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Frame Element Stiffness Matrices

CEE 421L. Matrix Structural Analysis

Department of Civil and Environmental EngineeringDuke University

Henri P. Gavin

Fall, 2014

Truss elements carry axial forces only. Beam elements carry shear forcesand bending moments. Frame elements carry shear forces, bending moments,and axial forces. This document picks up with the previously-derived trussand beam element stiffness matrices in local element coordinates and proceedsthrough frame element stiffness matrices in global coordinates.

1 Frame Element Stiffness Matrix in Local Coordinates,k

A frame element is a combination of a truss element and a beam element.The forces and displacements in the local axial direction are independent of theshear forces and bending moments.

N1

V1

M1

N2

V2

M2

=

q1

q2

q3

q4

q5

q6

=

EAL

0 12EI

L3

sym

0 6EI

L24EI

L

EAL

0 0 EA

L

0 12EI

L3

6EIL2

0 12EI

L3

0 6EI

L22EI

L 0

6EIL2

4EIL

u1

u2

u3

u4

u5

u6

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2 CEE 421L. Matrix Structural Analysis Duke University Fall 2014 H.P. Gavin

2 Relationships between Local Coordinates and Global Coordinates: T

The geometric relationship between local displacements, u, and global dis-placements, v, is

u1

=v1

cos +v2

sin u2

=

v1

sin +v2

cos u3

=v3

or, u= T v.

The equilibrium relationship between local forces, q, and global forces, f,is

q1=f1 cos +f2 sin q2= f1 sin +f2 cos q3=f3

or, q=T f, where, in both cases,

T=

c s 0

s c 0 00 0 1

c s 00 s c 0

0 0 1

c= cos = x2

x1L

s= sin =y2 y1

L

The coordinate transformation matrix, T, is orthogonal, T1 =TT.

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Frame Element Stiffness Matrices 3

3 Frame Element Stiffness Matrix in Global Coordinates: K

Combining the coordinate transformation relationships,

q = k u

T f = k T v

f = TT k T v

f = K v

which provides the force-deflection relationships in global coordinates. Thestiffness matrix in global coordinates is K= TT k T

K=

EA

L c2 EA

L cs

EA

L c2

EA

L cs+12EI

L3 s2 12EI

L3 cs 6EI

L2s 12EI

L3 s2 +12EI

L3 cs 6EI

L2s

EAL

s2 EAL

cs EAL

s2

+12EIL3

c2 6EIL2

c +12EIL3

cs 12EIL3

c2 6EIL2

c

4EIL

6EIL2

s 6EIL2

c 2EIL

EAL

c2 EAL

cs

+12EIL3 s2 12EIL3 cs

6EIL2 s

sym

EAL

s2

+12EIL3

c2 6EIL2

c

4EIL

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4 Frame Element Stiffness Matrices for Elements with End-Releases

Some elements in a frame may not be fixed at both ends. For example,an element may be fixed at one end and pinned at the other. Or, the element

may be guided on one end so that the element shear forces at that end are zero.Or, the frame element may be pinned at both ends, so that it acts like a trusselement. Such modifications to the frame element naturally affect the elementsstiffness matrix.

Consider a frame element in which a set of end-coordinates r are released,and the goal is to find a stiffness matrix relation for the primary p retainedcoordinates. The element end forces at the released coordinates, qr are all zero.One can partition the element stiffness matrix equation as follows

qp

qr

kpp kpr

krp krr

up

ur

The element displacement coordinates at the released coordinates do not equalthe structural displacements at the collocated structural coordinates, since thecoordinates r are released. Since the element end forces at the released coor-dinates are all zero (qr = 0), the element end displacements at the releasedcoordinates must be related to the displacements at the primary (retained)

coordinates as:

ur = k1

rr krpup

The element end forces at the primary coordinates The element stiffness matrixequation relating qp and up is

qp=kpp kprk

1

rr krp

up

The rows and columns of the released element stiffness matrix correspondingto the released coordinates, r, are set to zero. The rows and columns of thereleased element stiffness matrix corresponding to the retained coordinates, p,are [kpp kprk

1

rr krp]. This is the element stiffness matrix that should assembleinto the structural coordinates collocated with the primary (retained) coordi-natesp. The following sections give examples for pinned-fixed and fixed-pinnedframe elements. Element stiffness matrices for many other end-release casescan be easily computed.

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Frame Element Stiffness Matrices 5

4.1 Pinned-Fixed Frame Element in Local Coordinates,k . . . (r= 3)

k=

EAL

0 0 EAL

0 0

3EIL3 0 0

3EIL3

3EIL2

0 0 0 0

EAL

0 0sym

3EIL3

3EIL2

3EI

L

4.2 Pinned-Fixed Frame Element in Global Coordinates,K= TTkT

K=

EAL

c2 EAL

cs 0 EAL

c2 EAL

cs 3EIL2

s

+3EIL3

s2 3EIL3

cs 3EIL3

s2 +3EIL3

cs

EAL

s2 0 EAL

cs EAL

s2 3EIL2

c

+3EI

L3 c2

+3EI

L3 cs

3EI

L3 c2

0 0 0 0

EAL

c2 EAL

cs EIL2

s

+3EIL3

s2 3EIL3

cs

sym

EAL

s2 3EIL2

c3EIL3

c2

3EIL

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4.3 Fixed-Pinned Frame Element in Local Coordinates,k . . . (r= 6)

k=

EAL 0 0

EAL 0 0

3EIL3

3EIL2

0 3EIL3

0

3EIL

0 3EIL2

0

EAL

0 0sym

3EIL3

0

0

4.4 Fixed-Pinned Frame Element in Global Coordinates,K= TTkT

K=

EAL

c2 EAL

cs 3EIL2

s EAL

c2 EAL

cs 0+3EI

L3s2 3EI

L3cs 3EI

L3s2 +3EI

L3cs

EAL

s2 3EIL2

c EAL

cs EAL

s2 0+3EI

L3c2 +3EI

L3cs 3EI

L3c2

3EIL

3EIL2

s 3EIL2

c 0

EAL

c2 EAL

cs 0+3EI

L3s2 3EI

L3cs

sym

EAL

s2 0+3EI

L3c2

0

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Frame Element Stiffness Matrices 7

5 Notation

u = Element deflection vector in the Local coordinate systemq = Element force vector in the Local coordinate systemk = Element stiffness matrix in the Local coordinate system

... q=k uT = Coordinate Transformation Matrix

... T1 =TT

v = Element deflection vector in the Global coordinate system... u= T v

f = Element force vector in the Global coordinate system... q=T f

K = Element stiffness matrix in the Global coordinate system

... K= TT

k Td = Structural deflection vector in the Global coordinate systemp = Structural load vector in the Global coordinate system

Ks = Structural stiffness matrix in the Global coordinate system... p= Ks d

Local Global

Element Deflection u v

Element Force q fElement Stiffness k K

Structural Deflection - d

Structural Loads - p

Structural Stiffness - Ks

For frame element stiffness matrices including shear deformations, see:J.S. Przemieniecki, Theory of Matrix Structural Analysis, Dover Press, 1985.

(... a steal at $12.95)

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