ME 323: Mechanics of MaterialsFall 2020

Homework 7Due Wednesday, Oct 21

Problem 7.1 (10 Points)The beam ABC is loaded as shown below. Let the beam’s elastic modulus be E, the second moment ofarea be I, and the uniform distributed load be q = 4P/L.

Determine the following using the second-order integration method:

(a) The deflection at the free end A in terms of P ,L,E and I.(b) Non-dimensional plot of the deflection along the beam ABCD (i.e. x/L vs. v/ |vA|, where |vA| is the

magnitude of deflection at the free end A). You can use Matlab or other tools for the plot.(c) True or False: The maximum deflection (absolute value) occur at point A.

1

fBD_P gof b TITIA X IB ei t TRftp.RB ic

4LEquilibrium

of Efg o PatRB 914L Re o CD

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Substituting into

RB 431 8931for thedeflections we need section thebeaminto two partsSection 1 0 EXELmmmm

p

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Grn

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II

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Max PX 9214RB X D

Integrating MEX W r.tl

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boundary compatibility condition

Boundary conditionsa point B is a roller joint

241 4 0EIU X L EI 2k O

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EI U2CX L3 941 RBz.LI GL Cy 0

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EI Zz at 44 3231 323914 8kg3446344 0

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9 8279Et E PE Z RB E

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as Eff 9 B PLZ RB EC 4 259Et log Pet RB B

Using 3 and substituting for RB we get

a 49.8 Z PE2 12914 43 PLS

C 3 739.23 13PLZ

at 9Et HqPR

The final deflection equation

IIe IIe EIE t3E t EEE4PLSBEI

for OfXp LIEEE.EE

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a The deflection at the free end

ax 07 2 07 ff tHaEEP

Given 9 41

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b The deflection of the beam is

c false which is obvious fromthedisplacement plot

ME 323: Mechanics of MaterialsFall 2020

Homework 7Due Wednesday, Oct 21

Problem 7.2 (10 Points)The cantilever beam ABC, shown below, is fixed to ground at A and pinned at B to the rod BD. A uniformload wo is distributed along the beam between A and B. The beam is made from a material with an elasticmodulus E1 and a cross-sectional second area moment I1, with a length L1 + L2. The rod has a uniformcross-sectional area A2, length L3, and modulus of elasticity E2. For this problem, use the following pa-rameters: L1 = 2L2 = 2 m, L3 = 1.5 m, E1 = 3E2 = 70 GPa, A2 = 25 cm2, I1 = 85 cm4, and wo = 1 kN/m.

Determine the following using the second-order integration method:

(a) The reactions at point A.(b) The tension in the rod BC.(c) The deflection at the free end C.

2

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ME 323: Mechanics of MaterialsFall 2020

Homework 7Due Wednesday, Oct 21

Problem 7.3 (10 Points)A cantilever beam ABC is loaded as shown below. The beam is made from a material with a modu-lus of elasticity E and has a circular cross-sectional area with a diameter d. For this problem, use theparameters: L1 = 1 m, L2 = 2 m, d = 40 mm, E = 200 GPa, P = 150 N,Mo = 100 N.m, and wo = 80 N/m.

Determine the following using the superposition method:

(a) An analytical expression for the deflection curve, vx, and the deflection at point C.(b) The reaction at point B.

Note: Please refer to the attached table for the necessary beam deflection equations.

3

By inspection the beam is staticallyindeterminate

Roller supportatB redundant

principle of superpositionWo Wo

m WuA B C A Bf at A B C

mo TF Mo TP 193gWo

IA Btfb

Mo MpIl Wo

ftp.t 0IxIz8fEIL54xIzxIx42wot2Ex7tEtEEx4t4oi 180 3

ATIA B e zExg fEf

6 YXtx7OEXFI

1 8eI4X D LEX

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Y E it

A B C 031 2 1502

1 Ap6

94

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100 1FIR X DHEX

2 74

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BIX BZ I

To find By

2131 293 050

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By 5

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a

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UCXmFO0l03MT

ME 323: Mechanics of MaterialsFall 2020

Homework 7Due Wednesday, Oct 21

Problem 7.4 (5 points)

(A) Indicate which of the schematics presented below depicts the deflection curve of the following beam:(2 points)

Briefly justify your answer!!

4

O

ME 323: Mechanics of MaterialsFall 2020

Homework 7Due Wednesday, Oct 21

(B) Beam (i) and (ii) shown below are identical, except that beam (i) is made of steel, and beam (ii) ismade of aluminum. Note that Esteel > Ealuminum.

(a) Beam (i) - steel. (b) Beam (ii) - aluminum.

1. Let v⇤1 be the maximum deflection in beam (i) and v⇤2 the maximum deflection in beam (ii). Which

of the following is a true statement about the maximum deflection in each beam? (1.5 points)(a) v⇤1 = v

⇤2

(b) v⇤1 > v⇤2

(c) v⇤1 < v⇤2

2. Let vmax be the maximum deflection in beam (i). If the length of beam (i) increases from its originalvalue L to a new value 2L, and the same load is applied at the free end. The new value of themaximum deflection becomes v⇤max. Circle the correct answer: (1.5 points)(a) v⇤max = vmax(b) v⇤max = 2vmax(c) v⇤max = 4vmax(d) v⇤max = 8vmax(e) v⇤max = 16vmax

Briefly justify your answers!!

5

O0

IEM 00 6

Vmax PIO SEIvmax PULI

SEIMax