method of joints
DESCRIPTION
Statics and DynamicsTRANSCRIPT
=">²±}z63+/r9TS�/�14/9TS�*�CD*Z[!^P*<:0;&9+369+*h8³AZ.08WO".´AZ9+/�9+S�/�[*-(+8W/�:0;�/³A-8�Sµ1B/W1�,!/2(6*Z[¶9TS�/.X*&A-C�/2C·9+(+O�3P3�:X.0.0O"3P9T(+AZ9T/2CJAZ;�CJ3y9�AZ9+/4¥¦S�/9TS�/2(�/³A-8�SJ14/21�,!/2(�:03r:0;¸9+/W;�3P:0*<;*-(�8W*<1Bc�(+/23+3P:0*<;¹>)ºe.03P*B8WA-.082O�.´A�9T/Q9TS"/r(P/³A-89T:X*<;�[*<(+82/W36AZ9¦fjA-;�C�`»>A
B
C
2 m
2 m
2 m
60°
1000 N
vwIyxrze{e|}']=7>F±5�*<.0O"9+:0*<;�~�¼�S�/�[(TA-1B/Q[*<(P143�AZ;D/2V�O":0.´A�9T/W(+A-.!9+(+:0A-;�M<.X/Q3+*»9TSNA�9�9TS�/rA-;�M-.0/W3
B
v�Iyx�zU{6|�57=">F±AZ(+/�A-.X.£= �<½ >)ºb[(P/W/�,!*�C"�hC�:´A-M-(TA-1¾*-[�^P*-:0;&9¦m¿�7:X/W.XC�3
�em�`J82*<3�= � ½ ��G �-�<�<À ª�fUm 8W*<3�= � ½ �¨�[(P*<1Á9TS"/�3PO�1�*-[¶9TS"/r[*<(P8W/23U:0;�9TS"/���C�:X(+/W89T:X*<;�AZ;�CD[(P*<1Á9TS�/�3+O�1�*-[w9TS�/[*-(+8W/23¦:0;�9TS�/o��C�:0(P/W829+:0*<;�¥\/rM</29
�6fQm 82*<3� � ½ ��m�`J82*<3� � ½ ���7à fUm � �Um�`r¡¼�S�O�3¦[(+*-1¾9TS"/�L�(+3P96/2V�O�AZ9T:X*<;
G �<�<��� ±ZfQm 8W*-3�= � ½?&=<Ä
*-( fQm � G �<�-�qÅ pÆ9T/2;�3+:X*<;�tAZ;�C m�` � �rG �<�<�»Å p8W*<1Bc�(P/W3+3P:0*<;Nt¡Ç�*�d�:X;�Mq9T*U^P*<:0;&96` à � ��§Q��� ��:0/W.XC�3¦` §U� ��py�rG �<�<� t"82*<3N �-½ *<(
` §U� °-=<= Å p%(P:0M<S&9�t ¡¼�S"/ � ���e��� :X14c�.X:0/23
fQ` � G �-�<� 8W*<3"= � ½ � E �<�<Å p9+/W;�3P:0*<;Nt¡¼�S"/\/2V�O":0.0:X,�(+:XO�1¨*-[�^P*<:X;&9wf ��:0/2.0C�3 � ��§Q��� :X14c�.X:0/23¹f § ª4G �<�-� 82*<3� �<½]���*-( f §Q� �U°<=-= Å p%.X/2[Æ9utAZ;�C � ���6�¨�"à ��:0/W.XC�3
f � ��G �-�<� 8W*<3N= � ½ ��E �<�<ÅÈ���*-( f �6� G �<�<�-Å p%O�c�t ¡
?�s �
=">É®z63+/r9TS�/�14/9TS�*�CD*Z[!^P*<:0;&9+369+*h8³AZ.08WO".´AZ9+/�9+S�/�[*-(+8W/�:0;�/³A-8�Sµ1B/W1�,!/2(6*Z[¶9TS�/9+(+O�3P3o:0.X.0O�3y9T(TA�9T/WC¯A-3Q¥_/2.0.)AZ3�9+S�/q(+/WA-829+:0*<;¯[*-(+8W/23rAZ9Q9TS"/qc�:0;·mÊA-;�C�(P*<.0.X/W(`»>A
B C
2 m
2 m
45°
500 N
vwIyxrze{e|}']=7>²Â5�*<.0O"9+:0*<;�~¶ºj[(P/W/�,�*7C"��C�:´A-M-(TA-1¾*-[¶/WA-8�S»^P*<:X;&9³kNA-3P3+O�1B:0;"M»9+/W;�3P:0*<;�:0;�/³AZ8�S1B/W1�,!/2(�:03�:0.X.0O�3y9T(+AZ9T/2C¹>
BC
v�Iyx�zU{6|�57=">²Â¼�S"/r/2V�O�:0.X:0,�(P:0O�1Ë/WV�ONAZ9+:0*<;�[*<(w^P*-:0;&9¦fb:0;�9TS"/���C�:0(P/W89T:0*-;�:X3
E �<� ª�fQ`J82*<3N?�E ½ ���*-( fQ` � �Qs � s Å p%82*<14c"(+/W3P3+:X*<;�t ¡¼�S"/r/2V�O�:0.X:0,�(P:0O�1Ë/WV�ONAZ9+:0*<;�[*<(w^P*-:0;&9¦fb:0;�9TS"/r�hC�:0(P/W829+:0*<;�:X;�3
��py�Us � s<t"82*<3N?�E ½ ��fUm ���?�s7G
*-( fQm � E �<�qÅ p9+/W;�3P:0*<;Nt ¡Å /��968W*<;"3+:0C"/W(�9TS�/_^P*<:X;&96AZ96`B~
Ì ��§o���:X14c".0:0/23 �Um�`©�Ípy�Us � s<t2p%8W*-3N?�E ½ t ���*-( m�` � E �<�»Å p9+/W;�3P:0*<;�t
Ì ���6�¨�:X14c".0:0/23 ` � ª�py�Us � s-t"8W*<3N?�E ½ ���*-( ` �6� E �-�<Å p%O�cNt ¡vw:0;NA-.X.X�4AZ9¦c�*<:0;&9¦f�k m § ª}E �<�r���*-( m §o� �QE �-�<Å p%.0/[Æ9utAZ;�C m � ª®E �-�����*-( m �6� �UE �<�<Å pC�*�¥¦;�t ¡
?�s<±
=">É=®A<to�aA-.082O�.´AZ9+/�9TS"/»[*-(+8W/»:X;·/WA-8�S·14/21�,!/2(o*-[]9TS�/q9T(PO�3+3r:0.X.0O�3y9T(+AZ9T/2C�A-3Q¥_/2.0.AZ3U9TS�/q(P/³A-89T:X*<;�[*<(P8W/W3o*Z[]9+S�/�c�:X;�AZ9�`¾A-;�C¯9TS"/q(+*<.X.0/W(eAZ9of�>�,NteºUC�C·A-;AZC�C�:X9+:0*<;�A-.!G �<��Å [*-(+8W/�9+*�9+S�/¦c�:0;BAZ9]n¿:X;»9+S�/�d</2(P9T:X8³A-.NC":0(+/2829+:0*<;qc�*<:0;&9+:0;�MC"*�¥¦;&¥_A-(+C�A-;�C�(+/28³A-.X8WO�.0AZ9T/U9+S�/o[*<(+82/W36:0;�9+S�/�14/21�,!/2(+32>RîÈS�AZ9¦8�SNA-;�M-/W3�ïB
D500 N
1000 N
3 m
A
C
4 m
3 mvwIyxrze{e|}']=7>²=5�*<.0O"9+:0*<;�~£A&t¹¼�S�/R[(+/W/],�*7C"�QC�:´A-M-(TA-1B3£*-[�/WA-8�S¦^P*-:0;&9wA-;�Cr*-["9+S�/\/2;&9T:0(P/\,�*�C"�AZ(+/4L�(+3P9�8W*<;�3y9T(PO�829+/WC}A-3+3PO�14:X;�M¯/WA-8�S 14/21�,�/W(�9+*¯,�/�:0;J9+/W;�3P:0*<;�>µ¼�S�/W3P/AZ(+/oM<:Hd</W;�:0;�9+S�/oLNM<O�(P/->
D
3
5
v�Iyx�zU{6|�57=">²=Å *-9T/�9+SNAZ9»9TS�/�M</2*<1B/29T(y� :03»3PO�8�S}9+SNAZ9B/³A-8�S®A-;�M<.X/�:03»C�/LN;�/WC},&��A¸Â���?Z�E�9+(+:´AZ;�M<.0/Z> ¼�S�/�^P*<:0;&9q[(+/2/�,!*�C"� C�:0A-M<(+A-143�A-.0.a82*<;&9�A-:X;}1B*<(P/�9TSNA-;�9K¥\*O";�¤�;�*�¥¦;�3Wk�3+*»3P9TA-(P9+:0;�Mq¥¦:H9TS�9TS�/rv¶«_��ðÉ3�*-[w9TS�/o¥¦S�*-.0/o,�*7C"�h�7:X/W.XC�3Ì ã¸ñ �¨� ~�pò�U= å #�� ? åó taè�páE �<� å #7ª�f � åó t£ª�py�e å#%taè�py�rG �<�<� åó t ���
?�s-°
*-( �U=-f � åô ª}± �<�<� åô ª� �<�<� åô �¨� *-(6f �6� °-Â< ŠpO�c�tÌ ��§U��� ~N�e` § ª®E �<�r��� *<(¦` §Q� E �<� p%.0/[Æ9ut
Ì ���e��� ~��e` �e� G �<�-� ª�°<Â< ��� *<(6` �6� ��G³=&s Å pO�c�tÅ /��968W*<;"3+:0C"/W(�9TS�/�/2V�O":0.0:X,�(+:XO�1Ë/WV�ONAZ9+:0*<;�[*<(w^P*<:X;&96`Ì ��§Q��� ~N�Um�`���` §U��� *<(¦m�` � �e` §U� �QE �<�»Å p8W*<1Bc�(P/W3+3P:0*<;NtÌ ���6��� ~N�e`rn×��` �¦��� *<(+`rn � �e` �e� G�=&s Å p9+/W;�3P:0*<;Nt
¼�S"/�v�«a�g[*-(¦nÊ��:X/W.0C"3Ì ��§U��� ~N�6fQmg��m�n�Â&Ó-E ��� *<(¦fQn � �eÂ&Ó<E-m�n
Ì ���e��� ~&`�njª�m�nh?�Ó<E ���*-(m�` � E<ÓZ?Npá`rn�t � �U± � Ä Å p%82*<14c"(+/W3P3+:X*<;�t_3+*»9TS�AZ9¦fQn � G�±-E Å pÆ9T/W;"3+:0*-;�t¼�S"/�v�«a�g[*-(�fg��:X/W.0C"3Ì ��§o�¨� ~�E �-� ªQfQn�ªQfQm�páÂ&Ó<E-t ��� *<(¹fUm � �QE<ÓZÂNpá=&±-E<t � �rG � ?�± Å p%82*<1Bc�(+/23+3+:X*<;�t����� 8WA-;�,!/qO"3+/WC·A-3�A�8�S"/W8�¤�A-;�C·A-M<A-:0;���:X/W.0C"3Qf ��� °<Â< Š>¹¼�S�/»v¶«_�[*-(¦m×8WA-;�,!/�O�3P/WC¯AZ3¦AB8�S�/28�¤�>ºe.X9+/W(+;�AZ9T/2.X�<kw9TS�/�14AZ9T(P:H�JA-c�c"(+*&A-8�S 8WA-;�,!/hO�3+/2C¹>JIK;�9+S�:03�8WA-3+/�*-;�.X�·9TS�/[(P/W/�,�*7C"�h/WV�ONAZ9+:0*<;"3¦[*-(w^P*<:0;&9T3�f�k�mhk�`êA-;"C�n�A-(P/r;"/W/WC"/WC¹>vN(P*<1�º�~ páÂ&Ó-E<tyfUm�ª�fQnjª®E �<�r��� p�!tp%?�Ó-E<tyfUm�ª�f �6��� p��tvN(P*<1Á«U~ ��páÂ&Ó-E<tyfUm�ª�páÂ&Ó-E<tPm�ngª�m�` ��� pÆ�!t��p%?�Ó-E<tyfUmg�Íp%?�Ó<E-tPm�n � G �-�<� p%�"tvN(P*<1Ê�U~ �e` § ��m�` ��� pÆ�!t�e` � � `rn ��� p%�"t
?�s-Ä
vN(P*<1Á��~ �6fQn¿��m�nµp%Â&Ó<E<t ��� pÆ�!t`rngª}m�nµp?�Ó<E<t ��� p%�"t*-(�:0;�1hA�9T(+:Ø�h[*-(+1�~ÙÚÚÚÚÚÚÚÚÚÚÚÚÚÛ
Â&Ó<E G � � � � � �?�Ó<E � � � � G � ��eÂ&Ó<E � G Â"¡FE � � � ��6?�Ó<E � � �e?�Ó-E � � � �� � �rG � � � �rG �� � � � �rG � � �rG� �rG � �UÂ&Ó-E � � � �� � � ?�Ó<E G � � �
Ü ÝÝÝÝÝÝÝÝÝÝÝÝÝÞ
ÙÚÚÚÚÚÚÚÚÚÚÚÚÚÛ
fQmfQnm�`m�n`rnf �` §` �
Ü ÝÝÝÝÝÝÝÝÝÝÝÝÝÞ�
ÙÚÚÚÚÚÚÚÚÚÚÚÚÚÛ
�QE �-���G �<�<�����
Ü ÝÝÝÝÝÝÝÝÝÝÝÝÝÞ
5�*<.Xd�:0;"M�d7:0ABÇ�AZ9TS"8³A-C�1hA�9T(+:Ø��:X;&d</W(P3+:X*<;���:X/W.0C"3fQm � �rG � ?�± Å p%82*<1Bc�(+/23+3+:X*<;�t f �6� °<Â< ŠpO�c�tfQn � G�±-E Å pÆ9T/2;�3+:X*<;�t ` §Q� E �-�BÅ p.0/2[Æ9�tm�` � �UE �<�»Å p%82*<1Bc�(+/23+3+:X*<;�t ` �6� ��G�=-="¡Fs Å pO�c�tm�n � �U± � ° Å p%8W*-14c�(P/W3P3+:0*-;�t`�n � G�=<="¡²s Å pÆ9T/2;�3+:X*<;�t
,Ntr¼�S�:X3�:03r/�"A-89T.H�¸9+S�/h3+A-14/hA-3�c�A-(P9�A�/�78W/2c"9�9+SNAZ9�9+S�/h��/2V�O�AZ9T:X*<;J[*<(^P*-:0;&9�nÁ,�/W82*<14/23�`rn©ªÍp%?&Ó<E<tPm�nY��G �<�r��� >)¼�S�O�3�9+S�/Q14AZ9T(P:H�4/WV�ONAZ9+:0*<;O"3+/WC·:0;�cNAZ(P9�º×8³A-;·,�/q8W*<c�:X/WC¸A-;�C�9TS�/q.0A-3P9�/2.0/214/2;�9r:X;¯9TS�/».0*<A-Cµd</W89T*<(8�S�A-;�M</2C�[(P*<1 � 9T*�G �<� >)¼�S�/�;�/2¥b3P*<.0O79T:0*-;�:03fQm � �rG � ?�± Å p%82*<14c"(+/W3P3+:X*<;�t f �e� °<Â< Špõ&ö�tfQn � G�±<E Å pÆ9T/W;"3+:0*-;�t ` §Q� E �<�»Å pá÷ø�ù�úPtm�` � �UE �<�»Å p%82*<14c"(+/W3P3+:X*<;�t ` �6� �U±-=&s Å p%õ&ö�tm�n � �Q± � ° Å p8W*<1Bc�(P/W3+3P:0*<;Nt`�n � ±-=<s Å pÆ9T/2;�3+:X*<;�t
¼�S"/a*<;�.H��/2.0/W1B/W;&9+3�9TSNA�9)8�SNA-;�M-/WCBA-(+/]9TS�/a9+/W;�3P:0*<;�:0;»`rnêA-;�C�9+S�/a(+/WA-829+:0*<;` � >
?&° �
û-üÉýJþYÿ������������������� �����������������������uÿ������ �!"���#��$����������%����ü'&����(��)������%�*�����*+,!����-�uÿ��$.�������'�/���0�1���2��$3�4�5�����������0������ 6�����7��$8�-!"�:9����uÿyÿ��!"$6!�����$4�%��$�ÿ��!"$Nü»þ;�Øÿ%!�-!"�:93)��%�<�����0�������-���!"$ +,!����-�uÿ=���>�����<93��$*���>?@��$�� �����A��!"�������=���=91!"��$B�DC4üA B C D
E F
5 ft 5 ft 5 ft
500 lb 1500 lb
2 ft
vwIyxrze{e|}']=7>FsE !"��)����!"$GF;HI��yÿ��J�-!"$�ÿ%�%��)��-�0��+,�����:�1!K�� L���������M���N!�+O�����M��P%!"��$Q�A��ÿòÿ�)��/�($��R�����M��:���0�1���=�Øÿ5��$��%��$�ÿ��!"$�ü�þ>�Øÿ�!/�/����0�/HIS�TU!�+V�����A�5��!"���aÿ��%��)��-��)�����üE ����������$����5�����������AHIS�TU��ÿ5���5��!"���WK�����A��XY)����(�������)3�U��XQ)3�����!"$*����$ �1�<)�ÿ%�����%!!���������$������A�������-���!"$ +,!����-�uÿ5�3������-���
Ì Z8[]\_^ FB`bacId.e;` ^�^ af�g_h ^ a cId6e h ` ^�^ afDg_h `ia cId C � af \_^ F!��>C � \kj ` ^�^ g_h ` ^�^�^h ` \ h�h û�ý:���4lm)�9onÌ p § \_^ F"? § \q^
Ì p � \_^ F"? � g C � e j ^�^�^r\s^ !��5? � \st�u�u ��4lm)�9onH���!"�vP%!"��$Q�=?Ì p � \_^ F"? � e4?;wDÿ���$ j h�x t ½ \_^ !��>?>w \ t�u�uÿ���$ j t x t ½ \ jYy�jbz u ��4l,�%��$�ÿ��!"${nÌ p § \_^ F"?;| g ?>w��-!�ÿ j h�x t ½ \s^ !��5?;| \ e j ^"t�u ��4lm�-!"�/9����uÿòÿ���!"${nH���!"�vP%!"��$Q�=wÌ p § \_^ F�e>?>w��-!�ÿ j h�x t ½ g w p@\_^ !��>w p}\ jYy ^"t�u ���4l,�%��$�ÿ��!"${nÌ p � \_^ FQ|rw g ?;w�ÿ���$ j h�x t ½ \q^ !��>|�w \ e t�u�u ��4lm�-!"�/9����uÿòÿ���!"${nH���!"�vP%!"��$Q�=C~lm$�!��%�0|7W p ��$��*C��������r�:!����A���3��$*����!�)�$�K$�!i�5$�+�!��M�-�uÿ= ����Mn
Ì p � \_^ FBC � e4C p ÿ���$ j h�x t ½ \_^ !��5C p}\ h�h û�ýÿ���$ j h�x t ½ \su y h zBj ��4l,�%��$�ÿ��!"$on?&°"G
θ�=tan-1(2/5) = 21.8
� o� CD
v�Iyx�zU{6|�57=">FsÌ p § \s^ F�e5C p �-!�ÿ j h�x t ½ e��<C \s^ �<C \ e j�� h t ���4lm�-!"�:9����uÿòÿ��!"${nH���!"���($�ÿ%91���-���!"$7!�+�P%!"��$Q�;�� p@\ e h ` ^�^ ��4lm�-!"�:93���uÿòÿ��!"$L��$�� |/� \ e j�� h t ��4lm�-!"�:9����uÿyÿ��!"${nH���!"�vP%!"��$Q� p
Ì p § \_^ F�e5| p �-!�ÿ j h�x t ½ e8w p g C p �-!�ÿ j h�x t ½ \_^!�� | p \ e j ^"t�u g u h zBj �-!�ÿ j h�x t ½�-!�ÿ j h�x t ½ \st ��� ��4l,�%��$�ÿ��!"${n� !��%�J�����;�����/����$3��$��A��XY)������!"$7����P�!"��$Q� p l � p � \s^ n���$��7�����;����!���XQ)3�����!"$�ÿO!�+��XQ)��(��������()��v+�!��VP%!"��$Q�D|������A$�!��D)�ÿ����7��ÿ������� 7�����<����91��$�����$B���3)����/�� �$�!i�s�o�)�ÿ%�����%!��M������*�����09������K�!")�ÿD��������)��(�����!"$�ÿ�ü?&°&±
=">ÉÂ&±�ºY8W(+A-;�/�3PO�c�c�*<(P9+3oA�G �<�-�hÅ .X*&A-C¹>U�_*<1Bc�O"9T/�9TS�/�[*-(+8W/23U:X;�9TS�/�14/21�,!/2(+3`ri4k�` � AZ;�C�n � >D
A C E
B D F
0.5 m
0.5 m0.5 m
G
1000 N
0.5
m
vwIyxrze{e|}']=7>²Â&±5�*<.0O"9+:0*<;�~�57:0;"8W/]A682O"9�9TS�(P*<O�M<Sr9TS�/]C�/W3P:0(P/WCr;�O"1�,!/2(+3w8³A-;r,!/]14A-C�/)¥¦:X9+S�*<O"9L�(+3P968WA-.082O�.´AZ9+:0;�M�9+S�/�(+/WA-829+:0*<;�36AZ9¦fjA-;"C�mhkN*-;�.X��9+S�/o[(+/2/�K,�*�C"��C�:0A-M<(TAZ1*Z[w9+S�/�3+/W89T:X*<;�:03¦(+/2V�O":0(+/2C�AZ3�M<:Xd-/W;�:0;D5"=">ÉÂ&±7>
F
0.5
0.5
v�Iyx�zU{6|Í5"=7>²Â&±¼�S"/�9+S�(+/2/�3+8³AZ.´A-(�/2V�O�AZ9T:X*<;�3¦*-[w/WV�O�:0.X:0,�(P:0O�1¾A-(P/-~
Ì ��§U��� ~N�e`�ib� n � � ` � Ó Ò ± �¨�Ì ���e�¨� ~<` � Ó Ò ±o�ÍG �<�<�r��� ����` �b� G³?�GW? Å
Ì ãðï ��� ~N��py¡²E<tWpòG �<�<� t�ª�py¡²E<t2pá`ri�t ��� *<(¦`ri � G �<�<�qÅAZ;�C�9TS�/2;D[(+*-1¾9TS"/r/2V�O�:0.X:0,�(P:0O�1×/WV�ONA�9T:0*-;D:X;�9+S�/o���KC�:X(+/2829T:X*<;¹~
n �b� �U± �<�-�BÅ ¡
E-Â<°
=">ÉÂ<=}¼�S"/e*�¥¦;�/2(_*-[!9TS�/63P9+*<(TA-M-/e3PS�/WCh¥¦:X3+S�/23\9+*�SNAZ;�M�A�±<E � ¤�M�/W;"M<:0;�/¦[(+*-1¬9TS�/(P*7*Z[�A-3�:0.0.XO�3P9+(TAZ9+/WC·:0;�LNM<O�(P/�'R=">ÉÂ&E7k�A-;�CJ¥_A-;&9T3�9+*�/2;�3+O�(P/h9TS�AZ9�9+S�/h(P*7*-[¥¦:X.0.¹;"*-9U82*<.0.0A-c�3P/rC"O�/�9+*49TS"/�A-C"C�/WC�¥\/W:XM<S&9U*Z[¶9TS�/r1B*-9T*<(2>a¼�S�/�14/21�,!/2(+3AZ(+/�(+AZ9T/2C�9T*4S�*<.XCD9T/2;�3+:X*<;µA-;"C¯8W*<1Bc�(P/W3+3P:0*<;�.0*&A-C"3e*-[¶O�c�9T*h? �<�<�4Å >!ze3+/9+S�/614/9TS�*�C»*-[¹3P/W829+:0*<;"3\9T*rC�/9T/W(P14:X;�/�9TS�/6.0*<A-C�3R:0;B`ri4k7` � A-;"Chn�igA-;"C14A-¤Z/�AB(P/W8W*-141B/W;�C�AZ9T:X*<;�9T*q9TS"/r*�¥¦;�/2(W>5�*<.0O"9+:0*<;�>�v�:0(P3P9BAµ[(+/W/u�K,�*7C7� C�:´AZM<(TA-1ë*-[U9+S�/�8W(TA�9T/���:0/W.XC�3q9TS�/�.0*<A-C�3»AZ9c�*<:X;�9+3Qn AZ;�Cµi4>U¼�S�:03e:03U:X.0.XO�3P9+(TAZ9+/WC�:X;DLNM-O�(+/»57=">²Â-=<A">U¼�S"/q/WV�O�:0.X:0,�(P:0O�1/2V�ONAZ9T:X*<;�3�,�/W82*<14/
(a)
(b)
(c)v�Iyx�zU{6|Í5"=7>²Â<=
EZ?<?
Ì ��§Q��� ~N�AÈ Á GÒ ± ªÑÈ ÿ GÒ ± ���Ì ���6��� ~�ªrÈ Á GÒ ± ªÑÈ ï GÒ ± ��±Z?&E- À����
*<(JÈ Á � È ï � ±Z?�EZÂ3Ò ±± � G�s-Â-? źj[(+/2/�K,�*�C"��C�:´A-M-(TA-1�*-[¶9+S�/�¥¦S�*-.0/�3y9T(PO�829+O�(+/q:X3eLN(P3P9o(P/WV�O�:0(P/WC¯9+*�*<,"9TA-:0;9+S�/�(P/³A-89T:X*<;�AZ9hm A-;"C®9TS�:X34:X3hM<:Hd</2;®:0;®v�:0M-O�(+/�5"=">ÉÂ<=-,¹>¨¼�S�/¯1B*<1B/W;&9/2V�ONAZ9T:X*<;�A-,�*<O"9¦fb��:0/W.XC�3Ì ã [ � � pò�QE<t2pyG �-�<� t��pyG � tWpòG �<�<� tPª�G�E-m � � åô ª»plE å#áª�E åó tZèepyG�s � ?�tWp GÒ ± #ò� GÒ ± ó t
ªBpòG � å #"ª®E åó taè�G�s-Â-?�pò� GÒ ± å #!� GÒ ± ó t ���*-(em �6� ±<±-±-="¡²E ÅAZ3]/u�7c!/2829T/2C�[(+*<1¬3P��141B/29+(P�Dp:l>@/->Hk�pá± �<�<� ªµ±Z?�E-Â<t+Ó<±7> Å /u��9_82*<;�3+:XC�/W(\Ao[(+/W/,�*�C"�qC�:0A-M<(TAZ1¬*-[!9TS�/¦3+/2829+:0*<;»9T*�9TS�/6(+:XM<S&9R*-[£A�8WO79]9+S�(+*-O�M<S49+S�/¦.0:X;�/W3]`ri4k` � A-;�C�`�ig:0.X.0O�3y9T(TA�9T/WC»:X;hv�:XM<O�(P/U57=">²Â-=-8->)¼�S�/e/WV�ONAZ9+:0*<;"3\*-[¹/WV�O�:0.X:0,�(P:0O�1AZ(+/ p;�*-9+/r~¦3P:0; Ð � ±Ò ±ZÄ Ã 8W*<3 Ð � EÒ ±-Ä t
Ì ��§o�¨� ~N�e` � � `�ihp EÒ ±-Ä t���n�i©�ÈpòG�s-Â-?&tWp GÒ ± t ���Ì ���¦��� ~N��G �-�<� ª}`ri�p ±Ò ±-Ä t��®G�s-Â-?�p GÒ ± t£ª®±<±<±Z="¡FE �¨�
Ì ãðÿ ��� ~�pl±<t2pá` � tPª»plE<t2pl±<±<±Z="¡FE-t ��� *<(¹` �b� �QE<EZ=<= Å p�ûiü ¿ ö�ýZø"Ä"Ä æ üBþ¹t5�*»9+SNAZ9 n�i � ?&Â-? �qÅ púyø�þIÄ æ ü"þ£t¡5�:0;�82/h,�*-9TS .X*&A-C�3r:0; nhiíA-;�C�` � /�78W/2/WC�9TS�/�14A��7:01�O�1 14/21�,�/W(�.X*&A-CAZ.0.0*�¥\/WC¹kZ9TS�/aM<A-(TA-M-/a*�¥¦;�/W()3PS�*<O�.XCq,!/¦A-C7d7:X3+/2C4AZM&A-:0;"3P9¶S�A-;�M<:X;�MU9TS"/�14*-9+*<([(P*<1¾9+S�/�(TAZ[Æ9+/W(P3W>
EZ?�E
=">@?�s®�_*-14c�O79T/U9+S�/U[*<(P8W/W3�:X;�1B/W1�,�/W(P3afQ`»k�m�`YA-;�C�m�nD>)¼�S�/ å *�¥_/2(�c!*Z9aSNA-3Aq14A-3+3�*Z[R±<E�¤�M�>A
B
C
D
E
F
G0.1 m%
0.1 m%0.2 m
%0.2 m%
0.2
m
0.3
m
vwIyxrze{e|}']=7>É?�s5�*<.0O"9+:0*<;�>�ºb[(+/2/r,�*�C"��*-[�9TS�/�/2;�9+:0(P/o3P9T(PO�829+O�(+/r:03�M-:Xd</2;D:X;�=">É?&s-A">Ì ��§Q� f § ª}m §Q���7Ã Ì ã [ ���r� py¡ÉÂ&t2pám § t��Ípy¡É=&tWpá±Z?�E7¡ÉÂ&t ���
Ì ���6� f � ��±Z?�E�¡²Â ��� 3+*»9TS�AZ9 ¶·¸ ·¹ m §Q� ?&Ä � ¡É=f §o� �e?<Ä � ¡É=f �6� ±Z?&E7¡²Â
v�Iyx�zU{6|�5"=7>É?�sZAºë8WO"9µ9+S�(+*-O�M<S�9+S�/ C�/W3P:0(P/WCb1B/W1�,�/W(P3¯:03DM<:Xd-/W;b:0;�LNM<O�(P/ =">@?�sZ,¹> ¼�S�//2V�O�:0.X:0,�(P:0O�1×/WV�ONA�9T:0*-;�36A-(+/
Ì ã¸ä � fU`hpò¡²Â&t��Íp%?&Ä � ¡FE<t2py¡ÉÂ&t ��� *<(�fU` � ?&Ä � ¡FE ÅE-=&E
A
B
v�Iyxrze{e|®5"=">@?�sZ,Ì ��§Q���7 ?&Ä � ¡²E�ª}m�` ±Ò G� ª}m�n ±Ò E ��?&Ä � ¡²E¦ª�?&Ä � ¡FE ���
Ì ���¦��� ~αZ?�E7¡É¦ª}m�` ÂÒ G� ª�m�n GÒ E ���5�*<.Xd�:0;"M��7:X/W.XC�3 m�` � � � ¡ � ÄAZ;�C m�n � �QE�?&°"¡ÉÂ-? Å ¡Å *-9T/¦*<;"/e82*<O�.0C4A-.X3+*a^PO�3P9\O�3+/¦9TS"/Uv¶«_��*-[!9TS�/6(+/214*�d</2Ch3P/W829+:0*<;hA-;�ChA�d-*<:0C8WA-.082O�.´A�9T:0;"M�9TS�/�(P/³A-89T:0*-;�3W>
E-=<=
=">²E-Â��aAZ.08WO".´AZ9+/U9TS�/o[*<(P8W/�:0;�1B/W1�,�/W(A�1��kG�Jù¶k�AZ;�C ã �ë*-[�LNM<O�(P/r']=7>FE<±�>5�*<.0O"9+:0*<;�~o¼�S�/B[(P/W/u�K,�*7C"�¯C�:´A-M-(TA-1íA-;�C·(+/W3PO�.X9+:0;�M�/2V�O":0.0:X,�(+:XO�1Î/WV�ONAZ9+:0*<;�3[*-(68WA-.082O�.´A�9T:0;"M�9TS�/r3+O�c"c!*<(y9e[*-(+8W/23e:X3¦M<:Xd-/W;�:X;�9+S�/r3+*<.XO"9T:X*<;�9T*4=">FE-±7>_¼�S�/[(P/W/u�K,�*7C"�DC�:0A-M<(TAZ1 *-[\A�8WO79o9TS�(P*<O�M<S#�1� :X3oM<:Xd-/W;�:0;¯L�M<O�(+/B57=">FEZÂ">�¼�S�//2V�O�:0.X:0,�(P:0O�1×/WV�ONA�9T:0*-;D,�/W82*<14/23
v�Iyx�zU{6|Í5"=7>FE-Â�����e��� ~��1�Áª@�1� ®ñ Ô ë ª ã ��ª®E �-���������§o��� ~��'� Ññ Ô ë ���� ã�� �¨� ~�E �<� ª®± ã � ���3+*�9+SNAZ9 ã � � �U±<E � .X, ko�1� ��� k{�Jù � �Q±<E � .X, >
E<s<s
=">É=<°}ºj8³AZ;¯*-[)cNAZ:0;&9»pòG � .X,�t�A-;�CµAhc�A-:0;&9T/2(�pòG�° � .0,�t¦A-(+/rc�*<3+:H9T:0*-;�/WCD*<;¯Ah3y9T/2c.0AZ9+9+/W(2>¦�_*<1Bc�O"9T/�9TS"/�[*<(P8W/W3UAZ9e/³A-8�S¯82*<;�;�/2829+:0*<;¹>eÇD*�C�/2.�9+S�/�3y9T/2c¯.´A-C"C�/W(AZ369+S�/�ÂB14/21�,�/W(+363+S"*�¥¦;�k!¥¦:X9+SD[(P:0829+:0*<;�AZ9e3+O"c�c!*-(P9ef¬A-;"C¯AB[(P:089T:0*-;�.0/23+382*<;�;�/2829+:0*<;�AZ9emh>)¼�S�/rC�:X3P9�AZ;�8W/ri7ÌY:03oG<>²E�[Æ9UAZ;�C�` � :X3oG�[Æ9W>
A B
CF�
G
D
E�
W1W2
3 ft
0.6 ft
15°
vwIyxrze{e|}']=7>²=<°5�*<.0O"9+:0*<;�~�¼�S�/r/�7c�.X*7C�/2C�[(P/W/�,�*7C"�hC�:´AZM<(TA-1¾:03�M<:Hd</2;�:X;D5"=">É=<°Ä¸O�;�¤�;�*�¥¦3W~ f §&à f �Zà m �Zà i �Zà i §&Ã Ì �ZÃ Ì §<Ã6Îq��Ã6Î�§ /2V�O�AZ9T:X*<;�3�A-(P/-~»è¯Â � Ä5�*<.0O"9+:0*<;4[(+*<1ÁfrÌB~� ��§Q���"à f § � ` § ª@Ì §o�¨� pyG�t� ���6���7à f � ª}` � ªÏÌ �¦��� pl±<t� ã [ � �"à �U` § p%Â)3+:0;6s-E ½ tb� ` � p%ÂR82*<3�s<E ½ t�ª Ì § p%?"¡FE¶3+:X;¦s<E ½ tg�Ì � p%?�¡²E¶8W*<3Ns<E ½ t ��� páÂ&t[(P*<1Á`ri� ��§Q���"à ` § ��i §Q��� p%?�t� ���6���7à ��G � ��` � ��i �6��� plE<t� ã¸ñ �¨�"à �rG � ª}Â)8W*<3Ns<E ½ i �e��� pá=&t
="G�±
v�Iyx�zU{6|Í5"=7>²=<°[(P*<1Ám7Ì� ��§Q���"à �2Ì § ª�i §Q��� pls<t� ���6���7à �rÌ � ª�i � �ÍG�° � ª}m �6��� pá°&t� ã*) � �"à G-¡FE¶3+:X;6s<E ½ i § � G<¡²E)82*<3�s<E ½ i � ª pyG�° � tWpá±7¡XG�t"8W*<3�s<E ½ �p?�¡²E<t"8W*<3Ns<E ½ m �¦��� páÄ&tIK;�14AZ9T(P:H�4[*<(+1�k79+S�/W3P/�Äq/WV�ONA�9T:0*-;�3¦:0;�ÄqO�;�¤�;�*�¥¦;�3e9TA-¤-/o9+S�/o[*<(+1
ÙÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÛ
f §f �m �` §` �i §i �Ì §Ì �
Ü@ÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÞ
�
="G�Â
ÙÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÛ
G � � �rG � � � G �� G � � ª�G � � � ª�G� � � �eÂ)3+:0;6s-E ½ �eÂ)8W*<3�s-E ½ � � ?�¡²E¶3+:0;6s-E ½ �e?�¡²E¶8W*<3�s-E ½� � � G � ��G � � �� � � � �rG � �rG � �� � � � � � Â)8W*<3�s<E ½ � �� � � � � ��G � ª�G �� � G � � � G � �rG� � �6?�¡²E¶8W*<3�s<EZ½ � � G<¡FE¶3P:0;6s<E-½ �rG<¡²E)82*<3�s<E-½ � �
Ü ÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÞ
ÙÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÛ
����G �G ��G�° ��rG�° � pá±7¡XG�t"8W*<3�s-E-½
Ü ÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÞ
¥¦S":08�S��7:X/W.XC�3f §o��� ` §Q� �6?�G<¡²±-=-?».X, i �6� G�±7¡²°<s-Ä�.0,f �e� E-°"¡ÉÄ<=q.0, ` �¦� �Q±-±7¡²°<s-Äq.0, Ì §U� �6?�G<¡²±-=-?».X,m �e� G�Â"G<¡ � ?».0, i §Q� �e?"G<¡F±Z=-?».0, Ì �6� �UÂ-="¡ � °"G�.0,
Å *-9T/�A-3�A�V�O�:08�¤D8�S�/W8�¤¯9+SNAZ9of � ªÍm ��� G³° � ª©G � k¹A-;�Cµf §q�ê� k¹3P*�9+SNAZ99+S�/o�µA-;"C��h/WV�O�:0.X:0,�(P:0O�1Ë/WV�ONA�9T:0*-;�3�*-[�9TS�/�(+:XM<:0C�,�*�C"��A-(P/o3TAZ9+:03PL�/WC¹>
="G³?
=">²s<E®�aAZ.08WO".´AZ9+/}9TS"/�(+/WA-829+:0*<;ê[*<(P8W/23 AZ9 f AZ;�CË` A-;"C×9+S�/®[*<(P8W/23�AZ9 /³AZ8�S82*<;�;�/2829+:0*<;¹>
AB
C1000 N
4 m4 m
5 m
3 m
vwIyxrze{e|}']=7>Fs<E¼�S"/�[(P/W/�,�*7C"��C�:´A-M-(TA-1Ë*-[w/³A-8�SD14/21�,!/2(�:03�M<:Hd</2;�:X;�9TS�/oLNM<O"(+/
ByÖBx
CW
y]CW
x^3h
4i 3
jv�Iyx�zU{6|Í5"=7>Fs<E
Å *-9T/\m�`È:03wA69K¥\*Q[*<(+82/_1B/W1�,�/W(2>w¼�S"/W(+/aA-(+/\=U/WV�ONA�9T:0*-;�3�AZ;�Cq=eO";�¤�;�*�¥¦;�3[(P*<1¾9+S�/o[(+/2/�K,�*�C"��C�:0A-M<(TAZ1Ë[*<(�fUm� ��§Q��� ��:0/W.XC�3W~¦�6f § ª}m §o�¨� pyG�t�����6��� ��:0/2.0C�32~)f � ª�m � �ÍG �<�<�r��� pl±<t� ã [ ��� ��:0/W.XC�3W~]EZm � � ? �-�<�r��� páÂ&tvN(P*<1Ám�`� ��§Q��� ��:0/W.XC�3��Um § ª�` §o�¨� p%?�t�����6��� ��:0/2.0C�3o�em � ª�` �6��� plE<t� ã¸ñ �¨� ��:X/W.0C"36?<` � ª}Â-` §Q��� pá=&t
=&±-°
¥¦S":08�S�SNA³d</r3+*<.XO"9T:X*<;f §Q� �rG � =<s Å m §Q� �rG � =<s Å ` §U� �rG � =&s Åf �¦� ± �<�»Å m �¦� ° �<�»Å ` �6� ° �<�»Å ¡
ºe.X9+/W(+;�AZ9T/2.X�<k¹�-*<O·8³A-;·(+/28W*<M<;":0àW/»fU`ÊA-3rA�9K¥\*D[*-(+8W/»1B/W1�,�/W(2>q¼�S�/W;·9TS�/[(P/W/�,�*7C"�hC�:´AZM<(TA-1Ë��:0/W.XC�3Ì ��§Q��� ~ �6f § ��m 8W*<3�Â<=7¡²°&s ½ ���
Ì ���6��� ~Êf � ª�m 3+:0;¦Â-="¡²°-Ä ½ ��G �<�<�r���Ì ã [ �¨� ~ÊE-m 3+:0;¦Â-="¡²°<s ½ �¨p%?�t2pyG �<�-� t ���
mb��` ���¥¦S":08�S�SNA-3¦3+*<.XO"9T:X*<;
f §o� ��G � =&s Å m � ` � G�Â-Â< Åf �6� ± �<�qÅ
Å *-9T/Q9+SNAZ9em � ` � G³Â<Â Ò G � =<s Ô ª}° �<� Ô >
=&±-Ä