Building Design and Construction Handbook - Merritt - Ventech!

STRUCTURAL THEORY 5.113
independent forces acting. Hence, the force-displacement relationship can be written
for this element as
�� � �
j j
Mi �i
�� ��
�i Mi
� � � f � fM (5.153)
� M
M � � k � k� (5.154)
M �
j j
The flexibility matrix f then will be a 2 � 2 matrix. The first column can be
obtained by setting M i � 1 and M j � 0 (Fig. 5.92b). The resulting angular rotations
are given by Eqs. (5.107) and (5.108): For a beam with constant moment of inertia
I and modulus of elasticity E, the rotations are � � L/3EI and � � �L/6EI.
Similarly, the second column can be developed by setting M i � 0 and M j � 1.
The flexibility matrix for a beam in bending then is
� �
L L
�
�
�
3EI 6EI L 2 �1
f � � (5.155)
L L 6EI �1 2
�
6EI 3EI
The stiffness matrix, obtained in a similar manner or by inversion of f, is
� �
4EI 2EI
� � L L 2EI 2 1
k � �
2EI 4EI
(5.156)
L 1 2
L L
Beams Subjected to Bending and Axial Forces. For a beam subjected to nodal
moments M i and M j and axial forces P, flexibility and stiffness are represented by
3 � 3 matrices. The load-displacement relations for a beam of span L, constant
moment of inertia I, modulus of elasticity E, and cross-sectional area A are given
by
�� ���� ��
� Mi Mi �i
�j � f Mj Mj � k � j
(5.157)
e P P e
In this case, the flexibility matrix is
L
2
f � �1
6EI� 0
�1
2
0
0
0
��
(5.158)
where � � 6I/A, and the stiffness matrix is
where � � A/I.
EI
k �
4
2
2
4
0
0 (5.159)
0 0 �
L � �