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CIVL 3322 / MECH 3322 Deformation in Axially Loaded Members Loading.pptx.pdf · CIVL 3322 / MECH...

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Mechanics of Materials CIVL 3322 / MECH 3322 Deformation in Axially Loaded Members Design Principles I would suggest that you all look over chapter 4 although we will not spend a lot of time in class with those principles It will really help the civil students when they get to steel and reinforced concrete design classes Axial Deformation
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Page 1: CIVL 3322 / MECH 3322 Deformation in Axially Loaded Members Loading.pptx.pdf · CIVL 3322 / MECH 3322 Deformation in Axially Loaded Members Design Principles ! ... Axial Deformation

14 January 2011

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Mechanics of Materials CIVL 3322 / MECH 3322

Deformation in Axially Loaded Members

Design Principles

¢  I would suggest that you all look over chapter 4 although we will not spend a lot of time in class with those principles

¢  It will really help the civil students when they get to steel and reinforced concrete design classes

Axial Deformation

Page 2: CIVL 3322 / MECH 3322 Deformation in Axially Loaded Members Loading.pptx.pdf · CIVL 3322 / MECH 3322 Deformation in Axially Loaded Members Design Principles ! ... Axial Deformation

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Prismatic Homogenous Members

¢ Homogenous – made of the same material all through the member, same mechanical properties

¢ Prismatic – same cross-section

Axial Deformation

Saint-Venant’s Principle

¢ Equipollent loads will act the same at sufficient distance along an axially loaded member

Axial Deformation

Page 3: CIVL 3322 / MECH 3322 Deformation in Axially Loaded Members Loading.pptx.pdf · CIVL 3322 / MECH 3322 Deformation in Axially Loaded Members Design Principles ! ... Axial Deformation

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Saint-Venant’s Principle

¢ This is generally a valid assumption except in areas where something occurs to alter the flow of stress

Axial Deformation

Fundamentals

¢ For any homogenous prismatic section of an axially loaded member, the elongation (or compression) in that section can be found by the expression shown at the right.

Axial Deformation

σ = FA

σ = EεδL= ε

δL= σE

δ = LσE

δ = FLAE

Page 4: CIVL 3322 / MECH 3322 Deformation in Axially Loaded Members Loading.pptx.pdf · CIVL 3322 / MECH 3322 Deformation in Axially Loaded Members Design Principles ! ... Axial Deformation

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P05_008

A solid brass [E = 100 GPa] axial member is loaded and supported as shown in Fig. P5.8. Segments (1) and (2) each have a diameter of 25 mm and segment (3) has a diameter of 14 mm.

P5.8

Axial Deformation

What is the value of the axial load in segment (3) of the member shown in kN?

Axial Deformation

Page 5: CIVL 3322 / MECH 3322 Deformation in Axially Loaded Members Loading.pptx.pdf · CIVL 3322 / MECH 3322 Deformation in Axially Loaded Members Design Principles ! ... Axial Deformation

14 January 2011

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What is the value of the axial load in segment (2) of the member shown in kN?

Axial Deformation

What is the value of the axial load in segment (1) of the member shown in kN?

Axial Deformation

Page 6: CIVL 3322 / MECH 3322 Deformation in Axially Loaded Members Loading.pptx.pdf · CIVL 3322 / MECH 3322 Deformation in Axially Loaded Members Design Principles ! ... Axial Deformation

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What is the value of the average stress in segment (1) of the member shown in MPa?

Axial Deformation

What is the value of the average stress in segment (2) of the member shown in MPa?

Axial Deformation

Page 7: CIVL 3322 / MECH 3322 Deformation in Axially Loaded Members Loading.pptx.pdf · CIVL 3322 / MECH 3322 Deformation in Axially Loaded Members Design Principles ! ... Axial Deformation

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What is the value of the average stress in segment (3) of the member shown in MPa?

Axial Deformation

What is the value of the microstrain in segment (1)?

Axial Deformation

Page 8: CIVL 3322 / MECH 3322 Deformation in Axially Loaded Members Loading.pptx.pdf · CIVL 3322 / MECH 3322 Deformation in Axially Loaded Members Design Principles ! ... Axial Deformation

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What is the value of the microstrain in segment (2)?

Axial Deformation

What is the value of the microstrain in segment (3)?

Axial Deformation

Page 9: CIVL 3322 / MECH 3322 Deformation in Axially Loaded Members Loading.pptx.pdf · CIVL 3322 / MECH 3322 Deformation in Axially Loaded Members Design Principles ! ... Axial Deformation

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What is the elongation of segment (1) in mm?

Axial Deformation

What is the elongation of segment (2) in mm?

Axial Deformation

Page 10: CIVL 3322 / MECH 3322 Deformation in Axially Loaded Members Loading.pptx.pdf · CIVL 3322 / MECH 3322 Deformation in Axially Loaded Members Design Principles ! ... Axial Deformation

14 January 2011

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What is the elongation of segment (3) in mm?

Axial Deformation

How far down does point D move in mm after this loading is applied? (assume that the distances shown are unloaded distances.)

Axial Deformation

Page 11: CIVL 3322 / MECH 3322 Deformation in Axially Loaded Members Loading.pptx.pdf · CIVL 3322 / MECH 3322 Deformation in Axially Loaded Members Design Principles ! ... Axial Deformation

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If the load at C was acting upwards. What would the microstrain in section (2) be?

Axial Deformation

If the load at C was acting upwards. What would the elongation in mm in section (2) be? Be sure to include the sign.

Axial Deformation

Page 12: CIVL 3322 / MECH 3322 Deformation in Axially Loaded Members Loading.pptx.pdf · CIVL 3322 / MECH 3322 Deformation in Axially Loaded Members Design Principles ! ... Axial Deformation

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Fundamentals

¢  If either or both the load or cross section change as a function of distance along the axis, the expressions on the right are not used because they assume a constant load on the section and a constant cross sectional area.

Axial Deformation

σ = FA

σ = EεδL= ε

δL= σE

δ = LσE

δ = FLAE

Fundamentals

¢  Instead, we can use a differential form of the expression to estimate the strain and deflection.

¢ This is actually only an estimate limited to a small rate of change in the cross section.

Axial Deformation

0

( )( )

where x is the distance along the loading axis

L F x dxA x E

δ = ∫

Page 13: CIVL 3322 / MECH 3322 Deformation in Axially Loaded Members Loading.pptx.pdf · CIVL 3322 / MECH 3322 Deformation in Axially Loaded Members Design Principles ! ... Axial Deformation

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P5.12 ¢ A homogenous rod of length L and elastic

modulus E is a truncated cone with diameter that varies linearly from d0 at one end to 2d0 at the other end.

¢ A concentrated axial load P is applied to the ends of the rod, as shown in Fig. P5.12. Assume that the taper of the cone is slight enough for the assumption of a uniform axial stress distribution over a cross section to be valid.

Axial Deformation

P5.12

¢  (a) Determine an expression for the stress distribution on an arbitrary cross section at x.

¢  (b) Determine an expression for the elongation of the rod.

Axial Deformation

Page 14: CIVL 3322 / MECH 3322 Deformation in Axially Loaded Members Loading.pptx.pdf · CIVL 3322 / MECH 3322 Deformation in Axially Loaded Members Design Principles ! ... Axial Deformation

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Homework

¢ P 5.3 ¢ P 5.6 ¢ P 5.13

Axial Deformation


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