CN111460544B - Structure calculation method of elastic foundation beam considering shearing effect - Google Patents

Structure calculation method of elastic foundation beam considering shearing effect Download PDF

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CN111460544B
CN111460544B CN202010146802.9A CN202010146802A CN111460544B CN 111460544 B CN111460544 B CN 111460544B CN 202010146802 A CN202010146802 A CN 202010146802A CN 111460544 B CN111460544 B CN 111460544B
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elastic foundation
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杨成永
马文辉
韩薛果
费腾
程霖
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Beijing Jiaotong University
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Abstract

The invention belongs to the technical field of civil engineering, and particularly relates to a structure calculation method of an elastic foundation beam considering a shear effect, which is used for calculating the bending deformation, the shear deformation and the internal force of the elastic foundation beam. The method of the invention considers the structure calculation method of the elastic foundation beam of the shearing effect, deduces the calculation formulas of the beam deformation and the internal force according to the length and the end condition of the beam and the action position and the distribution range of the load, and comprises the calculation formulas of the bending deflection, the shearing deflection, the bending moment and the shearing force of the limited long beam and the infinite long beam of the elastic foundation under the action of concentrated force, fully distributed load and locally uniformly distributed load. Therefore, different contributions of bending and shearing to beam deformation can be evaluated, the structural calculation is more precise, measures for controlling the beam deformation can be accurately formulated, and the method has obvious technical and economic rationality.

Description

Structure calculation method of elastic foundation beam considering shear effect
Technical Field
The invention belongs to the technical field of civil engineering, and particularly relates to a structure calculation method of an elastic foundation beam considering a shear effect, which is used for calculating the bending deformation, the shear deformation and the internal force of the elastic foundation beam.
Background
Typically, the shear modulus of the beam is 0.4 times the modulus of elasticity, with the shear deformation of the beam not exceeding 3% of the total deformation. But when the shear stiffness of the beam is weakened for some reason, its shear deformation may become a significant part of the total deformation. For example, shield tunnels in underground works often have a shear stiffness that is one to two orders of magnitude less than a bending stiffness due to the presence of circumferential seams.
In the prior art, the elastic foundation beam shear deformation is processed by establishing a differential equation of a deflection line with respect to total deflection and calculating the bending deflection and the shear deflection together. This process requires a fourth derivative of shear deflection. It is known that the second derivative of shear deflection is load concentration, so the third and fourth derivatives of shear deflection are physically meaningless, resulting in incorrectness of the calculated results.
If the calculated value of the shear deformation of the beam is smaller than the actual value, the total deflection of the beam is smaller and the internal force is larger, so that the waste of tunnel materials and the insufficient estimation of the deflection are caused; otherwise, the beam strength is insufficient and the deflection estimation is conservative.
Therefore, the existing calculation methods cannot meet the practical requirements, and it is necessary to derive calculation formulas for giving bending deflection, shearing deflection and internal force of the elastic foundation beam.
Disclosure of Invention
In view of the above technical problems, the present invention provides a structural calculation method of an elastic foundation beam considering a shear effect, which can derive a calculation formula of beam deformation and internal force according to the length of the beam, end conditions, and an acting position and a distribution range of a load.
The invention is realized by the following technical scheme:
the structural calculation method of the elastic foundation beam considering the shearing effect deduces the calculation formulas of the beam deformation and the internal force according to the length and the end condition of the beam and the action position and the distribution range of the load, and comprises the calculation formulas of the bending deflection, the shearing deflection, the bending moment and the shearing force of the finite long beam and the infinite long beam of the elastic foundation under the action of concentrated force, fully distributed load and locally uniformly distributed load.
Further, the method specifically comprises the following steps:
(1) the bending deformation of the beam is solved by adopting a Fourier series method, and after the foundation reaction force caused by shearing deformation is considered, the bending deformation of the beam satisfies the differential equation of a bending line as follows:
Figure BDA0002401032410000021
in the formula, wb(x) For bending deformation, m, the deflection of the beam due to bending, is positive upwards;
x is a coordinate along the axis of the beam, m, and x is 0 at the middle point;
e is the modulus of elasticity, kPa, of the beam;
i is the moment of inertia of the beam cross section, m4；
K is a foundation coefficient, kPa/m;
d is the width of the beam, m;
g is the shear modulus of the beam, kPa;
a is the cross-sectional area of the beam, m2；
Eta is the shear coefficient of the beam;
q0distributing line load in kN/m;
q1locally and uniformly distributing line load, kN/m;
p is the concentration force, kN;
δ (x) is the pulse function, and h (x) is the Heaviside step function, introduced for the series solution;
the expression of the pulse function delta (x) is
Figure BDA0002401032410000031
The expression of the step function H (x) is
Figure BDA0002401032410000032
(2) Calculating the bending deflection and the internal force of the beam under the action of load:
deflection w of the beam due to bendingb(x)：
Figure BDA0002401032410000033
The calculation formula of the bending moment M (x) and the shearing force Q (x) generated by the bending deformation of the beam is
Figure BDA0002401032410000034
Figure BDA0002401032410000035
In the formula, a0And anThe bending deflection is a Fourier coefficient, N is 1,2, …, N;
θ0is the corner, rad, at both ends of the beam;
Q0shear forces at the two ends of the beam, kN;
n is the number of adopted series terms;
(3) calculating the shearing deflection w of the beam generated under the combined action of the external load and the foundation reaction forces(x)：
Figure BDA0002401032410000041
In the formula:
Figure BDA0002401032410000042
Figure BDA0002401032410000043
Figure BDA0002401032410000044
Figure BDA0002401032410000045
Figure BDA0002401032410000046
Figure BDA0002401032410000047
(4) calculating the total deflection w (x) of the beam:
w(x)＝wb(x)+ws(x)。
further, the bending deflection w of the beam is calculatedb(x) Bending moment M (x) and shearing force Q (x) generated by bending deformation, and shear deflection w of beams(x) To be determined parameter a in the formula (1)0、an(n＝1,2,…,N)、θ0And Q0The method is determined according to an equation system consisting of the following equations:
Figure BDA0002401032410000051
Figure BDA0002401032410000052
Figure BDA0002401032410000053
θ0when not equal to 0;
Figure BDA0002401032410000054
Q0when not equal to 0; in the formula:
Figure BDA0002401032410000055
Figure BDA0002401032410000056
Figure BDA0002401032410000057
Figure BDA0002401032410000058
Figure BDA0002401032410000059
Figure BDA0002401032410000061
Figure BDA0002401032410000062
Figure BDA0002401032410000063
e0、en、c0、cn、d0、dn、δ0、δn、H0、Hn、f0、fn、g0、gnall fourier coefficients.
x0The location of the concentrated force action, m;
x1is the starting position of locally and uniformly distributed load, m;
c is the distribution width of locally uniformly distributed load, m;
l is the half-length of the beam, m, and for an infinitely long elastic foundation beam, L is the half-width of the calculation range.
Further, solving the undetermined parameter a0、an(n＝1,2,…,N)、θ0And Q0The method comprises the following steps:
for the infinite length elastic foundation beam and the finite length elastic foundation beam fixedly supported at two ends, take theta00 and remove the formula
Figure BDA0002401032410000064
For the infinite length elastic foundation beam and the finite length elastic foundation beam with two free ends, Q is taken00 and remove the formula
Figure BDA0002401032410000065
The invention has the beneficial technical effects that:
the method provided by the invention can calculate the bending deflection, the shearing deflection and the internal force of the infinite-length elastic foundation beam and the finite-length elastic foundation beam under different supporting conditions, so that different contributions of bending and shearing to beam deformation can be evaluated, the structural calculation is more precise, measures for controlling the beam deformation can be accurately formulated, and the method has obvious technical and economic rationality.
Drawings
Fig. 1 is a schematic view of the load bearing of the elastic foundation beam in the embodiment of the invention.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention is described in further detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention.
On the contrary, the invention is intended to cover alternatives, modifications, equivalents and alternatives which may be included within the spirit and scope of the invention as defined by the appended claims. Furthermore, in the following detailed description of the present invention, certain specific details are set forth in order to provide a better understanding of the present invention. It will be apparent to one skilled in the art that the present invention may be practiced without these specific details.
The manner and parameters in which the resilient foundation beam is loaded are shown in figure 1. The meaning of the individual parameters is that q0Distributing line load in kN/m; q. q.s1Locally and uniformly distributing line load, kN/m; p is the concentration force, kN; the load is all positive in the upward direction. x is the number of0The location of the concentrated force action, m; x is the number of1Is the starting position of locally and uniformly distributed load, m; c. CThe distribution width of the locally and uniformly distributed load is m. L is the half length of the beam, m; for an infinitely long elastic foundation beam, L is the half width of the calculation range.
The total deformation of the beam includes deformation by bending and deformation by shearing. Bending deformation not only can generate deflection, but also can cause internal force (bending moment and shearing force) of the beam; shear deformation only increases the deflection of the beam on the basis of bending deformation, but does not generate internal forces. The bending deformation and the shearing deformation satisfy different differential equations and are calculated separately. For elastic foundation beams, shear deformation will affect the magnitude of the foundation reaction force in addition to increasing the deflection of the beam.
The bending deformation of the beam can be solved by adopting a Fourier series method, and after the foundation counter force caused by shearing deformation is considered, the bending deformation of the beam satisfies a differential equation of a deflection line of
Figure BDA0002401032410000081
In the formula, wb(x) For bending deformation, m, is positive in the upward direction;
x is the coordinate along the beam axis (x at the midpoint is 0), m.
E is the modulus of elasticity, kPa, of the beam;
i is the moment of inertia of the beam cross section, m4；
K is a foundation coefficient, kPa/m;
d is the width of the beam, m;
g is the shear modulus of the beam, kPa;
a is the cross-sectional area of the beam, m2；
η is the shear coefficient of the beam.
In the formula (1), delta (x) is a pulse function, H (x) is a Heaviside step function, and is introduced for solving series;
the expression of the pulse function delta (x) is
Figure BDA0002401032410000082
The expression of the step function H (x) is
Figure BDA0002401032410000083
According to the Fourier-grade numerical solution of the mechanical problem, a complementary term is introduced into the beam bending deflection expression to consider different supporting conditions of the short beam at the end part, and a calculation formula for calculating the bending deflection and the internal force of the beam under the action of load can be deduced.
Beam bending deflection wb(x) Is calculated by the formula
Figure BDA0002401032410000091
Figure BDA0002401032410000092
Figure BDA0002401032410000093
In the formula, a0And anThe Fourier coefficient of the bending deflection;
θ0is the corner, rad, at both ends of the beam;
Q0the shear at the two ends of the beam, kN.
The bending moment M (x) and the shearing force Q (x) generated by bending deformation are calculated by the formula
Figure BDA0002401032410000094
Figure BDA0002401032410000095
In the formula, N is the number of the adopted series terms (generally ≧ 40).
Load outside and foundationShear deflection w of beam produced under combined action of counter-forcess(x) Is calculated by the formula
Figure BDA0002401032410000096
In the formula (I), the compound is shown in the specification,
Figure BDA0002401032410000097
Figure BDA0002401032410000098
Figure BDA0002401032410000099
Figure BDA0002401032410000101
Figure BDA0002401032410000102
Figure BDA0002401032410000103
the undetermined parameters a in equations (4), (7), (8) and (9)0、an(n＝1,2,…,N)、θ0And Q0Determined by a system of equations consisting of:
Figure BDA0002401032410000104
Figure BDA0002401032410000105
Figure BDA0002401032410000106
Figure BDA0002401032410000111
in the formula (I), the compound is shown in the specification,
Figure BDA0002401032410000112
Figure BDA0002401032410000113
Figure BDA0002401032410000114
Figure BDA0002401032410000115
Figure BDA0002401032410000116
Figure BDA0002401032410000117
Figure BDA0002401032410000118
Figure BDA0002401032410000119
when solving the unknowns using the system of equations (13a), (13b), (13c), (13d)), θ is taken for the infinite beam and the finite beam clamped at both ends00 and formula (13c) is removed; for an infinite length beam andtaking Q from the limited length elastic foundation beam with two free ends00 and formula (13d) is removed.
It should be noted that, the method removes the items with KD in formula (1), formula (9), formula (13a) and formula (13b), and is also applicable to common inelastic foundation beams with simple and fixed two ends.

Claims (3)

1. The structural calculation method of the elastic foundation beam considering the shearing effect is characterized in that calculation formulas of beam deformation and internal force are deduced according to the length and the end condition of the beam and the action position and the distribution range of load, and the calculation formulas comprise the calculation formulas of bending deflection, shearing deflection, bending moment and shearing force of the limited long beam and the infinite long beam of the elastic foundation under the action of concentrated force, fully distributed load and locally uniformly distributed load;
the method specifically comprises the following steps:
(1) the bending deformation of the beam is solved by adopting a Fourier series method, and after the foundation reaction force caused by shearing deformation is considered, the bending deformation of the beam satisfies the differential equation of a bending line as follows:
Figure FDA0003628278630000011
in the formula, wb(x) For bending deformation, m, the deflection of the beam due to bending, is positive upwards;
x is a coordinate along the axis of the beam, m, and x is 0 at the middle point;
e is the modulus of elasticity, kPa, of the beam;
i is the moment of inertia of the beam cross section, m4；
K is a foundation coefficient, kPa/m;
d is the width of the beam, m;
g is the shear modulus of the beam, kPa;
a is the cross-sectional area of the beam, m2；
Eta is the shear coefficient of the beam;
p is the concentration force, kN;
q0distributing line load in kN/m;
q1locally and uniformly distributing line load, kN/m;
δ (x) is the pulse function, and h (x) is the Heaviside step function, introduced for the series solution;
the expression of the pulse function delta (x) is
Figure FDA0003628278630000021
The expression of the step function H (x) is
Figure FDA0003628278630000022
(2) Calculating the bending deflection and the internal force of the beam under the action of load:
deflection w of the beam due to bendingb(x)：
Figure FDA0003628278630000023
The calculation formula of the bending moment M (x) and the shearing force Q (x) generated by the bending deformation of the beam is
Figure FDA0003628278630000024
Figure FDA0003628278630000025
In the formula, a0And anN is the fourier coefficient of bending deflection, 1,2, …, N;
θ0is the corner, rad, at both ends of the beam;
Q0shear forces at the two ends of the beam, kN;
n is the number of adopted series terms;
(3) calculating the external load and the foundation reaction forceWith shear deflection w of the beam produced belows(x)：
Figure FDA0003628278630000031
In the formula:
Figure FDA0003628278630000032
Figure FDA0003628278630000033
Figure FDA0003628278630000034
Figure FDA0003628278630000035
Figure FDA0003628278630000036
Figure FDA0003628278630000037
(4) calculating the total deflection w (x) of the beam:
w(x)＝wb(x)+ws(x)；
in the above formula, x0The meaning of (A) is: the location of the concentrated force; x is the number of1The meaning of (A) is: locally and uniformly distributing the initial positions of the loads; the meaning of c is: the distribution width of locally and uniformly distributed loads; the meaning of L is: half the length of the beam, for an infinitely long elastic foundation beam, L is the half width of the calculated range.
2. According to claimThe method for calculating a structure of an elastic foundation beam considering a shearing effect as described in 1, wherein the bending deflection w of the beam is calculatedb(x) Bending moment M (x) and shearing force Q (x) generated by bending deformation, and shear deflection w of beams(x) A undetermined parameter a in the formula (1)0、an(n＝1,2,…,N)、θ0And Q0The method is determined according to an equation system consisting of the following equations:
Figure FDA0003628278630000041
Figure FDA0003628278630000042
Figure FDA0003628278630000043
θ0when not equal to 0;
Figure FDA0003628278630000044
Q0when not equal to 0;
in the formula:
Figure FDA0003628278630000045
Figure FDA0003628278630000046
Figure FDA0003628278630000047
Figure FDA0003628278630000051
Figure FDA0003628278630000052
Figure FDA0003628278630000053
Figure FDA0003628278630000054
Figure FDA0003628278630000055
x0m is the position of the concentrated force;
x1is the starting position of locally and uniformly distributed load, m;
c is the distribution width of locally uniformly distributed load, m;
l is the half-length of the beam, m, and for an infinitely long elastic foundation beam, L is the half-width of the calculation range.
3. The structural calculation method of an elastic foundation beam considering shear effect according to claim 2, wherein the undetermined parameter a is solved0、an(n＝1,2,…,N)、θ0And Q0The method comprises the following steps:
for the infinite length elastic foundation beam and the finite length elastic foundation beam with two fixed ends, take theta00 and remove the formula
Figure FDA0003628278630000056
For the infinite length elastic foundation beam and the finite length elastic foundation beam with two free ends, Q is taken00 and remove the formula
Figure FDA0003628278630000057
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