Dep.: CIVIL ENGINEERING

FACULTY OF CONSTRUCTIONS Dep.: CIVIL ENGINEERING

HOMEWORKS

Homework no. 1: Graphical Constructions

Homework no. 2: The Point

1. Represent in orthographic representation on two planes of projection, the folowing points: A(70,20,40); B(55,-20,45); C(45,10,-30); D(10,0,35); E(15,0,0); F(65,40,0); G(- 40,20-25); I(-50,-35,-10); K(25,-30,-50).

Obs.: Specify the space position of the points (the dihedral angles).

2. Represent in orthographic representation on three planes of projection, the folowing points: A(40,15,10); B(-30,20,5); C(-10,-25,-25); D(20,5,-30).

Obs.: Specify the space position of the points (the trihedral angles).

3. Represent the projections of the triangle ABC, on three planes of projection, knowing that the vertex A lies on the bisecting plane BI, the vertex B is situated in the quadrant III and the vertex C belongs to the horizontal plane [H].

Obs.: Specify the coordinates of the points.

4. One gives the point M(10,30,20). One requires to represent on three planes of projection the folowing points: - the point M; - the symmetrical M1 of the point M side [H]; - the symmetrical M2 of the point M side [V]; - the symmetrical M3 of the point M side [W].

Homework no. 3: The straight line

1. One gives the line (D) defined by the points A and B. One requires to find:

- The traces of the line: H(h,h’,h”), V(v,v’,v”) and W(w,w’,w”); - The regiones passed by the line.

A(15, 5,65); B(80,50,10).

2. One gives the straight line (D) defined by the points A and B, respective the point M(m,m’) not lying on that line. One requires to construct through the point M(m,m’), a horizontal line (G) and a frontal line (F) intersecting with (D). One works on three projections. M(10,40,10); A(-30,20,10) and B(25, 5,40).

3. One gives the straight line (D) defined by the points A and B, respective the point M(m,m’) not lying on that line. One requires to construct through the point M(m,m’), a line (Δ1) parallel to (D) and a line (Δ2) non intersecting with (D). M(25,20,20); A(50,30,15) and B(15, 50,40).

4. Represent in three projections all the types of particular segment-lines having the true length 30 mm.

5. Determine the true length of the segment line (AB) and the angles α and β between the line and the projection planes. A(50,10,25); B(20,20,10). Homework no. 4: THE PLANE

1. One gives three points (not lying on a line): A(40,14,25); B(28,6,40); C(45,6,35). One requires to determine the traces P,P,’P” of the plane defined by the previous points and the maximum angle α between the plane [P] and the horizontal projection plane [H].

2. Determine the line of intersection between two opaque plates: the plate [ABC] situated in a horizontal plane (or [H] parallel) having the coordinate +45 mm, respectively the oblique (skew) plate [MNP]. Study also, the visibility of the plates. A(103, 80, 45); B(55, 10, 45); C(15, 100, 45); M(110, 20, 30); N(60, 105, 90); P(15, 50, 10).

3. Determine the intersection (piercing) point between the straight line D(d,d') defined by the points M and N and the triangular plate [ABC]. Study the visibility of the line after the intersection.

A(15, 15, 35); B(50, 55, 60); C(85, 25, 10); M(18, 47, 15); N(70, 15, 48).

4. Determine the line of intersection between the plates [ABCD] and [PQR], using auxiliary projecting planes. Study also the visibility of the plates.

A(90, 10, 10); B(25, 5, 5); C(5, 40, 55); D(55, 45, z); P(80, 5, 40); N(30, 55, 0); C(5, 10, 65);

Not ă : All the plates are opaque. Homework no. 5: THE METHODS OF D.G. – The substitution of projection planes. The revolution.

1. Using the substitution of projection planes determine the true length of the line segment AB and the angles α and β between this line segment and the projection planes [H] and [V].

A(50, 5, 25); B(5, 35, 7).

2. Using the substitution of projection planes determine the true size of the angle between the intersecting lines (D) and (Δ) defined by (AC) and (BC).

A(48, 5,10); C(30,27,27); B(5,15, 10).

3. One gives the plane [P] by its traces: Px(80, 0, 0),

4. Using the method of revolution find the true length of the distance from the point M to the straight line (AB).

A(10, 25, 5); B(50, 5, 35); M(25, 50, 45).

5. Using the method of revolution, determine the true length of the distance between the parallel planes [P] and [Q] given by their traces: Px(15,0,0), the angle OPxP=135o, the angle OPxP’=150o, Qx(30,0,0). Homework no. 6: THE METHODS OF D.G. – The coincidence. The restouration.

1. Using the coincidence with a frontal plane determine the true size of the angle between the intersecting lines (D1) and (D2).

2. One gives the staigth line (D), defined by the points A and B, respectively a point M, not lying on that line. One requires to determine the distance from the point M to the straight line (AB).

A(10,8,12), B(65,27,35), M(30,30,26).

3. Determine the true length of the distance between two oblique parallel lines (D) and (Δ), given by their projections.

d' '

4. Represent a regular hexagon with the sides 25 mm contained in a vertical projecting plane, knowing that two sides are in frontal position.

Rx(60,0,0); the angle ORxR=90°, the angle ORxR’=45°.

5. Represent the projections of a cirle with the radius r=30 mm, contained in a horizontal projecting plane [Q] which have the angle b = 30o with [V]. Homework no. 7: THE AXONOMETRIC PROJECTIONS

1. Represent in isometric orthogonal axonometric projection the solids given in orthographic representation: 0 1 5 1 5 5 5 1 1 1 = = = r r r 0 6 0 3 10 30 10 30 10 30 10

x O 5 5 0 2 5 1 5

20 20 20 20 20 20 20

5 1.30 5

2. Represent in isometric oblique frontal axonometry the solids given in orthographic representation:

50 10 30 10 0 1 5 5

r= 25 5 =2 r=35 r 5 x r=3 O 0 2 5 4 5 1 0 1

35 50 35 3. Represent in dimetric orthogonal axonometry the solids given in orthographic representation:

66 20 30 16 0 1 5 2 0 6 10 2.5 r=1 5 2 0 x 1 O 7 1 0 6 5 1 7 1

4. Represent in isometric oblique horizontal axonometry the solids given in orthographic representation:

20 30 30 10 0 1

5 z z z z 2 5 1 0

10 15 20 5 Scara 1,5:1 3

Scara 2:1 5 1 0

z 15 5 30 10 z 1

0 R 0 1 1 5 5 2 1 1 R 5 35 0 5 35 R R5 5 5 5 5 1 1 1 0

O 1 O O O O 5

5 x x x x x 1

R 5 5 0 0 1 5 2 0 0 2 2 5 R 5

3 3 5 1 y 0 2 0 0

1 O 1 x 5 Scara 2:1 5 20 5 1 0 0 1 1 5 z 5 R40 1 R

1 0 1 0 1 5

0 10 25 10 30 10 15 10 0 2 1 5 3 0

O 1 x 5 1 10 20 15 25 10 0 1 5 5 y y y y 1

R 5 10 R1 5 0 2 y 5 15 15 5 10 5 15 15 5 5 y

z 10 5 10 5 10 60 0 1 0 1 0 1 5 x 1 O y 1

z R 2 5 5 2 x O 0 1 0 1 0 1

25 25 25 y Homework no. 8: POLYHEDRONS. Representation, plane sections with projecting planes and develpoments.

1. Represent a cube (hexahedron) with the side 35 mm, having a face contained in a vertical projecting plane [R]: Rx(60, 0, 0), the angle ORxR’=45.

2. One gives a right pyramid having as base a regular pentagon inscribed in a circle with the radius r=25 mm, situated in the plane [H]. The altitude of the pyramid is 75 mm. One requires the true size of the plane section in pyramid with a vertical projecting plane [Q] and the development of the pyramid frustum between the cutting plane and the base.

3. One gives the oblique pyramid ABCDV: A(80,25, 0); B(60,45, 0); C(40,20, 0); D(75, 7, 0); V(15,50,50). One requires the true size of the plane section in pyramid with a vertical projecting plane [P] and the development of the truncated pyramid between the cutting plane and the base. Px(25, 0, 0);

4. One gives the frontal prism ABCDA1B1C1D1. One requires to determine the plane section with a plane [P] perpendicular to the prism edges and the development of the prism portion between the cutting plane and the base. A(85, 7,0); B( 101,26, 0); C(78,46, 0); D(66,33, 0); A1(7, 7,66); Px(13, 0, 0). Homework no.9: POLYHEDRONS. Plane sections and develpoments. The intersection with a straight-line.

1. Determine the plane section with the oblique plane [P] in the pyramid ABCS and the development of the pyramid frustum between the cutting plane and the base.

A(83, 51, 0); B(67, 21, 0); C( 40, 30, 0); S( 23, 75, 63); Px(13, 0, 0);

2. One gives the quadrilateral right prism with the base ABCD contained in [H], having the altitude 65 mm. One requires to find the plane section with the oblique plane [P] in prism, the true size of the section and the development of the prism portion between the cutting plane and the upper base (inclusively the bases).

A(85, 28, 0); B( 50, 50, 0); C(36, 30, 0); D(65, 5, 0); Px(10, 0, 0);

3. One gives the oblique pyramid SABC with the base ABC contained in the plane [H]: A(125,45,0); B(62,80,0); C(90,8,0); S(7,40,75). Determine the points of intersection (piercing points) between the pyramid and the segment line (MN). Study also, the visibility of the line. M(105,70,55); N(34,20, 5).

4. One gives the triangular oblique prism ABCMNP with the base ABC lying in [H]. Determine the plane section in prism with the oblique (skew) plane [P], the true size of the section and the development of the prism portion between the cutting plane and the lower base (inclusively the bases). A(15, 10, 0); B( 20,30, 0); C(40,15, 0); M(65,40,50); Px(80, 0, 0);

1. One considers a cone of revolution having the base a circle with radius R=30 mm, contained in the plane [V], the vertex S(s,s’) and the altitude 70 mm (as perpendicular line to [V]). Determine the true size of an elliptical section by cutting the cone with a horizontal projecting plane [Q] and the development of the cone frustum between the cutting plane and the base (inclusively the bases).

2. One gives a cone of revolution having the base a circle with radius R=30 mm, contained in the plane [H], the vertex V(v,v’) and the altitude 70 mm. Cutting the cone with a vertical projecting plane [P], one requires the true size of a parabolical section in cone and the development of the truncated cone between the plane [P] and the base (inclusively the bases).

3. Determine the true size of a plane section with a vertical projecting plane [P] in the cylinder of revolution, having the base a circle with radius R=30 mm, contained in the plane [H], and the altitude 75 mm. Represent also the development of the truncated cylinder between the plane [P] and the base (inclusively the bases).

4. One gives the right cylinder having the base a circle lying in [H] with the centre (85,30,0), the radius R=25 mm and the altitude 70 mm. One requires to find the plane section with the oblique plane [P], the true size of the section and the development of the cylinder portion between the cutting plane and the lower base (inclusively the bases).

Px(0, 0, 0);

5. One considers the oblique plane, having the directrix a circle lying in [H], with radius r=20 mm, the centre (40,30,0) and the vertex S(0,50,50). One requires the true size of a section with the vertical plane [Q] (horizontal projecting plane) given by its traces and the development of the lateral surface of the cone.

Qx(-15,0,0);

1. Determine the intersection edges of the roof slopes (sides) given in the Figures, knowing that: - the inclinations of the slopes are: 30° (for the roof no.1), 45° (for the roof no.2), 60° (for the roof no.3) şi 50° (for the roof no.4). - the drawing scale is 1:100.

2. Represent the developments of three slopes (arbitrary selected by the students) for the roof no.4 and two views for the roof no.3.

9.00 6.00 3.00 8.00 3.00 2.00 0 0 0

. 90° 0 5 . 6 0 0 . 7 ° 0 9 1 4 0 0 0 . 0 7 . 6 0 0 . 5

4 5 °

4.00 11.00 5.00 9.00 6.00

6.00 6.00 3.00 0 0 . 3 0 0 . 6.00 10.00 4

2 0 0 . 4 0 0 . 4 0 0 . 5 4.00 6.00 0 0 . 2

4.00 4.00 7.00 0 0 . 5 0 0 0 0 . . 4 4

4.00 4.00 5.00 1 0 0 . 3 0 0 . 4 5 0 0 0 0 . .

3 5 5 0 0 0 0 . . 4 2

9.00 5.00 2.00 0 0 . 1 3.00 10.00 Homework no. 12: PROJECTIONS WITH ELEVATIONS. Solving the platforms (construction sites).

1. Determine the boundary lines (intersection lines) of the embankments for a horizontal platform and for the acces road of the construction site given by its contour lines. The slopes of the embankments are: - for the cut = 1/1 - for the fill = 2/3 - the slope of the acces road is: 1/5 The drawing scale is 1:200. Represent also a longitudinal section in the topographic surface with the frontal plane [F].

8 7 6 9 5 4 3 9 9 9 9 9 9 9

0 0 6.00 11.00 1 0 0 0 0 . . 6 6

+100.00 1 0 1 F 0 F 0 . 0 9 8 7 6 6 0 9 9 9 9 1 0 0 . 2 1 0 0 2 . 0 6 1

10.00 7.00 3 0 1

5 0 1 2. Determine the boundary lines of the embankments for a horizontal platform and for the acces road of the construction site given by its contour lines. The slopes of the embankments are: - for the cut = 1/1 - for the fill = 2/3 - the slope of the acces road is : 1/7 The drawing scale is 1:200.

4 3 2 0 5 5 5 51 5

49 21.50

48 R =7 .50 m +50.00

7.50 8.00 6.00 46