1. Beam buckling, beam bracing, ideal stiffness

 

This simple video illustrates lateral-torsional buckling for beams and explores the concept of the “ideal stiffness”, which is the stiffness at which buckling occurs between the brace points. In this case, a lateral spring is implemented at midspan at the top flange to web junction.

Ref.: Biju-Duval, Paul (2017). Development of three-dimensional finite element software for plate girders and tub girders during construction, PhD dissertation, The University of Texas at Austin

2. Column bracing, lateral buckling, torsional buckling

 

This simple video revisits fundamental concepts such as the Euler buckling load and illustrates the two different modes of buckling for columns: lateral buckling and torsional buckling. By adding two lateral springs along the beam at mid-depth of the web, the unbraced length for lateral buckling gets divided by three, hence torsional buckling controls.

Ref.: Biju-Duval, Paul (2017). Development of three-dimensional finite element software for plate girders and tub girders during construction, PhD dissertation, The University of Texas at Austin

3. Curved bridge - lateral bending diagram

 

Due to their geometry, curved bridges exhibit a combination of bending and torsion. The latter is composed of uniform torsion (St Venant torsion) - twisting of the girder cross-section - and warping - cross-sectional deformation. In design, it is essential to estimate the principal moment diagram as well as the torque diagram and the flange lateral bending diagrams. Lateral bending is directly dependent upon the unbraced length and cannot be captured by the traditional 6-DOF beam element. Shell elements in comparison are able to capture lateral bending stresses but plotting the lateral bending diagrams requires integration of the stresses across the flange width, which manually would be a tedious operation. mBrace3D automatically conducts this operation. In addition, mBrace3D also computes the principal bending, torque and shear diagrams along the girders.

Ref.: Biju-Duval, Paul (2017). Development of three-dimensional finite element software for plate girders and tub girders during construction, PhD dissertation, The University of Texas at Austin

Ref.: Davidson, J.S. and Yoo, C.H. (1996). "Local buckling of curved I-girder flanges", Journal of Structural Engineering, Vol. 122, No. 8, August

4. Curved bridge - global buckling, local buckling

 

This video illustrates the different buckling modes that affect steel bridges during erection: local buckling and global buckling. In general, many modes are a combination of global and local buckling, which makes the distinction between those modes less straightforward. Beam elements, even 7-DOF elements with warping DOF, are unable to capture all those modes. mBrace3D, by explicitly modelling all flanges and webs with shell elements, is able to compute all of them via an eigenvalue buckling analysis. For curved bridges, pre-buckling deformations are substantial and eigenvalue buckling analyses are therefore less meaningful, although they do give an upper bound on the bridge critical load.

Ref.: Biju-Duval, Paul (2017). Development of three-dimensional finite element software for plate girders and tub girders during construction, PhD dissertation, The University of Texas at Austin

5. Curved bridge - geometric nonlinear analysis (large displacement)

 

Geometric nonlinear analyses provide meaningful insight into the structural pre-buckling response, especially for curved bridges with large unbraced lengths. mBrace3D is able to conduct such large displacement analyses for plate girder bridges. By selecting before the analysis a number of DOFs along the bridge, the software also automatically computes the load vs. displacement curve at those DOFs.

Ref.: Biju-Duval, Paul (2017). Development of three-dimensional finite element software for plate girders and tub girders during construction, PhD dissertation, The University of Texas at Austin

6. Wind response - Eigenvalue frequency analysis

 

Bridge erectors rightfully worry about the influence of wind on the bridge stability during erection. In mBrace3D, wind loads are defined as uniformly distributed loads on the web and the structural response may be easily verified looking at the deflected shape but also at the flange lateral bending diagrams. The software is also able to compute the natural modes of the steel superstructure by solving the eigenvalue frequency analysis, using either the consist mass matrix or the lumped mass matrix - results using both methods are very close anyways. In this simple example, the first three modes consist of two bending modes followed by a torsional mode.

7. Tub girder buckling (Marcy Bridge), top lateral bracing

 

In the past two decades, a few bridges have collapsed due to global buckling during deck placement (or even simply under gravity loading), for example the Marcy Bridge in the state of New York in the early 2000s. Those failures have triggered significant research into global buckling, resulting for example in the derivation of closed-form equations. Here, the very structure of the Marcy Bridge, a straight tub girder bridge, is modelled. Wet concrete is modelled as top flange uniform loads and an eigenvalue buckling analysis is performed, quickly assessing the inaccuracy of the bracing under the loading applied. The video continues by showing how a few top lateral braces strategically located at the girder ends - where warping deformations are maximal - would have critically enhanced the bridge stability.

Ref.: Biju-Duval, Paul (2017). Development of three-dimensional finite element software for plate girders and tub girders during construction, PhD dissertation, The University of Texas at Austin

Ref.: Popp, D.R. (2004). UTrAp 2.0: Linearized buckling analysis of steel trapezoidal girders, Master's thesis, The University of Texas at Austin

Ref.: Yura, J.A. and Widianto (2005). "Lateral buckling and bracing of beams - A re-evaluation after the Marcy Bridge collapse", Structural Stability Research Council, Annual Stability Conference, Montreal, QC, Canada, April 6-9, 2005

8. Beam on elastic foundation

 

Closed-form solutions for beams on elastic foundation do exist but are quite complex. The finite element method, which is by nature an extension of the stiffness method, is well suited to address those situations. This video illustrates how the shear and moment diagrams, in addition to the deflected shape, may be quickly estimated by the software.

9. Curved bridge - erection sequence

 

In the US, bridge designers are required to provide at least one erection and placement sequence. This means that at all those stages, the bridge structural response - displacements, stresses - and stability must be checked. mBrace3D allows engineers to quickly define an erection or placement sequence in a single model, which is illustrated here on a four-girder three-span curved plate girder bridge.

Ref.: Steel Bridge Design Handbook, Design Example 3: Three-span continuous horizontally curved composite steel I-girder bridge, Publication No. FHWA-HIF-16-002 - Vol.23, December 2015

10. Exporting to LUSAS - results validation

 

Although mBrace3D is meant to be as user-friendly as possible, it is based on a series of algorithms that are intrinsically complex. As for any other software, results validation - deflections, stresses, cross-frame forces, buckling eigenvalues, etc. - is essential. mBrace3D automatically produces output files which are immediately exploitable in LUSAS to regenerate the mesh and reassign all the different attributes - cross-sectional profiles, boundary conditions, etc. This video illustrates how to easily import those output files into LUSAS to check the structural response, for example on a complex four-girder three-span curved plate girder bridge.

11. Support settlements

 

Support settlements - or supports not located at their theoretical elevation - may occur during construction. Here, a quick video shows how to simulate those settlements and evaluate their response on the bridge behavior.

12. Tub girder (trapezoidal box girder) - deck placement - lateral brace forces

 

Adequately bracing a curved tub girder during deck placement is not a straightforward task. It involves the estimation of lateral brace forces, which are due both to bending and warping of the open cross-section. 6-DOF beam models are unable to estimate those forces. Thanks to the explicit shell modelling of all steel plates, mBrace3D is able to accurately estimate whether bracing members are well designed and how they impact the overall response of the bridge.

Fan, Z. and Helwig, T.A. (1999). "Behavior of steel box girders with top flange bracing", Journal of Structural Engineering, Vol. 125, No. 8, August

Fan, Z. and Helwig, T.A. (2002). "Distortional loads and brace forces in steel box girders", Journal of Structural Engineering, Vol. 128, No. 6, June

Ref.: Yoo, C.H., Kim, K., and Choi, B.H. (2005). Research on horizontally curved steel box girders, Highway Research Center, Auburn University, Auburn, Alabama

13. Semi-circular beam - stress validation in LUSAS

 

This video is more applicable to architectural engineering than bridge design as such but gives meaningful insight into the lateral bending diagrams of a semi-circular beam under gravity loading. Displacements and stresses are verified in LUSAS.

14. Variable depth plate girder bridge - parabolic haunch - drop-in segment

 

In most cases, it is difficult to find in the public domain the detailed structural design of a bridge. Here, a variable plate girder bridge that was built in the state of Iowa a few years ago is explicitly modelled. Using beam elements, it is relatively easy to model parabolic haunches. However, this task is much more complex when using shell elements. mBrace3D is able to model those systems, which are often awarded national prizes thanks to their aesthetic appeal - they are a reminiscence of arch bridges. Unlike conventional plate girders, haunched girders carry significant amounts of shear stresses through their inclined flanges - the web alone does not carry the entirety of the shear flow. Although no information was found regarding the erection sequence of this particular bridge, it is decided to select one that implies the use of drop-in segments - to illustrate that feature in mBrace3D.

Ref.: Nielsen, S. and Olson, K. (2013). The New Keosauqua Bridge - Construction of the new bridge, NieKo

15. Skewed plate girder bridge - lean-on bracing - temperature loads

 

This video shows how to define lean-on cross-frames on a skewed plate girder bridge. Boundary conditions are also defined such that the bridge movement under thermal expansion/ contraction is unrestrained, resulting in zero stresses and cross-frame forces.

Ref.: Steel Bridge Design Handbook, Bracing System Design, Publication No. FHWA-HIF-16-002 - Vol. 13, December 2015

16. Cross-frame fatigue analysis - influence surfaces

 

mBrace3D is able to compute influence surfaces for any type of variable - displacement, stress, etc. Here, three traffic lanes are defined on a skewed plate girder bridge and influence surfaces for the axial forces in the cross-frames most closely located to the first abutment are derived. Based on those influence surfaces, the software then calculates the vehicle load configurations leading to the maximal stress range in each of those cross-frame members, keeping in mind that the maximal and minimal effects are to be obtained for the same lateral positioning of the truck load.

17. Cross-frame fatigue analysis validation - discrete loads on deck

 

This quick video illustrates how to verify the magnitude of the maximal and minimal cross-frame forces calculated during a previously defined fatigue analysis via the definition of discrete point loads over the bridge deck. This approach is applicable to any other quantity addressed by the vehicle load optimizer.

18. Wearing surface load, barrier line load

 

In design, a part of the composite load consists of the wearing surface and the barrier loads. This video shows how to simulate those loads in mBrace3D - wearing surface loads are modelled as uniformly distributed loads over the full deck area, barrier loads as distributed loads over the outer shells of the overhangs. Support reactions are checked and moment diagrams compared with the ones obtained from a pseudo-3D analysis in LUSAS - where the deck is modelled with shell elements and the girders with beam elements offset from the deck.

19. Cross-frame member eccentricity, stiffness reduction

 

Researchers at the University of Texas at Austin have shown that eccentric cross-frame connections - for example, single angle members attached on connecting plates - result in an overestimation of the cross-frame axial stiffness. mBrace3D is able to automatically implement stiffness reduction factors based on the closed-form equation that was found to estimate that stiffness reduction. This concept is applied to a curved, skewed plate girder bridge, where cross-frames are primary structural members and the structural behaviour therefore largely dependent upon their stiffness.

Ref.: Battistini, A.D., Wang, W., Helwig, T.A. and Engelhardt, M.D. (2016). "Stiffness behavior of cross frames in steel bridge systems", Journal of Bridge Engineering, Vol. 21, No. 6, June

20. Influence surfaces, vehicle load optimization

 

mBrace3D is able to find critical live load configurations based on influence surfaces. This is illustrated here on a simply supported plate girder bridge for three quantities: vertical support reaction, vertical displacement, composite bending moment. The software allows for the definition of any number and type of truck loads and traffic lanes. The moving load analysis parameters closely follow AASHTO terminology but are intended to cover any other design code.

21. Live load envelope shear and moment diagrams, AASHTO HL-93 truck loading

 

Provided the composite moment and shear are investigated at enough locations along the bridge length, mBrace3D is able to automatically plot the envelope live load moment and shear diagrams, which are immediately applicable to design. Here, a two-span continuous four-girder bridge directly taken from an AISC/NSBA manual is analyzed, and the envelope live load moment and shear diagrams for an exterior girder derived.

Ref.: Steel Bridge Design Handbook, Design Example 2A: Two-span continuous straight composite steel I-girder bridge, Publication No. FHWA-IF-12-052 - Vol.21, November 2012

22. Geometric nonlinear analysis of a straight column with an initial imperfection

 

An eigenvalue buckling analysis and a geometric nonlinear analysis (large displacement analysis) are conducted on a 50-ft. long W14x53 straight column. An initial imperfection (pure sweep) of L/500 is included via the modelling of an initial curvature. Simple geometric considerations show that the magnitude of the imperfection is here equivalent to a radius of curvature of 3,125-ft. The P vs. delta curve for the lateral displacement at midspan is automatically calculated. Results from mBrace3D are checked against a similar analysis in LUSAS. For a steel column, the large displacements would result in stresses well over the yield strength, hence a further analysis is conducted in LUSAS, this time including material nonlinearities. At the moment, mBrace3D is able to include geometric nonlinearities only.

23. Placement analysis - Camber - Differential deflection due to early concrete stiffening

 

A balanced deck placement sequence is defined on a two-span four-girder plate girder bridge. mBrace3D is able to capture the partially composite action between the deck and the steel superstructure via the modelling of shear studs as link elements. Those link elements have a time-dependent stiffness, whose magnitude was determined experimentally by earlier tests at the University of Texas at Austin by Topkaya. As the concrete deck hardens and the bridge gradually stiffens, even a balanced placement sequence can lead to an unbalanced camber diagram, which is the object of this video.

Ref.: Biju-Duval, Paul (2017). Development of three-dimensional finite element software for plate girders and tub girders during construction, PhD dissertation, The University of Texas at Austin

Ref.: Topkaya, C. (2002). Behavior of Curved Steel Trapezoidal Box Girders During Construction, Ph.D. Dissertation, The University of Texas at Austin, Austin, TX

Ref.: Engineering for Structural Stability in Bridge Construction, Publication No. FHWA-NHI-15-044, April 2015

24. Column lateral, torsional, local buckling - validation with LUSAS

 

This video explores the different buckling modes for columns: lateral buckling about the weak axis, torsional buckling, lateral buckling about the strong axis, local buckling. An unstiffened 50-ft. long W14x53 is taken as an example. The first 14 modes derived during the eigenvalue buckling analysis are checked in LUSAS. This video is intended to demonstrate the accuracy of the state-of-the-art eigensolver implemented in mBrace3D, called FEAST.

Ref.: Biju-Duval, Paul (2017). Development of three-dimensional finite element software for plate girders and tub girders during construction, PhD dissertation, The University of Texas at Austin

Ref.: Polizzi, E. (2009). “Density-Matrix-Based Algorithm for Solving Eigenvalue Problems”, Physical Review B, Vol. 79, No. 11, March

25. Straddle bent and dapped ends

 

This video quickly explains the easy implementation of dapped ends end straddle bents in mBrace3D. Straddle bents are modelled as vertical springs elements, whose axial stiffness corresponds to the flexural stiffness of the straddle bent. Dapped ends on the other hand only require the definition of one additional cross-sectional profile.

26. Global lateral buckling - geometric nonlinear analysis with an initial imperfection

 

This video explores the concept of global lateral buckling for a straight twin-girder bridge. Point loads are applied at the ends to simulate a uniform moment along the girders. An eigenvalue buckling analysis is conducted first. Then, a geometric nonlinear analysis is performed, with an initial imperfection entered as a fraction of the global lateral buckling mode.

Ref.: Yura, J., Helwig, T., Herman, R. and Zhou, C. (2008). "Global lateral buckling of I-shaped girder systems", Journal of Structural Engineering, Vol. 134, No. 9