Theoretical calculation force of beam on elastic foundation
Publish: 2021-04-23 04:15:13
1. PKPM has no special calculation of strip foundation under column, but for frame structure, if strip foundation is used under column, it can be calculated by foundation beam, that is, it can bear foundation reaction, and elastic foundation beam is used for calculation. The steps are as follows: 1. Input geological data; 2. Input parameters including basic parameters (mainly characteristic value of foundation bearing capacity) and ground beam raft parameters (mainly reaction coefficient of foundation bed, relevant material parameters of ground beam, reinforcement adjustment parameters and beam rib orientation); 3. Input grid (axis line extension command is modified to form cantilever foundation beam axis); 4. Modify load parameters, read load; 5 Define the foundation beam (the beam rib height and width must be defined, and the flange width of the ground beam can be given at will, but should be greater than the beam rib width, because the program will give the opportunity to adjust the flange width when exiting the interactive step) and arrange the foundation beam 6. Exit the interactive step: pay attention to the first modification of the flange width of the ground beam, and the second check whether the data file for elastic foundation beam calculation is generated (that is, the relevant load value appears, the width of the flange is changed) 7. Settlement calculation of elastic foundation beam / foundation: 7-01: check whether geological data are correct 7-02: set calculation parameters (Note: fully flexible assumption should be adopted, groundwater height needs to be modified) 7-03: enter additional reaction diagram, select settlement calculation menu to calculate settlement, After that, you can check the relevant required data 8. Elastic foundation beam / structure calculation 8-01: select whether to carry out cross bottom area reuse calculation, modify foundation beam parameters (Note: internal force used in ground beam calculation), select the model used in calculation (upper foundation stiffness generated by SATWE and Tat can be used for calculation 8-02: check internal force figure 9 Elastic foundation beam / refer to results (normal operation) 10. Construction drawing of elastic foundation beam (normal operation)
2. In the process of deformation of foundation beam under the action of external load, the bottom of beam is always close to the surface of foundation, that is, the settlement or uplift of foundation is equal to the deflection of beam everywhere! Because the friction between the beam and the foundation has little influence on the calculation results, it can be ignored, so the foundation reaction is perpendicular to the contact surface everywhere. 3. The height span ratio of foundation beam is small, and the assumption of composite plane section can be directly applied to the calculation theory of deformation and internal force of beam in mechanics of materials.
3. Share to:
4. No matter what kind of foundation model assumption is used in the calculation of beams on elastic foundation, the following two basic solving conditions should be satisfied:
1) the deformation coordination condition between the foundation and the foundation beam, that is, the foundation and the foundation beam must be kept in contact before and after the calculation, and there must be no separation phenomenon
2) satisfy the static equilibrium condition, that is, the foundation beam must be in the static equilibrium state under the combined action of external load and base reaction. The analysis of beam on foundation is a classic subject, which is still in development e to the emergence of new foundation models, analysis methods and calculation methods. The
theoretical analysis and calculation method of elastic foundation beam is a very important problem in construction engineering, which needs further
improvement 1) based on Winkler model
1) initial parameter method... 1. An initial section of a beam is selected, and the four physical
quantities of the section, i.e. deflection w, rotation angle 0, bending moment M and shear force Q, are called initial parameters. By using the differential relationship between the deflection equation of the foundation
beam and the four physical quantities, the four
parameters in the deflection equation are expressed by the above four physical quantities, which is called initial parameter method. This method can make
integral constant have clear physical meaning, and simplify
a
calculation according to the physical meaning of parameters
2) finite difference method. The elastic foundation beam is divided into sections, the reaction force
P of each section is evenly distributed, and the resultant force R is at the midpoint of each section. The differential equations and boundary conditions are approximately replaced by the difference expression, and the discrete deflection is used to express the external force of each section. However, the deflection value of each discrete point can be obtained by solving the linear equations combined with the boundary conditions
there are resial value method, variable coefficient of subgrade
method, modified stiffness matrix method and so on
2.2 based on semi infinite body model
1) Guo's method l3j
the foundation reaction P (z) is approximately expressed as a power series of finite terms, that is:
P (x) = Ao + alz + a2x +... + n (2)
where + 1 undetermined coefficient n is the basic unknown quantity
by substituting equation (2) into equation (1), the polynomial expression of deflection at any point of the beam is obtained; Then (2) is substituted into the integral of the equilibrium differential equation of the foundation beam to obtain the polynomial expression of the settlement function of any
point of the foundation; Then, according to the compatibility condition that the deflection of the beam and the settlement of the foundation are equal, let the same power of Z in the two formulas take the same coefficient,
obtain a group of equations about inkstone, and then, from the static equilibrium condition of the beam, that is, the vertical force equilibrium and the moment equilibrium of a point on the beam, we can obtain two equations containing basic
unknowns. By solving the undetermined coefficient, the foundation reaction function is determined, thus solving the
problem. When the number of series terms in Guo's method is more, the accuracy of the results is better
2) chain link method_ 4J
the continuous beam supported on the foundation is simplified to a finite number of equidistant continuous beams on elastic supports, so that the original infinitely multiple statically indeterminate structure is simplified to a finite number of statically indeterminate
structure; With cantilever beam as basic system, the vertical displacement of fixed end is w. The sum angle displacement
is unknown. Assuming that the foundation reaction is uniform in each segment, the displacement coordination condition of the contact surface is realized by the rigid chain bar hinged at the center of each segment, and the internal force of the i-th chain
bar represents the resultant force of the foundation reaction of the i-th segment
the reaction forces XL, X2, X3,..., in these bars constitute the basic unknowns for solving the problem, and the actual displacement and rotation angle at the free end of the beam are additional unknowns. According to the continuity of displacement between the rigid bar and the semi infinite foundation, an equation can be obtained:
one
two Wo ～ akoo + △ help = 0 (3)
z = 1
Where is the relative displacement at point K when only x = 1; tile =
+, is the deflection of cantilever beam at point K, and is the foundation settlement at point K; Is the distance from the fixed end of the beam to the K point; Δ is the relative displacement of the external load at point K, that is, the negative displacement of the cantilever beam at point K
combined with two static equilibrium conditions:
one
∑ Mo + ∑ P = 0, one ∑ Aixi + ∑ MP = 0 (4)
I = 1, I 1
there are + 2 equations to solve all the above unknowns
the chain link method is widely used, regardless of the nature of the foundation, load types and member cross-section deformation. The more the number of links, the more accurate the solution is, but the more the workload is
large. Generally, 6-10 links can meet the accuracy requirements of the project
3) Cai Siwei's method
the foundation reaction is determined by using the deformation coordination condition between foundation beam and foundation, that is, the condition that they should still contact each other after deformation
the foundation beams are divided into equal parts, and the length of each section is B. assuming that the foundation reaction on each section is evenly distributed
the foundation reaction under the foundation beam is distributed in a ladder shape. Make the degree of foundation reaction strength
of each section as PL, P2,..., P. Under the action of these reaction forces, the foundation settlement occurs at all points
let the settlement at the midpoint of each section be expressed as WL, W2,..., W. According to the deformation compatibility condition of foundation and foundation beam, these settlements should be equal to the deflection of corresponding points on the foundation beam
the basic equations of the beam are:
a
etc.: (+ 2 wi 1 2WI + wi 1) (5) a + 111 L)
where I is the center of the i-th segment; Mi is the bending moment of section I
according to the number of sections, the settlement on the right side of the equation is listed as the function of PL, P2,...,
P by formula (1), and the settlement on the left side of the equation is the function of external load and foundation reaction, which can be written directly. From equation (5), n-2 equations can be listed, plus two equilibrium formulas ∑ = 0 and ∑ M = 0, then n values can be determined
other solutions, such as finite element method, trigonometric series method, distributed base reaction method, etc.
refer to relevant literature
1) the deformation coordination condition between the foundation and the foundation beam, that is, the foundation and the foundation beam must be kept in contact before and after the calculation, and there must be no separation phenomenon
2) satisfy the static equilibrium condition, that is, the foundation beam must be in the static equilibrium state under the combined action of external load and base reaction. The analysis of beam on foundation is a classic subject, which is still in development e to the emergence of new foundation models, analysis methods and calculation methods. The
theoretical analysis and calculation method of elastic foundation beam is a very important problem in construction engineering, which needs further
improvement 1) based on Winkler model
1) initial parameter method... 1. An initial section of a beam is selected, and the four physical
quantities of the section, i.e. deflection w, rotation angle 0, bending moment M and shear force Q, are called initial parameters. By using the differential relationship between the deflection equation of the foundation
beam and the four physical quantities, the four
parameters in the deflection equation are expressed by the above four physical quantities, which is called initial parameter method. This method can make
integral constant have clear physical meaning, and simplify
a
calculation according to the physical meaning of parameters
2) finite difference method. The elastic foundation beam is divided into sections, the reaction force
P of each section is evenly distributed, and the resultant force R is at the midpoint of each section. The differential equations and boundary conditions are approximately replaced by the difference expression, and the discrete deflection is used to express the external force of each section. However, the deflection value of each discrete point can be obtained by solving the linear equations combined with the boundary conditions
there are resial value method, variable coefficient of subgrade
method, modified stiffness matrix method and so on
2.2 based on semi infinite body model
1) Guo's method l3j
the foundation reaction P (z) is approximately expressed as a power series of finite terms, that is:
P (x) = Ao + alz + a2x +... + n (2)
where + 1 undetermined coefficient n is the basic unknown quantity
by substituting equation (2) into equation (1), the polynomial expression of deflection at any point of the beam is obtained; Then (2) is substituted into the integral of the equilibrium differential equation of the foundation beam to obtain the polynomial expression of the settlement function of any
point of the foundation; Then, according to the compatibility condition that the deflection of the beam and the settlement of the foundation are equal, let the same power of Z in the two formulas take the same coefficient,
obtain a group of equations about inkstone, and then, from the static equilibrium condition of the beam, that is, the vertical force equilibrium and the moment equilibrium of a point on the beam, we can obtain two equations containing basic
unknowns. By solving the undetermined coefficient, the foundation reaction function is determined, thus solving the
problem. When the number of series terms in Guo's method is more, the accuracy of the results is better
2) chain link method_ 4J
the continuous beam supported on the foundation is simplified to a finite number of equidistant continuous beams on elastic supports, so that the original infinitely multiple statically indeterminate structure is simplified to a finite number of statically indeterminate
structure; With cantilever beam as basic system, the vertical displacement of fixed end is w. The sum angle displacement
is unknown. Assuming that the foundation reaction is uniform in each segment, the displacement coordination condition of the contact surface is realized by the rigid chain bar hinged at the center of each segment, and the internal force of the i-th chain
bar represents the resultant force of the foundation reaction of the i-th segment
the reaction forces XL, X2, X3,..., in these bars constitute the basic unknowns for solving the problem, and the actual displacement and rotation angle at the free end of the beam are additional unknowns. According to the continuity of displacement between the rigid bar and the semi infinite foundation, an equation can be obtained:
one
two Wo ～ akoo + △ help = 0 (3)
z = 1
Where is the relative displacement at point K when only x = 1; tile =
+, is the deflection of cantilever beam at point K, and is the foundation settlement at point K; Is the distance from the fixed end of the beam to the K point; Δ is the relative displacement of the external load at point K, that is, the negative displacement of the cantilever beam at point K
combined with two static equilibrium conditions:
one
∑ Mo + ∑ P = 0, one ∑ Aixi + ∑ MP = 0 (4)
I = 1, I 1
there are + 2 equations to solve all the above unknowns
the chain link method is widely used, regardless of the nature of the foundation, load types and member cross-section deformation. The more the number of links, the more accurate the solution is, but the more the workload is
large. Generally, 6-10 links can meet the accuracy requirements of the project
3) Cai Siwei's method
the foundation reaction is determined by using the deformation coordination condition between foundation beam and foundation, that is, the condition that they should still contact each other after deformation
the foundation beams are divided into equal parts, and the length of each section is B. assuming that the foundation reaction on each section is evenly distributed
the foundation reaction under the foundation beam is distributed in a ladder shape. Make the degree of foundation reaction strength
of each section as PL, P2,..., P. Under the action of these reaction forces, the foundation settlement occurs at all points
let the settlement at the midpoint of each section be expressed as WL, W2,..., W. According to the deformation compatibility condition of foundation and foundation beam, these settlements should be equal to the deflection of corresponding points on the foundation beam
the basic equations of the beam are:
a
etc.: (+ 2 wi 1 2WI + wi 1) (5) a + 111 L)
where I is the center of the i-th segment; Mi is the bending moment of section I
according to the number of sections, the settlement on the right side of the equation is listed as the function of PL, P2,...,
P by formula (1), and the settlement on the left side of the equation is the function of external load and foundation reaction, which can be written directly. From equation (5), n-2 equations can be listed, plus two equilibrium formulas ∑ = 0 and ∑ M = 0, then n values can be determined
other solutions, such as finite element method, trigonometric series method, distributed base reaction method, etc.
refer to relevant literature
5. This paper introces several theoretical analysis methods for calculating beams on elastic foundation. It is necessary to "try to prove theoretically that the Grignard method is too large". Above (i.e. "Analysis on calculation method of beam on elastic foundation", the same below.) Therefore, this paper intends to analyze and compare the theoretical basis and accuracy of several calculation methods for beams on elastic foundation. 1、 There are two famous theories about the calculation of beams on elastic foundation: the local deformation theory (winkle hypothesis) and the semi infinite elastic body theory. The starting point of the former is to assume that the pressure on the foundation per unit area is proportional to the foundation settlement, i.e. P 21 K. (1) according to this assumption, it is easy to establish the differential equation of elastic foundation beam, and the solution of the differential equation can be obtained strictly; According to the elastic theory and compression test, the settlement also occurs in the vicinity of the compression range. Therefore, the settlement of the same foundation is related to the size of the foundation (the larger the foundation is, the larger the settlement is).
6. Therefore, the key problem in the calculation theory of elastic foundation beam is how to determine the relationship between foundation reaction and foundation settlement, or how to select the calculation model of elastic foundation (Fig. 8-1)< Local elasticity
7. Therefore, the key problem in the calculation theory of elastic foundation beam is how to determine the relationship between foundation reaction and foundation settlement, or how to select the calculation model of elastic foundation (Fig. 8-1)< Local elasticity
8. This paper introces several theoretical analysis methods for calculating beams on elastic foundation. It is necessary to "try to prove theoretically that the Grignard method is too large". Above (i.e. "Analysis on calculation method of beam on elastic foundation", the same below.) Therefore, this paper intends to analyze the theoretical basis and accuracy of several calculation methods for beams on elastic foundation
9. The beam slab raft foundation is similar to the inverted floor, the beam is turned up, and the bottom of the beam is flat
the foundation beam is arranged under the wall column. When the beam width is less than the column width, the beam needs to be widened locally
in order to force the raft evenly, the raft divided by the foundation beam should not be too large, and the ratio of the slab thickness to the slab span should not be too small.
the foundation beam is arranged under the wall column. When the beam width is less than the column width, the beam needs to be widened locally
in order to force the raft evenly, the raft divided by the foundation beam should not be too large, and the ratio of the slab thickness to the slab span should not be too small.
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