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There are a couple of ways to analyze suspension bridges. As previously discussed (Uniqueness and Difficulties of Suspension Bridge Analysis), the deflection theory proposed by Melan and solved by Moisseiff is one of the first. The second one may be trigonometric methods proposed by Timoshenko (Theory of suspension bridges, Journal of the Franklin Institute, Volume 235, Issue 4, April 1943, Pages 327-349). In this article, a brief explanation of trigonometric methods and related excel, and the example bridge dimensions will be provided.

**How to Perform Time-dependent Analysis?**

Mainly there are two ways to perform the time-dependent analysis. One is the time-step analysis and the other is the age-adjusted method. The age-adjusted method is a simplified one and can consider the long-term effects with only a one-step analysis. Of course, this is an approximate way to consider..

In the blog article intro to Time-dependent Analysis for Concrete Structures, we have touched upon the importance of construction stage analysis for concrete structures. The material time function can be plotted and inputted into analysis software like midas Civil to simulate their changing material behavior in various stages of the construction. This article will go over the process of calculating various parameters that contribute to the shape and location of the material's time functions.

It is easy to obtain the result from bridge finite element analysis, but to get more accurate results requires extra efforts. Even the most robust finite element analysis solvers adopts the method that approximates the structural behavior, by minimizing the associated error function compared with the complex function that represents the realistic structural behavior.

**Why is cable-stayed bridge difficult to analyze? **

Cable-stayed bridges, including so-called Extradosed bridges, do not have any classical solutions. Computer based displacement method is the only way to analyze cable-stayed bridges and it is hard to check the output. Also, in the cable-stayed bridge analysis, the following three nonlinearities should be considered.

From the previous tip basic nonlinear analysis explained, Dr. Seungwoo Lee talked about some fundamental differences between linear and nonlinear analysis in structural engineering. In a linear analysis, the relationship between the stress and strain of a model is held constantly, and the stiffness matrix of the model stays the same throughout the analysis. For a nonlinear analysis, there can be various factors that contributes to its nonlinearities, for example, material yielding, nonlin,,,

Generally a load rating is the bridge's ability to carry the live load for design and allowable legal vehicles, overloads, and determine weight postings. But how to perform the load rating of Curved and Complex Geometry Composite Steel Bridges in Midas Civil? Tom Less will expand the knowledge of Two-Dimensional/Grillage Modeling and Three-Dimensional Modeling and show how he sets up the inputs like rating design code, rating parameters, unbraced length, fatigue parameter, etc., for bridge rating design.

In this example, we are designing a pile cap for a bridge. We have already calculated the forces at the top of the footing, but unclear of the footing dimensions along with the forces at its bottom face.

Curved girder analysis is not easy like straight girders, and only limited software can handle this problem. Dr. Watanabe developed a full influence line theory for three-span continuous curved girders. The detailed theory is quite interesting, but only the methodology of how to use his methods is discussed in the article. His methods give an exact solution for a symmetric three-span curved girder, and the warping effects are not included. As always, start with an example. The input and output are self-explanatory and detailed discussion is not provided.

We are designing a pile cap for a bridge. We already calculated to forces at the top of the footing (we have not designed the footing yet and do not know the footing dimensions or the forces at the bottom). The factored forces at the footing top are given as Pu = 4212 kips, Vu longitudinal = 108.4 kips, Vu transverse = 112.4 kips, Mu Longitudinal = 7340 ft-kips, Mu transverse = 16480 ft-kips.

Iteration is just a repeated calculation. If we want to solve an equation x + 1 = 5 , we can assume x and check whether x + 1 = 5 or not. If not, we can try another value of x and repeat this calculation until x + 1 = 5. In this simple equation, we can solve x = 5 - 1 = 4 easily. But some engineering problems are more complicated and iteration is more efficient and/or sometimes iteration is the only way to find the solution.

One of the best examples of understanding creep behavior was developed by Dr. El-Badry. This example is from Dr. El-Badry’s paper with some modifications. At time t0 = 7 days, the cantilever is subjected to a uniform load q = 0.225 klf (section is 12”×18”). At time t1 = 30 days, simple support is introduced at B, thus preventing the increase in deﬂection at B due to creep. Determine the end reaction RB at t2 = ∞. Ignore the diﬀerence between Ec(t0) and Ec(t1). RH (percent) is relative humidity and assumed 70%. f’c = 5 ksi (=34.47 MPa). L = 5ft.

We know the fixed end force changes between time t_{1} and t_{2}, and we can calculate the reaction changes between the same time intervals. Not all commercial programs can handle these fixed end forces, but we can convert fixed end forces into nodal loads and do the post-processing easily.

For creep analysis, the most common problem in the real-world design is continuous girders built as span by span. This example is very well explained by Dr. Ghali et al. (Concrete structures, Stresses, and deformations, 4^{th} ed., CRC press, Example 4-2). Dr. Ghali et al. explained this problem by flexibility methods. The author will solve this same problem by stiffness methods. The programs do the matrix formulation and equation solve, and only the load matrix formulation and post-processing are our concern in the stiffness methods. Two MIDAS files are attached.

To better understand the creep behavior, solve the previous example in a less efficient way. Here, different sign conventions will be applied.

At time t_{0}, the immediate displacements/rotations for member 1 due to uniform loads are

In this article, the previous method explained in Creep Analysis 4 is applied to a real-world bridge and compares the output with MIDAS’.

In their paper, Dr. Elbadry et al. explained how to calculate long-term effects using conventional programs (Analysis of time-dependent effects in concrete structures using traditional linear computer programs, Canadian Journal of Civil Engineering, February 2001.). In this paper, Dr. Elbadry et al. modeled post-tensioning tendons with truss elements. A very detailed explanation is discussed in the paper, and the readers should not have any difficulty understanding the methodology. (Refer to MIDAS file ex6.5)

Think about a two-span continuous bridge, as shown in Fig 1. Let’s calculate the secondary creep moments. Detailed dimensions and creep factors, etc., are not important, and the MIDAS file “Creep 2ndary Check” is attached for the reader’s reference. Perform only the creep analysis, and the 2ndary creep moments at time=infinity are shown in Fig 2. We got 351 ft-kips at the pier location.

Influence line analysis is essential for bridge design. Classical influence line theory for continuous girders goes back to Müller-Breslau in the late 18th century. Currently, iterative analysis is commonly used since this method is rather simple to implement with the displacement methods. Both methods give the same results and we can use either of these methods.

The effects of the curvature sometimes can not be ignored in the design process. The following example uses a three-span continuous curved composite steel bridge. You will see a comparison result of the straight beam and the curved beam in this article. It will show you how the curvature may influence the design.

It looks not easy even to calculate section properties. But midas software has a powerful function to calculate these section properties for us. Just select PSC-Value>Define by Coordinates, then midas Civil will do the rest, or at least most of them. The methodology to calculate section properties from coordinates requires repeated calculation.

We have structural analysis programs. Can we make the model (like other conventional bridges) and perform the analysis (as usual)?

In eq(3), C1 and C2 are integral constants, Mo(x) is the moment as a simple span for the given loading p(x), y(x) is cable coordinates from the pylon top, and c is defined as follows.

Although the main span of 1480 ft may not be impressive, the Manhattan Bridge is the first bridge designed by a non-linear theory and called the first modern suspension bridge.

With the success of the Manhattan Bridge, Moisseiff became confident with his solution and designed the George Washington Bridge, a main span of 3500 ft, which is more than twice the Manhattan bridge. The George Washington Bridge was opened in 1931 and is the first bridge with a main span of more than 1 km (3000 ft).

Dr. Hirai was a Japanese engineer, professor, and researcher. In 1940, Dr. Hirai went to a movie theater and watched the news about the failure of the old Tacoma bridge. He was very inspired and started his research on wind engineering for suspension bridges by himself. His papers had been published since 1942, but they were rarely introduced into the Western world because Japan was isolated during World War II.

There are many programs that can check the strength of members with axial force and moment. However, not many programs can check the serviceability and strength with prestressing. In this and the next article, we will discuss the details of these two cases.

The previous article (Curvature effects on a medium-span curved bridge) showed that we should be cautious to get reasonable torsional moments through simple beam analysis. One of the easiest ways to refine the results is to add more nodes at the inner support locations, however, we still have a question about “how many?”. Now we are reviewing the effect of curvature for bending moments.

We are trying to find bending moments for three spans continuous curved girder, 150

ft + 223 ft + 150 ft = 523 ft, the radius is 1182’-6” as shown.

Finally we came to the composite section analysis. One of the best examples is from Dr. Gilbert and Dr. Ranzi (Example 5.10, Time-dependent behavious of concrete structures, CRC Press, 2010.)

Traditionally AASHTO and ACI gave us β=2 and θ=45° and these values are also allowed at the current AASHTO LRFD for some limited cases as defined in 5.7.3.4.1.

Our fore-engineers were aware of these values are very rough estimates but had to wait until the mid-80s before professors at the University of Toronto fully developed the modified compression field theory through extensive theoretical and experimental research (Prestressed concrete structures by Collins and Mitchell, 1991).

Believe it or not, longitudinal forces are caused by shear, and we do need to design/check for these longitudinal forces even under the pure shear condition. “Unfortunately, this concept was not included when the shear design procedures were originally developed. This omission can be a serious shortcoming”. (Design of highway bridges, An LRFD approach, 3^{rd} ed. by Baker and Puckett, 2013)

From the previous example, we can catch that there are some possible crack angle ranges for the given εx and vu/f’c. Now our question is which values of θ and β are the optimums? The previous example shows that, without considering longitudinal reinforcements, mostly (not always) the lowest crack angle results in the least number of stirrups. However, with considering longitudinal reinforcements, the optimum crack angle increases. The methodology to find out the optimum crack angle is proposed by Rahal and Collins (Background to the general method of shear design in the 1994 CSA-A23.3 standard, Canadian Journal of Civil Engineering, February 2011).

We can calculate the feasible crack angle ranges for each εx and vu/f’c combination. θ min corresponds to concrete crush and θ max corresponds to stirrup yielding.

Now, it’s time to solve the previous example from full iteration. For simplicity, interaction with flexure is not considered. In other words, it is assumed that the status is in a pure shear condition which rarely exists in the real world.

Sometimes we need tiresome calculations, even though they are not critical nor difficult, but they are essential, and they take time if we do not have proper tools. One of these is the moment of inertia calculation for cracked, circular concrete sections. We need this calculation when we perform service stress checks for flexure and deflection checks.

The previous article discussed how to solve a dynamics problem by hand (OK by Excel). In this article, the basics of dynamics will be reviewed.

The scary part of FEM is sometimes FEM gives wrong results without any error message. The analysis may be meaningless if an engineer cannot check or interpret the results. Let’s consider a simple example similar to the case from Dr. Gallagher (Finite Element Analysis: Fundamentals, 1975).

For the previous example, we can use high-order triangular elements. This element has six nodes per element and assumes the displacement is quadratic within an element. Also, each side edge can be curved, as shown.

Continuing on to the third part of this multi-part blog, another option is a quadrilateral element. As always, let’s start with an example.

In the FEM, the most critical question is “how many meshes do we need to get the correct (or reasonable) results?”

Most elements within the commercial programs have been tested and verified that they are converged with element number increases. This statement seems to be given, but not true for all elements. Theoretically, we can get the correct results if we have an infinite number of elements, but this is not feasible. So we have to decide the number of elements with the allowable error.

Dr. Seungwoo Lee has talked about some fundamental differences between linear and nonlinear analysis in structural engineering in the past. In linear analysis, the relationship between the stress and strain of a model is held constant, and the stiffness matrix of the model stays the same throughout the analysis. For a nonlinear analysis, there can be various factors that contribute to its nonlinearities, for example, material yielding, nonlinearities in the boundary conditions, and various forms of geometric nonlinearities. In this article, Dr. Lee will elaborate more on geometric nonlinearities and show how different approaches to approximate geometric nonlinearities can vary the structural analysis results.