1

1

Given one form of the element force-displacement relationship, it is possible to obtain alternate forms through simple operations. These transformations rely only on general mathematical mani-pulations and the principles of equilibrium and energy. They apply not only to a single element but also to complete systems, the members of which undergo com-patible deformation. They are developed here because they are particularly useful in forming the framework element stiffness matrix.

Flexibility-Stiffness Transformations

2

Consider the three bar truss assemblage shown in Fig. 4.5. Stiffness equations can be generated for the unsupported structure of Fig. 4.5(a) that can be expressed as

(4.11)f ff fs f

s sf ss s

{F } [k ] [k ] { }

{F } [k ] [k ] { }

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where subscript f designates components associated with the free dof and subscript s designates components associated with the support dof. Considering Fig. 4.5(b), the vectors in (4.11) are

Tf x1 y1 x2{F } F F F

Ts y2 x3 y3{F } F F F

Tf 1 1 2{ } u v u

Ts 2 3 3{ } v u v

(4.12)

Since the support displacementsare zero, (4.11) can be rewrittenas

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f fff

s sf

{F } [k ]{ }

{F } [k ]

(4.13)

The unknowns in (4.13) are the nodal displacements . These displacements can be calculated in terms of the applied loads at the free dof by solving the upper partition of (4.13)

f{ }

1f ff f f{ } [k ] {F } [d]{F } (4.14)

Solving the lower partition of (4.13)

s sf f sf f{F } [k ]{ } [k ][d]{F }

will provide the support reactions for zero support displacements.

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5

Equation (4.14) shows that the element flexibility matrix [d] equals the inverse of the stiffness matrix associated with the free displace-ment and force variables. Such a restriction is necessary for flexibility equations since such equations are only defined for stable structures and normally restrict attention to statically determinate structures. A stable structure is necessary since flexibility coefficient dij equals the release i displacement due to a unit virtual force at release j. If the structure is not stable, then infinite, rigid body displacements will result. 6

This is not the case for stiffnesscoefficient kij since it representsthe force at displacement dof i dueto a unit virtual displacement at dofj with all other displacements equalto zero. Since all displacementsare prescribed in the stiffnessanalysis, rigid body motion is pre-vented in determining the stiffnesscoefficients.

Note also that when an element orstructure is supported in a staticallydeterminate fashion, the comple-mentary strain energy has a valuethat is independent of the specificform of the support system.

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However, if a statically indeter-minate support condition is used,then the complementary strainenergy is dependent on upon thesupport condition. This meansthat the complementary strainenergy is different for each suchsupport system. Since the globalanalysis equations can be obtainedthrough a summation of elementstrain energies (see Chapter 6,MGZ), element strain energiesshould be independent of localelement support conditions.

8

Equations (4.11) – (4.14) led to the flexibility equations of (4.14) by starting with the stiffness equations and reducing the equations to a stable, statically determinate structure. Reversing the procedure and making use of equilibrium can be used to construct the complete stiffness equations of (4.11). This process begins by pre-multiplying both sides of (4.14) by 1[d] .

1f f ff f{F } [d] { } [k ]{ } (4.15)

Recalling that the columns of the stiffness matrix are in equilibrium, the lower partition of the stiffness

3

9

(4.22)

matrix [ksf] can be obtained as

s sf f ff f1

f

{F } [k ]{ } [ ][k ]{ }

[ ][d] { }

1

sf[k ] [ ][d]

where [] = equilibrium matrix and is generated from

s f{F } [ ]{F } (4.19)

s{ }

To complete the construction of (4.11), the matrices that pre-multiply

must be constructed, i.e.,and . Remembering that the stiffness matrix is symmetric:

fs[k ]

ss[k ]

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T 1 T Tfs sf

1 T

[k ] [k ] ([d] ) [ ]

[d] [ ]

(4.23)

Note the transpose of a symmetric matrix equals the original matrix. Since [d]-1 is symmetric, its trans-pose equals itself. Also recall that the transpose of a matrix product is the reverse product of each matrix transposed.

Construction of [kss] follows the same procedure used to construct [ksf]:

1 Tss fs[k ] [ ][k ] [ ][d] [ ] (4.24)

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Combining (4.11), (4.16), (4.22), and (4.24), the stiffness matrix can be expressed as

ff fs

sf ss

1 1 T

1 1 T

[k ] [k ][k]

[k ] [k ]

[d] [d] [ ]

[ ][d] [ ][d] [ ]

(4.25)

Equation (4.25) will be used in the next section to construct the stiffness matrix for a three-dimensional frame element.

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However, the best application of (4.25) is in generating “exact” stiffness equations for problems that are difficult to construct “exact” kinematic fields but for which the generation of exact equilibrium equations is fairly easy. Examples include non-prismatic beams, circular beams, flexibly connected beams, etc. These topics will be covered in subsequent chapters.

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In summary, (4.25) shows that the stiffness matrix can be constructed from the inverse of the flexibility matrix [d] and a matrix that derives from the element static equilibrium relationships – the equilibrium matrix []. The property of symmetry was invoked in constructing [kfs] from [ksf]. Equation (4.25) shows that matrix [kss] is obtained through a matrix triple product in which the pre-multiplier of the central matrix is equal to the transpose of the post-multiplier matrix. This triple product is known as a congruent transformation,

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which produces a symmetric matrix when the central matrix is sym-metric. Thus, [kss] is assured to be symmetric. We have made extensive use of congruent transformations in this course.

Equation (4.25) is a general formula for transformation from flexibility to stiffness form that includes rigid body motion degrees of freedom. The number s of support forces is dictated by the requirements of stable, statically determinate support. However, the number f of free degrees of freedom is essentially limitless.

3D Framework Element Stiffness Matrix

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In this section, we shall ignore deformations resulting from shear since for most frame members such deformations are small (less than 1% for sections with a length to depth ratio of 20). Furthermore, displacements resulting from out-of-plane (longitudinal) warping of a cross section that torsional forces may cause will also be ignored since such deformations are typically small for most steel members and essentially zero for all solid

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members. Figure 4.6 illustrates the right-hand sign convention for both the displacements and forces. The local x-axis coincides with the centroidal axis of the element. The local y-axis is taken as the weak or minor principal cross section axis (same as used in the AISC steel design manual) and the local z-axis is taken to be the strong or major principal cross section axis (note:this is the x-axis in the AISC steel design manual).

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The 12 degree-of-freedom element shown in Fig. 4.6 is straight, prismatic, and symmetrical about both principal cross section axes, referred to as a bisymmetrical section. For such a member, the shear center coincides with the centroid. In addition, four distinct and uncoupled deformation modes are appropriate: axial, flexural about each principal cross section

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axis, and torsion. Uncoupled mode means that a particular force vector only causes a displacement in the same vector direction. For example, Fx1 and Fx2 in Fig. 4.6 can only cause axial deformation and hence only axial displacements u1

and u2 can be produced by these two forces. Also, the torsional moments (torques) Mx1 and Mx2

only cause rotational displacements x1 and x2 in the element. Similarly, for bending about the y-axis, only forces Fz1, Fz2, My1 and My2 cause displacements w1, w2,y1 and y2 in the element.

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Lastly, for bending about the z-axis only forces Fy1, Fy2, Mz1 and Mz2

cause displacements v1, v2, z1 and z2 in the element. Uncoupled response for the bending deforma-tion modes is known from elemen-tary strength of materials where we learned that bending moments and shearing forces in one principal plane do not cause any deforma-tion outside the principal plane.

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In generating the stiffness matrix for the 12 dof of Fig. 4.6, we will develop four distinct submatrices: axial, torsion, bending about the z-axis, and bending about the y-axis. This will result in a 12x12 stiffness matrix that contains zeros in the locations where the various forces and displacements do not interact, i.e., are uncoupled.

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Axial Force Element Stiffness Equations

21

Fig. 4.7 Axial Force Element. (a)Support system.

(b) Free-body diagram.

Consider the axial force element shown in Fig. 4.7. The axial displacement at element node 1 in Fig. 4.7(a) is calculated as

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L L Lx1 x1

10 0 0

F F Lu edx dx dx

E EA EA

For this case, [d] = and {Ff} =

Fx1. Thus, [kff] = [d]-1 = The

equilibrium matrix is obtained from

Fig. 4.7(b): Fx2 = - Fx1 which shows

{Fs} = Fx2 and [] = -1. Substituting

[d]-1 and [] into (4.25):

LEA

EA.

L

a1 1EA

[k ]1 1L

(4.26a)

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x1 1

x2 2

and

F u1 1EAF u1 1L

(4.26b)

where subscript “a” has been added to the stiffness matrix of (4.26a) to emphasize axial deformation.

Pure Torsion Element Stiffness Equations

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Fig. 4.8 Pure Torsion Element. (a)Support system.

(b) Free-body diagram.

Mathematically, the pure torsion element is nearly identical to the axial force element since only two forces and displacements are involved in defining the response of each mode of deformation (compare Figs. 4.7 and 4.8).

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25

L Lx1 x1

x10 0

M M Ldx dx

GJ GJ

LGJ

Obviously, there are differences in response since the axial force element experiences a change in element length whereas the pure torsion element twists. From strength of materials, we know that the torsional moment Mx1 is related to the angle of twist as

J = torsion constantsection, b = component width and t = component thickness; = polar moment of inertia for closed

sections). The expression for x1

shows that [d] = and from Fig.

31bt for an open

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t1 1GJ

[k ]1 1L

(4.27a)

4.8(b), [] = -1, i.e., Mx2 = -Mx1. Substituting into (4.25):

x1 x1

x2 x2

and

M 1 1GJM 1 1L

(4.27b)

where subscript “t” on the element stiffness matrix signifies torsion.

Element Bending Stiffness Equations about z-Axis

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The stresses and strains at any cross section caused by bending are directed along the x-axis of the element, vary linearly with respect to the y-axis, and are constant in the x direction for a fixed value of y. In elementary mechanics, it is shown that the strain ex is expressed as

2

x 2z

y d ve y

dx

(4.28)

where z = radius of curvature about the z-axis, which is approximated as 28

1/(d2v/dx2). (NOTE: Positive curvature z = 1/z causes tension in the bottom fibers and compression in the top fibers.) From Hooke’s law

2

x x 2d v

Ee Eydx

(4.29)

Considering the pure bending case results in no resultant axial force, only the resultant bending moment Mz:

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z x 2A A

2 22

z2 2A

d vydA E y dA

dx

d v d vE y dA EI

dx dx

M

(4.31)

where Iz = second moment of area about the z-axis.

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Fig. 4.9 z-Axis Bending Element. (a)Support system.

(b) Free-body diagram.

Consider the beam of Fig. 4.9. The flexibility equations are expressed as

y11

z1 z1

Fv[d]

M

(bz.1)

Calculation of the flexibility coefficients can be expressed as

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vi vjij

zL

M (x)M (x)d dx

EI (bz.2)

where Mvi and Mvj are the virtual moment equations for dof i and j, respectively. (A curvature moment expression is included in your class notes.) For the cantilever beam of Fig. 4.9(a), the virtual moment equations are

Mv1 = xMv2 = -1

where x is measured from element node 1. Substituting (bz.3) into

(bz.3)

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(bz.2) for i, j = 1,2 and those results into (bz.1) leads to the following flexibility and inverse flexibility matrices

3 2

2z

L L1 3 2[d]

EI L L2

(bz.4)

1 z23

z z3 2

z z2

L1 212EI[d]

L LL2 3

12EI 6EI

L L6EI 4EI

LL

(bz.5)

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To complete the stiffness matrix calculations of (4.25), we need to calculate the equilibrium matrix. Expressing the support forces {Fs} (= <Fy2 Mz2>T) of Fig. 4.9(b) in terms of the free forces {Ff} (= <Fy1 Mz1>T) leads to

y y2 y1

2 z2 y1 z1

F 0 F F

M 0 M F L M

In matrix form:

y2 y1

z2 z1

F F1 0

L 1M M

1 0[ ]

L 1

(bz.6)

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Substituting (bz.5) and (bz.6) into(4.25) gives

y1 12 2

z1 z1z3

y2 22 2 z2z2

12 6L 12 6LF v

M 6L 4L 6L 2LEIF v12 6L 12 6LL

M 6L 2L 6L 4L

(4.32)

2 2z

bz 3

2 2

12 6L 12 6L

6L 4L 6L 2LEI[k ]

12 6L 12 6LL

6L 2L 6L 4L

subscript bz emphasizes that the stiffness matrix is for bending about the z-axis.

Element Bending Stiffness Equations about y-Axis

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Fig. 4.10 y-Axis Bending Element.

Figure 4.10 shows the displace-ment and force variables for bending about the y-axis. Com-paring this figure with Fig. 4.9 shows that the rotational displace-ments about the y-axis are the

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negative of the transverse displace-ment gradient whereas the rota-tional displacements about the z-axis equals the transverse dis-placement gradient, i.e.,

ydwdx

zdvdx

and

This minor complication is unavoid-able if a consistent right-hand-rule for displacements is to be adopted. Since y-axis bending is the same as z-axis bending, with the excep-tion stated, the stiffness equations for y-axis bending can be expressed as

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z1 12 2

y1 y1y3

z2 22 2y2 y2

12 6L 12 6LF w

M EI 6L 4L 6L 2L

F 12 6L 12 6L wLM 6L 2L 6L 4L

(4.33)

2 2y

by 3

2 2

12 6L 12 6L

EI 6L 4L 6L 2L[k ]

12 6L 12 6LL

6L 2L 6L 4L

subscript by emphasizes bending about the y-axis.

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Complete Element Stiffness Equations

37

Combining the stiffness equations of (4.26b), (4.27), (4.32), and (4.33) into a 12 x 12 system of stiffness equations results in (4.34). Zeros in this matrix identify that the local coordinate stiffness equation (equilibrium) does not interact with the degree of freedom that is multiplied by the zero terms.

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Such uncoupling results from using principal cross section coordinates and is true even for unsymmetricsections. However, for unsym-metric or singly symmetric sections, force variables Fx, My, and Mz as well as displacement variables u, y, and z are expressed in terms of the cross section centroidal axis. Force variables Fy, Fz, and Mx as well as displacement variables v, w, and x are expressed in terms of the shear center axis for the element. See 3D rotation notes on how to obtain all the variables at the element centroid.

Exact Nonprismatic Beam Stiffness Equations

40

Recall that the element stiffness matrix can be expressed in terms of the element flexibility matrix [d] and equilibrium matrix [] as shown in (4.25). Exact stiffness equations can be generated provided the

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flexibility and equilibrium matrices are exact. Considering the generic nonprismatic beam element of Fig. 1, the free and support forces vectors are

T2z1z1xf MMF}F{

T2y1y2xs FFF}F{

(2a)

(2b)

An exact flexibility matrix for the element of Fig. 1 can be construct-ed using the principle of virtual forces (see Section 7.6 in McGuire, Gallagher and Ziemien, 2000)

L

0

1T dx]Q[]C[]Q[]d[ (3)

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L

x1

L

x0

001]Q[where = force

interpolation matrix (i.e., interpo-

lation of the internal member axial

force Fx and bending moment Mz),

= inverse material

stiffness matrix or material compliance matrix; A(x) = cross section variation of area; and Iz(x) = z-axis moment of inertia varia-tion. Evaluating the matrix triple product of (3) leads to

)x(EI10

0)x(EA1

]C[

z

1

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11

22 23

32 33

1

2 3

3 4

d 0 0

[d] 0 d d

0 d d

d 0 0

0 d d

0 d d

(4)

where the unique flexibility coefficients di are calculated as

L

01 dx

)x(EA

1d

L

0 z

Lx

Lx2

2 dx)x(EI

)1(d

22

L

0 z

Lx

Lx

3 dx)x(EI

)(d

22

L

0 z

Lx

4 dx)x(EI

d2

2

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Closed-form evaluation of (4) is not possible for arbitrary cross sections, e.g., local buckling of component plates making up a cold-form section which are dependent on the compressive stress levels. Consequently, the flexibility coefficients of (4) must be evaluated numerically. A reliable integration scheme is Labatto quadrature, in which the abscissa (-1 xi 1) and weighting coefficients (Ai) are given in Table 1 for N = 2 to N = 10 integration points.

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45

Table 1. Labatto Quadrature Data

46

Once the flexibility coefficients of (4) are evaluated, the inverse flexibility matrix is simply

43

32

1

Dd

Dd

Dd

Dd

d1

1

kk0

kk0

00k

0

0

00

]d[

23

34

1

(5)

where D = d2 d4 – d3d3. The complete stiffness matrix also requires calculation of the equilibrium matrix

s f

x2 x1

y1 z1

z2y2

{F } [ ]{F }

F L 0 0 F1

F 0 1 1 ML

0 1 1 MF

(6)

Exact Fixed-End Force Vector for the

Nonprismatic Beam

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Using the results of (5) and (6) in (4.25) results in an “exact” stiffness matrix formulation for the nonprismatic element of Fig. 1.

Exact fixed-end forces at the free and support dof are generated as

L

0

1TT2z1z1f dx)}x(P{]C[]Q[u}{ (7)

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where {P(x)} = <N(x) Mz(x)>T = internal force vector distribution for the axial stress resultant N(x) and moment stress resultant Mz(x).

Fig. 2. Nonprismatic Beam Subjected to Uniform Load

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49

Consider the uniform load case depicted in Fig. 2, the axial force distribution is zero and the internal bending moment is

Substituting these force resultants in (7) gives

2

zqL q x

M (x) x2 2

dxEI

)xLx(

2

q

dxEI

)xLx()1(

2

q

0u

L

0 z

2L

x

2z

L

0 z

2L

x

1z

1

(8)

Using the flexibility equations

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}{}F{]d[ ff (9)

where for this case {Ff} = {FfE} =

equivalent element nodal forces at the free force dof. Substituting this observation into (9) and solving

E F 1f f f{F } {F } [d] { } (10)

The fixed-end forces at the support degrees of freedom {Fs

F} are obtained by simply considering equilibrium

}{}F{][}F{ qFf

Fs (12)

where { } = qL/2 <0 1 1>T =

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= uniform distributed load contri-bution to equilibrium as shown in Fig. 2. Again, for arbitrary moment of inertia variation, the free dofdisplacements should be evaluated using Labatto quadrature.

Rearranging the displacements such that they are ordered in the usual fashion ( ) leads to the following representation of the element stiffness matrix in terms of the unique stiffness coefficients

T2z221z11 vuvu}{

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(13)

where k1 = k1, k2 = (k1 + 2k3 + k4)/L2, k3 = (k2 + k3)/L, k4 = k2, k5 = (k3 + k4)/L, k6 = k3, k7 = k4; k1, k2, k3, and k4, are the stiffness coefficients in (5). Notice in (13) that the rotational stiffness coefficients k3

and k5 associated with the transverse displacement dof are not equal as is the case for a prismatic element; end rotational

1 1

2 3 2 5

3 3 6

1 1

2 3 2 5

5 6 5 7

0 0 0 0

0 0

0 4 0[k]

0 0 0 0

0 0

0 0

k -kk k -k kk k -k k

-k k-k -k k -kk k -k k

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53

stiffness coefficients k4 and k7 are not equal; and k6 ≠ k4/2 or ≠ k7/2.

Extension of the above concepts to three dimensions should be fairly straightforward. You need to consider torsion and bending about the weak axis separately. The equations for torsion will mirror the axial force case except that GJ(x) will replace EA(x). For bending about the y-axis (weak), replace Iz(x) with Iy(x) and take into account the sign change in the off-diagonal stiffness coefficients (see earlier section of these slides).