题目 矩形 1 的长为 197.6 mm,宽为 5.6 mm 矩形 2 的长为 300 mm, 宽为 5.6 mm 矩形 3 的长为 197.6 mm, 宽为 5.6 mm 黑点分别为它们的形心位置,x-x 轴已知,y-y 轴未知位置(需求) 以矩形 2 的左边沿为基准线 求此复杂截面的形心
Δ
x
=
2
×
197.6
×
5.6
×
197.6
2
+
5.6
×
300
×
5.6
2
2
×
197.6
×
5.6
+
5.6
×
300
=
57.37
m
m
Δx=\frac{2\times 197.6\times 5.6\times \frac{197.6}{2}+5.6\times 300\times \frac{5.6}{2}}{2\times 197.6\times 5.6+5.6\times 300}=57.37 mm
Δx=2×197.6×5.6+5.6×3002×197.6×5.6×2197.6+5.6×300×25.6=57.37mm 求此复杂截面的惯性矩 ① 对 x - x 轴(x-x 轴在矩形1下方,穿过矩形 2 的形心,在矩形 3 的上方)
I
x
1
=
(
150
+
2.8
)
2
×
5.6
×
197.6
I_{x1}=(150+2.8)^2\times5.6\times197.6
Ix1=(150+2.8)2×5.6×197.6
I
x
2
=
5.6
×
30
0
3
12
I_{x2}=\frac{5.6\times300^3}{12}
Ix2=125.6×3003
I
x
3
=
(
150
+
2.8
)
2
×
5.6
×
197.6
I_{x3}=(150+2.8)^2\times5.6\times197.6
Ix3=(150+2.8)2×5.6×197.6
I
x
=
I
x
1
+
I
x
2
+
I
x
3
=
6.43
×
1
0
7
m
m
4
I_{x}=I_{x1}+I_{x2}+I_{x3}=6.43\times10^7 mm^4
Ix=Ix1+Ix2+Ix3=6.43×107mm4 ① 对 y - y 轴(y-y 轴穿过矩形 1 但不过形心,在矩形 2 的右侧,穿过矩形 3 但不过形心)
I
y
1
=
5.6
×
197.
6
3
12
+
(
57.37
−
197.6
2
)
2
×
5.6
×
197.6
I_{y1}=\frac{5.6\times197.6^3}{12}+(57.37-\frac{197.6}{2})^2\times5.6\times197.6
Iy1=125.6×197.63+(57.37−2197.6)2×5.6×197.6
I
y
2
=
(
57.37
−
2.8
)
2
×
5.6
×
300
I_{y2}=(57.37-2.8)^2\times5.6\times300
Iy2=(57.37−2.8)2×5.6×300
I
y
3
=
5.6
×
197.
6
3
12
+
(
57.37
−
197.6
2
)
2
×
5.6
×
197.6
I_{y3}=\frac{5.6\times197.6^3}{12}+(57.37-\frac{197.6}{2})^2\times5.6\times197.6
Iy3=125.6×197.63+(57.37−2197.6)2×5.6×197.6
I
y
=
I
y
1
+
I
y
2
+
I
y
3
=
8.03
×
1
0
7
m
m
4
I_{y}=I_{y1}+I_{y2}+I_{y3}=8.03\times10^7 mm^4
Iy=Iy1+Iy2+Iy3=8.03×107mm4
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