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

2.1 Introduction Chapter 2: Axially Loaded Members Axially loaded members are structural components subjected only to tension or compression Sections 2.2 and 2.3 deal with the determination of changes in lengths caused by loads Section 2.4 is dealing with statically indeterminate structures Section 2.5 introduces the effects of temperature on the length of a bar Section 2.6 deals with stresses on inclined sections Section 2.7: Strain energy Section 2.8: Impact loading Section 2.9: Fatigue, 2.10: Stress concentration Sections 2.11 & 2.12: Non-linear behaviour

2.3: Changes in length under nonuniform conditions A prismatic bar of linearly elastic material loaded only at the ends changes in length by: This equation can be used in more general situations

2.3: Bars with intermediate axial loads A prismatic bar is loaded by one or more axial loads acting at intermediate points b and c We can determine the change in length of the bar by adding the elongations and shortenings algebraically

2.3: Bars with intermediate axial loads - Procedure First identify the segments of the bar. Segments are AB, BC, and CD as segments 1,2, and 3

2.3: Bars with intermediate axial loads - Procedure Then, determine the internal axial forces N 1, N 2, and N 3 in segments 1, 2, and 3 respectively Internal forces are denoted by the letter N and external loads are denoted by P

2.3: Bars with intermediate axial loads - Procedure By summing forces in the vertical direction we have: N 1 + P B = Pc + P D => N 1 = - P B + P C + P D N 2 = P C + P D N 3 = P D

2.3: Bars with intermediate axial loads - Procedure Then, determine the changes in the lengths of each segment: Segment 1Segment 2Segment 3

2.3: Bars with intermediate axial loads - Procedure Finally, add δ 1, δ 2 and δ 3 in order to obtain δ which is the change in length of the entire bar:

2.3: Bars consisting of prismatic segments Using the same procedure we can determine the change in length for a bar consisting of different prismatic segments Where; i is a numbering index and n is the total number of segments

2.3: Bars with continuously varying loads or dimensions Sometimes the axial force N and the cross-sectional area can vary continuously along the axis of the bar Load consists of a single force P B (acting at B) and distributed forces p(x) acting along the axis Therefore, we must determine the change in length of a differential element (fig 2-11 c) of the bar and then integrate over the length of the bar

2.3: Bars with continuously varying loads or dimensions The elongation dδ of the differential element can be obtained from the equation δ = (PL)/(EA) by substituting N(x) for P, dx for L and A(x) for A … and integrating over the length… integrating