CN112446069A - Pre-internal force of structural member and calculation method thereof - Google Patents
Pre-internal force of structural member and calculation method thereof Download PDFInfo
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Abstract
The invention discloses a pre-internal force of a structural member and a calculation method thereof, which comprises the steps of removing all or part of constraint of at least one node of the structural member so as to adjust the connection state of the at least one node of the structural member to a first connection state and applying preloading on an engineering structure with the structural member; calculating a first internal force of the structural member in the first connection state based on the preload; adding not less than all or part of released constraints to at least one node of the structural member, adjusting the node from the first connection state to the second connection state, and unloading the preloading of the engineering structure with the structural member; calculating a second internal force of the structural member in the second connected state based on the removal preload; and superposing the internal force to obtain a first target internal force. By adopting the method of the invention, the internal force on the homogenized structural member can be applied and removed by utilizing the preloading, so that the stress performance and the economical efficiency of the structural member in an engineering structure are improved, and the condition that the misjudgment of the structure is infeasible is avoided.
Description
Technical Field
The invention relates to the technical field of structural engineering, in particular to a pre-internal force of a structural member and a calculation method thereof.
Background
The engineering structure consists of a plurality of structural members, wherein the engineering structure also comprises a structure consisting of one structural member. Under the action of load, the forced deformation performance of the structural member is mainly determined by the forced effect of the structural member. The stress in the stress effect is mainly internal force, such as axial force, bending moment, shearing force, torque and the like. It is known that the acting and reacting forces between the engineering structure and the supporting body outside the engineering structure are the load of the engineering structure on the supporting body and the reacting force of the supporting body on the engineering structure. If the support body of the engineering structure is regarded as a part of the engineering structure, the support body of the support body to the engineering structure becomes a node of the engineering structure, and the counter force of the support body to the engineering structure becomes the internal force of the engineering structure. Thus, in a broad concept, the node comprises a seat and the internal force comprises a counter force. The invention adopts the concept that the node also comprises a support and the internal force also comprises a counter force.
Currently, in order to ensure the performance of structural members during construction (manufacturing) and use, the stress condition of structural members is usually analyzed before construction (manufacturing). At present, a common stress analysis method for a structural member usually calculates the stress of the structural member by assuming that the connection state of a node connected with the structural member is a stiffness state such as a fixed support or rigid connection, a hinged support or hinge connection, or an incomplete hinge support or incomplete hinge connection generated at one time. That is, the support or node state of the member structure is a single state generated at a time, and the entire load is received in the single state.
However, in actual design and construction (manufacturing), it is found that the reaction force generated under the action of load, i.e. the force transmitted by the structural member to the support, is often large in amplitude and uneven in distribution, the main internal force of the structure itself, such as bending moment, is also large in amplitude and uneven in distribution, which is likely to cause low engineering performance, the structural member and its support are low in standardization, and the economy is poor.
Disclosure of Invention
The embodiment of the invention discloses a pre-internal force of a structural member and a calculation method thereof, which can effectively homogenize the internal force of the structural member, reduce the amplitude of the internal force applied to the structural member, and are beneficial to saving the construction cost and improving the economy.
In order to solve the technical problem, the invention provides a pre-internal force of a structural member and a calculation method thereof, wherein the method comprises the following steps:
releasing all or part of the constraint of at least one node of a structural member to adjust the connection state of the at least one node of the structural member to a first connection state, applying a preload on an engineering structure having the structural member;
calculating a first internal force of the structural member in the first connection state based on the preload;
adding the total or partial constraint which is not less than the release at the at least one node of the structural member, adjusting the constraint from the first connection state to a second connection state, and unloading the preload applied to the engineering structure with the structural member;
calculating a second internal force of the structural member in the second connected state based on removing the preload;
and superposing the first internal force and the second internal force to obtain a first target internal force.
As an alternative, in the embodiment of the present invention, the structural member is a bending member, an axial tension member, a shear member, a torsion member or a multi-internal force member in the engineering structure.
As an alternative implementation, in an embodiment of the present invention, the structural member includes a middle support and two end supports, and the at least one node is the middle support.
As an optional implementation manner, in an embodiment of the present invention, the method further includes:
calculating the load borne by the engineering structure with the structural members in an initial connection state;
applying the load on the engineered structure having the structural members while the structural members are in the second connection state according to the calculated load;
calculating a third internal force of the structural member based on the load;
and superposing the third internal force and the first target internal force to obtain a second target internal force.
As an alternative implementation manner, in the embodiment of the present invention, the load is a distributed load and/or a concentrated load, and the preload is a load and an action with an effect direction consistent with the load effect direction, including any one or a combination of any more of a distributed load, a concentrated load, a hanging load, a pressure force, a tensile force, a tension force, a compression force, a tension force, a support displacement and a temperature action.
As an alternative implementation, in the embodiment of the present invention, the load is q, the preload is p, and p ═ μ q, where μ is a coefficient, and μ ≦ 1.
As an alternative implementation, in an embodiment of the invention, in the second connection state, the connection stiffness of the at least one node of the structural member is greater than the connection stiffness of the at least one node of the structural member in the first connection state.
As an alternative implementation, in an embodiment of the present invention, before the releasing all or part of the constraint of the at least one node of the structural member to adjust the connection state of the at least one node of the structural member to the first connection state and applying the preload on the engineering structure having the structural member, the method further includes:
adjusting the connection state of the at least one node of the structural member to form the initial connection state.
As an optional implementation manner, in an embodiment of the present invention, the releasing all or part of the constraint of at least one node of the structural member to adjust the connection state of the at least one node of the structural member to the first connection state and applying a preload on the engineering structure having the structural member specifically includes:
calculating the constraint of the at least one node of the structural member in the second connected state;
releasing the at least one node of the structural member from the full or partial restraint to which it is subjected, such that the connection state of the at least one node of the structural member is adjusted to the first connection state;
calculating the preload and taking the value of the preload;
applying the preload on the engineered structure having the structural members.
As an optional implementation manner, in an embodiment of the present invention, the adding, at least the released all or part of the constraint at the at least one node of the structural member to adjust the constraint from the first connection state to a second connection state, and removing the preload applied to the engineering structure with the structural member specifically includes:
adding the full or partial constraint that is not less than released at the at least one node of the structural member to cause the at least one node of the structural member to adjust from the first connection state to the second connection state;
applying a reverse preload on the engineered structure having the structural members to unload the preload from the engineered structure having the structural members;
the magnitude of the reverse preload is equal to the magnitude of the preload, and the direction of the reverse preload is opposite to the direction of the preload;
the structural member is constrained in the second connection state by no less than its constraint in the initial connection state.
Compared with the prior art, the invention has the beneficial effects that:
the embodiment of the invention provides a pre-internal force of a structural member and a calculation method thereof, which are characterized in that a constraint formed by at least one node of the structural member is generated in stages, so that the node forms two different connection states in different stages, wherein the different connection states comprise a first connection state and a second connection state, firstly, when the structural member is in the first connection state, preloading is applied to an engineering structure with the structural member, and the internal force of the structural member is calculated, then when the structural member is in the second connection state, reverse preloading which is equal to the preloading in magnitude and opposite to the preloading in direction is applied to the engineering structure with the structural member, so that the preloading applied to the engineering structure is removed, and the internal force of the structural member is calculated; and then all the internal forces are superposed to obtain the target internal force. By adopting the method, the preloading application and unloading can be utilized to homogenize the structural member and the internal force, such as bending moment, counter force and shearing force, so that the amplitude and amplitude difference of the internal force can be effectively reduced, the stress performance and the economical efficiency of the structural member in the engineering structure are improved, the condition that the misjudgment of the engineering structure is not feasible is avoided, and a direction is provided for an engineering structure scheme.
Drawings
In order to more clearly illustrate the technical solutions in the embodiments of the present invention, the drawings needed to be used in the embodiments will be briefly described below, and it is obvious that the drawings in the following description are only some embodiments of the present invention, and it is obvious for those skilled in the art that other drawings can be obtained according to these drawings without creative efforts.
FIG. 1 is a flow chart of a method for calculating the pre-internal force of a structural member according to an embodiment of the present invention;
FIG. 2 is a detailed flowchart of step 101 disclosed in the present invention;
FIG. 3 is a flowchart illustrating the step 103 disclosed in the present invention;
FIG. 4 is a flowchart detailing step 106 disclosed herein;
FIG. 5 is an internal force diagram of a conventional two-span continuous beam load bearing application;
FIG. 6 is an internal force diagram under preload with the center support of a structural member unconnected in accordance with the first embodiment of the present invention;
FIG. 7 is an internal force diagram for unloading a structural member in an incomplete articulation of a mid-support of the structural member in accordance with the first embodiment of the present invention;
FIG. 8 is a pre-internal force diagram of FIGS. 6 and 7 after superposition of internal forces;
FIG. 9 is an internal force diagram of the load applied to a structural member in an incomplete articulation of the central support of the structural member in accordance with the first embodiment of the present invention;
FIG. 10 is an internal force diagram under pretension of a structural member of case two of the present invention in a state where the middle support is not connected;
fig. 11 is an internal force diagram under tension of a central support of a structural member in a second embodiment of the invention in an incompletely hinged condition.
Detailed Description
The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
In the present invention, the terms "upper", "lower", "left", "right", "front", "rear", "top", "bottom", "inner", "outer", "center", "vertical", "horizontal", "lateral", "longitudinal", and the like indicate an orientation or positional relationship based on the orientation or positional relationship shown in the drawings. These terms are used primarily to better describe the invention and its embodiments and are not intended to limit the indicated devices, elements or components to a particular orientation or to be constructed and operated in a particular orientation.
Moreover, some of the above terms may be used to indicate other meanings besides the orientation or positional relationship, for example, the term "on" may also be used to indicate some kind of attachment or connection relationship in some cases. The specific meanings of these terms in the present invention can be understood by those skilled in the art as appropriate.
Furthermore, the terms "mounted," "disposed," "provided," "connected," and "connected" are to be construed broadly. For example, it may be a fixed connection, a removable connection, or a unitary construction; can be a mechanical connection, or an electrical connection; may be directly connected, or indirectly connected through intervening media, or may be in internal communication between two devices, elements or components. The specific meanings of the above terms in the present invention can be understood by those of ordinary skill in the art according to specific situations.
Furthermore, the terms "first," "second," and the like, are used primarily to distinguish one device, element, or component from another (the specific nature and configuration may be the same or different), and are not used to indicate or imply the relative importance or number of the indicated devices, elements, or components. "plurality" means two or more unless otherwise specified.
The following detailed description is made with reference to the accompanying drawings.
Referring to fig. 1, fig. 1 is a flow chart illustrating a pre-internal force of a structural member and a calculation method thereof according to an embodiment of the present invention. As shown in fig. 1, a method for calculating a pre-internal force of a structural member may include:
101. releasing all or part of the constraint of the at least one node of the structural member to adjust the connection state of the at least one node of the structural member to the first connection state, and applying a preload on the engineered structure having the structural member.
In this embodiment, the structural member may be a flexural member, an axial force tension member, a shear member, a torsion member, or a multi-internal force member in an engineered structure. Specifically, the bending member refers to a structural member which mainly bears bending moment and neglects axial force on the cross section, and the shearing member refers to a structural member which mainly bears shearing force and neglects axial force on the cross section. The bending member mainly refers to a structural beam, and the shearing member mainly refers to a member bearing shearing force, such as a joint connecting plate, a bracket, a buttress, a retaining wall, a shear wall and the like. Wherein, the shear wall structure comprises shear wall and floor. Under the action of horizontal force, the floor slab can be used as a horizontal beam (a bending component), the shear wall can be used as a support of the horizontal beam, and the counter force of the horizontal support of the horizontal beam is the horizontal shear force transmitted to the shear wall by the horizontal beam. Similarly, in the retaining wall structure of the buttress columns, the retaining wall can be used as a horizontal beam (a bending member) for bearing horizontal force, and the buttress columns can be used as a support of the retaining wall, wherein the horizontal counter force of the retaining wall is the horizontal shearing force transmitted from the retaining wall to the buttress columns. Therefore, the shear force borne by the shear wall in the shear wall structure or the shear force borne by the retaining wall in the buttress retaining wall structure can also be calculated according to a horizontal beam model, namely, simplified into a horizontal beam calculation model, wherein the horizontal beam can comprise a single-span beam or a multi-span continuous beam. The multi-span continuous beam comprises an equal-span continuous beam or an unequal-span continuous beam.
Axial tension members are members in which a resultant force acts on a centroid of a cross section (i.e., the resultant force coincides with an axis), and mainly include tension members and compression members. The tension members mainly include an axial tension member and an eccentric tension member, wherein the axial tension member mainly bears tension, and the bending moment, the shearing force and the torque are relatively minor and even negligible. Such as the lower chord of a roof truss, a tension web member or the wall of a round pool, etc. An eccentric tension member is a member in which the tensile force is deviated from the centroid of the section or the tensile force and the bending moment are existed on the section, and the shearing force and the torque are relatively minor or even negligible. Such as the lower chord of a roof truss bearing internode loads, the upper chord of a cantilever truss or the wall of a rectangular pool, etc. Common compression members are those in which bending moments, shear forces and torques are relatively minor and negligible. Mainly comprises eccentric compression components (also called bending components), wherein the eccentric compression components refer to components which are subjected to compression and bending when the action point of the pressure is deviated from the axis of the components, such as roof truss upper chords, frame structure columns, brick walls, brick piles and the like which are subjected to internode load.
The torsion member is a member which mainly bears torque on the cross section, and the bending moment, the axial force and the shearing force are relatively minor and even negligible, such as a rain tent beam and the like.
The multi-internal-force component refers to a structural component which mainly bears two or more than two of four internal forces of bending moment, shearing force, axial force and torque on a cross section, such as a component of a frame structure, particularly a frame column. .
Further, the structural member is a two-span continuous beam and comprises a middle support and two end supports, and the at least one support is the middle support of the two-span continuous beam. It is understood that the member may also be a continuous beam with three or more spans, including two end supports and a plurality of middle supports disposed between the two end supports. That is, as an alternative embodiment, the structural member may be a multi-span continuous beam, including two end supports and one or more middle supports, wherein the middle support is disposed between the two end supports. Specifically, if the rigidity of each span beam of the multi-span continuous beam is equal, and the rigidity of the plurality of middle supports are equal, at least one node is any one support in the plurality of middle supports; if the rigidity of each span beam of the multi-span continuous beam is not equal and the rigidity of the middle supports are not equal, namely the internal force of at least one span beam of the multi-span continuous beam is greater than the internal force of other span beams of the multi-span continuous beam, and the rigidity of the two supports connected to the span beam is not equal, at least one node is the support with higher rigidity in the middle support of one span beam with higher rigidity in the multi-span continuous beam.
As another alternative, the structural member may be a bridge, and the structural member may include two end mounts. Specifically, if the rigidity of the supports at the two ends of the structural member is equal, at least one node is any one of the supports at the two ends; if the rigidity of the supports at the two ends of the structural member is not equal to each other, at least one node is the support with higher rigidity in the supports at the two ends.
It will be appreciated that the first connected state includes, but is not limited to, any of an unconnected state, a hinged state, or a semi-fixed state.
In this embodiment, in order to analyze the stress of the structural member, the connection state of at least one node of the structural member should be adjusted to an initial connection state before being adjusted to the first connection state, where the initial connection state is a connection state when the stiffness state of the structural member and the support is generated once.
Specifically, the initial connection state is a state different from the first connection state, and in the initial connection state, the connection rigidity of at least one node of the structural member is larger than the connection rigidity of at least one node of the structural member in the first connection state.
In this embodiment, as shown in fig. 2, the step 101 may specifically include the following steps:
1011. a constraint of at least one node of the structural member in an initial connected state is calculated.
In this step, at least one node of the structural member is mainly subjected to constraint calculation when the rigidity is high, and then the structural member is in a statically indeterminate structural state in the initial connection state.
1012. And releasing all or part of the constraint on at least one node of the structural member so as to adjust the connection state of at least one node of the structural member to the first connection state.
In this embodiment, specifically, when the structural member is adjusted to the first connection state, the constraint of the structural member may be completely released, so that at least one node of the structural member is in the first connection state with weak rigidity due to the constraint being completely released, and in the first connection state, the structural member may be in a statically determined structural state.
1013. Calculating the preload and taking the value of the preload.
1014. A preload is applied to the engineered structure having the structural member.
102. Based on the preload, a first internal force of the structural member in the first connection state is calculated.
103. Adding not less than released all or part of constraint on at least one node of the structural member, adjusting the constraint from the first connection state to the second connection state, and unloading the preload applied to the engineering structure with the structural member.
It is understood that the number of constraints of the structural member in the second connection state is not less than the number of constraints of the structural member in the initial connection state, and the case of the present invention is exemplified by the structural member in the second connection state being equal to the number of constraints of the structural member in the initial connection state. When the number of constraints of the structural member in the second connection state is equal to the number of constraints thereof in the initial connection state, the second connection state is the initial connection state.
In this embodiment, as shown in fig. 3, the step 103 specifically includes the following steps:
1031. adding at least one node of the structural member with at least one released constraint to adjust the at least one node of the structural member from the first connection state to the second connection state.
As can be seen from the above step 101, the total or partial constraint is calculated for at least one node of the structural member in the initial connection state.
1032. Applying a reverse preload on the engineered structure having the structural member to unload the preload from the engineered structure having the structural member.
In the present embodiment, since the preload is applied to the engineering structure having the structural member when the at least one node of the structural member is in the first connection state, and the preload applied to the engineering structure having the structural member is removed when the at least one node of the structural member is adjusted from the first connection state to the second connection state, it is equivalent to applying a reverse preload, which is equal to the preload but opposite to the preload, to the engineering structure having the structural member in the second connection state. For example, if the preload is applied as a downward preload, then the preload can be removed by applying a pull equal in magnitude to the preload in an upward direction.
Specifically, with the load q and the preload p, since the preload p is removed, which is equivalent to a reverse preload p' having the same magnitude and the opposite direction to the preload p, applied to the flexural member, the load q and the preload p satisfy the following relationship:
p+p’+q＝q； (1)
p’＝-p； (2)
that is, p is μ q, where μ is a coefficient and μ ≦ 1.
Taking the preload as the pretension load as an example, in this way, the process of applying the pretension load and the process of removing the pretension load are equivalent to "pretension" and "release", and from pretension to release, the pretension load is completely zero in the process, but due to the different states of the two stages (the first connection state is different from the second connection state), the member is superposed and stored with a certain amount of bending moment, and the part of bending moment is called pretension type pretension bending moment.
It can be seen that in the present invention, the connection state of at least one node of the structural member is generated in stages to form two different connection states, and then the preload is applied in the first connection state to generate an internal force, which can generate a smaller preload internal force at a large magnitude of the conventional internal force and a larger preload internal force at a small magnitude of the conventional internal force. The removal of the preload applied in the first connection state in the second connection state, which is equivalent to the application of a load equal in magnitude and opposite in direction to the preload, may be referred to as "reverse preload", which generates an internal force in a direction completely opposite to the conventional internal force, so that all of the conventional internal force, regardless of magnitude, is uniformly absorbed, thereby reducing the internal force of the structural member.
Therefore, the internal force generated by applying the preload in the first connection state and the internal force generated by removing the preload in the second connection state are superposed, and the preload removal returns to zero, but the internal force cannot be completely offset due to different states and different size distributions of the internal force generated by the preload and the unload, and the residual internal force after partially offsetting in the superposition is pre-established before the second connection state of the structural member is formed and the load is applied, so that the internal force is called as the "pre-internal force".
That is, the key point of the present invention is that the connection rigidity of the node of the structural member which bears a large internal force is generated in different stages, and a measure of the internal force is applied according to the different connection rigidities. By utilizing the pre-internal force measure, the internal force distribution of the node of the structural member is retransferred and adjusted, so that part of the internal force of the node with concentrated internal force is transferred to the node or position with smaller internal force, and the internal force of the structural member is homogenized.
The measure of the internal force is a measure for unloading in the second stage (namely, the second connecting state is called as the state 2) by applying the preload in the first stage (namely, the first connecting state is called as the state 1) by utilizing the characteristic that the two connecting states of the structural member are different in the two stages.
The degree of the measure of the preload force is controlled by controlling the magnitude of the preload, i.e., controlling the preload at a certain ratio corresponding to the applied load, i.e., the ratio μ of the preload to the load is p/q, which is referred to as a "preload coefficient".
That is, with the solution of the invention, the actual internal forces of the structural members can be analyzed by analyzing the load versus preload to determine the internal forces of the members in the first connection state and the internal forces of the members in the second connection state.
104. Based on the removed preload, a second internal force of the structural member in the second connected state is calculated.
105. And superposing the first internal force and the second internal force to obtain a first target internal force.
106. And applying a load to the engineering structure having the structural members in the initial connection state of the structural members.
In this embodiment, as shown in fig. 4, the step 106 specifically includes the following steps:
1061. and calculating the load borne by the engineering structure with the structural member in the initial connection state.
In the present embodiment, the load refers to a load to be borne by an engineering structure having the structural members in an initial connection state. In the theory of structural engineering, the load may be a distributed load and/or a concentrated load. In particular, the load mainly comprises a constant load and a live load. The constant load comprises the self weight of the structure, a floor laminated layer, a floor surface layer and the like, and is determined by an engineering structure method. Live loads include loads of personnel, equipment, etc., as determined by the engineering structure's functions of use. That is, the constant load is generated by the engineering structure itself and the live load is generated by the user or the equipment used. Of course, under the influence of environmental factors, the structural member may also be subjected to dynamic loads such as wind loads, earthquake loads, and the like. In engineering theory, the specific values of these types of loads to which a structural member is subjected can be calculated according to the formula specified in the engineering structural specification.
Further, the preloading is a load and various actions whose effect direction is consistent with the load effect direction of the engineering structure with the structural member, and specifically includes any load and action consistent with and/or inconsistent with the load distribution of the engineering structure with the structural member, for example, may include any one or any combination of more of distributed load, concentrated load, hanging load, pressure, tension, compression, tension, support displacement and temperature action.
1062. According to the calculated load, a load is applied to the engineering structure having the structural members while the structural members are in the initial connection state.
107. From the load, a third internal force of the structural member is calculated.
108. And superposing the third internal force and the first target internal force to obtain a second target internal force.
It can be known that, after the measure of the internal force is taken, the internal force (second target internal force) of the structural member under the action of the load is the superposition of the internal force (first target internal force) and the traditional internal force (third internal force).
The difference between the bending moment, the shearing force and the reaction force of the structural member of the present invention and the bending moment, the shearing force and the reaction force of the conventional structural member will be described in detail below with reference to the examples and the drawings, and in this patent, the engineering structure having the structural member includes a structural member as an example, and at this time, the load borne by the structural member is the load borne by the engineering structure having the structural member:
case one
The uniformity of internal force was evaluated as "internal force uniformity index". The "internal force uniformity index" may be defined as the ratio of the minimum magnitude to the maximum magnitude of the internal force between the supports of the component, between sections of the component, or between two components being evaluated. The global range of values for the index is [0,1 ]. Where 1 represents perfect homogeneity and 0 represents complete heterogeneity. It is assumed that the index interval values are divided into 4 small intervals, each of which represents a uniformity of blur, as shown in table 1. The more uniform the index approaches 1, the more non-uniform the index approaches 0. Wherein [0.85,1) represents relatively uniform, [0.70,0.85) represents non-uniform, [0.30,0.70) represents very non-uniform, and [0,0.30) represents extreme non-uniformity.
TABLE 1 Uniform exponential table of internal force
| Uniformity of | Is totally heterogeneous | Extreme unevenness | Is very uneven | Unevenness of | Is relatively uniform | Is completely uniform |
| Index of refraction | 0 | (0,0.30) | [0.30,0.70) | [0.70,0.85) | [0.85,1) | 1 |
Referring to fig. 5, fig. 5 shows a graph of the bending moment and the reaction force of the structural members under load in the initial connection state. The traditional two-span continuous beam in fig. 5 comprises two end supports and a middle support arranged between the two end supports, the beam span is l, and the load born by the structural component can be vertical full-span uniform load.
Specifically, the structural member includes a first bridge portion and a second bridge portion, and a ratio of a span of the first bridge portion to a span of the second bridge portion is n ═ l2/l1The span of the structural member is the sum of the two spans of the two-span continuous beam, i.e., (n +1) l1。
As shown in fig. 5, according to the table of "maximum internal force coefficient under uniform load of unequal span beams" given by the first edition of manual for static calculation of building structure of 1975 of the china architecture industry publisher, 6 months, the internal force of a component under the load q can be calculated by looking up the internal force coefficient in the table:
bending moment M ═ K × ql1 2 (1)
Shear force Q is Kxql1 (2)
According to the force node balance principle, the shear force of the left and right cross sections of each support can be used for calculating each counter force as follows:
reaction force R ═ QRight side-QLeft side of (3)
Reaction force R ═ QRight side-QLeft side of)×ql1＝k×ql1 (4)
Therefore, the reaction force coefficient
And (5) calculating each reaction coefficient k of the conventional two-span continuous beam according to the formula (5). The bending moment coefficient, the shear coefficient and the reaction coefficient of the conventional two-span continuous beam are shown in table 2.
TABLE 2 moment coefficient, shear coefficient and reaction coefficient of the load q of the conventional two-span continuous beam
| n | MB | MAB | MBC | QA | QB left side | QB right side | QC | RA | RB | RC |
| 1.0 | -0.1250 | 0.0703 | 0.0703 | 0.3750 | -0.6250 | 0.6250 | -0.3750 | 0.3750 | 1.2500 | 0.3750 |
| 2.0 | -0.3750 | 0.0078 | 0.3301 | 0.1250 | -0.8750 | 1.1875 | -0.8125 | 0.1250 | 2.0625 | 0.8125 |
From the bending moment coefficient, the shear coefficient and the reaction force coefficient of the above table 2, the uniformity coefficients of the bending moment, the shear force and the reaction force can be calculated respectively, as shown in table 3.
TABLE 3 Uniform coefficient of bending moment, shearing force and counterforce of traditional two-span continuous beam loaded with load q
| n | MB | MAB | MBC | QA | QB left side | QB right side | QC | RA | RB | RC |
| 1.0 | 1.00 | 0.5624 | 0.5624 | 0.60 | 1.00 | 1.00 | 0.60 | 0.30 | 1.00 | 0.30 |
| 2.0 | 1.00 | 0.02 | 0.88 | 0.11 | 0.74 | 1.00 | 0.68 | 0.06 | 1.00 | 0.39 |
By combining tables 1 and 3, whether the stress condition of the structural member is uniform or not can be analyzed, and the analysis result is shown in table 4 below.
TABLE 4 homogeneity analysis of bending moment, shearing force and counterforce of traditional two-span continuous beam loaded with load q
| n | MB | MAB | MBC | QA | QB left side | QB right side | QC | RA | RB | RC |
| 1.0 | Uniformity | Is very uneven | Is very uneven | Is very uneven | Is very uneven | Uniformity | Is very uneven | Is very uneven | Uniformity | Is very uneven |
| 2.0 | Uniformity | Extreme unevenness | Is relatively uniform | Extreme unevenness | Unevenness of the flow of water | Uniformity | Is very uneven | Extreme unevenness | Uniformity | Is very uneven |
As can be seen from table 4, the bending moment, the shearing force and the reaction force of the conventional two-span continuous beam are very uneven, even very uneven.
Referring to fig. 6 to 9, the first connection state of the structural member is to release the constraint of the middle support of the two-span continuous beam, so that the two-span continuous beam is temporarily in the single-span simple beam state, the second connection state is in the incomplete hinge state, the applied preload is the load distributed in accordance with the load, the load is the load uniformly distributed across the full span, and the internal force is the bending moment, the shearing force and the counter force.
As shown in fig. 6, fig. 6 shows a graph of the reaction force and bending moment of the structural member under preload in the unconnected state of the center support. First, the coupled state of the middle mount of the structural member is adjusted to be in the uncoupled state, and a preload p is applied to the structural member with the direction of the preload p vertically downward. In this state, the reaction force and internal force calculation expressions are the same as the above expressions (1), (2), and (4), and the load q is simply changed to the preload p. The bending moment coefficient, shear coefficient and reaction force coefficient at this time were calculated as shown in table 5 below.
TABLE 5 moment coefficient, shear coefficient and reaction coefficient of the component bearing preload p at State 1
| n | MBp | MABp | MBCp | QAp | QB left p | QB right p | QCp | RAp | RBp | RCp |
| 1.0 | 0.5000 | 0.4375 | 0.4375 | 1.0000 | 0.0000 | 0.0000 | -1.0000 | 1.0000 | 0.0000 | 1.0000 |
| 2.0 | 1.0000 | 0.6250 | 1.0000 | 1.5000 | 0.5000 | 0.5000 | -1.5000 | 1.5000 | 0.0000 | 1.5000 |
In comparison with table 1, it can be seen from the bending moment coefficient, shear coefficient and reaction coefficient of table 5 that the structural member is in state 1 and is subjected to the internal force generated by the preload p. It is characterized in that:
compared with the traditional structural component, the maximum middle support of the structural component in the state 1 does not generate preloading counter force, and the smaller counter force of the supports at the two ends generates larger preloading counter force.
When the continuous beam spans in an equal span (n is 1), the transfer amount of the preloading counter force of the middle support of the two-span continuous beam to the supports at the two ends is equal; when the span is unequal (n is 2), the transfer amount of the preloading counter force of the middle support of the two-span continuous beam to the end support with small span is larger than that of the end support with large span, so that the homogenization is facilitated.
And thirdly, temporarily enabling the middle support B to be in an unconnected state, so that the middle support B becomes a midspan and generates a large midspan preloading positive bending moment, and enabling the negative bending moment peak value bending moment position of the traditional middle support B to generate a very small preloading bending moment and the positive bending moment peak value bending moment positions of the traditional two midspans to generate a large preloading bending moment.
And fourthly, generating a larger pre-load shearing force and peak value transfer at the position with small shearing force amplitude in the prior art.
FIG. 7 is a graph showing the reaction force and bending moment generated by the structural member after removal of the preload in the incompletely hinged condition of the mid-mount, as shown in FIG. 7. The unloading operation is performed by adjusting the central support of the structural member from the unconnected state to an incomplete articulation of the central support and removing the previously applied preload p. Compared with the first connection state, the operation is equivalent to that reverse preload p 'with the same magnitude and the opposite direction is applied in the second connection state, namely p' ═ p, the positive and negative distribution of the generated counter force is just opposite to that in the first connection state, and the positive and negative distribution of the generated bending moment is just opposite to that in the initial connection state, so that the operation has the effect of completely reducing the traditional internal force.
Since the preload p is applied in the first connection state and then adjusted to the second connection state, the preload p is completely zero in this process, but since the two phases differ, the structural members are superposed and store a certain amount of counter force, which is referred to as pre-counter force. The pre-reaction force happens to be mutually reduced with the negative reaction force at the two ends of the traditional component, so that the reaction force distribution of the structural component can be further reduced and homogenized.
Likewise, during the unloading process, the structural members may be allowed to stack up storing a certain amount of bending moment, referred to as pre-bending moment. The pre-bending moment is distributed linearly in a constant quantity, which happens to be mutually reduced with the negative bending moment at two ends of the traditional structural member, so that the bending moment distribution of the structural member can be further reduced and homogenized.
In state 2, the bending moment coefficient, the shear coefficient, and the reaction force coefficient of the structural member are shown in table 6. Similarly, the reaction force and internal force calculation formula of the structural member may be the same as the above formulas (1), (2) and (4), and only the load q may be changed to the reverse preload p'.
TABLE 6 moment coefficient, shear coefficient and reaction coefficient of reverse preload p' borne by the component in State 2
| n | MBp′ | MABp′ | MBCp′ | QAp′ | QB left p' | QB of right p' | QCp′ | RAp′ | RBp′ | RCp′ |
| 1.0 | 0.1250 | -0.0730 | -0.0730 | -0.3750 | 0.6250 | -0.6250 | 0.3750 | -0.3750 | -1.250 | -0.3750 |
| 2.0 | 0.3750 | -0.0078 | -0.3301 | -0.1250 | 0.8750 | -1.1875 | 0.8125 | -0.1250 | -2.0625 | -0.8125 |
Using the superposition principle of the structural theory, the effects of the superimposed preload p and the reverse preload p' applied to the two states, respectively, result in a first target reaction force (the force distribution is shown in fig. 8), as shown in table 7.
TABLE 7 target reaction force calculation Table
| n | RA | RB | RC |
| 1.0 | 0.3750q+0.625p | 1.2500q-1.25p | 0.3750q+0.625p |
| 2.0 | 0.1250q+1.375p | 2.0625q-2.0625p | 0.8125q+0.6875p |
Therefore, the difference between the reaction forces can be calculated as shown in table 8.
TABLE 8 Difference table for target reaction forces
| n | RB-RA | RB-RC |
| 1.0 | 0.875q-1.875p | 0.875q-1.875p |
| 2.0 | 1.9375q-3.4375p | 1.25q-2.75p |
It can be known that the loading effect of the ideal target reaction force is determined by calculating the difference between the target reaction forces, wherein the smaller the difference between the target reaction forces is, the better the loading effect of the ideal target reaction force is, and when the difference between the target reaction forces is zero, the better the loading effect of the ideal target reaction force is.
Let RB＝RAOr R isB＝RCSubstituting into Table 8, the ideal preload coefficient μ of the target reaction force loaded in stages can be obtained*As shown in table 9.
TABLE 9 ideal preload factor μ for target reaction force*
It will be appreciated that the use of the reaction coefficients of table 9 above to derive the magnitude of the preload applied in the first connection and the unload applied in the second connection is preferred, so that the moment, shear and reaction forces of the structural member are best balanced.
Therefore, by combining tables 2, 6 and 9 with equations (1), (2) and (3), the moment coefficient, the shear coefficient and the reaction force coefficient under the ideal preload coefficient of the reaction force can be calculated, as shown in table 10.
TABLE 10 Preload factor μ*Lower bending moment coefficient, shear coefficient and reaction coefficient
| n | MBpp′ | MABpp′ | MBCpp′ | QApp′ | QB left pp' | QB right pp' | QCpp′ | RApp′ | RBpp′ | RCpp′ |
| 1.0 | 0.2917 | 0.1701 | 0.1701 | 0.2917 | 0.2917 | -0.2917 | -0.2917 | 0.2917 | -0.5834 | 0.2917 |
| 2.0 | 0.6249 | 0.2805 | 0.3045 | 0.6249 | 0.6249 | -0.3125 | -0.3125 | 0.6249 | -0.9374 | 0.3125 |
As shown in fig. 9, fig. 9 shows a graph of the reaction force and bending moment of the structural member after unloading the preload in the incomplete hinge state of the mid-mount. Superimposing tables 2 and 10 results in the desired preload factor μ*Structural member bearing load after pre-internal force measures under staged preload and unload conditionsThe bending moment coefficient, shear coefficient and reaction force coefficient of q, i.e., p + p' + q ═ q, are shown in table 11.
TABLE 11 bending moment coefficient, shear coefficient and reaction coefficient under q action after ideal internal force measures
Comparing with table 1, from the bending moment coefficient, shear coefficient and reaction coefficient of table 11, it can be shown that the influence of pre-reaction, pre-bending moment and pre-shearing force on the traditional bending moment, shear and reaction is:
the counter force of the middle support is transferred to the supports at the two ends, so that the traditional maximum counter force of the middle support is reduced, and the traditional small counter force of the supports at the two ends is increased.
And the traditional large internal force bending moment of the structural member is transferred to the traditional small internal force bending moment, so that the traditional large internal force bending moment is reduced, and the traditional small internal force bending moment is increased.
And thirdly, the increase and decrease of the traditional shearing force are not obvious and are changed along with the ratio of the length span and the load distribution.
From the bending moment coefficient, the shear force coefficient and the reaction force coefficient of the upper table 11, the uniformity coefficients of the bending moment, the shear force and the reaction force after the ideal internal force measure can be respectively calculated, as shown in table 12.
TABLE 12 Uniform index of bending moment, shearing force and reaction force after ideal internal force measure
| n | MB | MAB | MBC | QA | QB left side | QB right side | QC | RA | RB | RC |
| 1.0 | 0.69 | 1.00 | 1.00 | 1.00 | 0.50 | 0.50 | 1.00 | 1.00 | 1.00 | 1.00 |
| 2.0 | 0.39 | 0.45 | 1.00 | 0.67 | 0.22 | 0.78 | 1.00 | 0.67 | 1.00 | 1.00 |
By combining the tables 1 and 12, relative analysis and evaluation can be made on the distribution uniformity of the bending moment, the shearing force and the reaction force of the two-span continuous beam after the ideal internal force pre-measure is adopted, and the analysis results are shown in the following table 13.
TABLE 13 homogeneity analysis of bending moment, shear force and counter force after ideal pre-internal force measure
| n | MB | MAB | MBC | QA | QB left side | QB right side | QC | RA | RB | RC |
| 1.0 | Is very uneven | Uniformity | Uniformity | Uniformity | Is very uneven | Is very uneven | Uniformity | Uniformity | Uniformity | Uniformity |
| 2.0 | Is very uneven | Is very uneven | Uniformity | Is very uneven | Extreme unevenness | Unevenness of the flow of water | Uniformity | Is very uneven | Uniformity | Uniformity |
Compared with the traditional method, the change of the bending moment, the shearing force and the uniformity of the counter force after the ideal internal force measure is shown as follows:
the counter force is improved from extremely uneven to uneven.
② the uniformity of the bending moment (main internal force) is unchanged.
③ the shearing force (secondary internal force) is changed from very uneven to even or to very uneven. But this deterioration in shear uniformity is not detrimental, as secondary internal forces, secondary to the shear reduction ratio.
By dividing table 11 by the ratio obtained in table 2, the conventional reduction ratios of bending moment, shear force and reaction force under the condition of applying preload unloading in stages and ideal target reaction force coefficient can be obtained, as shown in table 14.
TABLE 14 reduction ratio (%) -of bending moment, shearing force and counterforce amplitude after ideal internal force measure
| n | MB | MAB | MBC | QA | QB left side | QB right side | QC | RA | RB | RC |
| 1.0 | -33 | -93* | -93* | -7* | 47 | 47 | -7* | -78 | 47 | -78 |
| 2.0 | 34 | -3600 | -69* | -500 | 72 | 27 | -5* | -500 | 45 | -38 |
In table 14, a negative value indicates an increase ratio. The upper corner mark indicates the target peak after transfer, and the increase and decrease of the target peak are relative to the original peak. The preloading coefficients mu are all ideal preloading coefficients mu*Under the condition, the peak values of the sectional preloading and unloading, the bending moment, the shearing force and the counter force are changed relative to the traditional structural member as follows:
the reaction force is reduced and tends to be uniform, the reduction proportion reaches 45% -47%, specifically, the reaction force peak value of the middle support is reduced, and the reaction forces of the supports at the two ends are increased.
② the corresponding internal force is increased. Wherein, the bending moment (main internal force) of the middle support is partially transferred to the midspan, the position of the peak bending moment is transferred to the midspan, the new peak value is increased more than the original peak value, whether the new peak value is harmful or not is calculated, and the limitation is needed.
And thirdly, the peak shearing force (secondary internal force) of the right section of the middle support is partially transferred to the two end supports, the peak position of the middle support is possibly unchanged or transferred to the two end supports, and the peak value is possibly increased. If the position of the peak shear force is transferred from the middle support to the two end supports, and the new peak value is increased, the new peak value is almost negligible because the increase of the new peak value from the original peak value is very limited.
It will be appreciated that the above-described homogenisation effect is desirable if the variation in the bending moment to which the component is subjected under the above-described conditions is still within engineering tolerances. However, since the reaction force is reduced and equalized while the bending moment of the structural member is increased, it is necessary to reduce the equalized reaction force and to ensure that the bending moment of the structural member is increased and is within the engineering permissible range in order to ensure the feasibility of the structural member.
Therefore, the above-described ideal homogenization conditions for the reaction force can be appropriately reduced, for example: the peak value of the bending moment (main internal force) is not larger than the traditional peak value, and a relatively proper reaction force homogenization effect can be obtained. Then, on the condition that this is set, the preload coefficient μ is obtained.
The internal force coefficient of the table (table 8) of bending moment coefficient, shear coefficient and reaction coefficient under the action of the unloaded preload p and the internal force coefficient of the table (table 2) of bending moment coefficient, shear coefficient and reaction coefficient under the action of the traditional load q are superposed to obtain a bending moment (main internal force) calculation formula after the internal force pre-measure, as shown in table 15.
Table 15 bending moment calculation table after internal force measurement
When n is 1, after the internal force measures are taken, the new peak bending moment is equal to the traditional peak bending moment, namely:
|MAB|＝|MBi or I MBC|＝|MB| (6-1)
Then the process of the first step is carried out,
0.0703q+0.3675p＝0.125q (7-1)
from equation (7), the limit preload factor can be found:
when n is 2, after the internal force measures are taken, the new peak bending moment is equal to the traditional peak bending moment, namely:
|MAB|＝|MBi or I MBC|＝|MB| (6-2)
Then the process of the first step is carried out,
0.3301q+0.6699p＝0.3750q (7-2)
from equation (7-2), the limiting preload factor can be found:
the table (table 8) of the bending moment coefficient, the shear coefficient and the reaction coefficient under the condition that the peak bending moment is not increased is obtained by multiplying the preload coefficient by the bending moment coefficient, the shear coefficient and the reaction coefficient under the unload preload p, as shown in table 16.
TABLE 16 moment coefficient, shear coefficient and reaction coefficient under the condition of not increasing the moment
| n | MBpp′ | MABpp′ | MBCpp′ | QApp′ | QB left pp' | QB right pp' | QCpp′ | RApp′ | RBpp′ | RCpp′ |
| 1.0 | 0.0930 | 0.0547 | 0.0547 | 0.0930 | 0.0930 | -0.0930 | -0.0930 | 0.0930 | -0.1861 | 0.0930 |
| 2.0 | 0.0921 | 0.0414 | 0.0449 | 0.0921 | 0.0921 | -0.0461 | -0.0461 | 0.0921 | -0.1382 | 0.0461 |
Table 16 is superimposed on conventional table 2 to obtain a target reaction force coefficient and a target internal force coefficient under the total load q after the preliminary internal force measure under the condition that the bending moment peak is not increased, that is, p + p' + q ═ q, as shown in table 17.
TABLE 17 moment coefficient, shear coefficient and reaction coefficient after internal force measurement without increasing moment
The table 17 is divided by the table 2, and the complementary percentage is taken as the reduction ratio of the bending moment, the shearing force and the counter force after the internal force measures under the condition that the bending moment is not increased, as shown in the table 18.
After bending moment, shearing force and counter force reduction ratio (%) -of pre-internal force measure under the condition that bending moment of the meter 18 is not increased
| n | MB | MAB | MBC | QA | QB left side | QB right side | QC | RA | RB | RC |
| 1.0 | 97 | 0 | 0 | -25 | 15 | 15 | -25 | -25 | 15 | -25 |
| 2.0 | 25 | -530 | 0 | -74 | 11 | 4 | -6 | -74 | 7 | -6 |
As can be seen from table 18, even under the condition that the peak value of the bending moment is not increased, the reaction force and the shearing force are reduced by 15% to 25%. Although the peak reaction force is not at the preload coefficient μ ═ μ*The degree of attenuation and homogenization is good, but the attenuation and homogenization degree is 7% -15% lower than that of the traditional method, and the purposes of overall attenuation and overall homogenization of bending moment, shearing force and counter force are achieved.
It will be appreciated that M is mentioned aboveBBending moment of the support B, MABBending moment of the first beam-spanning part AB, MBCIs the bending moment of the second bridge portion BC. QAAs a shearing force of the support A, QB left sideShear force, Q, of the left cross-section of the support BB right sideThe shearing force of the right section of the support B. QCAs shear force of the support C, RAIs a counter-force of the support A, RBCounter-force of the support B, RCIs the counter force of the support C.
Case two
As shown in fig. 10 and 11, the preload in accordance with the direction and distribution of the load in the first case is replaced with the concentrated preload tension having the same direction and different distribution (as shown in fig. 6), the unload is replaced with the release tension (as shown in fig. 7), and other conditions are not described as an example. The main difference between the second case and the first case is that the preload is changed to the centralized preload, wherein the preload equivalent to the counter force of the centralized pretensioning type internal force method is called "equivalent preload", and the others are not essentially different, and therefore, the detailed explanation thereof is not necessary to be described herein.
It can be known that the centralized pretensioning type internal force method of the present embodiment can obtain the equivalent reaction force and internal force reduction and equalization effect to the reaction force of the preloading method of the first embodiment.
According to the pre-internal force of the structural member and the calculation method thereof provided by the embodiment of the invention, before the load of the engineering structure with the structural member is applied, the preloading application and the unloading are performed on the engineering structure with the structural member in a state by stages, and then the load is applied, so that the preload application and the unloading are used for homogenizing the stress of the structural member, the bending moment, the shearing force and the counter force of the middle support and the two end supports of the structural member are effectively reduced and homogenized, the amplitude difference of the bending moment is effectively reduced, the stress performance and the economical efficiency of the structural member in the structure are further improved, and the condition that the misjudgment of the engineering structure is not feasible is avoided.
The method for fixing the preloading and loading of the engineering structure with the structural member in a segmented manner can effectively reduce the difference of the bending moment amplitudes between the two ends of the structural member and the span, thereby being beneficial to improving the stress performance of the structural member in the engineering structure and further being beneficial to improving the economy of the structural member in the engineering structure.
The detailed description is given above to the internal force of a structural member and the calculation method thereof disclosed in the embodiments of the present invention, and the principle and the implementation of the present invention are explained in this document by applying specific examples, and the description of the above embodiments is only used to help understanding the method and the core idea of the present invention; meanwhile, for a person skilled in the art, according to the idea of the present invention, there may be variations in the specific embodiments and the application scope, and in summary, the content of the present specification should not be construed as a limitation to the present invention.
Claims (10)
1. A method of calculating the pre-internal force of a structural member, the method comprising:
releasing all or part of the constraint of at least one node of a structural member to adjust the connection state of the at least one node of the structural member to a first connection state, applying a preload on an engineering structure having the structural member;
calculating a first internal force of the structural member in the first connection state based on the preload;
adding the total or partial constraint which is not less than the release at the at least one node of the structural member, adjusting the constraint from the first connection state to a second connection state, and unloading the preload applied to the engineering structure with the structural member;
calculating a second internal force of the structural member in the second connected state based on the removed preload;
and superposing the first internal force and the second internal force to obtain a first target internal force.
2. The method of claim 1, wherein the structural member is a flexural member, an axial tension member, a shear member, a torsion member, or a multi-internal force member in the engineered structure.
3. The method of claim 2, wherein the structural member includes a central support and two end supports, and the at least one node is the central support.
4. The method of any of claims 1 to 3, further comprising:
calculating the load borne by the engineering structure with the structural members in an initial connection state;
applying the load on the engineering structure with the structural members while the structural members are in the initial connection state, according to the calculated load;
calculating a third internal force of the structural member based on the load;
and superposing the third internal force and the first target internal force to obtain a second target internal force.
5. The method according to claim 4, wherein the load is a distributed load and/or a concentrated load, and the preload is a load and effect having an effect direction consistent with the load effect direction, and comprises any one or a combination of any more of a distributed load, a concentrated load, a hanging load, a pressure force, a tension force, a counter pressure, a counter tension, a support displacement and a temperature effect.
6. The method of claim 5 wherein the load is q, the preload is p, and p ═ μ q, where μ is a coefficient and μ ≦ 1.
7. A method according to any one of claims 1 to 3, wherein in the second connection state the connection stiffness of the at least one node of the structural member is greater than the connection stiffness of the at least one node of the structural member in the first connection state.
8. The method of claim 4, wherein prior to the releasing of all or a portion of the restraint of the at least one node of the structural member to adjust the connection state of the at least one node of the structural member to the first connection state, the method further comprises:
adjusting the connection state of the at least one node of the structural member to form the initial connection state.
9. The method of claim 8, wherein the releasing all or part of the constraint of the at least one node of the structural member to adjust the connection state of the at least one node of the structural member to the first connection state applies a preload on the engineered structure having the structural member, specifically comprising:
calculating the constraint of the at least one node of the structural member in the initial connected state;
releasing the at least one node of the structural member from the full or partial restraint to which it is subjected, such that the connection state of the at least one node of the structural member is adjusted to the first connection state;
calculating the preload and taking the value of the preload;
applying the preload on the engineered structure having the structural members.
10. The method according to claim 8, wherein said adding not less than said released all or part of said constraint at said at least one node of said structural member to adjust from said first connection state to a second connection state and removing said preload from said engineered structure having said structural member comprises:
adding the full or partial constraint that is not less than released at the at least one node of the structural member to cause the at least one node of the member to adjust from the first connection state to the second connection state;
applying a reverse preload on the engineered structure having the structural members to unload the preload from the engineered structure having the structural members;
the magnitude of the reverse preload is equal to the magnitude of the preload, and the direction of the reverse preload is opposite to the direction of the preload;
the structural member is constrained in the second connection state by no less than its constraint in the initial connection state.
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| 唐柏鉴;张福星;马;: "强迫位移式门式刚架的强迫位移量确定方法", 江苏科技大学学报(自然科学版), no. 06, pages 22 - 25 * |
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