재료역학 (Mechanics of Material) - ADOPT

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재료역학 (Mechanics of Material)

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Problems: Axially loaded members

[EX] A 1 in. diameter steel rod ( 30 10 must carry a load in tension of 30000 lb (see Figure).If the initial length of the stressed portion of the rodis 21.75 in. What is its final length?


Axially Loaded Member(prismatic bar)

15 in

8000 lb

0.5 in

0.35 in

ASTM‐36 structural steel

Will the rod deflect and if so, by how much?

Will the rod support the weight?

Will the rod plastically deform?

[EX] In the design of a mansion, you need to hang a8000 lb chandelier in the center of a large ballroom.To do this, you will attach the chandelier into a 15inch long steel rod ( 29 10 ) embedded inthe ceiling.

=58 ksi (in tension)=36 ksi



Steel rod (E = 29 x 106 psi )=58 ksi (in tension)=36 ksi




☞ σ >

15 in

8000 lb

0.5 in

0.35 in

=0.788

If safety factor < 1, plastic deformation occur

. .

=0.024in.

. . =45714 psi =45.714ksi

☞ σ <


Problems: Axially loaded members[EX] A two‐story building has comlumns AB at the first floorand BC at the second floor (see Figure). The columns areloaded as shown in Figure, with the roof load equal to100 kips and the load applied at the second floor equalto 180 kips. The cross‐sectional areas of the upper andlower columns are 5.9 and 17.1 , respectively, andeach column has length 12 . Assuming that

30 10 , determine the shortening of eachcolumn due to the applied loads



[EX] A prismatic bar ABCD is subjected to loads , , as shown in Figure.The bar is made of steel with modulus of elasticity E=200 GPa andcross section area A=225 . Determine the deflection at lower end of the bardue to the loads , , . Does the bar elongate or shorten?


A concrete pier of square cross section is 6m high (see Figure).The sides taper uniformly from a width of 0.5 m at the top to 1.0m at the bottom. Determine the shortening of the pier under a compressive load of 1400kN (disregard the weight of the pier itself).Assume that the modulus of elasticity of concrete is 24 GPa.




4

where, : 0.5:

,x xd A

x1L

x

2L


steel

concreteConcrete

Steel

Problem: Axially Loaded Members[EX] A square column is formed of a 25 mm thick metal casting (outside dimension 250mm x 250mm and inside dimension 200mmx200mm) that is filled with concrete (see Figure). The casting has modulus of Elasticity 84 and the concrete core has modulus of elasticity 14 . Find the maximum permissible load P on the column if the allowable stresses in the metal and the concrete 42 and 5.6 respectively. (Assume that the stresses in the metal and concrete are uniformly distributed).

allowable stress 5.6MPa

allowable stress 42 MPa


steel

concrete

Problem: Axially Loaded MembersAll derivations must satisfy the assumption =

Concrete

Steel

allowable stress = 5.6MPa14

allowable stress = 42 MPa84


steel

concrete

Problem: Axially Loaded Members

Concrete허용응력으로축력계산

Steel허용응력으로축력계산



In the design of a mansion, you need to hang a8000 lb chandelier in the center of a largeballroom. To do this, you will attach thechandelier into a 15 inch long steel rod (E = 29 x106 psi ) embedded in the ceiling.




Will the rod plastically deform?so σ >

0.5 in 0.55 in

8000 lb

Lower Cross Section

Upper Cross Section

0.55 in

1.0 in

ASTM‐36 structural steel

L=2 in



[SOL 1] Approximate the total deflection by breaking the part into subsections(elements) and calculating the deflection along each of the smaller subsections.Can we get an exact solution to this problem?

where,

P

li

1. Sove this !when i=4,6,8

L=2 in



[Sol 2] The exact end deflection of the rod is obtained by calculus

Let each sub‐length, li = dx, and let dx → 0:

P

Lx

dx



• Put the upper and lower cross sections into the equation for area integrate with respect to x.

• Simplify the equation above:

where:• elongation of dx

L

dxExA

P

0 )(

total

P

Lx

dx

• total elongation of bar2. Sove this !




=58 ksi (in tension) > (L) =29.1 ksi


=29.1 ksi, so σ <


0.5 in 0.55 in

8000 lb

Lower Cross Section

Upper Cross Section 0

0.55 inTherefore the maximum stress at bottom

L

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재료역학 (Mechanics of Material) - ADOPT