Worked out problems with solutions in pdf files

## Allowable Load (P_{allow}),Factor of Safety (FS), Axial Stresses/Elongation, and Strain

- Finding internal stress in an axial member Example
A tension test is performed on a circular cylinder shaped steel specimen as shown below, compute the axial stress in the member (in MPa).

*Figure 2:Tension test on a circular cylinder shaped steel specimen *

- Computing strain, stress and modulus of elasticity in an axial member Example
A specimen has a length of 25 mm and a cross sectional area of 30 mm^{2}, and is subjected to an axial load of 3000 N as shown below. If the specimen is stretched by 2 mm and is in the elastic region), compute the following:

(a) the axial stress (σ),

(b) the strain (ε) and,

(c) the modulus of elasticity (E).

*Figure 2: Axial member *

- Computing average shear strain, average shear strain and relative displacement Example
*Figure 3: *

- Finding allowable average shear stress Example
Find the allowable average shear stress (Τ_{allow}) for the bolted joint lap with diameter given (d_{b} = 10 mm), if the allowable load is 2 kN (P_{allow} = 2 kN).

*Figure 4: Shear in bolted joint*

- Finding allowable load Example
Find the allowable load (P_{allow}) for the bolted joint lap with diameter given (d_{b} = 0.5 in), if the allowable average shear stress is 23 ksi (Τ_{allow} = 23 ksi).

*Figure 5: Shear in bolted joint*

- Finding Allowable Forces/Stresses using Factor of Safety (FS) Example
*Figure 6: *

- Finding axial elongation in an axial structure Example 1
The axial structure depicted below consists of circular steel members of different areas. Compute the following:

(a) the stresses in each member, and indicate whether the stresses are tensile (T) or compressive (C),

(b) the axial displacement at B (U_{B}), at C (U_{C}) and at D (U_{D}).

*Figure 7: Axial displacement*

- Finding axial stress and elongation in an axial member Example
*Figure 8: *

- Finding compressive axial stress and displacement in a steel pipe column Example
*Figure 9: *

- Finding internal forces and displacement in a statically indeterminate structure (axial) Example
*Figure 10: *

## Torques, Shear Stress and Relative Rotation in a Shaft

- Finding internal torques, maximum shear stress and angle of rotation in a hollow shaft Example (statically determinate)
*Figure 11: *

- Finding internal torques, maximum shear stress, and relative rotation in a shaft Example 1 (statically determinate)
A uniform steel shaft with diameter of 5 mm is depicted as shown below to transmit torques from gear A to B to C and with a modulus of rigidity (G = 20 GPa). Determine:

(a) the maximum shear stress in element 1 and,

(b) the maximum shear stress in element 2

*Figure 12: Steel Shaft *

- Finding internal torques, maximum shear stress, and relative rotation in a shaft Example 2 (statically determinate)
*Figure 13: *

- Finding internal torques, maximum shear stress, and relative rotation in a shaft Example 1 (statically indeterminate)
*Figure 14: *

- Finding internal torques, maximum shear stress, and relative rotation in a shaft Example 2 (statically indeterminate)
*Figure 15: *

## Internal Forces, Shear Force Diagram (SFD) and Bending Moment Diagram (BMD) Review

- Finding internal shear and moment at a point (in a beam)
*Figure 16:*

- Deriving shear and moment equation in a simply supported beam with triangular Uniform Distributed Load (UDL)
*Figure 17:Simply Supported Beam with Triangular UDL *

- Drawing shear and moment diagrams Review 1
Draw the shear (V) and moment (M) diagram for this overhang beam with a triangular distributed load, point load and concentrated moment.

*Figure 18: Overhang beam with a triangular distributed load, point load and concentrated moment.*

- Drawing shear and moment diagrams Review 2
Draw the shear (V) and moment (M) diagram for this overhang beam.

*Figure 19: Overhang beam with rectangular distributed loads and a point load*

- Strain-displacement analysis for beams
*Figure 20: *

- Finding maximum tensile stress, radius of curvature and maximum deflection of a beam
*Figure 21: *

## Normal/Flexural/Bending Stress, Shear Stress and Shear Flow Distribution in a Beam

**Normal and Shear Stress Distribution in a rectangular beam**

*Figure 22: Normal and Shear Stress Distribution in a rectangular beam*

- Finding flexural stresses in beams Example 1
A fixed wide-flange beam has a uniform distributed loading acting throughout half of the span, a point load and concentrated moment. Cross-section dimensions of the beam is given. Compute the maximum flexural tensile and compressive stresses.

*Figure 23: Fixed I-beam with UDL, concentrated load and moment *

- Finding flexural stresses in beams 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. Cross-section dimensions of the beam is given. Compute the maximum flexural tensile and compressive stresses.

*Figure 24: Simply supported beam with triangular distributed loading*

- Finding flexural stresses in beams Example 3
This overhang beam is supported by a pin at A , a rocker at B, and has a uniform distributed load acting in between the supports. Cross-section dimensions of the beam is given. Compute:

(a)the flexural/bending stress at C, D, E and F;

(b)compute the flexural/bending stress and draw the stress distribution at the left support (A) ,

(c) from part (b), what are the maximum flexural tensile and compressive stresses at support A

(d) what are the maximum flexural tensile and compressive stresses that occurs throughout the entire span of the beam?

*Figure 25: Overhang beam with a UDL, point load and concentrated moment *

- Finding maximum compressive and tensile flexural stresses in beams
*Figure 26: *

- Design problem: choosing lightest wide flange beam given allowable stress
*Figure 27: *

- Finding maximum flexural stress and maximum shear stress in a beam
*Figure 28: *

- Finding shear stress at different locations in an inverted C channel shape
Compute the shear stress, τ, at points A, B, C, D and E for the figure shown below.Then, draw the shear stress and shear flow distribution. Also, indicate where the maximum shear stress occurs (V = 25 kN).

*Figure 29: Inverted C-Channel Section *

- Finding shear stress at different locations in a C channel shape
Compute the shear stress, τ, at points A, B, C, and D for the figure shown below.Then, draw the shear stress and shear flow distribution. Also, indicate where the maximum shear stress occurs (V = 65 kN).

*Figure 30: C-Channel Section *

- Finding shear stress at different locations in an unsymmetrical I shape beam and drawing shear stress distribution
Compute the shear stress, τ, at points A, B, C, D and E for the figure shown below.Then, draw the shear stress and shear flow distribution. (V = 50 kN)

*Figure 31: Unsymmetrical Wide Flange Beam*

- Finding maximum shear stress in a hollow circle and drawing shear flow distribution
Compute the maximum shear stress, τ_{Max}, for the tubular steel beam as shown below. Then, draw the shear flow distribution(q).(V = 75 kN).

*Figure 32: Tubular Steel Beam *

- Finding shear stress at different locations in a hollow rectangular shape beam and drawing shear stress distribution
Compute the shear stress, τ, at points A, B, C, D and E for the figure shown below.Then, draw the shear stress and shear flow distribution. (V = 80 kN)

*Figure 33: Hollow Rectangular Shaped Beam*

- Finding shear stress at different locations in a T shape beam
Compute the shear stress, τ, at points A, B, C and D for the figure shown below. Then, draw the shear stress and shear flow distribution. Also, indicate where the maximum shear stress occurs (V = 40 kN)

*Figure 34: T beam *

- Given allowable shear stress in a T shape beam, find maximum load intensity(Wo) and drawing shear flow/shear stress/flexural stress distribution
*Figure 35: *

- Finding allowable shear given allowable shear flow in a symmetrical I shape beam and drawing shear flow distribution
*Figure 36: *

- Boundary Conditions
*Figure 37:*

- Continuity Conditions
*Figure 38: *

## Deflection, Slope, Curvature Equations (Double Integration Method, Euler Bernoulli Beam Equation), Deflection Shape and Superposition Method

- Drawing the deflection shape
Draw the deflection shape and indicate where the maximum deflection will occur for this simply supported beam with Uniformly Distributed Load (UDL) as shown below.

*Figure 39: Simply supported beam with Uniformly Distributed Load (UDL)*

- Finding slope and deflection in a cantilever beam with triangular distributed load
Determine:

(a)the equation of the slope, Θ(x), and deflection, v(x), and

(b)the maximum deflection, v_{Max} and draw the deflected shape,

for the fixed beam with triangular distributed loading as shown below.

*Figure 40: Fixed Beam With Triangular Distributed Loading*

- Finding slope, deflection equation and maximum deflection in a simply supported beam with triangular distributed load
Determine:

(a)the equation of the slope, Θ(x), and deflection, v(x), and

(b)the maximum deflection, v_{Max}, and draw the deflected shape,

for the simply supported beam with triangular distributed loading as shown below.

*Figure 41: Simply Supported Beam With Triangular Distributed Loading *

- Finding deflection equation and reactions in a propped cantilever beam with rectangular distributed load
Using superposition method, determine:

(a)the reactions,

(b)the equation of the slope, Θ(x), and deflection, v(x), and

(c) the location where the maximum deflection occurs (in meters) and the maximum deflection (in mm) if E = 200 GPa and I = 600 x 10^{5} mm^{4},

(d) draw the shear, moment diagrams and the deflection shape

for the propped cantilever beam with rectangular distributed loading as shown below.

*Figure 42: Propped Cantilever Beam with Rectangular Distributed Load*

- Finding deflection equation and reactions in a continuous beam with rectangular distributed load
Using superposition method, determine:

(a)the reactions,

(b)the equation of the slope, Θ(x), and deflection, v(x), and

(c) the location where the maximum deflection occurs (in meters) and the maximum deflection (in mm) if E = 200 GPa and I = 450 x 10^{5} mm^{4},

(d) draw the shear, moment diagrams and the deflection shape

for the continuous beam with rectangular distributed loading as shown below.

*Figure 43: Continuous Beam with Rectangular Distributed Load*

- Finding deflection equation and reactions in a double propped beam with triangular distributed load
Using superposition method, determine:

(a)the reactions,

(b)the equation of the slope, Θ(x), and deflection, v(x), and

(c) the location where the maximum deflection occurs (in meters) and the maximum deflection (in mm) if E = 200 GPa and I = 700 x 10^{5} mm^{4},

(d) draw the shear, moment diagrams and the deflection shape

for the cantilever double propped beam with triangular distributed loading as shown below.

*Figure 44: cantilever double propped beam with triangular distributed load*

- Finding deflection equation and reactions in a fixed-fixed beam with rectangular distributed load
Using superposition method, determine:

(a)the reactions,

(b)the equation of the slope, Θ(x), and deflection, v(x), and

(c) the location where the maximum deflection occurs ,

(d) draw the shear, moment diagrams and the deflection shape

for the fixed-fixed beam with rectangular distributed loading as shown below.

*Figure 45: Fixed-fixed beam with rectangular distributed load*

- Deflections and slopes of beams;Fixed-End Actions page 1
*Figure 46: *

- Deflections and slopes of beams;Fixed-End Actions page 2
*Figure 47: *

- Deflections and Slopes of Simply-Supported Uniform Beams page 1
*Figure 48: *

- Deflections and Slopes of Simply-Supported Uniform Beams page 2
*Figure 49: *

- Fixed-End Actions for Uniform Beams
*Figure 50: *

- Finding slope and deflection using superposition for a fixed beam with a uniform distributed load (UDL), concentrated moment, and a point load
*Figure 51: *

- Finding deflection using superposition example 2
*Figure 52: *

- Solving statically indeterminate beam (superposition),finding slope and deflection
*Figure 53: *

## Stress Block, Stress Resultants, Mohr's Circle, and Principal Stresses

Stress block sign convention

*Figure 54: Stress block sign convention*

- Finding stress resultants Example 1
For the given stress block below, find the principal stresses, and draw Mohr's Circle.

*Figure 55: Drawing Mohr's Circle for a stress block example 2*

- Finding stress resultants (stress block) Example 2 and drawing Mohr's Circle
For the given stress block below, find the principal stresses, and draw Mohr's Circle.

*Figure 56: Drawing Mohr's Circle for a stress block example 2 *

- Finding stress resultants and drawing Mohr's circle Example 3
For the given stress block below, find the principal stresses, and draw Mohr's Circle.

*Figure 57: Drawing Mohr's Circle for a stress block example 3 *

- Finding stress resultants and drawing Mohr's circle Example 4
*Figure 58: *

## Axial Stress and Hoop Stress in Pressure Vessels

- Pressure vessels Example 1
For the cylindrical thin-walled steel pressure vessel with inner diameter, d_{i} = 10 in, and a wall thickness, t = 0.75 in, calculate:

(a) the axial stress,σ_{a},and,

(b) the hoop stress, σ_{h}.

*Figure 59:Cylindrical thin-walled steel pressure vessel*

- Pressure vessels Example 2

*Figure 60: *

- Pressure vessels Example 3
*Figure 61:*

## Column Buckling

- Buckling of Columns Example 1
*Figure 62: *

- Buckling of Columns Example 2
*Figure 63: *

- Buckling of Columns Example 3
*Figure 64: *

- Buckling of Columns Example 4
*Figure 65: *