There is a typo in the SELECT code, needs to be

SELECT Particulars,
     LTRIM(RTRIM(SUBSTRING(
       /* <text>> */ Particulars,
       /* <start > */   CHARINDEX(';',Particulars,1) + 1,
       /* <length> */   CHARINDEX(';',
          SUBSTRING(Particulars, CHARINDEX(';', Particulars, 1) + 1, 99) + ';') - 1
      ) ) ) AS Result_string
FROM Sample_table

The innermost SUBSTRING will yield this:

 LUC20031026901 ; Vehicle
 Consolidated
 SUB35030172701 ; Building
 LIQ200310121 ; Liquor

and when the semicolon is added, it will look like this:

 LUC20031026901 ; Vehicle                                                                          ;
 Consolidated                                                                                      ;
 SUB35030172701 ; Building                                                                         ;
 LIQ200310121 ; Liquor                                                                             ;

drop table Sample_table
CREATE TABLE #Sample_table (Particulars varchar(120))
GO
INSERT INTO #Sample_table VALUES('LDR ; LUC20031026901 ; Vehicle')
INSERT INTO #Sample_table VALUES('LDR ; Consolidated                                                     ')
INSERT INTO #Sample_table VALUES('LDR ; SUB35030172701 ; Building')
INSERT INTO #Sample_table VALUES('LDRR ; LIQ200310121 ; Liquor')

select * from #Sample_table

SELECT Particulars,
--    quotename( -- enable this to see, that's just the string without spaces around
      LTRIM(RTRIM(
  SUBSTRING(Particulars,
   /*from*/CHARINDEX(';',Particulars,1)+1,   -- search first delimiter and then offset 1 from first deliminter

   /* to */CHARINDEX(';',                    -- search for the next delimter as the end-of-string
     SUBSTRING (Particulars,
         /*from*/    (CHARINDEX (';', Particulars,1)+1) -- the first character after first delimiter
         /* to */   , datalength(Particulars)     -- use datalength for not beeing bound to specific length
         ) + ';'  -- for any case add a delimiter (this was a good idea from Denis Oliynik !!), if it's an additonal, it's just lost
      ) - 1  -- delimiter Position minus one    
        )
     ) -- rigth parenthesis for lrim
       ) -- rigth parenthesis for rtrim
--  rigth parenthesis for quotename
FROM #Sample_table

/* commet out ltrim/rtrim to extract the whole string with spaces
This is a very general select not delimited to programmed stringlenthes or varchar's etc. It just extracts the first delimited string within another string, not more or less.

sincerely
Norbert L.Muth, M.A.
DBA of KNO-VA.de
Norbert.Muth@gmx.de
*/