## Mechanics of Materials Examples

Axial Loads

**Example 1 - Calculation of average normal stress on reinforced concrete column under compression**

The reinforced concrete column shown below consists of six A-36 steel reinforcing rods. It experiences an axial force of 50 kips. Find the average normal stress in the concrete and in each rod.

Steel rod diameter = 0.75 in

Modulus of Elasticity of Steel, Es = 29000 ksi

Modulus of Elasticity of Concrete, Ec = 4200 ksi (high strength concrete)

**Example 2 - Calculation of minimum required steel reinforcement diameter of reinforced concrete column under compression**

The reinforced concrete column shown below consists of six A-36 steel reinforcing rods. It experiences an axial force of 50 kips. Find the required diameter of each steel rod so that 75% of the load is carried by the steel and 25% of the load is carried by the concrete .

Modulus of Elasticity of Steel, Es = 29000 ksi

Modulus of Elasticity of Concrete, Ec = 4200 ksi (high strength concrete)

Stress

**Example 1 - Calculation of average normal stress **

Find the average normal stress at points A, B, and C. The diameter of each segment is shown in the figure below.

**Example 2 - Average normal stress and shear stress on inclined plane of a butt weld **

The two plates shown below are connected through a single V butt weld. The weld is used to transmit a force of 100 kips from one plate to the other. Find the average normal and average shear stress components due to this loading on the weld face, section AB.

**Example 3 - Calculation of average shear stress in bolts of a double shear simple connection**

For the simple double shear connection, find the maximum average shear stress acting on each 1/2" diameter bolt

Torsion

**Example 1 - Calculating the shear stress due to torsional loading on a solid circular shaft**

The solid circular shaft shown below experiences an internal torque of T = 10 kN - m. Find the shear stress at points A and B.

**Example 2 - Calculating the shear stress due to torsional loading on a tubular shaft**

The hollow circular shaft experiences an internal torque of T = 10 kN - m. Find the shear stress at points A and B

**Example 3 - Calculating angle of twist of circular shaft subjected to concentrated torque loading**

The A-36 steel shaft shown in the figure below experiences the torques shown. The diameter of the shaft is 30 mm. Calculate the angle of twist at end B.

The Shear Modulus of Elasticity, G = 75 GPa

**Example 4 - Calculating angle of twist of circular shaft subjected to uniform distributed torque loading**

The A-36 steel shaft shown in the figure below experiences a uniform distributed torque. The diameter of the shaft is 100 mm. Calculate the angle of twist at end B with respect to A

The Shear Modulus of Elasticity, G = 75 GPa

Combined Loadings

**Example 1 - Calculating the maximum internal pressure for a cylindrical and spherical pressure vessel**

A cylindrical pressure vessel has an inner diameter of 4 ft and a wall thickness of 0.5 in. If the hoop and longitudinal stress cannot exceed 25 ksi, find the maximum internal pressure that the vessel can sustain.

Also, find the maximum internal pressure that a spherical vessel of similar size can sustain.

Stress Transformation

**Example 1 - Calculating transformed stress for a differential plane stress element**

The differential element shown in the figure below experiences a state of plane stress.

Normal Stress in the normal x-direction: -100 MPa

Normal Stress in the normal y-direction: 50 MPa

Shear Stress in the x-y direction: -25 MPa

Find the equivalent stress on this element if it is oriented 30 degrees clockwise from the original position.

**Example 2 - Calculating principal stresses and associated orientation for a differential plane stress element**

The differential element shown in the figure below experiences a state of plane stress.

Normal Stress in the normal x-direction: -25 MPa

Normal Stress in the normal y-direction: 100 MPa

Shear Stress in the x-y direction: 50 MPa

Find the principal stresses at this point and the associated orientation of the element

**Example 3 - Calculating maximum in-plane shear stress and associated orientation for a differential plane stress element**

The differential element shown in the figure below experiences a state of plane stress.

Normal Stress in the normal x-direction: -25 MPa

Normal Stress in the normal y-direction: 100 MPa

Shear Stress in the x-y direction: 50 MPa

Find the maximum in-plane shear stress at this point and the associated orientation of the element

Strain Transformation

**Example 1 - Calculating transformed strain for a differential plane strain element**

The differential element shown in the figure below experiences a state of plane strain.

Normal Strain in the normal x-direction: 600 x 10^-6

Normal Strain in the normal y-direction: -400 x 10^-6

Shear Strain in the x-y direction: 300 x 10^-6

Find the equivalent strain on this element if it oriented 30 degrees clockwise from the original position.

**Example 2 - Calculating principal strains and associated orientation for a differential plane strain element**

The differential element shown in the figure below experiences a state of plane stress.

Normal Stress in the normal x-direction: -25 MPa

Normal Stress in the normal y-direction: 100 MPa

Shear Stress in the x-y direction: 50 MPa

Find the principal stresses at this point and the associated orientation of the element

**Example 3 - Calculating maximum in-plane shear strain and associated orientation for a differential plane strain element**

The differential element shown in the figure below experiences a state of plane strain.

Normal Strain in the normal x-direction: -400 x 10^-6

Normal Strain in the normal y-direction: 200 x 10^-6

Shear Strain in the x-y direction: 100 x 10^-6

Find the maximum in-plane shear strains at this point and the associated orientation of the element

Column Buckling

**Example 1 - Calculating maximum axial capacity for pinned steel column**

The W8 x 31 column shown below is pinned at both ends. Find the largest axial load that it can support before it buckles or the steel materials begins to yield.

The column material is A-36 steel

Modulus of Elasticity, Es = 29000 ksi

**Example 2 - ****Calculating maximum axial capacity for steel column braced about the weak axis**

The W6 x 15 steel column shown below is 30 ft long and fixed at both ends. It is braced at midheight about the y-y axis. The bracing is assumed to be pin connected to the column. Find the largest axial load that it can support before it buckles or the steel materials begins to yield.

Yield Stress, Fy = 60 ksi

Modulus of Elasticity, Es = 29000 ksi

**Example 3 - ****Calculating critical buckling load for pinned rectangular wood column**

The 14 ft wooden rectangular column is pinned at both ends. Assuming that yielding does not occur, find the critical buckling load.

Modulus of Elasticity, E = 1600 ksi

**Example 4 - Calculating critical buckling load for rectangular wood column fixed at the bottom and free at the top**

The 14 ft wooden rectangular column shown below is fixed at the bottom and free at the top. Assuming that yielding does not occur, find the critical buckling load.

Modulus of Elasticity, E = 1600 ksi

**Example 5 - Calculating critical buckling load for rectangular wood column fixed at both ends**

The 14 ft wooden rectangular column shown below is fixed at both ends. Assuming that yielding does not occur, find the critical buckling load.

Modulus of Elasticity, E = 1600 ksi

**Example 6 - ****Calculating critical buckling load for rectangular wood column fixed at the bottom and pinned at top**

The 14 ft wooden rectangular column is fixed at the bottom and pinned at the top. Assuming that yielding does not occur, find the critical buckling load.

Modulus of Elasticity, E = 1600 ksi