Page 150 - Handbook of Civil Engineering Calculations, Second Edition

STRUCTURAL STEEL DESIGN 1.133

Since A is the center of rotation of ABD, DD
is normal to AD and HD; since G is the
center of rotation of GF, FF
is normal to GF and HF. Therefore, H is the instantaneous
center of rotation of DF.
2. Record the pertinent dimensions and rotations
Record the dimensions a, b, and c in Fig. 33, and express 2 and 3 in terms of 1 . Thus,
2 / 1 HD/AD; 2 1 . Also, 3 / 1 HF/GF 49/25; 3 1.96 1 .
3. Determine the angular displacement, and evaluate the
internal work
Determine the angular displacement (in absolute value) at D and F, and evaluate the inter-
nal work in terms of 1 . Thus, D 1 + 2 2 1 ; F 1 + 3 2.96 1 . Then W I M p
( D + F ) 4.96M p 1 .
4. Apply the theorem of virtual displacements to determine the
displacement of each applied load
Determine the displacement of each applied load in the direction of the load. Multiply the
displacement by the load to obtain the external work. Record the results as shown:

Displacement in direction
Load of load External work
Section kips kN ft m ft·kips kN·m
B 4 17.8 h 25 2 25 1 7.6 1 100 1 135.6 1
C 34 151.2 v 10 2 10 1 3.0 1 340 1 461.0 1
D 25 111.2 v 20 1 6.1 1 500 1 678.0 1
E 22 97.9 v 10 1 3.0 1 220 1 298.3 1
________ ___________
Total 1160 1 1572.9 1

5. Equate the external and internal work to find M p
Thus, 4.96M p 1 1160 1 ; M p 234 ft·kips (317.3 kN·m).
Other modes of failure may be assumed and the corresponding value of M p computed
in the same manner. The failure mechanism analyzed in this procedure (plastic hinges at
D and F) yields the highest value of M p and is therefore the true mechanism.

REDUCTION IN PLASTIC-MOMENT
CAPACITY CAUSED BY AXIAL FORCE

A W10 45 beam-column is subjected to an axial force of 84 kips (373.6 kN) at ultimate
load. (a) Applying the exact method, calculate the plastic moment this section can devel-
op with respect to the major axis. (b) Construct the interaction diagram for this section,
and then calculate the plastic moment by assuming a linear interaction relationship that
approximates the true relationship.

Calculation Procedure:
1. Record the relevant properties of the member
Let P applied axial force, kips (kN); P y axial force that would induce plastification if
acting alone, kips (kN) Af y ; M p
plastic-moment capacity of the section in combina-
tion with P, ft·kips (kN·m).