Worked out problems with solutions in pdf files

## Magnitude and Direction of a Resultant Force

- Finding resultant force, magnitude and direction Example 1
Find the magnitude and direction of the resultant force for the figure shown below.

*Figure 2: Finding resultant force *

- Spring problem Example
For the figure shown below, F2 has a spring stiffness, k= 0.4 kN/m, with a stretched length of s = 0.5 m; F2 and F3 are regular forces. Find the magnitude of the force F3.

*Figure 3:Resultant forces with a spring *

- Moment of force Example
For the figure given below, compute the resultant moment about point A and B due to the forces acting on the structure.

*Figure 4:Finding resultant moment *

- Couple Moment Example
*Figure 5: *

- Support Reactions
*Figure 6:Reaction supports *

- Types of beams
*Figure 7:Types of beams *

- Types of loads
*Figure 8:Types of loads *

## Reactions

- Finding reactions Example 1
This simply supported beam is supported by a pin at A, a roller at B, and has a point load at the midspan, find the reactions at A and B.

*Figure 7: Simply supported beam with a point load at the midspan*

- Finding reactions Example 2
This simply supported beam is supported by a pin at A , a roller at B, and has a uniformly distributed load acting throughout the entire span, find the reactions at A and B.

*Figure 7: Simply supported beam with a uniform distributed load *

- Finding reactions Example 3
The beam is supported by a rocker at A, roller at B and is subjected to a point load and concentrated moment. Calculate the reactions (in kN).

*Figure 7:Rocker Roller beam with a point load and concentrated moment *

- Finding reactions Example 4
The beam is supported by a fixed vertical roller at A, is pulled by a tension cable at B, and has no support at C. Find the reactions at A.

*Figure 8:Fixed Vertical Roller Beam *

- Finding reactions Example 5
Find the reactions for this beam with angled supports.

*Figure 9: Beam with angled supports*

- Finding reactions Example 6
The beam is supported by a fixed horizontal roller at A and is pulled by a tension cable at B. Find the reactions at A.

*Figure 10: Fixed Horizontal Roller Beam with a vertical Tension cable*

## Trusses

Forces in a truss are transmitted axially through the joints, each member of the truss consists of a "2 force member". In order to find the internal force in a truss, 2 methods can be applied:

*Method of joints*: ideal when used to find internal forces in all members of the truss,
*Method of sections*: ideal when used to find internal forces in only some of the members of the truss (can only cut 3 members at once).

- Finding internal forces in trusses (using Method of Sections Example)
The truss shown below is supported by a pin at A and a roller at H. Compute the forces in member CE, BE and BD and indicate whether the members are in tension (T) or compression (C).

*Figure : Method of Sections Example 1*

- Finding internal forces in trusses (using Method of Joints Example)
Compute the forces in each member and indicate whether the members are in tension (T) or compression (C).

*Figure : Method of Joints Example 1*

- Finding internal forces in trusses (using Method of Sections Example)
The horizontal bowstring truss shown below is supported by a pin at A and a roller at L. Compute the forces in member DF, EF and EG and indicate whether the members are in tension (T) or compression (C).

*Figure : Bowstring truss *

- Finding internal forces in trusses (using Method of joints Example)
The horizontal bowstring truss shown below is supported by a pin at A and a roller at L. Compute the forces in each member and indicate whether the members are in tension (T) or compression (C).

*Figure : Bowstring truss *

- Finding internal forces in trusses (using Method of Sections Example)
The horizontal fink truss shown below is supported by a pin at A and a roller at G. Compute the forces in member CD, BF and BD and indicate whether the members are in tension (T) or compression (C).

*Figure : Fink truss *

- Finding internal forces in trusses (using Method of joints Example)
The horizontal fink truss shown below is supported by a pin at A and a roller at G. Compute the forces in each member and indicate whether the members are in tension (T) or compression (C).

*Figure : Fink truss *

- Finding internal forces in trusses (using Method of Sections Example)
The horizontal double howe truss shown below is supported by a pin at A and a roller at L. Compute the forces in member GI, HF and FI and indicate whether the members are in tension (T) or compression (C).

*Figure : Double Howe truss *

- Finding internal forces in trusses (using Method of joints Example)
The horizontal double howe truss shown below is supported by a pin at A and a roller at L. Compute the forces in each member and indicate whether the members are in tension (T) or compression (C).

*Figure : Double Howe Truss *

- Finding internal forces in trusses (using Method of Sections Example)
The vertical truss shown below is supported by a pin at A and a roller at B. Compute the forces in member EC, CF and DF and indicate whether the members are in tension (T) or compression (C).

*Figure : Vertical Truss *

- Finding internal forces in trusses (using Method of joints Example)
The vertical truss shown below is supported by a pin at A and a roller at B. Compute the forces in each member and indicate whether the members are in tension (T) or compression (C).

*Figure : Vertical Truss *

## Frames and Machines

- Finding internal forces in frames and machines Example 1
The structure shown below has a roller support at A and is pinned at B and D, consisting of rigid frames and pulleys. Find the reactions at A, B and D for the frames and machines if m1 = 100 kg and m2 = 150 kg.

*Figure 11: Frames and Machines Example 1 *

- Finding internal forces in frames and machines Example 2
*Figure 12: *

## Shear and Moment diagrams

Relationships between Loading, Shear and Moment diagram

*Figure : Relationships between Loading, Shear and Moment diagram *

Derivation of Shear (V(x)) and Moment (M(x)) Equations for a Simply Supported Beam With Uniform Distributed Loading (UDL)

*Figure : Derivation of Shear (V(x)) and Moment (M(x)) Equations for a Simply Supported Beam With Uniform Distributed Loading (UDL)*

- Drawing Shear (V) and Moment (M) diagrams Example 1
This simply supported beam is supported by a pin at A , a roller at B, and has a uniformly distributed load acting throughout the entire span, draw the shear and moment diagrams.

*Figure : Simply supported beam with a uniform distributed load *

- Drawing Shear (V) and Moment (M) diagrams Example 2
This simply supported beam is supported by a pin at A , a roller at B, and has a triangular distributed load acting throughout the entire span, draw the shear and moment diagrams.

*Figure : Simply supported beam with triangular UDL *

- Drawing Shear (V) and Moment (M) diagrams Example 3
This beam is fixed at A, free at C, has a uniform distributed load acting throughout half of the span, has a concentrated load and a concentrated moment, draw the shear and moment diagrams.

*Figure : Fixed beam with UDL, concentrated load and moment *

- Drawing Shear (V) and Moment (M) diagrams Example 4
This beam is supported by a pin at A, and a roller at C. It has a uniform distributed load acting throughout half of the span from B to C, and has a concentrated moment at midspan from A to B, draw the shear and moment diagrams.

*Figure : Simply Supported Beam with UDL and concentrated moment *

- Drawing Shear (V) and Moment (M) diagrams Example 5
This beam is supported by a rocker at A, and a roller at C. A triangular distributed load is distributed from A to B, and a point load halfway from B to C, draw the shear and moment diagrams.

*Figure : Rocker-roller support beam with a triangular distributed load and point load *

- Drawing Shear (V) and Moment (M) diagrams Example 6
This beam is supported by a rocker at A, is fixed at C, and has a hinge at B. A concentrated moment is applied at C, and 2 point loads acting from A to B and B to C, draw the shear and moment diagrams.

*Figure : Fixed-rocker beam with hinge at midspan, 2 point loads and a concentrated moment *

- Drawing Shear (V) and Moment (M) diagrams Example 7
Draw the shear and moment diagram for this overhang beam with a Uniformly Distributed Load (UDL) and a point load as shown below.

*Figure 13:Overhang beam with a Uniformly Distributed Load (UDL) and a point load *

- Drawing Shear (V) and Moment (M) diagrams Example 8
The beam is fixed at A, has a hinge at B and has a roller support at C. Draw the shear and moment diagram for this beam.

*Figure 14:Fixed Roller Beam With Hinge *

- Drawing Shear (V) and Moment (M) diagrams Example 9
Draw the shear and moment diagram for this overhang beam as shown below.

*Figure 15:Overhang Beam With Uniformly Distributed Load (UDL) *

- Drawing Shear (V) and Moment (M) diagrams Example 10
The beam is fixed at A, has hinges and continuous supports.Draw the shear and moment diagram for the beam.

*Figure 16: Fixed Continuous Beam With Hinges *

## Friction

- Solving a friction Example 1
*Figure 17: *

- Solving a friction Example 2
*Figure 18: *

## Inertia

- Calculating inertia Example 1
Compute the moment of inertia about the centroid (Ix) for the hollow rectangular section given below.

*Figure 19:Finding the inertia for a rectangular hollow section *

- Calculating inertia Example 2
Compute the moment of inertia about the centroid (Ix) for the T-shape section given below.

*Figure 20:Finding the inertia for a T-shape beam *

- Calculating inertia Example 3
Compute the moment of inertia about the centroid (Ix) for the channel section given below.

*Figure 19:Finding the inertia for a rectangular hollow section *