CN106372305B - Method for calculating length coefficient of unequal-division crossed inclined timber of steel structure - Google Patents

Method for calculating length coefficient of unequal-division crossed inclined timber of steel structure Download PDF

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CN106372305B
CN106372305B CN201610775991.XA CN201610775991A CN106372305B CN 106372305 B CN106372305 B CN 106372305B CN 201610775991 A CN201610775991 A CN 201610775991A CN 106372305 B CN106372305 B CN 106372305B
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赵怀宇
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PowerChina Hebei Electric Power Engineering Co Ltd
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Abstract

The invention discloses a method for calculating a length coefficient of a steel structure unequal crossing diagonal member, which comprises the following steps of A) deducing a yield condition theoretical formula of an axial compression rod member of a single elastic support, namely deducing a buckling equation and then calculating buckling critical load; B) deducing a yield condition theoretical formula of the axial tension member of the single elastic support, namely deducing a buckling equation and then solving a buckling critical load; C) derivation of yield equation and simplified calculation formula under one-pull-one-pressure working condition, namely according to Euler critical force formula Pcr＝π2EI/(μl)2And performing reverse extrapolation to obtain a calculated length coefficient mu. The invention starts from the buckling equation of the unequal cross inclined timber, calculates the calculated length coefficient according to the buckling load, and finally provides a simplified formula of the calculated length coefficient of the unequal cross inclined timber. Thus, the problem of calculating the length coefficient when the intersection point of the inclined timber is not at the midpoint can be simply solved.

Description

Method for calculating length coefficient of unequal-division crossed inclined timber of steel structure
Technical Field
The invention relates to the technical field of steel structure design, in particular to a method for calculating length coefficients of unequal-division crossed inclined bars of a steel structure.
Background
In GB 50017-2003 Steel Structure design Specification, the calculation method of the calculated length coefficient of the cross members is that when the lengths of two cross rods are equal, the calculated length l of the compression rod is equal when the other cross rod is pulled and the cross sections of the two cross rods are the same and are not interrupted at the cross point, and then the calculated length l of the compression rod is equal0As shown in formula (a), the calculated length coefficient μ of the plunger is shown in formula (b),
Figure BDA0001098178660000011
Figure BDA0001098178660000012
in the formula, l is the distance between the centers of the truss nodes (the intersection points are not considered as nodes);
n is the internal force of the calculated rod, N0The internal forces of the other rod are absolute values, and when both rods are pressed, N is taken0Less than or equal to N, and the sections of the two rods are the same.
The above formula only considers the situation when the intersection point is at the midpoint of the oblique timber, and no reasonable formula expression is given for the calculated length coefficient of the unequal intersection timber.
Disclosure of Invention
The technical problem to be solved by the invention is to provide a method for calculating the calculated length coefficient of the unequal-division cross diagonal members of the steel structure, so that the calculation of the calculated length coefficient of the unequal-division cross diagonal members is realized.
In order to solve the technical problems, the technical scheme adopted by the invention is as follows:
a method for calculating the length coefficient of an unequal crossing diagonal material of a steel structure comprises a pressed rod piece and a pulled rod piece which have the same length, the same section and the same elastic modulus, wherein the pressed rod piece and the pulled rod piece are uninterrupted continuous rod pieces and have the same length proportion after being divided by a cross point, the pressed rod piece and the pulled rod piece are both of a connecting structure with two hinged ends and one end capable of displacing along the rod axis direction, and the method for calculating the length coefficient of the unequal crossing diagonal material comprises the following steps under the working condition of pulling and pressing,
A) deducing a yield condition theoretical formula of the axial compression rod piece of the single elastic support, namely deducing a buckling equation and then solving buckling critical load;
B) deducing a yield condition theoretical formula of the axial tension member of the single elastic support, namely deducing a buckling equation and then solving a buckling critical load;
C) derivation of yield equation and simplified calculation formula under one-pull-one-pressure working condition, namely according to Euler critical force formula Pcr＝π2EI/(μl)2And performing reverse extrapolation to obtain a calculated length coefficient mu.
The technical scheme of the invention is further improved as follows: the single elastically supported axial compression bar member in the step a) is specifically that a support member with a spring constant c is arranged on the compression bar member, the compression bar member can freely rotate at the support member, the bending rigidity of the compression bar member is EI, the bending line is continuous, and y' (l)1) Not equal to 0, the concrete process derived from the theoretical formula of the yield condition of the pressed rod piece is as follows,
let the deflection of the pressed rod at the supporting member be v, the generated spring force be cv, establish a balance equation, when x is less than or equal to l1Then, the formula (1) is obtained,
Figure BDA0001098178660000021
order to
Figure BDA0001098178660000022
Then the formula (2) is obtained,
Figure BDA0001098178660000023
the deflection line of the pressure receiving rod member is shown in formula (3),
Figure BDA0001098178660000024
in the formula, constant A1And B1The boundary conditions y (0) of the pressed rod can be equal to 0 and y (l)1) The average of the values obtained for v, i.e.,
Figure BDA0001098178660000025
B1＝0，
therefore, the temperature of the molten steel is controlled,
Figure BDA0001098178660000031
Figure BDA0001098178660000032
Figure BDA0001098178660000033
when l is1When x is less than or equal to l,
Figure BDA0001098178660000034
or,
Figure BDA0001098178660000035
to obtain the formula (7),
Figure BDA0001098178660000036
using the boundary conditions y (l) 0 and y (l) of the compression rod1) Given as v, a constant a in formula (7)2And B2，
Figure BDA0001098178660000037
Figure BDA0001098178660000038
Therefore, the temperature of the molten steel is controlled,
Figure BDA0001098178660000039
Figure BDA00010981786600000310
Figure BDA0001098178660000041
the two equations of equation (6) and equation (10) are equal, and the yield condition of the pressed rod piece can be obtained as follows,
Figure BDA0001098178660000042
that is to say that the first and second electrodes,
Figure BDA0001098178660000043
the technical scheme of the invention is further improved as follows: the axial pulled rod element elastically supported in the step B) is specifically that a pressing member with the spring constant of c is arranged on the pulled rod element, the pulled rod element can freely rotate at the pressing member, and the theoretical formula of the yield condition of the pulled rod element is deduced in a specific process,
let the deflection of the pull rod at the supporting member be v, the generated spring force be cv, establish a balance equation, when x is less than or equal to l1Then, the formula (13) is obtained,
Figure BDA0001098178660000044
order to
Figure BDA0001098178660000045
The formula (14) is obtained,
Figure BDA0001098178660000046
the deflection line of the pulled rod member is shown in formula (15),
Figure BDA0001098178660000047
in the formula, constant A1And B1The boundary conditions y (0) of the pulled member can be equal to 0 and y (l)1) The average of the values obtained for v, i.e.,
Figure BDA0001098178660000048
Figure BDA0001098178660000049
therefore, it is
Figure BDA0001098178660000051
Figure BDA0001098178660000052
Figure BDA0001098178660000053
When l is1When x is less than or equal to l,
Figure BDA0001098178660000054
or,
Figure BDA0001098178660000055
the formula (19) is obtained,
Figure BDA0001098178660000056
using the boundary conditions y (l) 0 and y (l) of the pulled member1) Given as v, a constant a in formula (19)2And B2，
Figure BDA0001098178660000057
Figure BDA0001098178660000058
Therefore, the temperature of the molten steel is controlled,
Figure BDA0001098178660000059
Figure BDA00010981786600000510
Figure BDA00010981786600000511
the equation (18) is equal to the equation (22), and the yield condition of the pulled rod element can be obtained as follows,
Figure BDA0001098178660000061
when P is equal to P1When the temperature of the water is higher than the set temperature,
Figure BDA0001098178660000062
the technical scheme of the invention is further improved as follows: the concrete process of deriving the yield equation and the simplified calculation formula under the condition of one-pull-one-press in the step C) is that,
obtained from the formula (18) and the formula (22),
Figure BDA0001098178660000063
wherein,
Figure BDA0001098178660000064
Figure BDA0001098178660000065
the maximum value of P is such that,
Figure BDA0001098178660000066
solving the P value by adopting a numerical iterative algorithm, and obtaining the P value according to an Euler critical force formula Pcr＝π2EI/(μl)2And performing reverse extrapolation to obtain a calculated length coefficient mu.
The technical scheme of the invention is further improved as follows: for different P1The ratio/P and different inequality degrees correct the calculated length coefficient mu by the correction value,
Figure BDA0001098178660000067
due to the adoption of the technical scheme, the invention has the technical progress that:
the invention starts from the buckling equation of the unequal cross inclined timber, calculates the calculated length coefficient according to the buckling load, and finally provides a simplified formula of the calculated length coefficient of the unequal cross inclined timber. Thus, the problem of calculating the length coefficient when the intersection point of the inclined timber is not at the midpoint can be simply solved.
Drawings
FIG. 1 is a schematic view of a cross diagonal calculation model according to the present invention;
FIG. 2 is a schematic view of a computational model of an axial compression rod with a single resilient support according to the present invention;
FIG. 3 is a schematic view of a mechanical model of an axial compression rod with a single elastic support according to the present invention;
FIG. 4 is a schematic view of the mechanical model of an axial tension member with a single resilient support of the present invention;
FIG. 5 is a comparison graph of theoretical values and correction values under a pull-push condition according to the present invention.
Detailed Description
The invention is described in further detail below:
the invention discloses a method for calculating a length coefficient of a steel structure unequal-division crossed diagonal material. The schematic diagram of the cross diagonal material calculation model is shown in FIG. 1.
In the calculation method of the invention, the following assumed conditions are introduced:
(1) the lengths of the pressed rod piece and the pulled rod piece in the crossed oblique material are the same;
(2) the sections of the pressed rod piece and the pulled rod piece in the crossed oblique material are the same, and the elastic modulus is the same;
(3) the length proportion of the pressed rod piece and the pulled rod piece after the intersection in the crossed diagonal material is divided is the same;
(4) the compression rod piece and the tension rod piece in the crossed oblique material are uninterrupted continuous rod pieces, namely, compression bending or stretch bending components;
(5) both ends of a pressed rod piece and a pulled rod piece in the crossed oblique material are hinged, and one end of the pressed rod piece and the pulled rod piece can displace along the rod axis direction.
Under the working condition of one pulling and one pressing, the method for calculating the length coefficient of the unequal-division cross inclined timber comprises the following steps,
A) deducing a yield condition theoretical formula of the axial compression rod piece of the single elastic support, namely deducing a buckling equation and then solving buckling critical load;
B) deducing a yield condition theoretical formula of the axial tension member of the single elastic support, namely deducing a buckling equation and then solving a buckling critical load;
C) derivation of yield equation and simplified calculation formula under one-pull-one-pressure working condition, namely according to Euler critical force formula Pcr＝π2EI/(μl)2And performing reverse extrapolation to obtain a calculated length coefficient mu.
Specifically, the single elastically supported axial pressure receiving rod member in step a) is specifically that a support member with a spring constant c is arranged on the pressure receiving rod member, the pressure receiving rod member is freely rotatable at the support member, a schematic diagram of a computational model is shown in fig. 2, and a schematic diagram of a mechanical model is shown in fig. 3.
At this time, the flexural rigidity of the compression rod member is EI, the deflection line is continuous and y' (l)1) Not equal to 0, the concrete process derived from the theoretical formula of the yield condition of the pressed rod piece is as follows,
let the deflection of the pressed rod at the supporting member be v, the generated spring force be cv, establish a balance equation, when x is less than or equal to l1Then, the formula (1) is obtained,
Figure BDA0001098178660000081
order to
Figure BDA0001098178660000082
Then the formula (2) is obtained,
Figure BDA0001098178660000083
the deflection line of the pressure receiving rod member is shown in formula (3),
Figure BDA0001098178660000084
in the formula, constant A1And B1The boundary conditions y (0) of the pressed rod can be equal to 0 and y (l)1) The average of the values obtained for v, i.e.,
Figure BDA0001098178660000085
B1＝0，
therefore, the temperature of the molten steel is controlled,
Figure BDA0001098178660000086
Figure BDA0001098178660000087
Figure BDA0001098178660000091
when l is1When x is less than or equal to l,
Figure BDA0001098178660000092
or,
Figure BDA0001098178660000093
to obtain the formula (7),
Figure BDA0001098178660000094
using the boundary conditions y (l) 0 and y (l) of the compression rod1) Given as v, a constant a in formula (7)2And B2，
Figure BDA0001098178660000095
Figure BDA0001098178660000096
Therefore, the temperature of the molten steel is controlled,
Figure BDA0001098178660000097
Figure BDA0001098178660000098
Figure BDA0001098178660000099
the two equations of equation (6) and equation (10) are equal, and the yield condition of the pressed rod piece can be obtained as follows,
Figure BDA00010981786600000910
that is to say that the first and second electrodes,
Figure BDA0001098178660000101
specifically, the single elastically supported axial pulled rod in step B) is to arrange a pressing member with a spring constant c on the pulled rod, where the pulled rod can freely rotate, and the mechanical model diagram is shown in fig. 3.
The concrete process deduced from the theoretical formula of the yield condition of the tension member is that,
let the deflection of the pull rod at the supporting member be v, the generated spring force be cv, establish a balance equation, when x is less than or equal to l1Then, the formula (13) is obtained,
Figure BDA0001098178660000102
order to
Figure BDA0001098178660000103
The formula (14) is obtained,
Figure BDA0001098178660000104
the deflection line of the pulled rod member is shown in formula (15),
Figure BDA0001098178660000105
in the formula, constant A1And B1The boundary conditions y (0) of the pulled member can be equal to 0 and y (l)1) The average of the values obtained for v, i.e.,
Figure BDA0001098178660000106
Figure BDA0001098178660000107
therefore, it is
Figure BDA0001098178660000108
Figure BDA0001098178660000109
Figure BDA0001098178660000111
When l is1When x is less than or equal to l,
Figure BDA0001098178660000112
or,
Figure BDA0001098178660000113
the formula (19) is obtained,
Figure BDA0001098178660000114
using the boundary conditions y (l) 0 and y (l) of the pulled member1) Given as v, a constant a in formula (19)2And B2，
Figure BDA0001098178660000115
Figure BDA0001098178660000116
Therefore, the temperature of the molten steel is controlled,
Figure BDA0001098178660000117
Figure BDA0001098178660000118
Figure BDA0001098178660000119
the equation (18) is equal to the equation (22), and the yield condition of the pulled rod element can be obtained as follows,
Figure BDA00010981786600001110
when P is equal to P1When the temperature of the water is higher than the set temperature,
Figure BDA0001098178660000121
specifically, the specific process of deriving the yield equation and the simplified calculation formula under the condition of one-pull-one-press in the step C) is that,
obtained from the formula (18) and the formula (22),
Figure BDA0001098178660000122
wherein,
Figure BDA0001098178660000123
Figure BDA0001098178660000124
the maximum value of P is such that,
Figure BDA0001098178660000125
the formula P is an unknown number, other parameters are known, a numerical iteration algorithm is adopted to solve the value P, and after the value P is obtained, the formula P is based on the Euler critical forcecr＝π2EI/(μl)2And performing reverse extrapolation to obtain a calculated length coefficient mu.
When l is1＝0.5l，l2When the length coefficient of the compression bar is 0.5l, the approximate solution of the calculated length coefficient of the compression bar is the formula (26) in the steel structure design specification:
Figure BDA0001098178660000126
for different P1the/P ratio and the different degrees of inequality are calculated as shown in Table 1 below.
As can be seen from table 1, under the working condition of one pull and one press, the difference between the calculated length coefficient result of the press rod and the time of equal division becomes larger and larger as the degree of unequal division is increased, so the correction of the calculated length coefficient needs to be considered on the basis of equal division cross diagonal materials in the specification of steel structure design.
Correcting the calculated length coefficient mu by the correction value,
Figure BDA0001098178660000127
correction in FIG. 5The value 0.5 "means l1:l2The value calculated by the correction formula (27) when the ratio is 0.5:0.5, and the "correction value 0.6" in fig. 5 means l1:l2The value calculated by the correction formula (27) when the ratio is 0.4:0.6, and the "correction value 0.7" in fig. 5 means l1:l2The "theoretical value" is a value calculated according to (25) when the value is 0.3:0.7 according to the correction formula (27).
As can be seen from Table 2 and FIG. 5, under the condition of one pull and one press, the corrected value is slightly conservative, and the difference from the theory is not more than 5% at most, so the invention is adopted
Figure BDA0001098178660000131
The calculated length coefficient of the unequal-division cross material under the working condition of one-pull-one-press is calculated, the standard requirement is met, and the actual operation is simplified.
TABLE 1 influence of unequal degrees on the calculated length coefficient of the compression bar under the working conditions of one pull and one press
Figure BDA0001098178660000132
TABLE 2 comparison table of theoretical value and correction value under one-pull-one-press working condition
Figure BDA0001098178660000133
Figure BDA0001098178660000141

Claims (1)

1. A design method of the length of the unequal cross diagonal timber comprises the steps of firstly calculating the calculated length coefficient of the unequal cross diagonal timber, and then designing the length of the unequal cross diagonal timber according to the obtained calculated length coefficient, and is characterized in that: the cross diagonal material comprises a pressed rod piece and a pulled rod piece which are the same in length, the same in cross section and the same in elastic modulus, the pressed rod piece and the pulled rod piece are uninterrupted continuous rod pieces, the length proportion of the pressed rod piece and the pulled rod piece is the same after being divided by a cross point, the pressed rod piece and the pulled rod piece are both of a connecting structure with two hinged ends, one end of the connecting structure can displace along the rod shaft direction, and the calculation method for calculating the length coefficient of the unequal cross diagonal material under the working condition of one pulling and one pressing comprises the following steps,
A) deducing a yield condition theoretical formula of the axial compression rod piece of the single elastic support, namely deducing a buckling equation and then solving buckling critical load;
B) deducing a yield condition theoretical formula of the axial tension member of the single elastic support, namely deducing a buckling equation and then solving a buckling critical load;
C) derivation of yield equation and simplified calculation formula under one-pull-one-pressure working condition, namely according to Euler critical force formula Pcr＝π2EI/(μl)2Performing reverse-deducing to obtain a calculated length coefficient mu;
the single elastically supported axial compression bar member in the step a) is specifically that a support member with a spring constant c is arranged on the compression bar member, the compression bar member can freely rotate at the support member, the bending rigidity of the compression bar member is EI, the bending line is continuous, and y' (l)1) Not equal to 0, the concrete process derived from the theoretical formula of the yield condition of the pressed rod piece is as follows,
let the deflection of the pressed rod at the supporting member be v, the generated spring force be cv, establish a balance equation, when x is less than or equal to l1Then, the formula (1) is obtained,
Figure FDA0002532632260000011
order to
Figure FDA0002532632260000012
Then the formula (2) is obtained,
Figure FDA0002532632260000013
the deflection line of the pressure receiving rod member is shown in formula (3),
Figure FDA0002532632260000014
in the formula, constant A1And B1The boundary conditions y (0) of the pressed rod can be equal to 0 and y (l)1) The average of the values obtained for v, i.e.,
Figure FDA0002532632260000021
B1＝0，
therefore, the temperature of the molten steel is controlled,
Figure FDA0002532632260000022
Figure FDA0002532632260000023
Figure FDA0002532632260000024
when l is1When x is less than or equal to l,
Figure FDA0002532632260000025
or,
Figure FDA0002532632260000026
to obtain the formula (7),
Figure FDA0002532632260000027
using the boundary conditions y (l) 0 and y (l) of the compression rod1) Given as v, a constant a in formula (7)2And B2，
Figure FDA0002532632260000028
Figure FDA0002532632260000029
Therefore, the temperature of the molten steel is controlled,
Figure FDA0002532632260000031
Figure FDA0002532632260000032
Figure FDA0002532632260000033
the two equations of equation (6) and equation (10) are equal, and the yield condition of the pressed rod piece can be obtained as follows,
Figure FDA0002532632260000034
that is to say that the first and second electrodes,
Figure FDA0002532632260000035
the axial pulled rod element elastically supported in the step B) is specifically that a pressing member with the spring constant of c is arranged on the pulled rod element, the pulled rod element can freely rotate at the pressing member, and the theoretical formula of the yield condition of the pulled rod element is deduced in a specific process,
let the deflection of the pull rod at the supporting member be v, the generated spring force be cv, establish a balance equation, when x is less than or equal to l1Then, the formula (13) is obtained,
Figure FDA0002532632260000036
order to
Figure FDA0002532632260000037
The formula (14) is obtained,
Figure FDA0002532632260000038
the deflection line of the pulled rod member is shown in formula (15),
Figure FDA0002532632260000039
in the formula, constant A1And B1The boundary conditions y (0) of the pulled member can be equal to 0 and y (l)1) The average of the values obtained for v, i.e.,
Figure FDA0002532632260000041
Figure FDA0002532632260000042
therefore, it is
Figure FDA0002532632260000043
Figure FDA0002532632260000044
Figure FDA0002532632260000045
When l is1When x is less than or equal to l,
Figure FDA0002532632260000046
or,
Figure FDA0002532632260000047
the formula (19) is obtained,
Figure FDA0002532632260000048
using the boundary conditions y (l) 0 and y (l) of the pulled member1) Given as v, a constant a in formula (19)2And B2，
Figure FDA0002532632260000049
Figure FDA00025326322600000410
Therefore, the temperature of the molten steel is controlled,
Figure FDA0002532632260000051
Figure FDA0002532632260000052
Figure FDA0002532632260000053
the equation (18) is equal to the equation (22), and the yield condition of the pulled rod element can be obtained as follows,
Figure FDA0002532632260000054
when P is equal to P1When the temperature of the water is higher than the set temperature,
Figure FDA0002532632260000055
the concrete process of deriving the yield equation and the simplified calculation formula under the condition of one-pull-one-press in the step C) is that,
obtained from the formula (12) and the formula (24),
Figure FDA0002532632260000056
wherein,
Figure FDA0002532632260000057
Figure FDA0002532632260000058
the maximum value of P is such that,
Figure FDA0002532632260000059
solving the P value by adopting a numerical iterative algorithm, and obtaining the P value according to an Euler critical force formula Pcr＝π2EI/(μl)2Performing reverse-deducing to obtain a calculated length coefficient mu;
for different P1The ratio/P and different inequality degrees correct the calculated length coefficient mu by the correction value,
Figure FDA0002532632260000061
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