常用截面几何特性计算公式

A0
=
3 2
a
4l 2 − a 2
An = A + A0
h ZG = 4
V
=
hA 3
�
�1 + � �
a1 a
+
� �� �
a1 a
� �� �
2
� � � �
A1
=
33 2
a2
1
A = 3 3 a2 2
A0 = 3g(a1 + a)
An = A + A1 + A0
h(a 2 + 2a1a + 3a 2 )
号
表 G1-4 常用静定梁的支点反力、弯矩和变形计算公式
支点反力
弯矩方程
挠度曲线方程
0 ≤x ≤ l / 2 :
0≤x ≤l / 2 :
1
F FA = FB = 2
Fx M (x) =
2
y
=
− Fl3 48EI
� � � �
3x l
−
4x3 l3
� � � �
最大挠度
在x = l / 2处 :
ymax
在x = l / 2处 :
ymax
=
− Fa 24EI
(3l 2
−
4a 2 )
θA = −θB = − Fa2 (l − a) 2 EI
9
10
序 载荷情况及剪力图弯矩图
号
4
5
支点反力
弯矩方程
挠度曲线方程
M FA = FB = l
M (x)
=
M ��1 −
x� �
� l�
y
=
− Ml2 6EI
� � � �
iX = 0.32h iY = 0.20b
iX = 0.28h iY = 0.24b
iX = 0.30h iY = 0.17b
iX = 0.26h iY = 0.21b
iX = 0.21h iY = 0.21b iZ = 0.185h
iX = 0.43h iY = 0.43b
iX = 0.42h iY = 0.22b
在x = 1 处 : 3
ymax = − Ml2 9 3EI
在x = l / 2处 :
y = − Ml2 16EI
(续) 10
梁端转角
−Ml θA = 3EI
Ml θB = 6EI
−Ml θA = 6EI
iX = 0.39h iY = 0.20b
iX = 0.35h iY = 0.56b
iX = 0.38h iY = 0.60b
8
截面形状
回转半径
iX = 0.38h iY = 0.44b
截面形状
(续)
回转半径
iX = 0.45h iY = 0.24b
iX = 0.35dcp
d cp
=
D+d 2
iX = 0.40h iY = 0.21b
iX = 0.44h iY = 0.38b
iX = 0.45h iY = 0.235b
iX = 0.37h iY = 0.54b
iX = 0.44h iY = 0.32b
iX = 0.37h iY = 0.45b
G2.2 受静载荷梁的支点反力、弯矩和变形计算公式�表 G1-4、表 G1-5�
序 载荷情况及剪力图弯矩图
a ZG = 2
V = abh A = ab A0 = 2h(a + b) An = 2(ab + ah + bh)
d = a2 +b2 + h2 (d为对角线)
h ZG = 2
体积 V、底面积 A、侧面积 A0、全 面积 An、重心位置 G 的计算公式
V = p h(3a 2 + h 2 ) 6
= p h 2 (3r − h) 3
第 1 章 常用资料、数据和一般标准
G1 常用几何体的体积、面积及重心位置�表 G1-1�
图
形
表 G1-1 常用几何体的体积、面积及重心位置
体积 V、底面积 A、侧面积 A0、全 面积 An、重心位置 G 的计算公式
图
形
V = a3 A = a2 A0 = 4a2 An = 6a2 d = 3a (d为对角线 )
2x l
−
3x2 l2
+
x3 l3
� � � �
M FA = FB = l
ห้องสมุดไป่ตู้
Mx M (x) =
l
y
=
− Ml 2 6EI
� � � �
x l
−
x3 l3
� � � �
最大挠度
在x
=
�
�1 �
−
�
1
� �
3
� �
l处 :
ymax = −Ml2 9 3EI
在x = l / 2处 :
y = −Ml2 16EI
V = 1 pr2h 3
A = pr 2 A0 = prl An = pr (r + l )
l = r2 + h2 h
ZG = 4
图
形
(续)
体积 V、底面积 A、侧面积 A0、全面积 An、重心位置 G 的计算公式
h V = 6 (2ab + ab1 + a1b + 2a1b1) A1 = a1b1 A = ab
��
b ��
y = − Fb(3l2 − 4b2) 48EI
0≤x ≤ l :
3
FA = FB = F
0 ≤x ≤a : M (x) = Fx a ≤x ≤l − a : M = Fa
y = − Fx [3a(l − a) − x2 ] 6EI
0≤x ≤l − a :
y = − Fa [3x(l − x) − a2 ] 6EI
A = pa2
A0 = 2 prh = p(a 2 + h 2 )
An = p(2rh + a 2 ) = p(h 2 + 2a 2 )
ZG
=
h(4r − h) 4(3r − h)
V = 4 pabc 3
重心G在椭球中心
V = 4p r3 3
An = 4p r 2
重心G与球心重合
V = 2p2Rr2 = p2 Dd2 4
a2
a4
12
a2 Wx = 6 Wx1 = 0.1179a3
a = 0.289a
12
a ex = 2 ex1 = 0.7071a
a2 −b2
a4 −b4 12
Wx
=
a4 −b4 6a
Wx1
=
0.1179
a4
− b4 a
0.289 a2 + b2
a ex= 2 ex1 = 0.7071a
ab
ab 3
12
cosα�J x
=
J x1
−
Ays2
Iy
=
r4 8
� απ � � 180°
− sinα
−
2 3
sinα
sin 2
α 2
���Wx �
=
r
Jx − ys
ix = 0.1319d d
iy = 4
ex = 0.2878d ys = 0.2122d
ix =
Ix A
iy =
Iy F
1 =
D2 + d2
4
ys = 2(D2 + Dd + d 2 )
Ix
p =
4
(ab3
− a1b13
)
Iy
=
p 4
(a3b − a13b1)
Wx
=
p (ab3 − a1b13) 4b
Wy
=
p(
a3b − 4a
a13b1)
ix =
Ix A
iy =
Iy A
ex = b ey = a
BH − b(e2 + h)
Ix
=
Be13
+ ae23 3
− bh3
Wx1
=
Ix e1
Wx 2
ZG =
1
4(a 2 + a1a + a 2 )
1
( A1为顶面积, g 为斜高)
V = 1 abh 3
A = ab
A0
=
1 2
(b
4h2 + a2
+ a 4h2 + b2 )
An
=
ab +
1 (b 2
4h2 + a2
+ a 4h2 + b2 )
h ZG = 4
V = 3 3 a2h 2
A = 3 3 a2 2
bH 2 Wxa = 24
bH 2
H = 0.236H
H
32
3
Wxb = 12
4
截面形状
面积 A
惯性矩 I
截面系数W = I e
回转半径 i = I A
(续)
形心距离 e
A = 2.598C2 C=R
I x = 0.5413R4 I y =I x
Wx = 0.625R3 Wy = 0.5413R3
=
Ix e2
Be13 + ae23 − bh3 3[HB − b(e2 + h)]
e1
=
aH 2 + bt 2 2(aH + bt)
e2 = H − e1
6
截面形状
面积 A
惯性矩 I
截面系数W = I e
回转半径 i = I A
(续)
形心距离 e
BH + bh
Ix
=
BH 3 + 12
bh3
Wx
=
BH 3 + bh3 6H
ix = 0.4566R
ex = 0.866R ey = R
pd 2
pd 4
pd 3
4
64
32
d
d
4
2
p (D2 − d2) 4
p (D4 − d4) 64
p
� �
D4
−
d
4
� �
32
� �
D
� �
D4 + d 4 4
D 2
a2 − pd 2 4
1
� �
a
4
−
3pd 4
� �
12
� �
16
� �
1
� �
A0 = 6ah An = 3 3a2 + 6ah
d = h2 + 4a2 (d为对角线)
ZG
=
h 2
3
G2 常用力学公式
G2.1 常用截面的力学特性(表 G1-2、表 G1-3)
表 G1-2 常用截面的几何及力学特性
截面形状
面积 A
惯性矩 I
截面系数W = I e
回转半径 i = I A
形心距离 e
a2 + b 2 + 4ab H 3 36(a + b)
Wxa
=
H 2 (a 2 + 4ab + b2 ) 12(a + 2b)
Wxb
=
H 2 (a2 + 4ab + b2 ) 12(2a + b)
H ×
3(a + b) a2 + 4ab + b2 2
H (2a + b) 3(a + b)
bH 2
bH 3 36
An = 4p2Rr = p2Dd 重心G在圆环中心
V
=
2 p
r3
3
A =p r2
A0 = 2p r 2
An = 3p r 2
ZG
=
3 8
r
V=πr2h A0=2πrh An=2πr(r+h) ZG= h
2
2
图形
体积 V、底面积 A、侧面积 A0、全面积 An、重心位置 G 的计算公式
V = ph(R 2 − r 2 )
Wy
=
pd 3 64
� �1 − � �
d4 D4
� � � �
A = 1 [rl − c(r − h)]�l = 0.01745α 2
c = 2 h(2r − h)�α = 57.296l r
r = c2 + 4h2 �h = r − 1 4r 2 − c2
8h
2
I x1
=
lr 3 8
−
r4 8
sinα
A0
=
1 2
[(b1
+
b)
4h2 + (a − a1)2
+ (a1 + a) 4h2 + (b − b1)2 ]
An = A + A1 + A0
ZG
=
h(ab + ab1 + a1b + 3a1b1) 2(2ab + ab1 + a1b + 2a1b1)
V = 1 Ah = 3 a 2 h
3
2
A = 3 3 a2 2
3p(D + d )
ix =
Ix A
c3 ys = 12A
截面形状
面积 A
惯性矩 I
截面系数W = I e
回转半径 i = I A
5
(续)
形心距离 e
pab
Ix
=
pab3 4
Iy
=
pa 3b 4
Wx
=
pab2 4
Wy
=
pa 2 b 4
b ix = 2
a iy = 2
ex = b ey = a
p(ab − a1b1)
A = p(R2 − r 2 )
A0 = 2ph(R + r)
An = 2p(R + r)(R − r + h)
ZG
=
h 2
V = p h(R2 + r 2 + Rr) 3
A0 = pl(R + r) An = p(R 2 + r 2 ) + A0
l = (R − r)2 + h2
ZG
=
h(R 2 + 2Rr + 3r 2 ) 4(R 2 + Rr + r 2 )
=
− Fl3 48EI
9
梁端转角 − Fl 2
θA = −θB = 16EI
0≤x ≤a :
0≤x ≤a :
2
Fb FA = l
Fa FB = l
Fbx M (x) =
1 a ≤x ≤1:
Fbx M (x) = − F(x − a)
1
y = − Fbx (l 2 − x2 − b2 ) 6EIl
0≤x ≤l :
BH 3 + bh3
H
12(BH + bh)
2
BH − bh
Ix
=
BH 3 − bh3 12
Wx
=
BH 3 − bh3 6H
ix =
BH 3 − bh3 12(BH − bh)
H 2
截面形状
表 G1-3 主要组合截面的回转半径
回转半径
截面形状
iX = 0.30h iY = 0.215h
7
回转半径
iX = 0.21h iY = 0.21b
y = − Fb 6EIl
若a > b,在x = l2 − b2 处 : 3
ymax = − Fb(l 2 − b2 )3 / 2 9 3EIl
在x = l / 2处 :
−Fab(l + b) θA = 6EIl
Fab(l + a) θB = 6EIl
×
� �(l
2
−
b2)x
−
x3
+
(x
−
a)3
� �
a
4
−
3pd
4
� �
6a
� �
16
� �
16a4 − 3pd 4 48(4a2 − pd 2 )
a 2
pd 2 8
Ix = 0.00686d 4
Iy
=
pd 4 128
Wx = 0.0239d 4
Wy
=
pd 3 64
p(D2 − d 2) 8
Ix = 0.00686(D4
− d4)
Iy
=
p(D4 − d 4) 128
ab 2
b = 0.289b
b
6
12
2
b(H − h)
Ix
=
b(H 3 − h3 ) 12
Iy
=
b3 (H − h) 12
Wx
=
b(H 3 − h3) 6H
Wy
=
b2 (H 6
− h)
ix =
H 2 + Hh + h2 12
i y = 0.289b
H ex = 2
b ey = 2
H (a + b) 2