CVEN 345 -- Lecture 14

Analysis of statically indetrminate stuctures
Force Method -- Trusses

Welcome to the page on lecture 14, which I am setting up as an outline of the material in lecture 14 on March 2, 2000. The information will help you to focus on what is important. If you have questions, email me!

Analysis of statically indeterminate structure

A structure is statically indeterminate when the number of unknowns exceeds the number of equilibrium equations in the analysis. Indeterminancy may arise as a result of added supports or members.

Although analysis is more involved than statically determinate structures. There are some major advantages to statically indeterminate structure.

Advantages

• The maximum stresses and deflections in indeterminate structures are smaller than its statically determinate counterpart.
• Statically determinate structures have a tendency to redistribute its load to redundant supports in the case of faulty design or overloading occurs.
• Statically indeterminate structures can support loads with support loading on thinner members with increased stability. (Cost $ saving)

Disadvantages

• More costly to make an indeterminate structure
• Redundant structures can induce problems such as differential displacement i.e. A stress created by settlement of a support.

Method of Analysis of indeterminate structures

It is necessary to satisfy equilibrium, compatibility, and force displacement requirements.

• Equilibrium reaction forces
• Compatibility for multiple supports.
• Force Displacement requirements (two methods).
  1. Force method with consistant deformation
  2. Displacement method with consistant force.

We will look at the force method first using virtual work to analysis structures.

Force Method

1. Establish the degree of indeterminacy of the structure.
2. Remove sufficient number of unknown forces fro the indeterminate structure to make the structure determinate and stable. This structure is known as the primary structure and the extra forces that are removed are known as redundants.
3. Compute the unrestrained deformation of the primary structure at each redundant location.
4. Find the compatibility conditions for solving the structure.
5. Solve for the compatibility conditions.

For a single redundant reaction component we can use:

dD + VDddd	=	0
VD	=	- dD / ddd
		&
Fi	=	F'iprimary + VD * Fvi

where,

dD	--	Displacement in the redundant location
ddd	--	Displacement in the force applied at the redundant location
VD	--	Magnitude of the force required to obtain the compatibility condition.
Fi	--	Internal force in the member
Fvi	--	Virtual internal force in the truss
F'primaryi	--	Internal forces due to the primary loading in the member

Example problem of statically indeterminate truss -- External Redundant Support (1)

The truss has an additional support and uses virtual work to find the force required to meet the compatibility condition, i.e. the deflection at the redundant force is zero.

Example problem of statically indeterminate truss -- Internal Redundant Member (1)

The truss is statically determinate however the interior is indeterminate. If the problem solves it similar to external redundant supports.

More than one redundant member

The problem requires that you remove the redundant structures. For this example setup, we will deal with two redundant members. Remove the two redundant members from the truss and solve for the primary structure. There will be two compatibility conditions:

dB + VBdbb + VCdbc	=	0
dC + VBdcb + VCdcc	=	0

where,

dB	--	Displacement at B due to the primary forces
dC	--	Displacement at C due to the primary forces
dbb	--	Displacement at B due to the force applied at B
dbc	--	Displacement at B due to the force applied at C
dcb	--	Displacement at C due to the force applied at B
dcc	--	Displacement at C due to the force applied at C
VB	--	Magnitude of the force applied at B
VC	--	Magnitude of the force applied at C

We can substitute for the d terms in summations and two compatibility conditions.

S F'i Fvbi Li / Ai Ei	+	VB S Fvbi Fvbi Li / Ai Ei	+	VC S Fvbi Fvci Li / Ai Ei	=	0
S F'i Fvci Li / Ai Ei	+	VB S Fvci Fvbi Li / Ai Ei	+	VC S Fvci Fvci Li / Ai Ei	=	0

This can be written in a matrix vector type format. So it resembles:

	Fvb	Fcb
Fvb	A11	A12	VB		- S F'i Fvbi Li / Ai Ei
				=
Fcb	A21	A22	VC		- S F'i Fvci Li / Ai Ei

A11	=	S Fvbi Fvbi Li / Ai Ei
A12	=	S Fvbi Fvci Li / Ai Ei
A21	=	S Fvci Fvbi Li / Ai Ei
A22	=	S Fvci Fvci Li / Ai Ei