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ADOPT, GNUMechanics of Materials
재료역학 (Mechanics of Material)
http://www.adopt.kr/lecture/lecture01/index.html과목 홈페이지 주소 :
ADOPT, GNUMechanics of Materials
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?
ADOPT, GNUMechanics of Materials
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
ADOPT, GNUMechanics of Materials
Axially Loaded Member(prismatic bar)
Steel rod (E = 29 x 106 psi )=58 ksi (in tension)=36 ksi
Will the rod deflect and if so, by how much?
Will the rod support the weight?
Will the rod plastically deform?
☞ 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 <
ADOPT, GNUMechanics of Materials
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
ADOPT, GNUMechanics of Materials
Problems: Axially loaded members
[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?
ADOPT, GNUMechanics of Materials
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.
Problems: Axially loaded members
ADOPT, GNUMechanics of Materials
Problems: Axially loaded members
4
where, : 0.5:
,x xd A
x1L
x
2L
ADOPT, GNUMechanics of Materials
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
ADOPT, GNUMechanics of Materials
steel
concrete
Problem: Axially Loaded MembersAll derivations must satisfy the assumption =
Concrete
Steel
allowable stress = 5.6MPa14
allowable stress = 42 MPa84
ADOPT, GNUMechanics of Materials
steel
concrete
Problem: Axially Loaded Members
Concrete허용응력으로축력계산
Steel허용응력으로축력계산
ADOPT, GNUMechanics of Materials
Axially Loaded Member(prismatic bar)
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.
=58 ksi (in tension)=36 ksi
Will the rod deflect and if so, by how much?
Will the rod support the weight?
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
ADOPT, GNUMechanics of Materials
Axially Loaded Member(prismatic bar)
[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
ADOPT, GNUMechanics of Materials
Axially Loaded Member(prismatic bar)
[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
ADOPT, GNUMechanics of Materials
Axially Loaded Member(prismatic bar)
• 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 !
ADOPT, GNUMechanics of Materials
Axially Loaded Member(prismatic bar)
Will the rod support the weight?
=58 ksi (in tension) > (L) =29.1 ksi
Will the rod plastically deform?
=29.1 ksi, so σ <
=58 ksi (in tension)=36 ksi
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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