CN108959692B - A method for calculating average shear stress under multiaxial non-proportional loads - Google Patents

Abstract

The invention discloses a method for calculating the average shear stress under multi-axis non-proportional load. Under the multi-axis non-proportional load, the direction and magnitude of the shear stress vector on the material plane are constantly changing with time. Under cyclic loading, the end of the shear stress vector in the material plane will trace a trace. Therefore, under multiaxial non-proportional loads, it is very difficult to determine not only the magnitude of the shear stress, but also the average shear stress. The present invention obtains through coordinate transformation and projection

Shear stress time history in a plane; described by discrete points

The trace drawn by the end of the shear stress vector on the plane; each discrete shear stress end point is projected to the local coordinate system that passes through the coordinate origin and is orthogonal to the material plane; determined according to the average value of the projected lengths on the two coordinate axes The center of the trace graph of the shear stress; calculate the distance from the origin of the coordinates to the center of the graph, and obtain the material in

Average shear stress in the plane.

Description

A method for calculating average shear stress under multiaxial non-proportional loads

technical field

The invention belongs to the technical field of aviation systems, and specifically refers to a method for calculating average shear stress under multi-axis non-proportional loads.

Background technique

With the development of aviation industry, more and more new types of aircraft have the characteristics of vast flight space, increased flight speed, and extended flight life, and the requirements for safety and economy are also getting higher and higher. Studies have shown that fatigue failure is one of the main causes of structural failure, especially multiaxial fatigue failure. About 80% of the failures of mechanical parts belong to fatigue failure. Because there are often no obvious signs of deformation before fatigue failure, it is often easy to cause major accidents. At present, the multi-axial fatigue problem in engineering applications is often handled by simplifying it into uniaxial fatigue, which greatly simplifies the calculation steps, but also leads to a large difference between the fatigue life prediction and the actual life, reducing the fatigue life. Predicted reference value. Therefore, studying the fatigue life of structural parts under multiaxial loading and finding a more accurate and streamlined fatigue life prediction method are still the main issues in current engineering.

In multiaxial fatigue damage prediction, calculating the stress amplitude and average shear stress on the material plane is one of the most basic and critical issues in fatigue life analysis. In the real multiaxial fatigue loading process, the loads on the components are often variable amplitude loads or even random loads. As shown in Fig. 1, in a cyclic complex loading process, the stress vector _Sn depicts a closed curve Ψ', where the stress vector _{Sn can be decomposed into a normal stress vector σ n and} a shear stress vector τ. The normal stress σ _n only changes its magnitude, but does not change its direction, so its magnitude and mean value are relatively easy to determine. However, the magnitude and direction of the shear stress vector τ tends to vary with time, and the tip of τ depicts a closed curve Ψ. Since the shape of this curve changes with the external load, the magnitude of the shear stress and Means are a complex problem.

SUMMARY OF THE INVENTION

In view of the above situation, the purpose of the present invention is to calculate the average shear stress by considering the comprehensive influence of each point on the end trace of the shear stress vector on the average stress value, so as to make it more sensitive to the loading history.

For achieving the above object, the technical scheme adopted in the present invention is as follows:

A method for calculating the average shear stress under a multi-axial non-proportional load of the present invention includes the following steps:

(1) Define the fatigue risk point O and set it as the coordinate origin; define the rectangular coordinate system Oxyz; define the plane Δ; define the local coordinate system Ouv on the plane Δ;

(2) Input the multiaxial stress loading history;

经过坐标转换，将O点上的应力的空间矢量投影到所研究的面Δ内，得到在多轴非比例载荷下材料在面Δ上的剪应力矢量随时间变化历程；(3) For a given plane

After coordinate transformation, the space vector of the stress on the O point is projected into the studied surface Δ, and the time course of the shear stress vector of the material on the surface Δ under the multi-axis non-proportional load is obtained;

(4) Use discrete points to describe the trace drawn by the shear stress vector end of the above material on the surface Δ under a multi-axis non-proportional cyclic load;

(5) Project each discrete shear stress to the local coordinate system Ouv which passes through the coordinate origin and is orthogonal on the material plane respectively;

(6) Determine the graph center of the shear stress trace according to the average value of the projected lengths on the two coordinate axes;

(7) Calculate the distance from the origin of the coordinates to the center of the above graph, which is the mean value of the shear stress of the material on the plane.

θ表示：

为面Δ的法线在x-y平面上的投影与x轴的夹角，θ为面Δ的法线与z轴的夹角；并定义面Δ上的局部坐标系Ouv。Preferably, the step (1) further includes: taking the fatigue hazard point O as the origin of the coordinates, and defining the natural coordinate system Oxyz; setting the surface to be determined as Δ, the positional relationship between the surface Δ and the natural coordinate system Oxyz is given by

θ means:

is the angle between the projection of the normal of the surface Δ on the xy plane and the x-axis, and θ is the angle between the normal of the surface Δ and the z-axis; and defines the local coordinate system Ouv on the surface Δ.

Preferably, the step (2) further comprises: expressing the stress loading history in the form of a matrix:

Among them, σ _xx (t) is the normal stress of the action surface perpendicular to the x-axis and along the x-axis; σ _xy (t) is the shear stress of the action surface perpendicular to the x-axis and along the y-axis; σ _xz (t ) is the shear stress of the action surface perpendicular to the x-axis and along the z-axis; σ _yx (t) is the shear stress of the action surface perpendicular to the y-axis and along the x-axis; σ _yy (t) is the action surface perpendicular to the y-axis, the normal stress along the y-axis; σ _yz (t) is the shear stress of the action surface perpendicular to the y-axis and the direction along the z-axis; σ _zx (t) is the action surface perpendicular to the z-axis, the direction along the The shear stress of the x-axis; _σzy (t) is the shear stress of the action surface perpendicular to the z-axis and the direction along the y-axis; σ _zz (t) is the normal stress of the action surface perpendicular to the z-axis and the direction along the z-axis.

确定；假设面Δ的单位法向向量n与x，y，z轴的方向余弦分别n_x，n_y，n_z，用θ，

表示为：Preferably, the step (3) further includes: the object is subjected to a multi-axis fatigue load, the surface to be determined is a surface Δ, and the natural coordinate system x, y, z is projected onto the local coordinate system u, v on the surface Δ, Then the positional relationship between the surface Δ and the natural coordinate system x, y, z is through the angle between its unit normal vector n and the x, z axis

Determine; Assuming that the unit normal vector n of the surface Δ and the direction cosines of the x, y, and z axes are respectively n _x , _ny , n _z , use θ,

Expressed as:

The stress on the face Δ is expressed as:

Among them, _Snx is the stress along the x-axis direction on the surface Δ; _Sny is the stress along the y-axis direction on the surface Δ; S _nz is the stress along the z-axis direction on the surface Δ;

The stress _Sn on this Δ surface is expressed in tensor form as:

Sn =σ(t)· _n

The vector Sn is decomposed into a normal stress _σn normal to the plane Δ, that is, the projection of _Sn on _n :

σ _n =(n·S _n )n=(n·σ(t)·n)n

and the shear stress vector τ in the face Δ:

τ=S _n −σ _n =σ(t)·n−(n·σ(t)·n)n.

Preferably, the step (4) further includes: the trace at the end of the material plane Δ shear stress vector is Ψ, which is composed of a series of discrete points, where shear stress τ _j =τ(t _j ), 0≤t _j ≤T, t _j represents a time point, T is a cycle, j=1, 2...k, the larger the value of k, the closer the shape of the polygon is to the real curve Ψ, the coordinates of these discrete points are defined as: (τ _u (t _j ), τ _v (t _j )).

Preferably, the step (5) further comprises: projecting the discrete point described in the step (4) onto the local coordinate system Ouv on the plane Δ, and the projected value is the coordinate of the point (τ _u (t _j ) , τ _v (t _j )), the direction vector of the u and v axes on the local coordinate system Ouv is defined as:

Then the projections of shear stress in the u and v directions τ _u (t _j ) and τ _v (t _j ) are respectively:

τ _u (t _j )=u·τ=u·[σ(t _j )·n-(n·σ(t _j )·n)n]=u·σ(t _j )·n

τ _v (t _j )=v·τ=v·[σ(t _j )·n−(n·σ(t _j )·n)n]=v·σ(t _j )·n.

Preferably, the step (6) further comprises: defining the graph center of the shear stress trace as the average of the projections of all discrete points on the trace Ψ in the u and v directions respectively, that is, the coordinates of the shear stress center point are:

Preferably, the step (7) further comprises: defining the average shear stress as a vector value from point O to center C, namely:

Beneficial effects of the present invention:

The calculation method of the present invention is suitable for determining the average shear stress of metal materials under multi-axial non-proportional loads. In view of the problem that the previous method is not sensitive to the shape of the trace at the end of the shear stress vector, a detailed consideration is given to the trace of the end of the shear stress vector. The influence of each node on the overall amplitude provides basic support for structural design of aeronautical structures and fatigue life analysis under service loads.

Description of drawings

Figure 1 is a schematic diagram of the decomposition of stress on the material plane;

Fig. 2 is the principle diagram of the method of the present invention;

Figure 3a is the true stress decomposition diagram;

Figure 3b is a schematic diagram of projecting the stress vector to the plane to be determined;

Figure 4 is a schematic diagram of describing the end trace of the shear stress vector with discrete points;

θ＝84°下剪应力曲线以及剪应力中心示意图。Figure 5 is

The shear stress curve and the schematic diagram of the shear stress center at θ=84°.

Detailed ways

In order to facilitate the understanding of those skilled in the art, the present invention will be further described below with reference to the embodiments and the accompanying drawings, and the contents mentioned in the embodiments are not intended to limit the present invention.

Referring to Figure 2, a method for calculating the average shear stress under a multi-axial non-proportional load of the present invention includes the following steps:

(1) Define the fatigue risk point O and set it as the coordinate origin; define the rectangular coordinate system Oxyz; define the plane Δ; define the local coordinate system Ouv on the plane Δ;

θ表示：

为面Δ的法线在x-y平面上的投影与x轴的夹角，θ为面Δ的法线与z轴的夹角；并定义面Δ上的局部坐标系Ouv。Referring to Figure 3a, the object is subjected to multi-axis fatigue loads, the fatigue hazard point O is taken as the origin of the coordinates, and the natural coordinate system Oxyz is defined; set the surface to be determined as Δ, the positional relationship between the surface Δ and the natural coordinate system Oxyz is given by

θ means:

is the angle between the projection of the normal of the surface Δ on the xy plane and the x-axis, and θ is the angle between the normal of the surface Δ and the z-axis; and defines the local coordinate system Ouv on the surface Δ.

(2) Input the multiaxial stress loading history;

The stress loading history is expressed in the form of a matrix:

Among them, σ _xx (t) is the normal stress of the action surface perpendicular to the x axis and the direction along the x axis; σ _yx (t) is the shear stress of the action surface perpendicular to the x axis and the direction along the y axis; σ _xz (t ) is the shear stress of the action surface perpendicular to the x-axis and along the z-axis; σ _yx (t) is the shear stress of the action surface perpendicular to the y-axis and along the x-axis; σ _yy (t) is the action surface perpendicular to the y-axis, the normal stress along the y-axis; σ _yz (t) is the shear stress of the action surface perpendicular to the y-axis and the direction along the z-axis; σ _zx (t) is the action surface perpendicular to the z-axis, the direction along the The shear stress of the x-axis; _σzy (t) is the shear stress of the action surface perpendicular to the z-axis and the direction along the y-axis; σ _zz (t) is the normal stress of the action surface perpendicular to the z-axis and the direction along the z-axis.

经过坐标转换，将O点上的应力的空间矢量投影到所研究的面Δ内，得到在多轴非比例载荷下材料在面Δ上的剪应力矢量随时间变化历程；(3) For a given plane

After coordinate transformation, the space vector of the stress on the O point is projected into the studied surface Δ, and the time course of the shear stress vector of the material on the surface Δ under the multi-axis non-proportional load is obtained;

确定；假设面Δ的单位法向向量n与x，y，z轴的方向余弦分别n_x，n_y，n_z，用

表示为：Referring to Figure 3b, the object is subjected to multi-axial fatigue loads, and the surface to be determined is the surface Δ, and the natural coordinate system x, y, z is projected onto the local coordinate system u, v on the surface Δ, then the surface Δ and the natural coordinate The positional relationship of the system x, y, z is through the angle between its unit normal vector n and the x, z axis

Determine; assuming that the unit normal vector n of the surface Δ and the direction cosines of the x, y, and z axes are respectively n _x , _ny , and n _z , use

Expressed as:

The stress on the face Δ is expressed as:

Among them, _Snx is the stress along the x-axis direction on the surface Δ; _Sny is the stress along the y-axis direction on the surface Δ; S _nz is the stress along the z-axis direction on the surface Δ;

The stress _Sn on this Δ surface is expressed in tensor form as:

Sn =σ(t)· _n

The vector Sn is decomposed into a normal stress _σn normal to the plane Δ, that is, the projection of _Sn on _n :

σ _n =(n·S _n )n=(n·σ(t)·n)n

and the shear stress vector τ in the face Δ:

τ=S _n −σ _n =σ(t)·n−(n·σ(t)·n)n.

(4) Use discrete points to describe the trace drawn by the shear stress vector end of the above material on the surface Δ under a multi-axis non-proportional cyclic load;

Referring to FIG. 4 , the trace at the end of the Δ shear stress vector on the material plane is Ψ, which is composed of a series of discrete points, where the shear stress τj=τ(t _j ), 0≤t _j ≤T, t _j represents Time point, T is a cycle, j=1,2...k, the larger the value of k, the closer the shape of the polygon is to the real curve Ψ, the coordinates of these discrete points are defined as: (τ _u (t _j ), τ _v (t _j )).

(5) Project each discrete shear stress to the local coordinate system Ouv which passes through the coordinate origin and is orthogonal on the material plane respectively;

Project the discrete point described in step (4) to the local coordinate system Ouv on the surface Δ, and the projected value is the coordinate of the point (τ _u (t _j ), τ _v (t _j )), the local coordinate system The direction vector of the u, v axis on Ouv is defined as:

Then the projections of shear stress in the u and v directions τ _u (t _j ) and τ _v (t _j ) are respectively:

τ _u (t _j )=u·τ=u·[σ(t _j )·n-(n·σ(t _j )·n)n]=u·σ(t _j )·n

τ _v (t _j )=v·τ=v·[σ(t _j )·n−(n·σ(t _j )·n)n]=v·σ(t _j )·n.

(6) Determine the graph center of the shear stress trace according to the average value of the projected lengths on the two coordinate axes;

The graphic center of the shear stress trace is defined as the average of the projections of all discrete points on the trace Ψ in the u and v directions, that is, the coordinates of the shear stress center point are:

(7) Calculate the distance from the origin of the coordinates to the center of the above figure, which is the mean value of the shear stress of the material on the plane;

Define the average shear stress magnitude as the vector value from point O to center C, namely:

In this example, the combination of tension and torsion is used for loading, and the load spectrum is shown in Table 1 as follows (other stress values: σ _yy , σ _zz , σ _xz , σ _yz are all 0MPa):

Table 1

S1: Define the fatigue hazard point O and set it as the coordinate origin;

S2: Input the load spectrum of Table 1;

θ＝84°；S3: take

θ=84°;

S4: The time history of shear stress on the Δ surface of the material is obtained through coordinate transformation;

S5: Discrete the end traces of the shear stress vector on the Δ surface, and the discrete results and the schematic diagram of the shear stress center are shown in Figure 5;

S6: Calculate the graph center of the shear stress trace, the coordinates are (0.41,-3.53);

S7: Calculate the average shear stress value

There are many specific application ways of the present invention, and the above are only the preferred embodiments of the present invention. It should be pointed out that for those skilled in the art, without departing from the principles of the present invention, several improvements can be made. These Improvements should also be considered as the protection scope of the present invention.

Claims (3)

1. A method for calculating average shear stress under multi-axis non-proportional load is characterized by comprising the following steps:

(1) defining a fatigue danger point O and setting the fatigue danger point O as a coordinate origin; defining a rectangular coordinate system Oxyz; defining a plane delta; defining a local coordinate system Ouv on the plane Δ;

(2) inputting a multi-axis stress loading process;

(3) for a given plane delta, projecting a space vector of stress on the point O into the studied plane delta through coordinate conversion to obtain a time-varying course of a shear stress vector of the material on the plane delta under multi-axis non-proportional load;

(4) describing, with discrete points, a trace of the material described above at a multiaxial non-proportional cyclic load at the end of a shear stress vector at plane Δ;

(5) projecting each discrete shear stress to a local coordinate system Ouv which passes through the origin of coordinates and is orthogonal on the material plane respectively;

(6) determining the graph center of the shear stress trace according to the average value of the projection lengths on the two coordinate axes;

(7) calculating the distance from the origin of coordinates to the center of the graph, namely the mean value of the shearing stress of the material on the plane;

the (1) further comprises: taking the fatigue danger point O as a coordinate origin, and defining a natural coordinate system Oxyz; let the surface to be solved be Δ, and the position relationship between the surface Δ and the natural coordinate system Oxyz is determined by

θ represents:

is the included angle between the projection of the normal of the surface delta on the x-y plane and the x axis, and theta is the included angle between the normal of the surface delta and the z axis; and defines a local coordinate system Ouv on the plane Δ;

the (4) further comprises: the material has a trajectory at the end of the plane Δ shear stress vector of Ψ, which is formed by a series of discrete points, wherein the shear stress τ is_j＝τ(t_j)，0≤t_j≤T，t_jRepresenting the time point, T is a cycle period, j is 1,2 … k, the larger the value of k, the closer the shape of the polygon is to the real curve Ψ, and the coordinates of these discrete points are defined as: (τ)_u(t_j)，τ_v(t_j))；

The (5) further includes: projecting the discrete point in the step (4) to a local coordinate system Ouv on the surface delta, wherein the projected value is the coordinate (tau) of the point_u(t_j)，τ_v(t_j) The direction vector of the u, v axes on local coordinate system Ouv is defined as:

the projection τ of the shear stress in the u and v directions_u(t_j)，τ_v(t_j) Respectively as follows:

τ_u(t_j)＝u·τ＝u·[σ(t_j)•n-(n·σ(t_j)·n)n]＝u·σ(t_j)·n

τ_v(t_j)＝v·τ＝v·[σ(t_j)·n-(n·σ(t_j)·n)n]＝v·σ(t_j)·n；

wherein tau is a shear stress vector within the plane delta, and n is a unit normal vector of the plane delta;

the (6) further includes: the graph center of the shear stress trace is defined as the average value of the projections of all discrete points on the trace Ψ in the u and v directions, that is, the coordinates of the shear stress center point are:

the (7) further includes: defining the magnitude of the average shear stress as the vector value from point O to center C, i.e.:

2. the method of claim 1, wherein the step (2) further comprises: the stress loading course is expressed by a matrix form:

wherein t is a time point; sigma_xxNormal stress with the acting surface perpendicular to the x-axis and the direction along the x-axis; sigma_xyThe acting surface is perpendicular to the x axis, and the direction is along the shear stress of the y axis; sigma_xzThe shear stress of the acting surface vertical to the x axis and the direction along the z axis; sigma_yxThe shear stress of the acting surface vertical to the y axis and the direction along the x axis; sigma_yyNormal stress with the action surface perpendicular to the y-axis and the direction along the y-axis; sigma_yzThe shear stress of the acting surface vertical to the y axis and the direction along the z axis; sigma_zxShear stress in the direction along the x-axis, with the active surface perpendicular to the z-axis; sigma_zyThe shear stress of the acting surface vertical to the z-axis and the direction along the y-axis; sigma_zzPositive stress directed along the z-axis for the active surface to be perpendicular to the z-axis; sigma_xx(t) is the normal stress in the direction along the x-axis with the active surface perpendicular to the x-axis at time t; sigma_xy(t) is the shear stress at time point t, with the active surface perpendicular to the x-axis and the direction along the y-axis; sigma_xz(t) is the shear stress at time point t, with the active surface perpendicular to the x-axis and oriented along the z-axis; sigma_yx(t) is the shear stress at time point t, with the active surface perpendicular to the y-axis and in the direction along the x-axis; sigma_yy(t) is the positive stress of the acting surface perpendicular to the y-axis and along the y-axis at the time point t; sigma_yz(t) is the shear stress at time point t, with the active surface perpendicular to the y-axis and oriented along the z-axis; sigma_zx(t) is the shear stress at time point t, with the active surface perpendicular to the z-axis and oriented along the x-axis; sigma_zy(t) shear stress in the direction along the y-axis at the time point t, with the active surface perpendicular to the z-axis; sigma_zz(t) is the positive stress in the direction along the z-axis with the active surface perpendicular to the z-axis at time t.

3. The method of claim 2, wherein the step (3) further comprises: the object is subjected to the action of multi-axis fatigue load, the surface to be solved is a surface delta, local coordinate systems u and v on the surface delta in the directions of x, y and z of a natural coordinate system are projected, and then the position relation between the surface delta and the natural coordinate system x, y and z is determined through the angle between a unit normal vector n and the x and z axes

Determining; assume that the unit normal vector n of the plane Δ and the directional cosines of the x, y, z axes, respectively, n_x，n_y，n_zThe amount of the solvent is determined, using theta,

expressed as:

the stress on the plane Δ is expressed as:

wherein S is_nxStress on the plane delta along the x-axis direction; s_nyStress on the plane delta along the y-axis direction; s_nzStress on the plane delta along the z-axis direction;

stress S on the delta plane_nExpressed in tensor form as:

S_n＝σ(t)·n

vector S_nDecomposed into normal stress σ perpendicular to the plane Δ_nI.e. S_nProjection on n:

σ_n＝(n·S_n)n＝(n·σ(t)·n)n

and shear stress vector τ within plane Δ:

τ＝S_n-σ_n＝σ(t)·n-(n·σ(t)·n)n。

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