Drawing BMD & SFD for TRAPEZOIDAL LOAD on a beam with fixed supports

[Kelles]

I am a 3rd year civil engineering student and this is the first time i come across this problem and i am struggling with it.

My problem here is that i don't know how to draw a shear force and bending moment diagram for a trapezoidal load on a beam with fixed supports at both ends.

I know how to find the support reactions but i need an equation to find the maximum bending moment at mid-span.

Are there any standard equations to determine the different points on the Bending moment digram for trapezoidal loading or do i have to divide my beam into 3 section (1. left triangular loading, 2. UDL, 3. right triangular loading ) and then superimpose the resulting diagrams for the 3 sections.

I really need some help here...i have searched for hours online and i couldn't find anything for this.

 

[Kelles]

I should have said that i am a 1st year i guess, anyways i think i have figured this out. i have separated the trapezoid into 3 sections and calculated the reactions for each one and then superimposed them. I think this works fine with the shear force diagram but i am unsure about the bending moment diagram since i have a beam with 2 different trapezoidal loadings on each side. Also i have another beam with a UDL one one side a trapezoidal on the other side.

 

[PhanthomJay]

Are you sure the beam is fixed at both ends (statically indeterminate to the 3rd degree) or is it pinned at both ends? Have you developed a shear diagram or have you just determined the end reactions? Does the max bending moment occur at mid point or elsewhere? To draw the bending moment diagram, the slope of the moment diagram at any point is equal to the value of the shear at that point. The moment reaches a maximum at a point of zero shear. Rather tedious for a trapezoidal load.

 

[Kelles]

yes the beam is fixed at both ends...i am designing the beam, actually i am designing a whole steel frame structure with the aid of a professional software called SAP2000 and the joints need to be fixed because i am not using any vertical bracing.

I have already developed a shear force diagram by dividing the trapezoid into three sections and then superimposed the reactions.
If you could guide me on how to upload a picture i could show you the kind of trapezoidal loading i am talking about.

Yes the maximum bending moment is probably at mid-span, i haven't calculated it but from my experience i know its at midspan.

So my question now is about the BMD: If a calculate the bending moment that results at the supports and then multiply the shear force which i have calculated at 2 points between the beam with the distance from the support will i be able to draw a correct BMD?

 

[Kelles]

I am looking for a formula to find the maximum bending moment for a trapezoidal loading, can anyone point me out to it?

 

[pongo38]

You could refer to a book such as 'steel designers manual'.