ME 323: Mechanics of Materials

Fall 2020

Homework 9

Due Wednesday, November 11

Problem 9.1 (10 points)

A rod is made of three segments: BC, CD, and DH. All segments have length L and are made of a material

with Young’s modulus E. The cross-sectional area of segment BC decreases linearly from 2A at B to A at

C. The cross-sectional area of segment CD is constant. The cross-sectional area of segment DH increases

linearly from A at D to 3A at H. A force P acts to the left at point D.

Using a three-element (four-node) finite element model, do the following:

(a) Construct the global stiffness matrix [K] in terms of E, A and L.

(b) Construct the force vector [F] in terms of P.

(c) Enforce the displacement boundary conditions.

(d) Solve for the nodal displacements in terms of PL/EA.

(e) Determine the reactions at walls B and H in terms of P.

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ME 323: Mechanics of Materials

Fall 2020

Homework 9

Due Wednesday, November 11

Problem 9.2(10 points)

A rod is made up of four segments: AB, BC, CD and DE. Their geometric and material properties are

given below:

Cross section of AB: varies linearly from 6A to 4A. Elastic modulus: E

Cross section of BC: varies linearly from 4A to 2A. Elastic modulus: E

Cross section of CD: remains constant at 4A. Elastic modulus: 2E

Cross section of DE: varies linearly from 2A to 4A. Elastic modulus: 2E

(a) Use four finite elements as labelled (1,2,3,4) in the figure to write down the global stiffness matrix

[K] and the force vector [F] .

(b) Enforce the displacement boundary conditions and simplify the global stiffness matrix and the force

vector determined in (a).

(c) Use the MATLAB code (given in the last page) to solve for the displacements at B, C and D. State

your answers in terms of PL/EA.

(d) Modify your code to determine the internal force in each segment and the reaction forces at supports

A and E. Express your answers in terms of P.

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ME 323: Mechanics of Materials

Fall 2020

Homework 9

Due Wednesday, November 11

Problem 9.3(10 points)

Part A The thin-walled pressure vessel shown below has inner radius r = 25 in. and wall thickness

t = 2 in. It is made by welding two parts together along an inclined plane AB. The pressure vessel

contains a fluid that exerts an internal gauge pressure p = 1 ksi on the walls.

(a) Determine the axial and hoop stress (�a and �h) in the pressure vessel. Show these components

on a stress element (you can use the provided stress element below, or draw your own).

(b) Using the stress transformation equations, determine the normal stress �n and shear stress ⌧ntacting on the weld. Be careful when finding the correct angle to use in the stress transformation

equations.

Part B A point in a structure is found to be in a state of plane stress, with stress components �x

= -3 ksi, �y = 5 ksi, and ⌧xy = 3 ksi, as shown below.

(a) Determine the principal stresses �p1 and �p2 and the angles at which they occur (✓p1 and ✓p2).Show the principal stresses on a properly-oriented stress element.

(b) Determine the maximum in-plane shear stress ⌧max and the angles at which it occurs (✓s1 and

✓s2). Show the maximum shear stress on a properly-oriented stress element.

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ME 323: Mechanics of Materials

Fall 2020

Homework 9

Due Wednesday, November 11

Problem 9.4 Conceptual ( 2 + 1 + 2 points)

(a) Consider a rod BCD as shown in the figure below. It is fixed to walls at B and D, and a force

F0 is applied at point C. A finite element model of the rod consists of three nodes (B,C and D)

and two elements ((1) and (2)).

If k1 is the stiffness of element 1 and k2 is the stiffness of the second element, construct an expres-

sion for the global stiffness matrix [K]. Be sure to impose the displacement boundary conditions.

(b) Consider a rod ABCDE made of five nodes and four elements. It is fixed to the wall at node 1,

and a force F is applied at node 5. The size of the reduced global stiffness matrix [K] is 4x4.

The same rod is now split into 9 nodes and 8 elements. It is also subjected to the same loading

conditions. What is the new size of reduced global stiffness matrix [K]? Additionally, if the

displacement of the node at section B was calculated to be uB in the 5-node rod, compare its

displacement with a 9-node rod.

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ME 323: Mechanics of Materials

Fall 2020

Homework 9

Due Wednesday, November 11

(c) Consider the rod shown below, which consists of four nodes and three elements. The elements have

unknown areas.

Determine ↵1 and ↵2

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clear allclcclose all% set number of elementsN=4;%define elemental propertiesEA=[5;3;8;6]; L=[1;1;1;1];%set up force vectorF=[0;2;0;-1;0];%define boundary conditionsBC=[1;0;0;0;1];% SOLVER%set up stiffness matrixk=EA./L;K=zeros (N+1,N+1);for ii=1:N K(ii,ii) = K(ii,ii) + k(ii); K(ii+1,ii) = K(ii+1,ii) - k(ii); K(ii,ii+1) = K(ii,ii+1) - k(ii); K(ii+1,ii+1) = K(ii+1,ii+1) + k(ii);end%enforce BC?s on [K] andK_reduced = K;F_reduced = F;for jj= N+1:-1:1 if BC(jj)==1 K_reduced(jj,:)=[]; K_reduced(:,jj)=[]; F_reduced(jj)=[]; endend%solve reduced system of equationsu_reduced=inv(K_reduced)*F_reduced;%determine reaction at supports and plot results%Display Nodal Displacements%Display Nodal Forces%Display Stiffness Matrix

Published with MATLAB® R2019a

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