CN106557643A - Deep beam four-point bending calculation method for stress under a kind of concentration power effect - Google Patents

Abstract

The invention belongs to Vehicle Structure Strength technical field, the invention provides a kind of lower deep beam four-point bending calculation method for stress of concentration power effect, it is characterised in that comprise the following steps：Step 1：The stress field that concentration power is produced；Step 2：Stress field under shearing action；Step 3：The decomposition of direct stress when four_point bending beam moment of flexure and shearing collective effect；Step 4：Nondimensionalization analysis obtains direct stress.

Description

Deep beam four-point bending calculation method for stress under a kind of concentration power effect

Technical field

The invention belongs to Vehicle Structure Strength technical field, is related to a kind of lower 4 points of the deep beam of concentration power effect curved Transverse stress computational methods.

Background technology

In engineering structure, beam is one of the most frequently used component, considers in test the cost factor of beam, reduces beam Span-depth radio, generally less than 8, this short beam is referred to as deep beam in engineering structure.When concentrfated load is acted on beam, There is shear-bending coupling phenomenon, now do not only have direct stress on beam cross section and also have shear stress.Due to shear stress Exist, warping phenomenon can occur on the cross section of beam, and extruding phenomenon occurs in the section parallel with neutral line. In general, beam of the span-depth radio less than 5, is calculated with the pure bending theory and hypothesis of slender beam in the mechanics of materials When, error can reduce with span-depth radio and increase rapidly, and at this moment, elongated beam theory is just no longer adapted to.Deep beam should Power calculates major influence factors section, support restraint, span-depth radio.Bend under concentration power effect change During shape, the hypothesis that plane section assumption and longitudinal fiber are not mutually extruded differs too big with actual, and this just determines The complexity of deep beam stress analysis.Therefore, the mechanical characteristic of the deep beam under the effect of research four-point bending and simplification Computational methods are significant for engineering design, and for grasping, complex section deep beam stress distribution is also reasonable Value.

The content of the invention

The purpose of the present invention is：There is provided a kind of concentration power effect lower deep beam four-point bending calculation method for stress.

Technical scheme

Deep beam four-point bending calculation method for stress under a kind of concentration power effect, according to the characteristics of beam, when deep beam holds During by concentration power, span is little, deck-molding is big, and the amount of deflection of its spaning middle section is less.So, with the effect of power Point, can be according to Elasticity to consider the stress distribution that concentration power causes by Half-plane Body in the region in the center of circle One Half-plane Body on border by concentration power when calculation method for stress draw stress in beam under shearing action point Cloth.

According to principle of stacking, using coordinate transform, bending when deriving moment of flexure and shearing collective effect just should Power formula, meanwhile, the formula to deriving completes nondimensionalization analysis, has finally carried out analysis verification.

A kind of detailed process of the described lower deep beam four-point bending calculation method for stress of concentration power effect is as follows：

1. a kind of concentration power acts on lower deep beam four-point bending calculation method for stress, it is characterised in that including following Step：

Step 1：The stress field that concentration power is produced

It is analyzed by taking the beam in the case of rectangular cross-section as an example, according to Elasticity Half-plane Body on border By the direct stress that each concentration power P during concentration power causes in region of its application point for the center of circleIt is straight Angular coordinate expression formula is：

In formula, x is coordinate figure in X-axis, and y is coordinate figure in Y-axis；

Step 2：Stress field under shearing action

Beam two ends can also produce stress field in beam, according to superposition under the counter-force effect of concentration power R1 and R2 Principle, is obtained by the Stress field overlaping of 4 concentration power P beams internal stress generations, and 4 concentration power P are respectively F1, F2, R1, R2, need to carry out coordinate transform to the local coordinate system of 4 concentration points of force application：

A) stress field for F1, coordinate transform formula is：

B) stress field for F2, coordinate transform formula is：

C) stress field for R1, coordinate transform formula is：

D) stress field for R2, coordinate transform formula is：

During the above is various, (x_F1,y_F1), (x_F2,y_F2), (x_R1,y_R1), (x_R2,y_R2) be respectively concentration power Fl, F2, The application point of R1, R2 coordinate figure in a coordinate system；

F1=F2=R1=R2=F/2 under experimental conditions, the positive and negative general direction with beam of Fl, F2, R1, R2 For standard, (2) one formula (5) of formula is substituted into into formula (1), direct stress during shearing action is obtained

In formula, L_sFor the unilateral length of span, l is torque arm length, and h is depth of beam.

Step 3：Direct stress when four_point bending beam moment of flexure and shearing collective effect

In the presence of concentration power, only shearing does not have moment of flexure to beam yet, should so as to produce shearing stress and bending Power, according to the stress that elongated beam theory moment of flexure is producedThe stress that shearing is producedBy formula (6) really It is fixed, then according to principle of stacking under curved scissors collective effect, direct stress σ_xShould be：

Step 4：Nondimensionalization is analyzed

Calculate for convenience and analyze, induce one following dimensionless group：

OrderSpan-depth radioShear span ratio

Formula (7) is substituted into, is obtained

Order

Then

Formula (9) characteristic λ reflects deep beam calculation method for stress and mechanics of materials slender beam theoretical stress Difference, as it is drawn from deep beam infinitesimal force analysis, it is contemplated that curved scissors is to the coefficient knot of beam Really, so having the general character.

Further, it is further comprising the steps of：

In order to more intuitively represent the impact of warping stress that Fl, F2 cause, by (9) formula nondimensionalization, then Dimensionless direct stressIt is reduced to：

In formula,For dimensionless warping stress, different values of the dimensionless warping stress according to its position λ_σ1And λ_σ2,

For square-section, willFor people's dimensionless stress expression formula, obtain

Further, it is further comprising the steps of：

In order to more intuitively represent the impact of warping stress that Fl, F2 cause, by (9) formula nondimensionalization, then Dimensionless direct stressIt is reduced to：

In formula,For dimensionless warping stress, different values of the dimensionless warping stress according to its position λ_σ1And λ_σ2,

For the symmetrical I-shaped beam section of twin shaft, if upper edge strip, lower edge strip area are A, width is b, in Property wheelbase is from for h₂、h₁, web height is h₀, thickness is δ, and area is A_f, deck-molding is h,

For convenience of calculating, ifThen α₁=α₂=0.5, A=(1+2 β) A_f

Bring formula (15) into dimensionless stress expression formula, obtain：

Description of the drawings

Fig. 1 is the stress distribution schematic diagram that concentration power P of the present invention causes；

Wherein：P is concentrfated load, and X and Y is parallel with concentration power and vertical coordinate axess.

Fig. 2 is four-point bending coordinate schematic diagram of the present invention；

Fig. 3 is the symmetrical I-beam schematic cross-section of twin shaft of the present invention；

Fig. 4 is the rule that section stress of the present invention changes along beam length；

Wherein：1 is numerical analysis solution；2 is slender beam Theory Solution；3 is this method solution.

Specific embodiment

Inventive principle：

According to principle of stacking, using coordinate transform, bending when deriving moment of flexure and shearing collective effect just should The timely formula of power, meanwhile, the formula to deriving completes nondimensionalization analysis, has finally carried out analysis verification.

The present invention will be further described with reference to the accompanying drawings and examples：

Step 1：The stress field that concentration power is produced

By taking square-section as an example, the transversal force problem of four-point bending when studying two ends freely-supported.

For simply supported beam, beam two ends are by shearing and the collective effect of bending moment.The characteristics of from deep beam, work as beam When bearing concentration power, span is little, deck-molding is big, and the amount of deflection of its spaning middle section is less.So, with the work of power It is to consider the stress distribution (Fig. 1) that concentration power P causes by Half-plane Body in the region in the center of circle with point.According to elasticity When mechanics Half-plane Body receives concentration power on border, concentration power P causes in region of its application point for the center of circle Direct stressRectangular coordinate expression formula be：

In formula, x is coordinate figure in X-axis, and y is coordinate figure in Y-axis.

Step 2：Stress field under shearing action

Beam two ends can also produce stress field in beam, according to principle of stacking, beam under the counter-force effect of concentration power Stress field overlaping that internal stress is produced by 4 concentration powers and obtain.For this reason, it may be necessary to each concentration power F1, F2, The local coordinate system of R1, R2 application point carries out coordinate transform (Fig. 2).

A) stress field for F1, coordinate transform formula is：

B) stress field for F2, coordinate transform formula is：

C) stress field for R1, coordinate transform formula is：

D) stress field for R2, coordinate transform formula is：

During the above is various, (x_F1,y_F1), (x_F2,y_F2), (x_R1,y_R1), (x_R2,y_R2) concentration power Fl, F2 are respectively, Coordinate figure of the application point of support reaction R1, R2 in the global coordinate system shown in Fig. 2.

As when testing machine carries out four-point bending test, F1 and F2 is realized by loads fixture, and is ensured Fl=F2=F/2, knows F1=F2=R1=R2=F/2 by the equilibrium principle of power, Fl, F2, R1, R2 it is positive and negative with The general direction of beam is standard.(2) one formula (5) of formula is substituted into into formula (1), is solved：

In formula, L_sFor the unilateral length of span, l is torque arm length, and h is depth of beam.

Step 3：Bending normal stresses when 4 points of moments of flexure and shearing collective effect

In the presence of concentration power, only shearing does not have moment of flexure to beam yet, should so as to produce shearing stress and bending Power.According to the stress that elongated beam theory moment of flexure is producedShear the stress for producing to be determined by formula (6), Then according to principle of stacking under curved scissors collective effect, direct stress should be：

Step 4：Nondimensionalization is analyzed

Calculate for convenience and analyze, induce one following dimensionless group：

Shear span ratioSpan-depth radio

Formula (7) is substituted into, is obtained

Order

Then

Formula (12) characteristic λ reflects deep beam calculation method for stress and mechanics of materials slender beam theoretical stress Difference.As it is drawn from deep beam infinitesimal force analysis, it is contemplated that curved scissors is to the coefficient knot of beam Really, so having the general character.

In order to more intuitively represent the impact of warping stress that Fl, F2 cause, by (12) formula nondimensionalization, then Dimensionless direct stress is：

In formula (10),For dimensionless warping stress.

A) for square-section, willFor people's dimensionless stress expression formula, obtain

B) for the symmetrical I-shaped beam section of twin shaft (referring to such as Fig. 3), if upper edge strip, lower edge strip area are A,

Width is b, is h away from neutral axis distance₂、h₁, web height is h₀, thickness is δ, and area is A_f, deck-molding is h.

For convenience of calculating, ifThen α₁=α₂=0.5, A=(1+2 β) A_f

Bring formula (11) into dimensionless stress expression formula, obtain：

Step 5：Analysis verification

It is provided with a square-section deep beam, two ends freely-supported, h=30mm, l=50mm, L=100mm, E=210Gpa, υ=0.3.As F1==F2=1000N, respectively with solving herein, the elongated beam theory of the mechanics of materials, FInite Element Calculate the rule (Fig. 4) that section stress changes along beam length.It is apparent from, result of calculation is between FEM calculation As a result and mechanics of materials solution between, from Saint Venant's principle, illustrate that the method is effective.

Claims (3)

1. a kind of concentration power acts on lower deep beam four-point bending calculation method for stress, it is characterised in that including following step Suddenly：

Step 1：The stress field that concentration power is produced

It is analyzed by taking the beam in the case of rectangular cross-section as an example, according to Elasticity Half-plane Body on border By the direct stress that each concentration power P during concentration power causes in region of its application point for the center of circleIt is straight Angular coordinate expression formula is：

${\mathrm{\σ}}_x^p=-\frac{2p}{\mathrm{\π}}\frac{x^{2}y}{{(x^2+y^2)}^2}---\left(1\right)$

In formula, x is coordinate figure in X-axis, and y is coordinate figure in Y-axis；

Step 2：Stress field under shearing action

Beam two ends can also produce stress field in beam, according to superposition under the counter-force effect of concentration power R1 and R2 Principle, is obtained by the Stress field overlaping of 4 concentration power P beams internal stress generations, and 4 concentration power P are respectively F1, F2, R1, R2, need to carry out coordinate transform to the local coordinate system of 4 concentration points of force application：

A) stress field for F1, coordinate transform formula is：

$\left\{\begin{array}{c}{x}_{F1}=x+(L_S-l)\\ y_{F1}=y+h/2\end{array}\right.---\left(2\right)$

B) stress field for F2, coordinate transform formula is：

$\left\{\begin{array}{c}{x}_{F2}=x-(L_S-l)\\ y_{F2}=y+h/2\end{array}\right.---\left(3\right)$

C) stress field for R1, coordinate transform formula is：

$\left\{\begin{array}{c}{x}_{R1}=-x-L_S\\ y_{R1}=-y+h/2\end{array}\right.---\left(4\right)$

D) stress field for R2, coordinate transform formula is：

$\left\{\begin{array}{c}{x}_{R2}=-x+L_S\\ y_{R2}=-y+h/2\end{array}\right.---\left(5\right)$

During the above is various, (x_F1,y_F1), (x_F2,y_F2), (x_R1,y_R1), (x_R2,y_R2) be respectively concentration power Fl, F2, The application point of R1, R2 coordinate figure in a coordinate system；

F1=F2=R1=R2=F/2 under experimental conditions, the positive and negative general direction with beam of Fl, F2, R1, R2 For standard, (2) one formula (5) of formula is substituted into into formula (1), direct stress during shearing action is obtained

$\begin{array}{c}{\mathrm{\σ}}_x^F=\frac{F}{\mathrm{\π}}\frac{{\left(x+L_S-l\right)}^2\left(y+h/2\right)}{{\[{\left(x+L_S-l\right)}^2+{\left(y+h/2\right)}^2\]}^2}\\ +\frac{F}{\mathrm{\π}}\frac{{\left(x-L_S+l\right)}^2\left(y+h/2\right)}{{\[{\left(x-L_S+l\right)}^2+{\left(y+h/2\right)}^2\]}^2}\\ -\frac{F}{\mathrm{\π}}\frac{{\left(x+L_S\right)}^2\left(-y+h/2\right)}{{\[{\left(x+L_S\right)}^2+{\left(-y+h/2\right)}^2\]}^2}\\ -\frac{F}{\mathrm{\π}}\frac{{\left(x-L_S\right)}^2\left(-y+h/2\right)}{{\[{\left(x-L_S\right)}^2+{\left(-y+h/2\right)}^2\]}^2}\end{array}---\left(6\right)$

In formula, L_sFor the unilateral length of span, l is torque arm length, and h is depth of beam.

Step 3：Direct stress when four_point bending beam moment of flexure and shearing collective effect

In the presence of concentration power, only shearing does not have moment of flexure to beam yet, should so as to produce shearing stress and bending Power, according to the stress that elongated beam theory moment of flexure is producedThe stress that shearing is producedBy formula (6) really It is fixed, then according to principle of stacking under curved scissors collective effect, direct stress σ_xShould be：

$\begin{array}{c}{\mathrm{\σ}}_x={\mathrm{\σ}}_x^F+{\mathrm{\σ}}^M=\frac{My}{I}+\frac{F}{\mathrm{\π}}\frac{{\left(x+L_S-l\right)}^2\left(y+h/2\right)}{{\[{\left(x+L_S-l\right)}^2+{\left(y+h/2\right)}^2\]}^2}\\ +\frac{F}{\mathrm{\π}}\frac{{\left(x-L_S+l\right)}^2\left(y+h/2\right)}{{\[{\left(x-L_S+l\right)}^2+{\left(y+h/2\right)}^2\]}^2}\\ -\frac{F}{\mathrm{\π}}\frac{{\left(x+L_S\right)}^2\left(-y+h/2\right)}{{\[{\left(x+L_S\right)}^2+{\left(-y+h/2\right)}^2\]}^2}\\ -\frac{F}{\mathrm{\π}}\frac{{\left(x-L_S\right)}^2\left(-y+h/2\right)}{{\[{\left(x-L_S\right)}^2+{\left(-y+h/2\right)}^2\]}^2}\end{array}---\left(7\right)$

Step 4：Nondimensionalization is analyzed

Calculate for convenience and analyze, induce one following dimensionless group：

Order $\mathrm{\&xi;}=\frac{y}{h/2},\mathrm{\&eta;}=\frac{x}{L_s},(0<\mathrm{\&eta;}<1),$ Span-depth radio $\mathrm{\&alpha;}=\frac{2L_s}{h},$ Shear span ratio $k=\frac{l}{L_s}$

Formula (7) is substituted into, is obtained

$\begin{array}{c}{\mathrm{\&sigma;}}_x=\frac{My}{I}+\frac{F}{2\mathrm{\&pi;}h}\{\frac{4{(\mathrm{\&eta;}+1-k)}^2{\mathrm{\&alpha;}}^2(\mathrm{\&xi;}+1)}{{\&lsqb;{(\mathrm{\&eta;}+1-k)}^2{\mathrm{\&alpha;}}^2+{(\mathrm{\&xi;}+1)}^2\&rsqb;}^2}\\ +\frac{4{(\mathrm{\&eta;}-1+k)}^2{\mathrm{\&alpha;}}^2(\mathrm{\&xi;}+1)}{{\&lsqb;{(\mathrm{\&eta;}-1+k)}^2{\mathrm{\&alpha;}}^2+{(\mathrm{\&xi;}+1)}^2\&rsqb;}^2}\\ -\frac{4{(\mathrm{\&eta;}+1)}^2{\mathrm{\&alpha;}}^2(-\mathrm{\&xi;}+1)}{{\&lsqb;{(\mathrm{\&eta;}+1)}^2{\mathrm{\&alpha;}}^2+{(-\mathrm{\&xi;}+1)}^2\&rsqb;}^2}\\ -\frac{4{(\mathrm{\&eta;}-1)}^2{\mathrm{\&alpha;}}^2(-\mathrm{\&xi;}+1)}{{\&lsqb;{(\mathrm{\&eta;}-1)}^2{\mathrm{\&alpha;}}^2+{(-\mathrm{\&xi;}+1)}^2\&rsqb;}^2}\}\end{array}---\left(8\right)$

Order

$\begin{array}{c}\mathrm{\&lambda;}=\frac{4{(\mathrm{\&eta;}+1-k)}^2{\mathrm{\&alpha;}}^2(\mathrm{\&xi;}+1)}{{\&lsqb;{(\mathrm{\&eta;}+1-k)}^2{\mathrm{\&alpha;}}^2+{(\mathrm{\&xi;}+1)}^2\&rsqb;}^2}\\ +\frac{4{(\mathrm{\&eta;}-1+k)}^2{\mathrm{\&alpha;}}^2(\mathrm{\&xi;}+1)}{{\&lsqb;{(\mathrm{\&eta;}-1+k)}^2{\mathrm{\&alpha;}}^2+{(\mathrm{\&xi;}+1)}^2\&rsqb;}^2}\\ -\frac{4{(\mathrm{\&eta;}+1)}^2{\mathrm{\&alpha;}}^2(-\mathrm{\&xi;}+1)}{{\&lsqb;{(\mathrm{\&eta;}+1)}^2{\mathrm{\&alpha;}}^2+{(-\mathrm{\&xi;}+1)}^2\&rsqb;}^2}\\ -\frac{4{(\mathrm{\&eta;}-1)}^2{\mathrm{\&alpha;}}^2(-\mathrm{\&xi;}+1)}{{\&lsqb;{(\mathrm{\&eta;}-1)}^2{\mathrm{\&alpha;}}^2+{(-\mathrm{\&xi;}+1)}^2\&rsqb;}^2}\end{array}$

Then

${\mathrm{\&sigma;}}_x=\frac{My}{I}+\frac{F\mathrm{\&lambda;}}{2\mathrm{\&pi;}h}---\left(9\right)$

Formula (9) characteristic λ reflects deep beam calculation method for stress and mechanics of materials slender beam theoretical stress Difference, as it is drawn from deep beam infinitesimal force analysis, it is contemplated that curved scissors is to the coefficient knot of beam Really, so having the general character.

2. concentration power according to claim 1 acts on lower deep beam four-point bending calculation method for stress, its feature It is, it is further comprising the steps of：

In order to more intuitively represent the impact of warping stress that Fl, F2 cause, by (9) formula nondimensionalization, then Dimensionless direct stressIt is reduced to：

$\stackrel{\&OverBar;}{\mathrm{\&sigma;}}=\left\{\begin{array}{cc}1+{\mathrm{\&lambda;}}_{\mathrm{\&sigma;}1}& \mathrm{\&eta;}<1-k\\ 1+{\mathrm{\&lambda;}}_{\mathrm{\&sigma;}2}& \mathrm{\&eta;}\&GreaterEqual;1-k\end{array}\right.---\left(10\right)$

In formula,For dimensionless warping stress, different values of the dimensionless warping stress according to its position λ_σ1And λ_σ2,

For square-section, willFor people's dimensionless stress expression formula, obtain

$\left\{\begin{array}{cc}{\mathrm{\&lambda;}}_{\mathrm{\&sigma;}1}=\frac{\mathrm{\&lambda;}}{3\mathrm{\&pi;}k\mathrm{\&alpha;}\mathrm{\&xi;}}& \mathrm{\&eta;}<1-k\\ {\mathrm{\&lambda;}}_{\mathrm{\&sigma;}2}=\frac{\mathrm{\&lambda;}}{3\mathrm{\&pi;}\mathrm{\&alpha;}\mathrm{\&xi;}(1-\mathrm{\&eta;})}& \mathrm{\&eta;}\&GreaterEqual;1-k\end{array}\right.---\left(11\right).$

3. concentration power according to claim 1 acts on lower deep beam four-point bending calculation method for stress, its feature It is, it is further comprising the steps of：

In order to more intuitively represent the impact of warping stress that Fl, F2 cause, by (9) formula nondimensionalization, then Dimensionless direct stressIt is reduced to：

$\stackrel{\&OverBar;}{\mathrm{\&sigma;}}=\left\{\begin{array}{cc}1+{\mathrm{\&lambda;}}_{\mathrm{\&sigma;}1}& \mathrm{\&eta;}<1-k\\ 1+{\mathrm{\&lambda;}}_{\mathrm{\&sigma;}2}& \mathrm{\&eta;}\&GreaterEqual;1-k\end{array}\right.---\left(10\right)$

In formula,For dimensionless warping stress, different values of the dimensionless warping stress according to its position λ_σ1And λ_σ2,

For the symmetrical I-shaped beam section of twin shaft, if upper edge strip, lower edge strip area are A, width is b, in Property wheelbase is from for h₂、h₁, web height is h₀, thickness is δ, and area is A_f, deck-molding is h,

For convenience of calculating, ifThen α₁=α₂=0.5, A=(1+2 β) A_f

$I=\frac{A_f{h}^2}{12}\&CenterDot;\frac{1+8\mathrm{\&beta;}+12{\mathrm{\&beta;}}^2}{1+2\mathrm{\&beta;}}---\left(12\right)$

Bring formula (15) into dimensionless stress expression formula, obtain：

$\left\{\begin{array}{cc}{\mathrm{\&lambda;}}_{\mathrm{\&sigma;}1}=\frac{\mathrm{\&lambda;}(1+6\mathrm{\&beta;})}{3\mathrm{\&pi;}k\mathrm{\&alpha;}\mathrm{\&xi;}}& \mathrm{\&eta;}<1-k\\ {\mathrm{\&lambda;}}_{\mathrm{\&sigma;}2}=\frac{\mathrm{\&lambda;}(1+6\mathrm{\&beta;})}{3\mathrm{\&pi;}\mathrm{\&alpha;}\mathrm{\&xi;}(1-\mathrm{\&eta;})}& \mathrm{\&eta;}\&GreaterEqual;1-k\end{array}\right.---\left(13\right).$