Question

Use Castigliano's theories to obtain vertical displacement of the truss point C.

E 3 m B C A 3 m 3 m 3 m 20 kN 20 kN LLI LL

Answer 1

1. Calculate support reactions and member forces due to applied loads

  

-P F E -P/2 -Pv2 PO 3 m P B C A P P P OL

we used a method of sections and method of joints to determine the normal force in each member.These forces caused by the real loads acting on the truss.

There are upward forces at joint A and D and downward forces at joint B and C and P's represents the tensile or compressive forces. P's at AB, BC and CD are tension forces. At EC the P is internal force and negative signs indicates compressive.

Apply virtual load, remove real loads, and calculate support reactions and member forces

-1/3 F E - V2/3 :2/2/3 1 - V2/3 1/3 1/3 B 2/3 2/3 C D A 1 kN 1/3 2/3

An unit force 1kN is applied at point C

Apply virtual work and solve for deflection at C

Actual Load	20 kN
Virtual Load	1 kN
cross section area (A)	4.5m2
Young's Modulus E	2x108 kN/m2

Members	n (kN)	N (kN)	L (m)	nNL kN^2m
AB	0.333	20	3	19.98
BC	0.667	20	3	40.02
CD	0.667	20	3	40.02
DE	-0.943	-28.284	4.242	113.141827
FE	-0.333	-20	3	19.98
EB	0.471	0	4.242	0
BF	0.333	20	3	19.98
AF	-0.471	-28.284	4.242	56.5109229
CE	1	20	3	60
			ΣΜ	369.632749

369.63N2 m 1.A nN A.E AE

369.63kN2.m 4.1 x 10kN.m 1kN.Ac 4.5m2 x 2 x 108kN/m2

A=4.1 x 10 m Cu

The vertical displacement at joint C of the steel truss is 4.1 x 10⁷ m