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Brianne Daniels
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1 STRUTURL NLYSIS [SK 43] EXERISES Q. (a) Using basic concepts, members towrds local axes is, E and q L, prove that the equilibrium equation for truss f f E L E L E L q E q L With f and q are both force and displacement at node respectively, while and L are both cross sectional area and length as shown in Figure Q(a). q q f f L Figure Q(a) (b) Three steel rods having different cross-sectional area and length are assembled in a series as shown in Figure Q(b). Joints and are loaded with point loads of 50 kn and 00 kn in an opposite direction along the bars axis as shown. Using the stiffness method, calculate the reactions at the left and right supports, and internal forces in all rods. Taking E as 00 kn/mm kn 00 kn 4 3 L = 000 mm = 50 mm L = 000 mm = 5 mm L 3 = 000 mm 3 = 75 mm Figure Q(b)

2 STRUTURL NLYSIS [SK 43] Q. Three steel rods having different cross-sectional area and length are assembled in a series as shown in Figure Q. Joints and are loaded with point loads of 80 kn and 50 kn along the bars axis as shown in the figure. Using the stiffness method, calculate the reactions at the left and right supports, and internal forces in all rods. Taking E as 00 kn/mm. 80 kn 50 kn L = 50 mm = 80 mm L = 50 mm = 40 mm L 3 = 00 mm 3 = 0 mm Figure Q Q3. truss carries loads of W kn at as shown in Figure Q3. ll members have E of 00 kn/mm and sectional area, of 000 mm. y using the flexibility/stiffness method, determine the maximum value of W if the allowable displacement at joint is.5 mm at any direction of global axis, xy. Determine the internal forces of each member as well. W kn y x Global axis 3 m Figure Q3 Q4. truss carries loads of 50 kn and 0 kn at as shown in Figure Q4. ll members have E of 00 kn/mm and sectional area, of 800 mm. Using the flexibility method, determine the displacement at by assuming the vertical reaction at as redundant. Determine the internal forces of each member as well. 50 kn 0 kn y x Global axis 3 m Figure Q4

3 3 m 3 m STRUTURL NLYSIS [SK 43] Q5. truss carries loads of 80 kn and 40 kn at as shown in Figure Q5. ll members have E of 00 kn/mm and sectional area, of 650 mm. Support has a vertical displacement of 0 mm. Using the flexibility method, determine the internal forces of each member. 80 kn 40 kn y x Global axis Figure Q5 Q6. truss carries loads of 90 kn and 60 kn at as shown in Figure Q6. ll members have E of 00 kn/mm and sectional area, of 750 mm. Using the flexibility method, determine the displacement at. Determine the internal forces of each member as well. 90 kn 60 kn y x Global axis Figure Q6 Q7. Figure Q7 shows a truss pinned supported at node and roller at node. The truss carries loads of 80 kn and 40 kn at node 3. ll members have E of 00 kn/mm and sectional area, of 400 mm. If the allowable vertical displacement at node is 5 mm, use the stiffness method to determine: (a) maximum vertical force applicable to node, (b) reaction at node, (c) internal forces in member 3, and (d) without ammending the existing forces, propose one suitable alteration to the truss in achieving zero force in member. Solutions must be in accordance to existing point of origin, members numbering, nodes numbering as well as given displacement numbering.

4 m STRUTURL NLYSIS [SK 43] Figure Q7 Q8. truss as shown in Figure Q8 is supported as pin at and roller at, carries loads of 50 kn and 0 kn at. ll members have E of 00 kn/mm and sectional area, of 650 mm. y using the flexibility method, determine the internal force of members and horizontal displacement at. 50 kn 0 kn m Figure Q8 Q9. Figure Q9 shows a beam with load P kn acting at. The value for the beam is x 0 0 knmm. Using either the flexibility or the stiffness method, determine the maximum P value if the deflection and rotation is 0 mm dan rad respectively. Sketch the shear force and bending moment diagram for the beam as well. P kn m Figure Q9

5 STRUTURL NLYSIS [SK 43] Q0. Figure Q0 shows a beam with uniform load of 36 kn/m acting throughout point to and point load of 40 kn acting at. The value for the beam is x 0 0 knmm. Using the flexibility method, determine the fixed end moment for each member by assuming the reaction at as redundant. Sketch the shear force and bending moment diagram for the beam as well. 40 kn 36 kn/m m Figure Q0 Q. Figure Q shows a beam with uniform load of 50 kn/m acting throughout point the beam and point load of 60 kn acting at. The value for the beam is x 0 0 knmm. Using the flexibility method, determine the deflection and rotation at. Sketch the shear force and bending moment diagram for the beam as well. 60 kn 50 kn/m I I 5 m m Figure Q Q. Figure Q shows a beam with uniform load of 48 kn/m acting throughout point to and point load of 50 kn acting at. The value for the beam is.5 x 0 0 knmm. Using the flexibility method, determine the fixed end moment for each member by assuming the reaction at as redundant. Sketch the shear force and bending moment diagrams for the beam as well. 48 kn/m 50 kn m Figure Q Q3. Figure Q3 shows a beam D with uniform load of 50 kn/m acting throughout point to and point load of 40 kn acting at D. The value for the beam is.5 x 0 0 knmm. Using the flexibility method, determine: (i) the deflection and rotation at, (ii) the end moments for each member.

6 STRUTURL NLYSIS [SK 43] 40 kn 50 kn/m D 5 m 3 m m Figure Q3 Q4. eam is supported as fixed at, pinned at and roller at carries a uniform load of 40 kn/m and a point load of 5 kn as shown in Figure Q4. The value is constant throughout the beam. Using the stiffness method, determine: (a) the deflection and rotation at in value, and (b) the end moments at and. (c) If the deflection at joint is allowed to 0 mm, propose a suitable technique to apply to the beam without changing the supports to ensure that the allowable deflection is not exceeded. 40 kn/m 5 kn (Pin joint) 6 m m m Figure Q4 Q5. Figure Q0 shows a beam with uniform load of 60 kn/m acting throughout point to and point load of 40 kn acting at. Sagging of 0 mm occurs at support in consequence to the applied loads. The value for the beam is.8 x 0 4 knm. xial effect in the members is negligable. Using the flexibility method, determine: (a) the deflection at. (b) the end moments at and, (c) the reactions at each support. 60 kn/m 40 kn I I 6 m m (Note: Settlement of 0 mm occurred at support.) Figure Q5

7 6 m STRUTURL NLYSIS [SK 43] Q6. beam D is supported as fixed at and roller at as shown in Figure Q6. The beam carries the point loads of 80 kn and 40 kn at and D, respectively. Sectional property () of each part of the beam is shown in the figure. Supposed, the beam is analysed by using the flexibility method (matrix operation). 80 kn 40 kn D Figure Q6 (a) (b) (c) (d) (e) (f) (g) Determine the degree of indeterminacy and degree of freedom of the beam. If the beam is divided into two parts, proposed the elements and nodes for analysis. For the proposed elements and nodes as in (b), what the nodal forces of the system? Show in diagram. From (c), determine a force transformation matrix (T) of the system. Determine the end moments of each element of the beam. xial effect in the beam is neglected. Draw a shear force and bending moment diagram of the beam. If a downward settlement of 0 mm occurs at the support, determine the changes of the end moment at in term of percentage, decrease or increase? Given, as.8 x 0 4 knm. Q7. beam is fixed at and roller at, carrying an uniformly distributed load of 60 kn/m and a point load of 80 kn at mid-span as shown in Figure Q7. The beam has a constant flexural rigidity () along the span. Using the stiffness method, determine the reactions at supports. Draw the shear force and bending moment diagrams. 80 kn 60 kn/m 3 m 3 m Figure Q7 Q8. Figure Q8 shows a frame carries a point load of 60 kn at mid span. Using the flexibility method, determine the end moments for each member, considering value as constant. 60 kn Figure Q8

8 6 m 3 m STRUTURL NLYSIS [SK 43] Q9. Figure Q9 shows a frame carrying a uniform load of 96 kn/m along beam. Using the flexibility method, determine the end moments for each member by assuming the reactions at as redundants. Neglect the axial deformation of the members. Taking as constant. 96 kn/m 8 m Figure Q9 Q0. Figure Q0 shows a frame carrying a point load of 80 kn and applied moment of 60 knm at. Using the flexibility method, determine the end moments for each member by assuming the reactions at as redundants. xial deformation of the members is neglected. Taking as constant. 80 kn 60 knm Figure Q0 Q. Figure Q shows a frame carrying a uniformly distributed load of 7 kn/m along beam. Supports and are fixed and roller, respectively. Using the flexibility method, determine the horizontal displacement at by assuming the reactions at as redundants. alculate also the end moments for each member. xial deformation of the members is neglected. The value of is x 0 0 knmm.

9 STRUTURL NLYSIS [SK 43] 7 kn/m 5 m Q. Figure Q shows a frame fixed ended at, pinned jointed at and pinned supported at. The frame carries a point load of kn at point and unformly distributed load of 8 kn/m along. oth E and I values are constant. (a) Using felxibility method, determine the horizontal reaction at. ssume the horizontal reaction at node as redundant. (b) Determine the reaction at. (c) Figure Q Determine the effect of moment at and rotation at, if pinned jointed at is substituted with fixed joint. 3 m kn Pin b 8 kn/m a Figure Q Q3. Figure Q3 shows a frame D carrying a point load at D and uniformly distributed load along. The given characteristic values for the frame is as per follows: = 0 mm ; E = 30 kn/mm ; I = 5000 mm 4 ased on the global axis, direction of member and numbering system for degree of freedom, detemine the following values using stiffness method.

Figure Q4 shows a frame pinned supported at and, and fixed ended at carrying a point load of 0 kn at mid span and uniformly distributed load of 4

10 3 m 3 m STRUTURL NLYSIS [SK 43] (a) displacement and rotation at support and, (b) end forces for member, and (c) sketch the shear force and bending moment diagram for member only. Figure Q3 Q4. Figure Q4 shows a frame pinned supported at and, and fixed ended at carrying a point load of 0 kn at mid span and uniformly distributed load of 4 kn/m along beam. The values for E, I snd are 00 kn/mm,.6 x 0 5 mm 4 dan 500 mm respectively. Using stiffness method, determine: (a) (b) reaction at each support, sketch the bending moment diagram for the frame. 4 kn/m I 0 kn I 6 m m m Figure Q4

11 6 m m m STRUTURL NLYSIS [SK 43] Q5. Figure Q3 shows a frame which fixed at and roller at. The frame carries a uniform load of 4 kn/m on beam. Sectional property () of each member of the frame is shown in the figure. y using flexibility method, determine the end moments of frame and horizontal displacement at support. xial effect of members is neglected. Given as.5 x 0 4 knm. 4 kn/m Figure Q5 Q6. frame is pinned at and and fixed at carrying a uniform load of 48 kn/m and a point load of 50 kn as shown in Figure Q. The value of E, I and of both members are 00 kn/mm,.6 x 0 5 mm 4 and 500 mm, respectively. 48 kn/m 50 kn.5 m.5 m Figure Q6 Using the stiffness method, determine: (a) the rotation at and, (b) the end moments of each member, (c) the reactions at supports, and (d) draw the shear force and bending moment diagrams of beam. Show the important values on the diagram.

12 STRUTURL NLYSIS [SK 43] Q7. (a) Sketch the following elements: (i) Three nodal, one dimensional element (ii) Three nodal, triangular element (iii) Six nodal, triangular element (iv) Four nodal, square element (v) Four nodal, tetrahedral element (vi) Eight nodal, hexahedral element (vii) Ten nodal, triangular element (b) (c) Divide the structures in Figure Q7 into suitable elements and name the elements used. Explain the reasons for choosing the elements used. Discuss the factors that influence the results accuracy in finite element method. Figure Q7 Q8. (a) riefly explain the elements below and give an appropriate example for each. (i) line element (ii) surface element (iii) solid element (b) (c) Name three ways of obtaining a good meshing results in finite element method. Using an element dx and dy (based on Figure Q8) applying stresses x, y and xy individually, determine the stress-strain equations.

STRUTURL NLYSIS [SK 43] xy y x xy y x v v v v E 0 0 0 0 Figure Q8 Q9.

13 STRUTURL NLYSIS [SK 43] xy y x xy y x v v v v E Figure Q8 Q9. (a) ided with appropriate diagram, discuss the importance of element numbering system technique in maximising the effectiveness of finite element method. (b) ased on the structures shown in Figure Q9, determine the suitable elements for finite element method of analysis. Provide simple justifications for the chosen elements. (c) Explain the factors influencing the accuracy of finite element method of analysis.

14 STRUTURL NLYSIS [SK 43] (i) (ii) (iii) (v) (iv) Figure Q9

Introduction. Section 3: Structural Analysis Concepts - Review

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