## 单位载荷法中虚位移限制条件辨析1)

*北京航空航天大学航空科学与工程学院,北京 100191

**中国科学院力学研究所, 北京 100190

††中国科学院大学工学院, 北京 100049

## DISCUSSION ON LIMITATION OF VIRTUAL DISPLACEMENT IN UNIT-LOAD METHOD1)

LI Min*, LI Yilun, CHEN Weimin,**,††,2)

*School of Aeronautic Science and Engineering, Beihang University, Beijing 100191, China

Lab. MSSMat, CentraleSupélec, Université de Paris-Saclay, Paris 91190, France

**Institute of Mechanics, Chinese Academy of Sciences, Beijing 100190, China

††School of Engineering Sciences, University of Chinese Academy of Sciences, Beijing 100049, China

 基金资助: 1)中科院先导项目资助(XDA22000000)

Abstract

Unit-load method is a general method to solve displacement, which plays an important role in the energy method of material mechanics. When using virtual displacement mode in the unit-load method, the main limiting condition for virtual displacement is to satisfy the displacement boundary condition. For statically indeterminate problems, especially for the unit-load system construction with free boundary conditions, the understanding of the limiting condition determines the flexibility of using the unit-load method. In this paper, several classical examples are used to analyze the effect of this restriction, and to provide reference for explaining related problems in teaching activities.

Keywords： unit-load method; virtual work principle; boundary conditions

LI Min, LI Yilun, CHEN Weimin. DISCUSSION ON LIMITATION OF VIRTUAL DISPLACEMENT IN UNIT-LOAD METHOD1). Mechanics in Engineering, 2022, 44(2): 368-372 DOI:10.6052/1000-0879-21-358

## 1 单位载荷法中虚位移的定义

(1) 对于所研究的力系(外力与内力)必须满足平衡条件与静力边界条件;

(2) 对于所选择的虚位移是微小的,满足变形连续条件与位移边界条件。

### 图2

$1\times \theta_{B} =\sum\limits_{i=1}^2 \int_0^{l}\overline{M}\left(x_{i}\right)\frac{M\left(x_{i}\right)}{EI} {\rm d}x_{i}$

## 2 静不定结构单位载荷系统的构造

### 图4

$A$点：水平位移$H_{A} =0$,垂直位移$V_{A} =0$,

$C$点：垂直位移$V_{C} =0$;

$A$点：水平位移$H_{A} =0$,垂直位移$V_{A} =0$,

$C$点：水平位移$H_{C} =0$,垂直位移$V_{C} =0$;

## 3 具有对称性无约束静不定结构单位载荷系统的构造

### 图8

$W_{{\rm e}} =1\times \theta_{C} +\overline{{M}}_{{C}'} \times \theta_{{C}'}+\overline{{F}}_{{\rm N}{C}'} \times H_{{C}'} +\overline{{F}}_{{\rm S}{C}'} \times V_{{C}'}$

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