a displacement of 0.148014” (Fig. 4.8) which can be used to find the stiffness of the frame by taking the load divided by the displacement. Using the displacement of the frame to calculate the stiffness, a value of C = 1/0.148014 = 6.576 k/in is determined. This is the same stiffness that the equation from Appendix A yielded. The same procedure was then repeated except with entire pony truss modeled as a simply supported structure, which can be seen in Fig. 4.9. The load was applied at each frame location individually to get the stiffness at each restraint point. A summary of these results can be found in Table 4.2 which references the panel locations in Fig 4.9. The results in Table 4.2 were determined using a compression chord tangent modulus of 7344 ksi and the moment of inertia of 4

418.3 in . From this table, the compression chord seems to have an effect on the overall lateral stiffness against buckling. To prove this point, a transverse load was applied to the center panel, 4

4

similar to before, and the inertia of the compression chord was varied from 425 in to 0.001 in . The results of this process can be seen in Table 4.3. This table shows that the moment of inertia for the compression chord should be accounted for when determining the lateral stiffness at the frames locations and that the current determination used for the frame stiffness is an underestimation. In order to compare the procedures from the published works, the new stiffness should be used.

33

Fig. 4.7 Rigid Frame Boundary Conditions

Fig. 4.8 Rigid Frame Displacements

34

`

1 2

3 4 5 6 7 8 9 10 11

Fig. 4.9 Load Application of Pony Truss Fra mes

Table 4.2 Resultant Stiffness at Each Ba y Panel Location 1 2 3 4 5 6 7 8 9 10 11

Displacement, ∆ (in) 0.10612 0.07518 0.07249 0.07227 0.07226 0.07226 0.07226 0.07227 0.07249 0.07518 0.10612

Stiffness, C (k/in) 9.426 13.301 13.795 13.837 13.839 13.839 13.839 13.837 13.795 13.301 9.423

35

Fig. 4.10 Pony Truss Lateral Displacement at Frame Location

Table 4.3 Effects of Top Chord Inertia on the Lateral Stiffness 4

Moment of Inertia (in ) 425 325 225 125 25 0.001 418.3

Displacement, ∆ (in) 0.048816 0.052882 0.058840 0.069192 0.100502 0.141938

Stiffness, C (k/in) 13.89 12.98 11.87 10.45 8.08 7.04 6.75 (using Eq. A1.1)

4.5 �������� ���� �������� ������� ��������� The calculation for each of the methods in chapter 2 was redone using the new stiffness value, C = 13.8 found in the previous section, by completing the same procedure mentioned in section 4.2. Table 4.4 summaries the critical buckling loads of these calculations. Engesser’s approach was not valid for new analysis since his theory is based off a minimum k value of 1.3. With the new stiffness value, all of the methods underestimate the critical load of the

36

compression chord when analyzing the entire truss since the formulas do not account for the inertia stiffness of the compression chord and the increase in stiffness with respect to k is not a linear response.

Table 4.4 Critical Load Results with New Stiffness, C Method Engesser Bleich Timoshenko Lutz & Fisher Holt Energy Method

K NA 1.02 1.16 1.08 1.01 -

Pn NA 827 807 819 828 1032

F.S NA 2.30 2.24 2.28 2.30 2.87

Using ANSYS, the compression chord modeled with the new stiffness provided a large increase in the compression capacity. Figure 4.11 shows the compression chord modeled with rigid supports and the interior restraints having a stiffness of 13.8. The buckling load was found to be 979.03 kips. Although this load is closer to the pony truss analysis of 1121 kips, the end supports in this model were rigid which does not accurately account for the elasticity at these supports. The next figure, Fig. 4.12, shows the buckling shape with the elastic supports having the new stiffness values determined in Table 4.2. This buckled shape essentially shows the “exact” buckling load and shape of the compression chord treated as a single member.

37

Fig. 4.11 Compression Chord Analysis with New Stiffness

Fig. 4.12 Compression Chord Analysis with Elastic Ends

38

5. �������, �����������, ��� ��������������� 5.1 ������� Previously published work closely relates the critical buckling load of the compression chord to the actual load when the chord is taken as a single member with the stiffness provided according to Appendix A. However, the formulas in Appendix A ignore the contributions of the moment of inertia of the top chord. This discrepancy was examined using the finite element method. From the FEM, a new stiffness was calculated and used to calculate the new buckling load. A summary of these results can be seen in Table 5.1, which is compared to full pony truss model load of P = 1121 kips. This analysis shows that the omission of the top chord inertia when calculating the frame stiffness will cause an underestimation of the critical buckling load. It is evident that the capacity of the compression chord can be affected by including the inertia effects of the chord.

Table 5.1: Buckling Load Analysis Results Method Engesser Bleich Timoshenko Lutz & Fisher Holt Energy Method FEM Chord - Rigid Ends FEM Chord – Elastic Ends FEM Pony Truss

C = 6.75 C = 13.8 k Pn k Pn 1.28 788 NA NA 1.30 785 1.02 827 1.66 721 1.16 807 1.39 770 1.08 819 1.33 779 1.01 828 730 1032 719 979 644 886 Pcr = 1121 kips

39

5.2 ����������� The following are the conclusions that can be attained from the study presented herein: •

Current design of a compression chord for a pony truss bridge would be best accomplished by using the effective length factor, k, provided by Holt to determine the critical load from Chapter E in the AISC manual. However, such an effective length factor is dependent on the stiffness of the elastic lateral supports.

•

The energy method provides a close answer for calculating the critical buckling load for the truss top chords with rigid supports. However, its application depends on the number of the provided elastic supports.

•

The finite element method will provide satisfactory results when using the appropriate member rigidity, i.e., using the correct tangent modulus of elasticity.

•

All available procedures require the knowledge of the provided elastic stiffness. Hence, it is important to be able to calculate this stiffness factor. This means that one must consider the type of connection between the floor beams and the vertical as well as with diagonal members when calculating the lateral spring stiffness.

5.3 ��������������� Past testing on pony truss bridges is limited to Holt’s work (1957) which focused on the effective length factor on the compression chord and not the stiffness supplied by the frames. Empirical testing of a pony truss model could reveal a better understanding of the actual stiffness supplied by the compression chord inertia. Due to the complexity of a full pony truss and the necessity of empirical confirmation of any design model, testing of physical models is required before determining any definitive conclusions. •

Test a model to verify the effect of inertia on the compression chord stiffness.

•

Determine a method to verify the rigidity of the frame connection.

40

�������� � Calculations of the stiffness of the lateral support to a pony truss

A

h

B

Ic Ib

C

D

b The calculation of the stiffness for the lateral supports to the top chord of an unbraced truss can be calculated using the energy method. This can be accomplished by calculating the force, C, that is required to induce a unit displacement in the lateral direction at the panel point. For this purpose, the virtual work method was utilized. The following is the derivation of the spring constants of the lateral supports if: 1. The connection between the vertical and the floor beam is assumed rigid.

�

�

�

�

�� ��

��

Moment due to applied real load, C

� �

�

Moment due to virtual loads, Q, (Q was assumed = 1)

∗ = = ℎ ∗ℎ ∗ ℎ+ ℎ ∗ ∗ℎ

41

where, Q, is a virtual load that is applied in the horizontal direction at the points where the displacement is to be calculated. In these in calculations, a virtual load, Q, of unity was assumed. Notice that, ∆, is the relative displacement between points A and B. Therefore, to calculate the elastic constants, C, of the lateral spring, one needs to substitute a value of 2 for the displacement, ∆, in the equation above. This yields to:

Or;

∗ 1∗2 = + =

(A1.1)

3

2. To account for truss diagonals, a term of L d /3Id is added in the denominator where the additional stiffness is an addition to the vertical web.

=

(A1.2)

3. The effect if joints C & D are not rigid 1

1

A

D

θ k s B

k s

C

Where; k s is the joint rotational stiffness. The moment, M, at the vertical-floor beam connection is:

= = ∗ℎ = ∗ℎ/

Following the analysis that was summarized in the section above, the relative displacement between points A and D can be calculated as:

42

= 2 = ∗ + +2∗ ∗ℎ 1 = + + ∗ℎ 1 = + + 1 = + + =

(A1.3)

if k s = ∞, then it is a completely rigid connection

Notice that the above equations did not take into account the actual shape of the cross section of the top chord members. However, in most cases, the cross section of the chord consists of open, thin-walled sections having only one axis of symmetry, and hence the bending of the chord will be accompanied by twisting. Thus, the problem of bucking of the chord will be a caused by flexural and torsion. Therefore, the above relations for calculating the spring constant must be modified to account for such effects. This problem was studied in detail by Bleich (1952). Bleich determined that disregarding the torsion on the compression chord would lead to an unsafe design for members with only one axis of symmetry. Conversely, box sections, which have a high torsional rigidity, would underestimate the capacity of the chord.

43

�������� � Energy Method to determine the buckling load for a bar with 2 springs

1. Initial conditions

��

�

�/�

�

�/�

�/�

2. Displaced shape

� Δ�

y = asi n y = cos y = sin y = ∆= asin

Δ�

π

′

′′

π

π

π

π

π

Where:

and

3. External Work

V = y dx V = cos dx V = + sin ′

π

π

π

π

π

y = ∆= asin

π

44

V = π

4. Internal Energy

U = C∆ + C∆ + dx U = C∆ + C∆ + sin dx ′′

π

�=

π

π π π π π �� ��n ���n � − ��n

π

� = ��n ���n � π

π

5. Combine external work and internal energy and solve for P cr π

= � � �

π π π π π = � ��n ���n �

= �.�

∆π ∆

�.� =

π π ���n π � π �����n

π =

π

���n ���n �

�=

π

π

s i n ���n π

π �

π

(B1.1)

45

�������� � Engesser’s Method to determine the buckling load for a bar with 2 springs

1. Initial conditions

��

�

�/�

�

�/�

�/�

2. Displaced shape with continuous elastic medium

� Δ�

� = ���n π

�′ = ��� π

π

π

� = ��n π

3. External Work

y dx V = cos dx V = + sin V = �=

′

π

π

π

π

π

π

Δ�

46

4. Internal Energy

�=

ydx +

π π �x ��n

π

where: α = C/(L/3)

�= − ��n � − ��n π

�=

π

π

π

π �

5. Combine external work and internal energy and solve for P π

= � � �

π

= � �

π

π

= �.� �.� = π � � π ∆π

∆

�=

(C1.1)

+

6. Solve for required stiffness coefficient

P =

where: α = C/L and v = L/m (wavelength)

+

Note that Pcr will be at a minimum when

� = π C = �= −

= �

+

Substituting wavelength, v, back into the equation above:

(C1.2)

47

�������� � Schweda’s extension of Bleich’s analysis

= � −

sinh2 −1sin ±sinh sin2 −1 + 2ℎ 2 − ±

1 ±2 −1cosh = 2 −2+ = √ − = 1 + cos − − = 1− + 2cosh = +1 + + −1 + 2cos = +1 + + −1 + Where:

(D1.1)

48

Tables of Factor ψ: c

k

2n=6

1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5

1.2 1.00 1.25 1.41 1.39 1.35 1.65 2.01 2.41 2.79 2.98 2.93 2.69 2.27 1.99

1.3 0.78 0.93 1.04 1.06 1.10 1.30 1.54 1.77 1.97 2.12 2.10 1.97 1.76 1.71

1.4 0.65 0.75 0.84 0.87 0.94 1.09 1.25 1.40 1.55 1.64 1.66 1.60 1.48 1.48

1.5

1.6

1.7

1.8

1.9

0.82 0.94 1.06 1.18 1.29 1.36 1.38 1.35 1.27 1.32

0.94 1.03 1.11 1.17 1.19 1.17 1.12 1.17

0.90 0.97 1.02 1.04 1.04 1.00 1.06

0.85 0.90 0.94 0.94 0.91 0.96

0.85 0.83 0.89

1.5

1.6

1.7

1.8

1.9

0.81 0.90 0.99 1.06 1.08 1.08 1.20 1.32 1.46 1.58

0.87 0.92 0.95 0.96 1.05 1.16 1.26 1.36

0.85 0.87 0.94 1.03 1.11 1.20

0.85 0.92 1.00 1.07

0.90 0.97

c

k

2n=8

1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5

1.2 0.93 1.12 1.27 1.32 1.42 1.69 1.96 2.07 2.04 1.85 2.08 2.43 2.83 3.25

1.3 0.74 0.86 0.96 1.01 1.12 1.28 1.44 1.54 1.54 1.47 1.66 1.89 2.13 2.37

1.4 0.62 0.71 0.80 0.85 0.94 1.06 1.17 1.24 1.26 1.25 1.39 1.55 1.73 1.90

49

c

k

2n=10

1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5

1.2 0.99 1.06 1.21 1.26 1.43 1.64 1.74 1.72 1.85 2.14 2.42 2.66 2.74 2.65

1.3 0.75 0.83 0.93 1.00 1.11 1.24 1.32 1.34 1.46 1.63 1.80 1.92 1.99 2.00

1.4 0.63 0.70 0.78 0.84 0.93 1.03 1.09 1.13 1.22 1.34 1.45 1.55 1.61 1.63

1.5

1.6

1.7

1.8

1.9

0.81 0.88 0.94 0.98 1.05 1.14 1.23 1.34 1.36 1.39

0.83 0.87 0.93 1.00 1.07 1.14 1.18 1.21

0.83 0.89 0.95 1.01 1.05 1.08

0.86 0.91 0.94 0.97

0.86 0.89

1.5

1.6

1.7

1.8

1.9

0.80 0.87 0.92 0.98 1.05 1.12 1.17 1.20 1.27 1.36

0.82 0.87 0.93 0.99 1.03 1.06 1.12 1.19

0.83 0.88 0.92 0.95 1.00 1.06

0.84 0.87 0.91 0.96

0.82 0.87

c

k

2n=12

1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5

1.2 0.97 1.06 1.17 1.25 1.41 1.54 1.60 1.73 1.96 2.12 2.17 2.12 2.28 2.57

1.3 0.74 0.82 0.92 0.99 1.10 1.19 1.26 1.35 1.49 1.60 1.65 1.66 1.78 1.94

1.4 0.62 0.69 0.77 0.84 0.93 1.00 1.06 1.12 1.22 1.30 1.37 1.40 1.48 1.60

50

���������� American Institute of Steel Construction (AISC). 2011. Steel Construction Manual . American Institute of Steel Construction, Inc. American Association of State Highway and Transportation Official (AASHTO). 2007. AASHTO LRFD Bridge Design Specifications. AASHTO, Washington, D.C. Ballio, G. and Mazzolani, F. 1983. Theory and Design of Steel Structures. Chapman and Hall. New York. Bleich, F. 1952. Buckling Strength of Metal Structures. McGraw-Hill Book Company, Inc. New York. Brush, D.O. & Almroth, B.O. 1975. Buckling of Bars, Plates, and Shells. McGraw-Hill Book Company, Inc. New York. Galambos, T. 1988. Guide to Stability Design Criteria for Metal Structures 4 th Ed. John Wiley & Sons. New York. Holt, E. 1951. Buckling of a Continuous Beam-Column on Elastic Supports. Stability of Bridge Chords without Lateral Bracing, Column Res. Council Rep. No. 1. Holt, E. 1952. Buckling of a Pony Truss Bridge. Stability of Bridge Chords without Lateral Bracing, Column Res. Council Rep. No. 2. Holt, E. 1956. The Analysis and Design of Single Span Pony Truss Bridges. Stability of Bridge Chords without Lateral Bracing, Column Res. Council Rep. No. 3. Holt, E. 1957. Tests on Pony Truss Models and Recommendations for Design. Stability of Bridge Chords without Lateral Bracing, Column Res. Council Rep. No. 4. Lutz, L., and Fisher, J.M. 1985. A Unified Approach for Stability Bracing Requirements. AISC Eng. J. Vol. 22, No. 4. Salmon, C., Johnson, J. and Malhas, F. 2009. Steel Structures Design and Behavior 5 Ed. Pearson Prentice Hall. New Jersey. SAP Inc. 2009. ANSYS User's Manual for Release 12.1. Timoshenko, S. 1936. Theory of Elastic Stability. McGraw-Hill Book Company, Inc. New York Thandavamoorthy, T.S. 2005. Analysis of Structures: Strength & Behavior. Oxford University Press. New York.

View more...
by Derek Matthies

A study submitted in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE

Major: Civil Engineering (Structural Engineering)

Committee Members: Fouad Fanous - Major Professor Robert Abendroth - Committee Member Vernon Schaefer - Committee Member

Iowa State University of Science and Technology Ames, IA 2012

ii

Contents

List of Symbols and Abbreviations.................................................. Abbreviations.................................................. .............................................. iv List of Figures ...................................................... .............................................................................................................. ........................................................................... ................... v 1.

2.

Introduction Introduction and Objective ....................................................... ...................................................................................................... ............................................... 1 1.1

Introduction Introduction ..................................................... ............................................................................................................ ................................................................. .......... 1

1.2

Objectives Objectives ........................................................ ............................................................................................................... ................................................................. .......... 2

Background Background .................................................. ........................................................ ........................................................................... ................... 3 2.1

Buckling Behavior......................................................................................... ................... 3

2.2

Euler Buckling............................................................................................... ................... 4

2.3

Buckling of Bars on Elastic Supports .................................................. ............................ 6

2.4

Buckling of Un-braced Top Chord Truss Members......................................................... 7

2.4.1

Analysis according to Engesser .................................................... ................................................................................ ............................ 7

2.4.2

Buckling Load using the Energy Method....................................................... ................................................................. .......... 9

2.4.3

Buckling Solution with Variable Axial Load ................................................. ........ 10

2.4.4

Buckling of a Pony Truss Top Chord with Elastic Ends ........................................ 12

2.4.5

Analysis of a Pony Truss Top Chord According to Holt ....................................... 14

2.4.6

Buckling Load with Initial Initial Out-of-plane Out-of-plane Deformations Deformations ......................................... 16

2.5 3.

4.

Pony Truss Design according to AASHTO Specifications Specifications............................................ 17 17

Finite Element Analysis .................................................. ...................................................... 18 3.1

Finite Element Model of the Compression Compression Chord ................................................. ........ 18

3.2

Analysis of Top Chords as a Bar on Elastic Supports.................................................... 19

3.3

Finite Element Model of the Pony Truss ...................................................... ....................................................................... ................. 20

Discussion and Results Results of the Analysis Analysis of of a Pony Pony Truss Truss Top Chord .................................... 23 4.1

Effective Buckling Length Factor – Lateral Support Stiffness Relationships ........... ...... ......... .... 23

4.2

Example Calculations Calculations for the Buckling Load of a Pony Truss ...................................... 24

4.2.1

Calculations Calculations following Engesser’s Engesser’s Procedure ................................................ ........ 25

4.2.2

Calculations Calculations following Bleich’s Procedure .................................................... ............................................................ ........ 25

4.2.3

Calculations Calculations following Timoshenko’s Timoshenko’s Procedure Procedure ................................................... 26

4.2.4

Calculations Calculations following Lutz and Fisher’s Fisher’s Procedure.............................................. 26

4.2.5

Calculations Calculations following Holt’s Procedure Procedure ............................................... ................. 26

4.2.6

Calculations Calculations using the Energy Method .................................................. ................. 27

4.3

Analysis of the Pony Truss using Finite Element .................................................. ........ 28

ii

Contents

List of Symbols and Abbreviations.................................................. Abbreviations.................................................. .............................................. iv List of Figures ...................................................... .............................................................................................................. ........................................................................... ................... v 1.

2.

Introduction Introduction and Objective ....................................................... ...................................................................................................... ............................................... 1 1.1

Introduction Introduction ..................................................... ............................................................................................................ ................................................................. .......... 1

1.2

Objectives Objectives ........................................................ ............................................................................................................... ................................................................. .......... 2

Background Background .................................................. ........................................................ ........................................................................... ................... 3 2.1

Buckling Behavior......................................................................................... ................... 3

2.2

Euler Buckling............................................................................................... ................... 4

2.3

Buckling of Bars on Elastic Supports .................................................. ............................ 6

2.4

Buckling of Un-braced Top Chord Truss Members......................................................... 7

2.4.1

Analysis according to Engesser .................................................... ................................................................................ ............................ 7

2.4.2

Buckling Load using the Energy Method....................................................... ................................................................. .......... 9

2.4.3

Buckling Solution with Variable Axial Load ................................................. ........ 10

2.4.4

Buckling of a Pony Truss Top Chord with Elastic Ends ........................................ 12

2.4.5

Analysis of a Pony Truss Top Chord According to Holt ....................................... 14

2.4.6

Buckling Load with Initial Initial Out-of-plane Out-of-plane Deformations Deformations ......................................... 16

2.5 3.

4.

Pony Truss Design according to AASHTO Specifications Specifications............................................ 17 17

Finite Element Analysis .................................................. ...................................................... 18 3.1

Finite Element Model of the Compression Compression Chord ................................................. ........ 18

3.2

Analysis of Top Chords as a Bar on Elastic Supports.................................................... 19

3.3

Finite Element Model of the Pony Truss ...................................................... ....................................................................... ................. 20

Discussion and Results Results of the Analysis Analysis of of a Pony Pony Truss Truss Top Chord .................................... 23 4.1

Effective Buckling Length Factor – Lateral Support Stiffness Relationships ........... ...... ......... .... 23

4.2

Example Calculations Calculations for the Buckling Load of a Pony Truss ...................................... 24

4.2.1

Calculations Calculations following Engesser’s Engesser’s Procedure ................................................ ........ 25

4.2.2

Calculations Calculations following Bleich’s Procedure .................................................... ............................................................ ........ 25

4.2.3

Calculations Calculations following Timoshenko’s Timoshenko’s Procedure Procedure ................................................... 26

4.2.4

Calculations Calculations following Lutz and Fisher’s Fisher’s Procedure.............................................. 26

4.2.5

Calculations Calculations following Holt’s Procedure Procedure ............................................... ................. 26

4.2.6

Calculations Calculations using the Energy Method .................................................. ................. 27

4.3

Analysis of the Pony Truss using Finite Element .................................................. ........ 28

iii

5.

4.3.1

Two Dimensional Dimensional Analysis ................................................. ................................... 28

4.3.2

Three Dimensional Analysis ........................................................ .................................................................................. .......................... 30

4.4

Effects of Compression Chord Moment of Inertia on the Stiffness of the Elastic Supports ................................................. ....................................................... ........................................................................ ................. 32

4.5

Analysis with Modified Elastic Stiffness ...................................................... ....................................................................... ................. 35

Summary, Conclusions, Conclusions, and Recommendations Recommendations .................................................. ................. 38 5.1

Summary ............................................... ........................................................ ......................................................................... ................. 38

5.2

Conclusions Conclusions ..................................................... ............................................................................................................ ............................................................... ........ 39

5.3

Recommendations Recommendations .................................................... .......................................................................................................... ...................................................... 39

Appendix A ................................................. ........................................................ .................................................................................. .......................... 40 Appendix B ................................................. ........................................................ .................................................................................. .......................... 43 Appendix C ................................................. ........................................................ .................................................................................. .......................... 45 Appendix D ................................................. ........................................................ .................................................................................. .......................... 47 References References ................................................... ........................................................ .................................................................................. .......................... 50

iv

���� �� ������� ��� ������������� A b C Ce CE C0 c d E Et h I Ib Ic Id K k k s l L Ld Le M m n P Pcr Pd Q q qo r U v V x y α ∆ δ πp θ ϕ ψ

area of compression chord length of floor beams spring stiffness of interior supports spring stiffness of end supports Engesser’s spring stiffness required spring stiffness with rigid end supports ratio of C to C0 length of diagonal end members modulus of elasticity tangent modulus of elasticity height of truss moment of inertia moment of inertia of floor beam moment of inertia of vertical web member moment of inertia of diagonal web member 2 column stiffness: K = P/EI effective length coefficient joint spring stiffness distance between panels total length of the truss length of diagonal web member effective length bending moment number of buckling modes number of bays compression load critical buckling load axial load on diagonal end members virtual load distributed compressive force maximum compressive load with a varying load distribution distributi on 2 radius of gyration: r = I/A internal energy factor of safety external work distance from end support displacement of chord at point x elastic foundation constant maximum displacement of compression chord relative displacement of vertical web members potential energy angle of rotation at joints stiffness of compression chord Schweda’s elastic end support coefficient

v

���� �� ������� Figure 1.1

Pony Truss Bridge………...…...….… Bridge………...…...….……….……………………… …….……………………………...……….1 ……...……….1

Figure 2.1

Equilibrium Equilibrium Path for Initially Initially Straight Column…..…......………..……… Column…..…......………..……………….4 ……….4

Figure 2.2

Equilibrium Equilibrium Path for Slightly Crooked Column…….…......……..…… Column…….…......……..……………….4 ………….4

Figure 2.3

Euler Buckling………………………………...… Buckling………………………………...………..……...………………….5 ……..……...………………….5

Figure 2.4

Buckling Modes for a Bar with Pin Ends.....………………..……… Ends..... ………………..…………………….6 …………….6

Figure 2.5

Elastically Supported Bar......………….……… Bar..... .………….…………………………………………. ………………………………….7 7

Figure 2.6

Column on elastic supports…………………… supports…………………………………………….. ………………………...………... .………...9 9

Figure 2.7

Varying Axial Load Distribution………...……. Distribution ………...…….………………………………. ………………………………...10 ..10

Figure 2.8

Compression Compression Chord with Elastic Ends…..…………………………………........12 Ends…..…………………………………........12

Figure 3.1

Finite Element Idealization of the Top Chord as a Bar on Elastic Supports…….19

Figure 3.2

Finite Element Idealization of the Pony Truss……………………… Truss…………………………………...21 …………...21

Figure 3.3

Stress-Strain Stress-Strain Curve…..……………………………………… Curve…..………………………………………………………......22 ………………......22

Figure 4.1

Compression Compression Chord Design Curve……..…………… Curve……..……………………………………......23 ………………………......23

Figure 4.2

Energy Method Design Curve ……....………………………………………. ……....…… …………………………………......24 .....24

Figure 4.3

2-D Compression Chord Elements.....…….…………………………… Elements.....…….…………………………………......29 ……......29

Figure 4.4

3-D Compression Chord Elements………………..……………… Elements………………..…………………………......30 …………......30

Figure 4.5

Nonlinear Load vs. Displacement Curve.……….. Curve.………. . ………………….………. ………………….………...... .....30 30

Figure 4.6

Pony Truss Top Chord Analysis……..……………………… Analysis……. .……………………………………….... ………………......31 ..31

Figure 4.7

Rigid Frame Boundary Conditions.…..……… Conditions.…..……………...…………………… ……...………………………......33 …......33

Figure 4.8

Rigid Frame Displacements…..……...………………… Displacements…..……...………………………………………......33 ……………………......33

Figure 4.9

Load Application of Pony Truss Frames…..……...…………… Frames…..……. ..………………………….... ……………......34 ..34

Figure 4.10

Pony Truss Lateral Displacement at Frame.…………………………………... Frame.…………… ……………………......35 ...35

Figure 4.11

Compression Compression Chord with New Stiffness………………………..………… Stiffness………………………..……………......37 …......37

Figure 4.12

Compression Chord with Elastic Ends…………………………… Ends……………………………………….... …………......37 ..37

1

1. ������������ ��� ��������� 1.1 ������������ Lateral stability of steel members under compression has been of interest to researchers for years. Among these members: columns under axial compression load, unbraced compression flange of steel girders, and the top chord of a pony truss for which vertical clearance requirements prohibit direct lateral bracing. The pony truss, while no longer used in constructing new bridges, may find applications in similar situations such as a walkway for a conveyer system between grain elevators. The structural behavior of the previously listed members has been studied by several researchers. In the following chapter, the behavior of an axially loaded bar and the top chord of a pony truss are briefly summarized. The calculation of the critical load for a pony truss top chord using published relations has been examined and compared to the results obtained using an analytical method. The compression chord of the pony truss structure, where vertical clearance prohibits lateral bracing, is elastically supported in the horizontal plane by the truss vertical and diagonal web members, which together with the floor beams form rigid frames as show in Fig. 1.1c.

n Panels

��

Fig. 1.1a Pony Truss Elevation View A

Fig. 1.1c Section A-A A Fig. 1.1b Pony Truss Plan View

To analyze the compression chord of a pony truss, the chord can be treated as a bar on elastic supports (Ballio, 1983). This member with intermediate elastic restraints will buckle in half-waves depending on the stiffness of the elastic restraints. The buckled shape of the bar will fall somewhere between the extreme limits of a half-wave length of unity and the number of

2

spans between the end restraints. From the buckled shape, the effective length of the compression chord can be used to determine the critical load. The method on how to determine the effective length has long been the focus of compression chord buckling. The failure of several pony truss bridges at the end of the nineteenth century prompted the research of compression chord buckling. Engesser (sited in Galambos, 1988) was one of the first researchers to investigate the problem and develop an approximate formula to determine the required stiffness for the elastic restraints that corresponds to a specified effective wave length, k.

Engesser’s approach for determining the stiffness of the elastic restraints and its effects on

the compression chord was based off the assumption that the connection between the web members and the floor beam is rigid. This theory used the frame consisting of the floor beam and vertical and diagonal members at each panel point location to provide stiffness for the compression chord. However, the theory in question is if the idealized structure is a conservative approach of the actual frame stiffness. In other words, one may argue that investigating the behavior of the bridge as a three dimensional system may result in a higher stiffness coefficient of these lateral supports. To the writer’s knowledge, all of the research for determining the critical buckling load on a compression chord with elastic supports is based on Engesser’s assumption. From this assumption, others, such as Timoshenko (1936), Bleich (1952) and Holt (1952), provided methods of solving for the effective buckling length factor, k.

1.2 ���������� The objective of the work presented herein was to verify the results of the published solutions for determining the effective length factor using the finite element method. These objectives were accomplished by performing the following tasks: 1. Conduct a literature search to review available information that is related to the stability of the top chords in truss structures. 2. Verify the results of analyzing a top chord of a pony truss using the approaches given in published literature and the results obtained using the finite element method. 3. Recommend the most applicable published analysis technique for determining the critical load of an unbraced top chord of a truss system.

3

2. ���������� 2.1 �������� �������� The failure of an axially loaded bar in compression is defined by limit states which are an identifying condition of design criteria. Limit states for a structural member include strength limit states, which may result in yielding or rupture, or serviceability limits states (i.e. deflection, vibration, slenderness or clearance). Although not a limit state, buckling presents a failure mode due to high compressive stresses which causes the member to no longer be in equilibrium. Usually buckling occurs before the column reaches the full material strength. The buckling strength of compression members has long been studied to relate the empirical methods of analysis to the actual results. The elastic buckling of an axially loaded column in compression occurs when a certain critical load is reached causing the member to suddenly bow out. The deviation of the member axis will result in additional bending that gives rise to large deformations, which in turn cause the member to collapse. The load at which collapse occurs is referred to as the buckling load and is thus a design criterion for compression members. In linear mechanics of deformable bodies, displacements are proportional to the applied loads. The essence of buckling, however, is a disproportionate increase in displacement resulting from a small increase in load. For example, Fig. 2.1, from Brush (1975), shows the loaddisplacement relation (referred to equilibrium path) for an axially loaded column. Each point of this path represents an equilibrium configuration of the structure. However, as the applied load reaches a critical value, i.e., Euler Load, the equilibrium path will follow the secondary path shown in Fig. 2.1. Points along the primary (vertical) equilibrium path represent the configuration of a compressed, perfectly straight column, but as the critical load is reached, a secondary path is formed representing the bent equilibrium configurations. The critical load is defined as the minimum load for which the structure remains in equilibrium before instability is reached and failure occurs. Of course, no real column can be perfectly straight, and hence the load displacement relation will not follow that shown in the Fig. 2.1 but rather a different load displacement will be obtained. The load displacement relationship of an imperfect axially loaded column is shown in Fig. 2.2. When comparing Fig 2.1 and 2.2 for the straight and crooked columns, the figures show that the equilibrium paths generally converge as the lateral

4

displacements increase. Analyses for both columns, straight and slightly crooked, lead to large lateral displacements at the critical load. �

=

��

=

�

��������� ���� ������� ���� �Δ

Δ �� � � �/�

+Δ

Fig. 2.1 Equilibrium Paths for Initially Straight Column

Δ �� � � �/�

Fig. 2.2 Equilibrium Paths for Slightly Crooked Column

2.2 ����� �������� Buckling of axially loaded bars in compression was investigated by Leonhard Euler in 1744. His work was based on a straight, prismatic, concentrically-loaded column with pin-ended connections. In his work, Euler stated that if the applied load, P, was less than the critical value, the bar remained straight and underwent only axial compression. By that definition all fibers would remain elastic until buckling occurred. According to Salmon (2009), Euler’s formula was not widely accepted initially since the test results on columns did not agree with his theory. The discrepancies, however, were due to the fact that the elastic limit was exceeded before the elastic buckling was attained. Euler’s formula was finally validated in 1889 when Considère and Engesser independently published works showing that one must use the tangent elastic modulus, Et, to account for the fibers beyond the proportional limit. For the reader’s interest, the following summarizes the derivation of the Euler buckling load. Figure 2.3 illustrates the deflected shape on an axially loaded bar. The bending moment, M, at a distance, x, can be related to the curvature as follows:

= =

(2.1)

5

�

� � �/�

Δ

�

�/�

� Fig. 2.3 Euler Buckling (Thandavamoorthy 2005)

The solution for the linear differential equation above can be written as

= s i n + c o s / 0 = sin 0 = sin = / =

where, K, is equal to

(2.3)

. The constants A and B in Eq. 2.3 can be calculated utilizing the

support conditions at both ends of the bar. For a pinned-end column, the boundary conditions can be set as y = 0 at x = 0 and y = 0 at x = L. These conditions will result in: B=0

and

(2.4)

Equation 2.4 can then be written as

By substituting K =

(2.5)

into equation 2.5 and solving for P, Euler buckling equation yields

the critical buckling load.

(2.6)

6

where m = 0, 1, 2…… is referred to as the number of buckling modes. The deformed shapes for the first three buckling modes are shown in Fig. 2.4.

���

����

���

����

���

���

Fig. 2.4 Buckling Modes for a Bar with Pin Ends

2.3 �������� �� ���� �� ������� �������� A bar supported by rigid supports at the ends with equally spaced elastic restraints between the ends can have several modes of buckling depending on the stiffness of the supports. If the stiffness of the elastic support is sufficiently large, the bar will buckle in half-waves of a length equal to the distance between supports as shown by Fig. 2.5a. The bar will then behave similar to a bar on rigid supports. However, if the elastic supports are very flexible, then bar will behave similar to a bar not supported by restraints and deflect in one-half wave as shown by Fig. 2.5b. As the elastic stiffness varies between the two extreme limits, the bar will buckle somewhere between one half wave and the number of spans between the rigidly supported ends such as Fig. 2.5c. Therefore, the stiffness of the elastic supports that are provided by the vertical and diagonal members of a pony truss is vital in controlling the buckling load and the buckling length of the top chord.

7

(a)

P

P

P

P

(c) P

P

(b)

Figure 2.5 Elastically Supported Bar (Bleich, 1952)

2.4 �������� �� ��������� ��� ����� ����� ������� As mentioned above, the compression chord from the unbraced top chord in a steel truss, such as a pony truss, can be idealized for buckling analysis as a continuous beam that is braced by elastic springs, which correspond to the stiffness of the transverse frames at each panel point. Therefore, design of the transverse frames formed by the web members and floor beams will have a direct effect of the critical buckling load of the chord members. AASHTO section 6.14.2.9 (2007) addresses these issues and gives recommendations on the design of the vertical web as well as the connection to the floor beam. However, unless one considers the effect of imperfections of the compression chord, the calculated critical load is an upper limit. The following sections summarize some of the published work that is related to the analysis of the unbraced top chord of steel trusses.

2.4.1 �������� ��������� �� ��������

The analysis proposed by Engesser in the late 1800’s can be applied with some reasonable accuracy to analyze a bar that is pinned at its ends and is supported on equally spaced intermediate elastic springs provided that the half-wavelength of the buckled shape is at least 1.8 times the spring spacing (See Galambos 1988). However, one must realize that Engesser’s solution can only be used as a preliminary design tool and more comprehensive analysis is needed. Engesser examined the top-chord buckling problem of pony trusses and summarized his findings in a paper that was published in 1884. In the following years, he used his work to explain the failures of pony truss bridges and provide a rational method of design for similar structures. Engesser developed a simple formula to calculate the required stiffness, C req, of the

8

elastic support to reach the desired critical load that is based on a specific buckling length. In his work, Engesser suggested that one needs to assume an effective length factor, k, of 1.3. The top chord, including the end posts, is straight and of uniform cross section. Engesser also provided the following assumptions: 1. Its ends are taken as pin-connected and rigidly supported. 2. The equally spaced elastic supports have the same stiffness and can be replaced by a continuous elastic medium. 3. The axial compressive force is constant through the chord length. Engesser’s solution for the required stiffness of a pony-truss transverse frame which is derived in Appendix C is

=

(2.7)

If Creq is met at each frame location, the chord with the length between panels, l, will achieve the specified design load, P cr. Several researchers suggest that one needs to assume a factor of safety, v, of two when calculating the design load. In other words, the load P cr, can be taken as vP, where P is the calculated top chord member load. In addition, once the calculations show that the stress induced in the member exceeds the limit specified in the design specifications, the flexural rigidity EI should be modified using the tangent modulus, E tI. By combining Euler’s buckling equation and Eq. 2.7, the required spring constant is

=

(2.8)

The use of Engesser’s original approach in design is summarized as follows: 1. Carry out a structural analysis to calculate the maximum load in the top chord members. 2. Introduce a factor of safety not less than 2.0 and calculate the design load, P = v*load from step one above. 3. Use an admissible structural analysis technique to calculate the elastic constant, C, of the provided lateral restraint.

9

4. Utilize the information calculated above to estimate the ratio, Cl/P. One approach that can be used to calculate the provided lateral supports stiffness, C, is detailed in Appendix A. 5. Use the above calculated ratio and the number of panels, n, to calculate the effective length factor, k, using Table 2.2 from Holt (1956) or Fig. 4.1 citing other authors. 6. Apply k, found in step five, to the equations found in Chapter E of the AISC (2011) manual to determine the nominal compression capacity of the compression chord.

2.4.2 �������� ���� ����� ��� ������ ������

Similar to the approach for a simply supported column, a column simply supported at both ends with equally spaced interior elastic supports, as shown in Fig. 2.6, can be defined by an equation which represents the buckled shape. The total length, L, is the number of bays, n, multiplied by each bay length, l. Using the energy method, the deflected shape can be defined by the equation

= sin

(2.9)

where m is the mode number, and the number of modes can be related to n-1. Using the sin curve, the boundary conditions are y = 0 at x = 0 and y = 0 at x = L.

(a)

P

A

B

P

L = nl (b)

P

P

Figure 2.6 Column on Elastic Supports

To solve for the external work and internal energy of the member, the first and second differential equations for the line can be solved as

′ = cos

(2.10)

10

′′ = sin

(2.11)

With P as the axial load, the potential energy of the system can be set equal to the external work, V, plus the internal energy, U to find the critical buckling load. A full derivation of a bar on two elastic springs using the energy method is presented in Appendix B. Since the energy method uses an assumed buckled shape for the chord, the solution is obtained by some degree of approximation. If the assumed shape is properly chosen to satisfy the boundary conditions, the energy method provides a satisfactory approximation. For the case presented in Fig. 2.6, the potential energy can be represented as

= + = ′ + ′′ + ∑

(2.12)

The final term in equation 2.12 represents the energy of the elastic supports as a function of the spring constant C. By solving for P, the critical buckling can be found as a function of the mode number.

2.4.3 �������� �������� ���� �������� ����� ����

Timoshenko (1936) extended the work of Engessers’ to include the effects of a varying the axial load along the top chord of a truss structure. He assumed the compression load varies parabolically along the length of the chord with the load equal to zero at the ends then reaching a maximum value at the center (see Fig. 2.7). In addition, Timoshenko’s solution assumed that the ends were pin connections.

� �

�

�

Figure 2.7 Varying Axial Load Distribution (Timoshenko, 1936)

11

Similar to Engesser, Timoshenko assumed an equivalent elastic foundation, α, is related to the lateral support spring constant, C, and the distance between each lateral support, l, as

= = ∑ + ∑

(2.13)

The strain for the chord energy can be represented as

(2.14)

In his solution, Timoshenko also stated that if the bridge is uniformly loaded, the compressive forces that are transmitted to the chord by the diagonals are proportional to the distance from the middle of the bridge span as

= 1−

(2.15)

where x is the distance from the left support in the figure and q o is the maximum force of the axial load represented by

= = − −

(2.16)

The external work, V, can then be calculated using the information given above as follows

(2.17)

Substituting the information given in Eqs. 2.14 and 2.17 into the total potential energy relationship, one can then obtain the following:

=

(2.18)

Finally, since the elastic supports are treated as a continuous elastic medium, Eq. 2.19 can be developed to relate the critical buckling load to the effective length factor by combining Eqs. 2.8, 2.16 and 2.18.

=

(2.19)

12

2.4.4 �������� �� � ���� ����� ��� ����� ���� ������� ���� 2.4.4.1 �������� ��������� �� ������ (1952)

Bleich obtained his solution by using finite difference as an exact approach to quantify the buckling load of the chord. His solution was based on the ends being pin connections and equally spaced intermediate supports of equal rigidity. Bleich also assumed the chord had a constant moment of inertia and constant axial compressive force over the entire length. Later, Schweda extended Bleich’s results to include chords with elastic ends.

���

� � ��

�

� �� �

� ��

� � ��

�

���

���

�� �

Figure 2.8 Compression Chord with Elastic Ends (Bleich, 1952)

The theoretical exact solution proposed by Bleich for a chord supported on rigid ends was

where vP = Pc and

=

= = ϕ =

(2.20)

The stiffness of the chord is represented by ϕ which is equal to

(2.21)

To eliminate the tangent modulus, E t, which varies per the axial load, Euler’s buckling equation is substituted into equation 2.21, simplifying the stiffness to

ϕ =

(2.22)

For a chord with n spans in equation 2.20, there are n -1 different half-wave buckling configurations. As n increases, the spring constant required for an infinite number of spans, C∞, increases to a limiting value. Bleich (1952) showed that C n can be replaced by C ∞ for any

13

span where n is greater than six which corresponds to an error less than 1%. So, for trusses with more than six spans, equation 2.16 simplifies to

= = Φ

(2.23)

where Φ is given in Table 2.1 from Bleich (1952) and is valid for the elastic and plastic range of buckling. Table 2.1 Values of Φ in Eq. 2.23 (Bleich, 1952)

1/k

Φ

1/k

Φ

1/k

Φ

1/k

Φ

0.3

0.111

0.5

0.309

0.70

0.614

0.90

1.102

0.32

0.126

0.52

0.335

0.72

0.652

0.91

1.138

0.34

0.142

0.54

0.361

0.74

0.692

0.92

1.177

0.36

0.160

0.56

0.388

0.76

0.734

0.93

1.219

0.38

0.179

0.58

0.417

0.78

0.777

0.94

1.264

0.40

0.198

0.60

0.447

0.80

0.822

0.95

1.316

0.42

0.218

0.62

0.478

0.82

0.870

0.96

1.375

0.44

0.239

0.64

0.510

0.84

0.921

0.97

1.444

0.46

0.261

0.66

0.544

0.86

0.976

0.98

1.530

0.48

0.285

0.68

0.578

0.88

1.036

0.99

1.652

1.00

2.000

Since Bleich’s theory assumes a constant axial force, which is rarely the case in practice, the center bay of the chord should be designed with the appropriate k value for the maximum load and then used for the remaining bays. Although the previous assumption of designing the bridge for the center span only would yield conservative results, the assumption of rigid ends can result in unsafe buckling loads when using Bleich’s theory. Thus, in order to continue on Bleich’s exact buckling theory, Schweda provides results to determine the required stiffness for a chord supported elastically on the ends. The spring constant of the end supports is denoted by C e and the intermediate supports by C. Schweda assumed the load was a constant axial force throughout the length of the

14

chord with equally spaced elastic supports similar to Bleich’s theory. Figure 2.8 shows the compression chord with the diagonals extended a length, d , subjected to the compressive force vPd and pinned at points –n and +n. The spring constant must be larger the spring constant of

the chord with rigid end, C 0. Thus, the value of C = cC0 where c > 1.1. The required spring constant with the axial load in the diagonal is

= +

(2.24)

where CE is Engesser’s equation (Eq. 2.8). Schweda calculated ψ as a function of the number of bays with respect to c and k . These values are listed in Appendix D. If all of the crossframes are identical, C e in equation 2.24 is equal to C and the equation becomes

=

(2.25)

2.4.5 �������� �� � ���� ����� ��� ����� ��������� �� ����

Holt’s research in the 1950s tested pony truss bridges in an attempt to compare the actual buckling load of the compression chord with the design equations. His research not only tested the primary constraints mentioned above but also the effects of secondary factors. The following secondary factors were considered in his research (Holt, 1956): 1. Torsional stiffness of the chord and web members. 2. Lateral support given to the chord by the diagonals. 3. Effect of web-member axial stresses on the restraint provided by them. 4. Effect of non-parallel-chord trusses. 5. Error introduced by considering the chord and end post to be a single straight member. The results of Holt’s analysis proved that the error in determining the critically buckling load by neglecting the above factors was relatively small. His conclusion stated that the load capacity of a pony truss bridge would be satisfactorily predicted by previous buckling analyses mentioned. As shown by Bleich (1952), if the truss has at least ten panels, then the effective length factor depends only on the stiffness of the transverse frames. Thus, the appropriate

15

effective length factor is a function of Cl/P c which is shown in the results section. Holt noted that Bleich’s analysis showed adequate results for the entire range of effective length values where Timoshenko’s results show adequate results for k > 2. A summary of Holt’s results can be seen in Table 2.2. Based on the results of his research, Holt (1957) also recommended the following on the design of the end posts The end post should be designed as a cantilever to carry, in addition to its axial load, a transverse force of 0.3% of its axial load at its upper end.

Table 2.2 (Galambos, 1988)

1/k 1.00 0.980 0.960 0.950 0.940 0.920 0.900 0.850 0.800 0.750 0.700 0.650 0.600 0.550 0.500 0.450 0.400 0.350 0.300 0.293 0.259 0.250 0.200 0.180 0.150 0.139 0.114 0.100 0.097 0.085

4 3.686

3.352 2.961 2.448 2.035 1.750 1.232 0.121 0

1/k for Various Values of Cl/ P c and n n 6 8 10 12 3.616 3.660 3.714 3.754 3.284 2.944 2.806 2.787 3.000 2.665 2.542 2.456 2.595 2.754 2.303 2.252 2.643 2.146 2.094 2.593 2.263 2.045 1.951 2.460 2.013 1.794 1.709 2.313 1.889 1.629 1.480 2.147 1.750 1.501 1.344 1.955 1.595 1.359 1.200 1.739 1.442 1.236 1.087 1.639 1.338 1.133 0.985 1.517 1.211 1.007 0.860 1.362 1.047 0.847 0.750 1.158 0.829 0.714 0.624 0.886 0.627 0.555 0.454 0.530 0.434 0.352 0.323 0.187 0.249 0.170 0.203

14 3.785 2.771 2.454

16 3.809 2.774 2.479

2.254 2.101 1.968 1.681 1.456 1.273 1.111 0.988 0.878 0.768 0.668 0.537 0.428 0.292 0.183

2.282 2.121 1.981 1.694 1.465 1.262 1.088 0.940 0.808 0.708 0.600 0.500 0.383 0.280 0.187

0 0.135 0.045 0

0.107 0.068

0.103 0.055

0.121 0.053

0.112 0.070

0.017 0

0.031

0.029

0.025

0.003 0

0.010

0

0

16

2.4.6 �������� ���� ���� ������� ������������ ������������

Initial out-of-plane deformations of the compression chord can reduce the critical buckling load determined by the previously mentioned methods. There are two primary causes of out-ofplane deformations that need to be taken into consideration. A vehicle load on the floor beams would cause a displacement of the chord at the location of the load creating initial lateral displacements in the chord. The chord could also have initial crookedness and unintentional eccentricities due to manufacturing. Such lateral deflections would reduce the maximum load capacity of the chord. Lutz and Fisher (1985) addressed this issue in their publication to the Structural Stability Research Council in 1985. Their work was similar to the stiffness criteria George Winter proposed in 1960. Winter (1960) proposed the ideal stiffness, C, needed to fully brace the compression member over the length, l, is equal to ideal stiffness

where the stiffness required, C req, is usually twice the

. Lutz and Fisher used Engesser’s formula for a perfectly straight

compression chord and developed a factor of safety to account for the out-of-plane stiffness. The stiffness equation by Engesser was

2

where Le = k l and π /4

≅

= 2.5

(2.26)

2.5. For trusses with a small l relative to L e equation 2.26 provides an

accurate bracing stiffness for the solution. However, as l increases relative to L e, then equation 2.26 will result in unsafe errors. Thus, Lutz and Fisher proposed the following empirical equation to determine the required stiffness

= 2.5 +1.5

(2.27)

Then they extended the applicability of k factors to less than 1.3 going as low as 1.0. The fully braced case where k = 1 corresponds to Winter’s stiffness of

. Equation 2.27 provides a

minimum value of stiffness for the compression chord to reach the required critical load however; there are currently no design procedures available to account for initial imperfections. Design recommendations by AASHTO only make a note of the design vertical truss members and the connection to the floor beam.

17

2.5 ���� ����� ������ ��������� �� ������ �������������� AASHTO Specifications (2007) for the LRFD design of half-through trusses recommends design loads for both the top chord of the truss and the web verticals. AASHTO states in section 6.14.2.9 that The top chord shall be considered as a column with elastic lateral supports at the panel points. The vertical truss members and the floor beams and their connections in half-through truss spans shall be proportioned to resist a lateral force of not less than 300 pounds per linear foot applied at the top-chord panel points of each truss considered as a permanent load for Strength 1 Load Combination and factored accordingly. By applying the appropriate vehicle or other live load cases to the truss, the floor beam can be designed. The floor beams, in addition to the vertical truss members designed with the 300 plf applied load, provide the elastic lateral supports at the panel points.

18

3. ������ ������� �������� The following investigation focused on validating the published results for calculating the effective buckling length factor, k, that can be determined by the methods mentioned in the previous chapter. The example used for the analysis was studied by Galambos 1988 (see Fig. 3.2) in his book. The top chord of the truss consisted of a 10”x10”x5/8” box section that was designed for a maximum compressive force of 360 kips. The vertical and web members were composed of a W10x33 sections in addition to the floor beams for the rigid frame, which were W27x84 sections. The Young’s modulus for the members was assumed to be 29,000 ksi. The investigation by Galambos was carried out assuming a factor of safety of 2.

3.1 ������ ������� ����� �� ��� ����������� ����� The buckling analysis that is presented in this paper was conducted using the ANSYS 12.1 general purpose finite element program. ANSYS is a commercial engineering software that is capable of analyzing different engineering properties on the structure very quickly with a host of different elements available. Using ANSYS, the nonlinear material properties of the compression chord can also be investigated. In the following paragraphs, the element type and why it was used will be explained in more detail. A beam3 element was used in the compressions chord model and is a uniaxial element with tension, compression, and bending capabilities. Each node has three degrees of freedom: translations in the x and y axes and rotation about the z axis. This element is a 2-D element, which provided the analysis for the basic compression chord case. These results were compared to the results for a 3-D element, beam4, to show the adequacy of model. The beam4 element is similar to the beam3 element except the beam4 element has six degrees of freedom: translations in the x, y and z axes and rotations about the x, y and z axes. Stress stiffening and large deflection capabilities are also included. When using the beam4 element, the translations in the z direction must be restrained to simulate the actual buckling properties of the compression chord since the chord has considerably more stiffness against buckling in the z direction. The combin14 element is an element with no mass and was used as the spring in all of the analysis. This combination element has longitudinal or torsional capabilities in 1-D, 2-D or 3-D applications. The longitudinal spring-damper option, which was used in this analysis, is a

19

uniaxial tension-compression element with up to three degrees of freedom at each node: translations in the nodal x, y and z directions. When analyzing the nonlinear properties of the compression chord, the beam23 element was used. The beam23 element is a uniaxial element with tension-compression and bending capabilities which also has plastic, creep, and swelling capabilities. This element has three degrees of freedom at each node: translations in the x and y directions and rotation in the z direction. The element is defined by the area, moment of inertia, and height. A more in-depth description of each element is available from ANSYS 12.1 (SAP Inc., 2009).

3.2 �������� �� ��� ������ �� � ��� �� ������� �������� Figure 3.1 shows the boundary conditions of the 2-D model, which is supported in the x and y directions at the base and only in the x direction at the top to simulate the pin and roller connections. A beam3 element was used in the compressions chord model and is a uniaxial element with tension, compression, and bending capabilities.

Fig. 3.1 Finite Element Idealization of the Top Chord as a Bar on Elastic Supports

20

3.3 ������ ������� ����� �� ��� ���� ����� The pony truss described above was also modeled, as shown in Fig. 3.2. The model was composed of beam4 (3-D beam elements) and 3-D link8 elements. Link8 is a 3-D truss element with three translation degrees of freedom at each node. The diagonal members were modeled using these 3-D truss elements. This was done in order to compare the loads with the analysis mentioned above in chapter 2 since those authors neglected the effects of the diagonal web members. The truss was restrained in the x, y and z directions on one end and only the y and z directions on the other end for pin and roller connections as seen in the Fig 3.2. 3-D beam elements were used to model the top and bottom chords of the truss structure. 4

The moment of inertia for the 10x10 box section was calculated to be 418.3 in . By rearranging Eq. 2.7, the tangent modulus of elasticity can be calculated as 7344 ksi for the chord. This tangent modulus is used in the nonlinear model. The ANSYS program allows the user to carry out a nonlinear buckling analysis. In the work presented herein, the analysis was carried out considering the effects of the nonlinear material behavior only. For this purpose, the user needs to provide the stress-strain relationship of the material. In the nonlinear model of the compression chord, the material nonlinearity was modeled using a multilinear isotropic hardening option (MISO) for the material with the stressstrain profile as shown in Fig. 3.3. The proportional limit was specified as per the AISC manual as 0.4Fy, and the yield value for this model was 36 ksi. The tangent modulus was found above using Engesser’s equation and varies as the critical load changes. Notice that in Fig. 3.3, the portion as the material reaches yield was defined using very small slope. This was necessary to avoid overshooting and any problems that may cause non-convergence to the solution.

21

Fig. 3.2 Finite Element Idealization of the Pony Truss

The nonlinear solution of the compression chord in ANSYS used the Newton-Raphson option to converge on the displacements of the solution. An initial load larger than the predicted buckling load was applied to the chord and ANSYS then uses load steps to continuously apply the load in small increments to iterate the solution. For each iteration, the program calculates a new element stiffness matrix based on the element strains in the stress-strain profile provided. For the nonlinear model in this research, it was assumed that a converged solution was reached when the difference in displacements between load steps was equal to or less than 0.1%. The results of analyzing the truss described above using the different available analyses techniques are summarized in the following chapter.

22

������������� ����� �0 �� �0 � � � � � � � � � � �

��

�� � ���� ��� �0 �� �0 �

� � ��000 ���

0 0

0.00� 0.00� 0.00� 0.00� 0.00� 0.00� 0.00� 0.00� 0.00� 0.0�

������ �������

Fig. 3.3 Stress-Strain Curve

23

4. ���������� ��� ������� �� ��� �������� �� � ���� ����� ��� ����� 4.1 ��������� �������� ������ ������ � ������� ������� ��������� ������������� For the comparison of the equations in chapter two with the FEM, the equations needed to be rearranged for the calculation of the effective length factor, k. This factor is required to check the capacity of the top chord member following the recommendation used in AISC compression calculations. The relationships of the suggested procedures that can be used in calculating the effective length factor are shown in the Figures 4.1 and 4.2 below. Figure 4.1 relates the inverse of the effective length factor to the stiffness of the lateral support, and Fig. 4.2 depicts the energy relation between the stiffness of the lateral elastic support and Euler’s load, P e, to the critical compressive load, P cr, of the member.

��������� ������ ������� �.�00

�.000

0.�00

�������� � � �

0.�00

������ ����������

0.�00

�����������

0.�00

0.000 0.000

0.�00

�.000

�.�00

�.000

�.�00

�.000

�.�00

�.000

����

Fig. 4.1 Compression Chord Design Curves

�.�00

24

�������� �������� ���� ��0

�00

�0 � � � � � �

�0

�0

�0

0 0

�00

�000

��00

�000

��00

�000

�������

Fig. 4.2 Energy Method Design Curve

4.2 ������� ������������ ��� ��� �������� ���� �� � ���� ����� Given below are the calculations that were used to develop the buckling load of the truss structure that was examined by Galambos (1988). The dimensions of the truss structure were previously listed in chapter three of this document. In this example, a structural analysis of the truss system illustrated that the top chord member being investigated was subjected to a compression force of 360 kips. A factor of safety of 2, as suggested in the literature, was used in the analysis. From the specified truss geometry, the provided lateral spring stiffness, C, was calculated using Eq. A1.1 as

C = = ∗ ∗ = 6.75 kip/in

25

Then, the ratio of C l /P was determined as

= .∗ = 1.50

Using the calculated ratio C l /P in conjunction with Fig 4.1, one can calculate the value of 1/k according to the different analysis techniques that were previously summarized. 4.2.1 ������������ ��������� ���������� ��������� 1/k = 0.78 (From Fig. 4.1) k = 1.28

= ..∗ = 50.3

Following the design steps that are given in the AISC (2011) manual, one can calculate the critical buckling stress, F e, and the nominal buckling load, P n, using the ASD and LRFD approaches, respectively as

F = = . = 113.2 ksi

F = 0.658 F = 0.658 . 36 = 31.5 ksi Hence, the critical buckling load is:

P = FA = 31.5∗25.0 =

This procedure is then repeated using the other methods mentioned in chapter two. 4.2.2 ������������ ��������� �������� ��������� 1/k = 0.768 (From Fig. 4.1) k = 1.30

= ..∗ = 51.1 F = = . = 109.8 ksi

F = 0.658 F = 0.658 . 36 = 31.4 ksi

26

P = FA = 31.4∗25.0 =

4.2.3 ������������ ��������� ������������ ��������� 1/k = 0.604 (From Fig. 4.1) k = 1.66

= ..∗ = 64.9 F = = . = 67.9 ksi

F = 0.658 F = 0.658 . 36 = 28.8 ksi

P = FA = 28.8∗25.0 =

4.2.4 ������������ ��������� ���� ��� �������� ��������� 1/k = 0.719 (From Fig. 4.1) k = 1.39

= ..∗ = 54.5 F = = . = 96.2 ksi

F = 0.658 F = 0.658 . 36 = 30.8 ksi

P = FA = 30.8∗25.0 =

4.2.5 ������������ ��������� ������ ��������� 1/k = 0.75 (From Table 2.2) k = 1.33

= ..∗ = 52.3 F = = . = 104.7 ksi

F = 0.658 F = 0.658 . 36 = 31.2 ksi

P = FA = 31.2∗25.0 =

27

4.2.6 ������������ ����� ��� ������ ������

P = = ∗∗. = 11.84 kips = ..∗ = 912.2 = 61.67 P = PR = 11.84∗61.67 = (From Fig. 4.2)

A summary of these results is shown in Table 4.1. As noted in Galambos (1988) the factor of safety for the compression chord on elastic supports was 2.0 when determining the C l/ P ratio. However, it was later addressed that the compression chord in Galambos example was designed for the maximum panel load using AASHTO’s 1983 formula which uses a factor of safety of 2.12 (Ziemian, 2010). In essence, the results should be compared to the design load of P=360*2.12 = 763 kips. The factor of safety listed in this table was calculated as the ratio between the estimated critical buckling load and the applied member load of 360 kips.

Table 4.1 Critical Load Results Method Engesser Bleich Timoshenko Lutz & Fisher Holt Energy Method

k 1.28 1.30 1.66 1.39 1.33 -

Pn 788 785 721 770 779 730

F.S 2.19 2.18 2.00 2.14 2.16 2.03

The results above show the methods of analysis in chapter two reasonably predict the critical load of the compression chord for the pony truss example. However, it may be noted that using the energy method underestimates the critical buckling load since it is based on an assumed deformed buckling shape.

28

4.3 �������� �� ��� ���� ����� ����� ������ ������� 4.3.1 ��� ����������� ��������

To verify the critical load calculations of the compression chord, the truss was analyzed in ANSYS to solve for the critical buckling load. As mentioned in chapter three, the validity of these tests was checked with both a 2-D and 3-D element model. Using a 2-D beam3 element with the tangent modulus, E t = 7344 ksi, the critical buckling load was calculated to be 719.213 kips as shown in Fig. 4.3. The 3-D element analysis of the compression yielded the exact same buckling load of 719.213 kips (Fig. 4.4) validating the model. The compression chord buckling load was also determined using a nonlinear approach. This buckling load was found by plotting the Load vs. Displacement in the vertical direction and then noting the load at which large displacements occur with only a small increase in load. Using Fig. 4.5 the buckling load in the nonlinear model was equal to 720 kips. As seen in the graph, there are two changes in slope which account for the change in modulus at the proportion limit and the yielding limit. All of the FEM solutions prove the reasonability of the calculations determined in the previous section with C = 6.75 and E t = 7344 since the three compression chords analysis all had approximately the same buckling load of 720 kips. The next step of the analysis was to check the critical load on the compression chord when the entire truss was modeled.

29

Fig. 4.3 2-D Compression Chord Elements

Fig. 4.4 3-D Compression Chord Elements

30

Fig. 4.5 Nonlinear Load vs. Displacement Curve

4.3.2 ����� ����������� ��������

The pony truss model was analyzed similar to the compression chords in that a compression load was applied to the top chord on each side of the truss. In order to only focus on the lateral displacements of the compression chord, the truss was restrained in a manner to create two symmetrical sides of the truss. To accomplish this symmetrical model, the truss was restrained against translation in the z direction and rotations in the x and y directions at the center of the floor beams. By applying these boundary conditions, the resultant load could be compared to the compression chord models. The critical buckling load, when modeling the entire pony truss, increased dramatically to a load of 1121 kips on the compression chord which can be seen in Fig. 4.6. It is the opinion of this author that the increase in critical buckling load is due to the effect of inertia from the compression chord. The spring stiffness, when analyzing the compression

31

chord by previous work, was composed only of the frame stiffness as calculated in Appendix A. However, the lateral stiffness of the chord itself provides an addition stiffness which explains the significant increase in load when modeling the entire truss. The goal in the preceding paragraphs is to analyze this theory using finite element models.

Fig. 4.6 Pony Truss – Top Chord Analysis

32

4.4 ������� �� ����������� ����� ������ �� ������� �� ��� ��������� �� ��� ������� �������� When calculating the elastic stiffness restraining the compression chord from buckling in Appendix A, the stiffness is a product of the rigid frame only and ignores the contributions of the stiffness from the top chord. This discrepancy was examined using the finite element method. To analyze this theory, several models of the truss structure considering different moments of inertia of the top chord were analyzed by applying two lateral loads at the panel points as seen in Fig 4.9. The results of this analysis were then used to calculate the stiffness of the lateral supports. In order to determine if the inertia of the compression chord has an effect on the stiffness of the compression chord, the frame itself was analyzed as a control procedure before examining the entire truss. The frame was analyzed as a simply supported structure with a load of 1 kip placed on each vertical web as seen in Fig. 4.7. The load causes a displacement of 0.148014” (Fig. 4.8) which can be used to find the stiffness of the frame by taking the load divided by the displacement. Using the displacement of the frame to calculate the stiffness, a value of C = 1/0.148014 = 6.576 k/in is determined. This is the same stiffness that the equation from Appendix A yielded. The same procedure was then repeated except with entire pony truss modeled as a simply supported structure, which can be seen in Fig. 4.9. The load was applied at each frame location individually to get the stiffness at each restraint point. A summary of these results can be found in Table 4.2 which references the panel locations in Fig 4.9. The results in Table 4.2 were determined using a compression chord tangent modulus of 7344 ksi and the moment of inertia of 4

418.3 in . From this table, the compression chord seems to have an effect on the overall lateral stiffness against buckling. To prove this point, a transverse load was applied to the center panel, 4

4

similar to before, and the inertia of the compression chord was varied from 425 in to 0.001 in . The results of this process can be seen in Table 4.3. This table shows that the moment of inertia for the compression chord should be accounted for when determining the lateral stiffness at the frames locations and that the current determination used for the frame stiffness is an underestimation. In order to compare the procedures from the published works, the new stiffness should be used.

33

Fig. 4.7 Rigid Frame Boundary Conditions

Fig. 4.8 Rigid Frame Displacements

34

`

1 2

3 4 5 6 7 8 9 10 11

Fig. 4.9 Load Application of Pony Truss Fra mes

Table 4.2 Resultant Stiffness at Each Ba y Panel Location 1 2 3 4 5 6 7 8 9 10 11

Displacement, ∆ (in) 0.10612 0.07518 0.07249 0.07227 0.07226 0.07226 0.07226 0.07227 0.07249 0.07518 0.10612

Stiffness, C (k/in) 9.426 13.301 13.795 13.837 13.839 13.839 13.839 13.837 13.795 13.301 9.423

35

Fig. 4.10 Pony Truss Lateral Displacement at Frame Location

Table 4.3 Effects of Top Chord Inertia on the Lateral Stiffness 4

Moment of Inertia (in ) 425 325 225 125 25 0.001 418.3

Displacement, ∆ (in) 0.048816 0.052882 0.058840 0.069192 0.100502 0.141938

Stiffness, C (k/in) 13.89 12.98 11.87 10.45 8.08 7.04 6.75 (using Eq. A1.1)

4.5 �������� ���� �������� ������� ��������� The calculation for each of the methods in chapter 2 was redone using the new stiffness value, C = 13.8 found in the previous section, by completing the same procedure mentioned in section 4.2. Table 4.4 summaries the critical buckling loads of these calculations. Engesser’s approach was not valid for new analysis since his theory is based off a minimum k value of 1.3. With the new stiffness value, all of the methods underestimate the critical load of the

36

compression chord when analyzing the entire truss since the formulas do not account for the inertia stiffness of the compression chord and the increase in stiffness with respect to k is not a linear response.

Table 4.4 Critical Load Results with New Stiffness, C Method Engesser Bleich Timoshenko Lutz & Fisher Holt Energy Method

K NA 1.02 1.16 1.08 1.01 -

Pn NA 827 807 819 828 1032

F.S NA 2.30 2.24 2.28 2.30 2.87

Using ANSYS, the compression chord modeled with the new stiffness provided a large increase in the compression capacity. Figure 4.11 shows the compression chord modeled with rigid supports and the interior restraints having a stiffness of 13.8. The buckling load was found to be 979.03 kips. Although this load is closer to the pony truss analysis of 1121 kips, the end supports in this model were rigid which does not accurately account for the elasticity at these supports. The next figure, Fig. 4.12, shows the buckling shape with the elastic supports having the new stiffness values determined in Table 4.2. This buckled shape essentially shows the “exact” buckling load and shape of the compression chord treated as a single member.

37

Fig. 4.11 Compression Chord Analysis with New Stiffness

Fig. 4.12 Compression Chord Analysis with Elastic Ends

38

5. �������, �����������, ��� ��������������� 5.1 ������� Previously published work closely relates the critical buckling load of the compression chord to the actual load when the chord is taken as a single member with the stiffness provided according to Appendix A. However, the formulas in Appendix A ignore the contributions of the moment of inertia of the top chord. This discrepancy was examined using the finite element method. From the FEM, a new stiffness was calculated and used to calculate the new buckling load. A summary of these results can be seen in Table 5.1, which is compared to full pony truss model load of P = 1121 kips. This analysis shows that the omission of the top chord inertia when calculating the frame stiffness will cause an underestimation of the critical buckling load. It is evident that the capacity of the compression chord can be affected by including the inertia effects of the chord.

Table 5.1: Buckling Load Analysis Results Method Engesser Bleich Timoshenko Lutz & Fisher Holt Energy Method FEM Chord - Rigid Ends FEM Chord – Elastic Ends FEM Pony Truss

C = 6.75 C = 13.8 k Pn k Pn 1.28 788 NA NA 1.30 785 1.02 827 1.66 721 1.16 807 1.39 770 1.08 819 1.33 779 1.01 828 730 1032 719 979 644 886 Pcr = 1121 kips

39

5.2 ����������� The following are the conclusions that can be attained from the study presented herein: •

Current design of a compression chord for a pony truss bridge would be best accomplished by using the effective length factor, k, provided by Holt to determine the critical load from Chapter E in the AISC manual. However, such an effective length factor is dependent on the stiffness of the elastic lateral supports.

•

The energy method provides a close answer for calculating the critical buckling load for the truss top chords with rigid supports. However, its application depends on the number of the provided elastic supports.

•

The finite element method will provide satisfactory results when using the appropriate member rigidity, i.e., using the correct tangent modulus of elasticity.

•

All available procedures require the knowledge of the provided elastic stiffness. Hence, it is important to be able to calculate this stiffness factor. This means that one must consider the type of connection between the floor beams and the vertical as well as with diagonal members when calculating the lateral spring stiffness.

5.3 ��������������� Past testing on pony truss bridges is limited to Holt’s work (1957) which focused on the effective length factor on the compression chord and not the stiffness supplied by the frames. Empirical testing of a pony truss model could reveal a better understanding of the actual stiffness supplied by the compression chord inertia. Due to the complexity of a full pony truss and the necessity of empirical confirmation of any design model, testing of physical models is required before determining any definitive conclusions. •

Test a model to verify the effect of inertia on the compression chord stiffness.

•

Determine a method to verify the rigidity of the frame connection.

40

�������� � Calculations of the stiffness of the lateral support to a pony truss

A

h

B

Ic Ib

C

D

b The calculation of the stiffness for the lateral supports to the top chord of an unbraced truss can be calculated using the energy method. This can be accomplished by calculating the force, C, that is required to induce a unit displacement in the lateral direction at the panel point. For this purpose, the virtual work method was utilized. The following is the derivation of the spring constants of the lateral supports if: 1. The connection between the vertical and the floor beam is assumed rigid.

�

�

�

�

�� ��

��

Moment due to applied real load, C

� �

�

Moment due to virtual loads, Q, (Q was assumed = 1)

∗ = = ℎ ∗ℎ ∗ ℎ+ ℎ ∗ ∗ℎ

41

where, Q, is a virtual load that is applied in the horizontal direction at the points where the displacement is to be calculated. In these in calculations, a virtual load, Q, of unity was assumed. Notice that, ∆, is the relative displacement between points A and B. Therefore, to calculate the elastic constants, C, of the lateral spring, one needs to substitute a value of 2 for the displacement, ∆, in the equation above. This yields to:

Or;

∗ 1∗2 = + =

(A1.1)

3

2. To account for truss diagonals, a term of L d /3Id is added in the denominator where the additional stiffness is an addition to the vertical web.

=

(A1.2)

3. The effect if joints C & D are not rigid 1

1

A

D

θ k s B

k s

C

Where; k s is the joint rotational stiffness. The moment, M, at the vertical-floor beam connection is:

= = ∗ℎ = ∗ℎ/

Following the analysis that was summarized in the section above, the relative displacement between points A and D can be calculated as:

42

= 2 = ∗ + +2∗ ∗ℎ 1 = + + ∗ℎ 1 = + + 1 = + + =

(A1.3)

if k s = ∞, then it is a completely rigid connection

Notice that the above equations did not take into account the actual shape of the cross section of the top chord members. However, in most cases, the cross section of the chord consists of open, thin-walled sections having only one axis of symmetry, and hence the bending of the chord will be accompanied by twisting. Thus, the problem of bucking of the chord will be a caused by flexural and torsion. Therefore, the above relations for calculating the spring constant must be modified to account for such effects. This problem was studied in detail by Bleich (1952). Bleich determined that disregarding the torsion on the compression chord would lead to an unsafe design for members with only one axis of symmetry. Conversely, box sections, which have a high torsional rigidity, would underestimate the capacity of the chord.

43

�������� � Energy Method to determine the buckling load for a bar with 2 springs

1. Initial conditions

��

�

�/�

�

�/�

�/�

2. Displaced shape

� Δ�

y = asi n y = cos y = sin y = ∆= asin

Δ�

π

′

′′

π

π

π

π

π

Where:

and

3. External Work

V = y dx V = cos dx V = + sin ′

π

π

π

π

π

y = ∆= asin

π

44

V = π

4. Internal Energy

U = C∆ + C∆ + dx U = C∆ + C∆ + sin dx ′′

π

�=

π

π π π π π �� ��n ���n � − ��n

π

� = ��n ���n � π

π

5. Combine external work and internal energy and solve for P cr π

= � � �

π π π π π = � ��n ���n �

= �.�

∆π ∆

�.� =

π π ���n π � π �����n

π =

π

���n ���n �

�=

π

π

s i n ���n π

π �

π

(B1.1)

45

�������� � Engesser’s Method to determine the buckling load for a bar with 2 springs

1. Initial conditions

��

�

�/�

�

�/�

�/�

2. Displaced shape with continuous elastic medium

� Δ�

� = ���n π

�′ = ��� π

π

π

� = ��n π

3. External Work

y dx V = cos dx V = + sin V = �=

′

π

π

π

π

π

π

Δ�

46

4. Internal Energy

�=

ydx +

π π �x ��n

π

where: α = C/(L/3)

�= − ��n � − ��n π

�=

π

π

π

π �

5. Combine external work and internal energy and solve for P π

= � � �

π

= � �

π

π

= �.� �.� = π � � π ∆π

∆

�=

(C1.1)

+

6. Solve for required stiffness coefficient

P =

where: α = C/L and v = L/m (wavelength)

+

Note that Pcr will be at a minimum when

� = π C = �= −

= �

+

Substituting wavelength, v, back into the equation above:

(C1.2)

47

�������� � Schweda’s extension of Bleich’s analysis

= � −

sinh2 −1sin ±sinh sin2 −1 + 2ℎ 2 − ±

1 ±2 −1cosh = 2 −2+ = √ − = 1 + cos − − = 1− + 2cosh = +1 + + −1 + 2cos = +1 + + −1 + Where:

(D1.1)

48

Tables of Factor ψ: c

k

2n=6

1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5

1.2 1.00 1.25 1.41 1.39 1.35 1.65 2.01 2.41 2.79 2.98 2.93 2.69 2.27 1.99

1.3 0.78 0.93 1.04 1.06 1.10 1.30 1.54 1.77 1.97 2.12 2.10 1.97 1.76 1.71

1.4 0.65 0.75 0.84 0.87 0.94 1.09 1.25 1.40 1.55 1.64 1.66 1.60 1.48 1.48

1.5

1.6

1.7

1.8

1.9

0.82 0.94 1.06 1.18 1.29 1.36 1.38 1.35 1.27 1.32

0.94 1.03 1.11 1.17 1.19 1.17 1.12 1.17

0.90 0.97 1.02 1.04 1.04 1.00 1.06

0.85 0.90 0.94 0.94 0.91 0.96

0.85 0.83 0.89

1.5

1.6

1.7

1.8

1.9

0.81 0.90 0.99 1.06 1.08 1.08 1.20 1.32 1.46 1.58

0.87 0.92 0.95 0.96 1.05 1.16 1.26 1.36

0.85 0.87 0.94 1.03 1.11 1.20

0.85 0.92 1.00 1.07

0.90 0.97

c

k

2n=8

1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5

1.2 0.93 1.12 1.27 1.32 1.42 1.69 1.96 2.07 2.04 1.85 2.08 2.43 2.83 3.25

1.3 0.74 0.86 0.96 1.01 1.12 1.28 1.44 1.54 1.54 1.47 1.66 1.89 2.13 2.37

1.4 0.62 0.71 0.80 0.85 0.94 1.06 1.17 1.24 1.26 1.25 1.39 1.55 1.73 1.90

49

c

k

2n=10

1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5

1.2 0.99 1.06 1.21 1.26 1.43 1.64 1.74 1.72 1.85 2.14 2.42 2.66 2.74 2.65

1.3 0.75 0.83 0.93 1.00 1.11 1.24 1.32 1.34 1.46 1.63 1.80 1.92 1.99 2.00

1.4 0.63 0.70 0.78 0.84 0.93 1.03 1.09 1.13 1.22 1.34 1.45 1.55 1.61 1.63

1.5

1.6

1.7

1.8

1.9

0.81 0.88 0.94 0.98 1.05 1.14 1.23 1.34 1.36 1.39

0.83 0.87 0.93 1.00 1.07 1.14 1.18 1.21

0.83 0.89 0.95 1.01 1.05 1.08

0.86 0.91 0.94 0.97

0.86 0.89

1.5

1.6

1.7

1.8

1.9

0.80 0.87 0.92 0.98 1.05 1.12 1.17 1.20 1.27 1.36

0.82 0.87 0.93 0.99 1.03 1.06 1.12 1.19

0.83 0.88 0.92 0.95 1.00 1.06

0.84 0.87 0.91 0.96

0.82 0.87

c

k

2n=12

1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5

1.2 0.97 1.06 1.17 1.25 1.41 1.54 1.60 1.73 1.96 2.12 2.17 2.12 2.28 2.57

1.3 0.74 0.82 0.92 0.99 1.10 1.19 1.26 1.35 1.49 1.60 1.65 1.66 1.78 1.94

1.4 0.62 0.69 0.77 0.84 0.93 1.00 1.06 1.12 1.22 1.30 1.37 1.40 1.48 1.60

50

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