# Topic: Beams

## Question 22

The beam below carries a load of triangular distribution with a maximum value of 3kN/m  at midspan. The  beam  is  made of Glulam with E=16GPa and has a second moment of  area Imajor =1×10^9 mm4

Determine the  deflection at mid-span due to the  applied  loads.

HINT (1) Determine the reactions due to applied loads (2) Determine the equation of the moment diagram. ( 3) use a unit load
(4) 1GPa is 1000N/mm2

Solution  coming  soon

## Question 20

Here is question for all of you retaking Structures 2 in September 2013

Imagine you are assessing the interesting cantilever beam arrangement shown below, you have conclude that the moment capacity of the supports is adequate but you are worried about the overall deflection at the very end of the structure. Both cantilever beams are of the same length and have the same EI value of 20 MN.m2.
By formulating a suitable compatibility equation and through using Flexibility analysis determine the deflection at C.
This is  a bit easier than a  standard  exam  question  but   would carry  about  50-60 % of the available marks.
Imagine you are assessing the interesting cantilever beam arrangement shown below, you have conclude that the moment capacity of the supports is adequate but you are worried about the overall deflection at the very end of the structure. Both cantilever beams are of the same length and have the same EI value of 20 MN.m2. - ( sorry  i cant get  wordpress to  make squared)
By formulating a suitable compatibility equation and through using Flexibility analysis determine the deflection at C.
This is  a bit easier than a  standard  exam  question  but   would carry  about  50-60 % of the available marks.
HINTS coming soon!
1. Firstly read through the solution for Q4! It  uses the same analysis technique.
2. Draw yourself a primary  structure and a Unit load structure

## Question 19

Judging from last weeks performance quite a few of you have forgotten  how to draw BMD. So the next few questions will focus on this  issue.

The question focusses on  a 3 span  bridge with a  drop-in span in the middle similar to the Ness Bridge in Inverness Scotland, check out the Happy pontist blog: a  great  resource from  civil engineering  students.

HINT. The  trick  with  such  'drop-in' structures is that the double pin in the middle isolates the effects of load ( moment) from  either side. That is to say that a load on the  first span  cannot produce any moment on the middle or  third span.UPDATE 15/7/13. Judging  from the number of incorrect answers this  question  has  received I will be leaving it  up  for  a couple of  days  more.

If you  draw the deflected shape for the case where there load is at any point between the first  support and the adjacent pin, you  should see that  the middle span(between pins ) is always straight just  rotated slightly)

Try the question again and attempt to draw the deflected shapes.

SOLUTION: 17/6/13. A,C,D are the INCORRECT BMDs (this is what  you were asked to submit)  the  rest of them (B,E and F) are the correct BMD for the structure.

## Question 18

📥  Beams, ST1

Here is a question from  structures 2 but i'm sure  first years will  get it  right.

Question:  Which  BMD is the  correct for the cantilever beam  shown?

Solution: Lets  start from the point  where the load is  applied. It might help  if you  sketch the problem and mark a  line across the structure showing the  direction of the  applied load

1. The  bending moment will increases  linearly when we move away from the load ( we are moving  left) - all  diagrams  show  this!
2. As we move  down the vertical member on the  left  hand-side the  moment is constant as we are always the same perpendicular distance  from  the direction of the load.
• Note that at the corner the BMD are both on the  outside as there is tension on this face.
3. As we move along the bottom beam ( left to midpoint) we get closer and closer to the  direction of the applied load so the  moment  decreases linearly
4. As we  go past the  midpoint on the bottom beam  the bottom beam we get further  away  form the direction of the applied load and hence the moment increases linearly-only diagrams C & D show this.
5. If  we  move  up  the  vertical member on the right hand side we  stay a constant perpendicular distance  form the direction of load and so the moment is constant - the moment is on the  inside of the member  as this is  the face where  we have  tension. – Only diagram D shows this.

Lastly as we move along  the  top beam (right to left)  the moment  must  become less negative and reach  zero at the  point where we are inline  with the direction of load, when  we  have travelled  past  this  towards the  support the moment the moment increases and becomes  positive indicating a region of hogging bending ( tension on top face) near the  support

## Question 17

📥  Beams, ST1

Question:  If x= L/3 what is the vertical reaction at support A?

Solution: To solve this  we  need to take moment about  a  point that we know  the  moment for. Point  B  is a pin  support for must have  zero moment.

1. So taking  moments about B we  have acting clockwise the moment  generated  by the vertical reaction A (Va * L).
2. This must be equal to the anticlockwise moments  generated  by  the  applied   loads, which are  equal to
• P*(2L/3) for the moment from the point  load
• (wL/2)*(L/4) for the uniform loading - the first  term in brackets is the total load the  second term is the lever arm to the centre of the load.

3. equating  clockwise and  anticlockwise and dividing  by L we get Va=(2P/3) +(wL/8)

## Question 15

Which plausible load case will produce the maximum hogging moment in the structure?

Which plausible load case will produce the maximum sagging moment in the structure?

If you are struggling with this then have a look at the Solution, it  involves influence lines

# Something for my Bridge Engineering Students

Moment at B is Diagram  3, when the load is between B and D the moment at B is always zero
Moment at C is Diagram 4, when the load is at A, the moment C is hogging
Reaction at B is Diagram 2, when the load is at A the  reaction at B is greater than when the load is placed directly on B
Reaction at D is diagram 1, when the load is at A the reaction at D is downwards, as the load  move between B and D the reaction at D acts upwards with increasing magnitude.

Well done to the 62 correct submissions, below are the solutions  for those of  you ( not very many) who  got it wrong.

Solution:

Moment at B is Diagram  3, when the load is between B and D the moment at B is always zero

Moment at C is Diagram 4, when the load is at A, the moment C is hogging

Reaction at B is Diagram 2, when the load is at A the  reaction at B is greater than when the load is placed directly on B

Reaction at D is diagram 1, when the load is at A the reaction at D is downwards, as the load  move between B and D the reaction at D acts upwards with increasing magnitude.

# Something for all my Bridge Engineering students.

Pre-stressing a simply supported beam

For the simply supported pre-stressed beam shown above, calculate the maximum allowable  length (in meters) if the maximum tensile stresses due to the unfactored dead weight are limited to 1N/mm2.

Take the density of concrete to be 24kN/m3 and the centroidal  axis is at mid-height of the section.Please ignore any reliving effects or the action of live loads

hint1-prestress

Hopefully you got an  answer of 36.7m, which is less than the transportable limit of 40m. If you got anything  other than 36.7m please read the solution. week13 solution

## Question 6

For the structure shown below calculate the position of the roller support from the free end of the beam (C) such that the moment at the fixed end (A) is zero kNm.

[Solution x=3.33m]

If  you didnt get it right or were unsure of the BMD or deflected shape check out this SOLUTION