Changing Cross Section. Mechanics of Materials. Non-Prismatic Bars. Changing Cross Section. Changing Cross Section

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Hope Stokes
6 years ago
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1 Mechanics of Materials Non-rismatic Bars I do not pretend to understand the universe. It s a great deal bigger than I am. Tom Stoppard In our previous work, we have looked at the stress and strain in bars with constant cross sectional areas and with point loads applied aially Now we will consider what is happening in the bar if the cross section changes and/or if the load is a aially distributed load Thursday, September 19, 2002 Meeting Ten - Nonrismatic Bars 2 In the first case, we will vary the cross section of the material and see how the stress and deformation varies with the variation in the cross section If we take a cross-section at some distance to the right from the support we can draw a free-body diagram of the right hand section W W Thursday, September 19, 2002 Meeting Ten - Nonrismatic Bars 3 Thursday, September 19, 2002 Meeting Ten - Nonrismatic Bars 4

2 The aial stress at a distance could then be calculated as W We may develop a function that relates the cross sectional area as a function of the distance W Thursday, September 19, 2002 Meeting Ten - Nonrismatic Bars 5 Thursday, September 19, 2002 Meeting Ten - Nonrismatic Bars 6 This allows us to calculate the stress on any face at any depth in the bar W Now we need to utilize this changing stress through the bar to determine the overall change in length of the bar W σ = Thursday, September 19, 2002 Meeting Ten - Nonrismatic Bars 7 Thursday, September 19, 2002 Meeting Ten - Nonrismatic Bars 8

3 W We can start by taking a differential section of the bar d W The stress on the differential area is d Thursday, September 19, 2002 Meeting Ten - Nonrismatic Bars 9 Thursday, September 19, 2002 Meeting Ten - Nonrismatic Bars 10 W In terms of the strain, the stress on the differential area is d ε = W If we replace the strain by the deformation divided by the original length we have d σ = = δ d Thursday, September 19, 2002 Meeting Ten - Nonrismatic Bars 11 Thursday, September 19, 2002 Meeting Ten - Nonrismatic Bars 12

4 W The deformation through this differential section represents a differential deformation within the bar. d dδ = d W Isolating this differential deformation we have d d d Thursday, September 19, 2002 Meeting Ten - Nonrismatic Bars 13 Thursday, September 19, 2002 Meeting Ten - Nonrismatic Bars 14 W To calculate the deformation along the length of the bar we would sum up the differential deformations d d d W Since we are looking at differential quantities, we make summations using the integral d d d Thursday, September 19, 2002 Meeting Ten - Nonrismatic Bars 15 Thursday, September 19, 2002 Meeting Ten - Nonrismatic Bars 16

5 W Since the aial load is constant along the length and the modulus of elasticity is constant we can bring those outside of the integral d d d W nd the integral of all the deformation along the length of the bar is equal to the total deformation we have d d Thursday, September 19, 2002 Meeting Ten - Nonrismatic Bars 17 Thursday, September 19, 2002 Meeting Ten - Nonrismatic Bars 18 W NOT: ven though we have been using deformation, with an aial load the deformation is an elongation or a shortening d d W Finally, the change in length of the bar is d 0 d Thursday, September 19, 2002 Meeting Ten - Nonrismatic Bars 19 Thursday, September 19, 2002 Meeting Ten - Nonrismatic Bars 20

Since is a function of, this is as far as we can go until we define the function of W d Thursday, September 19, 2002 Meeting Ten - Nonrismatic Bars 21 0 d Notice how the epression we just developed

6 Since is a function of, this is as far as we can go until we define the function of W d Thursday, September 19, 2002 Meeting Ten - Nonrismatic Bars 21 0 d Notice how the epression we just developed resembles the epression for a prismatic bar W d Thursday, September 19, 2002 Meeting Ten - Nonrismatic Bars 22 0 d If we were to take small differential lengths along the ais they would be d lengths and the elongation of these lengths would still be W Thursday, September 19, 2002 Meeting Ten - Nonrismatic Bars 23 roblem and Thursday, September 19, 2002 Meeting Ten - Nonrismatic Bars 24

Distributed ial oads In some cases we may have a constant cross section but we may have an aial load that varies with along the bar Thursday, September 19, 2002 Meeting Ten - Nonrismatic Bars 25

7 Distributed ial oads In some cases we may have a constant cross section but we may have an aial load that varies with along the bar Thursday, September 19, 2002 Meeting Ten - Nonrismatic Bars 25 Thursday, September 19, 2002 Meeting Ten - Nonrismatic Bars 26 Distributed ial oads For instance, we we have a bar loaded as shown Distributed ial oads If the weight of the bar is significant, we can look at a section of the bar and look at the forces on that section + Thursday, September 19, 2002 Meeting Ten - Nonrismatic Bars 27 Thursday, September 19, 2002 Meeting Ten - Nonrismatic Bars 28

8 Distributed ial oads In this case, the aial load at any distance up from the bottom of the beam is both the load and the weight of that section + Distributed ial oads This means that the stress varies as we travel up the beam and therefore the strain at each point as we move up the beam + Thursday, September 19, 2002 Meeting Ten - Nonrismatic Bars 29 Thursday, September 19, 2002 Meeting Ten - Nonrismatic Bars 30 Distributed ial oads If we label the loading ais as, the elongation epression again become a differential epression + Thursday, September 19, 2002 Meeting Ten - Nonrismatic Bars 31 d d Distributed ial oads In this case, it is that is a function of rather than so the total elongation of the beam would be + Thursday, September 19, 2002 Meeting Ten - Nonrismatic Bars 32 0 d

9 Distributed ial oads In the most general case,, and would all be functions of : W δ = ( ) d ( ) ( 0 ) Thursday, September 19, 2002 Meeting Ten - Nonrismatic Bars 33 Thursday, September 19, 2002 Meeting Ten - Nonrismatic Bars 34 Homework Read section 3-6 roblem roblem roblem (Both the cross sectional area and the loading are functions) Thursday, September 19, 2002 Meeting Ten - Nonrismatic Bars 35

PHY 133 Lab 6 - Conservation of Momentum

Stony Brook Physics Laboratory Manuals PHY 133 Lab 6 - Conservation of Momentum The purpose of this lab is to demonstrate conservation of linear momentum in one-dimensional collisions of objects, and to