Page 143 - Handbook of Civil Engineering Calculations, Second Edition

1.126 STRUCTURAL STEEL ENGINEERING AND DESIGN

ANALYSIS OF A RECTANGULAR PORTAL
FRAME BY THE STATIC METHOD

Compute the plastic moment of the frame in Fig. 28a by using the static method.

Calculation Procedure:
1. Determine the relative values of the bending moments
Consider a bending moment as positive if the fibers on the interior side of the neutral
plane are in tension. Consequently, as the mechanisms in Fig. 28 reveal, the algebraic
sign of the plastic moment at a given section agrees with that of its angular displacement
during collapse.
Determine the relative values of the bending moments at B, C, and D. Refer to Fig. 29.
As previously found by statics, V A 10.36 kips (46.081 kN), M B 24H A , M C 24H A +
10V A ; therefore, M C M B + 103.6, Eq. a. Also, M D 24H A + 20V A – 74(10); M D M B
– 532.8, Eq. b; or M D M C – 636.4, Eq. c.
2. Assume the mode the failure in Fig. 28b
This requires that M B M D –M p . This relationship is incompatible with Eq. b, and the
assumed mode of failure is therefore incorrect.
3. Assume the mode of failure in Fig. 28c
This requires that M B M p , and M C < M p ; therefore, M C < M B . This relationship is in-
compatible with Eq. a, and the assumed mode of failure is therefore incorrect.
By a process of elimination, it has been ascertained that the frame will fail in the man-
ner shown in Fig. 28d.
4. Compute the value of M p for the composite mode of failure
Thus, M C M p , and M D – M p . Substitute these values in Eq. c. Or, –M p M p – 636.4;
M p 318.2 ft·kips (431.48 kN·m).

THEOREM OF COMPOSITE MECHANISMS

By analyzing the calculations in the calculation procedure before the last one, establish a
criterion to determine when a composite mechanism is significant (i.e., under what condi-
tions it may yield an M p value greater than that associated with the basic mechanisms).

Calculation Procedure:
1. Express the external and internal work associated with a given
mechanism
Thus, W E e , and W I iM p , where the coefficients e and i are obtained by applying
the mechanism method. Then M p e/i.
2. Determine the significance of mechanism sign
Let the subscripts 1 and 2 refer to the basic mechanisms and the subscript 3 to their com-
posite mechanism. Then M p1 e 1 /i 1 ; M p2 e 2 /i 2 .
When the basic mechanisms are superposed, the values of W E are additive. If the two
mechanisms do not produce rotations of opposite sign at any section, the values of W I are
also additive, and M p3 e 3 /i 3 (e 1 + e 2 )/(i 1 + i 2 ). This value is intermediate between M p1