Lateral Buckling Analysis of a Steel Pony Truss

8/10/2019 Lateral Buckling Analysis of a Steel Pony Truss

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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 MemberVernon Schaefer - Committee Member

Iowa State University of Science and Technology

Ames, IA

2012


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Contents

List of Symbols and Abbreviations................................................................................................ iv

List of Figures ................................................................................................................................. v

1. Introduction and Objective ...................................................................................................... 11.1 Introduction ...................................................................................................................... 1

1.2 Objectives ......................................................................................................................... 2

2. 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 Out-of-plane Deformations ......................................... 16

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

3. Finite Element Analysis ........................................................................................................ 18

3.1 Finite Element Model of the 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

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

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

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

4.2.1 Calculations following Engessers Procedure ........................................................ 25

4.2.2 Calculations following Bleichs Procedure ............................................................ 25

4.2.3 Calculations following Timoshenkos Procedure ................................................... 26

4.2.4 Calculations following Lutz and Fishers Procedure .............................................. 26

4.2.5 Calculations following Holts Procedure ................................................................ 26

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

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


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4.3.1 Two 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

5. Summary, Conclusions, and Recommendations ................................................................... 38

5.1 Summary ........................................................................................................................ 38

5.2 Conclusions .................................................................................................................... 39

5.3 Recommendations .......................................................................................................... 39

Appendix A ................................................................................................................................... 40

Appendix B ................................................................................................................................... 43

Appendix C ................................................................................................................................... 45Appendix D ................................................................................................................................... 47

References ..................................................................................................................................... 50


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iv

A area of compression chord

b length of floor beamsC spring stiffness of interior supports

Ce spring stiffness of end supportsCE Engessers spring stiffness

C0 required spring stiffness with rigid end supportsc ratio of C to C0

d length of diagonal end members

E modulus of elasticityEt tangent modulus of elasticity

h height of truss

I moment of inertiaIb moment of inertia of floor beam

Ic moment of inertia of vertical web member

Id moment of inertia of diagonal web memberK column stiffness: K

2= P/EI

k effective length coefficient

ks joint spring stiffness

l distance between panelsL total length of the truss

Ld length of diagonal web member

Le effective lengthM bending moment

m number of buckling modes

n number of bays

P compression loadPcr critical buckling load

Pd axial load on diagonal end members

Q virtual loadq distributed compressive force

qo maximum compressive load with a varying load distribution

r radius of gyration: r2= I/A

U internal energy

v factor of safety

V external workx distance from end support

y displacement of chord at point x elastic foundation constant

maximum displacement of compression chord

relative displacement of vertical web members

p potential energy

angle of rotation at joints

stiffness of compression chord

Schwedas elastic end support coefficient


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Figure 1.1 Pony Truss Bridge............1

Figure 2.1 Equilibrium Path for Initially Straight Column...........4

Figure 2.2 Equilibrium Path for Slightly Crooked Column..........4

Figure 2.3 Euler Buckling.........5

Figure 2.4 Buckling Modes for a Bar with Pin Ends........6

Figure 2.5 Elastically Supported Bar........7

Figure 2.6 Column on elastic supports......9

Figure 2.7 Varying Axial Load Distribution.......10

Figure 2.8 Compression Chord with Elastic 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...21

Figure 3.3 Stress-Strain Curve........22

Figure 4.1 Compression Chord Design Curve........23

Figure 4.2 Energy Method Design Curve ..........24

Figure 4.3 2-D Compression Chord Elements............29

Figure 4.4 3-D Compression Chord Elements........30

Figure 4.5 Nonlinear Load vs. Displacement Curve... .......30

Figure 4.6 Pony Truss Top Chord Analysis........31

Figure 4.7 Rigid Frame Boundary Conditions............33

Figure 4.8 Rigid Frame Displacements...........33

Figure 4.9 Load Application of Pony Truss Frames...........34

Figure 4.10 Pony Truss Lateral Displacement at Frame.......35

Figure 4.11 Compression Chord with New Stiffness........37

Figure 4.12 Compression Chord with Elastic Ends......37


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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 fora 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.

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

Fig. 1.1c Section A-A

n Panels

A

A

Fig. 1.1b Pony Truss Plan View

Fig. 1.1a Pony Truss Elevation View


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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 theresearch 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. Engessers 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 writers knowledge, all of the research for determining the critical buckling load on a

compression chord with elastic supports is based on Engessers 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.


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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 load-

displacement 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


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displacements increase. Analyses for both columns, straight and slightly crooked, lead to large

lateral displacements at the critical load.

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), Eulers 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. Eulers formula was finally validated in 1889 when Considre 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 readers 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)

=

+ /

Fig. 2.1 Equilibrium Paths for

Initially Straight Column

Fig. 2.2 Equilibrium Paths for

Slightly Crooked Column

=

/


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

= sin + cos (2.3)

where, K, is equal to/. The constants A and B in Eq. 2.3 can be calculated utilizing thesupport conditions at both ends of the bar. For a pinned-end column, the boundary conditionscan be set as y = 0 at x = 0 and y = 0 at x = L. These conditions will result in:

B = 0 and 0 = sin (2.4)Equation 2.4 can then be written as

0 = sin = (2.5)

By substituting K = /into equation 2.5 and solving for P, Euler buckling equation yieldsthe critical buckling load.

= (2.6)

Fig. 2.3 Euler Buckling (Thandavamoorthy 2005)

/

/


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Fig. 2.4 Buckling Modes for a Bar with Pin Ends

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.

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.


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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 1800s 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 Engessers

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, Creq, of the

Figure 2.5 Elastically Supported Bar (Bleich, 1952)

P

P

P

PP

P(a)

(b)

(c)


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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.

Engessers solution for the required stiffness of a pony-truss transverse frame which is derived

in Appendix C is = (2.7)If Creqis met at each frame location, the chord with the length between panels, l, will

achieve the specified design load, Pcr. 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 Pcr, 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 Eulers buckling equation and Eq. 2.7, the required spring constant is

= (2.8)The use of Engessers 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.


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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.

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)

Figure 2.6 Column on Elastic Supports

A B

L = nl

P

P

P

P

(a)

(b)


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= 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, Timoshenkos solution assumed that

the ends were pin connections.

Figure 2.7 Varying Axial Load Distribution (Timoshenko, 1936)


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Similar to Engesser, Timoshenko assumed an equivalent elastic foundation, , 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 qois 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)


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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 Bleichs results to include chords with elastic ends.

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

was

= (2.20)where vP = Pcand = = The stiffness of the chord is represented by which is equal to

= (2.21)To eliminate the tangent modulus, Et, which varies per the axial load, Eulers buckling

equation is substituted into equation 2.21, simplifying the stiffness to = (2.22)For a chord with nspans in equation 2.20, there are n-1 different half-wave buckling

configurations. As nincreases, the spring constant required for an infinite number of spans,

C, increases to a limiting value. Bleich (1952) showed that Cncan be replaced by Cfor any

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


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span where nis greater than six which corresponds to an error less than 1%. So, for trusses

with more than six spans, equation 2.16 simplifies to =

= (2.23)where is given in Table 2.1 from Bleich (1952) and is valid for the elastic and plastic range

of buckling.

Since Bleichs 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 Bleichs theory. Thus, in order to

continue on Bleichs 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 Ceand the intermediate supports

by C. Schweda assumed the load was a constant axial force throughout the length of the

1/k 1/k 1/k 1/k

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

Table 2.1 Values of in Eq. 2.23 (Bleich, 1952)


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chord with equally spaced elastic supports similar to Bleichs theory. Figure 2.8 shows the

compression chord with the diagonals extended a length, d, subjected to the compressive force

vPdand pinned at points n and +n. The spring constant must be larger the spring constant of

the chord with rigid end, C0. Thus, the value of C = cC0where c > 1.1. The required spring

constant with the axial load in the diagonal is

= + (2.24)where CEis Engessers equation (Eq. 2.8). Schweda calculated as a function of the number

of bays with respect to cand k. These values are listed in Appendix D. If all of the cross-

frames are identical, Cein equation 2.24 is equal to C and the equation becomes

= (2.25)2.4.5

Holts 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 Holts 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


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effective length factor is a function of Cl/Pcwhich is shown in the results section. Holt noted

that Bleichs analysis showed adequate results for the entire range of effective length values

where Timoshenkos results show adequate results for k > 2. A summary of Holts 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.

1/k for Various Values of Cl/Pc and n

1/k

n

4 6 8 10 12 14 161.00 3.686 3.616 3.660 3.714 3.754 3.785 3.809

0.980 3.284 2.944 2.806 2.787 2.771 2.774

0.960 3.000 2.665 2.542 2.456 2.454 2.479

0.950 2.595

0.940 2.754 2.303 2.252 2.254 2.282

0.920 2.643 2.146 2.094 2.101 2.1210.900 3.352 2.593 2.263 2.045 1.951 1.968 1.981

0.850 2.460 2.013 1.794 1.709 1.681 1.694

0.800 2.961 2.313 1.889 1.629 1.480 1.456 1.465

0.750 2.147 1.750 1.501 1.344 1.273 1.262

0.700 2.448 1.955 1.595 1.359 1.200 1.111 1.088

0.650 1.739 1.442 1.236 1.087 0.988 0.9400.600 2.035 1.639 1.338 1.133 0.985 0.878 0.808

0.550 1.517 1.211 1.007 0.860 0.768 0.708

0.500 1.750 1.362 1.047 0.847 0.750 0.668 0.600

0.450 1.158 0.829 0.714 0.624 0.537 0.500

0.400 1.232 0.886 0.627 0.555 0.454 0.428 0.383

0.350 0.530 0.434 0.352 0.323 0.292 0.280

0.300 0.121 0.187 0.249 0.170 0.203 0.183 0.187

0.293 0

0.259 0

0.250 0.135 0.107 0.103 0.121 0.112

0.200 0.045 0.068 0.055 0.053 0.070

0.180 00.150 0.017 0.031 0.029 0.025

0.139 0

0.114 0

0.100 0.003 0.010

0.097 0

0.085 0

Table 2.2 (Galambos, 1988)


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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-of-

plane 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 where the stiffness required, Creq, is usually twice the

ideal stiffness . Lutz and Fisher used Engessers 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 = 2.5 (2.26)where Le= kland

2

/4 2.5. For trusses with a small lrelative to Le equation 2.26 provides anaccurate bracing stiffness for the solution. However, as lincreases relative to Le, then equation2.26 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 Winters stiffness of . Equation 2.27 provides aminimum 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.


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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 bedesigned. 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.


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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 10x10x5/8 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 Youngs 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


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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 bendingcapabilities 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 andy 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


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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 modeledusing 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.

The moment of inertia for the 10x10 box section was calculated to be 418.3 in4. By rearranging

Eq. 2.7, the tangent modulus of elasticity can be calculated as 7344 ksi for the chord. Thistangent 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 stress-

strain 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 Engessers 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.


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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.

Fig. 3.2 Finite Element Idealization of the Pony Truss


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0

0

0

0

0

0 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.0

000

Fig. 3.3 Stress-Strain Curve


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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 Eulers load, Pe, to the critical

compressive load, Pcr, of the member.

0.000

0.00

0.00

0.00

0.00

.000

.00

0.000 0.00 .000 .00 .000 .00 .000 .00 .000 .00

Fig. 4.1 Compression Chord Design Curves


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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

0

0

0

0

0

00

0

0 00 000 00 000 00 000

Fig. 4.2 Energy Method Design Curve


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Then, the ratio of Cl/P was determined as

=

. = 1.50

Using the calculated ratio Cl/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

= .. = 50.3Following the design steps that are given in the AISC (2011) manual, one can calculatethe critical buckling stress, Fe, and the nominal buckling load, Pn, using the ASD and

LRFD approaches, respectively as

F = = . = 113.2 ksiF= 0.658 F= 0.658. 36 = 31.5 ksi

Hence, the critical buckling load is:

P= FA= 31.525.0 = 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

= .. = 51.1F = = . = 109.8 ksiF= 0.658 F= 0.658. 36 = 31.4 ksi


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P= FA= 31.425.0 = 4.2.3

1/k = 0.604 (From Fig. 4.1)

k = 1.66

= .. = 64.9F = = . = 67.9 ksiF= 0.658 F= 0.658. 36 = 28.8 ksi

P= FA= 28.825.0 = 4.2.4

1/k = 0.719 (From Fig. 4.1)

k = 1.39

= .. = 54.5F = = . = 96.2 ksi

F= 0.658 F= 0.658. 36 = 30.8 ksiP= FA= 30.825.0 = 4.2.5

1/k = 0.75 (From Table 2.2)

k = 1.33

=

.. = 52.3F = = . = 104.7 ksi

F= 0.658 F= 0.658. 36 = 31.2 ksiP= FA= 31.225.0 =


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4.2.6

P = = . = 11.84 kips=

.. = 912.2

= 61.67 (From Fig. 4.2)P= PR = 11.8461.67 = 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 Cl/P ratio.

However, it was later addressed that the compression chord in Galambos example was designed

for the maximum panel load using AASHTOs 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.

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.

Method k Pn F.S

Engesser 1.28 788 2.19

Bleich 1.30 785 2.18

Timoshenko 1.66 721 2.00

Lutz & Fisher 1.39 770 2.14

Holt 1.33 779 2.16

Energy Method - 730 2.03

Table 4.1 Critical Load Results


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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, Et= 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 Et= 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.


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Fig. 4.4 3-D Compression Chord Elements

Fig. 4.3 2-D Compression Chord Elements


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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

Fig. 4.5 Nonlinear Load vs. Displacement Curve


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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


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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

418.3 in4. 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,

similar to before, and the inertia of the compression chord was varied from 425 in4to 0.001 in

4.

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.


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Fig. 4.7 Rigid Frame Boundary Conditions

Fig. 4.8 Rigid Frame Displacements


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`

Panel Location Displacement, (in) Stiffness, C (k/in)

1 0.10612 9.426

2 0.07518 13.301

3 0.07249 13.795

4 0.07227 13.837

5 0.07226 13.839

6 0.07226 13.839

7 0.07226 13.839

8 0.07227 13.837

9 0.07249 13.795

10 0.07518 13.301

11 0.10612 9.423

Fig. 4.9 Load Application of Pony Truss Frames

Table 4.2 Resultant Stiffness at Each Bay

1

2

5

34

67

89

10

11


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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. Engessers

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

Moment of Inertia (in4) Displacement, (in) Stiffness, C (k/in)

425 0.048816 13.89

325 0.052882 12.98225 0.058840 11.87

125 0.069192 10.45

25 0.100502 8.08

0.001 0.141938 7.04

418.3 6.75 (using Eq. A1.1)

Fig. 4.10 Pony Truss Lateral Displacement at Frame Location

Table 4.3 Effects of Top Chord Inertia on the Lateral Stiffness


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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.

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.

Method K Pn F.S

Engesser NA NA NA

Bleich 1.02 827 2.30

Timoshenko 1.16 807 2.24

Lutz & Fisher 1.08 819 2.28

Holt 1.01 828 2.30

Energy Method - 1032 2.87

Table 4.4 Critical Load Results with New Stiffness, C


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Fig. 4.11 Compression Chord Analysis with New Stiffness

Fig. 4.12 Compression Chord Analysis with Elastic Ends


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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.

Method

C = 6.75 C = 13.8

k Pn k Pn

Engesser 1.28 788 NA NA

Bleich 1.30 785 1.02 827

Timoshenko 1.66 721 1.16 807Lutz & Fisher 1.39 770 1.08 819

Holt 1.33 779 1.01 828

Energy Method - 730 - 1032

FEM Chord - Rigid Ends - 719 - 979

FEM Chord Elastic Ends - 644 - 886

FEM Pony Truss Pcr= 1121 kips

Table 5.1: Buckling Load Analysis Results


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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 theappropriate 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 Holts 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.


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

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

b

Ic

Ib

A

C D

B


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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:

1 2 = + Or; = (A1.1)

2.

To account for truss diagonals, a term of Ld3/3Idis 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

Where;

ksis the joint rotational stiffness. The moment, M, at the vertical-floor beam connection is: = = = /Following the analysis that was summarized in the section above, the relative displacement

between points A and D can be calculated as:

11

A

B C

D

ks ks


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= 2 = + + 2

1 =

+

+

1 = + + 1 = + + =

(A1.3)

if ks= , 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 bemodified 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.


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

1. Initial conditions

2. Displaced shape

y = a sin y=

cos

y= sin Where: y= = asin and y= = a sin

3. External Work

V = ydx V = cos dx V = + sin

/ //


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44

V = 4. Internal Energy

U = C+ C+ dxU = C+ C+ sin dx =

n n

n

=

n

n

5. Combine external work and internal energy and solve for Pcr

=

= n

n

= .

. =

n

n

= n

n

= sin n (B1.1)


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

1. Initial conditions

2.

Displaced shape with continuous elastic medium

= n

=

=

n

3.

External Work

= ydx

V = cos

dx

V = + sin

V =

/ //


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4. Internal Energy

= ydx + n x where: = C/(L/3)=

n

n

=

5.

Combine external work and internal energy and solve for P

=

=

= .

. =

= + (C1.1)6. Solve for required stiffness coefficient

P = + where: = C/L and v = L/m (wavelength)Note that Pcrwill be at a minimum when

=

= + = Substituting wavelength, v, back into the equation above:

C= (C1.2)


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Schwedas extension of Bleichs analysis

=

sinh2 1sin sinh sin2 1 + 2 2 1 2 1 cosh (D1.1)Where:

= 2 2 + =

= 1 + cos

= 1 + 2 cosh = + 1+ + 1+ 2 cos = + 1+ + 1+


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Tables of Factor :

kc

1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

2n=6

1.2 1.00 0.78 0.651.3 1.25 0.93 0.75

1.4 1.41 1.04 0.84

1.5 1.39 1.06 0.87

1.6 1.35 1.10 0.94 0.82

1.7 1.65 1.30 1.09 0.94

1.8 2.01 1.54 1.25 1.06 0.94

1.9 2.41 1.77 1.40 1.18 1.03 0.90

2.0 2.79 1.97 1.55 1.29 1.11 0.97 0.85

2.1 2.98 2.12 1.64 1.36 1.17 1.02 0.90

2.2 2.93 2.10 1.66 1.38 1.19 1.04 0.94

2.3 2.69 1.97 1.60 1.35 1.17 1.04 0.94 0.85

2.4 2.27 1.76 1.48 1.27 1.12 1.00 0.91 0.83

2.5 1.99 1.71 1.48 1.32 1.17 1.06 0.96 0.89

kc

1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

2n=8

1.2 0.93 0.74 0.62

1.3 1.12 0.86 0.71

1.4 1.27 0.96 0.80

1.5 1.32 1.01 0.85

1.6 1.42 1.12 0.94 0.81

1.7 1.69 1.28 1.06 0.901.8 1.96 1.44 1.17 0.99 0.87

1.9 2.07 1.54 1.24 1.06 0.92

2.0 2.04 1.54 1.26 1.08 0.95 0.85

2.1 1.85 1.47 1.25 1.08 0.96 0.87

2.2 2.08 1.66 1.39 1.20 1.05 0.94 0.85

2.3 2.43 1.89 1.55 1.32 1.16 1.03 0.92

2.4 2.83 2.13 1.73 1.46 1.26 1.11 1.00 0.90

2.5 3.25 2.37 1.90 1.58 1.36 1.20 1.07 0.97


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kc

1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

2n=10

1.2 0.99 0.75 0.63

1.3 1.06 0.83 0.70

1.4 1.21 0.93 0.781.5 1.26 1.00 0.84

1.6 1.43 1.11 0.93 0.81

1.7 1.64 1.24 1.03 0.88

1.8 1.74 1.32 1.09 0.94 0.83

1.9 1.72 1.34 1.13 0.98 0.87

2.0 1.85 1.46 1.22 1.05 0.93 0.83

2.1 2.14 1.63 1.34 1.14 1.00 0.89

2.2 2.42 1.80 1.45 1.23 1.07 0.95 0.86

2.3 2.66 1.92 1.55 1.34 1.14 1.01 0.91

2.4 2.74 1.99 1.61 1.36 1.18 1.05 0.94 0.86

2.5 2.65 2.00 1.63 1.39 1.21 1.08 0.97 0.89

kc

1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

2n=12

1.2 0.97 0.74 0.62

1.3 1.06 0.82 0.69

1.4 1.17 0.92 0.77

1.5 1.25 0.99 0.84

1.6 1.41 1.10 0.93 0.80

1.7 1.54 1.19 1.00 0.87

1.8 1.60 1.26 1.06 0.92 0.82

1.9 1.73 1.35 1.12 0.98 0.872.0 1.96 1.49 1.22 1.05 0.93 0.83

2.1 2.12 1.60 1.30 1.12 0.99 0.88

2.2 2.17 1.65 1.37 1.17 1.03 0.92 0.84

2.3 2.12 1.66 1.40 1.20 1.06 0.95 0.87

2.4 2.28 1.78 1.48 1.27 1.12 1.00 0.91 0.82

2.5 2.57 1.94 1.60 1.36 1.19 1.06 0.96 0.87


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Documents

Lateral Buckling Analysis of a Steel Pony Truss