土方方格网计算公式图示及推导

方格网土方计算公式推导：

1、两点开挖工程量计算公式：

如上图示：d=A*h2/(h1+h2); e=A*h3/(h3+h4); S1=d*h2/2; S2=e*h3/2

S0=(d+e)/2*(h2+h3)/2/2

根据拟柱体体积计算公式：V=B/6*(S1+4*S0+S2)将上面已知数代入公式可得：V=B/6*{A*h2/(h1+h2)*h2/2+4*[A*h2/(h1+h2)+A*h3/(h3+h4)]/2*(h2+h3)/2/2

+h3*A*h3/(h3+h4)/2}

=A*B/6*{h2*h2/(h1+h2)+ h2*(h2+h3)/(h1+h2)+ h3*(h2+h3)/(h3+h4)

+h3*h3/(h3+h4)}/2

=A*B/12*{(2h2^2+h2*h3)/ (h1+h2)+(2*h3^2+h2*h3)/(h3+h4)}

2、三点开挖的挖方量计算公式：

由图分解可得，挖方体积=v1+v2-(v3-v4),由拟柱体体积计算公式可以得出：

V1={A*(h3+h4)/2+4*A/2*(h3+h2+h2+h4)/4}*B/6

=A*B/12*{h3+h4+2*h2+h3+h4}=A*B*(h2+h3+h4)/6

V2、V3、V4分别按四棱锥、三棱锥、三棱锥体积计算公式进行计算（体积=底面积*高/3）

V2= [√(A^2+B^2)]*1/2*1/3*[√(A^2+B^2)]*(h2+h4)/2

= (A^2+B^2)*(h2+h4) /12

V3=A*B/2/3*h1=A*B*h1/6

V4=h1/3*(B*h1/(h1+h4)*A*h1/(h1+h2)/2=A*B/6*h1^3/(h1+h2)/(h1+h4)

V=V1+V2-V3+V4

= A*B*(h2+h3+h4)/6+(A^2+B^2)*(h2+h4) /6+A*B/6*h1^2/(h1+h2)/(h1+h4)

- A*B*h1/6

= A*B /6*[ h2+h3+h4-h1+h1^3/(h1+h2)/(h1+h4)] +(A^2+B^2)*(h2+h4) /12

3、不机邻两点回填方量计算公式推导：

如图示：从h1和h3处将图形分成平面为两个直角三角形体：h4侧的体积公式如下：

Vh4=V1+V3-V2根据锥体体积公式：底面积*高/3可得

V1=(h1+h3)/2*[√(A^2+B^2)] /3*[√(A^2+B^2)]/2=(h1+h3)*(A^2+B^2) /12 V2=A*B/2*h4/3= A*B*h4/6

V3= h4/3*(B*h4/(h4+h1)*A*h4/(h4+h3)/2=A*B/6*h4^3/(h4+h1)/(h4+h3) V=(h1+h3)*(A^2+B^2) /12- A*B*h4/6+ A*B/6*h4^3/(h4+h1)/(h4+h3)

= A*B/6*[h4^3/(h4+h1)/(h4+h3)-h4]+ (h1+h3)*(A^2+B^2) /12

h2侧的体积公式推导方法h4侧的体积公式：

Vh2=A*B/6*[h2^3/(h2+h1)/(h2+h3)-h2]+ (h1+h3)*(A^2+B^2) /12

V=Vh2+Vh4

=A*B/6*[h2^3/(h2+h1)/(h2+h3)-h2]+ (h1+h3)*(A^2+B^2) /12

+ A*B/6*[h4^3/(h4+h1)/(h4+h3)-h4]+ (h1+h3)*(A^2+B^2) /12

= A*B/6*[ h2^3/(h2+h1)/(h2+h3) + h4^3/(h4+h1)/(h4+h3) -h2-h4]

+ (h1+h3)*(A^2+B^2) /6