Question B.1: The plane truss shown in Figure B-1 has four joints and five...
The plane truss shown in Figure B-1 has four joints and five members. Find support reactions at A and B and then use the methods of joints and sections to find all member forces. Let P = 35 kips and c =10 ft.
In many cases, the problem involves the analysis of a model of a real physical structure, such as this truss structure representing the fuselage of a model air plane (Fig. B-1b). Begin by sketching the portion of the structure of interest showing relevant supports, dimensions and loadings. This Conceptualization step in the analysis often
leads to a free-body diagram in mechanics problems.
Use the following four-step problem-solving approach.
1. Conceptualize [hypothesize, sketch]: First sketch a free-body diagram of the entire truss model (Figure B-2*). Only known applied forces at C and unknown reaction forces at A and B are shown and then are used in an equilibrium analysis to find the reactions.
2. Categorize [simplify, classify]: Overall equilibrium requires that the force components in x and y directions and the moment about the z axis must sum to zero; this leads to reaction force components A_{x} , A_{y}, and B_{y}. The truss is statically determinate (unknowns: m + r = 5 + 3 = 8, knowns: 2 j = 8), so all member forces can be obtained using the method of joints. If only a few selected member forces are of interest, the method of sections can be used. Use a statics sign convention when computing external reactions and a deformation sign convention when solving for member forces.
3. Analyze [evaluate: select relevant equations, carry out mathematical solution]**: First find the lengths of members AC and BC that are needed to compute distances to lines of action of forces.
Law of sines to find member lengths a and b: Use known angles \theta_{A} , \theta_{B} , and \theta_{C}*** and c = 10 ft to find lengths a and b:
b=c \frac{\sin \left(\theta_{B}\right)}{\sin \left(\theta_{C}\right)}=(10 ft ) \frac{\sin \left(40^{\circ}\right)}{\sin \left(80^{\circ}\right)}=6.527 ft,
a=c \frac{\sin \left(\theta_{A}\right)}{\sin \left(\theta_{C}\right)}=(10 ft ) \frac{\sin \left(60^{\circ}\right)}{\sin \left(80^{\circ}\right)}=8.794 ftCheck that computed lengths a and b give length c by using the law of cosines:
c=\sqrt{(6.527 ft )^{2}+(8.794 ft )^{2}-2(6.527 ft )(8.794 ft ) \cos \left(80^{\circ}\right)}=10 ftSupport reactions: Using the truss model free-body diagram in Figure B-2, sum forces in x and y directions and moments about joint 4:
\Sigma M_{A}=0 \quad B_{y}=\frac{1}{c}\left[P\left(b \cos \left(\theta_{A}\right)\right)+2 P\left(b \sin \left(\theta_{A}\right)\right]=51 kips \right.\Sigma F_{x}=0 \quad A_{x}=-2 P=-70 \text { kips }
\Sigma F_{y}=0 \quad A_{y}=P-B_{y}=-16 \text { kips }
The support reactions are computed using a statics sign convention (forces in the +x direction are positive). However, it is common to use a deformation sign convention (tension is positive) for member force calculations.
Member forces using method of joints: Begin by drawing free-body diagrams of the pin at each joint (Figure B-3)****. Use a deformation sign convention in which each member is assumed to be in tension (so the member force arrows act away from the two joints to which each member is connected). The forces are concurrent at each joint, so use force equilibrium at each location to find the unknown member forces.
First sum forces in the y direction at joint A to find member force AC, and then sum forces in the x direction to get member force AD:
\Sigma F_{y}=0 \quad A C=\frac{-1}{\sin \left(60^{\circ}\right)} A_{y}=18.46 \text { kips }\Sigma F_{x}=0 \quad A D=-A_{x}-A C \cos \left(60^{\circ}\right)=60.8 kips
Summing forces at joint B gives member forces BC and BD as
\Sigma F_{y}=0 \quad B C=\frac{-1}{\sin \left(40^{\circ}\right)} B_{y}=-79.3 \text { kips }\Sigma F_{x}=0 \quad B D=-B C \cos \left(40^{\circ}\right)=60.8 kips
The minus sign means that member BC is in compression, not in tension as assumed. Finally, observe that CD is a zero-force member because forces in the y direction must sum to zero at joint D.
Selected member forces using method of sections:***** An alternative approach is to make a section cut all the way through the structure to expose member forces of interest, such as AD, CD, and BC in Figure 1-9. Summing moments about joint B confirms that the force in member CD is zero.
4. Finalize [conclude; examine answer—Does it make sense? Are units correct?****** How does it compare to similar problem solutions?]: There are 2 j = 8 equilibrium equations for the simple plane truss considered, and using the method of joints, these are obtained by applying \Sigma F_{x}=0 \text { and } \Sigma F_{y}=0 at each joint in succession. A computer solution of these simultaneous equations leads to the three reaction forces and five member forces. The method of sections is an efficient way to find selected member forces. A key step is the choice of an appropriate section cut, which isolates the member of interest and eliminates as many unknowns as possible. This is followed by construction of a free-body diagram for use in the static equilibrium analysis to compute the member force of interest. A combination of the methods of sections and joints was used, here, which is a common solution approach in plane and space truss analysis.
*The next step is to simplify the problem, identify all unknowns, and make necessary assumptions to create a suitable model for analysis. This is the Categorize step.
**Use appropriate mathematical and computational techniques to solve the equations and obtain results, either in the form of mathematical formulas or numerical values. In this example, the equilibrium of the truss is of interest so all member lengths and orientations, coordinates of joints, and so on are needed to solve the problem. One or more methods of analysis are selected and a mathematical solution is carried out in numerical or symbolic form. This is the Analysis step.
***List the major steps in your analysis procedure so that it is easy to review or check at a later time.
****Additional sketches, such as this free body diagram of each joint in the truss (Fig. B-3), are often needed as the analysis proceeds.
*****In the full Example 1-1 solution in Chapter 1, a second analysis for member forces, the Method of Sections, is described but is not repeated here.
******In the Finalize step, the solution is reviewed to make sure that it is presented in a clear fashion so that it can be easily reviewed and checked by others. Are the expressions and numerical values obtained reasonable? Do they agree with your initial expectations? The results must be checked in as many ways as possible because errors can be disastrous and expensive so engineers should never rely on a single solution. The results are interpreted in terms of the physical behavior of the mechanical or structural system to give meaning to the results and draw conclusions about the behavior of the system. Is further analysis required perhaps using other loadings or support conditions?
