Structural Analysis Method of Joints / Method of Pins Twenty.pdf · Structural Analysis Method of...

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Structural Analysis Method of Joints / Method of Pins

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Wednesday, October 21, 2009 Method of Joints 2

Trusses

 Trusses are common means of transferring loads from their points of application to the supports

 They vary in design but their analysis always follows the same pattern


Tools

 Equations of equilibrium  Free Body Diagrams  Trig and algebra  Visualization

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Trusses

 A simple truss analysis, which we will do in this class, does not consider the weight of the members which make up the truss

 A simple truss analysis considers that all loading is made at connections

 A simple truss analysis considers that all truss members are two-force members


Trusses

 This means that we can break a truss down into a collection of elements for analysis


Trusses

 An example truss

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Trusses

 The red arrows are the loads (applied externally)


Trusses

 The dark black lines are the members


Trusses

 Each member goes from connection to connection

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Trusses

 The black line along the bottom of the truss represents three separate members


Trusses

 The support conditions are the pin at the left and the roller at the right


Trusses

 At each intersection of two or more members, we consider that the members are pinned together

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Trusses  For example, at the point labeled G on the

diagram, we have four members coming to a point connected by a single pin


Trusses

 To simplify our analysis, we assign each connection point with a letter


Trusses   I avoid using F and I as connection point

assignments but your text does not

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Trusses  Pick a starting point and label each connection, I

chose to start at the lower left and move across the top and then across the bottom


Trusses  Each of the members can now be labeled using

the points of connection


Trusses   In this truss we have the following members: AB,

BC, CD, DE, CE, CG, EG, BG, and AG

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Trusses  The order of the letters is not important, I choose

to put them in the order they appear in the alphabet


Method of Joints  To begin the analysis of the truss using this

method, we start by identifying all external forces acting on the truss


Method of Joints   In this case it would be the forces represented by

the red lines  The forces (loads) must always be applied at the

connections (joints)

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Method of Joints

 This is usually given in the problem


Method of Joints  Then we remove any supports and replace them

with reactions provided by the supports


Method of Joints  We have a pin at the left so we will have an x

and a y reaction

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Method of Joints  And we have a roller at the right so we only have

a y reaction


Method of Joints  We are also usually given the dimensions and

geometry of the truss in the given


Method of Joints  Now we can solve for the reactions at the

supports using the methods developed previously

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Method of Joints  We will use the entire truss as the FBD and

ignore the internal members (for the moment)


Method of Joints  Summing forces in the x-direction


Method of Joints  Summing moments about D

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Method of Joints  Summing forces in the y-direction


Method of Joints

 Redrawing our system with the known reactions, we have


Method of Joints  Now for the problems that we had done earlier,

we would be finished

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Method of Joints   In a truss analysis, we want to know what force

is being carried in each of the members


Method of Joints

 Each member of the truss can be in one of three conditions:   It can be in COMPRESSION, that means that it

will be pushing on each end where it is connected

  It can be in TENSION, that means that it will be pulling on each end where it is connected

  It can have no force in the member


Method of Joints

  In the method of joints, we are going to assume the condition of each of the members,

  draw a free body diagram of the connection points, and

  apply two of our three equilibrium conditions to solve for the force actually in the members

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Method of Joints

 When we used a pin connection previously, we used an x and a y reaction to show what was being provided by the pin

 Here we are actually drawing a FBD of the pin itself and showing the forces from the members onto the pin itself


Method of Joints

 You can assume any condition that you want for the members, I usually choose to assume that they are all in either tension or compression and then let the signs of the solutions dictate their actual condition


Method of Joints

 Since we can only use two of the equations of equilibrium, the sum of the forces in the x and the sum of the forces in the y direction, our analysis pattern will be dictated by the makeup of the truss itself

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Method of Joints

 First, we can look at all the forces acting on each pin of the truss

  It is critical to the analysis that you do not exclude either external forces or support reactions at each pin


Method of Joints

  If a member is pushing on a pin at one end, it is always pushing on the pin at its other end


Method of Joints

 Conversely, if a member is pulling at a pin on one end, it is pulling on the pin at its other end

 Never push pull!

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Method of Joints

 At any joint, you can only solve for two unknowns!!!!


Method of Joints

 Now to decide where to start the analysis, we need to find a joint where there are only two unknowns


Method of Joints

  In our case, the pins at A and at D both only have two unknowns, so we can start at either point

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Method of Joints

  If you thought that there were three unknowns at A and D because of ABx and ABy, remember that the force in the member has a line of action along the member


Method of Joints

 We choose A to start our analysis  We now draw a FBD of the pin at A


Method of Joints

  I always assume that the members are in compression so our FBD would look like

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Method of Joints

 Since I assume compression, the forces in the members are pushing on the connection (pin)


Method of Joints

 Now we can use our two equilibrium conditions to solve for the forces in the members


Method of Joints

 We report all the magnitudes of the forces as positive numbers and the forces themselves as either compression or tension

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Method of Joints

 Solving for AG


Method of Joints

 Now that AB and AB are known forces, we can revisit the FBD of the complete truss and find another joint/pin that has only two unknowns


Method of Joints

 Usually this will be adjoining the pin that we just solved for

 For this truss it could be either pin B or G

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Method of Joints

 At G, we have BG, CG, and EG as unknowns  Three unknowns, so this pin won’t work


Method of Joints

 At B, we have two unknowns, BC, and BG  We can proceed to work at BG


Method of Joints

 We draw a FBD of the pin at B

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Method of Joints

 We know that AB is in compression, so we are not making an assumption about AB


Method of Joints

 We are assuming that BC and BG are in compression for this FBD


Method of Joints

 Now we use our conditions of equilibrium to solve for BC and BG

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Method of Joints

 Now we use our conditions of equilibrium to solve for BC and BG


Method of Joints

 You might be asking yourself why BG is even included in the truss

 This is a very preliminary analysis of one loading condition, there is actually much more to truss design than this


Method of Joints

 Now we could go to either joint C or joint G  At C, we have CG, CE, and CD as unknowns  At G, we have CG and EG  Our next point is G

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Method of Joints

 The FBD at G  Notice that we have drawn AG as a tension

member, we know that from our earlier solution, it is no longer an assumption


Method of Joints

 We also have no longer included the force from BG, we know that it carries no force under there loading conditions


Method of Joints

 Using our equilibrium conditions

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Method of Joints

 Using our equilibrium conditions


Method of Joints

  You can continue on with the pattern until all the members in the truss are solved for

  You will always have one extra pin/joint available after all the members in the truss have been solved

  Check the sum of the forces in the x and y direction at that point to see if the truss closes

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Structural Analysis Method of Joints / Method of Pins Twenty.pdf · Structural Analysis Method of...

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