Alexandria University ‫جامعة اإلسكندرية‬

Faculty of Engineering ‫كلية الهندسة‬

Structural Engineering Department ‫قسم الهندسة اإلنشائية‬
Mechanics of Structures ‫مادة ميكانيكا االنشاءات‬
Fall 2022 2022 ‫خريف‬

Solution of Assignment No.1

1 – (I)
90 cm
y
C2 15 cm

x C x
52.5 cm C1 45 cm
y = 40.5 cm
22.5 cm

X y X
20 cm

A1 = 20 * 45 = 900 cm2

A2 = 90 * 15 = 1350 cm2

A = 900 + 1350 = 2250 cm2

∑ Marea @ X-X

900 * 22.5 + 1350 * 52.5 = 2250 y̅

y̅ = 40.5 cm
20 * 453 90 * 153
Ix = + 900*( − 18)2 + + 1350*122 = 663187.5 cm4
12 12

45 * 203 15 * 903
Iy = + = 941250 cm4
12 12

Iz = 663187.5 + 941250 = 1604437.5 cm4

rx = 17.17 cm, ry = 20.45 cm, rz = 26.70 cm

1
(II)

2 cm
y

1 cm 60 cm
x C x

y
32 cm

A1 = 32 * 60 = 1920 cm2

A2 = 30 * 56 = 1680 cm2

A = 1920 – 1680 = 240 cm2

32 * 603 30 * 563
Ix = – = 136960 cm4
12 12

60 * 323 56 * 303
Iy = – = 37840 cm4
12 12

I‫ = ئ‬136960 + 37840 = 174800 cm4

rx = 23.89 cm, ry = 12.56 cm, rz = 26.99 cm

2
(III)
y

C2

14 mm

C
x C1 x 450 mm

26 mm C2

y
300 mm y

A1 = 14 * 398 = 5572 mm2

A2 = 300 * 26 = 7800 mm2

A = 5572 + 2 * 7800 = 21172 mm2

14 * 3983 300 * 263
Ix = + 2∗ + 2 ∗ 7800 ∗ 2122 = 775.6 * 106 mm4
12 12

398 * 143 26 * 3003

Iy = + 2∗ = 117.1 * 106 mm4
12 12

Iz = 775.6 * 106 + 117.1 * 106 = 892.6 * 106 mm4

rx = 191.4 mm, ry = 74.4 mm, rz = 205.3 mm

Another Solution:
3
 y

14 mm

C
C2 x C1 x
C2 450 mm

26 mm
y
300 mm
y

A1 = 300 * 450 = 135000 mm2

A2 = 143 * 398 = 56914 mm2

A = 135000 – 2 * 56914 = 21172 mm2

300 * 4503 143 * 3983
Ix = − 2∗ = 775.6 * 106 mm4
12 12

450 * 3003 398 * 1433

Iy = − 2∗ − 2 ∗ 56914 ∗ 78.52 = 117.1 * 106 mm4
12 12

(IV)

4
 x = 29.49 mm
Y y

10 mm
300 mm x C C1 C2 x

16 mm

Y y

100 mm

A1 = 100 * 300 = 30000 mm2

A2 = 90 * 268 = 24120 mm2

A = 30000 – 24120 = 5880 mm2

∑ Marea @ Y-Y

30000 * 50 − 24120 * 55 = 5880 x̅

x̅ = 29.49 mm
100 * 3003 90 * 2683
Ix = − = 80.6 * 106 mm4
12 12

300 * 1003 268 * 903

Iy = + 30000 ∗ 20.512 − − 24120 ∗ 25.512 = 5.64 * 106 mm4
12 12

Iz = 80.6 * 106 + 5.64 * 106 = 86.3 * 106 mm4

rx = 117.1 mm, ry = 31.0 mm, rz = 121.1 mm

(V)

5
 8m
x = 4.4 m
4m
0.576 m

Y y
1m

x C1 C2 C C3 x 1m

Y y
1m 6m 3m

π * 12
A1 = = 1.571 m2
2

A2 = 6 * 2 = 12 m2
1
A 2 = * 3 * 2 = 3 m2
2

A = 1.571 + 12 + 3 = 16.571 m2

∑ Marea @ Y-Y

1.571 * 0.576 + 12 * 4 + 3 * 8 = 16.571 x̅

x̅ = 4.40 m
π * 14 6 * 23 3 * 23
Ix = + + = 4.89 m4
8 12 48

𝜋 8 2 * 63 2 * 33
Iy = ( – ) ∗ 14 + 1.571 ∗ (−3.824)2 + + 12 ∗ (−0.4)2 + + 3 ∗ 3.62
8 9π 12 36

= 101.38 m4

Iz = 4.89 + 101.38 = 106.28 m4

rx = 0.543 m, ry = 2.473 m, rz = 2.533 m

(VI)
6
 y 16mm plate

2 UPN300

x x
300 mm

y
154 mm 73 mm

A = 2 * 5880 + 2 * 300 * 16 = 21360 mm2

300 * 163
Ix = 2 ∗ 80.3 ∗ 106 + 2 ∗ + 2 ∗ 4800 ∗ 1582 = 400.5 * 106 mm4
12

16 * 3003
Iy = 2 ∗ 3.95 ∗ 106 + 2 ∗ 5880 ∗ 772 + 2 ∗ = 149.6 * 106 mm4
12

Iz = 400.5 * 106 + 149.6 * 106 = 550.1 * 106 mm4

rx = 136.9 mm, ry = 83.7 mm, rz = 160.5 mm

(VII)

7
 x = 9.03 mm
Y y
IPE600

UPN300
x x

Y y

33 mm

A = 15600 + 5880 = 21480 mm2

∑ Marea @ Y-Y

15600 * 0 + 5880 * 33= 21480 x̅

x̅ = 9.03 mm

Ix = 920.8 * 106 + 80.3 * 106 = 1001.1 * 106 mm4

Iy = 33.87 * 106 + 15600 * (–9.03)2 + 3.95 * 106 + 5880 * 23.972 = 42.5 * 106 mm4

Iz = 1001.1 * 106 + 42.5 * 106 = 1043.6 * 106 mm4

rx = 215.9 mm, ry = 44.5 mm, rz = 220.4 mm

2 – (I)

8
 y y20 mm

2
200 mm C2 45°
C
110 mm x x
20 mm
y = 57.37 mm
10 mm C1
X Y X
y
10 mm

x = 57.37 mm

100 mm

200 mm

A1 = 200 * 20 = 4000 mm2

A2 = 20 * 180 = 3600 mm2

A = 4000 + 3600 = 7600 mm2

∑ Marea @ X-X

4000 * 10 + 3600 * 110 = 7600 y̅

y̅ = 57.37 mm

∑ Marea @ Y-Y

4000 * 100 + 3600 * 10 = 7600 x̅

x̅ = 57.37 mm
200 * 203 20 * 1803
Ix = + 4000 * ( − 47.37)2 + + 3600 * 52.632 = 28.8 * 106 mm4
12 12

20 * 2003 180 * 203

Iy = + 4000 * 42.632 + + 3600 * ( − 47.37)2 = 28.8 * 106 mm4
12 12

Ixy = 0 + 4000 ∗ −47.37 * 42.63 + 0 + 3600 * 52.63 * ( − 47.37) = –17.1 * 106 mm4
−2∗ –17.1∗ 106
tan (2α) = =∞
28.8 ∗ 106 − 28.8 ∗ 106

α1 = 45o

28.8 ∗ 106 + 28.8 ∗ 106 28.8 ∗ 10 6 − 28.8 ∗ 106 2

I1 = + √( ) + (– 17.1 ∗ 106 )2 = 45.9 * 106 mm4
2 2
9
 28.8 ∗ 106 + 28.8 ∗ 106 28.8 ∗ 10 6 − 28.8 ∗ 106 2
I2 = − √( ) + (– 17.1 ∗ 106 )2 = 11.7 * 106 mm4
2 2

Iz = 28.8 * 106 + 28.8 * 106 = 57.6 * 106 mm4

r1 = 77.7 mm, r2 = 39.2 mm, rz = 87.1 mm

Another Solution:

y 20 mm
y

200 mm C1 C2

C
110 mm 100 mm x x
20 mm
y = 57.37 mm

X Y X
y

x = 57.37 mm

100 mm

110 mm

200 mm

A1 = 2002 = 40000 mm2

A2 = 1802 = 32400 mm2

A = 40000 – 32400 = 7600 mm2

∑ Marea @ X-X

40000 * 100 – 32400 * 110 = 7600 y̅

x̅ = y̅ = 57.37 mm
2004 1804
Ix = Iy = + 40000 * 42.632 − − 32400 * 52.632 = 28.8 * 106 mm4
12 12

Ixy = 0 + 40000 ∗ 42.63 * 42.63 − 0 − 32400 * 52.63 * 52.63 = –17.1 * 106 mm4

(II)

10
 133.2 mm
y

2 L200

x
x 133.2 mm
C
-45°

A = 2 * 7630 = 15260 mm2

Ix = 2 * 28.51 * 106 + 2 * 7630 * 66.62 = 124.7 * 106 mm2

Iy = 2 * 28500000 + 2 * 7630 * 66.62 = 124.7 * 106 mm2

Ixy = 2 * –16.8 * 106 + 2 * 7630 * 66.6 * 66.6 = 34.1 * 106 mm4

−2∗ 34.1 ∗ 106
tan (2α) = =–∞
124.7 ∗ 106 − 124.7 ∗ 106

α1 = – 45o

124.7 ∗ 106 + 124.7 ∗ 106 124.7 ∗ 106 − 124.7 ∗ 106 2

I1 = + √( ) + (34.1 ∗ 106 ) 2 = 158.8 * 106 mm4
2 2

124.7 ∗ 106 + 124.7 ∗ 106 124.7 ∗ 106 − 124.7 ∗ 106 2

I2 = − √( ) + (34.1 ∗ 106 ) 2 = 90.62 * 106 mm4
2 2

Iz = 158.8 * 106 + 90.62 * 106 = 294.4 * 106 mm4

r1 = 102.0 mm, r2 = 77.1 mm, rz = 127.8 mm

(III)

11
 y Y 1

-41.24°
C3 40 cm

y = 4.689 cm

X X
x C x 80 cm

2
y Y
x = 4.019 cm

A1 = 1202 = 14400 cm2

A2 = π * 202 = 1256.64 cm2

1
A3 = * 60 * 40 = 1200 cm2
2

A = 14400 – 1256.64 – 1200 = 11943.36 cm2

∑ Marea @ X-X

14400 * 0 – 400π * 0 – 1200 * 46.67 = 11943.36 y̅

y̅ = – 4.689 cm

∑ Marea @ Y-Y
12
14400 * 0 – 400π * 0 – 1200 * 40 = 11943.36 x̅

x̅ = – 4.019 cm
1204 𝜋 ∗ 204 60∗403
Ix = + 14400 * 4.6892 − − 400π ∗ 4.6892 − − 1200 ∗ 51.3552
12 4 36

= 14.17 * 106 cm4

1204 𝜋 ∗ 204 40∗603
Iy = + 14400 * 4.0192 − − 400π ∗ 4.0192 − − 1200 ∗ 44.0192
12 4 36

= 14.80 * 106 cm4

602 ∗402
Ixy = 0 + 14400 * 4.689 ∗ 4.019 − (0 + 400π ∗ 4.689 ∗ 4.019 + (− ) + 1200 ∗
72
51.355 ∗ 44.019) = –2.39 * 106 cm4

tan (2α) = –7.5757

α2 = – 41.24o

I1 = 16.89 * 106 cm4

I2 = 12.08 * 106 cm4

Iz = 14.17 * 106 + 14.80 * 106 = 28.97 * 106 cm4

r1 = 37.61 cm, r2 = 31.80 cm, rz = 49.25 cm

(IV)

13
 -28.53°

C2 y
1

C C1
x x

y
2
C3

A1 = 100 * 10 = 1000 cm2

A2 = 10 * 50 = 500 cm2

A3 = 10 * 50 = 500 cm2

A = 2000 cm2
100 * 103 10 * 503
Ix = + 2∗ + 2 ∗ 500 ∗ 302 = 111.67 * 104 cm4
12 12

10 * 1003 50 * 103
Iy = + 2∗ + 2 ∗ 500 ∗ 452 = 286.67 * 104 cm4
12 12

Ixy = 0 + 0 + 500 * 30 * (–45) + 0 + 500 * (–30) * 45 = –135 * 104 cm4

−2∗ –1350000
tan (2α) = = –1.5429
1116667 − 2866667

α2 = – 28.53o

I1 = 360.04 * 104 cm4

I2 = 38.29 * 104 cm4

Iz = 111.67 * 104 + 286.67 * 104 = 398.33 * 104 cm4

r1 = 42.4 cm, r2 = 13.8 cm, rz = 44.6 cm

14