CN113282987A - Non-equation Q + M graphical method for calculating strength of multi-load beam - Google Patents

Non-equation Q + M graphical method for calculating strength of multi-load beam Download PDF

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CN113282987A
CN113282987A CN202110583743.6A CN202110583743A CN113282987A CN 113282987 A CN113282987 A CN 113282987A CN 202110583743 A CN202110583743 A CN 202110583743A CN 113282987 A CN113282987 A CN 113282987A
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axis
shear
line
section
load
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赵巨才
赵阳
赵亮
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Xian University of Technology
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    • G06F30/00—Computer-aided design [CAD]
    • G06F30/10—Geometric CAD
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    • G06—COMPUTING; CALCULATING OR COUNTING
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    • G06F2119/14—Force analysis or force optimisation, e.g. static or dynamic forces

Abstract

The invention discloses a multi-load beam strength calculation equation-free 'Q + M' graphical method, wherein a datum line parallel to the axis of a beam is drawn as the axis, the length is represented in the direction of the axis, and the shear force value is represented in the direction perpendicular to the axis; drawing shear lines of the load on the beam at the corresponding cross section positions on the axis sequentially from left to right on the axis by taking the load on the beam as the direction from the left end of the beam, wherein the shear lines are sequentially connected; directly obtaining the maximum shearing force value Q according to the drawn picturemaxThe size, orientation and cross-sectional location of the probe; the bending moment of each section is equal to the algebraic sum of the area of all left closed figures of the section and the moment of all left concentrated force couples of the section, the area of the closed figures above the axis is positive, the area of the closed figures below the axis is negative, and the maximum positive bending moment is calculated
Figure DDA0003087250380000011
And maximum negative moment
Figure DDA0003087250380000012
Thus, the "Q + M" illustrated method is completed. The invention reflects the shearing force and the bending moment in the same figure, and can quickly obtain the maximum shearing force and the maximum bending moment.

Description

Non-equation Q + M graphical method for calculating strength of multi-load beam
Technical Field
The invention belongs to the technical field of mechanical analysis, and particularly relates to an equation-free 'Q + M' graphical method for calculating the strength of a multi-load beam.
Background
The strength calculation of the beam is very important in material mechanics and engineering design, the key point of the strength calculation of the beam is to calculate the maximum shearing force and the maximum bending moment of the beam, a sectional column equation is generally needed, an internal force diagram is drawn in a sectional mode, a shearing force (Q) diagram needs to be drawn, a bending moment (M) diagram needs to be drawn, and for a multi-load beam, the steps are complicated and complex, and the error rate is high.
Disclosure of Invention
The invention aims to provide an inequality Q + M graphical method for calculating the strength of a multi-load beam, which reflects shearing force and bending moment in the same graph and can quickly obtain the maximum shearing force and the maximum bending moment.
The technical scheme adopted by the invention is that the inequality 'Q + M' graphical method for calculating the strength of the multi-load beam comprises the following steps:
step 1) drawing a datum line parallel to the axis of the beam as an axis, wherein the direction of the axis represents the length, the direction perpendicular to the axis represents a shear force value, the shear force value above the axis is positive, and the shear force value below the axis is negative;
step 2) drawing shear lines of the load on the beam at the corresponding cross section position on the axis in sequence from left to right on the axis in sequence, wherein the shear lines are connected in sequence, for the unloaded beam section, a parallel line parallel to the axis is drawn by taking the tail end of the adjacent shear line on the left side as a starting point and is taken as the shear line of the unloaded beam section, the tail end of the parallel line is the starting point of the adjacent shear line on the right side, the starting point of the shear line on the left end of the beam is positioned on the axis, and the end point of the shear line on the right end of the beam is positioned on the axis;
step 3) directly obtaining the maximum shearing force value Q according to the graph drawn in the step 2maxThe size, orientation and cross-sectional location of the probe;
step 4) calculating the area of each closed figure formed by the shearing line and the axis, marking the direction and the size of the concentrated couple above the position of the corresponding axis of the section where the concentrated couple is located, wherein the bending moment of each section is equal to the algebraic sum of the area of all left closed figures and the moment of all left concentrated couples of the section, the area of the closed figure above the axis is positive, the area of the closed figure below the axis is negative, and calculating the maximum positive bending moment
Figure BDA0003087250360000021
And maximum negative moment
Figure BDA0003087250360000022
Thus, the "Q + M" illustrated method is completed.
The present invention is also characterized in that,
the load on the beam comprises a concentrated force, a concentrated force couple and a distributed load;
the concentrated force is upward, the corresponding shear line is drawn from bottom to top and is perpendicular to the axis, and the length of the shear line is the size of the upward concentrated force;
the concentrated force is downward, the corresponding shear line is drawn upward and downward and is vertical to the axis, and the length of the shear line is the magnitude of the downward concentrated force;
the concentrated couple does not draw a shear line;
the shearing force lines of the uniformly distributed load sections are inclined by taking the load concentration as a slope, and the change quantity of the shearing force value of two end points of the shearing force lines is equal to the area of the uniformly distributed load graph of the beam;
the shearing force line of the non-uniform load section is a curve, the slope of the tangent line of each point on the curve is equal to the load concentration of the corresponding cross-section position on the beam, and the change amount of the shearing force value of the two end points of the shearing force line is equal to the area of the non-uniform load graph of the beam.
And in the step 2), the shearing force value of each inflection point of the shearing force line is marked in the graph.
And 3), marking the area of each closed graph in the closed graphs.
The unit for length in the direction of the axis is m and the unit for shear force in the direction perpendicular to the axis is KN.
The invention has the beneficial effects that:
the invention relates to a method for calculating an equation-free 'Q + M' diagram of strength of a multi-load beam, which has the advantages of simple and clear steps, no need of an internal force equation, no need of a segmented internal force diagram, a shear force diagram and a bending moment diagram, even no need of solving constraint force, direct drawing of shear lines in sequence according to the external force (load on the beam), marking of the shear force value of each inflection point of the shear line, and marking of the bending moment in the diagram, so that the shear force and the bending moment are reflected in the same diagram, namely the 'Q + M' diagram, and the maximum shear force, the maximum bending moment and the section where the maximum bending moment is located can be obtained through simple calculation; the complex calculation steps are not involved in the graphic process, the analysis and calculation process is simple and quick, and the calculation speed and efficiency of the strength of the multi-load beam are greatly improved.
Drawings
FIG. 1 is an external force diagram of a multi-load beam according to example 1;
FIG. 2 is a "Q + M" diagram of the multi-load beam of example 1;
FIG. 3 is an external force diagram of the multi-load beam of the embodiment 2;
FIG. 4 is a "Q + M" diagram of the multi-load beam of example 2;
FIG. 5 is an external force diagram of the multi-load beam of the embodiment 3;
FIG. 6 is a "Q + M" diagram of the multi-load beam of example 3;
FIG. 7 is an external force diagram of the multi-load beam of the embodiment 4;
FIG. 8 is a "Q + M" diagram of the multi-load beam of example 4.
Detailed Description
The present invention will be described in detail with reference to the following embodiments.
The invention discloses a method for calculating an equation-free 'Q + M' diagram of the strength of a multi-load beam, which comprises the following steps:
step 1) drawing a datum line parallel to the axis of the beam as an axis, wherein the axis direction represents the length, the unit of the length is m, the direction perpendicular to the axis represents a shear force value, the unit of the shear force value is KN, the shear force value above the axis is positive, and the shear force value below the axis is negative;
step 2) drawing shear lines of the load on the beam at the corresponding section positions on the axis sequentially from left to right on the axis in the direction of the load on the beam from the left end of the beam, wherein the shear lines are sequentially connected, the starting point of the shear line at the left end of the beam is positioned on the axis, the ending point of the shear line at the right end of the beam is positioned on the axis, and the shear values of the inflection points of the shear line are marked in the drawing;
the load on the beam comprises a concentrated force, a concentrated force couple and a distributed load;
the concentrated force is upward, the corresponding shear line is drawn from bottom to top and is perpendicular to the axis, and the length of the shear line is the size of the upward concentrated force;
the concentrated force is downward, the corresponding shear line is drawn upward and downward and is vertical to the axis, and the length of the shear line is the magnitude of the downward concentrated force;
the concentrated couple does not draw a shear line;
the shearing force lines of the uniformly distributed load sections are inclined by taking the load concentration as a slope, and the change quantity of the shearing force value of two end points of the shearing force lines is equal to the area of the uniformly distributed load graph of the beam;
the shearing force line of the non-uniform load section is a curve, the slope of the tangent line of each point on the curve is equal to the load concentration of the corresponding cross-section position on the beam, and the change amount of the shearing force value of the two end points of the shearing force line is equal to the area of the non-uniform load graph of the beam;
for the unloaded beam section, drawing a parallel line parallel to the axis as a shear line of the unloaded beam section by taking the tail end of the adjacent shear line on the left side of the unloaded beam section as a starting point, wherein the tail end of the parallel line is the starting point of the adjacent shear line on the right side;
the shear line can also be drawn from the right end to the left end of the beam, but the drawn shear line is drawn along the reverse direction of the load on the beam, the end point of the shear line at the left end of the beam is located on the axis, and the start point of the shear line at the right end of the beam is located on the axis.
Step 3) directly obtaining the maximum shear force value Q according to the drawing drawn in the step 2 and the labeled shear force valuemaxThe size, orientation and cross-sectional location of the element.
Step 4) calculating the area of each closed graph formed by the shearing line and the axis, marking the area of each closed graph in the closed graph, marking the direction and the size of the concentrated couple above the position of the axis corresponding to the cross section where the concentrated couple is located, wherein the graph is a 'Q + M' graph, the bending moment of each cross section is equal to the algebraic sum of the area of all closed graphs on the left side of the cross section and the moment of all concentrated couples on the left side of the cross section when calculating the bending moment, the area of the closed graph above the axis is positive, the area of the closed graph below the axis is negative, and the maximum positive bending moment is calculated
Figure BDA0003087250360000051
And maximum negative moment
Figure BDA0003087250360000052
Thus, the "Q + M" illustrated method is completed.
Example 1
Calculate Q of the Beam as shown in FIG. 1max、
Figure BDA0003087250360000053
And
Figure BDA0003087250360000054
in order to perform strength calculations on the beam.
Firstly, the counter force R of the support at A is calculatedA＝10KN
Then, Q of the beam is obtained according to the method for calculating the strength of the multi-load beam by using the inequality Q + M graphic methodmax、
Figure BDA0003087250360000055
And
Figure BDA0003087250360000056
the method comprises the following steps:
step 1) drawing a datum line parallel to the axis of the beam as an axis, wherein the axis direction represents the length, the unit of the length is m, the direction perpendicular to the axis represents a shear force value, the unit of the shear force value is KN, the shear force value above the axis is positive, and the shear force value below the axis is negative;
and 2) sequentially drawing the shear lines of the load on the beam at the corresponding section positions on the axis from left to right on the axis in sequence from the left end of the beam, wherein the shear lines are sequentially connected, the starting point of the shear line at the left end of the beam is positioned on the axis, the ending point of the shear line at the right end of the beam is positioned on the axis, and the shear values of the inflection points of the shear line are marked in the graph as shown in FIG. 2.
Step 3) directly obtaining the maximum shear force value Q according to the drawing drawn in the step 2 and the labeled shear force valuemax29KN, and the maximum shear force appears in sections E and H, section E and section QmaxVertical axis down, H section QmaxIs vertically orientedThe axis is upward.
Step 4) calculating the area of each closed graph formed by the shearing line and the axis, marking the area of each closed graph in the closed graph, and marking the direction and the size of the concentrated couple above the position of the axis corresponding to the cross section where the concentrated couple is located, as shown in fig. 2, the graph is a 'Q + M' graph, when calculating the bending moment, the bending moment of each cross section is equal to the algebraic sum of the area of all closed graphs on the left side of the cross section and the moment of all concentrated couples on the left side of the cross section, the area of the closed graph above the axis is positive, and the area of the closed graph below the axis is negative, and the calculation can be carried out:
maximum positive bending moment
Figure BDA0003087250360000061
Is positioned on the C section;
maximum negative moment
Figure BDA0003087250360000062
Located in section H.
From the above, it can be seen that, for the simply supported beam, the "Q + M" diagram can be drawn by simply obtaining the reaction force of the support at one end, and Q is obtained by simple calculation according to the "Q + M" diagrammax、
Figure BDA0003087250360000063
And
Figure BDA0003087250360000064
example 2
Calculate Q of the Beam as shown in FIG. 3max、
Figure BDA0003087250360000065
And
Figure BDA0003087250360000066
in order to perform strength calculations on the beam.
Firstly, the counter force R of the support at A is calculatedA＝18KN
Then, the method is obtained according to the method for calculating the strength of the multi-load beam by using the inequality Q + M graphic methodQ of the beammax、
Figure BDA0003087250360000067
And
Figure BDA0003087250360000068
the method comprises the following steps:
step 1) drawing a datum line parallel to the axis of the beam as an axis, wherein the axis direction represents the length, the unit of the length is m, the direction perpendicular to the axis represents a shear force value, the unit of the shear force value is KN, the shear force value above the axis is positive, and the shear force value below the axis is negative;
and 2) sequentially drawing the shear lines of the load on the beam at the corresponding section positions on the axis from left to right on the axis in sequence from the left end of the beam, wherein the shear lines are sequentially connected, the starting point of the shear line at the left end of the beam is positioned on the axis, the ending point of the shear line at the right end of the beam is positioned on the axis, and the shear values of the inflection points of the shear line are marked in the figure as shown in fig. 4.
Step 3) directly obtaining the maximum shear force value Q according to the drawing drawn in the step 2 and the labeled shear force valuemax19KN and the maximum shear occurs at section E, B, section E QmaxVertical axis down, H section QmaxThe direction is vertical to the axis and upward.
Step 4) calculating the area of each closed graph formed by the shearing line and the axis, marking the area of each closed graph in the closed graph, and marking the direction and the size of the concentrated couple above the position of the axis corresponding to the cross section where the concentrated couple is located, as shown in fig. 4, the graph is a 'Q + M' graph, when calculating the bending moment, the bending moment of each cross section is equal to the algebraic sum of the area of all closed graphs on the left side of the cross section and the moment of all concentrated couples on the left side of the cross section, the area of the closed graph above the axis is positive, and the area of the closed graph below the axis is negative, and the calculation can be carried out:
maximum positive bending moment
Figure BDA0003087250360000071
Located on the O section.
Example 3
ComputingQ of the beam shown in FIG. 5max、
Figure BDA0003087250360000072
And
Figure BDA0003087250360000073
in order to perform strength calculations on the beam.
According to the method for calculating the strength of the multi-load beam by using the inequatless Q + M graphic method, the Q of the beam is obtainedmax、
Figure BDA0003087250360000074
And
Figure BDA0003087250360000075
the method comprises the following steps:
step 1) drawing a datum line parallel to the axis of the beam as an axis, wherein the axis direction represents the length, the unit of the length is m, the direction perpendicular to the axis represents a shear force value, the unit of the shear force value is KN, the shear force value above the axis is positive, and the shear force value below the axis is negative;
and 2) sequentially drawing the shear lines of the load on the beam at the corresponding section positions on the axis from left to right on the axis in sequence from the left end of the beam, wherein the shear lines are sequentially connected, the starting point of the shear line at the left end of the beam is positioned on the axis, the ending point of the shear line at the right end of the beam is positioned on the axis, and the shear values of the inflection points of the shear line are marked in the figure as shown in fig. 6.
Step 3) directly obtaining the maximum shear force value Q according to the drawing drawn in the step 2 and the labeled shear force valuemax25KN, and the maximum shear force occurs in sections E and B, section E QmaxVertical axis down, B section QmaxThe direction is vertical to the axis and upward.
Step 4) calculating the area of each closed graph formed by the shearing line and the axis, marking the area of each closed graph in the closed graph, and marking the direction and the size of the concentrated couple above the position of the axis corresponding to the cross section where the concentrated couple is located, as shown in fig. 2, the graph is a 'Q + M' graph, when calculating the bending moment, the bending moment of each cross section is equal to the algebraic sum of the area of all closed graphs on the left side of the cross section and the moment of all concentrated couples on the left side of the cross section, the area of the closed graph above the axis is positive, and the area of the closed graph below the axis is negative, and the calculation can be carried out:
maximum positive bending moment
Figure BDA0003087250360000081
Is positioned on the O section;
maximum negative moment
Figure BDA0003087250360000082
Located at section B.
According to the above contents, the cantilever beam can be directly drawn without calculation, the 'Q + M' diagram is drawn, the supporting reaction force of the fixed end can be calculated according to the internal force of the beam as zero after the shear line slides to the fixed end, and the Q is obtained by simple calculation according to the 'Q + M' diagrammax、
Figure BDA0003087250360000083
And
Figure BDA0003087250360000084
example 4
Calculate Q of the Beam as shown in FIG. 7max、
Figure BDA0003087250360000085
And
Figure BDA0003087250360000086
in order to perform strength calculations on the beam.
Firstly, the counter force R of the support at B is calculatedBAnd 29KN, the direction is vertical to the axial direction and upward.
Then, Q of the beam is obtained according to the method for calculating the strength of the multi-load beam by using the inequality Q + M graphic methodmax、
Figure BDA0003087250360000087
And
Figure BDA0003087250360000088
the method comprises the following steps:
step 1) drawing a datum line parallel to the axis of the beam as an axis, wherein the axis direction represents the length, the unit of the length is m, the direction perpendicular to the axis represents a shear force value, the unit of the shear force value is KN, the shear force value above the axis is positive, and the shear force value below the axis is negative;
and 2) sequentially drawing the shear lines of the load on the beam at the corresponding section positions on the axis from the right end of the beam from right to left in the direction of the direction opposite to the load on the beam, wherein the shear lines are sequentially connected, the starting point of the shear line at the left end of the beam is positioned on the axis, the ending point of the shear line at the right end of the beam is positioned on the axis, and the shear values of the inflection points of the shear line are marked in the figure as shown in fig. 8.
Step 3) directly obtaining the maximum shear force value Q according to the drawing drawn in the step 2 and the labeled shear force valuemax81KN and the maximum shear occurs at section A, section QmaxThe direction is vertical to the axis and upward.
Step 4) calculating the area of each closed graph formed by the shearing line and the axis, marking the area of each closed graph in the closed graph, and marking the direction and the size of the concentrated couple above the position of the axis corresponding to the section where the concentrated couple is located, as shown in fig. 8, the graph is a 'Q + M' graph, and according to the AD beam sigma M D0 according to fig. 8, i.e. MA+81+15.5 ═ 0, the bending moment of the A section is MA-96.5, then:
maximum negative moment
Figure BDA0003087250360000091
Is positioned on the section D;
maximum positive bending moment
Figure BDA0003087250360000092
A cross-section located 1.55m to the right of point E.

Claims (5)

1. The method for calculating the strength of the multi-load beam by using the inequality Q + M graphic method is characterized by comprising the following steps of:
step 1) drawing a datum line parallel to the axis of the beam as an axis, wherein the direction of the axis represents the length, the direction perpendicular to the axis represents a shear force value, the shear force value above the axis is positive, and the shear force value below the axis is negative;
step 2) drawing shear lines of the load on the beam at the corresponding cross section position on the axis in sequence from left to right on the axis in sequence, wherein the shear lines are connected in sequence, for the unloaded beam section, a parallel line parallel to the axis is drawn by taking the tail end of the adjacent shear line on the left side as a starting point and is taken as the shear line of the unloaded beam section, the tail end of the parallel line is the starting point of the adjacent shear line on the right side, the starting point of the shear line on the left end of the beam is positioned on the axis, and the end point of the shear line on the right end of the beam is positioned on the axis;
step 3) directly obtaining the maximum shearing force value Q according to the graph drawn in the step 2maxThe size, orientation and cross-sectional location of the probe;
step 4) calculating the area of each closed figure formed by the shearing line and the axis, marking the direction and the size of the concentrated couple above the position of the corresponding axis of the section where the concentrated couple is located, wherein the bending moment of each section is equal to the algebraic sum of the area of all left closed figures and the moment of all left concentrated couples of the section, the area of the closed figure above the axis is positive, the area of the closed figure below the axis is negative, and calculating the maximum positive bending moment
Figure FDA0003087250350000011
And maximum negative moment
Figure FDA0003087250350000012
Thus, the "Q + M" illustrated method is completed.
2. The method of claim 1, wherein the on-beam loads comprise concentrated forces, concentrated force couples, and distributed loads;
the concentrated force is upward, the corresponding shear line is drawn from bottom to top and is perpendicular to the axis, and the length of the shear line is the size of the upward concentrated force;
the concentrated force is downward, the corresponding shear line is drawn upward and downward and is vertical to the axis, and the length of the shear line is the magnitude of the downward concentrated force;
the concentrated couple does not draw a shear line;
the shearing force lines of the uniformly distributed load sections are inclined by taking the load concentration as a slope, and the change quantity of the shearing force value of two end points of the shearing force lines is equal to the area of the uniformly distributed load graph of the beam;
the shearing force line of the non-uniform load section is a curve, the slope of the tangent line of each point on the curve is equal to the load concentration of the corresponding cross-section position on the beam, and the change amount of the shearing force value of the two end points of the shearing force line is equal to the area of the non-uniform load graph of the beam.
3. The method for graphically calculating the inequality Q + M of the strength of the multi-load beam as claimed in claim 1, wherein in the step 2), the shearing force value of each inflection point of the shearing line is marked in the figure.
4. The method for calculating the inequality Q + M of the multi-load beam strength as claimed in claim 1, wherein in the step 3), the area of each closed figure is marked in the closed figure.
5. The method of claim 3, wherein the length is expressed in M in the axial direction and the shear force is expressed in KN in the vertical direction.
CN202110583743.6A 2021-05-27 2021-05-27 Non-equation Q + M graphical method for calculating strength of multi-load beam Pending CN113282987A (en)

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Application publication date: 20210820