# Chapter 6: Structural Analysis

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Chapter 6: Structural Analysis

Goals and Objectives Determine the forces in members of a truss using the method ofjoints Determine zero-force members Determine the forces in members of a truss using the method ofsections

Simple trussesTrusses are commonly used to supportroofs.A more challenging question is, that fora given load, how can we design thetrusses’ geometry to minimize cost?

ScaffoldingAn understanding of statics iscritical for predicting andanalyzing possible modes offailure.Buckling of slender members incompression is always aconsideration in structuralanalysis.

Simple trussesTruss: Structure composed of slendermembers joined together at endpoints Transmit loads to supportsAssumption of trusses Loading applied at joints, withnegligible weight (If weight included,vertical and split at joints) Members joined by smooth pinsResult: all truss members are twoforce members, and therefore theforce acting at the end of each memberwill be directed along the axis of themember

Roof trussesLoad on roof transmitted to purlins, and from purlins to roof trusses at joints.Bridge trussesLoad on deck transmitted to stringers, and from stringers to floor beams, and from floorbeams to bridge trusses at joints.

Truss joints Bolting or welding of the ends of themembers to a gusset plates or passing alarge bolt through each of the membersProperly aligned gusset plates equivalent topins (i.e., no moments) from coplanar,concurrent forcesSimple trusses built from triangularmembers

Method of joints Entire truss is in equilibrium if and only if all individual pieces (trussmembers and connecting pins) are in equilibrium.Truss members are two-force members: equilibrium satisfied by equal,opposite, collinear forces. Tension: member has forces elongating. Compression: member has forces shortening.Pins in equilibrium: 0 and 0Procedure for analysis: Free-body diagram for each joint Start with joints with at least 1 known forceand 1-2 unknown forces. Generates two equations, 1-2 unknowns foreach joint. Assume the unknown force members to be in tension; i.e.the forces “pull” on the pin. Numerical solutions will yieldpositive scalars for members in tension and negative scalarfor members in compression.

Example 1)The truss, used to support a balcony, is subjectedto the loading shown. Approximate each joint as apin and determine the force in each member. Statewhether the members are in tension orcompression.

We will determine the force in each member of the truss andindicate whether the members are in tension or compression.

Example 2)Determine the force in member FG of the truss andstate if the member is in tension or compression.

Zero-force members Particular members in a structure may experience no force for certain loads. Zero-force members are used to increase instability Identifying members with zero-force can expedite analysis.Two situations: Joint with two non-collinear members, no external or support reaction applied to thejoint Both members are zero-force members. Joint with two collinear member, plus third non-collinear, no loads applied to the joint Non-collinear member is a zero-force member.

Example 3)Determine the force in member GC of thetruss and state if the member is in tension orcompression.

Method of sections Determine external support reactions“Cut” the structure at a section of interest into two separatepieces and set either part into force and moment equilibrium(your cut should be such that you have up to three unknowns) Determine equilibrium equations (e.g.,moment around point of intersection of two lines) Assume all internal loads are tensile.

Example 4)Determine the force in member GC of thetruss and state if the member is in tension orcompression.

Frames and machinesFrames and machines are two common types of structures that have at least onemulti-force member (Recall that trusses have nothing but two-force members).Frames are generally stationaryand used to support variousexternal loads.Machines contain moving parts andare designed to alter the effect of forces

Frames and machinesThe members can be truss elements, beams, pulleys,cables, and other components. The general solutionmethod is the same:1.Set the entire structure into external equilibrium.This step will generally produce more unknownsthan there are relevant equations of equilibrium.2.Identify two-force members3.Isolate various part(s) of the structure, setting eachpart into equilibrium. The sought forces or couplesmust appear in one or more free-body diagrams.4.Solve for the requested unknowns. Look for ways toform single equations and single unknowns.Problems are going to be challenging since there areusually several unknowns. A lot of practice is needed todevelop good strategies and ease of solving theseproblems.

For the frame, draw the free-body diagram of (a) each member, (b) the pins at B and A, and(c) the two members connected together.

A 500 kg elevator car is being hoisted by a motor using a pulley system. Ifthe car travels at a constant speed, determine the force developed in thecables. Neglect the cable and pulley masses.T2T1

Example 5)The frame supports a 50kg cylinder. Determinethe horizontal and vertical components ofreaction at A and the force at C

Example 6)The compound beam shown is pin-connected atB. Determine the components of reaction at itssupports. Neglect its weight and thickness.

Particular members in a structure may experience no force for certain loads. Zero-force members are used to increase instability Identifying members with zero-force can expedite analysis. Two situations: Joint with two non-collinear members, no external or support reaction applied to the joint Both members are zero-force members. Joint with two collinear member,

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