# Lecture 8

```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 2.9: 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:
Elongation of a
prismatic bar in tension
FIG. 2-5
• This equation can be used in more general
situations
Copyright 2005 by Nelson, a division of Thomson Canada Limited
2.3: Bars with intermediate axial loads
(a) Bar with external loads acting at
intermediate points; (b), (c), and (d) freebody diagrams showing the internal axial
forces N1, N2, and N3
FIG. 2-9
Copyright 2005 by Nelson, a division of Thomson Canada Limited
• 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
(a) Bar with external loads acting at
intermediate points; (b), (c), and (d) freebody diagrams showing the internal axial
forces N1, N2, and N3
FIG. 2-9
Copyright 2005 by Nelson, a division of Thomson Canada Limited
• 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
(a) Bar with external loads acting at
intermediate points; (b), (c), and (d) freebody diagrams showing the internal axial
forces N1, N2, and N3
FIG. 2-9
Copyright 2005 by Nelson, a division of Thomson Canada Limited
• 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
(a) Bar with external loads acting at
intermediate points; (b), (c), and (d) freebody diagrams showing the internal axial
forces N1, N2, and N3
FIG. 2-9
Copyright 2005 by Nelson, a division of Thomson Canada Limited
• 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
(a) Bar with external loads acting at
intermediate points; (b), (c), and (d) freebody diagrams showing the internal axial
forces N1, N2, and N3
FIG. 2-9
Copyright 2005 by Nelson, a division of Thomson Canada Limited
• Then, determine the changes in the lengths of each segment:
Segment 1
Segment 2
Segment 3
2.3: Bars with intermediate axial loads - Procedure
(a) Bar with external loads acting at
intermediate points; (b), (c), and (d) freebody diagrams showing the internal axial
forces N1, N2, and N3
FIG. 2-9
Copyright 2005 by Nelson, a division of Thomson Canada Limited
• 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
FIG. 2-10
Bar consisting of prismatic
segments having different
axial forces, different
dimensions, and different
materials
Copyright 2005 by Nelson, a division of Thomson Canada Limited
• 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
Bar with varying cross-sectional
area and varying axial force
FIG 2-11
Copyright 2005 by Nelson, a division of Thomson Canada Limited
• 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
Bar with varying cross-sectional
area and varying axial force
FIG 2-11
Copyright 2005 by Nelson, a division of Thomson Canada Limited
• 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
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