Horizontal reactions for a simple span, inclined beam with vertical loads

[jochav5280]

Dear All:

My colleagues and I are having difficulty understanding if an inclined, simple span beam with vertical loads requires a horizontal reaction for static equilibrium. Our software tells us there is no reaction, which makes sense from a statics perspective since there are no horizontal external loads, however, this doesn't make intuitive sense when one considers a simple ladder example. Without a horizontal reaction at the base of the ladder, it would just fall over. Could someone help us clear this up please?

Thank you,

jochav5280

 

[JAE]

Check these out:

thread507-326814

thread507-189340

thread507-280567

 

[slickdeals]

If there is no vertical deflection allowed at the ridge (top of ladder), then the ladder on roller cannot slide.

 

[BAretired]

Depends on the supports. An inclined, simple span beam capable of developing vertical reactions at each end, when loaded vertically does not require horizontal reactions at either end.

In the case of a simple ladder, assuming no friction between the ladder and the wall, the upper reaction is horizontal. The lower reaction must have a vertical component equal to the total vertical load on the ladder and a horizontal component to match the upper reaction.

BA

 

[XR250]

As BA says, depends on the supports. A sloped rafter supported on sloped plates can slide. A sloped rafter bearing on horizontal plates (with the rafter birdsmouthed) cannot slide - assuming gravity loads only. The ladder is also an excellent example.
Sounds like you and your colleagues better brush up on your statics and free body diagrams before you design anything else

 

[racookpe1978]

Sure: Continuing what was said above:

If the upper connection is a perfect pin that cannot itself move, then the pin cannot go down, and the bottom of the ladder has no horizontal component, only a vertical component.

If the top connection is a real-world (sliding) connection, then bottom of the ladder (its feet) MUST have enough friction force (horizontal component force) with the pavement to equal the force against the wall at the top of the ladder.

Hint: Redo your calculation with the ladder sloped only 15 degrees from the horizontal. Now, put a ladder on the floor and try to leave it leaned over at 15 degrees.

MODELS AND TEXTBOOKS ARE NOT THE REAL WORLD. (But "sometimes" they approximate it close enough to use under limited circumstances.)

 

[JLNJ]

Have a look at the internal forces in your inclined beam. When you cut a section normal to the beam you will see the there is some axial load.

The internal statics work out, but it is not as intuitive as a beam parallel to the ground.

 

[jochav5280]

Dear All:

Thank you for your posts.

We obviously needed to brush up on our free body diagrams as it became quite clear once we did so.

With an inclined, simple-span beam that supports vertical loads, it becomes apparent that since there are no applied horizontal loads as well as only (1) pin-support, there's no way for a horizontal reaction at the pin-support to be balanced.

Many thanks again,

jochav5280

 

[BAretired]

With an inclined, simple-span beam that supports vertical loads, it becomes apparent that since there are no applied horizontal loads as well as only (1) pin-support, there's no way for a horizontal reaction at the pin-support to be balanced.

Not true. If one support is a pin and the other support is a roller permitting translation in any direction other than horizontal, there will be equal and opposite horizontal reactions at each end of the beam.

BA

 

[jochav5280]

Hello BA:

Agreed, that is why I wrote "simple span", which I believe normally implies that the roller support is a horizontal roller, unless specifically noted that the support is an inclined roller.

Thank you,

jochav5280

 

[Jerehmy]

I feel like this conversation is had once a year.

 

[racookpe1978]

Now, about that simple inclined beam against a fixed wall ...

Because the upper point is not pinned with a friction-free roller nor restrained vertically , there is both a horizontal and a vertical force on the lower end of the ladder.

http://content4.viralnova.com/wp-content/uploads/2014/06/stupid-people-14.jpg

 

[jochav5280]

Hello racookpe1978:

Agreed! The vertical reaction resists the vertically applied load and the horizontal reaction balances the stability reaction at the vertical roller support.

Thank you,

jochav5280

 

[BAretired]

So far so good. Now, what is the horizontal reaction and maximum moment on a ladder with horizontal span L and height H when a gravity load P is applied a horizontal distance 'a' away from the bottom support?

BA

 

[jochav5280]

Hello BAretired:

Summing the moments about the pinned support yields a horizontal reaction of (Pa)/H.

Best regards,

jochav5280

 

[BAretired]

Correct, but that is only half of my question. The maximum moment occurs at the point of load and is equal to P*a(L-a)/L , precisely the same value as that of a simple beam. The difference is that the ladder has an axial force whereas a simple beam does not.

BA

 

[jochav5280]

Hello BAretired:

My apologies, forgot the second part. Yep, I totally agree. Thank you for your help!

Best regards,

jochav5280

 

[BAretired]

Interestingly, we can conclude that the bending moment on an inclined beam is the same as that of a simple beam for any combination of gravity loads and for any inclination of roller.

BA

 

[jochav5280]

Hello BAretired:

That doesn't really make sense to me. I think picturing the extremes helps illustrate the point. As the ladder inclination approaches vertical, there will be less and less bending in the member and more axial load, which suggests that the bending moment is not the same as the simple span beam bending moment. As the ladder nears vertical, nearly all of the applied gravity loads will be resisted axially, not via bending. That said, the bending moment in the member would be proportional to the ladder inclination, which would always be less than that of a level simple span beam.

Did I miss something or misinterpret your response? Thanks for the follow-up.

Best regards,

jochav5280

 

[BAretired]

Perhaps I didn't express it correctly.

The bending moment on an inclined beam will be the same as that of a simple beam whose span is equal to the horizontal span of the inclined beam. This is true for any combination of gravity load and for any inclination of the roller support.

BA