*3.2.1 Structural analysis and design*

To comply with the previous qualities, the general statement to ensure safety on the structural design of any bridge should follow the next equation:

$$\text{Resistance} \ge \text{Effects of the loads} \tag{1}$$

*3.2.1.3 Torsion stress*

Where:

Where:

Where:

Where:

**101**

*3.2.1.5 Shear stress due bending*

*3.2.1.4 Bending stress*

Applied to elements with torsional moments.

*Bridges: Structures and Materials, Ancient and Modern DOI: http://dx.doi.org/10.5772/intechopen.90718*

τ: Torsional stress. Units: lb./in<sup>2</sup> (N/mm<sup>2</sup>

T: Torsional moment. Units: lb-in (N-mm) A: Cross-sectional area. Units: in<sup>2</sup> (mm<sup>2</sup>

Applied to elements with bending moments.

Applied to elements with bending moments.

*<sup>τ</sup>* <sup>¼</sup> *Tr*

*<sup>σ</sup><sup>b</sup>* <sup>¼</sup> *Mc*

σb: Bending stress. Units: lb./in<sup>2</sup> (N/mm<sup>2</sup>

*<sup>τ</sup><sup>b</sup>* <sup>¼</sup> *VQ*

For structures with combination of forces applied at the same times, all stresses are present and interact in the same time. Therefore, the stress combination can be represented with maximum and minimum principal stresses, as shown below:

r

) According to Eq. (7), we should note that on maximum and minimum principal

> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *σ<sup>x</sup>* � *σ<sup>y</sup>* 2 � �<sup>2</sup>

� *τxy*<sup>2</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *σ<sup>x</sup>* � *σ<sup>y</sup>* 2 � �<sup>2</sup>

� *τxy*<sup>2</sup>

�

σx, σy: Normal stress on x or y direction due axial or bending forces. τxy : Shear stress due direct shear forces, torsion or bending forces.

stress, the shear stress is always zero. Now, the maximum shear can be found

r

*τmax* ¼

τb: Shear stress due bending. Units: lb./in<sup>2</sup> (N/mm<sup>2</sup>

Q: Moment of area. Units: in<sup>3</sup> (mm<sup>3</sup>

I: Moment of inertia of the cross-sectional area. Units: in<sup>4</sup> (mm<sup>4</sup>

b: Width of the cross-sectional area. Units: in (mm)

*<sup>σ</sup>*1,2 <sup>¼</sup> *<sup>σ</sup><sup>x</sup>* <sup>þ</sup> *<sup>σ</sup><sup>y</sup>* 2 � �

σ1, σ2: Maximum and minimum principal stress.

Units for all stresses: lb./in<sup>2</sup> (N/mm<sup>2</sup>

following the next equation:

c: Distance between neutral axis and external fiber. Units: in (mm) I: Moment of inertia of the cross-sectional area. Units: in<sup>4</sup> (mm<sup>4</sup>

)

)

*<sup>J</sup>* (4)

*<sup>I</sup>* (5)

*Ib* (6)

)

)

)

(7)

(8)

)

)

The structural design process includes two general ways to comply with Eq. (1) and develop safety structures [1]:

a. Allowable Strength Design (ASD)

	- i. This procedure reduces the resistance multiplying a resistance factor φ, usually less than 1; and the load is multiplied by a load factor γ, usually greater than 1.
	- ii. Since each load has different levels of recurrence, these factors will vary depending on the load type.

The general way to obtain the stresses depends directly on the applied force, the internal force and the geometry of the structural element [8]. The behavior of each load applied can be listed as follows:

## *3.2.1.1 Axial stress*

Applied to elements with tension or compression forces.

$$
\sigma = \frac{P}{A} \tag{2}
$$

Where: σ: Axial stress. Units: lb./in2 (N/mm<sup>2</sup> ) P: Internal axial force. Units: lb. (N) A: Cross-sectional area. Units: in<sup>2</sup> (mm<sup>2</sup> )

#### *3.2.1.2 Direct shear stress*

A to elements with direct shear forces.

$$
\pi = \frac{V}{A} \tag{3}
$$

Where:

τ: Direct shear stress. Units: lb./in2 (N/mm<sup>2</sup> ) V: Internal shear force. Units: lb. (N) A: Cross-sectional area. Units: in<sup>2</sup> (mm<sup>2</sup> )

*Bridges: Structures and Materials, Ancient and Modern DOI: http://dx.doi.org/10.5772/intechopen.90718*

### *3.2.1.3 Torsion stress*

*3.2.1 Structural analysis and design*

*Infrastructure Management and Construction*

and develop safety structures [1]:

a. Allowable Strength Design (ASD)

usually greater than 1.

load applied can be listed as follows:

*3.2.1.1 Axial stress*

Where:

Where:

**100**

*3.2.1.2 Direct shear stress*

depending on the load type.

To comply with the previous qualities, the general statement to ensure safety on

The structural design process includes two general ways to comply with Eq. (1)

i. This procedure uses the linear behavior of the materials with a defined

i. This procedure reduces the resistance multiplying a resistance factor φ, usually less than 1; and the load is multiplied by a load factor γ,

ii. Since each load has different levels of recurrence, these factors will vary

The general way to obtain the stresses depends directly on the applied force, the internal force and the geometry of the structural element [8]. The behavior of each

*<sup>σ</sup>* <sup>¼</sup> *<sup>P</sup>*

)

*<sup>τ</sup>* <sup>¼</sup> *<sup>V</sup>*

)

)

)

*<sup>A</sup>* (2)

*<sup>A</sup>* (3)

ii. Safety is obtained specifying the effects of the loads should produce

yield strength which is located below the ultimate strength.

*Resistance* ≥ *Effects of the loads* (1)

the structural design of any bridge should follow the next equation:

stresses as a fraction of yielding stress.

Applied to elements with tension or compression forces.

σ: Axial stress. Units: lb./in2 (N/mm<sup>2</sup>

P: Internal axial force. Units: lb. (N) A: Cross-sectional area. Units: in<sup>2</sup> (mm<sup>2</sup>

A to elements with direct shear forces.

V: Internal shear force. Units: lb. (N) A: Cross-sectional area. Units: in<sup>2</sup> (mm<sup>2</sup>

τ: Direct shear stress. Units: lb./in2 (N/mm<sup>2</sup>

b. Load and Resistance Factor Design (LRFD)

Applied to elements with torsional moments.

$$
\pi = \frac{Tr}{J} \tag{4}
$$

Where:


### *3.2.1.4 Bending stress*

Applied to elements with bending moments.

$$
\sigma\_b = \frac{\mathcal{M}c}{I} \tag{5}
$$

Where:


#### *3.2.1.5 Shear stress due bending*

Applied to elements with bending moments.

$$
\pi\_b = \frac{VQ}{Ib} \tag{6}
$$

Where:


For structures with combination of forces applied at the same times, all stresses are present and interact in the same time. Therefore, the stress combination can be represented with maximum and minimum principal stresses, as shown below:

$$
\sigma\_{1,2} = \left(\frac{\sigma\_x + \sigma\_y}{2}\right) \pm \sqrt{\left(\frac{\sigma\_x - \sigma\_y}{2}\right)^2 - \tau\_{xy}^2} \tag{7}
$$

Where:

σ1, σ2: Maximum and minimum principal stress. σx, σy: Normal stress on x or y direction due axial or bending forces. τxy : Shear stress due direct shear forces, torsion or bending forces.

Units for all stresses: lb./in<sup>2</sup> (N/mm<sup>2</sup> )

According to Eq. (7), we should note that on maximum and minimum principal stress, the shear stress is always zero. Now, the maximum shear can be found following the next equation:

$$
\tau\_{\text{max}} = \sqrt{\left(\frac{\sigma\_{\text{x}} - \sigma\_{\text{y}}}{2}\right)^2 - \tau\_{\text{xy}}^2} \tag{8}
$$

In the case of the maximum shear stress, the normal stress is not zero and can be found as follows:

$$
\sigma\_{\rm prom} = \left(\frac{\sigma\_{\rm x} + \sigma\_{\rm y}}{2}\right) \tag{9}
$$

*3.3.1 Mexicali bridges: solution of the road distributor*

*Bridges: Structures and Materials, Ancient and Modern DOI: http://dx.doi.org/10.5772/intechopen.90718*

overpass section. This structure is shown in **Figure 8**.

seismic loads with good lateral displacement performance.

*3.3.2 Tacoma narrows bridge: a lesson learned*

movements of the bridge can be observed.

the world.

**Figure 8.**

**103**

*Mexicali bridges, road distributor [9].*

The Mexicali road distributor consists of a series of bridges connecting the main

The underpass bridge section, both for vehicular and a railway line, consists of a

The suspension bridge located at Tacoma Narrows consists of two main structural steel towers supporting a main cable and the main deck is stiffened by two steel girders. A total length of 5905 feet (1800 m) and a span of 2930 feet (893 m) were covered. It was inaugurated in 1940 and became one of the largest bridges in

The main feature of this bridge was the dramatically collapse of the main deck after a few months of inauguration, due to the oscillating movement with the action of the wind flow. These forces were considered for structural design; however, with a much slower wind velocity, the vibration movement increased with enough

Under research, the main reason for the collapse of the bridge was the concept of resonance, which means, a range of coincidence between the natural frequency of the structure and the frequency of wind thrust loads. The concept of vibration and

speed to make the structure collapse. Looking into **Figure 9**, the oscillating

resonance is not visible easily and many factors influenced on the event:

reinforced concrete bridges as underpass section and two structural steel bridges as

roads with the purpose of traffic flow continuity. Basically consists of two

reinforced concrete slab supported at the ends by retaining walls and circular columns of reinforced concrete supporting the center of the span. The main structure of this bridge section works as continuous girder. The spans are relatively short and the structural slab depth is enough to fulfill the flexure stresses requirements. The structure also has two structural steel decks as an overpass bridges, consisting of continuous girder and supported by steel columns. The length of the spans is small since they do not exceed 165 feet (50 m). A particular feature of this particular structure is the energy dissipation device located below the girder supports. Mexicali has high seismic activity and the structure needs to withstand the

*3.2.2 Bridge categories according the location of the main deck*

The bridges can be classified according to its size, geometry, main function and structure type. As a general way, the bridges can be divided into three categories:

Category 1: The structure is located below the main deck.


The geometry of this type of structures allows the user to have a clean view of the road. In addition, most of the structural elements of the geometry have to work under compression stress.

Category 2: The structure is located above the main deck.

a. Trusses.

b. Suspension bridges.

c. Cable-stayed bridges.

For these types of structures, the geometry is fully visible and for large bridges, such as suspended or cable-stayed, the user can appreciate the architecture. Most of the structural elements have to work under axial loads, mainly tension.

Category 3: The structure coincides with the main deck.

a. Girder-based bridges.


These types of structures work mainly under bending and shear stresses. Bridges of this type are the most used for short span.

#### **3.3 Special bridges: Mexicali, Tacoma, Coatzacoalcos, Calatrava Jerusalem**

Due to the great imagination of design and construction process, there are a large number of bridges in operation with a wide variety of geometries. Therefore, below are a few examples to show.

In the case of the maximum shear stress, the normal stress is not zero and can be

2 

(9)

*<sup>σ</sup>prom* <sup>¼</sup> *<sup>σ</sup><sup>x</sup>* <sup>þ</sup> *<sup>σ</sup><sup>y</sup>*

The bridges can be classified according to its size, geometry, main function and structure type. As a general way, the bridges can be divided into three

The geometry of this type of structures allows the user to have a clean view of the road. In addition, most of the structural elements of the geometry have to work

For these types of structures, the geometry is fully visible and for large bridges, such as suspended or cable-stayed, the user can appreciate the architecture. Most of

These types of structures work mainly under bending and shear stresses. Bridges

**3.3 Special bridges: Mexicali, Tacoma, Coatzacoalcos, Calatrava Jerusalem**

Due to the great imagination of design and construction process, there are a large number of bridges in operation with a wide variety of geometries. Therefore,

the structural elements have to work under axial loads, mainly tension.

Category 3: The structure coincides with the main deck.

ii. Girders with varieties of cross sections.

*3.2.2 Bridge categories according the location of the main deck*

Category 1: The structure is located below the main deck.

Category 2: The structure is located above the main deck.

found as follows:

categories:

a. Straight trusses.

d. Rigid frames.

under compression stress.

b. Suspension bridges.

c. Cable-stayed bridges.

a. Girder-based bridges.

i. Lightened and solid slabs.

of this type are the most used for short span.

below are a few examples to show.

**102**

a. Trusses.

b. Trusses with arch geometry.

*Infrastructure Management and Construction*

c. Arches with stone or masonry material.
