Solid Mechanics

Md. Mohiuddin

Lecturer

Department of Mechanical Engineering

ME 2213

Stress on Inclined Section

Stress on Inclined Section

Stress on Inclined Section

• This shows that a surface can be subjected to share stress and normal stress at the same time

Stress at a Point

• Stress over a surface is usually determined by dividing the force acting on the surface by the area.
• When the stress is constant over the area, the surface is said to be at uniform stress
• The state of stress at any point on such a surface will be the same.

• If stress is not uniform, the state of stress will vary on different points of the surface.
• Stress at any point is the stress distribution over a differential area enclosing the point of interest.

Plane Stress

• When the material is in plane stress in the xy plane, only the x and y faces of the element are subjected to stresses, and all stresses act parallel to the x and y axes

• What’s the significance?

It is important to find the surface where the maximum stress is acted upon and the corresponding stress values.

Plane Stress

Plane Stress

Plane Stress

By introducing the above trigonometric identities, the above equations become

Using a Similar approach it can be shown that

Transformation Equations

The addition of the first two equations provide

Plane Stress

Principal Stresses

• The maximum and minimum normal stresses are called principal stresses
• The planes on which principle stresses act are called the principal planes.

For Principle stresses:

Principal Stresses

This equation is shown in a previous slide

After simplification

One of the principal stresses

Principal Stresses

To obtain another principal stresses use the following formula

For Maximum Shear stresses:

Maximum Shear Stress

From a previous slide

This equation shows that the planes of maximum shear stress occur at 45° to the principal planes.

Maximum Share Stresses

This equation was shown in a previous slide

After simplification

Put these values

By comparing with the equations of principle stresses

Mohr’s Circle

• The transformation equations for plane stress can be represented in graphical form by a plot known as Mohr’s circle.

………..(i)

………..(ii)

………..(iii)

By rearranging equation (i)

Squaring both sides of the equations and then adding equations (ii) and (iii)

Mohr’s Circle

Mohr’s Circle

Procedures of Drawing Mohr’s Circle

Mohr’s Circle

Mohr’s Circle-Problem

Mohr’s Circle-Problem

Combined Stress

• Possible Combinations
1. Axial & Flexural
2. Axial & Torsional
3. Flexural & Torsional
4. Axial, Flexural & Torsional

• Combines only normal stresses
• Simplest

Combined Stress (Axial & Flexural)

Determine the resultant normal stresses at A and B at the wall.

Combined Stress (Axial & Flexural)

To avoid interference, a link in a machine is designed so that its cross-sectional area is reduced one half at section A-B as shown in the Fig. If the thickness of the link is 50 mm, compute the maximum force P that can be applied if the maximum normal stress on section A-B is limited to 80 MPa.

Combined Stress (Axial & Flexural)

A wooden beam 100 mm by 200 mm, supported as shown in the Figure, carries a load P. What is the largest safe value of P, if the maximum stress is not to exceed 10 MPa?

Combined Stress (Axial & Flexural)

Determine the largest load P that can be supported by the circular steel bracket shown in the Figure, if the normal stress on section A-B is limited to 80 MPa.