Mechanics of Materials – MAE 243 (Section 002) Spring 2008

1. Mechanics of Materials – MAE 243 (Section 002) Spring 2008 Dr. Konstantinos A. Sierros

2. Problem 2.6-7 During a tension test of a mild-steel specimen (see figure), the extensometer shows an elongation of 0.00120 in. with a gage length of 2 in. Assume that the steel is stressed below the proportional limit and that the modulus of elasticity E = 30 (10^6) psi. (a) What is the maximum normal stress σmax in the specimen? (b) What is the maximum shear stress τmax? (c) Draw a stress element oriented at an angle of 45° to the axis of the bar and show all stresses acting on the faces of this element.

3. Problem 2.6-15 Acting on the sides of a stress element cut from a bar in uniaxial stress are tensile stresses of 10,000 psi and 5,000 psi, as shown in the figure. (a) Determine the angle θ and the shear stress τ θand show all stresses on a sketch of the element. (b) Determine the maximum normal stress σmax and the maximum shear stress τmax in the material.

4. 2.6: Strain energy • Strain energy is a fundamental concept in applied mechanics • Consider axially loaded structural members subjected to static loads • Consider a prismatic bar subjected to a static load P • During the loading process, the load P moves slowly though the distance δ and does a certain amount of work • To find the work done by load P, we need to know the manner in which the force varies. Therefore we need to use a load-displacement diagram

5. 2.6: Strain energy: load – displacement diagram ‘ The work done by the load is equal to the area below the load – displacement curve’ Strain energy: The energy absorbed by the bar during the loading process

6. 2.6: Elastic and inelastic strain energy • Recall loading – unloading of a prismatic bar • The strain energy that recovers during unloading is called the elastic strain energy (triangle BCD) • Area OABDO represents energy that is lost in the process of permanently deforming the bar. This energy is called inelastic strain energy

7. 2.6: Linearly elastic behaviour • If the material is linearly elastic (i.e. follows Hooke’s law) • Load – displacement curve is a straight line and the strain energy stored in the bar is: we also know… • Therefore we can express the strain energy of a linearly elastic bar in either of the following forms: …and for linearly elastic springs (replacing EA/L by k)…

8. 2.6: Nonuniform bars • The total strain energy U of a bar consisting of several segments is equal to the sum of the strain energies of the individual segments • The strain energy of a nonprismatic bar with continuously varying axial force can be calculated by using equation 1 for the differential element dx and then integrating for the whole length of the bar. (1)

9. 2.6: Comments • Strain energy is not a linear function of the loads applied • Therefore, we cannot obtain the strain energy of a structure supporting more than one load by combining the strain energies obtained from the individual loads acting separately • Instead we must evaluate the strain energy with all the loads acting simultaneously (see example 2-13 page 124)

10. 2.6: Displacements caused by a single load • The displacement of a linearly elastic structure supporting only one load can be determined from its strain energy. Condition 1: Structure must behave in a linearly elastic manner Condition 2: Only one load may act on the structure

11. 2.6: Strain – energy density Replace P/A by σ divide by volume of bar (V = AL) Replace δ/L by ε In many situations it is convenient to use a quantity called strain energy density, defined as the strain energy per unit volume of material

12. Quiz 2 this Friday 15 February 2008 (1 question – 20 minutes) during class Statically indeterminate structures Homework 2 due next Wednesday 20 February 2008 It is already posted on the website …plus 1st midterm Friday 22nd February…