挠度计算公式的基本推导

2025-03-14 07:23:11
推荐回答(1个)
回答1:

     随着科学技术的进步以及建筑设计的发展,力学建筑不仅坚固,而且给人一种踏实舒服的感觉,那么一些工程建设就需要精确的科学计算之后,然后才开始进行工程的开发,下面小编就为大家简单的叙述一下挠度计算公式,以帮助一些建筑的设计完成。



第一步:


  1. 当荷载的力作用在跨中时挠度的计算方式是:fmax=(P·L3)/(48×E·I)

  2. 当荷载作用在任意一点时挠度的计算方式:fmax={P·L1·L2(L+L2)·[3×L1·(L+L2)]1/2}/(27×E·I·L)。

也就是说这两种情况我们如果进行分析的话,我们会发现集中荷载作用在任意一点时,也就是说任意一点可以是中点,那么上面的‚式就会包含式,而式知识挠度公式中的一个特例,当然也就是L1=L2= L/2这种情况。那么我们就可以这样思考了,将L1=L2= L/2代入‚式中,max={P·L1·L2(L+L2)·[3×L1·(L+L2)]1/2}/(27×E·I·L)。

     ={P·L/2·L/2(L+L/2)·[3×L/2·(L+L/2)]1/2}/(27×E·I·L)

={P·L2/4·(3L/2)·[9×L2/4]1/2}/(27×E·I·L)

={P·(3L2/8)·[3×L/2] }/(27×E·I)

=  P·(9L3/16)/(27×E·I)

=(P·L3)/(48×E·I)

这样也就验算了以上的思想了。



第二步:


简单的推导过程:

我们以简支梁来为例:全粱应将其分为两段

对于梁的左段来说,则当0≤X1≤L1时,其弯矩方程可以表示为:

Mx1=(P·L2/L)·X;设f1为梁左段的挠度,则由材料力学。

E·I·f1//=(P·L2/L)·X

积分得E·I·f1/=(P·L2/L)·X2/2+C1   

二次积分:E·I·f1=(P·L2/L)·X3/6+C1X+D1   ‚

因为X1等于零时:

简支梁的挠度f1等于零(边界条件)

将X1=0代入(2)得D1=0



而对于梁的右段,即当L1≤X2≤L时,其弯矩方程可以表现为:

MX2=(P·L2/L)·X-P·(X-L1);

设f2为梁右段的挠度,则由材料力学

E·I·f2//=(P·L2/L)·X-P·(X-L1)

积分得E·I·f2/=(P·L2/L)·X2/2-[P(X-L1)2/2]+C2      ƒ

二次积分:E·I·f2=[(P·L2/L)·X3/6]-[P·(X-L1)3/6]+C2X+D2   ④

将左右段连接,则可以

①在X=0处,f1=0;

②在X=L1处,f1/= f2/(f1/、 f2/为挠曲线的倾角);

③在X=L1处,f1= f2;

④在X=L处,f2=0;

由以上四条件求得(过程略):C1= C2= -[(P·L2)/6 L]·(L2-L22);D1=D2=0。

代入公式、‚、ƒ、④整理即得:

对于左段   0≤X≤L1


E·I·f1/=(P·L2/L)·X2/2+C1            (1)

          = P·L2/6L ·[3X2-(L2-L22)]          (5)

E·I·f1=(P·L2/L)·X3/6+C1X+D1          (2)

= (P·L2/6×L)·[X3-X(L2-L22)]               (6)

对于右段  L1≤X≤L

E·I·f2/=(P·L2/L)·X2/2-[P·(X-L2)2/2]+C2         (3)

= (P·L2/6×L)·[3X2-(L2-L22)]-[ P/2·(X-L1)2]        (7)

E·I·f2=[(P·L2/L)·X3/6]-[P·(X-L1)3/6]+C2X+D2         (4)

= (P·L2/6L)·[X3-X(L2-L22)] -[P/6·(X-L1)3]          (8)

等一一对应的过程式。


第三步:按以上基础继续进行:


     若L1>L2,则最大挠度就显然在左段内,命左段的倾角方程(5)f /等于零,即得最大挠度所在之位置,于是令:

P·L2 /6L·[3X2-(L2-L22)] =0

则:3X2-(L2-L22)= 0

得:X=[(L2-L22)/3]1/2                        (9)

将(9)式代入(6)式即得最大挠度

fmax= -[P·L2·(L2-L22)3/2]/ [9×31/2×L·E·I]                  (10)

展开即得:

fmax=-{(P·L1·L2·(L+L2)·[3×L1·(L+L2)]1/2)}/(27×E·I·L)。

  这就是公式的推导过程,对于非专业人士可能不会十分清楚,小编这样希望给专业人士一个帮助性的指引,希望有关人士可以在建筑上能够得以应用。以上就是有关挠度计算公式的内容,希望能对大家有所帮助!

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