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

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**Mechanics of Materials – MAE 243 (Section 002)**Spring 2008 Dr. Konstantinos A. Sierros**2.1 Introduction**Chapter 2: Axially Loaded Members • Axially loaded membersare 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 N1, N2, and N3 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: • N1 + PB = Pc + PD => N1 = - PB + PC + PD • N2 = PC + PD • N3 = PD**2.3: Bars with intermediate axial loads - Procedure**• Then, determine the changes in the lengths of each segment: Segment 1 Segment 2 Segment 3**2.3: Bars with intermediate axial loads - Procedure**• Finally, add δ1, δ2 and δ3 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 PB (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 integrating … and integrating over the length…