University of Massachusetts - AmherstDepartment of Civil & Environmental Engineering

CEE 331: Structural Analysis (Fall 2008)

Final Exam, December 17Notes:1. All problems weighted equally2. Equation sheet at end3. All answers and work must be shown on these pages. Box answers.

Problem 1: Calculate the moment function for the beam part of this frame. Make use of the given reactionsAx =−1.3k, Ay = 2.2k, Ma = 5.4k-ft andEx = −0.7k, Ey = 2.8k, Me = 4k-ft. Use thex3 coordinate given. Draw themoment diagram for the beam part of the frame.

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Workspace

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Problem 2: The solution given on the following page is an attempt to solve for the axial force in members ABC, DEF,and GHI using the cantilever method. The cross sectional areas are as given in the diagram. The solution is incorrect.Identify any and all errors made in the solution. Circle themand explain what has been done wrong.

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Problem 3: Using the portal method of approximate analysis:(a) calculate the internal forces at point B.(b) Draw a free body diagram of the segment AB, indicate and labeling all internal forces and reactions.(c) Using the free body diagram from (b), calculate the moment reaction at point A.

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Problem 4: (a) What would the size of theK11 stiffness matrix be for this truss structure? Note that the diagonalsare not connected at the center of the truss panel.

(b) Provide a labeling of the degrees of freedom for this frame structure that is consistent with our rules for numberingof degrees of freedom.

Problem 5: For the three structures shown below indicate whether the moment diagram shown could be correct ornot. If not, state why.

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Problem 6: Consider the three reactions at B as redundants.(a) Sketch deflected shapes and label displacements for the primary and three redundants.(b) Can you determine the horizontal and moment reactions atB without calculation? What are they? Explain youranswer making reference to the diagrams from (a).

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Problem 7: Use Castigliano’s theorem to calculate the vertical displacement at point C. Use the coordinates given.Sketch the deflected shape.

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Workspace

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Problem 8: Draw the free body diagram you would use in performing an approximate analysis of the truss to deter-mine the forces in panel ABCD. Assume that diagonals can carry only tension.

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Reference equationsEquilibrium equations

∑

Fx = 0,∑

Fy = 0,∑

M = 0

Strain Energy

Ui,bending =

∫

M(x)2

2EIdx

Ui,axial =

∫

N(x)2

2EAdx =

N2L

2EAif N(x) is constant

Work done by loading

Ue =1

2P∆ for a point load

Ue =

∫

1

2w(x)δ(x)dx for a distributed load

Castigliano’s theorem

∆i =∂Ui

∂Pi

Θi =∂Ui

∂M ′

i

Useful extra formulas for using Castigliano

∂

∂Pi

∫

M(x)2

2EIdx =

∫

M(x)

EI

∂M(x)

∂Pi

dx

∂

∂Pi

∫

N(x)2

2EAdx =

∫

N(x)

EA

∂N(x)

∂Pi

dx

∂

∂M ′

i

∫

M(x)2

2EIdx =

∫

M(x)

EI

∂M(x)

∂M ′

i

dx

∂

∂M ′

i

∫

N(x)2

2EAdx =

∫

N(x)

EA

∂N(x)

∂M ′

i

dx

Direction cosines

λx = cos(θx), λy = cos(θy)

Truss element stiffness matrix in local/element coordinates

k′ =

EA

L

Nx′ Fx′

[

1 −1−1 1

]

Nx′

Fx′

Truss element stiffness matrix in global coordinates

k =EA

L

Nx Ny Fx Fy

λ2

x λxλy −λ2

x −λxλy

λxλy λ2

y −λxλy −λ2

y

−λ2

x −λxλy λ2

x λxλy

−λxλy −λ2

y λxλy λ2

y

Nx

Ny

Fx

Fy

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Matrix stiffness equations for a truss element in global coordinates

q = kd

Marix stiffness equations for a truss in global coordinates

Q = KD

Partitioned matrix stiffness equations for a truss[

Qk

Qu

]

=

[

K11 K12

K21 K22

] [

Du

Dk

]

Matrix inverse[

a b

c d

]

−1

=1

ad − bc

[

d −b

−c a

]

Design wind pressureps = λKztIps30

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