4-1 b）求形心的位置
yC = 0
Ⅰ
Ⅱ
A1 = 500 × 600mm,zC = 500mm
2
1
A2 = (500? 36 × 2 ) × (600? 40 )mm
260
y
2
2
Ⅲ
zC 2 = 480mm
A3 = 200 × 600mm,zC3 = 100mm
zC
∑Az
=
∑A
i Ci
i
A1 z1 + A2 z2 + A3 z3
= 260mm
=
A1 + A2 + A3
1
4-4 b）试计算平面图形对形心轴yC的惯性矩。
I yC1
500 × 6003
2?
=?
+ 500 × 600 × (500? 260)? mm4
12
=2.63×1010mm4
I yC
260
2
(500? 72)×5603
2?
=?
+ (500? 72)×560× (480? 260)?mm4
12
=1.79×1010mm4
600 × 200 3
2?
I yC3 =?
+ 200 × 600 × (260? 100 )? mm 4
12
=0.347×1010mm4
I yC = I yC + I yC? I yC = (2.63 + 0.347?1.79)×10 mm = 1.19×10 mm
2
10
4
10
1
3
2
4
4-1 c）求形心的位置
C1
yC=0
查型钢表，槽钢No.14b，z01 = 1.67cm
y1
A1=21.3cm2，zC1 = (20 + 1.67)cm
工字钢 A2
=39.6cm2，
h
zC2 = = 10cm
2
C2
14.1
y2
zC =
A1 zC1 + A2 zC2
A1 + A2
= 14.1cm
4-4 c）试计算平面图形对形心轴yC的惯性矩。
槽钢 I y1 = 61.1cm,工字钢 I y = 2500cm
4
2
4
I yC = I y1 + A1(h + z01? zC )
[
2
]
h?
+?I y2 + A2? zC
2?
2
5
4
= 4.45 × 10 m
3
5.1 试求梁中截面1-1、2-2、3-3上的剪力和弯矩。设P、q、a
均为已知。
(a)
MA
FS1 = P + qa = 2qa
12
32
M 1 =? Pa? qa =? qa
2
2
RA
FS 2 = P + qa = 2qa
12
12
M 2 =? Pa? qa + m =? qa
2
2
4
5.1 试求梁中截面1-1、2-2、3-3上的剪力和弯矩。设P、q、a
均为已知。
(c)
RA
RB
B
∑M
∑F
= 0,R A × 0.6? 10 × 0.4 × 0.2 = 0,R A = 1.33kN
= 0,R A + RB? 10 × 0.4 = 0,RB = 2.67 kN
y
FS 1 = R A = 1.33kN,M 1 = RA × 0.2 = 267 N? m
FS 2 = R A? 10 × 0.2 =?0.67 kN,
M 2 = RA × 0.4? 10 × 0.2 × 0.1 = 333N? m
5
5.3 已知P、q、m、a，（1）列出梁的剪力方程和弯矩方程；
（2）作剪力图和弯矩图；（3）确定?FS?max及?M?max。
(a)
MA
RA
解,∑ Fy = 0,RA = 2 P
∑M
FS ( x ) =
A
= 0,M A + m? 2 Pa = 0
FS
M A = Pa
2P
0
(0 < x < a )
(a < x ≤ 2a )
P( 2 x? a )
M (x ) =
Pa
(0 < x ≤ a )
(a ≤ x < 2a )
6
5.3(b)
解：
FS
FS ( x ) =
(0 ≤ x ≤ a )
qa (a ≤ x < 2a )
qx
12
qx
(0 ≤ x ≤ a )
2
M (x ) =
a?
qa? x (a ≤ x < 2a )
2?
7
5.3 (f)
解,∑ M B = 0,P? 3a? RC? 2a + 6 Pa = 0
RC = 4.5Pa
RC
RB
∑F
y
= 0,RC + RB? P? 6 P = 0
FS
RB = 2.5Pa
FS ( x ) =
(0 < x < a )
3.5P (a < x < 2a )
2.5P (2a < x < 3a )
P
(0 ≤ x ≤ a )
M ( x ) = 0.5P(7 x? 9a ) (0 ≤ x ≤ 2a )
2.5P(3a? x ) (2a ≤ x8≤ 3a )
Px
5.3(b)
解：
FS
1、作剪力图
2、作弯矩图
9
5.3 (d)
解：1、确定支座反力
∑M
RA
RB
B
= 0,RB? 2a + 2m + m = 0
FS
3m
RB =?
2a
∑ Fy = 0,RA + RB = 0
3m
RA =
2a
2、作剪力图
3、作弯矩图
M C?
m
= R Aa? m =
2
10
5.3 (g)
解：1、确定支座反力
33
∑ M B = 0,q? 2 a? 4 a? RC? a = 0
9
RC = qa
8
3
∑ Fy = 0,RC + RB? 2 qa = 0
3
RB = qa
8
2、作剪力图
RC
RB
FS
3
a
8
3、作弯矩图
aa
12
M C =? q? =? qa
24
8
3a
3a 3a
9
M D = RB q?
qa 2
=
8
8 16 128
5.3 (h)
解：1、确定支座反力
1
RC = RE = (30 × 1× 2 + 20 ) = 40kN
2
RC
RE
2、作剪力图
FS
（kN）
3、作弯矩图
MC? =?30 ×1 × 0.5 =?15kN? m
M D =?30 ×1 ×1.5 + RC?1 =?5kN? m
(kN? m)
M E + =?30×1× 0.5 =?15kN? m
5-4 载荷、剪力和弯矩之间的关系
5-8 (a)试根据弯矩、剪力和载荷集度的导数关系，改正所示Q
图和M图中的错误。
a/4
FS
E
FS
qa?
a?
a 1 a 49qa
2
2a +? + qa? q =
ME =
4?
4?
424
32
2
5-4 载荷、剪力和弯矩之间的关系
5-8 (b)试根据弯矩、剪力和载荷集度的导数关系，改正所示Q
图和M图中的错误。
FS
FS
14
5.11 a) 作刚架弯矩图。
解：1、确定支座反力
2、作弯矩图
∑ M = 0，R
∑ F = 0，R
∑ F = 0，R
A
x
y
D
= 5kN
= 3kN
= 3kN
M C = 1× 3 = 3kN? m
Ax
Ay
M E = 5 × 2?1× 3 = 7kN? m
M B = 3× 4?1× 4 × 2 = 4kN? m
15
5.11 a) 作刚架弯矩图。
RC
RAx
RAy
解：1、确定支座反力
2、作弯矩图
∑F
x
= 0,q? 3a? R Ax = 0,R Ax = 3qa
C
Ay
∑ F = 0,R? R = 0
3
q
∑ M = 0,? 3a? 2 a = R
y
C
2a,RC = R Ay
Ay
9
= qa
4
16
5.11 b) 作刚架弯矩图。
MA
RA
2、作弯矩图解：1、确定支座反力
12
a qa
∑ M A = 0,qa? 2? 4? a? M A = 0,M A = 4 qa
5
qa
∑ Fy = 0,RA? qa? 4 = 0,RA = 4 qa
17
yC = 0
Ⅰ
Ⅱ
A1 = 500 × 600mm,zC = 500mm
2
1
A2 = (500? 36 × 2 ) × (600? 40 )mm
260
y
2
2
Ⅲ
zC 2 = 480mm
A3 = 200 × 600mm,zC3 = 100mm
zC
∑Az
=
∑A
i Ci
i
A1 z1 + A2 z2 + A3 z3
= 260mm
=
A1 + A2 + A3
1
4-4 b）试计算平面图形对形心轴yC的惯性矩。
I yC1
500 × 6003
2?
=?
+ 500 × 600 × (500? 260)? mm4
12
=2.63×1010mm4
I yC
260
2
(500? 72)×5603
2?
=?
+ (500? 72)×560× (480? 260)?mm4
12
=1.79×1010mm4
600 × 200 3
2?
I yC3 =?
+ 200 × 600 × (260? 100 )? mm 4
12
=0.347×1010mm4
I yC = I yC + I yC? I yC = (2.63 + 0.347?1.79)×10 mm = 1.19×10 mm
2
10
4
10
1
3
2
4
4-1 c）求形心的位置
C1
yC=0
查型钢表，槽钢No.14b，z01 = 1.67cm
y1
A1=21.3cm2，zC1 = (20 + 1.67)cm
工字钢 A2
=39.6cm2，
h
zC2 = = 10cm
2
C2
14.1
y2
zC =
A1 zC1 + A2 zC2
A1 + A2
= 14.1cm
4-4 c）试计算平面图形对形心轴yC的惯性矩。
槽钢 I y1 = 61.1cm,工字钢 I y = 2500cm
4
2
4
I yC = I y1 + A1(h + z01? zC )
[
2
]
h?
+?I y2 + A2? zC
2?
2
5
4
= 4.45 × 10 m
3
5.1 试求梁中截面1-1、2-2、3-3上的剪力和弯矩。设P、q、a
均为已知。
(a)
MA
FS1 = P + qa = 2qa
12
32
M 1 =? Pa? qa =? qa
2
2
RA
FS 2 = P + qa = 2qa
12
12
M 2 =? Pa? qa + m =? qa
2
2
4
5.1 试求梁中截面1-1、2-2、3-3上的剪力和弯矩。设P、q、a
均为已知。
(c)
RA
RB
B
∑M
∑F
= 0,R A × 0.6? 10 × 0.4 × 0.2 = 0,R A = 1.33kN
= 0,R A + RB? 10 × 0.4 = 0,RB = 2.67 kN
y
FS 1 = R A = 1.33kN,M 1 = RA × 0.2 = 267 N? m
FS 2 = R A? 10 × 0.2 =?0.67 kN,
M 2 = RA × 0.4? 10 × 0.2 × 0.1 = 333N? m
5
5.3 已知P、q、m、a，（1）列出梁的剪力方程和弯矩方程；
（2）作剪力图和弯矩图；（3）确定?FS?max及?M?max。
(a)
MA
RA
解,∑ Fy = 0,RA = 2 P
∑M
FS ( x ) =
A
= 0,M A + m? 2 Pa = 0
FS
M A = Pa
2P
0
(0 < x < a )
(a < x ≤ 2a )
P( 2 x? a )
M (x ) =
Pa
(0 < x ≤ a )
(a ≤ x < 2a )
6
5.3(b)
解：
FS
FS ( x ) =
(0 ≤ x ≤ a )
qa (a ≤ x < 2a )
qx
12
qx
(0 ≤ x ≤ a )
2
M (x ) =
a?
qa? x (a ≤ x < 2a )
2?
7
5.3 (f)
解,∑ M B = 0,P? 3a? RC? 2a + 6 Pa = 0
RC = 4.5Pa
RC
RB
∑F
y
= 0,RC + RB? P? 6 P = 0
FS
RB = 2.5Pa
FS ( x ) =
(0 < x < a )
3.5P (a < x < 2a )
2.5P (2a < x < 3a )
P
(0 ≤ x ≤ a )
M ( x ) = 0.5P(7 x? 9a ) (0 ≤ x ≤ 2a )
2.5P(3a? x ) (2a ≤ x8≤ 3a )
Px
5.3(b)
解：
FS
1、作剪力图
2、作弯矩图
9
5.3 (d)
解：1、确定支座反力
∑M
RA
RB
B
= 0,RB? 2a + 2m + m = 0
FS
3m
RB =?
2a
∑ Fy = 0,RA + RB = 0
3m
RA =
2a
2、作剪力图
3、作弯矩图
M C?
m
= R Aa? m =
2
10
5.3 (g)
解：1、确定支座反力
33
∑ M B = 0,q? 2 a? 4 a? RC? a = 0
9
RC = qa
8
3
∑ Fy = 0,RC + RB? 2 qa = 0
3
RB = qa
8
2、作剪力图
RC
RB
FS
3
a
8
3、作弯矩图
aa
12
M C =? q? =? qa
24
8
3a
3a 3a
9
M D = RB q?
qa 2
=
8
8 16 128
5.3 (h)
解：1、确定支座反力
1
RC = RE = (30 × 1× 2 + 20 ) = 40kN
2
RC
RE
2、作剪力图
FS
（kN）
3、作弯矩图
MC? =?30 ×1 × 0.5 =?15kN? m
M D =?30 ×1 ×1.5 + RC?1 =?5kN? m
(kN? m)
M E + =?30×1× 0.5 =?15kN? m
5-4 载荷、剪力和弯矩之间的关系
5-8 (a)试根据弯矩、剪力和载荷集度的导数关系，改正所示Q
图和M图中的错误。
a/4
FS
E
FS
qa?
a?
a 1 a 49qa
2
2a +? + qa? q =
ME =
4?
4?
424
32
2
5-4 载荷、剪力和弯矩之间的关系
5-8 (b)试根据弯矩、剪力和载荷集度的导数关系，改正所示Q
图和M图中的错误。
FS
FS
14
5.11 a) 作刚架弯矩图。
解：1、确定支座反力
2、作弯矩图
∑ M = 0，R
∑ F = 0，R
∑ F = 0，R
A
x
y
D
= 5kN
= 3kN
= 3kN
M C = 1× 3 = 3kN? m
Ax
Ay
M E = 5 × 2?1× 3 = 7kN? m
M B = 3× 4?1× 4 × 2 = 4kN? m
15
5.11 a) 作刚架弯矩图。
RC
RAx
RAy
解：1、确定支座反力
2、作弯矩图
∑F
x
= 0,q? 3a? R Ax = 0,R Ax = 3qa
C
Ay
∑ F = 0,R? R = 0
3
q
∑ M = 0,? 3a? 2 a = R
y
C
2a,RC = R Ay
Ay
9
= qa
4
16
5.11 b) 作刚架弯矩图。
MA
RA
2、作弯矩图解：1、确定支座反力
12
a qa
∑ M A = 0,qa? 2? 4? a? M A = 0,M A = 4 qa
5
qa
∑ Fy = 0,RA? qa? 4 = 0,RA = 4 qa
17
