Fundamentals of Structural Analysis Episode 1 Part 6 doc

Beam and Frame Analysis: Force Method, Part I by S. T. Mau

95

Improper internal connections.

Statical determinacy. A stable beam or frame is statically indeterminate if the number

of force unknowns is greater than the number of equilibrium equations. The difference

between the two numbers is the degree of indeterminacy. The number of force unknowns

is the sum of the number of reaction forces and the number of internal member force

unknowns. For reaction forces, a roller has one reaction, a hinge has two reactions and a

clamp has three reactions as shown below.

Reaction forces for different supports.

To count internal member force unknowns, first we need to count how many members

are in a frame. A frame member is defined by two end nodes. At any section of a member

there are three internal unknown forces, T , V, and M. The state of force in the member is

completely defined by the six nodal forces, three at each end node, because the three

internal forces at any section can be determined from the three equilibrium equations

taken from a FBD cutting through the section as shown below, if the nodal forces are

known.

Internal section forces are functions of the nodal forces of a member.

Thus, each member has six nodal forces as unknowns. Denoting the number of members

by M and the number of reaction forces at each support as R, the total number of force

unknowns in a frame is then 6M+ΣR.

x

T

V

M

Beam and Frame Analysis: Force Method, Part I by S. T. Mau

96

On the other hand, each member generates three equilibrium equations and each node

also generates three equilibrium equations. Denoting the number o f nodes by N, The

total number of equilibrium equations is 3M+3N.

FBDs of a node and two members.

Because the number of members, M, appears both in the count for unknowns and the

count for equations, we can simplify the expression for counting unknowns as shown

below.

Counting unknowns against available equations.

The above is equivalent to considering each member having only three force unknowns.

The other three nodal forces can be computed using these three nodal forces and the three

member equilibrium equations. Thus, a frame is statically determinate if 3M+ΣR= 3N.

If one or more hinges are present in a frame, we need to consider the conditions

generated by the hinge presence. As shown in the following figure, the presence of a

hinge within a member introduces one more equation, which can be called the condition

of construction. A hinge at the junction of three members introduces two conditions of

construction. The other moment at a hinge is automatically zero because the sum of all

moments at the hinge (or any other point) must be zero. We generalize to state that the

conditions of construction, C, is equal to the number of joining members at a hinge, m,

minus one, C=m-1. The conditions of construction at more than one hinges is ΣC.

Nodal

Equilibrium

Member

Equilibrium

Member

Equilibrium

Number of unknowns=6M+ΣR

Number of equations= 3M+3N

Number of unknowns=3M+ΣR

Number of equations= 3N