#stiffness matrix

Q. Solve for the internal forces and displacements of joint 2 of the truss in example. Let H = 0:75L and assign area A to members 3 and 5, and 0:5A to members 1, 2, and 4. Let f5 be the redundant force, and use the [C1] and [C2] matrices previously derived.

Solution:

Assembly of Structure Stiffness Matrix

Consider part of a structure with four externally applied forces, F1, F2, F4, and F5, and two applied moments, M3 and M6, acting at the two joints p and q connecting three members A, B, and C as shown in Figure. The freedom codes at joint p are {1, 2, 3} and at joint q are {4, 5, 6}. The structure stiffness matrix [K] is assembled on the basis of two…

Temperature effects

Most materials expand when subject to temperature rise. For a steel member in a structure, the expansion due to temperature rise is restrained by the other members connected to it. The restraint imposed on the heated member generates internal member forces exerted on the structure.

Uniform Temperature

The fixed-end force vector {PF} for a steel member shown in Figure subject…

Member Stiffness Matrix in Global Coordinate System

From

where

= member stiffness matrix in the global coordinate system.

An explicit expression for [Kge ] is

Coordinates Transformation

In order to establish the equilibrium conditions between the member forces in the local coordinate system and the externally applied loads in the global coordinate system, the member forces are transformed into the global coordinate system by force resolution. Figure shows a member inclined at an angle α to the horizontal.

Load Transformation

The forces in the global…

Example 1

Analyse the truss shown in Fig. 25.1a and evaluate reactions. Assume EA to be constant for all the members.

Solution

The numbering of joints and members are shown in Fig. Also, the possible displacements (degrees of freedom) at each node are indicated. Here lower numbers are used to indicate unconstrained degrees of freedom and higher numbers are used for constrained degrees of…

Q. Determine the member forces and plot the shear force and bending moment diagrams for the structure shown in Figure. The structure with a pin at D is subject to a vertical force of 100 kN being applied at C. For all members, E = 2×108kN/m 2 , A = 0.2 m 2 , and I = 0.001 m 4.

Sol.

The freedom codes for the whole struct ure are shown in Figure. There are four members separated by joints B, C, and…

In structural analysis an influence coecient Cij can be dened as the eect on d.o.f. i due to a unit action at d.o.f. j for an individual element or a whole structure.  It should be recalled that infuence lines are associated with the analysis of structures subjected to moving loads (such as bridges), and that theexibility and stiffness coecients are components of matrices used in structural…