Analysis of Perfect Frames (Graphical Method) Chapter 14 Analysis of Perfect Frames (Graphical Method)

Learning Objectives Introduction Construction of Space Diagram Construction of Vector Diagram Force Table Magnitude of Force Nature of Force Cantilever Trusses Structures with One End Hinged (or Pin-jointed) and the Other Freely Supported on Rollers and Carrying Horizontal Loads Structures with One End Hinged (or Pin-jointed) and the Other Freely Supported on Rollers and Carrying Inclined Loads Frames with Both Ends Fixed Method of Substitution

Construction of Space Diagram Introduction In the previous chapter, we have discussed the analytical methods for determining the forces in perfect frames. We have seen that the method of joints involves a long process, whereas the method of sections is a tedious one. Moreover, there is a possibility of committing some mathematical mistake, while finding out the forces in the various members of truss. The graphical method, for determining the forces in the members of a perfect frame, is a simple and comparatively fool-proof method. The graphical solution of a frame is done in the following steps: Construction of space diagram Construction of vector diagram Preparation of the table Construction of Space Diagram Fig. 14.1.

Construction of Vector Diagram It means the construction of a diagram of the given frame to a suitable linear scale, alongwith the loads it carries. The magnitude of support reactions is also found out and shown in the space diagram. Now name the various members and forces according to Bow’s notations as shown in Fig. 14.1 (a). In the space diagram of the truss ABC shown in Fig. 14.1 (a), the members AB, BC and CA are represented by SR (or RS), SQ (or QS) and PS (or SP) respectively. Similarly, load at C and reactions at A and B are represented by PQ, RP and QR respectively. Construction of Vector Diagram After drawing the space diagram and naming the various members of the frame according to Bow’s notations, as discussed in the last article, the next step is the construction of vector diagram. It is done in the following steps :   Select a suitable point p and draw pq parallel to PQ (i.e., vertically downwards) and equal to the load W at C to some suitable scale. Now cut off qr parallel to QR (i.e., vertically upwards) equal to the reaction RB to the scale.

Similarly, cut off rp parallel to RP (i. e Similarly, cut off rp parallel to RP (i.e., vertically upwards) equal to the reaction RA to the scale. Thus we see that in the space diagram, we started from P and returned to P after going for P-Q-R-P (i.e., considering the loads and reactions only). Now through p draw a line ps parallel to PS and throgh r draw rs parallel to RS, meeting the first line at s as shown in Fig. 14.1 (b). Thus psrp is the vector diagram for the joint (A). Similarly, draw the vector diagram qrsq for the joint (B) and pqsp is the vector diagram for the joint (C) as shown in Fig. 14.1 (b). Force Table After drawing the vector diagram, the next step is to measure the various sides of the vector diagram and tabulate the forces in the members of the frame. For the preparation of the table, we require :   Magnitude of forces Nature of forces

Magnitude of Force Measure all the sides of the vector diagram, whose lengths will give the forces in the corresponding members of the frame to the scale e.g., the length ps of the vector diagram will give the force in the member PS of the frame to the scale. Similarly, the length sr will give the force in the member SR to the scale and so on as shown in Fig. 14.2. (b). If any two points in the vector diagram coincide in the each other, then force in the member represented by the two letters will be zero. Fig. 14.2.

Nature of Force The nature of forces in the various members of a frame is determined by the following steps: In the space diagram, go round a joint in a clockwise direction and note the order of the two letters by which the members are named e.g., in Fig. 14.2 (a) the members at joint (A) are RP, PS and SR. Similarly, the members at joint (B) are QR, RS and SQ. And the members at joint (C) are PQ, QS and SP. Now consider a joint of the space diagram and note the order of the letters of all the members (as stated above). Move on the vector diagram in the order of the letters noted on the space diagram. Make the arrows on the members of the space diagram, near the joint, under consideration, which should show the direction of movement on the vector diagram. Put another arrow in the opposite direction on the other end of the member, so as to indicate the equilibrium of the method under the action of the internal stress. Similarly, go round all the joints and put arrows. Since these arrows indicates the direction of the internal forces only, thus the direction of the actual force in the member will be in opposite direction of the arrows, e.g., a member with arrows pointing outwards i.e., towards the joints [as member PS and SQ of Fig. 14.2 (a)] will be in compression; whereas a member with arrow pointing inwards i.e., away from the joints [as member SR in Fig. 14.2 (b)] will be in tension.

Example Construct a vector diagram for the truss shown in Fig. 14.17. Determine the forces in all the members of this truss. Fig. 14.17. Solution Since the truss is symmetrical in geometry and loading, therefore the reaction at the left hand support, First of all, draw the space diagram and name the members and forces according to Bow’s notations as shown in Fig. 14.18 (a).

Fig. 14.18. Now draw the vector (i.e., stress) diagram as shown in Fig. 14.18 (b). Measuring the various sides of the vector diagram, the results are tabulated here :

Cantilever Trusses We have already discussed that a truss which is connected to walls or columns etc., at one end, and free at the other is known as a cantilever truss. In the previous articles, we have noticed that the determination of the support reactions was absolutely necessary to draw a vector diagram. But in the case of cantilever trusses, determination of support is not essential, as we can start the construction of vector diagram from the free end. In fact this procedure, actually gives us the reactions at the connected ends of the truss.

Example A cantilever truss of span 2l is carrying loads as shown in Fig. 14.33. Determine graphically, or otherwise forces in all the members of the truss. Solution First of all, draw the space diagram, and name all the members according to Bow’s notations as shown in Fig. 14.34 (a).

Now draw the vector diagram as shown in Fig. 14. 34 (b) Now draw the vector diagram as shown in Fig. 14.34 (b). Measuring the various sides of the vector diagram, the results are tabulated here : S.No. Member Magnitude of force in kN Nature of force 1 HB 1.2 Tension 2 CH 0.6 Compression 3 GH 4 AG 5 GF 2.3 6 DF 7 EF 8 AE 2.9