37-Solved problem 8-33 for Plastic nominal Uniform Load

Solved problem 8-33 for a plastic nominal uniform load.

Solved problem 8-33 estimate the nominal load Wn for the end span.

In solved problem 8-33. The given section of the beams is W16x26. It is required to get the value of W_n, the plastic nominal uniform load,  from table 1-1, we get the value of the plastic section modulus Zx, which is 44.20 inch3.

The given yield stress is 50 ksi for steel. The nominal, which is the plastic moment value Mn is equal to Mp=Fy*Zx= =50*44.20/12=154.166 Ft.kips. Because we will use MASTAN 2 for comparison with our estimate, the Mn value is written as equal to 2210 kips.

We need to create Four hinges to make the system unstable.

What is the X distance value for shear for the first span?

The solution for the first span nominal Wn, is to be done by using the upper bound theorem. We have proved, earlier, that for the end span, the Plastic hinge is present at a distance =0.4142 *L from the left hinge, where L is the span length. I have included the detailed calculations in the next two slides to get that value.

We make a relation between Mp and Wn for the point of zero shear.

We write the expression for, Mx and equate it to the Mp value, and substitute the Mp and Mp/L values in terms of Wn.

Check the value of Wn using the static or the lower bound method.

We use the shear equation and substitute the value of x as equal to 0.4142L and we know the value of Mp as equal to 184.166 inch3 and span length which is equalt o 16 feet we can find the value of Wn.

The proof of the distance value for the plastic hinge is shown in a previous video by using both the lower bound and upper bound theorem.

Check the value of Wn using the upper bound method.

We have a span distance L=16′, then the x distance to the plastic hinge=0.414*(16)=6.624′, due to the mechanism of failure, a deflection will occur by a value=Δ.

For the first span, the distance to the maximum deflection of 6.624′ from the left support.
The remaining distance is (16-6.624)= 9.376′ to the right support, we two angles that are θ1 and θ2. θ1= tan θ1=Δ/6.624, for θ2= tan θ2=Δ/9.376. We add θ1+θ2, we will have (A)*((1/6.624)+(1/9.376))=0.25762*Δ.

The external work=Load*span*average of the deflection value, We=2*Wn*16*(Δ/2)=16*W_n, that is the external work value.
The internal work, for the same span, Mp*(θ1+θ2) due to the plastic hinge at 0.414L.

There will be a negative moment due to continuity at the right support plastic hinge will be created, for that joint, the internal work=Mp*θ2.

 The overall internal work for the first span will be =Mp*(θ1+θ2)+Mp*θ2=Mp*(θ1+θ2)+Mp*θ2=Mp(0.25762*Δ)+Mp*θ2.

For the second term, it is equal to Mp*(Δ/9.376), which is  Mp*0.1076 *Δ.
The total internal work=Mp*Δ((0.2576)+0.1076 )) =Mp*0.3643*Δ.

We will equate to the external work, which is =(32/2)*(W_n)*Δ, Δ delta goes with delta Δ. W_n*16= Mp*0.1067.
We have estimated Mp as 184.1666 ft. kips,

We can get the value of the plastic nominal uniform load Wn from the relation, Wn=0.02277Mp, this is not the final value, since we will estimate from the second span, another value of Wn, we will select the lesser value of Wn. Wn=0.02277 Mp.

This is the value of Wn as estimated from the first span plastic joint.

Solved problem 8-33 estimate the plastic nominal load Wn for the 2nd span.

For the second span, it can be considered as a continuous beam from both sides, we have 2*Mp at the two edges and one Mp at the mid-span due to symmetry. We will check the internal work. The second span =16′. We have three Mp acting as shown in the sketch, due to deflection, we will have an angle θ at both sides. For θ =tan θ =Δ/8.The value of 2*Δ=Δ/4.

The external work is =Load* span * average deflection=Wn*16*Δ*0.50. The internal work= Mp(θ) +Mp(θ)+Mp(2*θ).

The total value of the internal work =4*Mp*θ. The external work=internal work, then Wn*16*Δ*0.50=4*Mp*θ, θ=Δ/8. W_n*16*Δ*0.50=4*Mp*(Δ/8). delta Δ goes with delta Δ.

W_n=Mp*(1/16). The nominal Wn is estimated from the second span=Mp/16=0.0625 *MP. Mp is the M-nominal of the given section W16*26.
We have two values for Wn, the first one for Wn=0.02277*Mp, while the second value for Wn=0.0625*Mp.

We cannot select the bigger value for Wn, since Mp is reached under a lighter value of load Wn, the lesser value for the plastic nominal uniform load Wn will be selected, it will as Wn=0.02277*Mp, Since Mn=Mp=184.166 Ft.kips, then Wn=0.02277*184.166, Wn=4.1934 Kips/ft.

That is the value for the nominal uniform load for the three spans. But the first span and the third span will carry a nominal uniform load=2*41934=8.3868 Kips/Ft.

Use MASTAN 2 to estimate the value of Wn.

I have put a load of 2 kips/inch for the first and third spans and a load of 1 kip/inch for the mid-span.

Establish 40 increments each one is 0.05 and the max load is twice the applied load.

After making the first inelastic analysis and at the eighth increment, we find the value of Wn as equal to 0.349944 kip/inch if multiplied by 12 we get a value of 4.19328 kips /ft. We can get the value of Mp as equal to 2210 Kips. inch.

We can check the shear value by using MASTAN 2 for the first span to check the value of 2Wn, please find a detailed illustration in the next slide.

In the next slide, we can check the shear value of the mid-span by using MASTAN 2.

Thanks a lot and I hope this post is useful. I have included more details in the published post.

This is the pdf file used in the illustration of this post and the next post.

Have more information about the structural analysis –III.
This is the next post, 37a-solved problem 8-34– How to get Wn value for three-span beam?