# Simplified Calculation of Shear Rotations for First-Order Shear Deformation Theory in Deep Bridge Beams

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

_{b}), in this approach (known as Timoshenko’s beam theory or first-order shear deformation theory), the total rotation is composed of two different rotations: rotation due to the bending w

_{b}and rotation due to the shear w

_{s}. The rotations considered in the Euler–Bernoulli’s and Timoshenko’s beam theories in a support are illustrated in Figure 1a,b, respectively.

## 2. Materials and Methods

_{v}is the shear area and G is the shear modulus written as

#### 2.1. Example 1: Simply Supported Beam with a Concentrated Load

_{v}, cross-sectional area A and inertia I of the beam are 30 GPa, 0.833 m

^{2}, 1 m

^{2}and 2.083 m

^{4}, respectively. Dimensions of the beam cross-section are presented in Figure 2b and its mechanical properties are listed in Table 1. The boundary conditions of the structure were horizontal and vertical displacements restricted in node 1 and vertical displacement restricted in node 3 (this is to say u

_{1}= v

_{1}= v

_{3}= 0). The beam was subjected to a concentrated vertical force at mid-span (node 2) of 100 kN (V

_{2}= 100 kN).

_{b}) and due to the bending plus shear/Thimoshenko model (v

_{b}+ v

_{s}) are summarized in Figure 3a, while the rotations for such models are presented in Figure 3b. Please note that in Figure 3b, the terms w

_{(b)}and w

_{(b+s)}stand for the rotations obtained with the Bernoulli and Timoshenko models of the structural software, respectively. Figure 3b also includes the rotations that were numerically calculated using the derivation of the vertical total deflections, including shear effects (v

_{b}+ v

_{s}) throughout the beam direction (axis x). This information is referenced as d(v

_{b}+ v

_{s})/dx in the figure.

_{i}being the shear force at the ith node, the bending vertical deflection at mid-span (node 2) v

_{b}

_{,2}, the vertical deflection due to shear at mid-span v

_{s}

_{,2}, the bending rotation at the left support (node 1) w

_{b}

_{,1}and the shear rotation at the same support w

_{s}

_{,1}might be expressed using the Equations (5)–(8), respectively:

_{b}

_{,2}(33.34 mm) and w

_{b}

_{,1}(0.010 rad). When the shear deformations were considered in Midas/Civil, additional deformations v

_{s}

_{,2}were obtained. The values of these deformations (20.00 mm) corresponded with those calculated using Timoshenko’s beam theory. Nevertheless, this was not the case for the shear rotations, as no additional rotations (w

_{s}

_{,1}) were obtained in Midas/Civil when the shear effects were included. The error in this rotation (0.005 rad) represented the 50% bending rotation at the support (w

_{b}

_{,1}).

#### 2.2. Calculation of Total Rotations with Structural Software

- -
**Step 1**: calculate the bending rotations (w_{b}_{,i,j}) and the shear forces (S_{i}_{,j}) at each node (i) for each element (j) with the simulation software.- -
**Step 2**: apply Timoshenko’s beam theory to calculate the shear rotations (w_{s}_{,i,j}) at each node (i) for each element (j). According to this theory, the shear rotations can be calculated from G, A_{v}and S_{i}_{,j}. The former two terms can be directly obtained from the mechanical properties of the structure, while S_{i}_{,j}corresponds with the shear forces calculated in step 1.- -
**Step 3**: obtain the real rotations at the ith node at element j by summing up the bending (w_{b}_{,i}) and the shear (w_{s}_{,i,j}) rotations calculated in steps 1 and 2, respectively.

_{i}

_{,j}is not null. Steps 2 and 3 of this flow chart can be easily implemented and programmed.

#### 2.3. Example 2: Continuous Beam with a Concentrated Load

_{b}) and bending plus shear (v

_{b}+ v

_{s}) obtained throughout the beam using Midas/Civil. Figure 5b,c include the shear diagram (S) and rotations (w) obtained using this software when bending (S

_{(b)}and w

_{(b)}) or bending plus shear strains (S

_{(b+s)}and w

_{(b+s)}) are considered. Finally, Figure 5d presents a comparison of the rotations calculated using Midas/Civil with those obtained using the proposed methodology when shear deformation effects were considered. This information is presented in the figure as the sum of the bending rotations obtained when the shear deformation was considered (w

_{(b+s)}) plus the shear rotations obtained using the proposed methodology (w(Emadi’s)). This figure also includes the rotations that were numerically calculated using the derivation of the vertical deflections, including shear effects throughout the beam directrix (axis x). These rotations are referenced as d(v

_{b}+ v

_{s})/dx.

_{(b+s)}) did not correspond with those obtained from the derivation of the beam deflections throughout the beam directrix (x-axis). For example, the rotations obtained using the numerical derivation at the left support (0.0029 rad) were 266.76% higher than those obtained using the computer software (0.0011 rad). Figure 5d also shows that the rotations obtained using the numerical derivation corresponded exactly with those obtained using the proposed method, where the shear rotations (w(Emadi’s)) were added to the bending rotations calculated using the computer software.

## 3. Results

#### 3.1. Direct Analysis

_{b}

_{,1}) and shear (w

_{s}

_{,1}) rotations obtained at the left support for different beam slenderness values (L/h). Figure 6 presents the percentage error that would result if the shear rotations were neglected. To do so, the shear rotations at node 1 were divided by the total rotations at that node (w

_{b}

_{,1}+ w

_{s}

_{,1}). For illustrative purposes, the slenderness limits for deep beams proposed by the Eurocode EN [41] and by the ACI Committee 318 [42] are highlighted in this figure, together with the error limits of 2% and 5%.

_{s}

_{,1}) did not depend on the beam length, as its value (0.005 rad) remained constant for all the analyzed slenderness values. This was not the case for the bending rotations (w

_{b}

_{,1}), as they increased nonlinearly with the length and with the slenderness ratio for a beam of constant height. In fact, these rotations varied from 0.0025 rad (L/h = 1) to 0.5625 rad (L/h = 15). On the other hand, Figure 6b shows that the smaller the L/h ratio, the higher the error of neglecting shear rotations. In fact, shear rotations represented 68.2% of the total rotations when L/h = 1. This figure also shows that a percentage error of 2% was reached for an L/h ratio of 9.9. This error was increased to 5% for an L/h ratio of 6.2. The geometrical definitions of deep beams in the Eurocode EN and ACI Committee 318 led to errors of 18.2% and 11.12%, respectively, when shear rotations were neglected in this example.

_{b}

_{,1}) and shear (w

_{s}

_{,1}) rotations obtained at the left support for different beam slenderness values (L/h). Figure 7b presents the percentage of the rotations represented by the shear rotation. This figure also includes the geometrical boundary for deep beams proposed by the Eurocode EN and by the ACI Committee 318, as well as the error limits of 2% and 5%.

#### 3.2. Inverse Analysis

^{4}). For illustrative purposes, the slenderness limits for deep beams proposed by the Eurocode EN and by the ACI Committee 318 are highlighted in this figure.

#### 3.3. Application on the Real Bridge

_{v}, as shown in this sub-figure.

## 4. Discussion

_{b}+ v

_{s})/dx. The results of the shear diagram are also presented in Figure 13b. As the structure is isostatic, these diagrams were not affected by the effects of the shear deformations.

_{b}) calculated using the computer software. This figure also shows the effect of the shear diagrams in Figure 13b in the total rotations throughout the bridge deck.

_{b}) and the shear rotations (w

_{s}) on the left-hand side of the node are summarized in Table 4. This table also includes the percentage of shear rotations in terms of the total rotations (w

_{b}+ w

_{s}) in the different nodes in the cantilever.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Support rotation of a beam support according to Euler–Bernoulli’s (

**a**) and Timoshenko’s beam theories (

**b**).

**Figure 5.**Example 2: continuous beam with and without shear effects: (

**a**) vertical deflections, (

**b**) shear diagrams, (

**c**) rotations obtained with the commercial software, and (

**d**) rotations including shear effects.

**Figure 6.**Parametric analysis of the slenderness ratio on example 1: (

**a**) bending (w

_{b}

_{,1}) and shear (w

_{s}

_{,1}) rotations at node 1 and (

**b**) percentage error of rotations in node 1 when ws,1 was neglected.

**Figure 7.**Parametric analysis of the effect of the slenderness ratio on example 2: (

**a**) bending (w

_{b}

_{,1}) and shear (w

_{s}

_{,1}) rotations at node 1 and (

**b**) percentage error of rotations in node 1 when w

_{s}

_{,1}was neglected.

**Figure 8.**Parametric analysis of the effect of the slenderness ratio on a cantilever beam with a concentrated load: (

**a**) bending (w

_{b}

_{,3}) and shear (w

_{s}

_{,3}) rotations at node 3 and (

**b**) percentage error of rotations in node 3 when ws,3 was neglected.

**Figure 9.**Percentage error between the estimated inertia using the OM method $\overline{I}$ with the actual one I for three beams in terms of the slenderness (L/h ratio).

**Figure 10.**Example of application: (

**a**) Yunbao Bridge under construction [52] and (

**b**) a sketch of the analyzed construction stage.

**Figure 13.**Yunbao bridge analysis: (

**a**) vertical deflections, (

**b**) shear diagram, and (

**c**) rotations with and without shear effects.

Properties (Unit) | Value |
---|---|

Area (m^{2}) | 1.000 |

Shear area (m^{2}) | 0.833 |

Inertia (m^{4}) | 2.083 |

Young’s modulus (GPa) | 30.000 |

Poisson’s ratio $v$ | 0.250 |

Method | v_{b}_{,2} (mm) | v_{s}_{,2} (mm) | w_{b}_{,1} (rad) | w_{s}_{,1} (rad) |
---|---|---|---|---|

Timoshenko’s beam theory | 33.34 | 20.00 | 0.010 | 0.005 |

Midas/Civil without shear | 33.34 | 0.00 | 0.010 | 0.000 |

Midas/Civil with shear | 33.34 | 20.00 | 0.010 | 0.000 |

Parameters | Elements 1 and 7 | Elements 2 and 6 | Elements 3 and 5 | Element 4 |
---|---|---|---|---|

EA (N) | 4.33 × 10^{11} | 4.38 × 10^{11} | 4.43 × 10^{11} | 4.46 × 10^{11} |

GA_{v} (N) | 1.10 × 10^{10} | 1.25 × 10^{11} | 1.43 × 10^{11} | 1.53 × 10^{11} |

EI (N·m^{2}) | 1.34 × 10^{12} | 1.69 × 10^{12} | 2.13 × 10^{12} | 2.38 × 10^{12} |

Rotation | w_{b} (rad) | w_{s} (rad) | w_{s}/(w_{b} + w_{s}) |
---|---|---|---|

w1,1 | −6.6 × 10^{−5} | −2.3 × 10^{−4} | 77.7% |

w2,2 | −5.6 × 10^{−5} | −1.7 × 10^{−4} | 74.8% |

w3,3 | −3.5 × 10^{−5} | −1.7 × 10^{−4} | 82.7% |

w4,4 | −0.8 × 10^{−5} | −1.5 × 10^{−4} | 94.5% |

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## Share and Cite

**MDPI and ACS Style**

Emadi, S.; Ma, H.; Lozano-Galant, J.A.; Turmo, J.
Simplified Calculation of Shear Rotations for First-Order Shear Deformation Theory in Deep Bridge Beams. *Appl. Sci.* **2023**, *13*, 3362.
https://doi.org/10.3390/app13053362

**AMA Style**

Emadi S, Ma H, Lozano-Galant JA, Turmo J.
Simplified Calculation of Shear Rotations for First-Order Shear Deformation Theory in Deep Bridge Beams. *Applied Sciences*. 2023; 13(5):3362.
https://doi.org/10.3390/app13053362

**Chicago/Turabian Style**

Emadi, Seyyedbehrad, Haiying Ma, Jose Antonio Lozano-Galant, and Jose Turmo.
2023. "Simplified Calculation of Shear Rotations for First-Order Shear Deformation Theory in Deep Bridge Beams" *Applied Sciences* 13, no. 5: 3362.
https://doi.org/10.3390/app13053362