弹性地基梁

BULETINUL INSTITUTULUI POLITEHNIC DIN IA?I

Publicat de

Universitatea Tehnic? …Gheorghe Asachi” din Ia?i

Tomul LV (LIX), Fasc. 4, 2009

Sec?ia

CONSTRUC?II. ?RHITECTUR?

BEAMS ON ELASTIC FOUNDATION

THE SIMPLIFIED CONTINUUM APPROACH

BY

IANCU-BOGDAN TEODORU

Abstract. The key aspect in the design of flexible structural elements in contact with bearing soils is the way in which soil reaction, referred to qualitatively as soil’s reactive pressure (p), is assumed or accounted for in analysis. A magnitude and distribution of p might be preliminary assumed, or some mathematical relationships could be incorporated into the analysis itself, so that p is calculated as part of the analysis. In order to eliminate the bearing soil reaction as a variable in the problem solution, the simplified continuum approach is presented. This idealization provides much more information on the stress and deformation within soil mass compared to ordinary Winkler model, and it has the important advantage of the elimination of the necessity to determine the values of the foundation parameters, arbitrarily, because these values can be computed from the material properties (deformation modulus, E s, Poisson ratio, νs and depth of influence zone, H, along the beam) for the soil. A numerical investigation of the simplified-continuum approach is also presented.

Key words: beams; elastic foundations; Winkler foundation; Vlasov foundation;

two-parameter elastic foundation; EBBEF2p computer code.

1. Introduction

Generally, the analysis of bending of beams on an elastic foundation is developed on the assumption that the reaction forces of the foundation are

38 Iancu-Bogdan Teodoru

proportional at every point to the deflection of the beam at that point. The vertical deformation characteristics of the foundation are defined by means of identical, independent, closely spaced, discrete and linearly elastic springs. The constant of proportionality of these springs is known as the modulus of subgrade reaction, k s. This simple and relatively crude mechanical representation of soil foundation was firstly introduced by Winkler, in 1867 [3], [1].

The Winkler model, which has been originally developed for the analysis of railroad tracks, is very simple but does not accurately represents the characteristics of many practical foundations. One of the most important deficiencies of the Winkler model is that a displacement discontinuity appears between the loaded and the unloaded part of the foundation surface. In reality, the soil surface does not show any discontinuity (Fig. 1).

a b

Fig. 1 – Deflections of elastic foundations under uniform pressure: a – Winkler

foundation; b – practical soil foundations.

Historically, the traditional way to overcome the deficiency of Winkler model is by introducing some kind of interaction between the independent springs by visualising various types of interconnections such as flexural elements (beams in one-dimension (1-D), plates in 2-D), shear-only layers and deformed, pretensioned membranes [3]. The foundation model proposed by Filonenko and Borodich in 1940 [3] acquires continuity between the individual spring elements in the Winkler model by connecting them to a thin elastic membrane under a constant tension. In the model proposed by Hetényi in 1950 [3], interaction between the independent spring elements is accomplished by incorporating an elastic plate in 3-D problems, or an elastic beam in 2-D problems, that can deforms only in bending. Another foundation model, proposed by Pasternak in 1954, acquires shear interaction between springs by connecting the ends of the springs to a layer consisting of incompressible vertical elements which deform only by transverse shearing [3]. This class of mathematical models have another constant parameter which characterizes the interaction implied between springs and hence are called two-parameter models or, more simply, mechanical models (Fig. 2).

Another approach to develop, and also to improve foundation models starts with the three complex sets of differential equations with partial derivatives (compatibility, constitutive, equilibrium) governing the behavior of the soil as a semi-infinite continuum, and then introduce simplifying assumptions with respect to displacements or/and stresses in order to render the

Bul. Inst. Polit. Ia ?i, t. LV (LIX), f. 4, 2009 39 remaining equations fairly easy to solve in an exact, closed-form, manner. These are referred to as simplified-continuum models.

Fig. 2 – Beam resting on two-parameter elastic foundation.

Reissner [4], [5] pioneered a straightforward application of the simplified-continuum concept to produce what is referred to as the Reissner Simplified Continuum model. Assuming a foundation layer in which all in-plane stresses are negligibly small,

(1) 0x y xy σστ===,

and the horizontal displacements at the top and bottom surfaces of the foundation are zero, he obtain the analytical solution for elastic foundation given by

(2) ()()()()222121

4c q x,y q x,y c w x,y c w x,y c ??=??, where q (x,y ) is a distributed load acting on the foundation surface, w (x,y ) – the displacement of the foundation surface in the z -direction, and

(3) 123s s E HG c =; c H

=,

were E s , G s and H are deformation modulus, shear modulus and depth of the foundation, respectively.

A consequence of assumption (1) is that the shear stresses in the zx - and zy -planes are independent of the z -coordinate and thus are constant through the depth of the foundation, for a given surface point, (x,y ). Therefore, this model may be applied only to study the response near loading contact area and not to study stresses inside the foundation [3].

Vlasov, in 1960, adopted the simplified-continuum approach based on the variation principle and derived a two-parameter foundation model [2]. In his method the foundation was treated as an elastic layer and the constraints were

40 Iancu-Bogdan Teodoru

imposed by restricting the deflection within the foundation to an appropriate mode shape, φ(z). The two-parameters Vlasov model (Fig. 3) accounts for the effect of the neglected shear strain energy in the soil and shear forces that come from surrounding soil by introducing an arbitrary parameter, γ, to characterize the vertical distribution of the deformation in the subsoil [2]; the authors did not provide any mechanism for the calculation of γ. Jones and Xenophontos [2] established a relationship between the parameter γ and the displacement characteristics, but did not suggest any method for the calculation of its actual value. Following Jones and Xenophontos, Vallabhan and Das [7] determined the parameter γ as a function of the characteristic of the beam and the foundation, using an iterative procedure. They named this model a modified Vlasov model [7], [8].

Fig. 3 – Beam resting on two-parameters Vlasov foundation.

2. Governing Differential Equation for Beams on Two-Parameters Elastic

Foundation

All foundation models shown foregoing lead to the same differential equation. Basically, all these models are mathematically equivalent and differ only in the definition of their parameters [9]. The various two-parameters elastic foundation models define the reactive pressure of the foundation, p(x), as [9]

(4)

22

1

122

d()d() ()()()

d d

s

w x w x

p x k Bw x k B kw x k

x x =?=?,

where: B is the width of the beam cross section; w – deflection of the centroidal line of the beam; k1 – the second foundation parameter with a different definition for each foundation model. As a special case, if the second parameter

Bul. Inst. Polit. Ia ?i, t. LV (LIX), f. 4, 2009 41

k 1

, is neglected, the mechanical modelling of the foundation converges to the Winkler formulation.

Using the last relation and the beam theory, one can generate the governing differential equations for the centroidal line of the deformed beam resting on two-parameters elastic foundation as [9] (5) 42142

d ()d ()()()()()d d w x w x EI k x w x k x q x x x +?=, where: E is th

e modulus o

f elasticity for the constitutive material of the beam; I – the moment of inertia for the cross section of the beam; q (x ) – the distributed load on the beam.

2.1. Parameters Estimation

It is difficult to interpret exactly what subgrade material properties or characteristics are reflected in the various mechanical elements (springs, shears layers, etc.), thus evaluation on a rational, theoretical basis is cumbersome. The advantage of simplified-continuum approach is the elimination of the necessity to determine the values of the foundation parameters, arbitrarily, because these values can be computed from the material properties (deformation modulus, E s , Poisson number, νs , and depth of influence zone, H , along the beam) for the soil. Thus there is insight into exactly what each model assumes and implies in terms of subgrade behavior.

Therefore, with the assumptions of vertical displacement,

(6)

()()()v x,z w x z ?=,

and horizontal displacement,

(7)

()10u x,z =,

using variational calculus, Vlasov model parameters are expressed as [2], [7],

[8]

(8) ()()()2

01d d 112d s s s H s s E k z z ν?νν???=??+???∫, ()

210d 21H s s E k z ?ν=+∫, where

(9) ()sinh 1sinh z H z γ?γ

???????=

42 Iancu-Bogdan Teodoru

is a function defining the variation of the deflection, ν(x, z ), in the z-direction, which satisfy the boundary condition shown in Fig. 3, and (10) ()222d d d 1221d s s w x x H w x

νγν+∞

?∞+∞?∞?????????=??????

∫∫. Since γ is not known a priori , the solution technique for parameters evaluation is an iterative process which depends upon the value of the parameter γ. Therefore, by assuming an approximate value of γ initially, the values of k s and k 1 are evaluated using eq. (8). From the solution of the deflection of the beam, the value of γ is computed using eq. (10). The new γ value is again used to compute new values of k s and k 1. The proceeding is repeated until two succesive values of γ are approximately equal [7], [8].

3. Example

Even though the continuum mechanics approach may look laborious and difficult to use for a closed-form solution, the numerical model is quite simple and can be easily implemented in application-specific software. An example of such software is EBBEF2p [6] which is developed in MATLAB environment and can handle a wide range of static loading problems involving one-dimensional beams supported by one- or two-parameters elastic foundation,

for any loading and boundary condition. Using EBBEF2p computer code, the results for a sample beam resting on Vlasov Simplified Continuum (VSC) model are compared with solutions obtained by two-dimensional finite element

plane strain analyses (2-D FEM) (Fig. 4). a

b Fig. 4 – Geometry of the considered example: a – VSC model; b – 2-D FEM model.

Bul. Inst. Polit. Ia ?i, t. LV (LIX), f. 4, 2009 43

A beam of length L = 20 m, width b = 0.5 m and height h = 1.0 m, with modulus of elasticity E = 27,000 MPa, is considered to be suported by foundation having depth H = 5 m, deformation modulus E s = 20 MPa and Poissons ratio, νs = 0.25 (Fig. 4). The beam carries concentrated loads at ends, 250 kN each.

A total of 905, 15-noded triangular elements with a fourth order interpolation for displacements and twelve Gauss points for the numerical integration were used to define the mesh for the 2-D FEM model.

In both VSC and 2-D FEM models, the beam is modelled with flexure beam element (Table 1).

Table 1 Beam Modelling

VSC model 2-D FEM model Element type linear with 2 nodes linear with 2 nodes Total number of nodes 35 469 Total number of elements 34 117

a

b c Fig. 5 – VSC vs. 2-D FEM solution: a – settlement; b – bending moment;

c – shearing force.

44 Iancu-Bogdan Teodoru

The results from both 2-D FEM and EBBEF2p technique are shown for comparison in Fig. 5. It can be noted that both solution have almost the same shape and they are in good agreement. However, a full comparison between these two techniques is not fair, because in the 2-D finite element solution, complete compatibility of displacements at the beam–soil interface is assumed, but only vertical displacement compatibility exists in Vlasov model [8].

The results of the final computed values of the soil parameters are presented in Table 2. This demonstrate the versatility of the (Vlasov) simplified continuum foundation model: solve beam on elastic foundation problems without having a need to establish the values of foundation parameters.

Table 2

4. Conclusions

The simplified continuum approach, by Vlasov model viewpoint, for static structural analysis of foundation beams, is presented. To demonstrate the versatility of the Vlasov Simplified Continuum (VSC) foundation model, the results for a sample beam resting on VSC model are compared with solutions obtained by two-dimensional finite element plane strain analyses (2-D FEM). As a general observation, the obtained VSM solution are reasonably close to those from more sophisticated finite element solutions.

Received, October 27, 2009 “Gheorghe Asachi” Technical University of Ia?i,

Department of Transportation Infrastructure

and Foundations

e-mail: bteodoru@ce.tuiasi.ro

R E F E R E N C E S

1. Hetényi M., Beams on Elastic Foundation: Theory with Applications in the Fields of

Civil and Mechanical Engineering. Univ. of Michigan Press, Ann Arbor,

Michigan, 1964.

2. Jones R., Xenophontos J., The Vlasov Foundation Model. Internat. J. of Mech.

Science, 19, 6, 317–323 (1977).

3. Kerr A. D., Elastic and Viscoelastic Foundation Models. J. of Appl. Mech., 31, 3,

491–498 (1964).

4. Reissner E., A Note on Deflections of Plates on a Viscoelastic Foundation, J. of Appl.

Mech., 25, 144–155 (1958).

5. Reissner E., Note on the Formulation of the Problem of the Plate on an Elastic

Foundation, Acta Mechanica, 4, 1, 88–91 (1967).

Bul. Inst. Polit. Ia?i, t. LV (LIX), f. 4, 2009 45 6. Teodoru I.B., EBBEF2p – A Computer Code for Analysing Beams on Elastic Foun-

dations. Proc. of the 7th Internat. Symp. “Computational Civil Engineering”,

CCE, “Gheorghe Asachi” Techn. Univ., Faculty of Civil Engng., Jassy, 2009. 7. Vallabhan C.V.G., Das Y.C., A Parametric Study of Beams on Elastic Foundations.

J. of Engng. Mech. Div., 114, 12, 2072–2082 (1988).

8. Vallabhan C.V.G., Das Y.C., Modified Vlasov Model for Beams on Elastic Founda-

tions. J. of Geotechn. Engng., 117, 6, 956–966 (1991).

9. Zhaohua F., Cook D.R., Beam Elements on Two-Parameter Elastic Foundation. J. of

Engng. Mech., 109, 6, 1390-1402 (1983).

GRINZI REZEMATE PE MEDIU ELASTIC

Abordarea din perspectiva simplificat? a mecanicii continuului deformabil

(Rezumat)

Aspectul cheie ?n proiectarea elementelor structurale rezemate sau ?n contact cu masivele de p?mant se refer? la modalitatea prin care intervine ?n analiz? presiunea reactiv? a terenului, p, ca o ?nc?rcare de intensitate ?i distribu?ie compatibile cu poten?ialul masivului de p?mant de a prelua ?nc?rc?ri (cazul funda?iilor izolate sau metodelor simplificate pentru calculul grinzilor de fundare), sau ca o variabil?necunoscut?, prin ?ncorporarea sa ?ntr-o rela?ie conceptual? de propor?ionalitate cu tas?rile funda?iei (cazul masivelor de p?mant idealizate printr-o serie de resorturi a c?ror constant? de rigiditate este tocmai raportul ?n care se g?sesc presiunile reactive ?i tas?rile – modelul Winkler). Pentru a elimina presiunea reactiv? a terenului, ca o variabil? necunoscut? ?n solu?ia problemei, se introduce o abordare din perspectiva simplificat? a mecanicii continuului deformabil. Comparativ cu modelul Winkler sau alte modele mecanice, idealizarea prezentat? prezint? avantajul elimin?rii necesit??ii determin?rii, ?n mod arbitrar, a parametrilor caracteristici modelului avut ?n vedere; ace?tia sunt calcula?i ?n func?ie de caracteristicile de material ale masivului de p?mant (modulul de deforma?ie liniar?, E s, coeficientul lui Poisson, νs) ?i adancimea zonei de influen??, H. Performan?a ?i acurate?ea formul?rii, ?n descrierea r?spunsului ansamblului grind?–masiv de p?mant, sunt testate prin compararea rezultatelor cu solu?iile ob?inute pe modele numerice mai complexe, ?n ipoteza comport?rii liniar-deformabile a masivului de p?mant.

弹性地基梁法(“m”法)公式以及地下连续墙计算书

根据上海市标准《基坑工程设计规程》的规定，在施工临时工况下，地下连续墙的计算采用规范推荐的竖向弹性地基梁法(“m ”法)。弹性地基梁法取单位宽度的挡土墙作为竖向放置的弹性地基梁，支撑简化为与截面积、弹性模量、计算长度有关的弹簧单元，如图1为弹性地基梁法典型的计算简图。 图1 竖向弹性地基梁法计算简图 基坑开挖面或地面以下，水平弹簧支座的压缩弹簧刚度H K 可按下式计算： h b k K h H ..= z m k h .= 式中，H K 为土弹簧压缩刚度(kN/m)；h k 为地基土水平向基床系数(kN/m 3)；m 为基床系数的比例系数；z 为距离开挖面的深度；b 、h 分别为弹簧的水平向和垂直向计算间距(m)。 基坑内支撑的刚度根据支撑体系的布置和支撑构件的材质与轴向刚度等条件有关，按下式计算： B L A E K ....2α= 式中：K ——内支撑的刚度系数(kN/m/m)； α——与支撑松弛有关的折减系数，一般取0.5～1.0；混凝土支撑或钢支撑施加预压力时，取1.0； E ——支撑构件材料的弹性模量(kN/m 2)； A ——支撑构件的截面积(m 2)； L ——支撑的计算长度(m)； S ——支撑的水平间距(m)。 （2）水土压力计算模式 作用在弹性地基梁上的水土压力与土层分布以及地下水位有关系。水土压力计算采用水土分算，利用土体的有效重度和c 、?强度指标计算土压力，然后叠加水压力即得主动侧的水

土压力。土的c 、?值均采用勘察报告提供的固结快剪指标，地下连续墙变形、内力计算和各项稳定验算均采用水土分算原则，计算中地面超载原则上取为20kPa 。基坑周边地下连续墙配筋计算时分项系数取1.25。 ①土压力计算： 墙后主动土压力计算采用朗肯土压力计算理论，主动土压力强度（kPa ）计算公式如下： a a i i a K c K h r q p 2)(-+=∑ 其中，i r 为计算点以上各土层的重度，地下水位以上取天然重度，地下水位以下取水下重度； i h 为各土层的厚度； a K 为计算点处的主动土压力系数，)2 45(tan 2φ-= a K ； φ,c 为计算点处土的总应力抗剪强度指标。 按三轴固结不排水试验或直剪固快试验峰值强度指标取用。 ②水压力计算：作用在支护结构上主动土压力侧的水压力在基坑内地下水位以上按静水压力三角形分布计算；在基坑内地下水位以下水压力按矩形分布计算(水压力为常量)，并不计算作用于支护结构被动土压力侧的水压力，见下图所示。其中， w h ?为基坑内外水位差，w r 为水的重度，取为10kN/m 3。 图2 静水压力分布模式

弹性地基梁计算模型的选择

pkpm弹性地基梁5种模式的选择 pkpm弹性地基梁结构在进行计算时，程序给出了5种计算模式，现对这5种模式的计算和选择进行一些简单介绍。⑴按普通弹性地基梁计算：这种计算方法不考虑上部刚度的影响，绝大多数工程都可以采用此种方法，只有当该方法时基础设计不下来时才考虑其他方法。⑵按考虑等代上部结构刚度影响的弹性地基梁计算：该方法实际上是要求设计人员人为规定上部结构刚度是地基梁刚度的几倍。该值的大小直接关系到基础发生整体弯曲的程度。而上部结构刚度到底是地基梁刚度的几倍并不好确定。因此，只有当上部结构刚度较大、荷载分布不均匀，并且用模式1算不下来时方可采用，一般情况可不用选它。⑶按上部结构为刚性的弹性地基梁计算：模式3与模式2的计算原理实际上最一样的，只不过模式3自动取上部结构刚度为地基梁刚度的200倍。采用这种模式计算出来的基础几乎没有整体弯矩，只有局部弯矩。其计算结果类似传统的倒楼盖法。该模式主要用于上部结构刚度很大的结构，比如高层框支转换结构、纯剪力墙结构等。⑷按SATWE或TAT的上部刚度进行弹性地基架计算：从理论上讲，这种方法最理想，因为它考虑的上部结构的刚度最真实，但这也只对纯框架结构而言。对于带剪力墙的结构，由于剪力墙的刚度凝聚有时会明显地出现异常，尤其是采用薄壁柱理论的TAT软件，其刚度只能凝聚到离形心最近的节点上，因此传到基础的刚度就更有可能异常。所以此种计算模式不适用带剪力墙的结构。另外，设计人员在采用《JCCAD 用户手册及技术条件》附录C中推荐的基床反力系数K时，该值已经包含上部刚度了，所以没有必要再考虑一次。⑸按普通梁单元刚度的倒楼盖方式计算：模式5是传统的倒楼盖模型，地基梁的内力计算考虑了剪切变形。该计算结果明显不同与上述四种计算模式，因此一般没有特殊需要不推荐使用。

弹性地基梁程序说明书

弹性地基梁程序研究 设计人员：郑楠 指导老师：刘川顺老师 武汉大学水利水电学院 2003年6月9日

1弹性地基梁基本原理 假定地基为半无限的连续弹性体，应用弹性理论进行计算。本程序中，弹性地基梁采用的是链杆法原理，并考虑边荷载的作用。其具体表述如下：链杆法的基本作法是：将地基梁分成若干段，并且在每一梁段的中心设置一根不可压缩的刚性链杆将梁和地基联系起来，这样，就可以把一个原有无限多个支承的地基梁的计算问题，转化为一个支承在有限个可沉陷支座上的连续梁的计算问题。通过梁的平衡条件和梁与地基的变形协调条件，就可以建立求解超静定结构的典型方程，从而计算出各个链杆的内力，进而计算地基梁各计算截面的内力。 采用混合法建立典型方程。为了使建立的联立方程的工作简化，混合法采用悬臂梁作为基本结构，也就是除了把n个竖向力链杆切断外，还在梁的一端附加一个竖向链杆和一个刚臂，控制梁在这端的移动和转动，使其成为固定端。因而，基本未知量不但包括n个未知力（链杆内力），而且还包括两个位移（梁端的挠度和转角）。如图所示： 基本结构和原结构相比，可以知道基本结构中每一个竖向链杆的切口处的 X链杆切口处的相对位移应该等于零的条件，即相对线位移应该为零，根据 k =0（设使切口张开的位移为正）可以成立下列方程： k

0001 =?+--∑=kp k n i ki i a y X ?σ 式中： ki σ ：对单位力分别作用在悬臂梁和地基上时, k X 链杆切口处引起 的相对位移 kp ? ：外荷载使悬臂梁在k X 链杆切口处方向引起的位移 0y ：悬臂梁固定端处的竖向位移，以0y 向下为正，所以沿k X 方向 的位移为-0y 0?：悬臂梁固定端处的转角，以顺时针方向为正，所以由0?引起 的沿方向的位移为-0?k a 显然，对于n 个竖向链杆的切口，可以列出n 个方程。另外，通过否定梁端附加 链杆和刚臂的存在（即0,000==M R ），还可以列出如下的两平衡方程，即由 ∑=0y ，有 021=-+++++∑P X X X X n i 由00=∑M ，有∑∑=-=01 M X a n i i i 式中： ∑P ：外荷载在竖直方向投影代数和，以向下为正； ∑M ：外荷载对梁左端弯矩的代数和，以顺时针为正。 这样一共可以列出n+2个方程，可以求解n+2个未知量，即 0021,,,,,,?y X X X X n i 。 在上述的方程中，系数地基沉陷）梁挠度）((ki ki ki y V +=σ (1) 梁挠度的计算：

ANSYS基坑弹性地基梁全程序即详解

/prep7 L1=30 !设置变量 L2=30 h=-25 K, 1, 0, 0, 0, K, 2, L1, 0, 0, K, 3, L1, L2, 0, K, 4, 0, L2, 0, KWPA VE, 1 !将工作平面原点定义在1号点RECTNG, 0, L1, 0, L2, wpro, , -90, !将工作平面绕X轴Z到Y方向90度RECTNG, 0, L1, 0, -h, KWPA VE, 4 !将工作平面原点定义在4号点RECTNG, 0, L1, 0, -h, wpro, , ,90 !将工作平面绕y轴x到z方向90度RECTNG, 0, L2, 0, -h, KWPA VE, 3 !将工作平面原点定义在3号点RECTNG, 0, L2, 0, -h, AGLUE, all !粘结所有面

ET, 1, SHELL43 !ET，ITYPE，Ename，KOPT1，~，KOPT6，INOPR（定义单元） !KOPT1～KOPT6为元素特性编码 !shell43 4 节点塑性大应变单元 ET, 2, COMBIN14 !COMBIN14弹簧-阻尼器Spring-Damper MPTEMP,,,,,,,, !删除系统中已存在的温度表 MPTEMP, 1, 0 !定义一个温度表 MPDA TA, EX, 1, , 2.4E10 !指定与温度相应的材料性能数据弹性模量 MPDA TA, PRXY, 1, , 0.15 !主泊松比 ESIZE, 1, 0 !指定单元边长 AMESH, ALL !划分面生成面单元 NSEL, S, LOC, Z, 0 !选择一组节点子集创建新集 ESLN, S !选择已选节点上的单元 NSEL, S, LOC, Z, -1 !选择z坐标值为-1的--- ESLN, U !从已选集中删除此时剩下只支撑板 CM, STRUT, ELEM !将选择集命名STRUT生成元件 alls !all sel 全选 CMSEL, U, STRUT !去除STRUT元件 CM, W ALL, ELEM !将选择集命名wall生成元件 NSEL, S, LOC, X, 0.1, L1-0.1 !选择一组节点子集创建新集 NPLOT !显示节点 NSEL, R, LOC, Y, 0 !从当前集选择一组节点子集 ESLN, S !从已选集中选择 NSEL, S, LOC, Y, 1 !从当前集选择一组节点子集 ESLN, U !从已选集中删除 ENSYM, , , , ALL !反转壳单元法线方向 NSEL, S, LOC, Y, 0.1, L2-0.1 !选择一组节点子集创建新集 NPLOT !显示节点 NSEL, R, LOC, X, 0 !从当前集选择一组节点子集 ESLN, S !从已选集中选择 NSEL, S, LOC, X, 1 !从当前集选择一组节点子集 ESLN, U !从已选集中删除 ENSYM , , , , ALL !反转壳单元法线方向 ALLS NUMCMP, ALL !所有实体进行重新编号 !直接生成节点 *DO, i, 1, L1-1 ! 从1到29进行循环

倒楼盖法与弹性地基梁法

倒楼盖法 在计算筏型基础时，假设基底净反力为直线分布，当地基比较均匀，上部结构刚度较好、梁板式筏型基础的高跨比或平板式筏型基础的高厚比不小于1/16，且相邻柱荷载及柱距变化不超过20%，筏型基础可仅考虑局部弯曲作用，按倒楼盖来计算，即为倒楼盖法。 倒楼盖模型和弹性地基梁板模型 桩筏筏板有限元计算筏板基础时，倒楼盖模型和弹性地基梁板模型计算结果差异很大的原因 这主要是因为二者的性质是截然不同的： （1）弹性地基梁板模型采用的是文克尔假定，地基梁内力的大小受地基土弹簧刚度的影响，而倒楼盖模型中的梁只是普通砼梁，其内力的大小只与筏板传递给它的荷载有关，而与地基土弹簧刚度无关。（2）由于模型的不同，实际梁受到的反力也不同，弹性地基梁板模型支座反力大，跨中反力小。而倒楼盖模型中的反力只是均布线载。（3）弹性地基梁板模型考虑了整体弯曲变形的影响，而倒楼盖模型的底板只是一块刚性板，不受整体弯曲变形的影响。 （4）由于倒楼盖模型的底板只是一块刚性板，因此各点的反力均相同，由此计算得到的梁端剪力无法与柱子的荷载相平衡，而弹性地基梁板模型计算出来的梁端剪力与柱子的荷载是相平衡的。

地基模型的选择 λ 地基计算模型，大致可分为不连续模型和连续性模型两大类。在基础设计时，如何选择相应的地基模型则是一个比较复杂的问题，很难给出一个统一的标准。在此，本人仅就上述地基计算模型的力学特点和适用范围做一些简单的介绍。 λ 1.文克尔地基模型的受力特点和适用范围 λ 文克尔地基模型实质上来源于阿基米德浮力定律的一个推论，比如浮桥结构是严格执行文克尔地基模型的。显然，力学性质与液体相近的地基，比较符合文克尔模型假定。 λ 因此，该模型主要用于抗剪强度极低的流态淤泥质土或地基土塑性区开展比较大的基础。另外，当厚度不超过基底短边之半的薄压缩层地基，因压力比较大，剪应力比较小，所以也比较符合文克尔模型假定。 λ 从地基土的分类角度上讲，地基土可粗略地分为非粘性土和粘性土。一般地说，当基础位于非粘性土上时，采用文克尔地基模型还是比较合适的。特别是当基础比较软的情况。 λ

弹性地基梁分析--midas 迈达斯

例题 弹性地基梁分析 1

例题弹性地基梁分析 2 例题. 弹性地基梁分析 概要 此例题将介绍利用MIDAS/Gen做弹性地基梁性分析的整个过程，以及查看分析结果的 方法。 此例题的步骤如下: 1.简要 2.设定操作环境及定义材料和截面 3.利用建模助手建立梁柱框架 4.弹性地基模拟 5.定义边界条件 6.输入梁单元荷载 7.定义结构类型 8.运行分析 9.查看结果

例题 弹性地基梁分析 3 1.简要 本例题介绍使用MIDAS/Gen 进行弹性地基梁的建模分析。(该例题数据仅供参考) 基本数据如下： ? 轴网尺寸：见平面图 ? 柱： 900x1000，800x1000 ? 梁： 500x1000，400x1000，1000x1000 ? 混凝土：C30 图1 弹性地基梁分析模型

例题弹性地基梁分析 4 2.设定操作环境及定义材料和截面 在建立模型之前先设定环境及定义材料和截面 1.主菜单选择 文件>新项目 2.主菜单选择 文件>保存: 输入文件名并保存 3.主菜单选择 工具>单位体系: 长度 m, 力 kN 图2. 定义单位体系 4.主菜单选择 模型>材料和截面特性>材料： 添加：定义C30混凝土 材料号：1 名称：C30 规范：GB(RC) 混凝土：C30 材料类型：各向同性 5.主菜单选择 模型>材料和截面特性>截面： 添加：定义梁、柱截面尺寸 注：也可以通 过程序右下角 随时更改单位。

例题 弹性地基梁分析 5 图3 定义材料 图4 定义梁、柱截面

例题弹性地基梁分析 6 3.用建模助手建立模型 1、主菜单选择模型>结构建模助手>框架: 输入：添加x坐标，距离8，重复1；距离10，重复2；距离8，重复1； 添加z坐标，距离8，重复1；距离6，重复1； 编辑： Beta角，0；材料，C30；截面，500x1000； 点击； 插入：插入点，0，0，0； 图5 建立框架

弹性地基梁结构5种计算模式的选择

弹性地基梁结构5种计算模式的选择 弹性地基梁结构在进行计算时，程序给出了5种计算模式，现对这5种模式的计算和选择进行一些简单介绍。 ⑴按普通弹性地基梁计算：这种计算方法不考虑上部刚度的影响，绝大多数工程都可以采用此种方法，只有当该方法时基础设计不下来时才考虑其他方法。 ⑵按考虑等代上部结构刚度影响的弹性地基梁计算：该方法实际上是要求设计人员人为规定上部结构刚度是地基梁刚度的几倍。该值的大小直接关系到基础发生整体弯曲的程度。而上部结构刚度到底是地基梁刚度的几倍并不好确定。因此，只有当上部结构刚度较大、荷载分布不均匀，并且用模式1算不下来时方可采用，一般情况可不用选它。 ⑶按上部结构为刚性的弹性地基梁计算：模式3与模式2的计算原理实际上最一样的，只不过模式3自动取上部结构刚度为地基梁刚度的200倍。采用这种模式计算出来的基础几乎没有整体弯矩，只有局部弯矩。其计算结果类似传统的倒楼盖法。 该模式主要用于上部结构刚度很大的结构，比如高层框支转换结构、纯剪力墙结构等。 ⑷按SATWE或TAT的上部刚度进行弹性地基架计算：从理论上讲，这种方法最理想，因为它考虑的上部结构的刚度最真实，但这也只对纯框架结构而言。对于带剪力墙的结构，由于剪力墙的刚度凝聚有时会明显地出现异常，尤其是采用薄壁柱理论的TAT软件，其刚度只能凝聚到离形心最近的节点上，因此传到基础的刚度就更有可能异常。所以此种计算模式不适用带剪力墙的结构。 另外，设计人员在采用《JCCAD用户手册及技术条件》附录C中推荐的基床反力系数K时，该值已经包含上部刚度了，所以没有必要再考虑一次。 ⑸按普通梁单元刚度的倒楼盖方式计算：模式5是传统的倒楼盖模型，地基梁的内力计算考虑了剪切变形。该计算结果明显不同与上述四种计算模式，因此一般没有特殊需要不推荐使用。

10-1 弹性地基梁的解析方法

2. 弹性地基梁法 弹性地基梁内力计算：基床系数法和半无限弹性体法。 基床系数法：采用文克勒（Winkler）地基模型，地基由许多互不联系的弹簧所组成，某点的地基沉降仅由该点上作用的压力所产生。通过求解弹性地基梁的挠曲微分方程，可求出基础梁的内力。 半无限弹性体法：假定地基为半无限弹性体，将柱下条形基础看作放在半无限弹性体表面上的梁，而基础梁在荷载作用下，满足一般的挠曲微分方程。应用弹性理论求解基本挠曲微分方程，并引入基础与半无限弹性体满足变形协调的条件及基础的边界条件，求出基础的位移和基底压力，进而求出基础的内力。 半无限弹性体法的求解一般采用有限单元法等数值方法。

，根据微分梁单元力的平衡，则： ∑ Y ＝

M x w EI -=22d d 由材料力学知，梁的挠曲微分方程为：或22 44d d d d x M x w EI -=根据截面剪力与弯矩的相互关系，即则：x x M d dQ d d 22=q bp x w EI +-=44d d q bkw x w EI =+44 d d 引入文克勒地基模型及地基沉降s 与基础梁的挠曲变形协调条件，可得：。 w s =kw ks p ==代入上式，可得文克勒地基上梁的挠曲微分方程为：当梁上的分布荷载q =0时，梁的挠曲微分方程变为 齐次方程：0d d 44=+bkw x w EI 有缘学习＋Ｖ星ｙｇｄ３０７６或关注桃报：奉献教育（店铺）

令，称为梁的柔度指标，其单位为(长度)-1。的倒数值称为特征长度，值愈大，梁对地基的相对刚度愈大。 4 4EI kb =λλλλ1λ104d d 444 =+w x w λ该微分方程的通解为 )sin cos ()sin cos (4321x C x C e x C x C e w x x λλλλλλ+++=-于是，梁的挠曲微分方程可进一步写成如下形式： 式中C 1、C 2、C 3、C 4为待定参数，根据荷载及边界条件定；为无量纲量，当x ＝L (L 为基础长度)，称为柔性指数，它反映了相对刚度对内力分布的影响。x λL λ

弹性地基梁计算图表

自摇式机械化滑道课程设计 附件： 弹性地基梁计算原理及图表 大连理工大学 2012.2

弹性地基梁计算原理及图表 弹性地基上的梁在荷载和地基反力共同作用下产生变形后处于平衡状态。梁上的荷载通常是已知的。因此弹性地基梁的计算，关键就在于设法求得梁底的反力。由于梁整体搁置在地基上，即地基反力是沿着梁的全长分布的。它的计算比支承在有限个支座上的梁困难得多，但是若能确定反力的规律，便可用材料力学的方法求得基础梁的内力和变形。目前有三种计算假设的方法：假设地基反力为直线分布、地基基床系数法（亦称文克勒假设）、理想弹性体假设。目前，我国大多采用地基基床系数假设的方法，因此本“弹性地基梁计算图表及原理”只介绍以文克勒假设为基础的计算方法。 一、地基基床系数法计算理论与方法：见教材322页及参考文献的有关内容。 二、弹性地基梁影响线 弹性地基梁在动荷载作用下的影响线，就是当弹性地基梁上受有一个指向不变的单位荷载（如单位集中荷载）在梁上移动时，在一特定截面上所产生的某项作用量值（诸如截面弯矩、剪力或地基反力等）变化规律的图形。这些图形分别被称为该截面的弯矩影响线、剪力影响线和地基反力影响线等等。象船台滑道工程等作用有移动动荷载的基础结构，利用影响线进行计算是最方便的。 弹性地基梁的计算与全梁的折算长度总λ的大小有关。在单位集中荷载作用下的弯矩、剪力和地基反力的影响线值ηm 、ηq 和ηn 见表一～十八。 式中：L-梁的长度（cm ）； E-轨道梁材料的弹性模量（Pa ） S-梁的弹性特征长度(cm)； I-轨道梁的截面惯性矩（cm 4 ） K-基床系数（N/cm 3 ） b-轨道梁的底宽（cm ） 根据总λ值的大小，在实际计算中一般将弹性地基梁划分为： ① 0＜总λ＜1.0 刚性梁； ② 1.0≤总λ≤4.5 有限长梁； ③ 总λ＞4.5 无限长梁。 弹性地基梁的截面Φ上的 弯矩 ⑴ 剪力 ⑵ 地基反力 ⑶ 三、 计算步骤及公式 1、梁的尺寸，材料弹性模量，地基系数以及各种荷载组合均为已知。 2、计算梁的弹性特征长度S 与梁的折算长度总λ＝L/S 。 4bk EI 4101S S L = =λ总 ∑ η=φm P S )(M ∑η=φq P )(Q ∑η=φn P Sb 1)(P

改良的Vlasov 基础模型：弹性地基梁

The Modified Vlasov Foundation Model: An Attractive Approach for Beams Resting on Elastic Supports Iancu-Bogdan Teodoru and Vasile Mu?at Gheorghe Asachi Technical University of Ia?i, Faculty of Civil Engineering and Building Services, Department of Transportation Infrastructure and Foundations 43 Dimitrie Mangeron Blvd., 700050, Ia?i, Romania email: bteodoru@ce.tuiasi.ro ABSTRACT This paper is intended to give a unified framework and to apply the modified Vlasov foundation model to static analysis of beams resting on elastic foundation whose concept is widely encountered in engineering practice. In the design of such structures, to describe the foundation response to applied loads, the mechanical model of Winkler is often used, for almost one and a half century. However, it has some shortcomings, mainly because it assumes no interaction between the adjacent springs and thus neglects the vertical shearing stress that occurs within subgrade materials. In this paper, the Vlasov approach, applied to static analysis of beams resting on elastic foundations, is presented as an alternative to the classical Winkler model. This idealization provides much more information on the stress and deformation within soil mass compared to the well-known Winkler model, and it has the important advantage of eliminating the necessity of arbitrarily determining the values of the foundation parameters. A numerical investigation on applying the Vlasov approach to the static analysis of beams resting on elastic supports is also presented. The solutions of sample problems, obtained by using the Vlasov model, are compared with results obtained on more complex numerical models. KEYWORDS: Beams, Vlasov elastic foundations, finite element method, structural analysis, foundation analysis INTRODUCTION In common engineering practice, the static analysis and design of beams on elastic foundation is developed on the assumption that the reaction forces of the foundation are proportional at every point to the deflection of the beam at that point. The vertical deformation characteristics of the foundation are defined by means of identical, independent, closely spaced, discrete and linearly elastic springs. The constant of proportionality of these springs is known as subgrade reaction coefficient, k s. This simple and relatively crude mechanical representation of soil foundation was firstly introduced by Winkler (1967) (Hetényi, 1946; Kerr, 1964). The Winkler model, which has been originally developed for the analysis of railroad tracks, is very simple but does not accurately represent the characteristics of practical foundation soils. One of the most important deficiencies of the Winkler model is that it assumes no interaction between the adjacent springs and thus neglects the vertical shearing stress that occurs within subgrade materials. In addition, a displacement discontinuity appears between the loaded and the unloaded part of the foundation surface but, in reality, the soil surface does not show any discontinuity.