Help with Solid Mechanics Assignment: Stress, Deflection, Beams, Macaulay's

[princes.fiona]

i have an assignment due this thursday, it has 9 questions, i was able to solve five of them. but for some reason i missed most of the classes in second term so i don't understand anything about the last four questions. the questions are:


Question 1:

A beam is simply supported and loaded as shown in figure 1a. the cross section of the beam is shown in figure 1b. the beam is made of steel with E= 205 GN/m^2
Find:
(a) The end reactions
(b) The second moment of area for the beam about its centroid
(c) The position and magnitude of the maximum deflection of the beam

Question 2:

Figure 2 shows a continuous beam with two point loads and a distributed load. The weight of the beam is negligible by comparison. The foundations at supports A and B were not absolutely rigid and there was some deflection at these positions when the loading was applied.
Support A settled by 5mm and support B settled by 2mm
The flexural rigidity of the beam is 72MN/m^2
(a) Calculate the reactions at A, B and C, including the applied moment at C.
(b) Find the deflection at 50kN

Question 3:
A semi-circular arch of uniform rigidity, having one end hinged and other end placed on roller, is subjected to a horizontal force P and a vertical downward load W at crown C in the middle. The radius of the arch is R and the modulus of rigidity of the cross section is EI.

Find a formula for the horizontal displacement of the roller end, B.

Question 4:

A 2380 Nm torque is applied to gear D due to input torque from the motor at A, as shown in figure 4. Shafts 1 and 2 are both solid steel, 44 mm in diameter and 600 mm long. Gear B has a diameter of 250 mm and gear C has a diameter of 350 mm. The shear modulus G for steel is 80GN/m^2

Find out:
(a) the maximum stress in shaft 2
(b) the maximum stress in shaft 1
(c) the rotation angle of gear D relative to the motor at A, due to the loading.

these problems may include the topics of stress, deflection, beams, macaulays method etc. i attended all the classes in first term and solved all the question about balancing, gear trains etc, but i couldn't attend many classes in second term where thiese topics were covered so i couldn't solve these one's. please help me with my assignment. i tried to do it with my 1st years knowledge about beams but couldn't do it. a attched 3 figures related to these questions.

thanks a lot for your help!:!)

 

[Dr.D]

Please show your work on these problems so that we can see where to help you.

 

[princes.fiona]

i tried to do number 1 and 2 but couldn't finish it. sorry about my handwritting...

 

[princes.fiona]

Please show your work on these problems so that we can see where to help you.

i attached my works here... if u can please show me what to do now...

 

[Dr.D]

"...but for some reason i missed most of the classes in second term so i don't understand anything about the last four questions."

For some reason you may just need to repeat the term and attend class the next time. You have some pretty good problems here, and if you don't know where to start, it seems to suggest that things are exactly as you said -- you did not go to class. I don't think I can help you very much in that case.

 

[princes.fiona]

if you can tell me how to make the equations and which formulae's to use... i think i can do it... or atleast tell me what's wrong in my attemps of the first two questions... pleeeaaase!

 

[nvn]

princes.fiona: On question 1(a), your answer is correct. On question 1(b), all the numbers in your ybar formula are correct. But it appears you perhaps typed the numbers into your calculator wrong, because your final answer for ybar is wrong. You listed two answers for ybar, which doesn't make sense; both ybar final answers are wrong. Try computing ybar again; then recompute your current I formula using the correct ybar value. Also, show your work, or your answer, for question 1(c).

For question 2, which is an advanced problem, it appears you ignored the given support settlements. Therefore, it appears your work on question 2 is currently irrelevant. Address the given question, and try it again. Question 3 is an advanced problem; list relevant equations and show your work. Question 4 appears slightly easier than questions 2 and 3; show your work for question 4.

 

[princes.fiona]

i have missed all the classes which covered the topics for question 3,4 and 5. but I am still trying to understand those problems by studying in internet... if you can PLEASE show me how to do it or if u can give me any idea... i think i will be able to do it myself... but right now i don't have any clue about q 3,4 and 5... please show me the way how to do it... then i'll do the rest.

 

[mathmate]

For 1C:
You have calculated the reactions from 1A, and the second moment of area (I) in 1B.
E has been given.
To find the deflections y as a function of x, you can integrate from the fundamental moment equation:

EIy(x) = $$\int\int$$ M(x) dxdx + C1 x + C2

where M(x) is the moment along the beam, C1 and C2 are constants of integration obtained from boundary conditions.

You should be able to find explanations and examples similar to the given problem from any Strength of Materials textbook.

For example, Strength of Materials by Ferdinand Singer / Andrew Pytel has a similar example on page 216, or chapter 6-2 in case page numbers have changed. It's worth a trip to the library.

 

[nvn]

princes.fiona: Go ahead and complete problem 1(b) and 1(c). For problem 1(c), you can use integration, which you seem to be familiar with. For question 4(c), just look up the formula for torsional deflection of a shaft. Assume the gears are rigid.

Regarding questions 2 and 3, PF rules state you, not us, must list relevant formulas. Figuring out which formulas to use is a major part of the assignment and knowledge. Respondents who give you the formulas and solution methods would be basically doing part of your assignment for you. Furthermore, we do not know which methods your professor wants you to use on questions 2 and 3. Can you give us the names of the possible methods your professor would consider valid?

 

[princes.fiona]

princes.fiona: For question 4(c), just look up the formula for torsional deflection of a shaft. Assume the gears are rigid.

QUOTE]

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i have done number 1c and number 3 but i have been trying to upload it in here just to see if i have done it correctly or not...but there is some problem with my pc...i can't upload it.

i think i found out how to do 4a and 4b as well... but in 4c, i found the formula for torsional deflection of a shaft but I am not sure where to use it... is torsional deflection and rotation angle are the same thing? or do i have to find torsional deflection first in order to determine rotation angle? please tell me how to use this formula...

thanks for letting me know the pf rules... i didn't know about it... sorry about what I've asked before.

 

[nvn]

Yes, torsional deflection and rotation angle are the same thing. Make sure your attached files do not exceed the maximum allowable file size.

 

[mathmate]

...i have done number 1c

If you have problems uploading here at PF, you can upload elsewhere and provide us with the link. You can even reproduce the solution in text form, although it might take a little more time if you are not familiar with Latex.

 

[princes.fiona]

can anyone tell me how to form the bending moment equation for question 2 if i turn the beam around? i mean if the built in side (c) is in right hand side then what is M(x) ? i was told that it will be easier if i do the problem turning the beam around. thanx

 

[princes.fiona]

can you suggest me a book or website where i can get similar problems and their answers for question 5 ?

 

[mathmate]

can anyone tell me how to form the bending moment equation for question 2

Princess.fiona,
Unfortunately you have not told us what course you are taking, and what methods you have been taught to solve these problems. This makes it a little more difficult to suggest to you solutions that relate to what you have learned.
The theory behind the solution to problem 1 can be found in almost any strength of materials book. However, there are also practising engineers' solutions, basically by superposition of the deflections of primary individual loads. These solutions can be found in engineers' references, one example of which is Belastungsglieder, a classic German text full of problems and solutions.
Problem 2 is a statically indeterminate problem, which I hope you have covered in your course. To solve this type of problems by classical means, we have to release the prescribed indeterminacies, in this case, it is to the second degree. For example, the pinned supports at both A and B can be removed, reducing the structure to a simple cantilever. The solutions to a simple cantilever is well known, well documented (for example, Strength of Materials by Den Hartog), and even easy to remember. Look up the reference and you will find that all you need to remember are the factors 1,2,2,3,6,8 for both displacements and rotations for moment, point and distributed loads. This reduces the problem to a determinate problem similar to problem 1. Apply a unit load Pa at A and find the deflections at A and B. Do the same for B. By solving a simultaneous equation of degree 2, you will find the values of Pa and Pb that reduce the deflections at both A and B to zero (or to the prescribed deflections in your problem). These are the respective reactions at A and B. Now by superimposing the given loadings, Pa and Pb on the RELEASED structure, you will get the moment diagram as you requested.
The above paragraph is essentially a quick course on statically indeterminate analysis of structures. I do not see an advantage having the fixed end at one end or the other, unless you described your proposed analysis strategy.

A standard and classical text to classical analytical methods can be found in Theory of Structures, by Stephen Timoshenko, the 'father' of structural analyses.

From what I can deduce from your problems, you course deals with classical analysis methods. It is also possisble to solve these problems by matrix methods, or even finite element methods, without any of the classical knowledge. A responsible engineer will always recheck answers from a computer analysis using classical approximations to ensure that he has obtained the correct results. So I congratulate your educational institution for prescribing a curriculum that prepares you for a responsible career.

 

[princes.fiona]

thanx mathmate... I am studying mechanical engineering ...
actually i got stuck with question number 5 now... can u give me a clue that what topics will i have to study to do it...?

 

[mathmate]

I am not sure if I am missing anything. I see problems 1 to 4 in your initial post, and I do not seem to find problem 5 in your subsequent posts. Do you mean problem 4 or I missed something?

 

[princes.fiona]

oooh its my mistake... I am sorry ... i didnt upload it by mistake...

Question 5:
The component shown on figure 5 is rigidly attached to a foundation. The cross section of the component is rectangular. A 5kN concentrated force is applied to the top surface of the component in the direction shown. Find out the stresses acting on the surface at position A and show them on an infinitesimal.

sorry about that

 

[nvn]

mathmate: What is the official name of the method you posted in post 16? Do you think the pin constraint at point A in question 2 is a typo by princes.fiona? I currently assume the pin constraint at point A in the second attachment is a typo, and should instead be a roller.

princes.fiona: Post your answers to question 2(a) and 2(b).

 

[mathmate]

mathmate: What is the official name of the method you posted in post 16?

It is a special case of the stiffness method, which in most cases used for application to computer analysis. The structure is broken up into standard pieces, in this case, three simply suppported beams. The rotations at each end of the simply supported beams would be calculated, and moments applied so that they would have the same rotations on each side of the support between adjacent beams. For the fixed end, the rotation is simply set to zero.

The method I described is inspired by the simple formulae presented by Den Hartog (see reference), adapted for hand calculations by making use of the known properties of the cantilever. The theory behind it is simply the classical approach for a statically indeterminate problem, namely compatibility of displacements (including rotations) and satisfaction of constraints under given loading conditions.

In fact, in real practice requiring manual verifications, if I am not using the table approach (e.g. Belastungsglieder), I would probably be doing moment distribution, another classical approach adapted for hand calculations. However, I realize that only civil engineers learn moment distribution, I was not about to introduce an approach not everyone has covered.

Today, most practising engineers would simply type in a few lines of data to Stress, Ansys, Nastran, Pro-Engineers, SDRC or other brands of analyses programs to get their results in a matter of seconds. What is missing (except in princess.fiona's case) the resources to provide the ability to do calculations/verifications by hand.

Do you think the pin constraint at point A in question 2 is a typo by princes.fiona? I currently assume the pin constraint at point A in the second attachment is a typo, and should instead be a roller.

I have made the same assumptions as you did. In most building structures, the beam flexural stiffness is much greater than the supporting columns (in translation), so that configuration is equivalent to a roller joint as the axial movement effect is negligible in most cases.