工程力学英文版课件15 Columns
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1
Columns
§17–1 Stable and Unstable Equilibrium
§17–2 Ideal Column with Pin Supports
§17–3 Effective Length of Columns
§17–4 Design Formulas
2
§17-1 Stable and Unstable Equilibrium
P a b b a
15
Summarizing the above discussion, the Euler formula for a pin-supported long slender column can be written, 2 EI Pcr 2 L Pcr: maximum axial load on the column just before it begins to buckle. This load must not cause the stress in the column to exceed the proportional limit.
When the column is in its deflected position as shown in figure (a), the internal bending moment can be determined by using the method of sections. The free-body diagram of a segment in the deflected position is shown in figure (b). Here both the deflection y and the internal moment M are shown in the positive direction according to the sign convention.
P 源自文库0 sin L EI
which is satisfied if
P L n EI
11
n EI P , n 1, 2, 3, 2 L The smallest value of P is obtained when n = 1, so the critical load for the column is therefore 2 EI Pcr 2 L This load is sometimes referred to as the Euler load, after the Swiss mathematician Leonhard Euler, who originally solved this problem in 1757. 12
§17-2 Ideal Column with Pin Supports
The critical buckling load for a column that is pin supported as shown in figure will be determined. An ideal column is perfectly straight before loading, is made of homogeneous material, and upon which the load is applied through the centroid of the cross section. The material behaves in a linear-elastic manner and the column buckles or bends in a single plane.
13
It should also be noted that the loadcarrying capacity of a column will increase as the moment of inertia of the cross section increases. Thus, efficient columns are designed so that most of the column’s crosssectional area is located as far away as possible from the principal centroidal axes for the section. This is why hollow sections such as tubes are more economical than solid sections.
Long slender members subjected to an axial compressive force are called columns, and the lateral deflection that occurs is called buckling. Quite often the buckling of a column can lead to a sudden and dramatic failure of a structure or mechanism, and as a result, special attention must be given to the design of columns so that they can safely support their intended loadings without buckling. The maximum axial load that a column can support when it is on the verge of buckling is called the critical load. 5
P y
x
This is a homogeneous, second-order, linear differential equation with constant coefficients. The general solution is:
9
P P C2 cos y C1 sin x x EI EI
C B C B C
B
(a) (b) Stable equilibrium Neutral equilibrium
(c) Unstable equilibrium
3
Stability: Capacity of a component or a structural element to retain the original state of equilibrium. In previous chapters we have discussed some of the methods used to determine a member’s strength and deflection, while assuming that the member was always in stable equilibrium. Some members, however, may be subjected to compressive loadings, and if these members are long and slender the loading may be large enough to cause the member to deflect laterally or sideway. 4
2 2
or
It should be noted that the critical load is independent of the strength of the material; rather it depends only on the column’s dimensions (I and L) and the material’s stiffness or modulus of elasticity E. For this reason, as far as elastic buckling is concerned, columns made, for example, of high-strength steel offer no advantage over those made of lower-strength steel, since the modulus of elasticity for both is approximately the same.
P P
6
Whether or not a column will remain stable or become unstable when subjected to an axial load will depend on its ability to restore itself, which is based on its resistance to bending. Hence, in order to determine the critical load and the buckled shape of the column, we will d2 y apply the equation EI 2 M, which relates the d x internal moment in the column to its deflected shape. Recall that this equation assumes that the slope of the elastic curve is small and that deflections occur only by bending. 7
The two constants of integration are determined from the boundary conditions at the ends of the column. Since y = 0 at x = 0, then C2 0 . And since y = 0 at x = L, then
14
It is also important to realize that a column will buckle about the principal axis of the cross section having the least moment of inertia (the weakest axis). For example, a column having a rectangular cross section, as shown in figure, will buckle about the a-a axis, not the b-b axis.
y M P P
P
x (b)
P y x
8
L
(a)
Summing moments, the internal moment is:
M Py
2
Therefore, we have:
y P x
M
d y EI 2 Py d x
d2 y P y 0 2 d x EI
Consider the three smooth surfaces shown in figure. The surfaces are all horizontal at point B. A sphere will be in equilibrium at B. If the sphere is now moved to point C, there are three possibilities.
P 0 C1 sin L EI
10
This equation is satisfied if C1 0; however, then y = 0, which is a trivial solution that requires the column to always remain straight. The other possibility is:
Columns
§17–1 Stable and Unstable Equilibrium
§17–2 Ideal Column with Pin Supports
§17–3 Effective Length of Columns
§17–4 Design Formulas
2
§17-1 Stable and Unstable Equilibrium
P a b b a
15
Summarizing the above discussion, the Euler formula for a pin-supported long slender column can be written, 2 EI Pcr 2 L Pcr: maximum axial load on the column just before it begins to buckle. This load must not cause the stress in the column to exceed the proportional limit.
When the column is in its deflected position as shown in figure (a), the internal bending moment can be determined by using the method of sections. The free-body diagram of a segment in the deflected position is shown in figure (b). Here both the deflection y and the internal moment M are shown in the positive direction according to the sign convention.
P 源自文库0 sin L EI
which is satisfied if
P L n EI
11
n EI P , n 1, 2, 3, 2 L The smallest value of P is obtained when n = 1, so the critical load for the column is therefore 2 EI Pcr 2 L This load is sometimes referred to as the Euler load, after the Swiss mathematician Leonhard Euler, who originally solved this problem in 1757. 12
§17-2 Ideal Column with Pin Supports
The critical buckling load for a column that is pin supported as shown in figure will be determined. An ideal column is perfectly straight before loading, is made of homogeneous material, and upon which the load is applied through the centroid of the cross section. The material behaves in a linear-elastic manner and the column buckles or bends in a single plane.
13
It should also be noted that the loadcarrying capacity of a column will increase as the moment of inertia of the cross section increases. Thus, efficient columns are designed so that most of the column’s crosssectional area is located as far away as possible from the principal centroidal axes for the section. This is why hollow sections such as tubes are more economical than solid sections.
Long slender members subjected to an axial compressive force are called columns, and the lateral deflection that occurs is called buckling. Quite often the buckling of a column can lead to a sudden and dramatic failure of a structure or mechanism, and as a result, special attention must be given to the design of columns so that they can safely support their intended loadings without buckling. The maximum axial load that a column can support when it is on the verge of buckling is called the critical load. 5
P y
x
This is a homogeneous, second-order, linear differential equation with constant coefficients. The general solution is:
9
P P C2 cos y C1 sin x x EI EI
C B C B C
B
(a) (b) Stable equilibrium Neutral equilibrium
(c) Unstable equilibrium
3
Stability: Capacity of a component or a structural element to retain the original state of equilibrium. In previous chapters we have discussed some of the methods used to determine a member’s strength and deflection, while assuming that the member was always in stable equilibrium. Some members, however, may be subjected to compressive loadings, and if these members are long and slender the loading may be large enough to cause the member to deflect laterally or sideway. 4
2 2
or
It should be noted that the critical load is independent of the strength of the material; rather it depends only on the column’s dimensions (I and L) and the material’s stiffness or modulus of elasticity E. For this reason, as far as elastic buckling is concerned, columns made, for example, of high-strength steel offer no advantage over those made of lower-strength steel, since the modulus of elasticity for both is approximately the same.
P P
6
Whether or not a column will remain stable or become unstable when subjected to an axial load will depend on its ability to restore itself, which is based on its resistance to bending. Hence, in order to determine the critical load and the buckled shape of the column, we will d2 y apply the equation EI 2 M, which relates the d x internal moment in the column to its deflected shape. Recall that this equation assumes that the slope of the elastic curve is small and that deflections occur only by bending. 7
The two constants of integration are determined from the boundary conditions at the ends of the column. Since y = 0 at x = 0, then C2 0 . And since y = 0 at x = L, then
14
It is also important to realize that a column will buckle about the principal axis of the cross section having the least moment of inertia (the weakest axis). For example, a column having a rectangular cross section, as shown in figure, will buckle about the a-a axis, not the b-b axis.
y M P P
P
x (b)
P y x
8
L
(a)
Summing moments, the internal moment is:
M Py
2
Therefore, we have:
y P x
M
d y EI 2 Py d x
d2 y P y 0 2 d x EI
Consider the three smooth surfaces shown in figure. The surfaces are all horizontal at point B. A sphere will be in equilibrium at B. If the sphere is now moved to point C, there are three possibilities.
P 0 C1 sin L EI
10
This equation is satisfied if C1 0; however, then y = 0, which is a trivial solution that requires the column to always remain straight. The other possibility is: