Last Updated on September 15, 2026 by Maged kamel
Solved problem 8-32 for plastic nominal Uniform Load.
Solved problem 8-32 for plastic Nominal Uniform Load using the lower-bound theorem.
We have a Statically determinate system: a Beam hinged at one support and roller-supported at the other. It is loaded under a partially Uniform Load. The Uniform Load occupies a length of 16′ out of the total spanof of32′.
The Beam section is W18x40 with Fy = 50 ksi. The Beam is statically determinate.
We have a W18x40 section. Use a table to get Zx, the plastic section Modulus for that section. The Zx Value is 78.40 inch3.
The Nominal Mn = Mp = FyZx = 50 × 78.40 in.kipss, then divide Mp by 12 to convert itto ft-kipss.Mn=Mp=50*78.40/12)=326.66 Ft.kips.
What is the number of hinges required for collapse?
The given beams have two vertical reactions. If we disregard the horizontal Load, we will use the system of equilibrium equations for ∑ Y=0 and ∑M=0. The system is statically determinate. We added a hinge in the middle of the Beam to create a mechanism.
We need to estimate the Beam’s positive Moment at the plastic hinge at C. To find the reactions and estimate the plastic Moment, the Load weight is Wn*16.
The Reaction at A takes half of this Value, RA = 0.50*16W = 8Wn, and the same reaction Value RB for the support at B.
To estimate the Moment at point C, we will multiply RA*8. the Moment at c=Moment at D=64*Wn.
Add a parabola of w*(16)^2/8 to get the Moment at point C, which is 64*Wn + 32*Wn = 96*Wn, the maximum positive Moment.
The plastic Moment at the plastic hinge is Mp = 96*Wn = 326.66 Ft-Kips. The plastic nominal Uniform Load Wn = Mp/96 = 326.66/96 = 3.40 kips/ft is the Value Wn estimated by the lower-bound theorem or the static method.
Solve the example using the upper-bound theorem.
Another method for estimating the plastic nominal Uniform Load is the upper-bound theorem, or the mechanism method.
Since we have a partially Uniform Load, the external work of the Uniform Load is the Value of this Load multiplied by the average Deflection underneath the Uniform Load.
This rule applies if the Uniform Load occupies the full length of the span; then, the external work will be Wn*0.50*Δ. However, we will treat it differently here. At support A, the Deflection at point C creates a slope; due to collapse, the angle at A = θ.
The same angle will be present at the roller support; its Value is =θ. The positive Plastic Moment at mid-span has an angle of 2θ, which represents internal work.
The external work can be estimated as the sum of case 1 for the fully loaded span, Wi=0.50*Wn*32*Δ, minus the external work for the two parts for the Uniform loading acting upwards.
The Deflection at the edge is Δ/2, since the distance from the edge to midspan is 0.50. The external work for that case =2*Wn*8*(0.50)*Δ/2.
The total external work for the system = 0.50*Wn*32*Δ- 2*Wn*8*(0.50)*Δ/2 = 16*Wn*Δ- 4*Wn*Δ = 12*Wn*Δ.
This is the external work for the system, while the internal work is Mp*2*θ, where θ = δ/16.(12*Wn*Δ)=Mp*2*( Δ/16)= W n, Δ goes wit h; Δ,;Wn=2*Mp/(16*and Wn = 2*Mp Wn = 2* Mp Wn=Mp/9 6, since Mp=Mn=326.66 Ft. kips, then wn=326.66/96=3.40 kips/ft.
The plastic nominal Uniform Load is 3.40 Kips/ft, the same Value obtained using the lower-bound method or the static method.
Solving problem 8-32 by MASTAN 2.
The Beam was modeled using MASTAN 2. In the first inelastic analysis, using a Uniform Load of 1 kip/inch, the node distances are all in inches.
A plastic hinge is located at the mid-span due to a Wn Load of 0.284 kip/inch. The Wn Value given by the program MASTAN 2 matches our estimated Wn Value of 3.402 Kip/ ft.
The next slide image shows the Moment distribution done by MASTAN 2 for the partially loaded Beam. The plastic Moment obtained is 3920 in-kips, which matches the earlier estimate. The Wn Value is 3.403 K/ft.
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More information on Structural Analysis III.
The next Post is the Plastic Nominal Uniform Load for a partially loaded Beam.
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Here is a Link to Chapter 8 – Bending Members, Section A, Beginner’s Guide to the Steel Construction Manual, 16th ed.