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10. Stress Failure

• We need to do an analysis of our designs for suitability. This involves looking at static forces in the design, and then analyzing the members to see if they will carry the prescribed loads.

• Although static values are usually quite accurate, the failure load for a member is hard to predict accurately, and impossible to predict exactly. This is because we are subject to the effects of geometry, material properties, unexpected loads, etc.

10.1 Factor of Safety
• When we do basic design work we typically use a process of the basic format,

• In particular the design process often involves a factor of safety that allows for,

variations in material properties

variations in expectations load

the non-uniform distribution of stress

cyclic loading

etc

• The factor of safety is applied as,

• The values vary for various systems. The values given below are reasonable for static systems.

• For example let us consider a two member mechanism 1.44 from [Beer and Johnston]

#### 10.1.1 Problems

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Problem 10.1 The mechanism below is to be analyzed. Before answering any of the questions, read the notes, and study the diagram.

- The cable is 1/8 inch in diameter, and made of 6061-T6.

- The upper bracket is made of cold rolled steel.

- The pulley is made of nylon: assume it is frictionless.

- The beam is made of oak.

- The pulley is supported by a brass pin. There are brackets on both sides of the pulley.

a) Find the new length of the cable (assume the cable does not stretch on the pulley). Is the factor of safety high enough?

b) Is the factor of safety for stress failure in the beam high enough?

c) What is the minimum pin diameter for the pulley?

e) How thick should the upper brackets be for a factor of safety of 3?

10.2 References
10.1 Soustas-Little, R.W. and Inman, D.J., Engineering Mechanics Statics, Prentice-Hall, 1997.

10.3 Strain Failure
Problem 10.2 If the rod is 6” before the load is applied, what is the new length? What load P would result in plastic deformation? What load would result in rupture? What is the factor of safety?

• When a load is applied and removed in cycles the material may fatigue. This means the ultimate strength is effectively reduced. Design curves that can take this into account are found in handbooks, and look like those shown below,

10.4 Poisson’s Ratio
• An isotropic material has the same properties in all directions. (materials such as fiber glass do not).

• When we apply a stress to an isotropic material in one direction, we induce stress in the perpendicular direction. The resulting ratio between perpendicular stresses, and strains, is called Poisson’s Ratio

• In physical terms: as we stretch a bar, it becomes a bit thinner.

Problem 10.3 What would the diameter of the bar become after the load P has been applied?

10.5 Generalized Hooke’s Law
• When we load a material in a single direction the effect of Poisson’s ratio is naturally included in Young’s Modulus. But, when there are multiple loads in multiple directions, we must uses Poisson’s ratio to determine how they intersect.

Problem 10.4 The cubical gage block to the right is loaded on two faces with 1 kip of compression. The modulus of elasticity for the material is 1 Mpsi, and the Poisson’s ratio is 0.3. What are the new outside dimensions?

10.6 Stress Concentrations
• Saint Venant’s Principle states that regardless of how a force is applied, when we move far enough away, the distribution becomes even. For example, if we are applying forces as point loads, they will have very high stress concentrations near the point of application. But, as we move away from the point of application, the force distribution evens out. Consider the pin in the hole where the pin applies a load of P.

• Stress concentrations are hard to predict, and this must often be done using experiments, Finite Element Analysis (FEA), or other techniques.

• During design we must pay attention to the stress concentrations. At some points the stress will be higher that the average stress.

• One technique for estimating maximum stress concentrations is to use tables derived experimentally.

Problem 10.5 Find the maximum stress that will be expected in this bar with a hole in it.

10.7 Torsion Stress Concentrations
• We can estimate the maximum stress for two shafts connected with a fillet using the following graph,

Problem 10.6 What is the maximum stress in the shaft shown. The shafts shown are joined with a fillet of radius 1/16”. What is the maximum shear stress if a torque of 100 lb.in. is applied?

10.8 References
10.2 Soustas-Little, R.W. and Inman, D.J., Engineering Mechanics Statics, Prentice-Hall, 1997.

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