When damage occurs, there are changes in the structural stiffness. This causes changes in the vibrational information, such as translation and rotation angle of the joints in the structure. This paper presents a sensitivity study of translation and slope of the first-order mode shape of the joints in the frame structure, and which will contribute to structural damage identification further. Starting from the equation of natural vibration of frame structure, and according to the characteristics of the stiffness matrix, derive sensitivity coefficient expression of translation and slope of first-order mode shape of the joints in the structure, and obtain the first-order relative sensitivity coefficient of the first story is always smaller than zero when single column of the first story damage occurs. The numerical analysis of the four stories three spans frame shows consistent results with the formula derivation

Hongping Zhu

Lin Li

Yi Chen Yuan

English

Vibroengineering PROCEDIA

  © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. SEP 2015, VOLUME 5. ISSN 2345-0533 281 
The sensitivity analysis of the translation and the 
rotation angle of the first-order mode shape of the joints 
in frame structures 
Yi Chen Yuan1, Lin Li2, Hongping Zhu3 
School of Civil Engineering and Mechanics, Huazhong University of Science and Technology,  
Wuhan 430074, China 
2Corresponding author 
E-mail: 1hustyyc@126.com, 2lilines@aliyun.com, 3hpzhu@mail.hust.edu.cn 
(Accepted 4 September 2015) 
Abstract. When damage occurs, there are changes in the structural stiffness. This causes changes 
in the vibrational information, such as translation and rotation angle of the joints in the structure. 
This paper presents a sensitivity study of translation and slope of the first-order mode shape of the 
joints in the frame structure, and which will contribute to structural damage identification further. 
Starting from the equation of natural vibration of frame structure, and according to the 
characteristics of the stiffness matrix, derive sensitivity coefficient expression of translation and 
slope of first-order mode shape of the joints in the structure, and obtain the first-order relative 
sensitivity coefficient of the first story is always smaller than zero when single column of the first 
story damage occurs. The numerical analysis of the four stories three spans frame shows consistent 
results with the formula derivation. 
Keywords: sensitivity analysis, translation, rotation angle. 
1. Introduction 
The modal parameters of the structure (such as frequency, modal shape and modal damping) 
are the function of structural physical parameters (such as mass, stiffness and damping), so the 
change of structural physical parameters will lead to the changes of vibration modal parameters 
of structure. In order to correctly establish the relationship between the dynamic properties of 
structure and damage of structure, it is necessary to conduct the damage sensitivity analysis of the 
dynamic characteristics. 
Zhao and DeWolf [1] derived sensitivity coefficients of natural frequency, vibration mode, 
and the modal flexibility, which was applied to the spring-mass model. It was concluded that the 
modal flexibility is more sensitive to the damage. Zhu et al. [2, 3] used optimization method to 
identify the damage of periodic structure based on sensitivity analysis of modal parameters of 
periodic structures. Li et al. [4] derived sensitivity coefficient expressions of structure natural 
frequency, mode shapes and mode shape slops based on the characteristics of shear frame 
structural stiffness matrix. Yang [5] presented a mixed sensitivity method that combines 
eigenvalue and flexibility sensitivity to identify structural damage. Chen [6] analyzed the 
effectiveness of damage localization method based on the change in the value of the slope of the 
first mode. Yam et al. [7] studied the sensitivity of static and dynamic parameters to damage 
occurring in plate-like structures and proposed corresponding damage indices. 
The first-order mode shape slope is a good index of damage detection for shear frames, which 
can be used to detect the damage of story stiffness [8]. As for the general frame, there is not only 
the translation but also rotation angle. Thus the damage of the beam and column will cause the 
changes of the slope and rotation angle.  
In this paper sensitivity study of the translation and slope of the first-order mode shape of first 
story is conducted for further damage detection of general frame structures.  
2. Sensitivity analyses 
In the sensitivity analysis, the sensitivity coefficient of the structure modal parameter is divided 
THE SENSITIVITY ANALYSIS OF THE TRANSLATION AND THE ROTATION ANGLE OF THE FIRST-ORDER MODE SHAPE OF THE JOINTS IN FRAME 
STRUCTURES. YI CHEN YUAN, LIN LI, HONGPING ZHU 
282 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. SEP 2015, VOLUME 5. ISSN 2345-0533  
into absolute sensitivity coefficient and relative sensitivity coefficient. Absolute sensitivity 
coefficient is defined as the derivative of structure modal parameters with respect to structural 
physical parameters, and relative sensitivity coefficient is defined as the ratio of the rate of change 
of structure modal parameter to the rate of change of structural physical parameter. 
2.1. Sensitivity analyses of translation of mode shapes 
The equilibrium equation for undamped structural vibration is: 
(ܭ − ߱ଶܯ)߮௜ = 0, (1)
where ܭ and ܯ are the stiffness and mass matrices, respectively, ߱௜ is the ݅th circular frequency, 
and ߮௜ is the ݅th mode shape corresponding to ߱௜, which is normalized to be the unit-mass mode 
shape, i.e.: 
߮௜் ܯ߮௜ = 1, (2)
the derivative of Eq. (1) with respect to any beam or column stiffness, ݇௧, is: 
൬∂ܭ∂݇௧ − 2߱௜
∂߱௜
∂݇௧ ܯ൰߮௜ + (ܭ − ߱௜
ଶܯ)∂߮௜∂݇௧ = 0. (3)
And based on vibration theory, the mode shapes of a structure are independent of each other, 
and thus a complete set of mode shapes can be used to constitute the sensitivity coefficients of the 
݅th mode shape. Therefore, the absolute sensitivity coefficients of the mode shapes can be defined 
as follows: 
∂߮௜
∂݇௧ =෍ߚ௥
ே
௥ୀଵ
߮௥, (4)
both sides of Eq. (3) are premultiplied by the transposed vector of the ݎth mode shape, ߮௥் .Making 
use of the orthogonal properties of the mode shapes, Eq. (3) can be rewritten as: 
߮௥்
∂ܭ
∂݇௧ ߮௜ + ߮௥் (ܭ − ߱௜
ଶܯ)∂߮௜∂݇௧ = 0, ݎ ≠ ݅, (5)
because ߮௥் ܭ = ߮௥் ߱௥ଶܯ, Eq. (5) is simplified: 
߮௥்
∂ܭ
∂݇௧ ߮௜ + (߱௥
ଶ − ߱௜ଶ)߮௥் ܯ
∂߮௜
∂݇௧ = 0, ݎ ≠ ݅. (6)
For the first story column, substituting Eq. (4) into Eq. (6) and making use of the orthogonal 
properties of the mode shapes leads to: 
ߚ௥ =
2ܧ൫6߮௜,ଵ߮௥,ଵ + 3߮௜,ଵ߮௥,௡ା௦݈ଵ + 3߮௜,ଵ߮ଵ,௡ା௦݈ଵ + 2߮௜,௡ା௦߮௥,௡ା௦݈ଵଶ൯
݈ଵଷ(−߱௥ଶ + ߱௜ଶ)
, ݎ ≠ ݅, (7)
where ݊ and ݉ are the number of stories and the number of spans respectively, ݏ is the ݏth column 
in the first story ,which is no more than (݉ + 1). ܧ and ݈ଵ are respectively the elastic modulus 
and the length of the column. And when ݎ = ݅: 
ߚ௥ = 0. (8)
THE SENSITIVITY ANALYSIS OF THE TRANSLATION AND THE ROTATION ANGLE OF THE FIRST-ORDER MODE SHAPE OF THE JOINTS IN FRAME 
STRUCTURES. YI CHEN YUAN, LIN LI, HONGPING ZHU 
 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. SEP 2015, VOLUME 5. ISSN 2345-0533 283 
Thus, the first-order absolute sensitivity coefficient of the mode shapes can be summarized as 
follows: 
௜ܵ௧ெ =
߲߮௜
߲݇௧ =෍ߚ௥
ே
௥ୀଵ
߮௥, (9)
where ߚ௥ = ൝
ଶா൫଺ఝ೔,భఝೝ,భାଷఝ೔,భఝೝ,೙శೞ௟భାଷఝ೔,భఝభ,೙శೞ௟భାଶఝ೔,೙శೞఝೝ,೙శೞ௟భమ൯
௟భయ(ିఠೝమାఠ೔మ)
, (ݎ ≠ ݅),
0, ݎ = ݅,
 and the first-order 
relative sensitivity coefficient of the mode shapes can be summarized as follows: 
ܵ௜̅௧ெ = ݈݅݉௱௞೟→଴
߂߮௜߮௜
߂݇௧݇௧
≈ ߲߮௜߲݇௧ ⋅
݇௧
߮௜. (10)
2.2. Sensitivity analyses of mode shape slopes 
For general frames, the mode shape slope of the ݆th story in the ݅th mode is defined as: 
߮′௜௝ =
߮௜௝ − ߮௜(௝ିଵ)
ℎ௝ , (11)
where ℎ௝ is the height of the ݆th story. Taking the first derivatives of Eq. (11) with respect to ݇௧ 
leads to: 
௜ܵ௝,௧ௌ =
߲߮′௜௝
߲݇௧ =
1
ℎ௝ ቆ
߲߮௜௝
߲݇௧ −
߲߮௜(௝ିଵ)
߲݇௧ ቇ, (12)
where ௜ܵ௝,௧ௌ  is the first-order absolute sensitivity coefficient of the mode shape slope of the ݆th story 
in the ݅th mode with respect to the ݐth column or beam stiffness. And the first--order relative 
sensitivity coefficient: 
ܵ௜̅௝,௧ௌ = ݈݅݉௱௞೟→଴
߂߮′௜௝
߮′௜௝
߂݇௧݇௧
≈ ߲߮
ᇱ
௜௝
߲݇௧ ⋅
݇௧
߮ᇱ௜௝
= ݇௧߲߮௜௝ − ߲߮௜(௝ିଵ) ቆ
߲߮௜௝
߲݇௧ −
߲߮௜(௝ିଵ)
߲݇௧ ቇ, 
(13)
when ݆ = 1: 
ܵ௜̅ଵ,௧ௌ ≈
߲߮௜ଵ
߲݇௧ ⋅
݇௧
߮௜ଵ, (14)
when ݅ = 1: 
߲߮ଵଵ
߲݇௧ ⋅
݇௧
߮ଵଵ =
݇௧
߮ଵଵ ⋅෍ߚ௥߮௥ଵ
ே
௥ୀଶ
= ݇௧߮ଵଵ ⋅ (ߚଶ߮ଶଵ + ߚଷ߮ଷଵ + ߚସ߮ସଵ+. . . ). (15)
According to the analysis of the examples, we can get nearly accurate answer of డఝభభడ௞೟  when 
ݎ = ݊. As for ߚ௥߮௥ଵ, suppose: 
THE SENSITIVITY ANALYSIS OF THE TRANSLATION AND THE ROTATION ANGLE OF THE FIRST-ORDER MODE SHAPE OF THE JOINTS IN FRAME 
STRUCTURES. YI CHEN YUAN, LIN LI, HONGPING ZHU 
284 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. SEP 2015, VOLUME 5. ISSN 2345-0533  
ߚ௥߮௥ଵ < 0, (16)
ߚ௥߮௥ଵ =
2ܧ൫6߮ଵ,ଵ߮௥,ଵ + 3߮ଵ,ଵ߮௥,௡ା௦݈ଵ + 3߮௥,ଵ߮ଵ,௡ା௦݈ଵ + 2߮ଵ,௡ା௦߮௥,௡ା௦݈ଵଶ൯
݈ଵଷ(−߱௥ଶ + ଵ߱ଶ) ⋅ ߮௥,ଵ < 0. 
(17)
Therefore: 
(3߮ଵ,ଵ߮௥,௡ା௦݈ଵ + 2߮ଵ,௡ା௦߮௥,௡ା௦݈ଵଶ)߮௥,ଵ
6߮ଵ,ଵ߮௥,ଵଶ + 3߮௥,ଵଶ ߮ଵ,௡ା௦݈ଵ > −1, 
(18)
when ߮௥,௡ା௧, φ୰,ଵ symbol are the same, inequality (18) is correct. And when ߮௥,௡ା௧, ߮௥,ଵ have 
different symbol: 
(3߮ଵ,ଵ݈ଵ + 2߮ଵ,௡ା௦݈ଵଶ)
6߮ଵ,ଵ + 3߮ଵ,௡ା௦݈ଵ < −
߮௥,ଵ
߮௥,௡ା௦. 
(19)
According to the analysis of the examples, | ఝభ,భఝభ,೙శೞ௟భ |  is the minimum in all of ฬ
ఝ೔,భ
ఝ೔,೙శೞ௟భ
ฬ  
(݅ = 1, 2, 3, ...). And making − ఝೝ,భఝೝ,೙శೞ௟భ equals to 
ఝభ,భ
ఝభ,೙శೞ௟భ, inequality (19) yields: 
(3 ߮ଵ,ଵ߮ଵ,௡ା௦݈ଵ + 2)
6 ߮ଵ,ଵ߮ଵ,௡ା௦݈ଵ + 3
< ߮ଵ,ଵ߮ଵ,௡ା௦݈ଵ, (20)
߮ଵ,ଵ
߮ଵ,௡ା௦݈ଵ >
√3
3 ≈ 0.577. (21)
Thus, inequality (16) is necessarily true when ఝభ,భఝభ,೙శೞ௟భ >
√ଷ
ଷ . And according to the analysis 
of cantilever column ,taking concrete frame columns minimum section 300 mm×300 mm, and the 
corresponding ఝభ,భఝభ,೙శೞ௟భ  is 0.669, which is more than 
√ଷ
ଷ . So inequality (16) is true, that is the 
first-order relative sensitivity coefficient of the first story is always smaller than zero when the 
first story column is damaged. 
3. Numerical example 
The proposed method is demonstrated using numerical example of three spans four stories 
frame, as shown in Fig. 1, which is referred to as model 1 hereinafter. All of the column and beam 
length is 3 m, and the cross section of the column is 500 mm×500 mm and cross section of the 
beam is 300 mm×500 mm. Other parameter information is in the Table 1. 
 
Fig. 1. Three spans four stories frame 
THE SENSITIVITY ANALYSIS OF THE TRANSLATION AND THE ROTATION ANGLE OF THE FIRST-ORDER MODE SHAPE OF THE JOINTS IN FRAME 
STRUCTURES. YI CHEN YUAN, LIN LI, HONGPING ZHU 
 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. SEP 2015, VOLUME 5. ISSN 2345-0533 285 
Table 1. Parameter information of the frame 
Elastic Modulus Poisson’s ratio Density 
2.486E+10 0.2 2400 kg/m3 
When the frame is damaged, stiffness of the column or beam will reduce. In the study, the 
reducing of the stiffness is simulated by the reducing of moment of inertia of column or beam. 
The following discussing the sensitivity of the first-order mode shape slope of first story when the 
first story column is damaged. 
Assumes that the damage degree is 20 %, and the translation of the first order mode shape 
before and after the damage is shown in Table 2, and the slope of the first order mode shape before 
and after the damage is shown in Table 3. 
Table 2. Translation of the first order mode shape before and after the damage 
Story The translation before damage The translation after damage 
1 0.001817 0.001849 
2 0.004335 0.004349 
3 0.006258 0.006252 
4 0.007249 0.007231 
Table 3. Slope of the first order mode shape before and after the damage 
Story The slope before damage The slope after damage 
1 0.000606 0.000616 
2 0.000839 0.000833 
3 0.000641 0.000634 
4 0.000330 0.000326 
The Table 3 shows that when the first story column is damaged the slope of the first order 
mode shape of first story will increase, which also means that the first-order relative sensitivity 
coefficient of mode shape slope of first story is smaller than zero. 
From the above numerical example we can obtain that the first-order relative sensitivity 
coefficient of the first story is smaller than zero when the first story column is damaged, and this 
conclusion is the same with the results of the formula derivation. 
4. Conclusions 
Starting from the equation of natural vibration of frame structure, and according to the 
characteristics of the stiffness matrix, the paper has derived sensitivity coefficient expression of 
translation and slope of first-order mode shape of the joints in the structure. And some important 
observations can be pointed out: 
1. When the single column of the first story damage occurs, the first-order relative sensitivity 
coefficient of the first story is always smaller than zero. 
2. The damage of column of the first story can be detected by observing the changes of 
first-order mode shape slope of the first story. 
Acknowledgements 
The authors would like to express their most sincere thanks for the support of the National 
Natural Science Foundation of China (Grant No. 51278214). 
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THE SENSITIVITY ANALYSIS OF THE TRANSLATION AND THE ROTATION ANGLE OF THE FIRST-ORDER MODE SHAPE OF THE JOINTS IN FRAME 
STRUCTURES. YI CHEN YUAN, LIN LI, HONGPING ZHU 
286 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. SEP 2015, VOLUME 5. ISSN 2345-0533  
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The sensitivity analysis of the translation and the rotation angle of the first-order mode shape of the joints in frame structures

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The sensitivity analysis of the translation and the rotation angle of the first-order mode shape of the joints in frame structures

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