AR30084

University of Bath

EXAMINATION FOR BEng DEGREE IN CIVIL ENGINEERING & MEng DEGREE IN CIVIL & ARCHITECTURAL ENGINEERING

Year 4/5

STRUCTURES 5

xxxxday xxth January 2005 xx.00 – xx.00 (3 hours) Instructions to candidates: Answer any four questions in total. Use a separate answer booklet for each of sections A and B. All questions carry equal marks. Full marks on this examination amount to 80 marks. University calculators may be used.

This is an Open Book Examination so that candidates may refer to their own copies of appropriate Codes-of-Practice and lecture notes.

AR30084/continued

2

SECTION A 1. The reinforced concrete beam-column shown in figure 1(a) forms part of a braced portal

frame in a building. It is partially restrained against rotation at its ends. A considerable factored axial load N = 1050kN is carried by the member, in addition to a factored uniformly distributed load wudl = 17kN/m (which includes self-weight and acts normal to the member). All loading results in the set of axial force N, shear force S and bending moment M resultants shown in figure 1(a). The end conditions of the member satisfy 'Condition 3' in Table 3.21 of BS8110.

The cross-section of the beam-column is square and is shown in figure 1(b). The member

is 6.5m long. The concrete grade is C35.

(a) Treating the member as a column, show that it is slender. (2 marks) (b) Demonstrate that this member is in equilibrium under the applied loading and stress

resultants shown. Hence, draw the bending moment and shear force diagrams for the member, showing salient points, and show that the maximum bending moment is 89.6kNm. (7 marks)

(c) Using an appropriate column design chart, design the necessary longitudinal reinforce-

ment, including allowance for the axial slenderness of the member. (6 marks) (d) Design transverse links by treating the member as a column. Check that this design is

sufficient in shear at a distance av = 2d away from the member ends by treating the member as a beam. (5 marks)

S = 61.5kN M = 21.6kNm wudl = 17kN/m N = 1050kN 300mm 6.5m S = 49kN M = 19kNm 300mm N = 1050kN

(a) (b) Figure 1

AR30084/continued

3

2. The simply-supported concrete slab shown in figure 2(a) is to be designed to carry a factored uniformly-distributed loading of 12kN/m2 at the ULS. This factored loading includes self-weight, screed and imposed live loads.

(a) If you were to design this slab using the Hillerborg Strip Method, describe clearly where

you might choose to place strong bands and how you would apportion loading to various strips of your choice. Do not carry out any calculations. (4 marks)

(b) The slab is eventually actually designed containing bottom steel only in both directions.

This results in constant sagging moment capacities mx = 19kNm/m and my = 17kNm/m across the entire slab area. If an accidental concentrated load, P, is applied to the slab at the apex of the cut-out (in addition to the uniformly-distributed loading of 12kN/m2), as shown in figure 2(b), estimate an upper bound on this load P such that collapse of the slab will just occur. For this, choose a sensible yield-line collapse pattern. (8 marks)

(c) Sketch another two distinct collapse mechanisms which could occur in preference to your

chosen pattern in part (b) above. (4 marks) (d) Explain why adequate ductility of collapse is so vital to the successful implementation of

yield-line theory. (4 marks) 5.5m Y 2m X (a) 3m 1m 4m 1.5m (b)

Figure 2 AR30084/continued

P

4

3. The Class 2 post-tensioned continuous concrete beam shown in figure 3(a) contains a tendon which follows a concordant profile. The cross-section of the beam at support point A is shown in figure 3(b), while figure 3(c) shows the equivalent cross-section at support point B. Note: the cross-sectional properties are provided below figures 3(b) and 3(c). The concrete grade is C60, but transfer occurs after 14 days when the cube strength is fcu = 50N/mm2. During this transfer, the tendon is locked off at point C and is jacked from point A, then locked off there and anchored. The maximum jacking force in the tendon during jacking at point A is 8.1MN.

(a) Show that the stresses at point A for this Class 2 member are within allowable limits

during transfer, assuming just 10% losses occur. (5 marks) (b) During service conditions (the concrete has fully hardened to grade C60), the beam carries

a maximum hogging dead moment at point B of 2610kNm, together with a maximum hogging live moment at point B of 4350kNm. The effective prestressing force in the tendon under such conditions, allowing for all losses, is 4.9MN at point B. Show that the serviceability limit state stresses at point B exceed allowable values. (6 marks)

(c) Describe briefly how the designer might have reduced the total maximum hogging moment

at point B in order to satisfy serviceability limit state requirements. Do not carry out any calculations. (2 marks)

(d) The beam is to be checked for ultimate flexural strength at point B for the same loading

case as in part (b) above. The area of tendon is 6350mm2 and fpu = 1700N/mm2. Take the partial load safety factor on the dead loading moment γdl = 1.40 and the partial load safety factor on the live loading moment γll = 1.60. Show that without any additional steel reinforcement present, this beam fails the ultimate limit state check. (7 marks)

29m 29m (a) 2000 2000 700 300 350 300 350 200 1100 1300 550 550 800 800 (b) Section at A (c) Section at B

Figure 3 Ac = 1.30 × 106 mm2, Ixx = 541 × 109 mm4, Zt = 773 × 106 mm3, Zb = 416× 106 mm3

AR30084/continued

A B

Centroidal Axis

C

5

SECTION B

θ2

θ1C

A

B

O

r

L, m, I = mL

S

4.

2

12Figure 4

Figure 4 shows a uniform slender beam AB of length

�

L supported by a massless inextensible rope of total length

�

S which passes over a small frictionless pulley at the fixed point O. Point C is halfway along the beam and the state of the system is defined by the two degrees of freedom

�

θ1 and

�

θ2 . The total mass of the beam is

�

m

and its moment of inertia about C is

�

I =mL2

12. The acceleration due to gravity is

�

g .

i. Demonstrate that the distance OC is given by

�

r =12

S2 S2 − L2⎛ ⎝ ⎜ ⎞

⎠ ⎟

S2 − L2 cos2 θ2. Note

that if you get stuck on this you can still do the remainder of the question. Hint: Use the cosine rule on triangles ACO and OCB. The standard form of the cosine rule is

�

c2 = a2 + b2 −2abcosC . (5 marks)

ii. Explain why the gravitational potential energy is

�

G = −mgr sin θ1 +θ2( ) (5 marks) and

iii. why the kinetic energy is

�

T =12

m r•2

+ r2 θ•

1+ θ•

2⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

2⎛

⎝

⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟

+12

mL2

12θ•1 +θ

•2

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

2

.

(5 marks)

iv. Hence find the terms in Lagrange’s equations of motion,

�

ddt

∂T

∂θ•

i

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ −∂T∂θi

+∂G∂θi

= 0 . (5 marks)

Note that part iv is difficult and good marks can be obtained from a partial answer. AR30084/continued

6

5.

a

b

Δ 2

Δ2

Δ 1 Δ 1EIBmB

mC

EIC

mC

EIC

Figure 5

Figure 5 shows a rigid jointed portal frame of span a and height b . The bending stiffnesses of the beam and columns are EIB and

�

EIC respectively. The mass per unit length of the structure and associated cladding etc. are mB and mC.

The structure is modelled with two degrees of freedom, Δ1 and Δ2 for sway deformation.

i. Write down the 2 × 2 stiffness and mass matrices for the structure. (12 marks)

ii. Hence estimate the fundamental natural frequency of the structure if a = 2b,

�

EIB = 2EIC and mB = 2mC . (8 marks)

You may usethe followinginformation:

δ4

δ3

δ1

δ 2

L

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−−−−

−−

=

22

22

3

2333636

3233636

2

matrix stiffness elastic

LLLLLLLLLLLL

LEI

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−−−−−−

=

22

22

422313221561354313422135422156

420

matrix mass

LLLLLLLLLLLL

mL

AR30084/continued

7

s

k

L

P

α

β L

6

Figure 6

A

B

C

Figure 6 shows a strut consisting of two rigid links, each of length

�

L, pinned together at B and pinned to a support at A.

At B there is a moment spring of stiffness

�

moment = k × rotation and a linear spring of stiffness

�

s whose other end is attached to a sliding bearing so that it remains horizontal.

Both springs are unstressed when the angles

�

α and

�

β are zero.

i. Write down an expression for the potential energy of the load

�

G (4 marks)

ii. and for the strain energy

�

U . (4 marks) iii. Write down the conditions for equilibrium in terms of the total potential

energy,

�

E = G + U . (4 marks)

iv. What are the conditions for this equilibrium to be stable? (4 marks) v. Write down the answers to iii and iv for the case when

�

α and

�

β are small. The resulting equations do not have to be solved. (4 marks)

TJI/CJKW AR30084