Structural mechanics

14.17. Spatial Frames

For each bar of the frame, the orientation of the axes of the local coordinate system (hereinafter referred to in small letters x, y, z) will be con-

sidered known. The axis ox is directed from node P1 to node P2 (from a node with a lower number to a node with a higher number). The axes oy and oz of the right Cartesian coordinate system are located in a plane perpendicular to the axis ox and passing through the point P1 . Since the

position of the cross section of the frame bar is taken to be predetermined, the position of the axes oy and oz is also established. The location of the

axes along each bar must be fixed unambiguously.

Let relative to the axes of the global coordinate system OXYZ :

Then the local coordinate system adopted for the bar is characterized by a matrix of direction cosines:

Using the matrix T , Cartesian rectangular coordinates are transformed when the axes are rotated.

Define the force and strain vectors in the bar of the spatial frame in the following form:

S [N, MT , M yb , M ye , M zb, M ze ]T ,

[ l, T , yb , ye , zb , ze ]T

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The efforts in the bar end sections, oriented along the axes of the local coordinate system, are expressed by the vector r* through the vector S using the equilibrium matrix a* :

r* a*S ,

where

r* [rxb ,ryb ,rzb ,mxb ,myb ,mzb ,rxe ,rye ,rze ,mxe ,mye ,mze ]T .

The components of the vector r* are shown in Figure 14.29. The positive components directions of the vector S are shown in Figure 14.30.

In these figures, a vector image of moments was used. The moment acting in a clockwise direction along a certain axis (when viewed from a point corresponding to the end of the coordinate axis) is depicted by a vector directed in the positive direction of the axis.

In Figure 14.29 notations accepted:

mxb, mxe are torques at the beginning and at the end of the bar;

myb , mye are bending moments at the beginning and at the end of the bar relative to the axis y ;

mzb , mze – bending moments at the beginning and at the end of the bar rel-

ative to the axis z .

The equilibrium conditions for the bar allow us to obtain the following relationships:

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The equilibrium matrices of rods and bars of plane and spatial trusses, plane frames, systems of cross beams are obtained as special cases from the written matrix a (14.22) by deleting the corresponding rows and columns.

The matrix of internal stiffness of the bar rigidly fixed at the ends has the following form:

The basic equations of structural mechanics for calculating the bars systems in the form of a displacement method are presented in the form:

R z F A K .

The matrix A of equilibrium equations of the calculated system is compiled element by element using the equilibrium matrices a (14.22) of the bars.

E x a m p l e. Plot the diagram of the longitudinal forces, torsional and bending moments in the frame (Figure 14.31). Accept the following stiffness ratios for all bars:

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Figure 14.31

The positions of the local coordinate axes for each bar of the frame shown in Figure 14.32, determine the matrix of direction cosines:

Figure 14.32

The matrix of frame equilibrium equations is given in table 14.6.

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Taking a load vector

F2 0; 0; 11; 3; 7.5; 0 T ,

where the dimension of forces in kN, moments in kNm, lengths in m, positive moments are directed relative to the axes of the general coordinate system in a clockwise direction, when viewed from a point corresponding to the end of the axis, we obtain:

z 1.87; 2.75; 17.54; 3.89; 2.64; 0.76 T EJ1 y ,

S [ 0.31; 0.65; 3.80; 4.68; 0.21; 0.05; | 0.92; 0.88; 6.51; 9.10;

0.24; 0.74;| 4.38; 0.19; 1.94; 0.62; 2.86; 0.91;]T.

The corresponding diagtams of efforts are shown in Figure 14.33. Figure 14.34 shows (in axonometric view) the bending moments and torques acting on the cut-out node 2. The moments are divided into groups according to their location with respect to the coordinate planes.

Figure 14.33

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For this node, the equations of projections of forces on the coordinate axes are also satisfied.

E x a m p l e. Construct efforts diagrams in the frame (Figure 14.35), taking for all bars

EA h2 10EJ y , GT 0.27EJ y , EJ z 0.5EJ y .

The frame is loaded with two generalized nodal forces:

F1 12.0,0, 98.0,40.0, 52.0,0 T ,

F2 0,0, 98.0,40.0,64.0,0 T .

Figure 14.35

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Dimension of forces – kN, moments – kNm, lengths – m.

The matrixes of the direction cosines of the axes of the local coordinate system for the frame bars are presented in the following forms:

Matrices of internal rigidity of the 4th and 5th bars coincide:

After the formation of the matrix of external rigidity

R AKAT ,

we solve the system of equations

Rz F

and determine the forces in the frame rods by expression

S KAT z .

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