For practical engineering structures, it is usually difficult to measure external load distribution in a direct manner, which makes inverse load identification important. Specifically, load identification is a typical inverse problem, for which the models (e.g., response matrix) are often ill-posed, resulting in degraded accuracy and impaired noise immunity of load identification. This study aims at identifying external loads in a stiffened plate structure, through comparing the effectiveness of different methods for parameter selection in regulation problems, including the Generalized Cross Validation (GCV) method, the Ordinary Cross Validation method and the truncated singular value decomposition method. With demonstrated high accuracy, the GCV method is used to identify concentrated loads in three different directions (e.g., vertical, lateral and longitudinal) exerted on a stiffened plate. The results show that the GCV method is able to effectively identify multi-source static loads, with relative errors less than 5%. Moreover, under the situation of swept frequency excitation, when the excitation frequency is near the natural frequency of the structure, the GCV method can achieve much higher accuracy compared with direct inversion. At other excitation frequencies, the average recognition error of the GCV method load identification less than 10%.

With increasing demands for high-performance engineering structures, structural health monitoring (SHM) has received widespread attention, where the importance of load identification, as a major SHM task, has been widely recognized. Specifically, external load information in engineering structures is important for the estimation of structural reliability, service life, etc. [

In actual engineering, there are many types of loads. Generally speaking, the method of determining the load acting on the structure is often to adopt a direct measurement method and an indirect identification method. The former is to determine the structural load through the measurement method or determine the size of the load by measuring the parameters related to the load. However, in many cases in actual engineering, it is difficult or impossible to determine the direct measurement of the external load acting on the structure. In this case, it is often only possible to determine the load by using relevant information such as the response of the structural system measured in practice, for example, from the displacement response, velocity response, acceleration response and strain response of the structure to obtain load information [

In order to improve the accuracy of load identification, a number of methods have been developed. Amiri et al. [

While load identification methods have been applied in practical engineering, most existing studies are focusing on the utilization of displacement response [

For load identification in linear systems, it is often assumed that the load positions are known, and the magnitude and direction of the loads need to be determined. According to the concept of classic transfer path analysis (TPA), the relationship between the loads and structural responses can be expressed as follows:

where n is the number of loads; m is the number of responses;

where

Based on direct matrix inversion, the force vector in

where

The TSVD is established based on conventional Singular Value Decomposition (SVD), and the essence of solving ill-posed problem is to modify the singular values of the response matrix

where

where u and v are the vectors in

Assuming that a vector of response

In applications, the measured response and response matrix both contain unknown errors, assuming that

In order to minimize the system error

where

where

Cross validation is based on the statistics concept that the value of the optimal regularization parameter could provide the most accurate results of prediction of the missing data. The original data set is divided into a training set and a test set. The training set is used for the calculation of parameters, and the test set is used for testing, so that the parameter with the smallest test error is selected as the optimal regularization parameter.

In the OCV method [

where m is the number of responses.

where

The GCV method, developed by Goulb et al. [

when

Assuming a matrix

where m is the number of response positions. Multiplying a matrix by

Both sides of

where

Applying OCV to

where

when

To compare the effectiveness of TSVD method, GCV method and OCV method in load identification, numerical study is carried out to identify external loads on a simply supported stiffened flat plate structure. The dimension of the plate is 600 × 180 × 3 cm. The material is steel (with Young’s modulus of 2.1 GPa, Poisson’s ratio of 0.3 and density of 7850 kg/m^{3}). The plate is orthogonally stiffened, using 600 cm long stiffeners along the length, equally spaced with interval of 60 cm. The height and width of the rectangular cross-section of the stiffeners are 4 cm and 3 cm, respectively. Five 180 cm stiffeners along the width direction are equanlly spaced with interval of 100 cm, with a rectangular cross-section with height and width of 4 cm and 5 cm, respectively. The finite element model of the plate is shown in

Force positions | Response positions | ||||||
---|---|---|---|---|---|---|---|

No. | x/cm | y/cm | z/cm | No | x/cm | y/cm | z/cm |

1 | 12 | 19 | 4 | 1 | 50 | 19 | 7 |

2 | 50 | 0 | 4 | 2 | 59.5 | 60.75 | 7 |

3 | 59.5 | 80.25 | 4 | 3 | 21.5 | 51 | 7 |

4 | –121.5 | 60.75 | 4 | 4 | –31 | 0 | 7 |

5 | 97.5 | –31.5 | 0 | 5 | 12 | −19 | 7 |

6 | 261 | 0 | 4 | 6 | –121.5 | 28.5 | 7 |

7 | 197.5 | 31.5 | 7 | ||||

8 | –197.5 | 28.5 | 7 |

Multi-source load identification is carried out under static load and sweep frequency excitation, respectively. The load identification results of static loads are shown in

Identification results of vertical concentrated load ( |
|||||||
---|---|---|---|---|---|---|---|

No. | Applied load |
GCV | Relative error |
OCV | Relative error |
TSVD | Relative error |

1 | 123 | 116.58 | 5.22 | 115.23 | 6.32 | 117.26 | 4.66 |

2 | 183 | 176.85 | 3.36 | 161.36 | 11.83 | 190.07 | 3.87 |

Identification results of lateral concentrated load ( |
|||||||

No. | Applied load |
GCV | Relative error |
OCV | Relative error |
TSVD | Relative error |

1 | 169 | 168.50 | 0.30 | 168.12 | 0.52 | 168.55 | 0.27 |

4 | 231 | 218.86 | 5.26 | 218.41 | 5.45 | 218.92 | 5.23 |

Identification results of longitudinal concentrated load ( |
|||||||

No. | Applied load |
GCV | Relative error |
OCV | Relative error |
TSVD | Relative error |

3 | 325 | 347.40 | 6.89 | 349.89 | 7.66 | 348.84 | 7.33 |

5 | 246 | 245.20 | 0.32 | 247.02 | 0.41 | 247.77 | 0.72 |

For multi-source static load identification, it can be seen from

In addition, dynamic loads are identified using GCV, OCV and TSVD, respectively. A vertical force of 124N is applied at point 1 (see

From

Frequency = 1 Hz | |||||||
---|---|---|---|---|---|---|---|

No. | Applied load |
GCV | Relative error |
OCV | Relative error |
TSVD | Relative error |

1 | 124 | 49.80 | 59.84 | 40.62 | 67.24 | 30.58 | 75.33 |

Frequency = 37.47 Hz | |||||||

No. | Applied load |
GCV | Relative error |
OCV | Relative error |
TSVD | Relative error |

1 | 124 | 119.90 | 3.31 | 86.97 | 29.86 | 118.98 | 4.05 |

Frequency = 58.32 Hz | |||||||

No. | Applied load |
GCV | Relative error |
OCV | Relative error |
TSVD | Relative error |

1 | 124 | 121.75 | 1.82 | 121.72 | 1.84 | 119.15 | 3.91 |

Frequency = 94.79 Hz | |||||||

No. | Applied load |
GCV | Relative error |
OCV | Relative error |
TSVD | Relative error |

1 | 124 | 124.23 | 0.19 | 128.58 | 3.69 | 118.59 | 4.36 |

It can be seen from

Therefore, we adopt the GCV method to calculate the regularization parameters in the following studies.

Concentrated forces in three directions are simultaneously applied at the six load application positions of the structure, namely the vertical force, the lateral force and the longitudinal force. The identification results are shown in

Vertical concentrated load | ||||||
---|---|---|---|---|---|---|

No. | 1 | 2 | 3 | 4 | 5 | 6 |

Applied load ( |
1022 | 995 | 1333 | 786 | 1000 | 1234 |

Identified load ( |
1062.48 | 1034.41 | 1385.79 | 817.13 | 1039.60 | 1282.87 |

Relative error (%) | 3.96 | 3.96 | 3.96 | 3.96 | 3.960 | 3.96 |

Lateral concentrated load | ||||||

No. | 1 | 2 | 3 | 4 | 5 | 6 |

Applied load ( |
654 | 941 | 200 | 528 | 445 | 111 |

Identified load ( |
679.90 | 978.30 | 207.90 | 548.90 | 462.60 | 115.40 |

Relative error (%) | 3.96 | 3.96 | 3.95 | 3.96 | 3.96 | 3.96 |

Longitudinal concentrated load | ||||||

No. | 1 | 2 | 3 | 4 | 5 | 6 |

Applied load ( |
128 | 569 | 845 | 662 | 369 | 100 |

Identified load ( |
133.10 | 591.50 | 878.50 | 688.20 | 383.60 | 104.00 |

Relative error (%) | 3.98 | 3.95 | 3.96 | 3.96 | 3.96 | 4.00 |

From

The frequency range used for the simulation of swept frequency is from 1 to 500 Hz, where the multi-source concentrated forces are applied in three different directions. The first 10 natural frequencies of the structure are shown in

No. | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|

Frequency (Hz) | 1.53 | 5.79 | 9.51 | 18.96 | 26.49 |

No. | 6 | 7 | 8 | 9 | 10 |

Frequency (Hz) | 36.69 | 38.10 | 51.66 | 60.09 | 84.78 |

Apply vertical loads at six force application points separately, as shown in

No. | 1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|---|

Load ( |
134 | 160 | 116 | 177 | 154 | 122 |

Frequency (Hz) | 1 | 9.12 | 17.24 | 25.37 | 37.55 |
---|---|---|---|---|---|

Direct inverse (%) | 4.48E + 3 | 7.28E + 3 | 1.04E + 3 | 449.24 | 2.09E + 3 |

GCV method (%) | 21.31 | 313.18 | 65.98 | 102.16 | 92.24 |

Frequency (Hz) | 1 | 9.12 | 17.24 | 25.37 | 37.55 |
---|---|---|---|---|---|

Direct inverse (%) | 33.14 | 118.32 | 36.42 | 23.52 | 10.91 |

GCV method (%) | 6.83 | 48.65 | 2.63 | 2.23 | 8.17 |

From

Four longitudinal loads are applied on the structure at the same time, as shown in

No. | 2 | 3 | 4 | 6 |
---|---|---|---|---|

Load ( |
158 | 107 | 149 | 200 |

Frequency (Hz) | 1 | 8.32 | 37.16 | 51.62 |
---|---|---|---|---|

Direct inverse (%) | 4.96E + 3 | 2.43E + 3 | 7.86E + 3 | 1.04E + 4 |

GCV method (%) | 62.14 | 78.07 | 110.35 | 86.86 |

Frequency (Hz) | 1 | 8.32 | 37.16 | 51.62 |
---|---|---|---|---|

Direct inverse (%) | 1.44E + 3 | 21.81 | 311.56 | 3.20E + 3 |

GCV method (%) | 11.81 | 25.57 | 1.83 | 18.53 |

From

Four lateral loads are applied on the structure at the same time, as shown in

No. | 1 | 2 | 3 | 6 |
---|---|---|---|---|

Load ( |
218 | 253 | 279 | 233 |

Frequency (Hz) | 1 | 8.32 | 37.16 | 51.62 |
---|---|---|---|---|

Direct inverse (%) | 8.10E + 3 | 4.98E + 3 | 3.40E + 6 | 1.05E + 5 |

GCV method (%) | 52.19 | 56.38 | 169.24 | 53.55 |

Frequency (Hz) | 1 | 8.32 | 37.16 | 51.62 |
---|---|---|---|---|

Direct inverse (%) | 91.01 | 366.50 | 7.68E + 4 | 5.05E + 3 |

GCV method (%) | 3.49 | 20.69 | 2.83 | 32.04 |

From

This paper aims at identifying static/dynamic multi-source loads in stiffened plate structures by using strain responses. The conclusions obtained are as follows:

(1) The GCV, OCV and TSVD methods are used to identify static and dynamic load distributions on a reinforced plate structure, relying on strain responses. The recognition accuracy of the GCV method is proven to be higher than that of the TSVD and OCV methods.

(2) The GCV method is then used to identify the multi-source static loads applied in three different directions in the stiffened plate structure. The GCV method is able to identify the magnitudes of the loads with high accuracy, with relative error less than 5%.

(3) Under swept frequency excitation, the GCV method can effectively identify the multi-source load with relative errors less than 10%. In particular, near the natural frequency of the structure, the identification errors are much larger. However, compared with direct inversion, the GCV method is able to largely reduce the error of load identification.

The authors are grateful for the financial support provided by National Key R&D Program of China and National Natural Science Foundation of China.