hw2
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7 The Stiffness MethodFramesThe Stiffness Method oJ analysis s similar in its aPproach o theflexibility method of analysis, he major difference being that the unknownsin the procedure in the stilfness method are the node disPlacements,whereas n the flexibility method they are the internal lorces, moments andreactions. The other major difference between the two methods is that thestiffness method requires no iudgement to be exercised by the analyst'whereas the flexibility method reguires a choice of releases. This makesthe stilfness method particularly suited to automatic computation and it isthe basis oi virtually all 'Plane Framesr and 'Space Frames'comPuterprograms.The stren8th of the stiflness method, i.e. its suitability as a basis ofautomatic computation, is its weakness as a method of hand analysis.Because he method requires little or no judgement to be exercised,practice in stiffness problems promotes little understanding of structuralbehaviour. On the other hand' the flexibility method requires a soundunderstanding of structural behaviour and statics in the first steps of theanalytical procedure.The reader should be aware that neither the flexibility nor stiffnessmethod are praclical manual methods oJ analysis n the desiSn oflicebecause real structures are too hiShly redundant and must be checked for anumber of dilferent load cases. The analytical Procedure of both methods,requiring the setting up and solving of large numbers of simultaneousequations s cumbersome, time_consuming and error-prone. It isalso illusory in terms oJ accuracy. lt may be na tile.4?a ica.l -19 accuratebut no more accurate in terms oJ the relationship of the analytical modelto the real structure than the method oI moment distribution. The latteris sometimes described as 'approximate' as if it were aPProximate o thebehaviour of the real structure. It is no more or less close to the realTRANSCRIPT
University of Massachusetts - AmherstDepartment of Civil & Environmental Engineering
CEE 331: Structural Analysis
Homework #2: Due September 23
Problem 1: For the truss element shown below, with properties E = 11, 000 ksi,A = 0.15 sq.in.
(a) calculate the element stiffness matrix in global coordinates(b) calculate using the matrix equation q = kd the element force vector in globalcoordinates q = [qNx, qNy, qFx, qFy]T that results from the element displacementvector
d =
dNx
dNy
dFx
dFy
=
.1
.10
−.05
in.
Sketch the element with the resulting forces shown acting on the nodes.Note: boldface letters are vectors and matrices.[·]T is the transpose of a vec-tor/matrix.
Problem 2: 14-8. Make sure to provide illustrations to go with your calculations andto clearly label the rows and columns of your matrices.
Problem 3: The Adaptive Use Bridge Project is an ongoing effort being conducted atUMass to adapt historic trusses for use as pedestrian bridges on campus. One bridgehas already been reconstructed down by McGuirk Stadium. The project has a websiteat http://www.ecs.umass.edu/adaptive bridge use. It is linked from the course website.This problem asks you to work with the ‘Southern Vermont Bridge’ which is locatedat the northwest corner of Lot 11 near McGuirk.(a) Visit the bridge. Think about how you would idealize it for analysis. What kinds ofloads act on the bridge? What material is the bridge made of? What are the boundaryconditions? What do the elements and internal connections look like?Take a photo-graph of yourself with the bridge and include it.(b) Consult the drawings of the bridge that can be found by going to the ‘data’ tab
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of the web page, selecting ‘stadium bridge’, and choosing the PDF of page 1 under‘Bridge Dimensions’ under ‘Documentation’. Assume the bridge is made of iron withE = 29, 000 ksi, and calculate the element stiffness matrix in global coordinates of theelement connecting nodes U2 and L2. Assume that the direction of the element is fromL2 towards U2, and that the global origin is at L0, with x positive to the right and y
positive upwards. You must calculate the cross sectional area and length of the elementfrom the drawings provided.Note: A clear presentation of the solution to this problem iscritical. Provide clearillustrations and labeling, and show clearly all your calculations of length, angles,area and stiffness coefficients. State clearly any assumptions you make along theway.
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