Handbook of Civil Engineering Calculations

placement. In Fig. 33, D is displaced to D' and F to F'. Draw a straight line through A and
D intersecting the prolongation of GF at H.
Since A is the center of rotation of ABD 9 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 S 2 and O 3 in terms of Q 1. Thus,
   O 2 IO 1 =HDIAD; :. O 2 = O 1. Also, O 3 IO 1 = HFIGF = 49125; :. O 3 = 1.96O 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, O 0 =O 1 -^O 2 = 2O 1 ; Op=O 1 -^O 3 = 2.96O 1. Then W 1 = Mp (QD

• Op) = 4.96MpO 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 AA = 2502 = (^250) l 1.6O 1 IQQO 1 135.60!
C 34 151.2 A^=IOa 2 =IOa 1 3.0^ 1 34Oa 1 461.Oa 1
D 25 111.2 A, = 20^ 6.Ia 1 50Oa 1 678.Oa 1
E 22 97.9 A^=IOa 1 3.Oa 1 22Oa 1 298Ja 1
Total 116Oa 1 1572.9^

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

REDUCTION IN PLASTIC-MOMENT

CAPACITY CAUSED BYAXIAL FORCE

A WlO x 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); Py = axial force that would induce plastification if