COMPLEX STRESS TUTORIAL 4 THEORIES OF FAILURE. You should judge your progress by completing the self assessment exercises.

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1 COMPLEX STRESS TUTORIAL 4 THEORIES OF FAILURE This short tutorial covers no known elements of the E.C. or Edexcel Exams but should be studied as part of complex stress, structures and materials. You should judge your progress by completing the self assessment exercises. These may be sent for marking or you may request copies of the solutions at a cost (see home page). On completion of this tutorial you should be able to do the following. Explain the greatest principal stress theory (Rankine) Explain the greatest principal strain theory (St. Venant) Explain the imum shear stress theory (Guest And Coulomb) Solve problems involving the above theories. It is assumed that students doing this tutorial are already familiar with complex stress theory. D.J.DUNN

INTRODUCTION Modern CADD systems allow the engineer to calculate stress levels in a component using finite stress analysis linked to the model.

2 INTRODUCTION Modern CADD systems allow the engineer to calculate stress levels in a component using finite stress analysis linked to the model. The reasons why a given material fails however, is not something a computer can predict without the results of research being added to its data bank. In some cases it fails because the imum tensile stress has been reached and in others because the imum shear stress has been reached. The exact combination of loads that makes a component fail depends very much on the properties of the material such as ductility, grain pattern and so on. This section is about some of the theories used to predict whether a complex stress situation is safe or not. There are many theories about this and we shall examine three. First we should consider what we regard as failure. Failure could be regarded as when the material breaks or when the material yields. If a simple tensile test is conducted on a ductile material, the stress strain curve may look like this. Figure The imum allowable stress in a material is σ. This might be regarded as the stress at fracture (ultimate tensile stress), the stress at the yield point or the stress at the limit of proportionality (often the same as the yield point). The Modulus of elasticity is defined as E stress/strain σ/ε and this is only true up to the limit of proportionality. Note that some materials do not have a proportional relationship at all. The imum allowable stress may be determined with a simple tensile test. There is only one direct stress in a tensile test (σ F/A) so it follows that σ σ and it will have a corresponding strain ε ε. Complex stress theory tells us that there will be a shear stress and strain γ that has a imum value on a plane at 45 o to the principal plane. It is of interest to note that in a simple tensile test on a ductile material, at the point of failure, a cup and cone is formed with the sides at 45 o to the axis. Brittle materials often fail with no narrowing (necking) but with a flat fail plane at 45 o to the axis. This suggests that these materials fail due to the imum shear stress being reached. Figure 2 D.J.DUNN 2 2

In a complex stress situation, there are principal stresses σ, σ 2 and σ. σ is the greatest and σ is the smallest. Remember that a negative stress is smaller than zero.

3 In a complex stress situation, there are principal stresses σ, σ 2 and σ. σ is the greatest and σ is the smallest. Remember that a negative stress is smaller than zero. There are corresponding principal strains ε, ε 2 and ε and shear strains. Figure. THE GREATEST PRINCIPAL STRESS THEORY (RANKINE) This simply states that in a complex stress situation, the material fails when the greatest principal stress equals the imum allowable value. σ σ σ could be the stress at yield or at fracture depending on the definition of failure. If σ is less than σ then the material is safe. Safety Factor σ /σ WORKED EXAMPLE No. A certain material fractured in a simple tensile test at a stress level of 800 MPa. The same material when used as part of a structure must have a safety factor of. Calculate the greatest principal stress that should be allowed to occur in it based on Rankine s theory. SOLUTION S.F. σ /σ 800/σ σ 800/ MPa D.J.DUNN

4 2. THE GREATEST PRINCIPAL STRAIN THEORY (St. VENANT) This states that in a complex stress situation, the material fails when the greatest principal strain reaches the imum allowable strain determined in a simple tensile test. ε ε ε is the value determined in a simple tensile test. If the imum allowable stress is taken as the value at the limit of proportionality, we may further develop the theory using the modulus of elasticity. ε σ /E From dimensional relationships (covered in other tutorials) we have: ε { σ ν( σ2 + σ )} E ε σ Safety factor and the material fails when ε σ ν σ + σ σ is less than { ( )} { σ ν( σ + σ )} 2 2 WORKED EXAMPLE No.2 A certain material fractured in a simple tensile test at a stress level of 600 MPa. The same material when used as part of a loaded structure must has principal stresses of 600, 400, and -200 MPa. Determine the safety factor at this load based on the greatest principal strain theory. Take Poisson s ratio as SOLUTION Safety factor σ 600 { σ ν( σ + σ )} { ( ) } 2 The component is just safe as the safety factor is larger than..0 D.J.DUNN 4 4

5 . THE MAXIMUM SHEAR STRESS THEORY (GUEST and COULOMB) This states that in a complex stress situation, the material fails when the greatest shear strain in the material equals the value determined in a simple tensile test. Applying complex stress theory to a tensile test gives this as ½ σ In a simple tensile test, σ could be what ever stress is regarded as the imum allowable. In a dimensional complex stress situation the imum shear strain is ½ (σ - σ ) If this is less than then the material is safe. Safety factor σ σ σ On the limit when the safety factor is it follows that σ (σ - σ ) Put into words, failure occurs when the imum allowable stress is equal to the difference between the greatest and the smallest principal stresses. Note negative stresses are smaller than zero. WORKED EXAMPLE No. Show that based on the greatest shear stress theory, the structure in W.E. No.2 should have failed. SOLUTION Safety factor σ σ σ ( 200) 0.75 The component should have failed as the safety factor is less than. D.J.DUNN 5 5

6 SELF ASSESSMENT EXERCISE No.. A certain steel fails in a simple tensile test when the stress is 0 MPa. The same steel is used in a complex stress situation and the principal stresses are MPa, MPa and 0 MPa. Determine the factor of safety based on the three theories. ν 0. (Answers 2.7, 2.97 and 2.7) 2. A certain steel fails in a simple tensile test when the stress is 0 MPa. The same steel is used in a complex stress situation and the principal stresses are MPa, 0 MPa and - MPa. Determine the factor of safety based on the three theories. ν 0. (Answers 2.7, 2.52 and 2.4). A certain steel failed in a simple tensile test when the stress is 460 MPa. The same steel is used in a complex stress situation and the principal stresses are 200 MPa, 50 MPa and -00 MPa. Determine the factor of safety based on the three theories. ν 0. (Answers 2.,.5 and 2.49) 4. The results from a 60o strain gauge rosette are εa 600 µε εb -200 µε εc 400 µε ν 0. E 205 GPa Determine the principal strains and stresses. The same material failed at a stress level of 00 MPa in a tensile test. Calculate the safety factor of the complex situation based on the three theories. (Answers ε 747 µε, ε2-24 µε, σ 57 MPa, σ2 0.6 MPa,.9,.92 and.9). 4. STRAIN ENERGY Another theory for failure of materials is based on strain energy. This simply states that a component will fail when the total strain energy reaches a critical level. This is not covered here but strain energy is covered in the tutorial of that name. D.J.DUNN 6 6

ENGINEERING SCIENCE H1 OUTCOME 1 - TUTORIAL 2 SHEAR FORCE IN BEAMS EDEXCEL HNC/D ENGINEERING SCIENCE LEVEL 4 H1 FORMERLY UNIT 21718P

ENGINEERING SCIENCE H1 OUTCOME 1 - TUTORIAL 2 SHEAR FORCE IN BEAMS EDEXCEL HNC/D ENGINEERING SCIENCE LEVEL 4 H1 FORMERLY UNIT 21718P This material is duplicated in the Mechanical Principles module H2 and